:: JORDAN10 semantic presentation

begin

registration
cluster V21() connected compact non horizontal non vertical for ( ( ) ( ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty ) set ) ) ;
end;

theorem :: JORDAN10:1
for k, n, i, j being ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) st 1 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) <= k : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & k : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) <= len (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty V20() V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) ( non empty non trivial V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() V20() V28() FinSequence-like FinSubsequence-like V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & [i : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ,j : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( tabular ) ( V13() V15() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(K324( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V28() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K324( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) : ( ( ) ( ) set ) & [i : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ,(j : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( tabular ) ( V13() V15() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(K324( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V28() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K324( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) : ( ( ) ( ) set ) & (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty V20() V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) ( non empty non trivial V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() V20() V28() FinSequence-like FinSubsequence-like V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) /. k : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V18() V28() V35(2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V142() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) = (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( tabular ) ( V13() V15() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(K324( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V28() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K324( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) * (i : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ,j : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V18() V28() V35(2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V142() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) & (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty V20() V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) ( non empty non trivial V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() V20() V28() FinSequence-like FinSubsequence-like V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) /. (k : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V18() V28() V35(2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V142() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) = (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( tabular ) ( V13() V15() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(K324( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V28() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K324( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) * (i : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ,(j : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V18() V28() V35(2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V142() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) holds
i : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) < len (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( tabular ) ( V13() V15() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(K324( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V28() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K324( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: JORDAN10:2
for k, n, i, j being ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) st 1 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) <= k : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & k : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) <= len (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty V20() V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) ( non empty non trivial V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() V20() V28() FinSequence-like FinSubsequence-like V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & [i : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ,j : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( tabular ) ( V13() V15() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(K324( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V28() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K324( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) : ( ( ) ( ) set ) & [i : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ,(j : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( tabular ) ( V13() V15() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(K324( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V28() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K324( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) : ( ( ) ( ) set ) & (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty V20() V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) ( non empty non trivial V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() V20() V28() FinSequence-like FinSubsequence-like V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) /. k : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V18() V28() V35(2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V142() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) = (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( tabular ) ( V13() V15() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(K324( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V28() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K324( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) * (i : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ,(j : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V18() V28() V35(2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V142() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) & (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty V20() V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) ( non empty non trivial V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() V20() V28() FinSequence-like FinSubsequence-like V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) /. (k : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V18() V28() V35(2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V142() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) = (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( tabular ) ( V13() V15() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(K324( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V28() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K324( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) * (i : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ,j : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V18() V28() V35(2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V142() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) holds
i : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) > 1 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: JORDAN10:3
for k, n, i, j being ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) st 1 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) <= k : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & k : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) <= len (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty V20() V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) ( non empty non trivial V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() V20() V28() FinSequence-like FinSubsequence-like V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & [i : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ,j : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( tabular ) ( V13() V15() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(K324( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V28() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K324( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) : ( ( ) ( ) set ) & [(i : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ,j : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( tabular ) ( V13() V15() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(K324( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V28() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K324( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) : ( ( ) ( ) set ) & (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty V20() V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) ( non empty non trivial V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() V20() V28() FinSequence-like FinSubsequence-like V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) /. k : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V18() V28() V35(2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V142() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) = (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( tabular ) ( V13() V15() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(K324( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V28() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K324( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) * (i : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ,j : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V18() V28() V35(2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V142() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) & (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty V20() V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) ( non empty non trivial V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() V20() V28() FinSequence-like FinSubsequence-like V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) /. (k : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V18() V28() V35(2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V142() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) = (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( tabular ) ( V13() V15() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(K324( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V28() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K324( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) * ((i : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ,j : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V18() V28() V35(2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V142() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) holds
j : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) > 1 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: JORDAN10:4
for k, n, i, j being ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) st 1 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) <= k : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & k : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) <= len (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty V20() V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) ( non empty non trivial V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() V20() V28() FinSequence-like FinSubsequence-like V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) & [i : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ,j : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( tabular ) ( V13() V15() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(K324( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V28() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K324( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) : ( ( ) ( ) set ) & [(i : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ,j : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ] : ( ( ) ( ) set ) in Indices (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( tabular ) ( V13() V15() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(K324( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V28() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K324( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) : ( ( ) ( ) set ) & (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty V20() V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) ( non empty non trivial V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() V20() V28() FinSequence-like FinSubsequence-like V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) /. k : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V18() V28() V35(2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V142() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) = (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( tabular ) ( V13() V15() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(K324( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V28() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K324( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) * ((i : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ,j : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V18() V28() V35(2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V142() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) & (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty V20() V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) ( non empty non trivial V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() V20() V28() FinSequence-like FinSubsequence-like V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) /. (k : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V18() V28() V35(2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V142() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) = (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( tabular ) ( V13() V15() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(K324( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V28() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K324( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) * (i : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ,j : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V18() V28() V35(2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V142() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) holds
j : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) < width (Gauge (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( tabular ) ( V13() V15() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17(K324( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) V18() V28() FinSequence-like FinSubsequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K324( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() FinSequence-membered ) M11( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) )) ) : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: JORDAN10:5
for n being ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) holds C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) misses L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty V20() V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) ( non empty non trivial V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() V20() V28() FinSequence-like FinSubsequence-like V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() closed boundary bounded compact ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: JORDAN10:6
for n being ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) holds N-bound (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty V20() V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) ( non empty non trivial V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() V20() V28() FinSequence-like FinSubsequence-like V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) ) : ( ( ) ( non empty V21() closed boundary bounded compact ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) = (N-bound C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) + (((N-bound C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) - (S-bound C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) / (2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) |^ n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) ;

theorem :: JORDAN10:7
for i, j being ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) st i : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) < j : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) holds
N-bound (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,j : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty V20() V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) ( non empty non trivial V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() V20() V28() FinSequence-like FinSubsequence-like V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) ) : ( ( ) ( non empty V21() closed boundary bounded compact ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) < N-bound (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,i : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty V20() V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) ( non empty non trivial V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() V20() V28() FinSequence-like FinSubsequence-like V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) ) : ( ( ) ( non empty V21() closed boundary bounded compact ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) ;

registration
let C be ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ;
let n be ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ;
cluster Cl (RightComp (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty ) set ) ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty V20() V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) ( non empty non trivial V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() V20() V28() FinSequence-like FinSubsequence-like V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) ) : ( ( ) ( non empty V21() ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V21() ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty ) set ) ) -> compact ;
end;

theorem :: JORDAN10:8
for n being ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) holds N-min C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) : ( ( ) ( V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V18() V28() V35(2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) FinSequence-like FinSubsequence-like V142() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) in RightComp (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty V20() V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) ( non empty non trivial V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() V20() V28() FinSequence-like FinSubsequence-like V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: JORDAN10:9
for n being ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) holds C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) meets RightComp (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty V20() V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) ( non empty non trivial V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() V20() V28() FinSequence-like FinSubsequence-like V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: JORDAN10:10
for n being ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) holds C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) misses LeftComp (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty V20() V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) ( non empty non trivial V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() V20() V28() FinSequence-like FinSubsequence-like V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: JORDAN10:11
for n being ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) holds C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) c= RightComp (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty V20() V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) ( non empty non trivial V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() V20() V28() FinSequence-like FinSubsequence-like V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: JORDAN10:12
for n being ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) holds C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) c= BDD (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty V20() V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) ( non empty non trivial V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() V20() V28() FinSequence-like FinSubsequence-like V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) ) : ( ( ) ( non empty V21() closed boundary bounded compact ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( V21() open ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: JORDAN10:13
for n being ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) holds UBD (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty V20() V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) ( non empty non trivial V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() V20() V28() FinSequence-like FinSubsequence-like V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) ) : ( ( ) ( non empty V21() closed boundary bounded compact ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty V21() open connected being_Region ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty ) set ) ) c= UBD C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) : ( ( ) ( non empty V21() open connected being_Region ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty ) set ) ) ;

definition
let C be ( ( compact non horizontal non vertical ) ( non empty V21() closed bounded compact non horizontal non vertical ) Subset of ) ;
func UBD-Family C -> ( ( ) ( ) set ) equals :: JORDAN10:def 1
{ (UBD (L~ (Cage (C : ( ( ) ( ) set ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty V20() V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) ( non empty non trivial V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() V20() V28() FinSequence-like FinSubsequence-like V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) ) : ( ( ) ( non empty V21() closed boundary bounded compact ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty V21() open connected being_Region ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty ) set ) ) where n is ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : verum } ;
func BDD-Family C -> ( ( ) ( ) set ) equals :: JORDAN10:def 2
{ (Cl (BDD (L~ (Cage (C : ( ( ) ( ) set ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty V20() V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) ( non empty non trivial V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() V20() V28() FinSequence-like FinSubsequence-like V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) ) : ( ( ) ( non empty V21() closed boundary bounded compact ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V21() open ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V21() ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty ) set ) ) where n is ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) : verum } ;
end;

definition
let C be ( ( compact non horizontal non vertical ) ( non empty V21() closed bounded compact non horizontal non vertical ) Subset of ) ;
:: original: UBD-Family
redefine func UBD-Family C -> ( ( ) ( ) Subset-Family of ) ;
:: original: BDD-Family
redefine func BDD-Family C -> ( ( ) ( ) Subset-Family of ) ;
end;

registration
let C be ( ( compact non horizontal non vertical ) ( non empty V21() closed bounded compact non horizontal non vertical ) Subset of ) ;
cluster UBD-Family C : ( ( compact non horizontal non vertical ) ( non empty V21() closed bounded compact non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) -> non empty ;
cluster BDD-Family C : ( ( compact non horizontal non vertical ) ( non empty V21() closed bounded compact non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) -> non empty ;
end;

theorem :: JORDAN10:14
for C being ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) holds union (UBD-Family C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ) : ( ( ) ( non empty ) Subset-Family of ) : ( ( ) ( V21() ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty ) set ) ) = UBD C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) : ( ( ) ( non empty V21() open connected being_Region ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: JORDAN10:15
for C being ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) holds C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) c= meet (BDD-Family C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ) : ( ( ) ( non empty ) Subset-Family of ) : ( ( ) ( V21() ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: JORDAN10:16
for n being ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) holds BDD C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) : ( ( ) ( V21() open ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty ) set ) ) misses LeftComp (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty V20() V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) ( non empty non trivial V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() V20() V28() FinSequence-like FinSubsequence-like V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: JORDAN10:17
for n being ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) holds BDD C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) : ( ( ) ( V21() open ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty ) set ) ) c= RightComp (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty V20() V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) ( non empty non trivial V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() V20() V28() FinSequence-like FinSubsequence-like V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: JORDAN10:18
for n being ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) )
for P being ( ( ) ( V21() ) Subset of )
for C being ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) st P : ( ( ) ( V21() ) Subset of ) is_inside_component_of C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) holds
P : ( ( ) ( V21() ) Subset of ) misses L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty V20() V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) ( non empty non trivial V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() V20() V28() FinSequence-like FinSubsequence-like V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() closed boundary bounded compact ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: JORDAN10:19
for n being ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) holds BDD C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) : ( ( ) ( V21() open ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty ) set ) ) misses L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V10() V11() V12() V26() ext-real V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) )) : ( ( non empty V20() V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) ( non empty non trivial V13() V16( NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) V17( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) V18() V20() V28() FinSequence-like FinSubsequence-like V208( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) special unfolded s.c.c. standard V226() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty V21() closed boundary bounded compact ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty ) set ) ) ;

theorem :: JORDAN10:20
for C being ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) holds COMPLEMENT (UBD-Family C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ) : ( ( ) ( non empty ) Subset-Family of ) : ( ( ) ( ) Element of K6(K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) ) = BDD-Family C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) : ( ( ) ( non empty ) Subset-Family of ) ;

theorem :: JORDAN10:21
for C being ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) holds meet (BDD-Family C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ) : ( ( ) ( non empty ) Subset-Family of ) : ( ( ) ( V21() ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty ) set ) ) = C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) \/ (BDD C : ( ( connected compact non horizontal non vertical ) ( non empty V21() closed connected bounded compact non horizontal non vertical ) Subset of ) ) : ( ( ) ( V21() open ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty V21() ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V26() ext-real positive non negative V103() V150() V151() V152() V153() V154() V155() ) Element of NAT : ( ( ) ( V150() V151() V152() V153() V154() V155() V156() ) Element of K6(REAL : ( ( ) ( V150() V151() V152() V156() ) set ) ) : ( ( ) ( non empty ) set ) ) ) ) : ( ( strict ) ( non empty TopSpace-like T_0 T_1 T_2 V116() V162() V163() V164() V165() V166() V167() V168() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V21() ) set ) ) : ( ( ) ( non empty ) set ) ) ;