environ
vocabularies HIDDEN, NUMBERS, TOPREAL2, SUBSET_1, EUCLID, RELAT_1, PRE_TOPC, FUNCT_1, STRUCT_0, JORDAN19, TARSKI, GOBOARD9, JORDAN9, CARD_1, JORDAN6, TOPREAL1, RELAT_2, PSCOMP_1, RCOMP_1, KURATO_2, XBOOLE_0, XXREAL_0, FINSEQ_1, JORDAN8, MATRIX_1, TREES_1, SPPOL_1, ARYTM_3, METRIC_1, ARYTM_1, NEWTON, ORDINAL2, RLTOPSP1, JORDAN1A, MCART_1, XXREAL_2, SEQ_4, GOBOARD2, JORDAN21, JORDAN2C, GOBOARD1, FINSEQ_6, GOBOARD5, PARTFUN1, PCOMPS_1, WEIERSTR, SEQ_1, SEQ_2, REAL_1, COMPLEX1, JORDAN10, JORDAN3, CONVEX1, NAT_1;
notations HIDDEN, TARSKI, XBOOLE_0, SUBSET_1, FUNCT_1, ORDINAL1, NUMBERS, XXREAL_0, XCMPLX_0, XREAL_0, REAL_1, COMPLEX1, NAT_1, NAT_D, RELSET_1, PARTFUN1, FUNCT_2, XXREAL_2, SEQ_4, NEWTON, FINSEQ_1, MATRIX_0, DOMAIN_1, STRUCT_0, PRE_TOPC, COMPTS_1, FINSEQ_6, CONNSP_1, METRIC_1, PCOMPS_1, PSCOMP_1, RLVECT_1, RLTOPSP1, EUCLID, TOPREAL1, TOPREAL2, GOBOARD1, GOBOARD2, WEIERSTR, SPPOL_1, GOBOARD5, TOPRNS_1, JORDAN2C, JORDAN6, GOBOARD9, JORDAN8, JORDAN9, GOBRD13, TOPREAL6, JORDAN10, JORDAN1K, JORDAN1A, KURATO_2, JORDAN19, JORDAN21;
definitions TARSKI, TOPRNS_1, XXREAL_2;
theorems XBOOLE_1, JORDAN1A, JORDAN6, XBOOLE_0, JORDAN1K, EUCLID, TOPREAL1, TARSKI, JORDAN16, METRIC_1, SPRECT_3, TOPMETR, JORDAN7, GOBRD14, JORDAN1I, INT_1, GOBOARD5, FINSEQ_6, REVROT_1, SPRECT_2, GOBOARD7, JGRAPH_1, JORDAN1H, JORDAN9, GOBRD13, NAT_1, JORDAN8, SUBSET_1, JORDAN2C, JORDAN10, PSCOMP_1, SPPOL_2, SPPOL_1, SEQ_4, FUNCT_2, WEIERSTR, JORDAN1G, TOPRNS_1, ABSVALUE, JORDAN19, KURATO_2, JORDAN21, XREAL_1, JORDAN1J, JORDAN1B, RELAT_1, JORDAN5D, TOPREAL3, JORDAN1F, NEWTON, SPRECT_1, XXREAL_0, PEPIN, TOPREAL6, JCT_MISC, MATRIX_0, XXREAL_2, NAT_D, XREAL_0, COMPTS_1, PRE_TOPC, RLTOPSP1, ORDINAL1;
schemes FUNCT_2;
registrations XBOOLE_0, RELSET_1, NUMBERS, XXREAL_0, XREAL_0, NAT_1, INT_1, MEMBERED, FINSEQ_6, STRUCT_0, COMPTS_1, EUCLID, TOPREAL1, TOPREAL2, GOBOARD2, SPPOL_2, TOPREAL5, SPRECT_2, JORDAN6, SPRECT_3, JORDAN2C, REVROT_1, TOPREAL6, JORDAN21, JORDAN8, JORDAN10, VALUED_0, FUNCT_1, ORDINAL1, RLTOPSP1, JORDAN1, XCMPLX_0, NEWTON;
constructors HIDDEN, REAL_1, RCOMP_1, NEWTON, BINARITH, CONNSP_1, MONOID_0, TOPRNS_1, TOPREAL4, GOBOARD2, PSCOMP_1, GOBOARD9, JORDAN5C, JORDAN6, JORDAN2C, TOPREAL6, JORDAN1K, JORDAN21, JORDAN8, GOBRD13, JORDAN9, JORDAN10, JORDAN1A, KURATO_2, JORDAN19, NAT_D, SEQ_4, FUNCSDOM, CONVEX1, JORDAN11;
requirements HIDDEN, SUBSET, BOOLE, REAL, NUMERALS, ARITHM;
equalities JORDAN6, JORDAN21;
expansions TARSKI, JORDAN6, XXREAL_2;
Lm1:
for r being Real
for X being Subset of (TOP-REAL 2) st r in proj2 .: X holds
ex x being Point of (TOP-REAL 2) st
( x in X & proj2 . x = r )
Lm2:
dom proj2 = the carrier of (TOP-REAL 2)
by FUNCT_2:def 1;
Lm3:
for R being non empty Subset of (TOP-REAL 2)
for n being Nat holds 1 <= len (Gauge (R,n))
theorem Th8:
for
i,
j,
k,
m being
Nat for
C being
compact non
horizontal non
vertical Subset of
(TOP-REAL 2) st
m > k &
[i,j] in Indices (Gauge (C,k)) &
[i,(j + 1)] in Indices (Gauge (C,k)) holds
dist (
((Gauge (C,m)) * (i,j)),
((Gauge (C,m)) * (i,(j + 1))))
< dist (
((Gauge (C,k)) * (i,j)),
((Gauge (C,k)) * (i,(j + 1))))
theorem Th9:
for
k,
m being
Nat for
C being
compact non
horizontal non
vertical Subset of
(TOP-REAL 2) st
m > k holds
dist (
((Gauge (C,m)) * (1,1)),
((Gauge (C,m)) * (1,2)))
< dist (
((Gauge (C,k)) * (1,1)),
((Gauge (C,k)) * (1,2)))
theorem Th10:
for
i,
j,
k,
m being
Nat for
C being
compact non
horizontal non
vertical Subset of
(TOP-REAL 2) st
m > k &
[i,j] in Indices (Gauge (C,k)) &
[(i + 1),j] in Indices (Gauge (C,k)) holds
dist (
((Gauge (C,m)) * (i,j)),
((Gauge (C,m)) * ((i + 1),j)))
< dist (
((Gauge (C,k)) * (i,j)),
((Gauge (C,k)) * ((i + 1),j)))
theorem Th11:
for
k,
m being
Nat for
C being
compact non
horizontal non
vertical Subset of
(TOP-REAL 2) st
m > k holds
dist (
((Gauge (C,m)) * (1,1)),
((Gauge (C,m)) * (2,1)))
< dist (
((Gauge (C,k)) * (1,1)),
((Gauge (C,k)) * (2,1)))
theorem
for
C being
Simple_closed_curve for
i being
Nat for
r,
t being
Real st
r > 0 &
t > 0 holds
ex
n being
Nat st
(
i < n &
dist (
((Gauge (C,n)) * (1,1)),
((Gauge (C,n)) * (1,2)))
< r &
dist (
((Gauge (C,n)) * (1,1)),
((Gauge (C,n)) * (2,1)))
< t )
theorem Th13:
for
C being
Simple_closed_curve for
n being
Nat st
0 < n holds
upper_bound (proj2 .: ((L~ (Cage (C,n))) /\ (LSeg (((Gauge (C,n)) * ((Center (Gauge (C,n))),1)),((Gauge (C,n)) * ((Center (Gauge (C,n))),(len (Gauge (C,n))))))))) = upper_bound (proj2 .: ((L~ (Cage (C,n))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,n)))) + (W-bound (L~ (Cage (C,n))))) / 2))))
theorem Th14:
for
C being
Simple_closed_curve for
n being
Nat st
0 < n holds
lower_bound (proj2 .: ((L~ (Cage (C,n))) /\ (LSeg (((Gauge (C,n)) * ((Center (Gauge (C,n))),1)),((Gauge (C,n)) * ((Center (Gauge (C,n))),(len (Gauge (C,n))))))))) = lower_bound (proj2 .: ((L~ (Cage (C,n))) /\ (Vertical_Line (((E-bound (L~ (Cage (C,n)))) + (W-bound (L~ (Cage (C,n))))) / 2))))
theorem
for
C being
Simple_closed_curve for
n being
Nat st
0 < n holds
UMP (L~ (Cage (C,n))) = |[(((E-bound (L~ (Cage (C,n)))) + (W-bound (L~ (Cage (C,n))))) / 2),(upper_bound (proj2 .: ((L~ (Cage (C,n))) /\ (LSeg (((Gauge (C,n)) * ((Center (Gauge (C,n))),1)),((Gauge (C,n)) * ((Center (Gauge (C,n))),(len (Gauge (C,n))))))))))]| by Th13;
theorem
for
C being
Simple_closed_curve for
n being
Nat st
0 < n holds
LMP (L~ (Cage (C,n))) = |[(((E-bound (L~ (Cage (C,n)))) + (W-bound (L~ (Cage (C,n))))) / 2),(lower_bound (proj2 .: ((L~ (Cage (C,n))) /\ (LSeg (((Gauge (C,n)) * ((Center (Gauge (C,n))),1)),((Gauge (C,n)) * ((Center (Gauge (C,n))),(len (Gauge (C,n))))))))))]| by Th14;
Lm4:
TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) = TopSpaceMetr (Euclid 2)
by EUCLID:def 8;