:: JORDAN1I semantic presentation

begin

theorem :: JORDAN1I:1
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) holds (W-min (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) .. (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) > 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN1I:2
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) holds (E-max (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) .. (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) > 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN1I:3
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) holds (S-max (L~ (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) .. (Cage (C : ( ( connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) > 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ;

begin

theorem :: JORDAN1I:4
for f being ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard ) special_circular_sequence)
for p being ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Point of ( ( ) ( non empty V2() ) set ) ) st p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Point of ( ( ) ( non empty V2() ) set ) ) in rng f : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard ) special_circular_sequence) : ( ( ) ( V2() ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) holds
left_cell (f : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard ) special_circular_sequence) ,(p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Point of ( ( ) ( non empty V2() ) set ) ) .. f : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard ) special_circular_sequence) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) = left_cell ((Rotate (f : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard ) special_circular_sequence) ,p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Point of ( ( ) ( non empty V2() ) set ) ) )) : ( ( ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ,1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1I:5
for f being ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard ) special_circular_sequence)
for p being ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Point of ( ( ) ( non empty V2() ) set ) ) st p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Point of ( ( ) ( non empty V2() ) set ) ) in rng f : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard ) special_circular_sequence) : ( ( ) ( V2() ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) holds
right_cell (f : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard ) special_circular_sequence) ,(p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Point of ( ( ) ( non empty V2() ) set ) ) .. f : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard ) special_circular_sequence) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) = right_cell ((Rotate (f : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard ) special_circular_sequence) ,p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Point of ( ( ) ( non empty V2() ) set ) ) )) : ( ( ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ,1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1I:6
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( non empty connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) holds W-min C : ( ( non empty connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) in right_cell ((Rotate ((Cage (C : ( ( non empty connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ,(W-min (L~ (Cage (C : ( ( non empty connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) )) : ( ( ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ,1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1I:7
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( non empty connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) holds E-max C : ( ( non empty connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) in right_cell ((Rotate ((Cage (C : ( ( non empty connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ,(E-max (L~ (Cage (C : ( ( non empty connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) )) : ( ( ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ,1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1I:8
for n being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( non empty connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) holds S-max C : ( ( non empty connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) in right_cell ((Rotate ((Cage (C : ( ( non empty connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ,(S-max (L~ (Cage (C : ( ( non empty connected compact non horizontal non vertical ) ( non empty connected bounded compact non horizontal non vertical ) Subset of ) ,n : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) )) : ( ( ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ,1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) ;

begin

theorem :: JORDAN1I:9
for f being ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence)
for p being ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Point of ( ( ) ( non empty V2() ) set ) ) st p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Point of ( ( ) ( non empty V2() ) set ) ) `1 : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) < W-bound (L~ f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) holds
p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Point of ( ( ) ( non empty V2() ) set ) ) in LeftComp f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) : ( ( ) ( non empty ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1I:10
for f being ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence)
for p being ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Point of ( ( ) ( non empty V2() ) set ) ) st p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Point of ( ( ) ( non empty V2() ) set ) ) `1 : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) > E-bound (L~ f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) holds
p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Point of ( ( ) ( non empty V2() ) set ) ) in LeftComp f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) : ( ( ) ( non empty ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1I:11
for f being ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence)
for p being ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Point of ( ( ) ( non empty V2() ) set ) ) st p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Point of ( ( ) ( non empty V2() ) set ) ) `2 : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) < S-bound (L~ f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) holds
p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Point of ( ( ) ( non empty V2() ) set ) ) in LeftComp f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) : ( ( ) ( non empty ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1I:12
for f being ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence)
for p being ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Point of ( ( ) ( non empty V2() ) set ) ) st p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Point of ( ( ) ( non empty V2() ) set ) ) `2 : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) > N-bound (L~ f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) holds
p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Point of ( ( ) ( non empty V2() ) set ) ) in LeftComp f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) : ( ( ) ( non empty ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: JORDAN1I:13
for f being ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence)
for G being ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) st f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) is_sequence_on G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) holds
for i, j, k being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) <= k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) & k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) <= len f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) & [i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ] : ( ( ) ( ) set ) in Indices G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( ) set ) & [(i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ] : ( ( ) ( ) set ) in Indices G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( ) set ) & f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) /. k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) = G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * ((i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) & f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) /. (k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) = G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * (i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) holds
j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) < width G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN1I:14
for f being ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence)
for G being ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) st f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) is_sequence_on G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) holds
for i, j, k being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) <= k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) & k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) <= len f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) & [i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ] : ( ( ) ( ) set ) in Indices G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( ) set ) & [i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,(j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ] : ( ( ) ( ) set ) in Indices G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( ) set ) & f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) /. k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) = G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * (i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) & f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) /. (k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) = G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * (i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,(j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) holds
i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) < len G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN1I:15
for f being ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence)
for G being ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) st f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) is_sequence_on G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) holds
for i, j, k being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) <= k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) & k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) <= len f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) & [i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ] : ( ( ) ( ) set ) in Indices G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( ) set ) & [(i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ] : ( ( ) ( ) set ) in Indices G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( ) set ) & f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) /. k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) = G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * (i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) & f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) /. (k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) = G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * ((i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) holds
j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) > 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN1I:16
for f being ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence)
for G being ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) st f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) is_sequence_on G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) holds
for i, j, k being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) <= k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) & k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) <= len f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) & [i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ] : ( ( ) ( ) set ) in Indices G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( ) set ) & [i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,(j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ] : ( ( ) ( ) set ) in Indices G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( ) set ) & f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) /. k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) = G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * (i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,(j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) & f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) /. (k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) = G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * (i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) holds
i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) > 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN1I:17
for f being ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence)
for G being ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) st f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) is_sequence_on G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) holds
for i, j, k being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) <= k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) & k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) <= len f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) & [i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ] : ( ( ) ( ) set ) in Indices G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( ) set ) & [(i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ] : ( ( ) ( ) set ) in Indices G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( ) set ) & f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) /. k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) = G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * ((i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) & f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) /. (k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) = G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * (i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) holds
(f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) /. k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) `2 : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) <> N-bound (L~ f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) ;

theorem :: JORDAN1I:18
for f being ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence)
for G being ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) st f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) is_sequence_on G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) holds
for i, j, k being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) <= k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) & k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) <= len f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) & [i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ] : ( ( ) ( ) set ) in Indices G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( ) set ) & [i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,(j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ] : ( ( ) ( ) set ) in Indices G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( ) set ) & f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) /. k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) = G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * (i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) & f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) /. (k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) = G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * (i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,(j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) holds
(f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) /. k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) `1 : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) <> E-bound (L~ f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) ;

theorem :: JORDAN1I:19
for f being ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence)
for G being ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) st f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) is_sequence_on G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) holds
for i, j, k being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) <= k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) & k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) <= len f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) & [i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ] : ( ( ) ( ) set ) in Indices G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( ) set ) & [(i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ] : ( ( ) ( ) set ) in Indices G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( ) set ) & f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) /. k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) = G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * (i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) & f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) /. (k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) = G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * ((i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) holds
(f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) /. k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) `2 : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) <> S-bound (L~ f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) ;

theorem :: JORDAN1I:20
for f being ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence)
for G being ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) st f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) is_sequence_on G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) holds
for i, j, k being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) <= k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) & k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) <= len f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) & [i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ] : ( ( ) ( ) set ) in Indices G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( ) set ) & [i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,(j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ] : ( ( ) ( ) set ) in Indices G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( ) set ) & f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) /. k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) = G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * (i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,(j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) & f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) /. (k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) = G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * (i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) holds
(f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) /. k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) `1 : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) <> W-bound (L~ f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) ;

theorem :: JORDAN1I:21
for f being ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence)
for G being ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board)
for k being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) st f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) is_sequence_on G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) & 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) <= k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) & k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) <= len f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) & f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) /. k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) = W-min (L~ f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) holds
ex i, j being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) st
( [i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ] : ( ( ) ( ) set ) in Indices G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( ) set ) & [i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,(j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ] : ( ( ) ( ) set ) in Indices G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( ) set ) & f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) /. k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) = G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * (i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) & f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) /. (k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) = G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * (i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,(j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ) ;

theorem :: JORDAN1I:22
for f being ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence)
for G being ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board)
for k being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) st f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) is_sequence_on G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) & 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) <= k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) & k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) <= len f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) & f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) /. k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) = N-min (L~ f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) holds
ex i, j being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) st
( [i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ] : ( ( ) ( ) set ) in Indices G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( ) set ) & [(i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ] : ( ( ) ( ) set ) in Indices G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( ) set ) & f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) /. k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) = G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * (i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) & f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) /. (k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) = G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * ((i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ) ;

theorem :: JORDAN1I:23
for f being ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence)
for G being ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board)
for k being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) st f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) is_sequence_on G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) & 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) <= k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) & k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) <= len f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) & f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) /. k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) = E-max (L~ f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) holds
ex i, j being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) st
( [i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,(j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ] : ( ( ) ( ) set ) in Indices G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( ) set ) & [i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ] : ( ( ) ( ) set ) in Indices G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( ) set ) & f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) /. k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) = G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * (i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,(j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) & f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) /. (k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) = G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * (i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ) ;

theorem :: JORDAN1I:24
for f being ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence)
for G being ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board)
for k being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) st f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) is_sequence_on G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) & 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) <= k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) & k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) <= len f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) & f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) /. k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) = S-max (L~ f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) holds
ex i, j being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) st
( [(i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ] : ( ( ) ( ) set ) in Indices G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( ) set ) & [i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ] : ( ( ) ( ) set ) in Indices G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) : ( ( ) ( ) set ) & f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) /. k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) = G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * ((i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) & f : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) special_circular_sequence) /. (k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) = G : ( ( non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) Go-board) * (i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ) ;

theorem :: JORDAN1I:25
for f being ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard ) special_circular_sequence) holds
( f : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard ) special_circular_sequence) is clockwise_oriented iff (Rotate (f : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard ) special_circular_sequence) ,(W-min (L~ f : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard ) special_circular_sequence) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) )) : ( ( ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) /. 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) in W-most (L~ f : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard ) special_circular_sequence) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN1I:26
for f being ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard ) special_circular_sequence) holds
( f : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard ) special_circular_sequence) is clockwise_oriented iff (Rotate (f : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard ) special_circular_sequence) ,(E-max (L~ f : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard ) special_circular_sequence) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) )) : ( ( ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) /. 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) in E-most (L~ f : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard ) special_circular_sequence) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN1I:27
for f being ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard ) special_circular_sequence) holds
( f : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard ) special_circular_sequence) is clockwise_oriented iff (Rotate (f : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard ) special_circular_sequence) ,(S-max (L~ f : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard ) special_circular_sequence) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) )) : ( ( ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) /. 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) in S-most (L~ f : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard ) special_circular_sequence) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: JORDAN1I:28
for i, k being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) )
for C being ( ( non empty being_simple_closed_curve compact non horizontal non vertical ) ( non empty connected bounded being_simple_closed_curve compact non horizontal non vertical ) Subset of )
for p being ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Point of ( ( ) ( non empty V2() ) set ) ) st p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Point of ( ( ) ( non empty V2() ) set ) ) `1 : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) = ((W-bound C : ( ( non empty being_simple_closed_curve compact non horizontal non vertical ) ( non empty connected bounded being_simple_closed_curve compact non horizontal non vertical ) Subset of ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) + (E-bound C : ( ( non empty being_simple_closed_curve compact non horizontal non vertical ) ( non empty connected bounded being_simple_closed_curve compact non horizontal non vertical ) Subset of ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) / 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) & i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) > 0 : ( ( ) ( empty V4() V5() V6() V8() V9() V10() V11() V12() Function-like functional V37() ext-real V92() V93() V94() V95() V96() V97() V98() V99() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) & 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) <= k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) & k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) <= width (Gauge (C : ( ( non empty being_simple_closed_curve compact non horizontal non vertical ) ( non empty connected bounded being_simple_closed_curve compact non horizontal non vertical ) Subset of ) ,i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) & (Gauge (C : ( ( non empty being_simple_closed_curve compact non horizontal non vertical ) ( non empty connected bounded being_simple_closed_curve compact non horizontal non vertical ) Subset of ) ,i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) * ((Center (Gauge (C : ( ( non empty being_simple_closed_curve compact non horizontal non vertical ) ( non empty connected bounded being_simple_closed_curve compact non horizontal non vertical ) Subset of ) ,i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) in Upper_Arc (L~ (Cage (C : ( ( non empty being_simple_closed_curve compact non horizontal non vertical ) ( non empty connected bounded being_simple_closed_curve compact non horizontal non vertical ) Subset of ) ,i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( non empty ) ( non empty ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) & p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Point of ( ( ) ( non empty V2() ) set ) ) `2 : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) = upper_bound (proj2 : ( ( Function-like V40( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) , REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) ) ( V13() V16( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) V17( REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) Function-like V40( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) , REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) ) Element of K6(K7( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ,REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) .: ((LSeg (((Gauge (C : ( ( non empty being_simple_closed_curve compact non horizontal non vertical ) ( non empty connected bounded being_simple_closed_curve compact non horizontal non vertical ) Subset of ) ,1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) * ((Center (Gauge (C : ( ( non empty being_simple_closed_curve compact non horizontal non vertical ) ( non empty connected bounded being_simple_closed_curve compact non horizontal non vertical ) Subset of ) ,1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ,((Gauge (C : ( ( non empty being_simple_closed_curve compact non horizontal non vertical ) ( non empty connected bounded being_simple_closed_curve compact non horizontal non vertical ) Subset of ) ,i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) * ((Center (Gauge (C : ( ( non empty being_simple_closed_curve compact non horizontal non vertical ) ( non empty connected bounded being_simple_closed_curve compact non horizontal non vertical ) Subset of ) ,i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,k : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) )) : ( ( ) ( boundary connected ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) /\ (Lower_Arc (L~ (Cage (C : ( ( non empty being_simple_closed_curve compact non horizontal non vertical ) ( non empty connected bounded being_simple_closed_curve compact non horizontal non vertical ) Subset of ) ,i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( non empty non constant circular special unfolded s.c.c. standard clockwise_oriented ) ( non empty V2() V13() V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard clockwise_oriented ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( non empty boundary connected bounded being_simple_closed_curve compact ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty ) ( non empty ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V93() V94() V95() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() V12() ext-real ) Element of REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) holds
ex j being ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) st
( 1 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) <= width (Gauge (C : ( ( non empty being_simple_closed_curve compact non horizontal non vertical ) ( non empty connected bounded being_simple_closed_curve compact non horizontal non vertical ) Subset of ) ,i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) & p : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Point of ( ( ) ( non empty V2() ) set ) ) = (Gauge (C : ( ( non empty being_simple_closed_curve compact non horizontal non vertical ) ( non empty connected bounded being_simple_closed_curve compact non horizontal non vertical ) Subset of ) ,i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) * ((Center (Gauge (C : ( ( non empty being_simple_closed_curve compact non horizontal non vertical ) ( non empty connected bounded being_simple_closed_curve compact non horizontal non vertical ) Subset of ) ,i : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( tabular ) ( V13() non empty-yielding V16( NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) V17(K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) Function-like FinSequence-like tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column ) FinSequence of K286( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) M11( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) )) ) ) : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( V4() V5() V6() V10() V11() V12() V37() ext-real V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V33(2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V84() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V4() V5() V6() V10() V11() V12() V37() ext-real positive V92() V93() V94() V95() V96() V97() V98() ) Element of NAT : ( ( ) ( non empty V4() V5() V6() V93() V94() V95() V96() V97() V98() V99() ) Element of K6(REAL : ( ( ) ( non empty V26() V93() V94() V95() V99() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V49() TopSpace-like V131() V156() V157() V158() V159() V160() V161() V162() strict add-continuous Mult-continuous ) RLTopStruct ) : ( ( ) ( non empty V2() ) set ) ) ) ;