:: SPRECT_4 semantic presentation

begin

theorem :: SPRECT_4:1
for f being ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2)
for Q being ( ( closed ) ( functional closed ) Subset of ) st L~ f : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2) : ( ( ) ( non empty functional closed V78( TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) ) ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) meets Q : ( ( closed ) ( functional closed ) Subset of ) & not f : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2) /. 1 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V13() Function-like V33(2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V196() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) in Q : ( ( closed ) ( functional closed ) Subset of ) holds
(L~ (R_Cut (f : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2) ,(First_Point ((L~ f : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2) ) : ( ( ) ( non empty functional closed V78( TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) ) ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ,(f : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2) /. 1 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V13() Function-like V33(2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V196() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ,(f : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2) /. (len f : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2) ) : ( ( ) ( V10() V11() V12() V91() ext-real V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V13() Function-like V33(2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V196() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ,Q : ( ( closed ) ( functional closed ) Subset of ) )) : ( ( ) ( V13() Function-like V33(2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V196() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) )) : ( ( ) ( V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like FinSequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( functional closed V78( TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) ) ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) /\ Q : ( ( closed ) ( functional closed ) Subset of ) : ( ( ) ( functional ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) = {(First_Point ((L~ f : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2) ) : ( ( ) ( non empty functional closed V78( TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) ) ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ,(f : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2) /. 1 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V13() Function-like V33(2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V196() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ,(f : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2) /. (len f : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2) ) : ( ( ) ( V10() V11() V12() V91() ext-real V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V13() Function-like V33(2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V196() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ,Q : ( ( closed ) ( functional closed ) Subset of ) )) : ( ( ) ( V13() Function-like V33(2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V196() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) } : ( ( ) ( non empty V2() functional ) set ) ;

theorem :: SPRECT_4:2
for f being ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty V2() functional ) set ) )
for p being ( ( ) ( V13() Function-like V33(2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V196() ) Point of ( ( ) ( non empty V2() functional ) set ) ) st f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty V2() functional ) set ) ) is being_S-Seq & p : ( ( ) ( V13() Function-like V33(2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V196() ) Point of ( ( ) ( non empty V2() functional ) set ) ) = f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty V2() functional ) set ) ) /. (len f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( V10() V11() V12() V91() ext-real V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V13() Function-like V33(2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V196() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) holds
L~ (L_Cut (f : ( ( non empty ) ( non empty V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like FinSequence-like ) FinSequence of ( ( ) ( non empty V2() functional ) set ) ) ,p : ( ( ) ( V13() Function-like V33(2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V196() ) Point of ( ( ) ( non empty V2() functional ) set ) ) )) : ( ( ) ( V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like FinSequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( functional closed V78( TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) ) ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) = {} : ( ( ) ( empty V2() Function-like functional V204() V205() V206() V207() V208() V209() V210() ) set ) ;

theorem :: SPRECT_4:3
for f being ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2)
for p being ( ( ) ( V13() Function-like V33(2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V196() ) Point of ( ( ) ( non empty V2() functional ) set ) )
for j being ( ( ) ( V10() V11() V12() V91() ext-real V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( V10() V11() V12() V91() ext-real V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( V10() V11() V12() V91() ext-real V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) < len f : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2) : ( ( ) ( V10() V11() V12() V91() ext-real V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) & p : ( ( ) ( V13() Function-like V33(2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V196() ) Point of ( ( ) ( non empty V2() functional ) set ) ) in L~ (mid (f : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2) ,j : ( ( ) ( V10() V11() V12() V91() ext-real V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ,(len f : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2) ) : ( ( ) ( V10() V11() V12() V91() ext-real V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like FinSequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( functional closed V78( TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) ) ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) holds
LE f : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2) /. j : ( ( ) ( V10() V11() V12() V91() ext-real V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V13() Function-like V33(2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V196() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ,p : ( ( ) ( V13() Function-like V33(2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V196() ) Point of ( ( ) ( non empty V2() functional ) set ) ) , L~ f : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2) : ( ( ) ( non empty functional closed V78( TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) ) ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ,f : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2) /. 1 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V13() Function-like V33(2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V196() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ,f : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2) /. (len f : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2) ) : ( ( ) ( V10() V11() V12() V91() ext-real V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V13() Function-like V33(2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V196() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ;

theorem :: SPRECT_4:4
for f being ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2)
for p, q being ( ( ) ( V13() Function-like V33(2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V196() ) Point of ( ( ) ( non empty V2() functional ) set ) )
for j being ( ( ) ( V10() V11() V12() V91() ext-real V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) st 1 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) <= j : ( ( ) ( V10() V11() V12() V91() ext-real V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) & j : ( ( ) ( V10() V11() V12() V91() ext-real V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) < len f : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2) : ( ( ) ( V10() V11() V12() V91() ext-real V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) & p : ( ( ) ( V13() Function-like V33(2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V196() ) Point of ( ( ) ( non empty V2() functional ) set ) ) in LSeg (f : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2) ,j : ( ( ) ( V10() V11() V12() V91() ext-real V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( functional ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) & q : ( ( ) ( V13() Function-like V33(2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V196() ) Point of ( ( ) ( non empty V2() functional ) set ) ) in LSeg (p : ( ( ) ( V13() Function-like V33(2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V196() ) Point of ( ( ) ( non empty V2() functional ) set ) ) ,(f : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2) /. (j : ( ( ) ( V10() V11() V12() V91() ext-real V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) + 1 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V10() V11() V12() V91() ext-real V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V13() Function-like V33(2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V196() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( functional closed V78( TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) ) ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) holds
LE p : ( ( ) ( V13() Function-like V33(2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V196() ) Point of ( ( ) ( non empty V2() functional ) set ) ) ,q : ( ( ) ( V13() Function-like V33(2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V196() ) Point of ( ( ) ( non empty V2() functional ) set ) ) , L~ f : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2) : ( ( ) ( non empty functional closed V78( TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) ) ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ,f : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2) /. 1 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V13() Function-like V33(2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V196() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ,f : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2) /. (len f : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2) ) : ( ( ) ( V10() V11() V12() V91() ext-real V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V13() Function-like V33(2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V196() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ;

theorem :: SPRECT_4:5
for f being ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2)
for Q being ( ( closed ) ( functional closed ) Subset of ) st L~ f : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2) : ( ( ) ( non empty functional closed V78( TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) ) ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) meets Q : ( ( closed ) ( functional closed ) Subset of ) & not f : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2) /. (len f : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2) ) : ( ( ) ( V10() V11() V12() V91() ext-real V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V13() Function-like V33(2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V196() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) in Q : ( ( closed ) ( functional closed ) Subset of ) holds
(L~ (L_Cut (f : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2) ,(Last_Point ((L~ f : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2) ) : ( ( ) ( non empty functional closed V78( TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) ) ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ,(f : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2) /. 1 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V13() Function-like V33(2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V196() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ,(f : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2) /. (len f : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2) ) : ( ( ) ( V10() V11() V12() V91() ext-real V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V13() Function-like V33(2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V196() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ,Q : ( ( closed ) ( functional closed ) Subset of ) )) : ( ( ) ( V13() Function-like V33(2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V196() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) )) : ( ( ) ( V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like FinSequence-like ) FinSequence of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ) : ( ( ) ( functional closed V78( TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) ) ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) /\ Q : ( ( closed ) ( functional closed ) Subset of ) : ( ( ) ( functional ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) = {(Last_Point ((L~ f : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2) ) : ( ( ) ( non empty functional closed V78( TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) ) ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ,(f : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2) /. 1 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V13() Function-like V33(2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V196() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ,(f : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2) /. (len f : ( ( being_S-Seq ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like one-to-one FinSequence-like special unfolded s.n.c. being_S-Seq ) S-Sequence_in_R2) ) : ( ( ) ( V10() V11() V12() V91() ext-real V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V13() Function-like V33(2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V196() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) ,Q : ( ( closed ) ( functional closed ) Subset of ) )) : ( ( ) ( V13() Function-like V33(2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like V196() ) Element of the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) } : ( ( ) ( non empty V2() functional ) set ) ;

theorem :: SPRECT_4:6
for f being ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard ) special_circular_sequence) holds LeftComp f : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard ) special_circular_sequence) : ( ( ) ( non empty functional being_Region ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) <> RightComp f : ( ( non empty non constant circular special unfolded s.c.c. standard ) ( non empty V2() V13() V16( NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) V17( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) Function-like non constant FinSequence-like circular special unfolded s.c.c. standard ) special_circular_sequence) : ( ( ) ( non empty functional being_Region ) Element of K6( the U1 of (TOP-REAL 2 : ( ( ) ( non empty V10() V11() V12() V91() ext-real positive V193() V204() V205() V206() V207() V208() V209() ) Element of NAT : ( ( ) ( V204() V205() V206() V207() V208() V209() V210() ) Element of K6(REAL : ( ( ) ( V204() V205() V206() V210() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( strict ) ( non empty V47() TopSpace-like T_2 V108() V139() V140() V141() V142() V143() V144() V145() strict ) RLTopStruct ) : ( ( ) ( non empty V2() functional ) set ) ) : ( ( ) ( ) set ) ) ;