environ
vocabularies HIDDEN, NUMBERS, NAT_1, XXREAL_0, ARYTM_1, ARYTM_3, CARD_1, STRUCT_0, BORSUK_1, PRE_TOPC, EUCLID, MCART_1, RLTOPSP1, REAL_1, RELAT_1, SUPINF_2, XXREAL_1, FINSEQ_1, XBOOLE_0, SUBSET_1, FUNCT_1, TOPREAL1, TOPS_2, PARTFUN1, TOPMETR, TREAL_1, VALUED_1, FUNCT_4, RCOMP_1, TARSKI, ORDINAL2, PCOMPS_1, METRIC_1, JORDAN3, RFINSEQ, FINSEQ_5, GROUP_2, ORDINAL4, GOBOARD5, GOBOARD2, TREES_1, TOPREAL4, GOBOARD1, MATRIX_1, FUNCT_2;
notations HIDDEN, TARSKI, XBOOLE_0, ORDINAL1, SUBSET_1, NUMBERS, XXREAL_0, XCMPLX_0, XREAL_0, REAL_1, NAT_1, NAT_D, RCOMP_1, RELAT_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, FINSEQ_1, FINSEQ_4, TOPMETR, FINSEQ_6, SEQ_4, JORDAN3, STRUCT_0, TOPREAL1, RFINSEQ, PRE_TOPC, TOPS_2, COMPTS_1, BORSUK_1, RLVECT_1, RLTOPSP1, EUCLID, TREAL_1, METRIC_1, GOBOARD1, GOBOARD2, MATRIX_0, TOPREAL4, GOBOARD5, FUNCT_4, FINSEQ_5, PCOMPS_1;
definitions TARSKI, JORDAN3, TOPREAL4, XBOOLE_0;
theorems BORSUK_1, COMPTS_1, EUCLID, FINSEQ_3, FINSEQ_4, FUNCT_1, FUNCT_2, RELAT_1, JORDAN3, GOBOARD1, NAT_1, GOBOARD9, FUNCT_4, TOPMETR2, GOBOARD2, PRE_TOPC, RCOMP_1, HEINE, PCOMPS_1, SPPOL_2, TARSKI, TOPMETR, TOPREAL1, TOPREAL3, RFINSEQ, FINSEQ_5, GOBOARD5, GOBOARD7, FINSEQ_1, TOPS_2, TREAL_1, JORDAN5A, ZFMISC_1, JORDAN4, PARTFUN2, INT_1, RELSET_1, FINSEQ_6, XBOOLE_0, XBOOLE_1, XCMPLX_1, FINSEQ_2, XREAL_1, XXREAL_0, PARTFUN1, CARD_1, MATRIX_0, XXREAL_1, XXREAL_2, NAT_D, RLTOPSP1, SEQ_4, RLVECT_1;
schemes NAT_1;
registrations FUNCT_1, RELSET_1, FUNCT_2, XREAL_0, MEMBERED, STRUCT_0, PRE_TOPC, METRIC_1, BORSUK_1, EUCLID, TOPMETR, TOPREAL1, GOBOARD2, GOBOARD5, VALUED_0, XXREAL_2, RLTOPSP1, SEQ_4, SPPOL_2, NAT_1, ORDINAL1;
constructors HIDDEN, FUNCT_4, REAL_1, SEQ_4, RCOMP_1, FINSEQ_4, RFINSEQ, NAT_D, FINSEQ_5, TOPS_2, COMPTS_1, TREAL_1, TOPREAL4, GOBOARD2, GOBOARD5, JORDAN3, GOBOARD1, CONVEX1, BINOP_2, PCOMPS_1, MONOID_0;
requirements HIDDEN, REAL, NUMERALS, BOOLE, SUBSET, ARITHM;
equalities XBOOLE_0, STRUCT_0, RELAT_1, RLTOPSP1, RLVECT_1;
expansions TARSKI, JORDAN3, XBOOLE_0;
theorem Th1:
for
i1 being
Nat st 1
<= i1 holds
i1 -' 1
< i1
theorem
for
i,
k being
Nat st
i + 1
<= k holds
1
<= k -' i
theorem
for
i,
k being
Nat st 1
<= i & 1
<= k holds
(k -' i) + 1
<= k
Lm1:
for r being Real st 0 <= r & r <= 1 holds
( 0 <= 1 - r & 1 - r <= 1 )
reconsider jj = 1 as Real ;
theorem
for
p1,
p2,
q1,
q2,
q3 being
Point of
(TOP-REAL 2) st
p1 <> p2 &
LE q1,
q2,
p1,
p2 &
LE q2,
q3,
p1,
p2 holds
LE q1,
q3,
p1,
p2
Lm2:
for f being FinSequence of (TOP-REAL 2)
for p, q being Point of (TOP-REAL 2) st p in L~ f & q in L~ f & p <> f . (len f) & f is being_S-Seq & not p in L~ (L_Cut (f,q)) holds
q in L~ (L_Cut (f,p))
Lm3:
for f being FinSequence of (TOP-REAL 2)
for p, q being Point of (TOP-REAL 2) st p in L~ f & q in L~ f & ( Index (p,f) < Index (q,f) or ( Index (p,f) = Index (q,f) & LE p,q,f /. (Index (p,f)),f /. ((Index (p,f)) + 1) ) ) & p <> q holds
L~ (B_Cut (f,p,q)) c= L~ f
theorem
for
f being
FinSequence of
(TOP-REAL 2) for
p,
q being
Point of
(TOP-REAL 2) st
p in L~ f &
q in L~ f & (
Index (
p,
f)
< Index (
q,
f) or (
Index (
p,
f)
= Index (
q,
f) &
LE p,
q,
f /. (Index (p,f)),
f /. ((Index (p,f)) + 1) ) ) &
p <> q holds
L~ (B_Cut (f,p,q)) c= L~ f by Lm3;