:: JORDAN22 semantic presentation

begin

theorem :: JORDAN22:1
for C being ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve)
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) holds (Upper_Appr C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ) : ( ( Function-like V46( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ,K636( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) Element of K6(K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) ( V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K636( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) Element of K6(K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V46( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ,K636( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) Element of K6(K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ,K636( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) Element of K6(K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( functional ) Element of K636( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) Element of K6(K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) c= Cl (RightComp (Cage (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V32() V49() V158() V159() V160() V161() V162() V163() V164() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V272() ) ( non empty V2() V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( functional open connected V224( TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional closed bounded compact ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN22:2
for C being ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve)
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) holds (Lower_Appr C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ) : ( ( Function-like V46( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ,K636( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) Element of K6(K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) ( V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K636( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) Element of K6(K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V46( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ,K636( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) Element of K6(K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ,K636( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) Element of K6(K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( functional ) Element of K636( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) Element of K6(K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) c= Cl (RightComp (Cage (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V32() V49() V158() V159() V160() V161() V162() V163() V164() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V272() ) ( non empty V2() V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( functional open connected V224( TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) ) ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional closed bounded compact ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ;

registration
let C be ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ;
cluster Upper_Arc C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( non empty ) ( non empty functional closed bounded compact with_the_max_arc ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) -> non empty connected ;
cluster Lower_Arc C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( non empty ) ( non empty functional closed bounded compact with_the_max_arc ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) -> non empty connected ;
end;

theorem :: JORDAN22:3
for C being ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve)
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( (Upper_Appr C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ) : ( ( Function-like V46( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ,K636( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) Element of K6(K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) ( V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K636( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) Element of K6(K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V46( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ,K636( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) Element of K6(K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ,K636( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) Element of K6(K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( functional ) Element of K636( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) Element of K6(K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) is compact & (Upper_Appr C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ) : ( ( Function-like V46( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ,K636( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) Element of K6(K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) ( V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K636( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) Element of K6(K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V46( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ,K636( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) Element of K6(K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ,K636( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) Element of K6(K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( functional ) Element of K636( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) Element of K6(K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) is connected ) ;

theorem :: JORDAN22:4
for C being ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve)
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( (Lower_Appr C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ) : ( ( Function-like V46( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ,K636( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) Element of K6(K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) ( V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K636( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) Element of K6(K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V46( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ,K636( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) Element of K6(K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ,K636( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) Element of K6(K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( functional ) Element of K636( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) Element of K6(K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) is compact & (Lower_Appr C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ) : ( ( Function-like V46( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ,K636( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) Element of K6(K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) ( V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K636( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) Element of K6(K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) Function-like V46( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ,K636( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) Element of K6(K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) Element of K6(K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ,K636( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) Element of K6(K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( functional ) Element of K636( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) Element of K6(K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) is connected ) ;

registration
let C be ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ;
cluster North_Arc C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) -> compact ;
cluster South_Arc C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( functional ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) -> compact ;
end;

theorem :: JORDAN22:5
for R being ( ( non empty ) ( non empty functional ) Subset of )
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) holds [1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ] : ( ( ) ( ) Element of K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ,NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V23( RAT : ( ( ) ( non empty V34() V158() V159() V160() V161() V164() ) set ) ) V23( INT : ( ( ) ( non empty V34() V158() V159() V160() V161() V162() V164() ) set ) ) V148() V149() V150() V151() ) set ) ) in Indices (Gauge (R : ( ( non empty ) ( non empty functional ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) : ( ( ) ( ) set ) ;

theorem :: JORDAN22:6
for R being ( ( non empty ) ( non empty functional ) Subset of )
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) holds [1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ,2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ] : ( ( ) ( ) Element of K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ,NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V23( RAT : ( ( ) ( non empty V34() V158() V159() V160() V161() V164() ) set ) ) V23( INT : ( ( ) ( non empty V34() V158() V159() V160() V161() V162() V164() ) set ) ) V148() V149() V150() V151() ) set ) ) in Indices (Gauge (R : ( ( non empty ) ( non empty functional ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) : ( ( ) ( ) set ) ;

theorem :: JORDAN22:7
for R being ( ( non empty ) ( non empty functional ) Subset of )
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) holds [2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ] : ( ( ) ( ) Element of K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ,NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V23( RAT : ( ( ) ( non empty V34() V158() V159() V160() V161() V164() ) set ) ) V23( INT : ( ( ) ( non empty V34() V158() V159() V160() V161() V162() V164() ) set ) ) V148() V149() V150() V151() ) set ) ) in Indices (Gauge (R : ( ( non empty ) ( non empty functional ) Subset of ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) : ( ( ) ( ) set ) ;

theorem :: JORDAN22:8
for i, m, k, j being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) st m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) > k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) & [i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ] : ( ( ) ( ) Element of K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ,NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V23( RAT : ( ( ) ( non empty V34() V158() V159() V160() V161() V164() ) set ) ) V23( INT : ( ( ) ( non empty V34() V158() V159() V160() V161() V162() V164() ) set ) ) V148() V149() V150() V151() ) set ) ) in Indices (Gauge (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) : ( ( ) ( ) set ) & [i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ,(j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ] : ( ( ) ( ) Element of K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ,NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V23( RAT : ( ( ) ( non empty V34() V158() V159() V160() V161() V164() ) set ) ) V23( INT : ( ( ) ( non empty V34() V158() V159() V160() V161() V162() V164() ) set ) ) V148() V149() V150() V151() ) set ) ) in Indices (Gauge (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) : ( ( ) ( ) set ) holds
dist (((Gauge (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) * (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ,((Gauge (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) * (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ,(j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) < dist (((Gauge (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) * (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ,((Gauge (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) * (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ,(j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) ;

theorem :: JORDAN22:9
for m, k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) st m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) > k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) holds
dist (((Gauge (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) * (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ,((Gauge (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) * (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ,2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) < dist (((Gauge (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) * (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ,((Gauge (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) * (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ,2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) ;

theorem :: JORDAN22:10
for i, m, k, j being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) st m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) > k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) & [i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ] : ( ( ) ( ) Element of K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ,NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V23( RAT : ( ( ) ( non empty V34() V158() V159() V160() V161() V164() ) set ) ) V23( INT : ( ( ) ( non empty V34() V158() V159() V160() V161() V162() V164() ) set ) ) V148() V149() V150() V151() ) set ) ) in Indices (Gauge (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) : ( ( ) ( ) set ) & [(i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ] : ( ( ) ( ) Element of K7(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ,NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V23( RAT : ( ( ) ( non empty V34() V158() V159() V160() V161() V164() ) set ) ) V23( INT : ( ( ) ( non empty V34() V158() V159() V160() V161() V162() V164() ) set ) ) V148() V149() V150() V151() ) set ) ) in Indices (Gauge (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) : ( ( ) ( ) set ) holds
dist (((Gauge (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) * (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ,((Gauge (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) * ((i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) < dist (((Gauge (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) * (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ,((Gauge (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) * ((i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) ;

theorem :: JORDAN22:11
for m, k being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) st m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) > k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) holds
dist (((Gauge (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) * (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ,((Gauge (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,m : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) * (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) < dist (((Gauge (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) * (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ,((Gauge (C : ( ( compact non horizontal non vertical ) ( non empty functional closed bounded compact non horizontal non vertical ) Subset of ) ,k : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) * (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) ;

theorem :: JORDAN22:12
for C being ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve)
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) )
for r, t being ( ( real ) ( V11() real ext-real ) number ) st r : ( ( real ) ( V11() real ext-real ) number ) > 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V32() V49() V158() V159() V160() V161() V162() V163() V164() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) & t : ( ( real ) ( V11() real ext-real ) number ) > 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V32() V49() V158() V159() V160() V161() V162() V163() V164() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) holds
ex n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) st
( i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) < n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) & dist (((Gauge (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) * (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ,((Gauge (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) * (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ,2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) < r : ( ( real ) ( V11() real ext-real ) number ) & dist (((Gauge (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) * (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ,((Gauge (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) * (2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) < t : ( ( real ) ( V11() real ext-real ) number ) ) ;

theorem :: JORDAN22:13
for C being ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve)
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) st 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V32() V49() V158() V159() V160() V161() V162() V163() V164() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) < n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) holds
upper_bound (proj2 : ( ( Function-like V46( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) , REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) ) ( V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) V23( REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) Function-like V46( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) , REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) V148() V149() V150() ) Element of K6(K7( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ,REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( V148() V149() V150() ) set ) ) : ( ( ) ( ) set ) ) .: ((L~ (Cage (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V272() ) ( non empty V2() V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) /\ (LSeg (((Gauge (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) * ((Center (Gauge (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ,((Gauge (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) * ((Center (Gauge (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ,(len (Gauge (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) )) : ( ( ) ( non empty functional closed connected bounded compact convex ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( functional ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V158() V159() V160() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) = upper_bound (proj2 : ( ( Function-like V46( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) , REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) ) ( V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) V23( REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) Function-like V46( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) , REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) V148() V149() V150() ) Element of K6(K7( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ,REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( V148() V149() V150() ) set ) ) : ( ( ) ( ) set ) ) .: ((L~ (Cage (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V272() ) ( non empty V2() V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) /\ (Vertical_Line (((E-bound (L~ (Cage (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V272() ) ( non empty V2() V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) + (W-bound (L~ (Cage (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V272() ) ( non empty V2() V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V34() V158() V164() ) set ) ) ) : ( ( ) ( functional ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( functional ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V158() V159() V160() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) ;

theorem :: JORDAN22:14
for C being ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve)
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) st 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V32() V49() V158() V159() V160() V161() V162() V163() V164() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) < n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) holds
lower_bound (proj2 : ( ( Function-like V46( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) , REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) ) ( V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) V23( REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) Function-like V46( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) , REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) V148() V149() V150() ) Element of K6(K7( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ,REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( V148() V149() V150() ) set ) ) : ( ( ) ( ) set ) ) .: ((L~ (Cage (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V272() ) ( non empty V2() V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) /\ (LSeg (((Gauge (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) * ((Center (Gauge (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ,((Gauge (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) * ((Center (Gauge (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ,(len (Gauge (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) )) : ( ( ) ( non empty functional closed connected bounded compact convex ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( functional ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V158() V159() V160() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) = lower_bound (proj2 : ( ( Function-like V46( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) , REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) ) ( V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) V23( REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) Function-like V46( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) , REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) V148() V149() V150() ) Element of K6(K7( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ,REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( V148() V149() V150() ) set ) ) : ( ( ) ( ) set ) ) .: ((L~ (Cage (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V272() ) ( non empty V2() V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) /\ (Vertical_Line (((E-bound (L~ (Cage (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V272() ) ( non empty V2() V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) + (W-bound (L~ (Cage (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V272() ) ( non empty V2() V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V34() V158() V164() ) set ) ) ) : ( ( ) ( functional ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( functional ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V158() V159() V160() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) ;

theorem :: JORDAN22:15
for C being ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve)
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) st 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V32() V49() V158() V159() V160() V161() V162() V163() V164() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) < n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) holds
UMP (L~ (Cage (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V272() ) ( non empty V2() V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) = |[(((E-bound (L~ (Cage (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V272() ) ( non empty V2() V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) + (W-bound (L~ (Cage (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V272() ) ( non empty V2() V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V34() V158() V164() ) set ) ) ,(upper_bound (proj2 : ( ( Function-like V46( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) , REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) ) ( V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) V23( REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) Function-like V46( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) , REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) V148() V149() V150() ) Element of K6(K7( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ,REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( V148() V149() V150() ) set ) ) : ( ( ) ( ) set ) ) .: ((L~ (Cage (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V272() ) ( non empty V2() V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) /\ (LSeg (((Gauge (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) * ((Center (Gauge (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ,((Gauge (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) * ((Center (Gauge (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ,(len (Gauge (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) )) : ( ( ) ( non empty functional closed connected bounded compact convex ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( functional ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V158() V159() V160() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) ]| : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ;

theorem :: JORDAN22:16
for C being ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve)
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) st 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V32() V49() V158() V159() V160() V161() V162() V163() V164() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) < n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) holds
LMP (L~ (Cage (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V272() ) ( non empty V2() V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) = |[(((E-bound (L~ (Cage (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V272() ) ( non empty V2() V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) + (W-bound (L~ (Cage (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V272() ) ( non empty V2() V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) / 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of COMPLEX : ( ( ) ( non empty V34() V158() V164() ) set ) ) ,(lower_bound (proj2 : ( ( Function-like V46( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) , REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) ) ( V19() V22( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) V23( REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) Function-like V46( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) , REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) V148() V149() V150() ) Element of K6(K7( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ,REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( V148() V149() V150() ) set ) ) : ( ( ) ( ) set ) ) .: ((L~ (Cage (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V272() ) ( non empty V2() V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) /\ (LSeg (((Gauge (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) * ((Center (Gauge (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ,((Gauge (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) * ((Center (Gauge (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ,(len (Gauge (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) )) : ( ( ) ( non empty functional closed connected bounded compact convex ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( functional ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V158() V159() V160() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) ]| : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ;

theorem :: JORDAN22:17
for C being ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve)
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) holds (UMP C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) < (UMP (L~ (Cage (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V272() ) ( non empty V2() V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) ;

theorem :: JORDAN22:18
for C being ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve)
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) holds (LMP C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) > (LMP (L~ (Cage (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V272() ) ( non empty V2() V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) ;

theorem :: JORDAN22:19
for C being ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve)
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) st 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V32() V49() V158() V159() V160() V161() V162() V163() V164() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) < n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) holds
ex i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) st
( 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Gauge (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) & UMP (L~ (Cage (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V272() ) ( non empty V2() V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) = (Gauge (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) * ((Center (Gauge (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) ;

theorem :: JORDAN22:20
for C being ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve)
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) st 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V32() V49() V158() V159() V160() V161() V162() V163() V164() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) < n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) holds
ex i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) st
( 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) <= i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) <= len (Gauge (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) & LMP (L~ (Cage (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V272() ) ( non empty V2() V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) = (Gauge (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) * ((Center (Gauge (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V19() non empty-yielding V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23(K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K285( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) M9( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) )) ) ) : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) ;

theorem :: JORDAN22:21
for C being ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve)
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) st 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V32() V49() V158() V159() V160() V161() V162() V163() V164() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) < n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) holds
UMP (L~ (Cage (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V272() ) ( non empty V2() V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) = UMP (Upper_Arc (L~ (Cage (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V272() ) ( non empty V2() V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty ) ( non empty functional closed connected bounded compact with_the_max_arc ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ;

theorem :: JORDAN22:22
for C being ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve)
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) st 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V32() V49() V158() V159() V160() V161() V162() V163() V164() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) < n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) holds
LMP (L~ (Cage (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V272() ) ( non empty V2() V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) = LMP (Lower_Arc (L~ (Cage (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V272() ) ( non empty V2() V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty ) ( non empty functional closed connected bounded compact with_the_max_arc ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ;

theorem :: JORDAN22:23
for C being ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve)
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) st 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V32() V49() V158() V159() V160() V161() V162() V163() V164() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) < n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(UMP C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) < (UMP (Upper_Arc (L~ (Cage (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V272() ) ( non empty V2() V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty ) ( non empty functional closed connected bounded compact with_the_max_arc ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) ;

theorem :: JORDAN22:24
for C being ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve)
for n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) st 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V32() V49() V158() V159() V160() V161() V162() V163() V164() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) < n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(LMP (Lower_Arc (L~ (Cage (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V272() ) ( non empty V2() V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty ) ( non empty functional closed connected bounded compact with_the_max_arc ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) < (LMP C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) ;

theorem :: JORDAN22:25
for C being ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve)
for i, j being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) st i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(UMP (L~ (Cage (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V272() ) ( non empty V2() V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) <= (UMP (L~ (Cage (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V272() ) ( non empty V2() V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) ;

theorem :: JORDAN22:26
for C being ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve)
for i, j being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) st i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(LMP (L~ (Cage (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V272() ) ( non empty V2() V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) <= (LMP (L~ (Cage (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V272() ) ( non empty V2() V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) ;

theorem :: JORDAN22:27
for C being ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve)
for i, j being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) st 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V32() V49() V158() V159() V160() V161() V162() V163() V164() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) < i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(UMP (Upper_Arc (L~ (Cage (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V272() ) ( non empty V2() V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty ) ( non empty functional closed connected bounded compact with_the_max_arc ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) <= (UMP (Upper_Arc (L~ (Cage (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V272() ) ( non empty V2() V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty ) ( non empty functional closed connected bounded compact with_the_max_arc ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) ;

theorem :: JORDAN22:28
for C being ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve)
for i, j being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) st 0 : ( ( ) ( empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real non positive non negative Function-like functional V32() V49() V158() V159() V160() V161() V162() V163() V164() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) < i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) & i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(LMP (Lower_Arc (L~ (Cage (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V272() ) ( non empty V2() V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty ) ( non empty functional closed connected bounded compact with_the_max_arc ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) <= (LMP (Lower_Arc (L~ (Cage (C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard V272() ) ( non empty V2() V19() V22( NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) V23( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard V272() ) FinSequence of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty ) ( non empty functional closed connected bounded compact with_the_max_arc ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) `2 : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) ;

begin

theorem :: JORDAN22:29
for C being ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) holds W-min C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) in North_Arc C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) : ( ( ) ( functional closed bounded compact ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN22:30
for C being ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) holds E-max C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) in North_Arc C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) : ( ( ) ( functional closed bounded compact ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN22:31
for C being ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) holds W-min C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) in South_Arc C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) : ( ( ) ( functional closed bounded compact ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN22:32
for C being ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) holds E-max C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) in South_Arc C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) : ( ( ) ( functional closed bounded compact ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN22:33
for C being ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) holds UMP C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) in North_Arc C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) : ( ( ) ( functional closed bounded compact ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN22:34
for C being ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) holds LMP C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) in South_Arc C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) : ( ( ) ( functional closed bounded compact ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN22:35
for C being ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) holds North_Arc C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) : ( ( ) ( functional closed bounded compact ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) c= C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ;

theorem :: JORDAN22:36
for C being ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) holds South_Arc C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) : ( ( ) ( functional closed bounded compact ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) c= C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ;

theorem :: JORDAN22:37
for C being ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) holds
( ( LMP C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) in Lower_Arc C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) : ( ( non empty ) ( non empty functional closed connected bounded compact with_the_max_arc ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) & UMP C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) in Upper_Arc C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) : ( ( non empty ) ( non empty functional closed connected bounded compact with_the_max_arc ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) or ( UMP C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) in Lower_Arc C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) : ( ( non empty ) ( non empty functional closed connected bounded compact with_the_max_arc ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) & LMP C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) : ( ( ) ( V19() Function-like V41(2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V148() V149() V150() ) Element of the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) in Upper_Arc C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) : ( ( non empty ) ( non empty functional closed connected bounded compact with_the_max_arc ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: JORDAN22:38
for C being ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) holds W-bound C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) = W-bound (North_Arc C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ) : ( ( ) ( functional closed bounded compact ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) ;

theorem :: JORDAN22:39
for C being ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) holds E-bound C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) = E-bound (North_Arc C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ) : ( ( ) ( functional closed bounded compact ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) ;

theorem :: JORDAN22:40
for C being ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) holds W-bound C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) = W-bound (South_Arc C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ) : ( ( ) ( functional closed bounded compact ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) ;

theorem :: JORDAN22:41
for C being ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) holds E-bound C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) = E-bound (South_Arc C : ( ( being_simple_closed_curve ) ( non empty functional closed connected bounded being_simple_closed_curve compact with_the_max_arc non horizontal non vertical ) Simple_closed_curve) ) : ( ( ) ( functional closed bounded compact ) Element of K6( the carrier of (TOP-REAL 2 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V32() V49() V158() V159() V160() V161() V162() V163() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V158() V159() V160() V161() V162() V163() V164() ) Element of K6(REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V69() TopSpace-like T_2 V105() V171() V203() V204() V205() V206() V207() V208() V209() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V34() V158() V159() V160() V164() ) set ) ) ;