:: Properties of the Upper and Lower Sequence on the Cage
:: by Robert Milewski
::
:: Received August 1, 2002
:: Copyright (c) 2002-2026 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies NUMBERS, SUBSET_1, EUCLID, XBOOLE_0, RELAT_1, PRE_TOPC, MCART_1,
FUNCT_1, TARSKI, JORDAN6, REAL_1, RCOMP_1, JORDAN2C, GOBOARD1, XXREAL_0,
FINSEQ_1, TREES_1, RLTOPSP1, SPPOL_1, JORDAN1A, RELAT_2, JORDAN8,
TOPREAL1, JORDAN1E, MATRIX_1, SEQ_4, PSCOMP_1, XXREAL_2, TOPREAL2,
ARYTM_3, CARD_1, JORDAN9, JORDAN3, PARTFUN1, GOBOARD5, ARYTM_1, FINSEQ_4,
GRAPH_2, FINSEQ_6, ZFMISC_1, GOBOARD9, GOBOARD2, FINSEQ_5, SPRECT_2,
GROUP_2, NAT_1, CONNSP_1;
notations TARSKI, XBOOLE_0, ENUMSET1, SUBSET_1, ORDINAL1, NUMBERS, XXREAL_0,
XCMPLX_0, XREAL_0, REAL_1, NAT_1, NAT_D, FUNCT_1, RELSET_1, PARTFUN1,
FINSEQ_1, FINSEQ_4, FINSEQ_5, XXREAL_2, SEQ_4, MATRIX_0, FINSEQ_6,
GRAPH_2, STRUCT_0, PRE_TOPC, RCOMP_1, TOPREAL2, CONNSP_1, COMPTS_1,
RLTOPSP1, EUCLID, MEASURE6, PSCOMP_1, SPRECT_2, GOBOARD1, TOPREAL1,
GOBOARD2, GOBOARD5, GOBOARD9, GOBRD13, SPPOL_1, JORDAN3, JORDAN8,
JORDAN2C, JORDAN6, JORDAN9, JORDAN1A, JORDAN1E;
constructors REAL_1, RCOMP_1, FINSEQ_4, NEWTON, BINARITH, CONNSP_1, REALSET2,
GOBOARD2, PSCOMP_1, GRAPH_2, GOBOARD9, JORDAN3, JORDAN5C, JORDAN6,
JORDAN2C, JORDAN8, GOBRD13, JORDAN9, JORDAN1A, JORDAN1E, NAT_D, SEQ_4,
RELSET_1, CONVEX1, MEASURE6;
registrations XBOOLE_0, RELSET_1, NAT_1, MEMBERED, FINSEQ_1, FINSEQ_6,
STRUCT_0, COMPTS_1, TOPREAL1, TOPREAL2, GOBOARD2, JORDAN1, SPPOL_2,
PSCOMP_1, GOBRD11, TOPREAL5, SPRECT_1, SPRECT_2, REVROT_1, TOPREAL6,
JORDAN8, JORDAN1E, JORDAN1J, FUNCT_1, EUCLID, RLTOPSP1, MEASURE6,
JORDAN5A, JORDAN2C, XREAL_0, ORDINAL1;
requirements NUMERALS, BOOLE, SUBSET, REAL, ARITHM;
definitions TARSKI, XBOOLE_0;
equalities PSCOMP_1;
expansions TARSKI, XBOOLE_0;
theorems NAT_1, FINSEQ_1, GOBOARD1, FINSEQ_4, EUCLID, FINSEQ_3, SPPOL_2,
TARSKI, JORDAN3, PSCOMP_1, FINSEQ_5, FINSEQ_6, GOBOARD7, TOPREAL1,
GOBOARD5, SPRECT_2, SPPOL_1, FUNCT_2, GOBOARD9, FINSEQ_2, SUBSET_1,
JORDAN4, SPRECT_3, TOPREAL3, JORDAN8, PARTFUN2, SPRECT_1, XBOOLE_0,
XBOOLE_1, ZFMISC_1, SEQ_4, GOBRD14, TOPREAL6, JORDAN2C, PRE_TOPC,
JORDAN6, JORDAN9, JORDAN1H, JORDAN1A, JORDAN1C, JORDAN1E, JORDAN10,
JGRAPH_1, REVROT_1, RCOMP_1, COMPTS_1, ENUMSET1, JORDAN1B, JORDAN1F,
JORDAN1G, JORDAN1I, JORDAN1J, GOBOARD3, TOPREAL8, GRAPH_2, SPRECT_5,
JORDAN1D, XREAL_1, XXREAL_0, JCT_MISC, PARTFUN1, MATRIX_0, XXREAL_2,
NAT_D, XREAL_0, RLTOPSP1;
begin
reserve n for Nat;
theorem :: Uogolnic i przenisc cmp. JORDAN1A:15
for A,B be Subset of TOP-REAL 2 st A meets B holds proj1.:A meets proj1.:B
proof
let A,B be Subset of TOP-REAL 2;
assume A meets B;
then consider e be object such that
A1: e in A and
A2: e in B by XBOOLE_0:3;
reconsider e as Point of TOP-REAL 2 by A1;
A3: e`1 = proj1.e by PSCOMP_1:def 5;
then
A4: e`1 in proj1.:B by A2,FUNCT_2:35;
e`1 in proj1.:A by A1,A3,FUNCT_2:35;
hence thesis by A4,XBOOLE_0:3;
end;
theorem
for A,B be Subset of TOP-REAL 2 for s be Real st A misses B & A
c= Horizontal_Line s & B c= Horizontal_Line s holds proj1.:A misses proj1.:B
proof
let A,B be Subset of TOP-REAL 2;
let s be Real such that
A1: A misses B and
A2: A c= Horizontal_Line s and
A3: B c= Horizontal_Line s;
assume proj1.:A meets proj1.:B;
then consider e be object such that
A4: e in proj1.:A and
A5: e in proj1.:B by XBOOLE_0:3;
reconsider e as Real by A4;
consider d be Point of TOP-REAL 2 such that
A6: d in B and
A7: e = proj1.d by A5,FUNCT_2:65;
A8: d`2 = s by A3,A6,JORDAN6:32;
consider c being Point of TOP-REAL 2 such that
A9: c in A and
A10: e = proj1.c by A4,FUNCT_2:65;
c`2 = s by A2,A9,JORDAN6:32;
then c = |[c`1,d`2]| by A8,EUCLID:53
.= |[e,d`2]| by A10,PSCOMP_1:def 5
.= |[d`1,d`2]| by A7,PSCOMP_1:def 5
.= d by EUCLID:53;
hence contradiction by A1,A9,A6,XBOOLE_0:3;
end;
theorem Th3:
for S be closed Subset of TOP-REAL 2 st S is bounded holds proj1
.:S is closed
proof
let S be closed Subset of TOP-REAL 2;
assume S is bounded;
then Cl(proj1.:S) = proj1.:Cl S by TOPREAL6:83
.= proj1.:S by PRE_TOPC:22;
hence thesis;
end;
theorem Th4:
for S be compact Subset of TOP-REAL 2 holds proj1.:S is compact
proof
let S being compact Subset of TOP-REAL 2;
proj1.:S is closed by Th3;
hence thesis by JORDAN1C:3,RCOMP_1:11;
end;
theorem Th5:
for G be Go-board for i,j,k,j1,k1 be Nat st 1 <= i & i
<= len G & 1 <= j & j <= j1 & j1 <= k1 & k1 <= k & k <= width G holds LSeg(G*(i
,j1),G*(i,k1)) c= LSeg(G*(i,j),G*(i,k))
proof
let G be Go-board;
let i,j,k,j1,k1 be Nat;
assume that
A1: 1 <= i and
A2: i <= len G and
A3: 1 <= j and
A4: j <= j1 and
A5: j1 <= k1 and
A6: k1 <= k and
A7: k <= width G;
A8: j1 <= k by A5,A6,XXREAL_0:2;
j <= k1 by A4,A5,XXREAL_0:2;
then
A9: 1 <= k1 by A3,XXREAL_0:2;
then
A10: G*(i,k1)`2 <= G*(i,k)`2 by A1,A2,A6,A7,SPRECT_3:12;
A11: 1 <= j1 by A3,A4,XXREAL_0:2;
1 <= j1 by A3,A4,XXREAL_0:2;
then
A12: 1 <= k by A8,XXREAL_0:2;
A13: k1 <= width G by A6,A7,XXREAL_0:2;
j <= k1 by A4,A5,XXREAL_0:2;
then
A14: j <= width G by A13,XXREAL_0:2;
then G*(i,j)`1 = G*(i,1)`1 by A1,A2,A3,GOBOARD5:2
.= G*(i,k)`1 by A1,A2,A7,A12,GOBOARD5:2;
then
A15: LSeg(G*(i,j),G*(i,k)) is vertical by SPPOL_1:16;
j1 <= k by A5,A6,XXREAL_0:2;
then
A16: j1 <= width G by A7,XXREAL_0:2;
then
A17: G*(i,j)`2 <= G*(i,j1)`2 by A1,A2,A3,A4,SPRECT_3:12;
A18: k1 <= width G by A6,A7,XXREAL_0:2;
then
A19: G*(i,j1)`2 <= G*(i,k1)`2 by A1,A2,A5,A11,SPRECT_3:12;
G*(i,j1)`1 = G*(i,1)`1 by A1,A2,A11,A16,GOBOARD5:2
.= G*(i,k1)`1 by A1,A2,A9,A18,GOBOARD5:2;
then
A20: LSeg(G*(i,j1),G*(i,k1)) is vertical by SPPOL_1:16;
G*(i,j)`1 = G*(i,1)`1 by A1,A2,A3,A14,GOBOARD5:2
.= G*(i,j1)`1 by A1,A2,A11,A16,GOBOARD5:2;
hence thesis by A15,A20,A17,A19,A10,GOBOARD7:63;
end;
theorem Th6:
for G be Go-board for i,j,k,j1,k1 be Nat st 1 <= i & i
<= width G & 1 <= j & j <= j1 & j1 <= k1 & k1 <= k & k <= len G holds LSeg(G*(
j1,i),G*(k1,i)) c= LSeg(G*(j,i),G*(k,i))
proof
let G be Go-board;
let i,j,k,j1,k1 be Nat;
assume that
A1: 1 <= i and
A2: i <= width G and
A3: 1 <= j and
A4: j <= j1 and
A5: j1 <= k1 and
A6: k1 <= k and
A7: k <= len G;
A8: j1 <= k by A5,A6,XXREAL_0:2;
j <= k1 by A4,A5,XXREAL_0:2;
then
A9: 1 <= k1 by A3,XXREAL_0:2;
then
A10: G*(k1,i)`1 <= G*(k,i)`1 by A1,A2,A6,A7,SPRECT_3:13;
A11: 1 <= j1 by A3,A4,XXREAL_0:2;
1 <= j1 by A3,A4,XXREAL_0:2;
then
A12: 1 <= k by A8,XXREAL_0:2;
A13: k1 <= len G by A6,A7,XXREAL_0:2;
j <= k1 by A4,A5,XXREAL_0:2;
then
A14: j <= len G by A13,XXREAL_0:2;
then G*(j,i)`2 = G*(1,i)`2 by A1,A2,A3,GOBOARD5:1
.= G*(k,i)`2 by A1,A2,A7,A12,GOBOARD5:1;
then
A15: LSeg(G*(j,i),G*(k,i)) is horizontal by SPPOL_1:15;
j1 <= k by A5,A6,XXREAL_0:2;
then
A16: j1 <= len G by A7,XXREAL_0:2;
then
A17: G*(j,i)`1 <= G*(j1,i)`1 by A1,A2,A3,A4,SPRECT_3:13;
A18: k1 <= len G by A6,A7,XXREAL_0:2;
then
A19: G*(j1,i)`1 <= G*(k1,i)`1 by A1,A2,A5,A11,SPRECT_3:13;
G*(j1,i)`2 = G*(1,i)`2 by A1,A2,A11,A16,GOBOARD5:1
.= G*(k1,i)`2 by A1,A2,A9,A18,GOBOARD5:1;
then
A20: LSeg(G*(j1,i),G*(k1,i)) is horizontal by SPPOL_1:15;
G*(j,i)`2 = G*(1,i)`2 by A1,A2,A3,A14,GOBOARD5:1
.= G*(j1,i)`2 by A1,A2,A11,A16,GOBOARD5:1;
hence thesis by A15,A20,A17,A19,A10,GOBOARD7:64;
end;
theorem
for G be Go-board for j,k,j1,k1 be Nat st 1 <= j & j <= j1
& j1 <= k1 & k1 <= k & k <= width G holds LSeg(G*(Center G,j1),G*(Center G,k1))
c= LSeg(G*(Center G,j),G*(Center G,k))
proof
let G be Go-board;
let j,k,j1,k1 be Nat;
assume that
A1: 1 <= j and
A2: j <= j1 and
A3: j1 <= k1 and
A4: k1 <= k and
A5: k <= width G;
A6: Center G <= len G by JORDAN1B:13;
1 <= Center G by JORDAN1B:11;
hence thesis by A1,A2,A3,A4,A5,A6,Th5;
end;
theorem
for G be Go-board st len G = width G for j,k,j1,k1 be Nat
st 1 <= j & j <= j1 & j1 <= k1 & k1 <= k & k <= len G holds LSeg(G*(j1,Center G
),G*(k1,Center G)) c= LSeg(G*(j,Center G),G*(k,Center G))
proof
let G be Go-board;
assume len G = width G;
then
A1: Center G <= width G by JORDAN1B:13;
let j,k,j1,k1 be Nat;
assume that
A2: 1 <= j and
A3: j <= j1 and
A4: j1 <= k1 and
A5: k1 <= k and
A6: k <= len G;
1 <= Center G by JORDAN1B:11;
hence thesis by A2,A3,A4,A5,A6,A1,Th6;
end;
theorem Th9:
for C be compact connected non vertical non horizontal Subset of
TOP-REAL 2 for i,j,k be Nat st 1 <= i & i <= len Gauge(C,n) & 1 <= j
& j <= k & k <= width Gauge(C,n) & Gauge(C,n)*(i,j) in L~Lower_Seq(C,n) ex j1
be Nat st j <= j1 & j1 <= k & LSeg(Gauge(C,n)*(i,j1),Gauge(C,n)*(i,k
)) /\ L~Lower_Seq(C,n) = {Gauge(C,n)*(i,j1)}
proof
let C be compact connected non vertical non horizontal Subset of TOP-REAL 2;
let i,j,k be Nat;
assume that
A1: 1 <= i and
A2: i <= len Gauge(C,n) and
A3: 1 <= j and
A4: j <= k and
A5: k <= width Gauge(C,n) and
A6: Gauge(C,n)*(i,j) in L~Lower_Seq(C,n);
set G = Gauge(C,n);
A7: k >= 1 by A3,A4,XXREAL_0:2;
then
A8: [i,k] in Indices G by A1,A2,A5,MATRIX_0:30;
set X = LSeg(G*(i,j),G*(i,k)) /\ L~Lower_Seq(C,n);
A9: G*(i,j) in LSeg(G*(i,j),G*(i,k)) by RLTOPSP1:68;
then reconsider X1=X as non empty compact Subset of TOP-REAL 2 by A6,
XBOOLE_0:def 4;
A10: LSeg(G*(i,j),G*(i,k)) meets L~Lower_Seq(C,n) by A6,A9,XBOOLE_0:3;
set s = G*(i,1)`1;
set e = G*(i,k);
set f = G*(i,j);
set w2 = upper_bound(proj2.:(LSeg(f,e) /\ L~Lower_Seq(C,n)));
A11: j <= width G by A4,A5,XXREAL_0:2;
then [i,j] in Indices G by A1,A2,A3,MATRIX_0:30;
then consider j1 be Nat such that
A12: j <= j1 and
A13: j1 <= k and
A14: G*(i,j1)`2 = w2 by A4,A10,A8,JORDAN1F:2,JORDAN1G:5;
set q = |[s,w2]|;
A15: j1 <= width G by A5,A13,XXREAL_0:2;
A16: G*(i,k)`1 = s by A1,A2,A5,A7,GOBOARD5:2;
then f`1 = e`1 by A1,A2,A3,A11,GOBOARD5:2;
then
A17: LSeg(f,e) is vertical by SPPOL_1:16;
take j1;
thus j <= j1 & j1 <= k by A12,A13;
consider pp be object such that
A18: pp in N-most X1 by XBOOLE_0:def 1;
reconsider pp as Point of TOP-REAL 2 by A18;
A19: pp in X by A18,XBOOLE_0:def 4;
then
A20: pp in L~Lower_Seq(C,n) by XBOOLE_0:def 4;
A21: 1 <= j1 by A3,A12,XXREAL_0:2;
then
A22: G*(i,j1)`1 = s by A1,A2,A15,GOBOARD5:2;
then
A23: q = G*(i,j1) by A14,EUCLID:53;
then
A24: q`2 <= e`2 by A1,A2,A5,A13,A21,SPRECT_3:12;
A25: q`2 = N-bound X by A14,A23,SPRECT_1:45
.= (N-min X)`2 by EUCLID:52
.= pp`2 by A18,PSCOMP_1:39;
pp in LSeg(G*(i,j),G*(i,k)) by A19,XBOOLE_0:def 4;
then pp`1 = q`1 by A16,A22,A23,A17,SPPOL_1:41;
then
A26: q in L~Lower_Seq(C,n) by A20,A25,TOPREAL3:6;
for x be object holds x in LSeg(e,q) /\ L~Lower_Seq(C,n) iff x = q
proof
let x be object;
thus x in LSeg(e,q) /\ L~Lower_Seq(C,n) implies x = q
proof
reconsider EE = LSeg(f,e) /\ L~Lower_Seq(C,n) as compact Subset of
TOP-REAL 2;
reconsider E0 = proj2.:EE as compact Subset of REAL by JCT_MISC:15;
A27: e in LSeg(f,e) by RLTOPSP1:68;
A28: f`2 <= q`2 by A1,A2,A3,A12,A15,A23,SPRECT_3:12;
f`1 = q`1 by A1,A2,A3,A11,A22,A23,GOBOARD5:2;
then q in LSeg(e,f) by A16,A22,A23,A24,A28,GOBOARD7:7;
then
A29: LSeg(e,q) c= LSeg(f,e) by A27,TOPREAL1:6;
assume
A30: x in LSeg(e,q) /\ L~Lower_Seq(C,n);
then reconsider pp = x as Point of TOP-REAL 2;
A31: pp in LSeg(e,q) by A30,XBOOLE_0:def 4;
then
A32: pp`2 >= q`2 by A24,TOPREAL1:4;
pp in L~Lower_Seq(C,n) by A30,XBOOLE_0:def 4;
then pp in EE by A31,A29,XBOOLE_0:def 4;
then proj2.pp in E0 by FUNCT_2:35;
then
A33: pp`2 in E0 by PSCOMP_1:def 6;
E0 is real-bounded by RCOMP_1:10;
then E0 is bounded_above by XXREAL_2:def 11;
then q`2 >= pp`2 by A14,A23,A33,SEQ_4:def 1;
then
A34: pp`2 = q`2 by A32,XXREAL_0:1;
pp`1 = q`1 by A16,A22,A23,A31,GOBOARD7:5;
hence thesis by A34,TOPREAL3:6;
end;
assume
A35: x = q;
then x in LSeg(e,q) by RLTOPSP1:68;
hence thesis by A26,A35,XBOOLE_0:def 4;
end;
hence thesis by A23,TARSKI:def 1;
end;
theorem
for C be compact connected non vertical non horizontal Subset of
TOP-REAL 2 for i,j,k be Nat st 1 <= i & i <= len Gauge(C,n) & 1 <= j
& j <= k & k <= width Gauge(C,n) & Gauge(C,n)*(i,k) in L~Upper_Seq(C,n) ex k1
be Nat st j <= k1 & k1 <= k & LSeg(Gauge(C,n)*(i,j),Gauge(C,n)*(i,k1
)) /\ L~Upper_Seq(C,n) = {Gauge(C,n)*(i,k1)}
proof
let C be compact connected non vertical non horizontal Subset of TOP-REAL 2;
let i,j,k be Nat;
assume that
A1: 1 <= i and
A2: i <= len Gauge(C,n) and
A3: 1 <= j and
A4: j <= k and
A5: k <= width Gauge(C,n) and
A6: Gauge(C,n)*(i,k) in L~Upper_Seq(C,n);
set G = Gauge(C,n);
A7: k >= 1 by A3,A4,XXREAL_0:2;
then
A8: [i,k] in Indices G by A1,A2,A5,MATRIX_0:30;
set X = LSeg(G*(i,j),G*(i,k)) /\ L~Upper_Seq(C,n);
A9: G*(i,k) in LSeg(G*(i,j),G*(i,k)) by RLTOPSP1:68;
then reconsider X1=X as non empty compact Subset of TOP-REAL 2 by A6,
XBOOLE_0:def 4;
A10: LSeg(G*(i,j),G*(i,k)) meets L~Upper_Seq(C,n) by A6,A9,XBOOLE_0:3;
set s = G*(i,1)`1;
set e = G*(i,k);
set f = G*(i,j);
set w1 = lower_bound(proj2.:(LSeg(f,e) /\ L~Upper_Seq(C,n)));
A11: j <= width G by A4,A5,XXREAL_0:2;
then [i,j] in Indices G by A1,A2,A3,MATRIX_0:30;
then consider k1 be Nat such that
A12: j <= k1 and
A13: k1 <= k and
A14: G*(i,k1)`2 = w1 by A4,A10,A8,JORDAN1F:1,JORDAN1G:4;
set p = |[s,w1]|;
A15: k1 <= width G by A5,A13,XXREAL_0:2;
f`1 = s by A1,A2,A3,A11,GOBOARD5:2
.= e`1 by A1,A2,A5,A7,GOBOARD5:2;
then
A16: LSeg(f,e) is vertical by SPPOL_1:16;
take k1;
thus j <= k1 & k1 <= k by A12,A13;
consider pp be object such that
A17: pp in S-most X1 by XBOOLE_0:def 1;
A18: 1 <= k1 by A3,A12,XXREAL_0:2;
then
A19: G*(i,k1)`1 = s by A1,A2,A15,GOBOARD5:2;
then
A20: p = G*(i,k1) by A14,EUCLID:53;
then
A21: f`2 <= p`2 by A1,A2,A3,A12,A15,SPRECT_3:12;
A22: f`1 = p`1 by A1,A2,A3,A11,A19,A20,GOBOARD5:2;
reconsider pp as Point of TOP-REAL 2 by A17;
A23: pp in X by A17,XBOOLE_0:def 4;
then
A24: pp in L~Upper_Seq(C,n) by XBOOLE_0:def 4;
A25: p`2 = S-bound X by A14,A20,SPRECT_1:44
.= (S-min X)`2 by EUCLID:52
.= pp`2 by A17,PSCOMP_1:55;
pp in LSeg(G*(i,j),G*(i,k)) by A23,XBOOLE_0:def 4;
then pp`1 = p`1 by A22,A16,SPPOL_1:41;
then
A26: p in L~Upper_Seq(C,n) by A24,A25,TOPREAL3:6;
for x be object holds x in LSeg(p,f) /\ L~Upper_Seq(C,n) iff x = p
proof
let x be object;
thus x in LSeg(p,f) /\ L~Upper_Seq(C,n) implies x = p
proof
reconsider EE = LSeg(f,e) /\ L~Upper_Seq(C,n) as compact Subset of
TOP-REAL 2;
reconsider E0 = proj2.:EE as compact Subset of REAL by JCT_MISC:15;
A27: f in LSeg(f,e) by RLTOPSP1:68;
A28: e`1 = p`1 by A1,A2,A5,A7,A19,A20,GOBOARD5:2;
A29: p`2 <= e`2 by A1,A2,A5,A13,A18,A20,SPRECT_3:12;
A30: f`2 <= p`2 by A1,A2,A3,A12,A15,A20,SPRECT_3:12;
f`1 = p`1 by A1,A2,A3,A11,A19,A20,GOBOARD5:2;
then p in LSeg(f,e) by A28,A30,A29,GOBOARD7:7;
then
A31: LSeg(p,f) c= LSeg(f,e) by A27,TOPREAL1:6;
assume
A32: x in LSeg(p,f) /\ L~Upper_Seq(C,n);
then reconsider pp = x as Point of TOP-REAL 2;
A33: pp in LSeg(p,f) by A32,XBOOLE_0:def 4;
then
A34: pp`2 <= p`2 by A21,TOPREAL1:4;
pp in L~Upper_Seq(C,n) by A32,XBOOLE_0:def 4;
then pp in EE by A33,A31,XBOOLE_0:def 4;
then proj2.pp in E0 by FUNCT_2:35;
then
A35: pp`2 in E0 by PSCOMP_1:def 6;
E0 is real-bounded by RCOMP_1:10;
then E0 is bounded_below by XXREAL_2:def 11;
then p`2 <= pp`2 by A14,A20,A35,SEQ_4:def 2;
then
A36: pp`2 = p`2 by A34,XXREAL_0:1;
pp`1 = p`1 by A22,A33,GOBOARD7:5;
hence thesis by A36,TOPREAL3:6;
end;
assume
A37: x = p;
then x in LSeg(p,f) by RLTOPSP1:68;
hence thesis by A26,A37,XBOOLE_0:def 4;
end;
hence thesis by A20,TARSKI:def 1;
end;
theorem Th11:
for C be compact connected non vertical non horizontal Subset of
TOP-REAL 2 for i,j,k be Nat st 1 <= i & i <= len Gauge(C,n) & 1 <= j
& j <= k & k <= width Gauge(C,n) & Gauge(C,n)*(i,j) in L~Lower_Seq(C,n) & Gauge
(C,n)*(i,k) in L~Upper_Seq(C,n) ex j1,k1 be Nat st j <= j1 & j1 <=
k1 & k1 <= k & LSeg(Gauge(C,n)*(i,j1),Gauge(C,n)*(i,k1)) /\ L~Lower_Seq(C,n) =
{Gauge(C,n)*(i,j1)} & LSeg(Gauge(C,n)*(i,j1),Gauge(C,n)*(i,k1)) /\ L~Upper_Seq(
C,n) = {Gauge(C,n)*(i,k1)}
proof
let C be compact connected non vertical non horizontal Subset of TOP-REAL 2;
let i,j,k be Nat;
assume that
A1: 1 <= i and
A2: i <= len Gauge(C,n) and
A3: 1 <= j and
A4: j <= k and
A5: k <= width Gauge(C,n) and
A6: Gauge(C,n)*(i,j) in L~Lower_Seq(C,n) and
A7: Gauge(C,n)*(i,k) in L~Upper_Seq(C,n);
set G = Gauge(C,n);
A8: j <= width G by A4,A5,XXREAL_0:2;
then
A9: [i,j] in Indices G by A1,A2,A3,MATRIX_0:30;
set s = G*(i,1)`1;
set e = G*(i,k);
set f = G*(i,j);
set w1 = lower_bound(proj2.:(LSeg(f,e) /\ L~Upper_Seq(C,n)));
A10: G*(i,k) in LSeg(G*(i,j),G*(i,k)) by RLTOPSP1:68;
then
A11: LSeg(G*(i,j),G*(i,k)) meets L~Upper_Seq(C,n) by A7,XBOOLE_0:3;
A12: k >= 1 by A3,A4,XXREAL_0:2;
then [i,k] in Indices G by A1,A2,A5,MATRIX_0:30;
then consider k1 be Nat such that
A13: j <= k1 and
A14: k1 <= k and
A15: G*(i,k1)`2 = w1 by A4,A11,A9,JORDAN1F:1,JORDAN1G:4;
A16: k1 <= width G by A5,A14,XXREAL_0:2;
A17: G*(i,j) in LSeg(G*(i,j),G*(i,k1)) by RLTOPSP1:68;
then
A18: LSeg(G*(i,j),G*(i,k1)) meets L~Lower_Seq(C,n) by A6,XBOOLE_0:3;
set X = LSeg(G*(i,j),G*(i,k1)) /\ L~Lower_Seq(C,n);
reconsider X1=X as non empty compact Subset of TOP-REAL 2 by A6,A17,
XBOOLE_0:def 4;
consider pp be object such that
A19: pp in N-most X1 by XBOOLE_0:def 1;
reconsider pp as Point of TOP-REAL 2 by A19;
A20: pp in X by A19,XBOOLE_0:def 4;
then
A21: pp in L~Lower_Seq(C,n) by XBOOLE_0:def 4;
set p = |[s,w1]|;
set w2 = upper_bound(proj2.:(LSeg(f,p) /\ L~Lower_Seq(C,n)));
set q = |[s,w2]|;
A22: pp in LSeg(G*(i,j),G*(i,k1)) by A20,XBOOLE_0:def 4;
A23: 1 <= k1 by A3,A13,XXREAL_0:2;
then
A24: G*(i,k1)`1 = s by A1,A2,A16,GOBOARD5:2;
then
A25: p = G*(i,k1) by A15,EUCLID:53;
[i,k1] in Indices G by A1,A2,A23,A16,MATRIX_0:30;
then consider j1 be Nat such that
A26: j <= j1 and
A27: j1 <= k1 and
A28: G*(i,j1)`2 = w2 by A9,A13,A25,A18,JORDAN1F:2,JORDAN1G:5;
take j1,k1;
thus j <= j1 & j1 <= k1 & k1 <= k by A14,A26,A27;
A29: j1 <= width G by A16,A27,XXREAL_0:2;
A30: 1 <= j1 by A3,A26,XXREAL_0:2;
then
A31: G*(i,j1)`1 = s by A1,A2,A29,GOBOARD5:2;
then
A32: q = G*(i,j1) by A28,EUCLID:53;
then
A33: q`2 <= p`2 by A1,A2,A16,A25,A27,A30,SPRECT_3:12;
A34: q`2 = N-bound X by A25,A28,A32,SPRECT_1:45
.= (N-min X)`2 by EUCLID:52
.= pp`2 by A19,PSCOMP_1:39;
A35: f`1 = p`1 by A1,A2,A3,A8,A24,A25,GOBOARD5:2;
then LSeg(f,p) is vertical by SPPOL_1:16;
then pp`1 = q`1 by A24,A25,A31,A32,A22,SPPOL_1:41;
then
A36: q in L~Lower_Seq(C,n) by A21,A34,TOPREAL3:6;
for x be object holds x in LSeg(p,q) /\ L~Lower_Seq(C,n) iff x = q
proof
let x be object;
thus x in LSeg(p,q) /\ L~Lower_Seq(C,n) implies x = q
proof
reconsider EE = LSeg(f,p) /\ L~Lower_Seq(C,n) as compact Subset of
TOP-REAL 2;
reconsider E0 = proj2.:EE as compact Subset of REAL by JCT_MISC:15;
A37: p in LSeg(f,p) by RLTOPSP1:68;
A38: f`2 <= q`2 by A1,A2,A3,A26,A29,A32,SPRECT_3:12;
f`1 = q`1 by A1,A2,A3,A8,A31,A32,GOBOARD5:2;
then q in LSeg(p,f) by A24,A25,A31,A32,A33,A38,GOBOARD7:7;
then
A39: LSeg(p,q) c= LSeg(f,p) by A37,TOPREAL1:6;
assume
A40: x in LSeg(p,q) /\ L~Lower_Seq(C,n);
then reconsider pp = x as Point of TOP-REAL 2;
A41: pp in LSeg(p,q) by A40,XBOOLE_0:def 4;
then
A42: pp`2 >= q`2 by A33,TOPREAL1:4;
pp in L~Lower_Seq(C,n) by A40,XBOOLE_0:def 4;
then pp in EE by A41,A39,XBOOLE_0:def 4;
then proj2.pp in E0 by FUNCT_2:35;
then
A43: pp`2 in E0 by PSCOMP_1:def 6;
E0 is real-bounded by RCOMP_1:10;
then E0 is bounded_above by XXREAL_2:def 11;
then q`2 >= pp`2 by A28,A32,A43,SEQ_4:def 1;
then
A44: pp`2 = q`2 by A42,XXREAL_0:1;
pp`1 = q`1 by A24,A25,A31,A32,A41,GOBOARD7:5;
hence thesis by A44,TOPREAL3:6;
end;
assume
A45: x = q;
then x in LSeg(p,q) by RLTOPSP1:68;
hence thesis by A36,A45,XBOOLE_0:def 4;
end;
hence
LSeg(Gauge(C,n)*(i,j1), Gauge(C,n)*(i,k1)) /\ L~Lower_Seq(C,n) = {Gauge
(C,n)*(i,j1)} by A25,A32,TARSKI:def 1;
set X = LSeg(G*(i,j),G*(i,k)) /\ L~Upper_Seq(C,n);
reconsider X1=X as non empty compact Subset of TOP-REAL 2 by A7,A10,
XBOOLE_0:def 4;
consider pp be object such that
A46: pp in S-most X1 by XBOOLE_0:def 1;
reconsider pp as Point of TOP-REAL 2 by A46;
A47: pp in X by A46,XBOOLE_0:def 4;
then
A48: pp in L~Upper_Seq(C,n) by XBOOLE_0:def 4;
f`1 = s by A1,A2,A3,A8,GOBOARD5:2
.= e`1 by A1,A2,A5,A12,GOBOARD5:2;
then
A49: LSeg(f,e) is vertical by SPPOL_1:16;
pp in LSeg(G*(i,j),G*(i,k)) by A47,XBOOLE_0:def 4;
then
A50: pp`1 = p`1 by A35,A49,SPPOL_1:41;
p`2 = S-bound X by A15,A25,SPRECT_1:44
.= (S-min X)`2 by EUCLID:52
.= pp`2 by A46,PSCOMP_1:55;
then
A51: p in L~Upper_Seq(C,n) by A48,A50,TOPREAL3:6;
for x be object holds x in LSeg(p,q) /\ L~Upper_Seq(C,n) iff x = p
proof
let x be object;
thus x in LSeg(p,q) /\ L~Upper_Seq(C,n) implies x = p
proof
A52: p`2 <= e`2 by A1,A2,A5,A14,A23,A25,SPRECT_3:12;
A53: f`2 <= p`2 by A1,A2,A3,A13,A16,A25,SPRECT_3:12;
A54: e`1 = p`1 by A1,A2,A5,A12,A24,A25,GOBOARD5:2;
f`1 = p`1 by A1,A2,A3,A8,A24,A25,GOBOARD5:2;
then
A55: p in LSeg(f,e) by A54,A53,A52,GOBOARD7:7;
A56: e`1 = q`1 by A1,A2,A5,A12,A31,A32,GOBOARD5:2;
j1 <= k by A14,A27,XXREAL_0:2;
then
A57: q`2 <= e`2 by A1,A2,A5,A30,A32,SPRECT_3:12;
A58: f`2 <= q`2 by A1,A2,A3,A26,A29,A32,SPRECT_3:12;
f`1 = q`1 by A1,A2,A3,A8,A31,A32,GOBOARD5:2;
then q in LSeg(f,e) by A56,A58,A57,GOBOARD7:7;
then
A59: LSeg(p,q) c= LSeg(f,e) by A55,TOPREAL1:6;
reconsider EE = LSeg(f,e) /\ L~Upper_Seq(C,n) as compact Subset of
TOP-REAL 2;
reconsider E0 = proj2.:EE as compact Subset of REAL by JCT_MISC:15;
assume
A60: x in LSeg(p,q) /\ L~Upper_Seq(C,n);
then reconsider pp = x as Point of TOP-REAL 2;
A61: pp in LSeg(p,q) by A60,XBOOLE_0:def 4;
then
A62: pp`2 <= p`2 by A33,TOPREAL1:4;
pp in L~Upper_Seq(C,n) by A60,XBOOLE_0:def 4;
then pp in EE by A61,A59,XBOOLE_0:def 4;
then proj2.pp in E0 by FUNCT_2:35;
then
A63: pp`2 in E0 by PSCOMP_1:def 6;
E0 is real-bounded by RCOMP_1:10;
then E0 is bounded_below by XXREAL_2:def 11;
then p`2 <= pp`2 by A15,A25,A63,SEQ_4:def 2;
then
A64: pp`2 = p`2 by A62,XXREAL_0:1;
pp`1 = p`1 by A24,A25,A31,A32,A61,GOBOARD7:5;
hence thesis by A64,TOPREAL3:6;
end;
assume
A65: x = p;
then x in LSeg(p,q) by RLTOPSP1:68;
hence thesis by A51,A65,XBOOLE_0:def 4;
end;
hence thesis by A25,A32,TARSKI:def 1;
end;
theorem
for C be compact connected non vertical non horizontal Subset of
TOP-REAL 2 for i,j,k be Nat st 1 <= j & j <= k & k <= len Gauge(C,n)
& 1 <= i & i <= width Gauge(C,n) & Gauge(C,n)*(j,i) in L~Lower_Seq(C,n) ex j1
be Nat st j <= j1 & j1 <= k & LSeg(Gauge(C,n)*(j1,i),Gauge(C,n)*(k,i
)) /\ L~Lower_Seq(C,n) = {Gauge(C,n)*(j1,i)}
proof
let C be compact connected non vertical non horizontal Subset of TOP-REAL 2;
let i,j,k be Nat;
assume that
A1: 1 <= j and
A2: j <= k and
A3: k <= len Gauge(C,n) and
A4: 1 <= i and
A5: i <= width Gauge(C,n) and
A6: Gauge(C,n)*(j,i) in L~Lower_Seq(C,n);
set G = Gauge(C,n);
A7: k >= 1 by A1,A2,XXREAL_0:2;
then
A8: [k,i] in Indices G by A3,A4,A5,MATRIX_0:30;
set X = LSeg(G*(j,i),G*(k,i)) /\ L~Lower_Seq(C,n);
A9: G*(j,i) in LSeg(G*(j,i),G*(k,i)) by RLTOPSP1:68;
then reconsider X1=X as non empty compact Subset of TOP-REAL 2 by A6,
XBOOLE_0:def 4;
A10: LSeg(G*(j,i),G*(k,i)) meets L~Lower_Seq(C,n) by A6,A9,XBOOLE_0:3;
set s = G*(1,i)`2;
set e = G*(k,i);
set f = G*(j,i);
set w2 = upper_bound(proj1.:(LSeg(f,e) /\ L~Lower_Seq(C,n)));
A11: len G = width G by JORDAN8:def 1;
then
A12: j <= width G by A2,A3,XXREAL_0:2;
then [j,i] in Indices G by A1,A4,A5,A11,MATRIX_0:30;
then consider j1 be Nat such that
A13: j <= j1 and
A14: j1 <= k and
A15: G*(j1,i)`1 = w2 by A2,A10,A8,JORDAN1F:4,JORDAN1G:5;
set q = |[w2,s]|;
A16: 1 <= j1 by A1,A13,XXREAL_0:2;
take j1;
thus j <= j1 & j1 <= k by A13,A14;
consider pp be object such that
A17: pp in E-most X1 by XBOOLE_0:def 1;
reconsider pp as Point of TOP-REAL 2 by A17;
A18: pp in X by A17,XBOOLE_0:def 4;
then
A19: pp in L~Lower_Seq(C,n) by XBOOLE_0:def 4;
A20: j1 <= width G by A3,A11,A14,XXREAL_0:2;
then
A21: G*(j1,i)`2 = s by A4,A5,A11,A16,GOBOARD5:1;
then
A22: q = G*(j1,i) by A15,EUCLID:53;
then
A23: q`1 <= e`1 by A3,A4,A5,A14,A16,SPRECT_3:13;
A24: G*(k,i)`2 = s by A3,A4,A5,A7,GOBOARD5:1;
then f`2 = e`2 by A1,A4,A5,A11,A12,GOBOARD5:1;
then
A25: LSeg(f,e) is horizontal by SPPOL_1:15;
A26: q`1 = E-bound X by A15,A22,SPRECT_1:46
.= (E-min X)`1 by EUCLID:52
.= pp`1 by A17,PSCOMP_1:47;
pp in LSeg(G*(j,i),G*(k,i)) by A18,XBOOLE_0:def 4;
then pp`2 = q`2 by A24,A21,A22,A25,SPPOL_1:40;
then
A27: q in L~Lower_Seq(C,n) by A19,A26,TOPREAL3:6;
for x be object holds x in LSeg(e,q) /\ L~Lower_Seq(C,n) iff x = q
proof
let x be object;
thus x in LSeg(e,q) /\ L~Lower_Seq(C,n) implies x = q
proof
A28: f`1 <= q`1 by A1,A4,A5,A11,A13,A20,A22,SPRECT_3:13;
f`2 = q`2 by A1,A4,A5,A11,A12,A21,A22,GOBOARD5:1;
then
A29: q in LSeg(e,f) by A24,A21,A22,A23,A28,GOBOARD7:8;
e in LSeg(f,e) by RLTOPSP1:68;
then
A30: LSeg(e,q) c= LSeg(f,e) by A29,TOPREAL1:6;
reconsider EE = LSeg(f,e) /\ L~Lower_Seq(C,n) as compact Subset of
TOP-REAL 2;
reconsider E0 = proj1.:EE as compact Subset of REAL by Th4;
assume
A31: x in LSeg(e,q) /\ L~Lower_Seq(C,n);
then reconsider pp = x as Point of TOP-REAL 2;
A32: pp in LSeg(e,q) by A31,XBOOLE_0:def 4;
then
A33: pp`1 >= q`1 by A23,TOPREAL1:3;
pp in L~Lower_Seq(C,n) by A31,XBOOLE_0:def 4;
then pp in EE by A32,A30,XBOOLE_0:def 4;
then proj1.pp in E0 by FUNCT_2:35;
then
A34: pp`1 in E0 by PSCOMP_1:def 5;
E0 is real-bounded by RCOMP_1:10;
then E0 is bounded_above by XXREAL_2:def 11;
then q`1 >= pp`1 by A15,A22,A34,SEQ_4:def 1;
then
A35: pp`1 = q`1 by A33,XXREAL_0:1;
pp`2 = q`2 by A24,A21,A22,A32,GOBOARD7:6;
hence thesis by A35,TOPREAL3:6;
end;
assume
A36: x = q;
then x in LSeg(e,q) by RLTOPSP1:68;
hence thesis by A27,A36,XBOOLE_0:def 4;
end;
hence thesis by A22,TARSKI:def 1;
end;
theorem Th13:
for C be compact connected non vertical non horizontal Subset of
TOP-REAL 2 for i,j,k be Nat st 1 <= j & j <= k & k <= len Gauge(C,n)
& 1 <= i & i <= width Gauge(C,n) & Gauge(C,n)*(k,i) in L~Upper_Seq(C,n) ex k1
be Nat st j <= k1 & k1 <= k & LSeg(Gauge(C,n)*(j,i),Gauge(C,n)*(k1,i
)) /\ L~Upper_Seq(C,n) = {Gauge(C,n)*(k1,i)}
proof
let C be compact connected non vertical non horizontal Subset of TOP-REAL 2;
let i,j,k be Nat;
assume that
A1: 1 <= j and
A2: j <= k and
A3: k <= len Gauge(C,n) and
A4: 1 <= i and
A5: i <= width Gauge(C,n) and
A6: Gauge(C,n)*(k,i) in L~Upper_Seq(C,n);
set G = Gauge(C,n);
A7: k >= 1 by A1,A2,XXREAL_0:2;
then
A8: [k,i] in Indices G by A3,A4,A5,MATRIX_0:30;
set X = LSeg(G*(j,i),G*(k,i)) /\ L~Upper_Seq(C,n);
A9: G*(k,i) in LSeg(G*(j,i),G*(k,i)) by RLTOPSP1:68;
then reconsider X1=X as non empty compact Subset of TOP-REAL 2 by A6,
XBOOLE_0:def 4;
A10: LSeg(G*(j,i),G*(k,i)) meets L~Upper_Seq(C,n) by A6,A9,XBOOLE_0:3;
set s = G*(1,i)`2;
set e = G*(k,i);
set f = G*(j,i);
set w1 = lower_bound(proj1.:(LSeg(f,e) /\ L~Upper_Seq(C,n)));
A11: len G = width G by JORDAN8:def 1;
then
A12: j <= width G by A2,A3,XXREAL_0:2;
then [j,i] in Indices G by A1,A4,A5,A11,MATRIX_0:30;
then consider k1 be Nat such that
A13: j <= k1 and
A14: k1 <= k and
A15: G*(k1,i)`1 = w1 by A2,A10,A8,JORDAN1F:3,JORDAN1G:4;
set p = |[w1,s]|;
A16: k1 <= width G by A3,A11,A14,XXREAL_0:2;
f`2 = s by A1,A4,A5,A11,A12,GOBOARD5:1
.= e`2 by A3,A4,A5,A7,GOBOARD5:1;
then
A17: LSeg(f,e) is horizontal by SPPOL_1:15;
take k1;
thus j <= k1 & k1 <= k by A13,A14;
consider pp be object such that
A18: pp in W-most X1 by XBOOLE_0:def 1;
A19: 1 <= k1 by A1,A13,XXREAL_0:2;
then
A20: G*(k1,i)`2 = s by A4,A5,A11,A16,GOBOARD5:1;
then
A21: p = G*(k1,i) by A15,EUCLID:53;
then
A22: f`1 <= p`1 by A1,A4,A5,A11,A13,A16,SPRECT_3:13;
A23: f`2 = p`2 by A1,A4,A5,A11,A12,A20,A21,GOBOARD5:1;
reconsider pp as Point of TOP-REAL 2 by A18;
A24: pp in X by A18,XBOOLE_0:def 4;
then
A25: pp in L~Upper_Seq(C,n) by XBOOLE_0:def 4;
A26: p`1 = W-bound X by A15,A21,SPRECT_1:43
.= (W-min X)`1 by EUCLID:52
.= pp`1 by A18,PSCOMP_1:31;
pp in LSeg(G*(j,i),G*(k,i)) by A24,XBOOLE_0:def 4;
then pp`2 = p`2 by A23,A17,SPPOL_1:40;
then
A27: p in L~Upper_Seq(C,n) by A25,A26,TOPREAL3:6;
for x be object holds x in LSeg(p,f) /\ L~Upper_Seq(C,n) iff x = p
proof
let x be object;
thus x in LSeg(p,f) /\ L~Upper_Seq(C,n) implies x = p
proof
reconsider EE = LSeg(f,e) /\ L~Upper_Seq(C,n) as compact Subset of
TOP-REAL 2;
assume
A28: x in LSeg(p,f) /\ L~Upper_Seq(C,n);
then reconsider pp = x as Point of TOP-REAL 2;
A29: pp in LSeg(p,f) by A28,XBOOLE_0:def 4;
then
A30: pp`1 <= p`1 by A22,TOPREAL1:3;
A31: p`1 <= e`1 by A3,A4,A5,A14,A19,A21,SPRECT_3:13;
A32: f`1 <= p`1 by A1,A4,A5,A11,A13,A16,A21,SPRECT_3:13;
A33: e`2 = p`2 by A3,A4,A5,A7,A20,A21,GOBOARD5:1;
reconsider E0 = proj1.:EE as compact Subset of REAL by Th4;
A34: f in LSeg(f,e) by RLTOPSP1:68;
f`2 = p`2 by A1,A4,A5,A11,A12,A20,A21,GOBOARD5:1;
then p in LSeg(f,e) by A33,A32,A31,GOBOARD7:8;
then
A35: LSeg(p,f) c= LSeg(f,e) by A34,TOPREAL1:6;
pp in L~Upper_Seq(C,n) by A28,XBOOLE_0:def 4;
then pp in EE by A29,A35,XBOOLE_0:def 4;
then proj1.pp in E0 by FUNCT_2:35;
then
A36: pp`1 in E0 by PSCOMP_1:def 5;
E0 is real-bounded by RCOMP_1:10;
then E0 is bounded_below by XXREAL_2:def 11;
then p`1 <= pp`1 by A15,A21,A36,SEQ_4:def 2;
then
A37: pp`1 = p`1 by A30,XXREAL_0:1;
pp`2 = p`2 by A23,A29,GOBOARD7:6;
hence thesis by A37,TOPREAL3:6;
end;
assume
A38: x = p;
then x in LSeg(p,f) by RLTOPSP1:68;
hence thesis by A27,A38,XBOOLE_0:def 4;
end;
hence thesis by A21,TARSKI:def 1;
end;
theorem Th14:
for C be compact connected non vertical non horizontal Subset of
TOP-REAL 2 for i,j,k be Nat st 1 <= j & j <= k & k <= len Gauge(C,n)
& 1 <= i & i <= width Gauge(C,n) & Gauge(C,n)*(j,i) in L~Lower_Seq(C,n) & Gauge
(C,n)*(k,i) in L~Upper_Seq(C,n) ex j1,k1 be Nat st j <= j1 & j1 <=
k1 & k1 <= k & LSeg(Gauge(C,n)*(j1,i),Gauge(C,n)*(k1,i)) /\ L~Lower_Seq(C,n) =
{Gauge(C,n)*(j1,i)} & LSeg(Gauge(C,n)*(j1,i),Gauge(C,n)*(k1,i)) /\ L~Upper_Seq(
C,n) = {Gauge(C,n)*(k1,i)}
proof
let C be compact connected non vertical non horizontal Subset of TOP-REAL 2;
let i,j,k be Nat;
assume that
A1: 1 <= j and
A2: j <= k and
A3: k <= len Gauge(C,n) and
A4: 1 <= i and
A5: i <= width Gauge(C,n) and
A6: Gauge(C,n)*(j,i) in L~Lower_Seq(C,n) and
A7: Gauge(C,n)*(k,i) in L~Upper_Seq(C,n);
set G = Gauge(C,n);
A8: len G = width G by JORDAN8:def 1;
then
A9: j <= width G by A2,A3,XXREAL_0:2;
then
A10: [j,i] in Indices G by A1,A4,A5,A8,MATRIX_0:30;
set s = G*(1,i)`2;
set e = G*(k,i);
set f = G*(j,i);
set w1 = lower_bound(proj1.:(LSeg(f,e) /\ L~Upper_Seq(C,n)));
A11: G*(k,i) in LSeg(G*(j,i),G*(k,i)) by RLTOPSP1:68;
then
A12: LSeg(G*(j,i),G*(k,i)) meets L~Upper_Seq(C,n) by A7,XBOOLE_0:3;
A13: k >= 1 by A1,A2,XXREAL_0:2;
then [k,i] in Indices G by A3,A4,A5,MATRIX_0:30;
then consider k1 be Nat such that
A14: j <= k1 and
A15: k1 <= k and
A16: G*(k1,i)`1 = w1 by A2,A12,A10,JORDAN1F:3,JORDAN1G:4;
A17: k1 <= width G by A3,A8,A15,XXREAL_0:2;
set p = |[w1,s]|;
set w2 = upper_bound(proj1.:(LSeg(f,p) /\ L~Lower_Seq(C,n)));
set q = |[w2,s]|;
A18: G*(j,i) in LSeg(G*(j,i),G*(k1,i)) by RLTOPSP1:68;
then
A19: LSeg(G*(j,i),G*(k1,i)) meets L~Lower_Seq(C,n) by A6,XBOOLE_0:3;
A20: 1 <= k1 by A1,A14,XXREAL_0:2;
then
A21: G*(k1,i)`2 = s by A4,A5,A8,A17,GOBOARD5:1;
then
A22: p = G*(k1,i) by A16,EUCLID:53;
f`2 = s by A1,A4,A5,A8,A9,GOBOARD5:1
.= e`2 by A3,A4,A5,A13,GOBOARD5:1;
then
A23: LSeg(f,e) is horizontal by SPPOL_1:15;
set X = LSeg(G*(j,i),G*(k1,i)) /\ L~Lower_Seq(C,n);
reconsider X1=X as non empty compact Subset of TOP-REAL 2 by A6,A18,
XBOOLE_0:def 4;
consider pp be object such that
A24: pp in E-most X1 by XBOOLE_0:def 1;
[k1,i] in Indices G by A4,A5,A8,A20,A17,MATRIX_0:30;
then consider j1 be Nat such that
A25: j <= j1 and
A26: j1 <= k1 and
A27: G*(j1,i)`1 = w2 by A10,A14,A22,A19,JORDAN1F:4,JORDAN1G:5;
A28: j1 <= width G by A17,A26,XXREAL_0:2;
reconsider pp as Point of TOP-REAL 2 by A24;
A29: pp in X by A24,XBOOLE_0:def 4;
then
A30: pp in L~Lower_Seq(C,n) by XBOOLE_0:def 4;
take j1,k1;
thus j <= j1 & j1 <= k1 & k1 <= k by A15,A25,A26;
A31: pp in LSeg(G*(j,i),G*(k1,i)) by A29,XBOOLE_0:def 4;
A32: 1 <= j1 by A1,A25,XXREAL_0:2;
then
A33: G*(j1,i)`2 = s by A4,A5,A8,A28,GOBOARD5:1;
then
A34: q = G*(j1,i) by A27,EUCLID:53;
then
A35: q`1 <= p`1 by A4,A5,A8,A17,A22,A26,A32,SPRECT_3:13;
A36: q`1 = E-bound X by A22,A27,A34,SPRECT_1:46
.= (E-min X)`1 by EUCLID:52
.= pp`1 by A24,PSCOMP_1:47;
A37: f`2 = p`2 by A1,A4,A5,A8,A9,A21,A22,GOBOARD5:1;
then LSeg(f,p) is horizontal by SPPOL_1:15;
then pp`2 = q`2 by A21,A22,A33,A34,A31,SPPOL_1:40;
then
A38: q in L~Lower_Seq(C,n) by A30,A36,TOPREAL3:6;
for x be object holds x in LSeg(p,q) /\ L~Lower_Seq(C,n) iff x = q
proof
let x be object;
thus x in LSeg(p,q) /\ L~Lower_Seq(C,n) implies x = q
proof
reconsider EE = LSeg(f,p) /\ L~Lower_Seq(C,n) as compact Subset of
TOP-REAL 2;
assume
A39: x in LSeg(p,q) /\ L~Lower_Seq(C,n);
then reconsider pp = x as Point of TOP-REAL 2;
A40: pp in LSeg(p,q) by A39,XBOOLE_0:def 4;
then
A41: pp`1 >= q`1 by A35,TOPREAL1:3;
A42: f`1 <= q`1 by A1,A4,A5,A8,A25,A28,A34,SPRECT_3:13;
reconsider E0 = proj1.:EE as compact Subset of REAL by Th4;
A43: p in LSeg(f,p) by RLTOPSP1:68;
f`2 = q`2 by A1,A4,A5,A8,A9,A33,A34,GOBOARD5:1;
then q in LSeg(p,f) by A21,A22,A33,A34,A35,A42,GOBOARD7:8;
then
A44: LSeg(p,q) c= LSeg(f,p) by A43,TOPREAL1:6;
pp in L~Lower_Seq(C,n) by A39,XBOOLE_0:def 4;
then pp in EE by A40,A44,XBOOLE_0:def 4;
then proj1.pp in E0 by FUNCT_2:35;
then
A45: pp`1 in E0 by PSCOMP_1:def 5;
E0 is real-bounded by RCOMP_1:10;
then E0 is bounded_above by XXREAL_2:def 11;
then q`1 >= pp`1 by A27,A34,A45,SEQ_4:def 1;
then
A46: pp`1 = q`1 by A41,XXREAL_0:1;
pp`2 = q`2 by A21,A22,A33,A34,A40,GOBOARD7:6;
hence thesis by A46,TOPREAL3:6;
end;
assume
A47: x = q;
then x in LSeg(p,q) by RLTOPSP1:68;
hence thesis by A38,A47,XBOOLE_0:def 4;
end;
hence
LSeg(Gauge(C,n)*(j1,i),Gauge(C,n)*(k1,i)) /\ L~Lower_Seq(C,n) = {Gauge(
C,n)*(j1,i)} by A22,A34,TARSKI:def 1;
set X = LSeg(G*(j,i),G*(k,i)) /\ L~Upper_Seq(C,n);
reconsider X1=X as non empty compact Subset of TOP-REAL 2 by A7,A11,
XBOOLE_0:def 4;
consider pp be object such that
A48: pp in W-most X1 by XBOOLE_0:def 1;
reconsider pp as Point of TOP-REAL 2 by A48;
A49: pp in X by A48,XBOOLE_0:def 4;
then
A50: pp in L~Upper_Seq(C,n) by XBOOLE_0:def 4;
pp in LSeg(G*(j,i),G*(k,i)) by A49,XBOOLE_0:def 4;
then
A51: pp`2 = p`2 by A37,A23,SPPOL_1:40;
p`1 = W-bound X by A16,A22,SPRECT_1:43
.= (W-min X)`1 by EUCLID:52
.= pp`1 by A48,PSCOMP_1:31;
then
A52: p in L~Upper_Seq(C,n) by A50,A51,TOPREAL3:6;
for x be object holds x in LSeg(p,q) /\ L~Upper_Seq(C,n) iff x = p
proof
let x be object;
thus x in LSeg(p,q) /\ L~Upper_Seq(C,n) implies x = p
proof
j1 <= k by A15,A26,XXREAL_0:2;
then
A53: q`1 <= e`1 by A3,A4,A5,A32,A34,SPRECT_3:13;
A54: e`2 = p`2 by A3,A4,A5,A13,A21,A22,GOBOARD5:1;
A55: f`1 <= p`1 by A1,A4,A5,A8,A14,A17,A22,SPRECT_3:13;
A56: f`1 <= q`1 by A1,A4,A5,A8,A25,A28,A34,SPRECT_3:13;
A57: p`1 <= e`1 by A3,A4,A5,A15,A20,A22,SPRECT_3:13;
f`2 = p`2 by A1,A4,A5,A8,A9,A21,A22,GOBOARD5:1;
then
A58: p in LSeg(f,e) by A54,A55,A57,GOBOARD7:8;
A59: e`2 = q`2 by A3,A4,A5,A13,A33,A34,GOBOARD5:1;
f`2 = q`2 by A1,A4,A5,A8,A9,A33,A34,GOBOARD5:1;
then q in LSeg(f,e) by A59,A56,A53,GOBOARD7:8;
then
A60: LSeg(p,q) c= LSeg(f,e) by A58,TOPREAL1:6;
reconsider EE = LSeg(f,e) /\ L~Upper_Seq(C,n) as compact Subset of
TOP-REAL 2;
reconsider E0 = proj1.:EE as compact Subset of REAL by Th4;
assume
A61: x in LSeg(p,q) /\ L~Upper_Seq(C,n);
then reconsider pp = x as Point of TOP-REAL 2;
A62: pp in LSeg(p,q) by A61,XBOOLE_0:def 4;
then
A63: pp`1 <= p`1 by A35,TOPREAL1:3;
pp in L~Upper_Seq(C,n) by A61,XBOOLE_0:def 4;
then pp in EE by A62,A60,XBOOLE_0:def 4;
then proj1.pp in E0 by FUNCT_2:35;
then
A64: pp`1 in E0 by PSCOMP_1:def 5;
E0 is real-bounded by RCOMP_1:10;
then E0 is bounded_below by XXREAL_2:def 11;
then p`1 <= pp`1 by A16,A22,A64,SEQ_4:def 2;
then
A65: pp`1 = p`1 by A63,XXREAL_0:1;
pp`2 = p`2 by A21,A22,A33,A34,A62,GOBOARD7:6;
hence thesis by A65,TOPREAL3:6;
end;
assume
A66: x = p;
then x in LSeg(p,q) by RLTOPSP1:68;
hence thesis by A52,A66,XBOOLE_0:def 4;
end;
hence thesis by A22,A34,TARSKI:def 1;
end;
theorem
for C be compact connected non vertical non horizontal Subset of
TOP-REAL 2 for i,j,k be Nat st 1 <= i & i <= len Gauge(C,n) & 1 <= j
& j <= k & k <= width Gauge(C,n) & Gauge(C,n)*(i,j) in L~Upper_Seq(C,n) ex j1
be Nat st j <= j1 & j1 <= k & LSeg(Gauge(C,n)*(i,j1),Gauge(C,n)*(i,k
)) /\ L~Upper_Seq(C,n) = {Gauge(C,n)*(i,j1)}
proof
let C be compact connected non vertical non horizontal Subset of TOP-REAL 2;
let i,j,k be Nat;
assume that
A1: 1 <= i and
A2: i <= len Gauge(C,n) and
A3: 1 <= j and
A4: j <= k and
A5: k <= width Gauge(C,n) and
A6: Gauge(C,n)*(i,j) in L~Upper_Seq(C,n);
set G = Gauge(C,n);
A7: k >= 1 by A3,A4,XXREAL_0:2;
then
A8: [i,k] in Indices G by A1,A2,A5,MATRIX_0:30;
set X = LSeg(G*(i,j),G*(i,k)) /\ L~Upper_Seq(C,n);
A9: G*(i,j) in LSeg(G*(i,j),G*(i,k)) by RLTOPSP1:68;
then reconsider X1=X as non empty compact Subset of TOP-REAL 2 by A6,
XBOOLE_0:def 4;
A10: LSeg(G*(i,j),G*(i,k)) meets L~Upper_Seq(C,n) by A6,A9,XBOOLE_0:3;
set s = G*(i,1)`1;
set e = G*(i,k);
set f = G*(i,j);
set w2 = upper_bound(proj2.:(LSeg(f,e) /\ L~Upper_Seq(C,n)));
A11: j <= width G by A4,A5,XXREAL_0:2;
then [i,j] in Indices G by A1,A2,A3,MATRIX_0:30;
then consider j1 be Nat such that
A12: j <= j1 and
A13: j1 <= k and
A14: G*(i,j1)`2 = w2 by A4,A10,A8,JORDAN1F:2,JORDAN1G:4;
set q = |[s,w2]|;
A15: j1 <= width G by A5,A13,XXREAL_0:2;
A16: G*(i,k)`1 = s by A1,A2,A5,A7,GOBOARD5:2;
then f`1 = e`1 by A1,A2,A3,A11,GOBOARD5:2;
then
A17: LSeg(f,e) is vertical by SPPOL_1:16;
take j1;
thus j <= j1 & j1 <= k by A12,A13;
consider pp be object such that
A18: pp in N-most X1 by XBOOLE_0:def 1;
reconsider pp as Point of TOP-REAL 2 by A18;
A19: pp in X by A18,XBOOLE_0:def 4;
then
A20: pp in L~Upper_Seq(C,n) by XBOOLE_0:def 4;
A21: 1 <= j1 by A3,A12,XXREAL_0:2;
then
A22: G*(i,j1)`1 = s by A1,A2,A15,GOBOARD5:2;
then
A23: q = G*(i,j1) by A14,EUCLID:53;
then
A24: q`2 <= e`2 by A1,A2,A5,A13,A21,SPRECT_3:12;
A25: q`2 = N-bound X by A14,A23,SPRECT_1:45
.= (N-min X)`2 by EUCLID:52
.= pp`2 by A18,PSCOMP_1:39;
pp in LSeg(G*(i,j),G*(i,k)) by A19,XBOOLE_0:def 4;
then pp`1 = q`1 by A16,A22,A23,A17,SPPOL_1:41;
then
A26: q in L~Upper_Seq(C,n) by A20,A25,TOPREAL3:6;
for x be object holds x in LSeg(e,q) /\ L~Upper_Seq(C,n) iff x = q
proof
let x be object;
thus x in LSeg(e,q) /\ L~Upper_Seq(C,n) implies x = q
proof
reconsider EE = LSeg(f,e) /\ L~Upper_Seq(C,n) as compact Subset of
TOP-REAL 2;
reconsider E0 = proj2.:EE as compact Subset of REAL by JCT_MISC:15;
A27: e in LSeg(f,e) by RLTOPSP1:68;
A28: f`2 <= q`2 by A1,A2,A3,A12,A15,A23,SPRECT_3:12;
f`1 = q`1 by A1,A2,A3,A11,A22,A23,GOBOARD5:2;
then q in LSeg(e,f) by A16,A22,A23,A24,A28,GOBOARD7:7;
then
A29: LSeg(e,q) c= LSeg(f,e) by A27,TOPREAL1:6;
assume
A30: x in LSeg(e,q) /\ L~Upper_Seq(C,n);
then reconsider pp = x as Point of TOP-REAL 2;
A31: pp in LSeg(e,q) by A30,XBOOLE_0:def 4;
then
A32: pp`2 >= q`2 by A24,TOPREAL1:4;
pp in L~Upper_Seq(C,n) by A30,XBOOLE_0:def 4;
then pp in EE by A31,A29,XBOOLE_0:def 4;
then proj2.pp in E0 by FUNCT_2:35;
then
A33: pp`2 in E0 by PSCOMP_1:def 6;
E0 is real-bounded by RCOMP_1:10;
then E0 is bounded_above by XXREAL_2:def 11;
then q`2 >= pp`2 by A14,A23,A33,SEQ_4:def 1;
then
A34: pp`2 = q`2 by A32,XXREAL_0:1;
pp`1 = q`1 by A16,A22,A23,A31,GOBOARD7:5;
hence thesis by A34,TOPREAL3:6;
end;
assume
A35: x = q;
then x in LSeg(e,q) by RLTOPSP1:68;
hence thesis by A26,A35,XBOOLE_0:def 4;
end;
hence thesis by A23,TARSKI:def 1;
end;
theorem
for C be compact connected non vertical non horizontal Subset of
TOP-REAL 2 for i,j,k be Nat st 1 <= i & i <= len Gauge(C,n) & 1 <= j
& j <= k & k <= width Gauge(C,n) & Gauge(C,n)*(i,k) in L~Lower_Seq(C,n) ex k1
be Nat st j <= k1 & k1 <= k & LSeg(Gauge(C,n)*(i,j),Gauge(C,n)*(i,k1
)) /\ L~Lower_Seq(C,n) = {Gauge(C,n)*(i,k1)}
proof
let C be compact connected non vertical non horizontal Subset of TOP-REAL 2;
let i,j,k be Nat;
assume that
A1: 1 <= i and
A2: i <= len Gauge(C,n) and
A3: 1 <= j and
A4: j <= k and
A5: k <= width Gauge(C,n) and
A6: Gauge(C,n)*(i,k) in L~Lower_Seq(C,n);
set G = Gauge(C,n);
A7: k >= 1 by A3,A4,XXREAL_0:2;
then
A8: [i,k] in Indices G by A1,A2,A5,MATRIX_0:30;
set X = LSeg(G*(i,j),G*(i,k)) /\ L~Lower_Seq(C,n);
A9: G*(i,k) in LSeg(G*(i,j),G*(i,k)) by RLTOPSP1:68;
then reconsider X1=X as non empty compact Subset of TOP-REAL 2 by A6,
XBOOLE_0:def 4;
A10: LSeg(G*(i,j),G*(i,k)) meets L~Lower_Seq(C,n) by A6,A9,XBOOLE_0:3;
set s = G*(i,1)`1;
set e = G*(i,k);
set f = G*(i,j);
set w1 = lower_bound(proj2.:(LSeg(f,e) /\ L~Lower_Seq(C,n)));
A11: j <= width G by A4,A5,XXREAL_0:2;
then [i,j] in Indices G by A1,A2,A3,MATRIX_0:30;
then consider k1 be Nat such that
A12: j <= k1 and
A13: k1 <= k and
A14: G*(i,k1)`2 = w1 by A4,A10,A8,JORDAN1F:1,JORDAN1G:5;
set p = |[s,w1]|;
A15: k1 <= width G by A5,A13,XXREAL_0:2;
f`1 = s by A1,A2,A3,A11,GOBOARD5:2
.= e`1 by A1,A2,A5,A7,GOBOARD5:2;
then
A16: LSeg(f,e) is vertical by SPPOL_1:16;
take k1;
thus j <= k1 & k1 <= k by A12,A13;
consider pp be object such that
A17: pp in S-most X1 by XBOOLE_0:def 1;
A18: 1 <= k1 by A3,A12,XXREAL_0:2;
then
A19: G*(i,k1)`1 = s by A1,A2,A15,GOBOARD5:2;
then
A20: p = G*(i,k1) by A14,EUCLID:53;
then
A21: f`2 <= p`2 by A1,A2,A3,A12,A15,SPRECT_3:12;
A22: f`1 = p`1 by A1,A2,A3,A11,A19,A20,GOBOARD5:2;
reconsider pp as Point of TOP-REAL 2 by A17;
A23: pp in X by A17,XBOOLE_0:def 4;
then
A24: pp in L~Lower_Seq(C,n) by XBOOLE_0:def 4;
A25: p`2 = S-bound X by A14,A20,SPRECT_1:44
.= (S-min X)`2 by EUCLID:52
.= pp`2 by A17,PSCOMP_1:55;
pp in LSeg(G*(i,j),G*(i,k)) by A23,XBOOLE_0:def 4;
then pp`1 = p`1 by A22,A16,SPPOL_1:41;
then
A26: p in L~Lower_Seq(C,n) by A24,A25,TOPREAL3:6;
for x be object holds x in LSeg(p,f) /\ L~Lower_Seq(C,n) iff x = p
proof
let x be object;
thus x in LSeg(p,f) /\ L~Lower_Seq(C,n) implies x = p
proof
reconsider EE = LSeg(f,e) /\ L~Lower_Seq(C,n) as compact Subset of
TOP-REAL 2;
reconsider E0 = proj2.:EE as compact Subset of REAL by JCT_MISC:15;
A27: f in LSeg(f,e) by RLTOPSP1:68;
A28: e`1 = p`1 by A1,A2,A5,A7,A19,A20,GOBOARD5:2;
A29: p`2 <= e`2 by A1,A2,A5,A13,A18,A20,SPRECT_3:12;
A30: f`2 <= p`2 by A1,A2,A3,A12,A15,A20,SPRECT_3:12;
f`1 = p`1 by A1,A2,A3,A11,A19,A20,GOBOARD5:2;
then p in LSeg(f,e) by A28,A30,A29,GOBOARD7:7;
then
A31: LSeg(p,f) c= LSeg(f,e) by A27,TOPREAL1:6;
assume
A32: x in LSeg(p,f) /\ L~Lower_Seq(C,n);
then reconsider pp = x as Point of TOP-REAL 2;
A33: pp in LSeg(p,f) by A32,XBOOLE_0:def 4;
then
A34: pp`2 <= p`2 by A21,TOPREAL1:4;
pp in L~Lower_Seq(C,n) by A32,XBOOLE_0:def 4;
then pp in EE by A33,A31,XBOOLE_0:def 4;
then proj2.pp in E0 by FUNCT_2:35;
then
A35: pp`2 in E0 by PSCOMP_1:def 6;
E0 is real-bounded by RCOMP_1:10;
then E0 is bounded_below by XXREAL_2:def 11;
then p`2 <= pp`2 by A14,A20,A35,SEQ_4:def 2;
then
A36: pp`2 = p`2 by A34,XXREAL_0:1;
pp`1 = p`1 by A22,A33,GOBOARD7:5;
hence thesis by A36,TOPREAL3:6;
end;
assume
A37: x = p;
then x in LSeg(p,f) by RLTOPSP1:68;
hence thesis by A26,A37,XBOOLE_0:def 4;
end;
hence thesis by A20,TARSKI:def 1;
end;
theorem
for C be compact connected non vertical non horizontal Subset of
TOP-REAL 2 for i,j,k be Nat st 1 <= i & i <= len Gauge(C,n) & 1 <= j
& j <= k & k <= width Gauge(C,n) & Gauge(C,n)*(i,j) in L~Upper_Seq(C,n) & Gauge
(C,n)*(i,k) in L~Lower_Seq(C,n) ex j1,k1 be Nat st j <= j1 & j1 <=
k1 & k1 <= k & LSeg(Gauge(C,n)*(i,j1),Gauge(C,n)*(i,k1)) /\ L~Upper_Seq(C,n) =
{Gauge(C,n)*(i,j1)} & LSeg(Gauge(C,n)*(i,j1),Gauge(C,n)*(i,k1)) /\ L~Lower_Seq(
C,n) = {Gauge(C,n)*(i,k1)}
proof
let C be compact connected non vertical non horizontal Subset of TOP-REAL 2;
let i,j,k be Nat;
assume that
A1: 1 <= i and
A2: i <= len Gauge(C,n) and
A3: 1 <= j and
A4: j <= k and
A5: k <= width Gauge(C,n) and
A6: Gauge(C,n)*(i,j) in L~Upper_Seq(C,n) and
A7: Gauge(C,n)*(i,k) in L~Lower_Seq(C,n);
set G = Gauge(C,n);
A8: j <= width G by A4,A5,XXREAL_0:2;
then
A9: [i,j] in Indices G by A1,A2,A3,MATRIX_0:30;
set s = G*(i,1)`1;
set e = G*(i,k);
set f = G*(i,j);
set w1 = lower_bound(proj2.:(LSeg(f,e) /\ L~Lower_Seq(C,n)));
A10: G*(i,k) in LSeg(G*(i,j),G*(i,k)) by RLTOPSP1:68;
then
A11: LSeg(G*(i,j),G*(i,k)) meets L~Lower_Seq(C,n) by A7,XBOOLE_0:3;
A12: k >= 1 by A3,A4,XXREAL_0:2;
then [i,k] in Indices G by A1,A2,A5,MATRIX_0:30;
then consider k1 be Nat such that
A13: j <= k1 and
A14: k1 <= k and
A15: G*(i,k1)`2 = w1 by A4,A11,A9,JORDAN1F:1,JORDAN1G:5;
A16: k1 <= width G by A5,A14,XXREAL_0:2;
A17: G*(i,j) in LSeg(G*(i,j),G*(i,k1)) by RLTOPSP1:68;
then
A18: LSeg(G*(i,j),G*(i,k1)) meets L~Upper_Seq(C,n) by A6,XBOOLE_0:3;
set X = LSeg(G*(i,j),G*(i,k1)) /\ L~Upper_Seq(C,n);
reconsider X1=X as non empty compact Subset of TOP-REAL 2 by A6,A17,
XBOOLE_0:def 4;
consider pp be object such that
A19: pp in N-most X1 by XBOOLE_0:def 1;
reconsider pp as Point of TOP-REAL 2 by A19;
A20: pp in X by A19,XBOOLE_0:def 4;
then
A21: pp in L~Upper_Seq(C,n) by XBOOLE_0:def 4;
set p = |[s,w1]|;
set w2 = upper_bound(proj2.:(LSeg(f,p) /\ L~Upper_Seq(C,n)));
set q = |[s,w2]|;
A22: pp in LSeg(G*(i,j),G*(i,k1)) by A20,XBOOLE_0:def 4;
A23: 1 <= k1 by A3,A13,XXREAL_0:2;
then
A24: G*(i,k1)`1 = s by A1,A2,A16,GOBOARD5:2;
then
A25: p = G*(i,k1) by A15,EUCLID:53;
[i,k1] in Indices G by A1,A2,A23,A16,MATRIX_0:30;
then consider j1 be Nat such that
A26: j <= j1 and
A27: j1 <= k1 and
A28: G*(i,j1)`2 = w2 by A9,A13,A25,A18,JORDAN1F:2,JORDAN1G:4;
take j1,k1;
thus j <= j1 & j1 <= k1 & k1 <= k by A14,A26,A27;
A29: j1 <= width G by A16,A27,XXREAL_0:2;
A30: 1 <= j1 by A3,A26,XXREAL_0:2;
then
A31: G*(i,j1)`1 = s by A1,A2,A29,GOBOARD5:2;
then
A32: q = G*(i,j1) by A28,EUCLID:53;
then
A33: q`2 <= p`2 by A1,A2,A16,A25,A27,A30,SPRECT_3:12;
A34: q`2 = N-bound X by A25,A28,A32,SPRECT_1:45
.= (N-min X)`2 by EUCLID:52
.= pp`2 by A19,PSCOMP_1:39;
A35: f`1 = p`1 by A1,A2,A3,A8,A24,A25,GOBOARD5:2;
then LSeg(f,p) is vertical by SPPOL_1:16;
then pp`1 = q`1 by A24,A25,A31,A32,A22,SPPOL_1:41;
then
A36: q in L~Upper_Seq(C,n) by A21,A34,TOPREAL3:6;
for x be object holds x in LSeg(p,q) /\ L~Upper_Seq(C,n) iff x = q
proof
let x be object;
thus x in LSeg(p,q) /\ L~Upper_Seq(C,n) implies x = q
proof
reconsider EE = LSeg(f,p) /\ L~Upper_Seq(C,n) as compact Subset of
TOP-REAL 2;
reconsider E0 = proj2.:EE as compact Subset of REAL by JCT_MISC:15;
A37: p in LSeg(f,p) by RLTOPSP1:68;
A38: f`2 <= q`2 by A1,A2,A3,A26,A29,A32,SPRECT_3:12;
f`1 = q`1 by A1,A2,A3,A8,A31,A32,GOBOARD5:2;
then q in LSeg(p,f) by A24,A25,A31,A32,A33,A38,GOBOARD7:7;
then
A39: LSeg(p,q) c= LSeg(f,p) by A37,TOPREAL1:6;
assume
A40: x in LSeg(p,q) /\ L~Upper_Seq(C,n);
then reconsider pp = x as Point of TOP-REAL 2;
A41: pp in LSeg(p,q) by A40,XBOOLE_0:def 4;
then
A42: pp`2 >= q`2 by A33,TOPREAL1:4;
pp in L~Upper_Seq(C,n) by A40,XBOOLE_0:def 4;
then pp in EE by A41,A39,XBOOLE_0:def 4;
then proj2.pp in E0 by FUNCT_2:35;
then
A43: pp`2 in E0 by PSCOMP_1:def 6;
E0 is real-bounded by RCOMP_1:10;
then E0 is bounded_above by XXREAL_2:def 11;
then q`2 >= pp`2 by A28,A32,A43,SEQ_4:def 1;
then
A44: pp`2 = q`2 by A42,XXREAL_0:1;
pp`1 = q`1 by A24,A25,A31,A32,A41,GOBOARD7:5;
hence thesis by A44,TOPREAL3:6;
end;
assume
A45: x = q;
then x in LSeg(p,q) by RLTOPSP1:68;
hence thesis by A36,A45,XBOOLE_0:def 4;
end;
hence
LSeg(Gauge(C,n)*(i,j1), Gauge(C,n)*(i,k1)) /\ L~Upper_Seq(C,n) = {Gauge
(C,n)*(i,j1)} by A25,A32,TARSKI:def 1;
set X = LSeg(G*(i,j),G*(i,k)) /\ L~Lower_Seq(C,n);
reconsider X1=X as non empty compact Subset of TOP-REAL 2 by A7,A10,
XBOOLE_0:def 4;
consider pp be object such that
A46: pp in S-most X1 by XBOOLE_0:def 1;
reconsider pp as Point of TOP-REAL 2 by A46;
A47: pp in X by A46,XBOOLE_0:def 4;
then
A48: pp in L~Lower_Seq(C,n) by XBOOLE_0:def 4;
f`1 = s by A1,A2,A3,A8,GOBOARD5:2
.= e`1 by A1,A2,A5,A12,GOBOARD5:2;
then
A49: LSeg(f,e) is vertical by SPPOL_1:16;
pp in LSeg(G*(i,j),G*(i,k)) by A47,XBOOLE_0:def 4;
then
A50: pp`1 = p`1 by A35,A49,SPPOL_1:41;
p`2 = S-bound X by A15,A25,SPRECT_1:44
.= (S-min X)`2 by EUCLID:52
.= pp`2 by A46,PSCOMP_1:55;
then
A51: p in L~Lower_Seq(C,n) by A48,A50,TOPREAL3:6;
for x be object holds x in LSeg(p,q) /\ L~Lower_Seq(C,n) iff x = p
proof
let x be object;
thus x in LSeg(p,q) /\ L~Lower_Seq(C,n) implies x = p
proof
A52: p`2 <= e`2 by A1,A2,A5,A14,A23,A25,SPRECT_3:12;
A53: f`2 <= p`2 by A1,A2,A3,A13,A16,A25,SPRECT_3:12;
A54: e`1 = p`1 by A1,A2,A5,A12,A24,A25,GOBOARD5:2;
f`1 = p`1 by A1,A2,A3,A8,A24,A25,GOBOARD5:2;
then
A55: p in LSeg(f,e) by A54,A53,A52,GOBOARD7:7;
A56: e`1 = q`1 by A1,A2,A5,A12,A31,A32,GOBOARD5:2;
j1 <= k by A14,A27,XXREAL_0:2;
then
A57: q`2 <= e`2 by A1,A2,A5,A30,A32,SPRECT_3:12;
A58: f`2 <= q`2 by A1,A2,A3,A26,A29,A32,SPRECT_3:12;
f`1 = q`1 by A1,A2,A3,A8,A31,A32,GOBOARD5:2;
then q in LSeg(f,e) by A56,A58,A57,GOBOARD7:7;
then
A59: LSeg(p,q) c= LSeg(f,e) by A55,TOPREAL1:6;
reconsider EE = LSeg(f,e) /\ L~Lower_Seq(C,n) as compact Subset of
TOP-REAL 2;
reconsider E0 = proj2.:EE as compact Subset of REAL by JCT_MISC:15;
assume
A60: x in LSeg(p,q) /\ L~Lower_Seq(C,n);
then reconsider pp = x as Point of TOP-REAL 2;
A61: pp in LSeg(p,q) by A60,XBOOLE_0:def 4;
then
A62: pp`2 <= p`2 by A33,TOPREAL1:4;
pp in L~Lower_Seq(C,n) by A60,XBOOLE_0:def 4;
then pp in EE by A61,A59,XBOOLE_0:def 4;
then proj2.pp in E0 by FUNCT_2:35;
then
A63: pp`2 in E0 by PSCOMP_1:def 6;
E0 is real-bounded by RCOMP_1:10;
then E0 is bounded_below by XXREAL_2:def 11;
then p`2 <= pp`2 by A15,A25,A63,SEQ_4:def 2;
then
A64: pp`2 = p`2 by A62,XXREAL_0:1;
pp`1 = p`1 by A24,A25,A31,A32,A61,GOBOARD7:5;
hence thesis by A64,TOPREAL3:6;
end;
assume
A65: x = p;
then x in LSeg(p,q) by RLTOPSP1:68;
hence thesis by A51,A65,XBOOLE_0:def 4;
end;
hence thesis by A25,A32,TARSKI:def 1;
end;
theorem Th18:
for C be compact connected non vertical non horizontal Subset of
TOP-REAL 2 for i,j,k be Nat st 1 <= j & j <= k & k <= len Gauge(C,n)
& 1 <= i & i <= width Gauge(C,n) & Gauge(C,n)*(j,i) in L~Upper_Seq(C,n) ex j1
be Nat st j <= j1 & j1 <= k & LSeg(Gauge(C,n)*(j1,i),Gauge(C,n)*(k,i
)) /\ L~Upper_Seq(C,n) = {Gauge(C,n)*(j1,i)}
proof
let C be compact connected non vertical non horizontal Subset of TOP-REAL 2;
let i,j,k be Nat;
assume that
A1: 1 <= j and
A2: j <= k and
A3: k <= len Gauge(C,n) and
A4: 1 <= i and
A5: i <= width Gauge(C,n) and
A6: Gauge(C,n)*(j,i) in L~Upper_Seq(C,n);
set G = Gauge(C,n);
A7: k >= 1 by A1,A2,XXREAL_0:2;
then
A8: [k,i] in Indices G by A3,A4,A5,MATRIX_0:30;
set X = LSeg(G*(j,i),G*(k,i)) /\ L~Upper_Seq(C,n);
A9: G*(j,i) in LSeg(G*(j,i),G*(k,i)) by RLTOPSP1:68;
then reconsider X1=X as non empty compact Subset of TOP-REAL 2 by A6,
XBOOLE_0:def 4;
A10: LSeg(G*(j,i),G*(k,i)) meets L~Upper_Seq(C,n) by A6,A9,XBOOLE_0:3;
set s = G*(1,i)`2;
set e = G*(k,i);
set f = G*(j,i);
set w2 = upper_bound(proj1.:(LSeg(f,e) /\ L~Upper_Seq(C,n)));
A11: len G = width G by JORDAN8:def 1;
then
A12: j <= width G by A2,A3,XXREAL_0:2;
then [j,i] in Indices G by A1,A4,A5,A11,MATRIX_0:30;
then consider j1 be Nat such that
A13: j <= j1 and
A14: j1 <= k and
A15: G*(j1,i)`1 = w2 by A2,A10,A8,JORDAN1F:4,JORDAN1G:4;
set q = |[w2,s]|;
A16: 1 <= j1 by A1,A13,XXREAL_0:2;
take j1;
thus j <= j1 & j1 <= k by A13,A14;
consider pp be object such that
A17: pp in E-most X1 by XBOOLE_0:def 1;
reconsider pp as Point of TOP-REAL 2 by A17;
A18: pp in X by A17,XBOOLE_0:def 4;
then
A19: pp in L~Upper_Seq(C,n) by XBOOLE_0:def 4;
A20: j1 <= width G by A3,A11,A14,XXREAL_0:2;
then
A21: G*(j1,i)`2 = s by A4,A5,A11,A16,GOBOARD5:1;
then
A22: q = G*(j1,i) by A15,EUCLID:53;
then
A23: q`1 <= e`1 by A3,A4,A5,A14,A16,SPRECT_3:13;
A24: G*(k,i)`2 = s by A3,A4,A5,A7,GOBOARD5:1;
then f`2 = e`2 by A1,A4,A5,A11,A12,GOBOARD5:1;
then
A25: LSeg(f,e) is horizontal by SPPOL_1:15;
A26: q`1 = E-bound X by A15,A22,SPRECT_1:46
.= (E-min X)`1 by EUCLID:52
.= pp`1 by A17,PSCOMP_1:47;
pp in LSeg(G*(j,i),G*(k,i)) by A18,XBOOLE_0:def 4;
then pp`2 = q`2 by A24,A21,A22,A25,SPPOL_1:40;
then
A27: q in L~Upper_Seq(C,n) by A19,A26,TOPREAL3:6;
for x be object holds x in LSeg(e,q) /\ L~Upper_Seq(C,n) iff x = q
proof
let x be object;
thus x in LSeg(e,q) /\ L~Upper_Seq(C,n) implies x = q
proof
A28: f`1 <= q`1 by A1,A4,A5,A11,A13,A20,A22,SPRECT_3:13;
f`2 = q`2 by A1,A4,A5,A11,A12,A21,A22,GOBOARD5:1;
then
A29: q in LSeg(e,f) by A24,A21,A22,A23,A28,GOBOARD7:8;
e in LSeg(f,e) by RLTOPSP1:68;
then
A30: LSeg(e,q) c= LSeg(f,e) by A29,TOPREAL1:6;
reconsider EE = LSeg(f,e) /\ L~Upper_Seq(C,n) as compact Subset of
TOP-REAL 2;
reconsider E0 = proj1.:EE as compact Subset of REAL by Th4;
assume
A31: x in LSeg(e,q) /\ L~Upper_Seq(C,n);
then reconsider pp = x as Point of TOP-REAL 2;
A32: pp in LSeg(e,q) by A31,XBOOLE_0:def 4;
then
A33: pp`1 >= q`1 by A23,TOPREAL1:3;
pp in L~Upper_Seq(C,n) by A31,XBOOLE_0:def 4;
then pp in EE by A32,A30,XBOOLE_0:def 4;
then proj1.pp in E0 by FUNCT_2:35;
then
A34: pp`1 in E0 by PSCOMP_1:def 5;
E0 is real-bounded by RCOMP_1:10;
then E0 is bounded_above by XXREAL_2:def 11;
then q`1 >= pp`1 by A15,A22,A34,SEQ_4:def 1;
then
A35: pp`1 = q`1 by A33,XXREAL_0:1;
pp`2 = q`2 by A24,A21,A22,A32,GOBOARD7:6;
hence thesis by A35,TOPREAL3:6;
end;
assume
A36: x = q;
then x in LSeg(e,q) by RLTOPSP1:68;
hence thesis by A27,A36,XBOOLE_0:def 4;
end;
hence thesis by A22,TARSKI:def 1;
end;
theorem
for C be compact connected non vertical non horizontal Subset of
TOP-REAL 2 for i,j,k be Nat st 1 <= j & j <= k & k <= len Gauge(C,n)
& 1 <= i & i <= width Gauge(C,n) & Gauge(C,n)*(k,i) in L~Lower_Seq(C,n) ex k1
be Nat st j <= k1 & k1 <= k & LSeg(Gauge(C,n)*(j,i),Gauge(C,n)*(k1,i
)) /\ L~Lower_Seq(C,n) = {Gauge(C,n)*(k1,i)}
proof
let C be compact connected non vertical non horizontal Subset of TOP-REAL 2;
let i,j,k be Nat;
assume that
A1: 1 <= j and
A2: j <= k and
A3: k <= len Gauge(C,n) and
A4: 1 <= i and
A5: i <= width Gauge(C,n) and
A6: Gauge(C,n)*(k,i) in L~Lower_Seq(C,n);
set G = Gauge(C,n);
A7: k >= 1 by A1,A2,XXREAL_0:2;
then
A8: [k,i] in Indices G by A3,A4,A5,MATRIX_0:30;
set X = LSeg(G*(j,i),G*(k,i)) /\ L~Lower_Seq(C,n);
A9: G*(k,i) in LSeg(G*(j,i),G*(k,i)) by RLTOPSP1:68;
then reconsider X1=X as non empty compact Subset of TOP-REAL 2 by A6,
XBOOLE_0:def 4;
A10: LSeg(G*(j,i),G*(k,i)) meets L~Lower_Seq(C,n) by A6,A9,XBOOLE_0:3;
set s = G*(1,i)`2;
set e = G*(k,i);
set f = G*(j,i);
set w1 = lower_bound(proj1.:(LSeg(f,e) /\ L~Lower_Seq(C,n)));
A11: len G = width G by JORDAN8:def 1;
then
A12: j <= width G by A2,A3,XXREAL_0:2;
then [j,i] in Indices G by A1,A4,A5,A11,MATRIX_0:30;
then consider k1 be Nat such that
A13: j <= k1 and
A14: k1 <= k and
A15: G*(k1,i)`1 = w1 by A2,A10,A8,JORDAN1F:3,JORDAN1G:5;
set p = |[w1,s]|;
A16: k1 <= width G by A3,A11,A14,XXREAL_0:2;
f`2 = s by A1,A4,A5,A11,A12,GOBOARD5:1
.= e`2 by A3,A4,A5,A7,GOBOARD5:1;
then
A17: LSeg(f,e) is horizontal by SPPOL_1:15;
take k1;
thus j <= k1 & k1 <= k by A13,A14;
consider pp be object such that
A18: pp in W-most X1 by XBOOLE_0:def 1;
A19: 1 <= k1 by A1,A13,XXREAL_0:2;
then
A20: G*(k1,i)`2 = s by A4,A5,A11,A16,GOBOARD5:1;
then
A21: p = G*(k1,i) by A15,EUCLID:53;
then
A22: f`1 <= p`1 by A1,A4,A5,A11,A13,A16,SPRECT_3:13;
A23: f`2 = p`2 by A1,A4,A5,A11,A12,A20,A21,GOBOARD5:1;
reconsider pp as Point of TOP-REAL 2 by A18;
A24: pp in X by A18,XBOOLE_0:def 4;
then
A25: pp in L~Lower_Seq(C,n) by XBOOLE_0:def 4;
A26: p`1 = W-bound X by A15,A21,SPRECT_1:43
.= (W-min X)`1 by EUCLID:52
.= pp`1 by A18,PSCOMP_1:31;
pp in LSeg(G*(j,i),G*(k,i)) by A24,XBOOLE_0:def 4;
then pp`2 = p`2 by A23,A17,SPPOL_1:40;
then
A27: p in L~Lower_Seq(C,n) by A25,A26,TOPREAL3:6;
for x be object holds x in LSeg(p,f) /\ L~Lower_Seq(C,n) iff x = p
proof
let x be object;
thus x in LSeg(p,f) /\ L~Lower_Seq(C,n) implies x = p
proof
reconsider EE = LSeg(f,e) /\ L~Lower_Seq(C,n) as compact Subset of
TOP-REAL 2;
assume
A28: x in LSeg(p,f) /\ L~Lower_Seq(C,n);
then reconsider pp = x as Point of TOP-REAL 2;
A29: pp in LSeg(p,f) by A28,XBOOLE_0:def 4;
then
A30: pp`1 <= p`1 by A22,TOPREAL1:3;
A31: p`1 <= e`1 by A3,A4,A5,A14,A19,A21,SPRECT_3:13;
A32: f`1 <= p`1 by A1,A4,A5,A11,A13,A16,A21,SPRECT_3:13;
A33: e`2 = p`2 by A3,A4,A5,A7,A20,A21,GOBOARD5:1;
reconsider E0 = proj1.:EE as compact Subset of REAL by Th4;
A34: f in LSeg(f,e) by RLTOPSP1:68;
f`2 = p`2 by A1,A4,A5,A11,A12,A20,A21,GOBOARD5:1;
then p in LSeg(f,e) by A33,A32,A31,GOBOARD7:8;
then
A35: LSeg(p,f) c= LSeg(f,e) by A34,TOPREAL1:6;
pp in L~Lower_Seq(C,n) by A28,XBOOLE_0:def 4;
then pp in EE by A29,A35,XBOOLE_0:def 4;
then proj1.pp in E0 by FUNCT_2:35;
then
A36: pp`1 in E0 by PSCOMP_1:def 5;
E0 is real-bounded by RCOMP_1:10;
then E0 is bounded_below by XXREAL_2:def 11;
then p`1 <= pp`1 by A15,A21,A36,SEQ_4:def 2;
then
A37: pp`1 = p`1 by A30,XXREAL_0:1;
pp`2 = p`2 by A23,A29,GOBOARD7:6;
hence thesis by A37,TOPREAL3:6;
end;
assume
A38: x = p;
then x in LSeg(p,f) by RLTOPSP1:68;
hence thesis by A27,A38,XBOOLE_0:def 4;
end;
hence thesis by A21,TARSKI:def 1;
end;
theorem Th20:
for C be compact connected non vertical non horizontal Subset of
TOP-REAL 2 for i,j,k be Nat st 1 <= j & j <= k & k <= len Gauge(C,n)
& 1 <= i & i <= width Gauge(C,n) & Gauge(C,n)*(j,i) in L~Upper_Seq(C,n) & Gauge
(C,n)*(k,i) in L~Lower_Seq(C,n) ex j1,k1 be Nat st j <= j1 & j1 <=
k1 & k1 <= k & LSeg(Gauge(C,n)*(j1,i),Gauge(C,n)*(k1,i)) /\ L~Upper_Seq(C,n) =
{Gauge(C,n)*(j1,i)} & LSeg(Gauge(C,n)*(j1,i),Gauge(C,n)*(k1,i)) /\ L~Lower_Seq(
C,n) = {Gauge(C,n)*(k1,i)}
proof
let C be compact connected non vertical non horizontal Subset of TOP-REAL 2;
let i,j,k be Nat;
assume that
A1: 1 <= j and
A2: j <= k and
A3: k <= len Gauge(C,n) and
A4: 1 <= i and
A5: i <= width Gauge(C,n) and
A6: Gauge(C,n)*(j,i) in L~Upper_Seq(C,n) and
A7: Gauge(C,n)*(k,i) in L~Lower_Seq(C,n);
set G = Gauge(C,n);
A8: len G = width G by JORDAN8:def 1;
then
A9: j <= width G by A2,A3,XXREAL_0:2;
then
A10: [j,i] in Indices G by A1,A4,A5,A8,MATRIX_0:30;
set s = G*(1,i)`2;
set e = G*(k,i);
set f = G*(j,i);
set w1 = lower_bound(proj1.:(LSeg(f,e) /\ L~Lower_Seq(C,n)));
A11: G*(k,i) in LSeg(G*(j,i),G*(k,i)) by RLTOPSP1:68;
then
A12: LSeg(G*(j,i),G*(k,i)) meets L~Lower_Seq(C,n) by A7,XBOOLE_0:3;
A13: k >= 1 by A1,A2,XXREAL_0:2;
then [k,i] in Indices G by A3,A4,A5,MATRIX_0:30;
then consider k1 be Nat such that
A14: j <= k1 and
A15: k1 <= k and
A16: G*(k1,i)`1 = w1 by A2,A12,A10,JORDAN1F:3,JORDAN1G:5;
A17: k1 <= width G by A3,A8,A15,XXREAL_0:2;
set p = |[w1,s]|;
set w2 = upper_bound(proj1.:(LSeg(f,p) /\ L~Upper_Seq(C,n)));
set q = |[w2,s]|;
A18: G*(j,i) in LSeg(G*(j,i),G*(k1,i)) by RLTOPSP1:68;
then
A19: LSeg(G*(j,i),G*(k1,i)) meets L~Upper_Seq(C,n) by A6,XBOOLE_0:3;
A20: 1 <= k1 by A1,A14,XXREAL_0:2;
then
A21: G*(k1,i)`2 = s by A4,A5,A8,A17,GOBOARD5:1;
then
A22: p = G*(k1,i) by A16,EUCLID:53;
f`2 = s by A1,A4,A5,A8,A9,GOBOARD5:1
.= e`2 by A3,A4,A5,A13,GOBOARD5:1;
then
A23: LSeg(f,e) is horizontal by SPPOL_1:15;
set X = LSeg(G*(j,i),G*(k1,i)) /\ L~Upper_Seq(C,n);
reconsider X1=X as non empty compact Subset of TOP-REAL 2 by A6,A18,
XBOOLE_0:def 4;
consider pp be object such that
A24: pp in E-most X1 by XBOOLE_0:def 1;
[k1,i] in Indices G by A4,A5,A8,A20,A17,MATRIX_0:30;
then consider j1 be Nat such that
A25: j <= j1 and
A26: j1 <= k1 and
A27: G*(j1,i)`1 = w2 by A10,A14,A22,A19,JORDAN1F:4,JORDAN1G:4;
A28: j1 <= width G by A17,A26,XXREAL_0:2;
reconsider pp as Point of TOP-REAL 2 by A24;
A29: pp in X by A24,XBOOLE_0:def 4;
then
A30: pp in L~Upper_Seq(C,n) by XBOOLE_0:def 4;
take j1,k1;
thus j <= j1 & j1 <= k1 & k1 <= k by A15,A25,A26;
A31: pp in LSeg(G*(j,i),G*(k1,i)) by A29,XBOOLE_0:def 4;
A32: 1 <= j1 by A1,A25,XXREAL_0:2;
then
A33: G*(j1,i)`2 = s by A4,A5,A8,A28,GOBOARD5:1;
then
A34: q = G*(j1,i) by A27,EUCLID:53;
then
A35: q`1 <= p`1 by A4,A5,A8,A17,A22,A26,A32,SPRECT_3:13;
A36: q`1 = E-bound X by A22,A27,A34,SPRECT_1:46
.= (E-min X)`1 by EUCLID:52
.= pp`1 by A24,PSCOMP_1:47;
A37: f`2 = p`2 by A1,A4,A5,A8,A9,A21,A22,GOBOARD5:1;
then LSeg(f,p) is horizontal by SPPOL_1:15;
then pp`2 = q`2 by A21,A22,A33,A34,A31,SPPOL_1:40;
then
A38: q in L~Upper_Seq(C,n) by A30,A36,TOPREAL3:6;
for x be object holds x in LSeg(p,q) /\ L~Upper_Seq(C,n) iff x = q
proof
let x be object;
thus x in LSeg(p,q) /\ L~Upper_Seq(C,n) implies x = q
proof
reconsider EE = LSeg(f,p) /\ L~Upper_Seq(C,n) as compact Subset of
TOP-REAL 2;
assume
A39: x in LSeg(p,q) /\ L~Upper_Seq(C,n);
then reconsider pp = x as Point of TOP-REAL 2;
A40: pp in LSeg(p,q) by A39,XBOOLE_0:def 4;
then
A41: pp`1 >= q`1 by A35,TOPREAL1:3;
A42: f`1 <= q`1 by A1,A4,A5,A8,A25,A28,A34,SPRECT_3:13;
reconsider E0 = proj1.:EE as compact Subset of REAL by Th4;
A43: p in LSeg(f,p) by RLTOPSP1:68;
f`2 = q`2 by A1,A4,A5,A8,A9,A33,A34,GOBOARD5:1;
then q in LSeg(p,f) by A21,A22,A33,A34,A35,A42,GOBOARD7:8;
then
A44: LSeg(p,q) c= LSeg(f,p) by A43,TOPREAL1:6;
pp in L~Upper_Seq(C,n) by A39,XBOOLE_0:def 4;
then pp in EE by A40,A44,XBOOLE_0:def 4;
then proj1.pp in E0 by FUNCT_2:35;
then
A45: pp`1 in E0 by PSCOMP_1:def 5;
E0 is real-bounded by RCOMP_1:10;
then E0 is bounded_above by XXREAL_2:def 11;
then q`1 >= pp`1 by A27,A34,A45,SEQ_4:def 1;
then
A46: pp`1 = q`1 by A41,XXREAL_0:1;
pp`2 = q`2 by A21,A22,A33,A34,A40,GOBOARD7:6;
hence thesis by A46,TOPREAL3:6;
end;
assume
A47: x = q;
then x in LSeg(p,q) by RLTOPSP1:68;
hence thesis by A38,A47,XBOOLE_0:def 4;
end;
hence
LSeg(Gauge(C,n)*(j1,i),Gauge(C,n)*(k1,i)) /\ L~Upper_Seq(C,n) = {Gauge(
C,n)*(j1,i)} by A22,A34,TARSKI:def 1;
set X = LSeg(G*(j,i),G*(k,i)) /\ L~Lower_Seq(C,n);
reconsider X1=X as non empty compact Subset of TOP-REAL 2 by A7,A11,
XBOOLE_0:def 4;
consider pp be object such that
A48: pp in W-most X1 by XBOOLE_0:def 1;
reconsider pp as Point of TOP-REAL 2 by A48;
A49: pp in X by A48,XBOOLE_0:def 4;
then
A50: pp in L~Lower_Seq(C,n) by XBOOLE_0:def 4;
pp in LSeg(G*(j,i),G*(k,i)) by A49,XBOOLE_0:def 4;
then
A51: pp`2 = p`2 by A37,A23,SPPOL_1:40;
p`1 = W-bound X by A16,A22,SPRECT_1:43
.= (W-min X)`1 by EUCLID:52
.= pp`1 by A48,PSCOMP_1:31;
then
A52: p in L~Lower_Seq(C,n) by A50,A51,TOPREAL3:6;
for x be object holds x in LSeg(p,q) /\ L~Lower_Seq(C,n) iff x = p
proof
let x be object;
thus x in LSeg(p,q) /\ L~Lower_Seq(C,n) implies x = p
proof
j1 <= k by A15,A26,XXREAL_0:2;
then
A53: q`1 <= e`1 by A3,A4,A5,A32,A34,SPRECT_3:13;
A54: e`2 = p`2 by A3,A4,A5,A13,A21,A22,GOBOARD5:1;
A55: f`1 <= p`1 by A1,A4,A5,A8,A14,A17,A22,SPRECT_3:13;
A56: f`1 <= q`1 by A1,A4,A5,A8,A25,A28,A34,SPRECT_3:13;
A57: p`1 <= e`1 by A3,A4,A5,A15,A20,A22,SPRECT_3:13;
f`2 = p`2 by A1,A4,A5,A8,A9,A21,A22,GOBOARD5:1;
then
A58: p in LSeg(f,e) by A54,A55,A57,GOBOARD7:8;
A59: e`2 = q`2 by A3,A4,A5,A13,A33,A34,GOBOARD5:1;
f`2 = q`2 by A1,A4,A5,A8,A9,A33,A34,GOBOARD5:1;
then q in LSeg(f,e) by A59,A56,A53,GOBOARD7:8;
then
A60: LSeg(p,q) c= LSeg(f,e) by A58,TOPREAL1:6;
reconsider EE = LSeg(f,e) /\ L~Lower_Seq(C,n) as compact Subset of
TOP-REAL 2;
reconsider E0 = proj1.:EE as compact Subset of REAL by Th4;
assume
A61: x in LSeg(p,q) /\ L~Lower_Seq(C,n);
then reconsider pp = x as Point of TOP-REAL 2;
A62: pp in LSeg(p,q) by A61,XBOOLE_0:def 4;
then
A63: pp`1 <= p`1 by A35,TOPREAL1:3;
pp in L~Lower_Seq(C,n) by A61,XBOOLE_0:def 4;
then pp in EE by A62,A60,XBOOLE_0:def 4;
then proj1.pp in E0 by FUNCT_2:35;
then
A64: pp`1 in E0 by PSCOMP_1:def 5;
E0 is real-bounded by RCOMP_1:10;
then E0 is bounded_below by XXREAL_2:def 11;
then p`1 <= pp`1 by A16,A22,A64,SEQ_4:def 2;
then
A65: pp`1 = p`1 by A63,XXREAL_0:1;
pp`2 = p`2 by A21,A22,A33,A34,A62,GOBOARD7:6;
hence thesis by A65,TOPREAL3:6;
end;
assume
A66: x = p;
then x in LSeg(p,q) by RLTOPSP1:68;
hence thesis by A52,A66,XBOOLE_0:def 4;
end;
hence thesis by A22,A34,TARSKI:def 1;
end;
theorem Th21:
for C be Simple_closed_curve for i,j,k be Nat st 1 <
i & i < len Gauge(C,n) & 1 <= j & j <= k & k <= width Gauge(C,n) & Gauge(C,n)*(
i,k) in L~Upper_Seq(C,n) & Gauge(C,n)*(i,j) in L~Lower_Seq(C,n) holds LSeg(
Gauge(C,n)*(i,j),Gauge(C,n)*(i,k)) meets Lower_Arc C
proof
let C be Simple_closed_curve;
let i,j,k be Nat;
assume that
A1: 1 < i and
A2: i < len Gauge(C,n) and
A3: 1 <= j and
A4: j <= k and
A5: k <= width Gauge(C,n) and
A6: Gauge(C,n)*(i,k) in L~Upper_Seq(C,n) and
A7: Gauge(C,n)*(i,j) in L~Lower_Seq(C,n);
consider j1,k1 be Nat such that
A8: j <= j1 and
A9: j1 <= k1 and
A10: k1 <= k and
A11: LSeg(Gauge(C,n)*(i,j1),Gauge(C,n)*(i,k1)) /\ L~Lower_Seq(C,n) = {
Gauge(C,n)*(i,j1)} and
A12: LSeg(Gauge(C,n)*(i,j1),Gauge(C,n)*(i,k1)) /\ L~Upper_Seq(C,n) = {
Gauge(C,n)*(i,k1)} by A1,A2,A3,A4,A5,A6,A7,Th11;
A13: k1 <= width Gauge(C,n) by A5,A10,XXREAL_0:2;
1 <= j1 by A3,A8,XXREAL_0:2;
then
LSeg(Gauge(C,n)*(i,j1),Gauge(C,n)*(i,k1)) meets Lower_Arc C by A1,A2,A9,A11
,A12,A13,JORDAN1J:58;
hence thesis by A1,A2,A3,A5,A8,A9,A10,Th5,XBOOLE_1:63;
end;
theorem Th22:
for C be Simple_closed_curve for i,j,k be Nat st 1 <
i & i < len Gauge(C,n) & 1 <= j & j <= k & k <= width Gauge(C,n) & Gauge(C,n)*(
i,k) in L~Upper_Seq(C,n) & Gauge(C,n)*(i,j) in L~Lower_Seq(C,n) holds LSeg(
Gauge(C,n)*(i,j),Gauge(C,n)*(i,k)) meets Upper_Arc C
proof
let C be Simple_closed_curve;
let i,j,k be Nat;
assume that
A1: 1 < i and
A2: i < len Gauge(C,n) and
A3: 1 <= j and
A4: j <= k and
A5: k <= width Gauge(C,n) and
A6: Gauge(C,n)*(i,k) in L~Upper_Seq(C,n) and
A7: Gauge(C,n)*(i,j) in L~Lower_Seq(C,n);
consider j1,k1 be Nat such that
A8: j <= j1 and
A9: j1 <= k1 and
A10: k1 <= k and
A11: LSeg(Gauge(C,n)*(i,j1),Gauge(C,n)*(i,k1)) /\ L~Lower_Seq(C,n) = {
Gauge(C,n)*(i,j1)} and
A12: LSeg(Gauge(C,n)*(i,j1),Gauge(C,n)*(i,k1)) /\ L~Upper_Seq(C,n) = {
Gauge(C,n)*(i,k1)} by A1,A2,A3,A4,A5,A6,A7,Th11;
A13: k1 <= width Gauge(C,n) by A5,A10,XXREAL_0:2;
1 <= j1 by A3,A8,XXREAL_0:2;
then
LSeg(Gauge(C,n)*(i,j1),Gauge(C,n)*(i,k1)) meets Upper_Arc C by A1,A2,A9,A11
,A12,A13,JORDAN1J:59;
hence thesis by A1,A2,A3,A5,A8,A9,A10,Th5,XBOOLE_1:63;
end;
theorem Th23:
for C be Simple_closed_curve for i,j,k be Nat st 1 <
i & i < len Gauge(C,n) & 1 <= j & j <= k & k <= width Gauge(C,n) & n > 0 &
Gauge(C,n)*(i,k) in Upper_Arc L~Cage(C,n) & Gauge(C,n)*(i,j) in Lower_Arc L~
Cage(C,n) holds LSeg(Gauge(C,n)*(i,j),Gauge(C,n)*(i,k)) meets Lower_Arc C
proof
let C be Simple_closed_curve;
let i,j,k be Nat;
assume that
A1: 1 < i and
A2: i < len Gauge(C,n) and
A3: 1 <= j and
A4: j <= k and
A5: k <= width Gauge(C,n) and
A6: n > 0 and
A7: Gauge(C,n)*(i,k) in Upper_Arc L~Cage(C,n) and
A8: Gauge(C,n)*(i,j) in Lower_Arc L~Cage(C,n);
A9: L~Lower_Seq(C,n) = Lower_Arc L~Cage(C,n) by A6,JORDAN1G:56;
L~Upper_Seq(C,n) = Upper_Arc L~Cage(C,n) by A6,JORDAN1G:55;
hence thesis by A1,A2,A3,A4,A5,A7,A8,A9,Th21;
end;
theorem Th24:
for C be Simple_closed_curve for i,j,k be Nat st 1 <
i & i < len Gauge(C,n) & 1 <= j & j <= k & k <= width Gauge(C,n) & n > 0 &
Gauge(C,n)*(i,k) in Upper_Arc L~Cage(C,n) & Gauge(C,n)*(i,j) in Lower_Arc L~
Cage(C,n) holds LSeg(Gauge(C,n)*(i,j),Gauge(C,n)*(i,k)) meets Upper_Arc C
proof
let C be Simple_closed_curve;
let i,j,k be Nat;
assume that
A1: 1 < i and
A2: i < len Gauge(C,n) and
A3: 1 <= j and
A4: j <= k and
A5: k <= width Gauge(C,n) and
A6: n > 0 and
A7: Gauge(C,n)*(i,k) in Upper_Arc L~Cage(C,n) and
A8: Gauge(C,n)*(i,j) in Lower_Arc L~Cage(C,n);
A9: L~Lower_Seq(C,n) = Lower_Arc L~Cage(C,n) by A6,JORDAN1G:56;
L~Upper_Seq(C,n) = Upper_Arc L~Cage(C,n) by A6,JORDAN1G:55;
hence thesis by A1,A2,A3,A4,A5,A7,A8,A9,Th22;
end;
theorem
for C be Simple_closed_curve for j,k be Nat holds 1 <= j &
j <= k & k <= width Gauge(C,n+1) & Gauge(C,n+1)*(Center Gauge(C,n+1),k) in
Upper_Arc L~Cage(C,n+1) & Gauge(C,n+1)*(Center Gauge(C,n+1),j) in Lower_Arc L~
Cage(C,n+1) implies LSeg(Gauge(C,n+1)*(Center Gauge(C,n+1),j), Gauge(C,n+1)*(
Center Gauge(C,n+1),k)) meets Lower_Arc C
proof
let C be Simple_closed_curve;
let j,k be Nat;
assume that
A1: 1 <= j and
A2: j <= k and
A3: k <= width Gauge(C,n+1) and
A4: Gauge(C,n+1)*(Center Gauge(C,n+1),k) in Upper_Arc L~Cage(C,n+1) and
A5: Gauge(C,n+1)*(Center Gauge(C,n+1),j) in Lower_Arc L~Cage(C,n+1);
A6: len Gauge(C,n+1) >= 4 by JORDAN8:10;
then len Gauge(C,n+1) >= 2 by XXREAL_0:2;
then
A7: 1 < Center Gauge(C,n+1) by JORDAN1B:14;
len Gauge(C,n+1) >= 3 by A6,XXREAL_0:2;
hence thesis by A1,A2,A3,A4,A5,A7,Th23,JORDAN1B:15;
end;
theorem
for C be Simple_closed_curve for j,k be Nat holds 1 <= j &
j <= k & k <= width Gauge(C,n+1) & Gauge(C,n+1)*(Center Gauge(C,n+1),k) in
Upper_Arc L~Cage(C,n+1) & Gauge(C,n+1)*(Center Gauge(C,n+1),j) in Lower_Arc L~
Cage(C,n+1) implies LSeg(Gauge(C,n+1)*(Center Gauge(C,n+1),j), Gauge(C,n+1)*(
Center Gauge(C,n+1),k)) meets Upper_Arc C
proof
let C be Simple_closed_curve;
let j,k be Nat;
assume that
A1: 1 <= j and
A2: j <= k and
A3: k <= width Gauge(C,n+1) and
A4: Gauge(C,n+1)*(Center Gauge(C,n+1),k) in Upper_Arc L~Cage(C,n+1) and
A5: Gauge(C,n+1)*(Center Gauge(C,n+1),j) in Lower_Arc L~Cage(C,n+1);
A6: len Gauge(C,n+1) >= 4 by JORDAN8:10;
then len Gauge(C,n+1) >= 2 by XXREAL_0:2;
then
A7: 1 < Center Gauge(C,n+1) by JORDAN1B:14;
len Gauge(C,n+1) >= 3 by A6,XXREAL_0:2;
hence thesis by A1,A2,A3,A4,A5,A7,Th24,JORDAN1B:15;
end;
theorem Th27:
for C be compact connected non vertical non horizontal Subset of
TOP-REAL 2 for i,j,k be Nat st 1 < j & k < len Gauge(C,n) & 1 <= i &
i <= width Gauge(C,n) & Gauge(C,n)*(k,i) in L~Upper_Seq(C,n) & Gauge(C,n)*(j,i)
in L~Lower_Seq(C,n) holds j <> k
proof
let C be compact connected non vertical non horizontal Subset of TOP-REAL 2;
let i,j,k be Nat;
assume that
A1: 1 < j and
A2: k < len Gauge(C,n) and
A3: 1 <= i and
A4: i <= width Gauge(C,n) and
A5: Gauge(C,n)*(k,i) in L~Upper_Seq(C,n) and
A6: Gauge(C,n)*(j,i) in L~Lower_Seq(C,n) and
A7: j = k;
A8: [j,i] in Indices Gauge(C,n) by A1,A2,A3,A4,A7,MATRIX_0:30;
Gauge(C,n)*(k,i) in L~Upper_Seq(C,n) /\ L~Lower_Seq(C,n) by A5,A6,A7,
XBOOLE_0:def 4;
then
A9: Gauge(C,n)*(k,i) in {W-min L~Cage(C,n),E-max L~Cage(C,n)} by JORDAN1E:16;
A10: len Gauge(C,n) = width Gauge(C,n) by JORDAN8:def 1;
len Gauge(C,n) >= 4 by JORDAN8:10;
then
A11: len Gauge(C,n) >= 1 by XXREAL_0:2;
then
A12: [len Gauge(C,n),i] in Indices Gauge(C,n) by A3,A4,MATRIX_0:30;
A13: [1,i] in Indices Gauge(C,n) by A3,A4,A11,MATRIX_0:30;
per cases by A9,TARSKI:def 2;
suppose
A14: Gauge(C,n)*(k,i) = W-min L~Cage(C,n);
Gauge(C,n)*(1,i)`1 = W-bound L~Cage(C,n) by A3,A4,A10,JORDAN1A:73;
then
(W-min L~Cage(C,n))`1 <> W-bound L~Cage(C,n) by A1,A7,A8,A13,A14,JORDAN1G:7;
hence contradiction by EUCLID:52;
end;
suppose
A15: Gauge(C,n)*(k,i) = E-max L~Cage(C,n);
Gauge(C,n)*(len Gauge(C,n),i)`1 = E-bound L~Cage(C,n) by A3,A4,A10,
JORDAN1A:71;
then
(E-max L~Cage(C,n))`1 <> E-bound L~Cage(C,n) by A2,A7,A8,A12,A15,JORDAN1G:7;
hence contradiction by EUCLID:52;
end;
end;
theorem Th28:
for C be Simple_closed_curve for i,j,k be Nat st 1 <
j & j <= k & k < len Gauge(C,n) & 1 <= i & i <= width Gauge(C,n) & LSeg(Gauge(C
,n)*(j,i),Gauge(C,n)*(k,i)) /\ L~Upper_Seq(C,n) = {Gauge(C,n)*(k,i)} & LSeg(
Gauge(C,n)*(j,i),Gauge(C,n)*(k,i)) /\ L~Lower_Seq(C,n) = {Gauge(C,n)*(j,i)}
holds LSeg(Gauge(C,n)*(j,i),Gauge(C,n)*(k,i)) meets Lower_Arc C
proof
let C be Simple_closed_curve;
let i,j,k be Nat;
set Ga = Gauge(C,n);
set US = Upper_Seq(C,n);
set LS = Lower_Seq(C,n);
set LA = Lower_Arc C;
set Wmin = W-min L~Cage(C,n);
set Emax = E-max L~Cage(C,n);
set Wbo = W-bound L~Cage(C,n);
set Ebo = E-bound L~Cage(C,n);
set Gij = Ga*(j,i);
set Gik = Ga*(k,i);
assume that
A1: 1 < j and
A2: j <= k and
A3: k < len Ga and
A4: 1 <= i and
A5: i <= width Ga and
A6: LSeg(Gij,Gik) /\ L~US = {Gik} and
A7: LSeg(Gij,Gik) /\ L~LS = {Gij} and
A8: LSeg(Gij,Gik) misses LA;
Gij in {Gij} by TARSKI:def 1;
then
A9: Gij in L~LS by A7,XBOOLE_0:def 4;
Gik in {Gik} by TARSKI:def 1;
then
A10: Gik in L~US by A6,XBOOLE_0:def 4;
A11: len Ga = width Ga by JORDAN8:def 1;
A12: j <> k by A1,A3,A4,A5,A9,A10,Th27;
A13: j <= width Ga by A2,A3,A11,XXREAL_0:2;
A14: 1 <= k by A1,A2,XXREAL_0:2;
A15: k <= width Ga by A3,JORDAN8:def 1;
A16: [j,i] in Indices Ga by A1,A4,A5,A11,A13,MATRIX_0:30;
A17: [k,i] in Indices Ga by A3,A4,A5,A14,MATRIX_0:30;
set go = R_Cut(US,Gik);
set co = L_Cut(LS,Gij);
A18: len US >= 3 by JORDAN1E:15;
then len US >= 1 by XXREAL_0:2;
then 1 in dom US by FINSEQ_3:25;
then
A19: US.1 = US/.1 by PARTFUN1:def 6
.= Wmin by JORDAN1F:5;
A20: Wmin`1 = Wbo by EUCLID:52
.= Ga*(1,k)`1 by A3,A14,JORDAN1A:73;
len Ga >= 4 by JORDAN8:10;
then
A21: len Ga >= 1 by XXREAL_0:2;
then
A22: [1,k] in Indices Ga by A14,A15,MATRIX_0:30;
then
A23: Gik <> US.1 by A1,A2,A17,A19,A20,JORDAN1G:7;
then reconsider go as being_S-Seq FinSequence of TOP-REAL 2 by A10,JORDAN3:35
;
A24: [1,j] in Indices Ga by A1,A13,A21,MATRIX_0:30;
A25: len LS >= 1+2 by JORDAN1E:15;
then
A26: len LS >= 1 by XXREAL_0:2;
then
A27: 1 in dom LS by FINSEQ_3:25;
len LS in dom LS by A26,FINSEQ_3:25;
then
A28: LS.len LS = LS/.len LS by PARTFUN1:def 6
.= Wmin by JORDAN1F:8;
A29: Wmin`1 = Wbo by EUCLID:52
.= Ga*(1,k)`1 by A3,A14,JORDAN1A:73;
A30: [j,i] in Indices Ga by A1,A4,A5,A11,A13,MATRIX_0:30;
then
A31: Gij <> LS.len LS by A1,A22,A28,A29,JORDAN1G:7;
then reconsider co as being_S-Seq FinSequence of TOP-REAL 2 by A9,JORDAN3:34;
A32: [len Ga,k] in Indices Ga by A14,A15,A21,MATRIX_0:30;
A33: LS.1 = LS/.1 by A27,PARTFUN1:def 6
.= Emax by JORDAN1F:6;
Emax`1 = Ebo by EUCLID:52
.= Ga*(len Ga,k)`1 by A3,A14,JORDAN1A:71;
then
A34: Gij <> LS.1 by A2,A3,A30,A32,A33,JORDAN1G:7;
A35: len go >= 1+1 by TOPREAL1:def 8;
A36: Gik in rng US by A4,A5,A10,A11,A14,A15,JORDAN1G:4,JORDAN1J:40;
then
A37: go is_sequence_on Ga by JORDAN1G:4,JORDAN1J:38;
A38: len co >= 1+1 by TOPREAL1:def 8;
A39: Gij in rng LS by A1,A4,A5,A9,A11,A13,JORDAN1G:5,JORDAN1J:40;
then
A40: co is_sequence_on Ga by JORDAN1G:5,JORDAN1J:39;
reconsider go as non constant s.c.c. being_S-Seq FinSequence of TOP-REAL 2
by A35,A37,JGRAPH_1:12,JORDAN8:5;
reconsider co as non constant s.c.c. being_S-Seq FinSequence of TOP-REAL 2
by A38,A40,JGRAPH_1:12,JORDAN8:5;
A41: len go > 1 by A35,NAT_1:13;
then
A42: len go in dom go by FINSEQ_3:25;
then
A43: go/.len go = go.len go by PARTFUN1:def 6
.= Gik by A10,JORDAN3:24;
len co >= 1 by A38,XXREAL_0:2;
then 1 in dom co by FINSEQ_3:25;
then
A44: co/.1 = co.1 by PARTFUN1:def 6
.= Gij by A9,JORDAN3:23;
reconsider m = len go - 1 as Nat by A42,FINSEQ_3:26;
A45: m+1 = len go;
then
A46: len go-'1 = m by NAT_D:34;
A47: LSeg(go,m) c= L~go by TOPREAL3:19;
A48: L~go c= L~US by A10,JORDAN3:41;
then LSeg(go,m) c= L~US by A47;
then
A49: LSeg(go,m) /\ LSeg(Gik,Gij) c= {Gik} by A6,XBOOLE_1:26;
m >= 1 by A35,XREAL_1:19;
then
A50: LSeg(go,m) = LSeg(go/.m,Gik) by A43,A45,TOPREAL1:def 3;
{Gik} c= LSeg(go,m) /\ LSeg(Gik,Gij)
proof
let x be object;
A51: Gik in LSeg(Gik,Gij) by RLTOPSP1:68;
assume x in {Gik};
then
A52: x = Gik by TARSKI:def 1;
Gik in LSeg(go,m) by A50,RLTOPSP1:68;
hence thesis by A52,A51,XBOOLE_0:def 4;
end;
then
A53: LSeg(go,m) /\ LSeg(Gik,Gij) = {Gik} by A49;
A54: LSeg(co,1) c= L~co by TOPREAL3:19;
A55: L~co c= L~LS by A9,JORDAN3:42;
then LSeg(co,1) c= L~LS by A54;
then
A56: LSeg(co,1) /\ LSeg(Gik,Gij) c= {Gij} by A7,XBOOLE_1:26;
A57: LSeg(co,1) = LSeg(Gij,co/.(1+1)) by A38,A44,TOPREAL1:def 3;
{Gij} c= LSeg(co,1) /\ LSeg(Gik,Gij)
proof
let x be object;
A58: Gij in LSeg(Gik,Gij) by RLTOPSP1:68;
assume x in {Gij};
then
A59: x = Gij by TARSKI:def 1;
Gij in LSeg(co,1) by A57,RLTOPSP1:68;
hence thesis by A59,A58,XBOOLE_0:def 4;
end;
then
A60: LSeg(Gik,Gij) /\ LSeg(co,1) = {Gij} by A56;
A61: go/.1 = US/.1 by A10,SPRECT_3:22
.= Wmin by JORDAN1F:5;
then
A62: go/.1 = LS/.len LS by JORDAN1F:8
.= co/.len co by A9,JORDAN1J:35;
A63: rng go c= L~go by A35,SPPOL_2:18;
A64: rng co c= L~co by A38,SPPOL_2:18;
A65: {go/.1} c= L~go /\ L~co
proof
let x be object;
assume x in {go/.1};
then
A66: x = go/.1 by TARSKI:def 1;
then
A67: x in rng go by FINSEQ_6:42;
x in rng co by A62,A66,FINSEQ_6:168;
hence thesis by A63,A64,A67,XBOOLE_0:def 4;
end;
A68: LS.1 = LS/.1 by A27,PARTFUN1:def 6
.= Emax by JORDAN1F:6;
A69: [len Ga,j] in Indices Ga by A1,A13,A21,MATRIX_0:30;
L~go /\ L~co c= {go/.1}
proof
let x be object;
assume
A70: x in L~go /\ L~co;
then
A71: x in L~co by XBOOLE_0:def 4;
A72: now
assume x = Emax;
then
A73: Emax = Gij by A9,A68,A71,JORDAN1E:7;
Ga*(len Ga,j)`1 = Ebo by A1,A11,A13,JORDAN1A:71;
then Emax`1 <> Ebo by A2,A3,A16,A69,A73,JORDAN1G:7;
hence contradiction by EUCLID:52;
end;
x in L~go by A70,XBOOLE_0:def 4;
then x in L~US /\ L~LS by A48,A55,A71,XBOOLE_0:def 4;
then x in {Wmin,Emax} by JORDAN1E:16;
then x = Wmin or x = Emax by TARSKI:def 2;
hence thesis by A61,A72,TARSKI:def 1;
end;
then
A74: L~go /\ L~co = {go/.1} by A65;
set W2 = go/.2;
A75: 2 in dom go by A35,FINSEQ_3:25;
A76: now
assume Gij`1 = Wbo;
then Ga*(1,j)`1 = Ga*(j,i)`1 by A1,A11,A13,JORDAN1A:73;
hence contradiction by A1,A16,A24,JORDAN1G:7;
end;
go = mid(US,1,Gik..US) by A36,JORDAN1G:49
.= US|(Gik..US) by A36,FINSEQ_4:21,FINSEQ_6:116;
then
A77: W2 = US/.2 by A75,FINSEQ_4:70;
A78: Wmin in rng go by A61,FINSEQ_6:42;
set pion = <*Gik,Gij*>;
A79: now
let n be Nat;
assume n in dom pion;
then n in {1,2} by FINSEQ_1:2,89;
then n = 1 or n = 2 by TARSKI:def 2;
hence ex j,i be Nat st [j,i] in Indices Ga & pion/.n = Ga*(j,i)
by A16,A17,FINSEQ_4:17;
end;
A80: Gik <> Gij by A12,A16,A17,GOBOARD1:5;
Gik`2 = Ga*(1,i)`2 by A3,A4,A5,A14,GOBOARD5:1
.= Gij`2 by A1,A4,A5,A11,A13,GOBOARD5:1;
then LSeg(Gik,Gij) is horizontal by SPPOL_1:15;
then pion is being_S-Seq by A80,JORDAN1B:8;
then consider pion1 be FinSequence of TOP-REAL 2 such that
A81: pion1 is_sequence_on Ga and
A82: pion1 is being_S-Seq and
A83: L~pion = L~pion1 and
A84: pion/.1 = pion1/.1 and
A85: pion/.len pion = pion1/.len pion1 and
A86: len pion <= len pion1 by A79,GOBOARD3:2;
reconsider pion1 as being_S-Seq FinSequence of TOP-REAL 2 by A82;
set godo = go^'pion1^'co;
A87: 1+1 <= len Cage(C,n) by GOBOARD7:34,XXREAL_0:2;
A88: 1+1 <= len Rotate(Cage(C,n),Wmin) by GOBOARD7:34,XXREAL_0:2;
len (go^'pion1) >= len go by TOPREAL8:7;
then
A89: len (go^'pion1) >= 1+1 by A35,XXREAL_0:2;
then
A90: len (go^'pion1) > 1+0 by NAT_1:13;
A91: len godo >= len (go^'pion1) by TOPREAL8:7;
then
A92: 1+1 <= len godo by A89,XXREAL_0:2;
A93: US is_sequence_on Ga by JORDAN1G:4;
A94: go/.len go = pion1/.1 by A43,A84,FINSEQ_4:17;
then
A95: go^'pion1 is_sequence_on Ga by A37,A81,TOPREAL8:12;
A96: (go^'pion1)/.len (go^'pion1) = pion/.len pion by A85,FINSEQ_6:156
.= pion/.2 by FINSEQ_1:44
.= co/.1 by A44,FINSEQ_4:17;
then
A97: godo is_sequence_on Ga by A40,A95,TOPREAL8:12;
LSeg(pion1,1) c= L~<*Gik,Gij*> by A83,TOPREAL3:19;
then LSeg(pion1,1) c= LSeg(Gik,Gij) by SPPOL_2:21;
then
A98: LSeg(go,len go-'1) /\ LSeg(pion1,1) c= {Gik} by A46,A53,XBOOLE_1:27;
A99: len pion1 >= 1+1 by A86,FINSEQ_1:44;
{Gik} c= LSeg(go,m) /\ LSeg(pion1,1)
proof
let x be object;
assume x in {Gik};
then
A100: x = Gik by TARSKI:def 1;
A101: Gik in LSeg(go,m) by A50,RLTOPSP1:68;
Gik in LSeg(pion1,1) by A43,A94,A99,TOPREAL1:21;
hence thesis by A100,A101,XBOOLE_0:def 4;
end;
then LSeg(go,len go-'1) /\ LSeg(pion1,1) = { go/.len go } by A43,A46,A98;
then
A102: go^'pion1 is unfolded by A94,TOPREAL8:34;
len pion1 >= 2+0 by A86,FINSEQ_1:44;
then
A103: len pion1-2 >= 0 by XREAL_1:19;
len (go^'pion1)+1-1 = len go+len pion1-1 by FINSEQ_6:139;
then len (go^'pion1)-1 = len go + (len pion1-2)
.= len go + (len pion1-'2) by A103,XREAL_0:def 2;
then
A104: len (go^'pion1)-'1 = len go + (len pion1-'2) by XREAL_0:def 2;
A105: len pion1-1 >= 1 by A99,XREAL_1:19;
then
A106: len pion1-'1 = len pion1-1 by XREAL_0:def 2;
A107: len pion1-'2+1 = len pion1-2+1 by A103,XREAL_0:def 2
.= len pion1-'1 by A105,XREAL_0:def 2;
len pion1-1+1 <= len pion1;
then
A108: len pion1-'1 < len pion1 by A106,NAT_1:13;
LSeg(pion1,len pion1-'1) c= L~<*Gik,Gij*> by A83,TOPREAL3:19;
then LSeg(pion1,len pion1-'1) c= LSeg(Gik,Gij) by SPPOL_2:21;
then
A109: LSeg(pion1,len pion1-'1) /\ LSeg(co,1) c= {Gij} by A60,XBOOLE_1:27;
{Gij} c= LSeg(pion1,len pion1-'1) /\ LSeg(co,1)
proof
let x be object;
assume x in {Gij};
then
A110: x = Gij by TARSKI:def 1;
pion1/.(len pion1-'1+1) = pion/.2 by A85,A106,FINSEQ_1:44
.= Gij by FINSEQ_4:17;
then
A111: Gij in LSeg(pion1,len pion1-'1) by A105,A106,TOPREAL1:21;
Gij in LSeg(co,1) by A57,RLTOPSP1:68;
hence thesis by A110,A111,XBOOLE_0:def 4;
end;
then LSeg(pion1,len pion1-'1) /\ LSeg(co,1) = {Gij} by A109;
then
A112: LSeg(go^'pion1,len go+(len pion1-'2)) /\ LSeg(co,1) = {(go^'pion1)/.
len (go^'pion1)} by A44,A94,A96,A107,A108,TOPREAL8:31;
A113: (go^'pion1) is non trivial by A89,NAT_D:60;
A114: rng pion1 c= L~pion1 by A99,SPPOL_2:18;
A115: {pion1/.1} c= L~go /\ L~pion1
proof
let x be object;
assume x in {pion1/.1};
then
A116: x = pion1/.1 by TARSKI:def 1;
then
A117: x in rng pion1 by FINSEQ_6:42;
x in rng go by A94,A116,FINSEQ_6:168;
hence thesis by A63,A114,A117,XBOOLE_0:def 4;
end;
L~go /\ L~pion1 c= {pion1/.1}
proof
let x be object;
assume
A118: x in L~go /\ L~pion1;
then
A119: x in L~pion1 by XBOOLE_0:def 4;
x in L~go by A118,XBOOLE_0:def 4;
then x in L~pion1 /\ L~US by A48,A119,XBOOLE_0:def 4;
hence thesis by A6,A43,A83,A94,SPPOL_2:21;
end;
then
A120: L~go /\ L~pion1 = {pion1/.1} by A115;
then
A121: (go^'pion1) is s.n.c. by A94,JORDAN1J:54;
rng go /\ rng pion1 c= {pion1/.1} by A63,A114,A120,XBOOLE_1:27;
then
A122: go^'pion1 is one-to-one by JORDAN1J:55;
A123: pion/.len pion = pion/.2 by FINSEQ_1:44
.= co/.1 by A44,FINSEQ_4:17;
A124: {pion1/.len pion1} c= L~co /\ L~pion1
proof
let x be object;
assume x in {pion1/.len pion1};
then
A125: x = pion1/.len pion1 by TARSKI:def 1;
then
A126: x in rng pion1 by FINSEQ_6:168;
x in rng co by A85,A123,A125,FINSEQ_6:42;
hence thesis by A64,A114,A126,XBOOLE_0:def 4;
end;
L~co /\ L~pion1 c= {pion1/.len pion1}
proof
let x be object;
assume
A127: x in L~co /\ L~pion1;
then
A128: x in L~pion1 by XBOOLE_0:def 4;
x in L~co by A127,XBOOLE_0:def 4;
then x in L~pion1 /\ L~LS by A55,A128,XBOOLE_0:def 4;
hence thesis by A7,A44,A83,A85,A123,SPPOL_2:21;
end;
then
A129: L~co /\ L~pion1 = {pion1/.len pion1} by A124;
A130: L~(go^'pion1) /\ L~co = (L~go \/ L~pion1) /\ L~co by A94,TOPREAL8:35
.= {go/.1} \/ {co/.1} by A74,A85,A123,A129,XBOOLE_1:23
.= {(go^'pion1)/.1} \/ {co/.1} by FINSEQ_6:155
.= {(go^'pion1)/.1,co/.1} by ENUMSET1:1;
co/.len co = (go^'pion1)/.1 by A62,FINSEQ_6:155;
then reconsider
godo as non constant standard special_circular_sequence by A92,A96,A97,A102
,A104,A112,A113,A121,A122,A130,JORDAN8:4,5,TOPREAL8:11,33,34;
A131: LA is_an_arc_of E-max C,W-min C by JORDAN6:def 9;
then
A132: LA is connected by JORDAN6:10;
A133: W-min C in LA by A131,TOPREAL1:1;
A134: E-max C in LA by A131,TOPREAL1:1;
set ff = Rotate(Cage(C,n),Wmin);
Wmin in rng Cage(C,n) by SPRECT_2:43;
then
A135: ff/.1 = Wmin by FINSEQ_6:92;
A136: L~ff = L~Cage(C,n) by REVROT_1:33;
then (W-max L~ff)..ff > 1 by A135,SPRECT_5:22;
then (N-min L~ff)..ff > 1 by A135,A136,SPRECT_5:23,XXREAL_0:2;
then (N-max L~ff)..ff > 1 by A135,A136,SPRECT_5:24,XXREAL_0:2;
then
A137: Emax..ff > 1 by A135,A136,SPRECT_5:25,XXREAL_0:2;
A138: now
assume
A139: Gik..US <= 1;
Gik..US >= 1 by A36,FINSEQ_4:21;
then Gik..US = 1 by A139,XXREAL_0:1;
then Gik = US/.1 by A36,FINSEQ_5:38;
hence contradiction by A19,A23,JORDAN1F:5;
end;
A140: Cage(C,n) is_sequence_on Ga by JORDAN9:def 1;
then
A141: ff is_sequence_on Ga by REVROT_1:34;
A142: right_cell(godo,1,Ga)\L~godo c= RightComp godo by A92,A97,JORDAN9:27;
A143: L~godo = L~(go^'pion1) \/ L~co by A96,TOPREAL8:35
.= L~go \/ L~pion1 \/ L~co by A94,TOPREAL8:35;
A144: L~Cage(C,n) = L~US \/ L~LS by JORDAN1E:13;
then
A145: L~US c= L~Cage(C,n) by XBOOLE_1:7;
A146: L~LS c= L~Cage(C,n) by A144,XBOOLE_1:7;
A147: L~go c= L~Cage(C,n) by A48,A145;
A148: L~co c= L~Cage(C,n) by A55,A146;
A149: W-min C in C by SPRECT_1:13;
A150: L~pion = LSeg(Gik,Gij) by SPPOL_2:21;
A151: now
assume W-min C in L~godo;
then
A152: W-min C in L~go \/ L~pion1 or W-min C in L~co by A143,XBOOLE_0:def 3;
per cases by A152,XBOOLE_0:def 3;
suppose
W-min C in L~go;
then C meets L~Cage(C,n) by A147,A149,XBOOLE_0:3;
hence contradiction by JORDAN10:5;
end;
suppose
W-min C in L~pion1;
hence contradiction by A8,A83,A133,A150,XBOOLE_0:3;
end;
suppose
W-min C in L~co;
then C meets L~Cage(C,n) by A148,A149,XBOOLE_0:3;
hence contradiction by JORDAN10:5;
end;
end;
right_cell(Rotate(Cage(C,n),Wmin),1) = right_cell(ff,1,GoB ff) by A88,
JORDAN1H:23
.= right_cell(ff,1,GoB Cage(C,n)) by REVROT_1:28
.= right_cell(ff,1,Ga) by JORDAN1H:44
.= right_cell(ff-:Emax,1,Ga) by A137,A141,JORDAN1J:53
.= right_cell(US,1,Ga) by JORDAN1E:def 1
.= right_cell(R_Cut(US,Gik),1,Ga) by A36,A93,A138,JORDAN1J:52
.= right_cell(go^'pion1,1,Ga) by A41,A95,JORDAN1J:51
.= right_cell(godo,1,Ga) by A90,A97,JORDAN1J:51;
then W-min C in right_cell(godo,1,Ga) by JORDAN1I:6;
then
A153: W-min C in right_cell(godo,1,Ga)\L~godo by A151,XBOOLE_0:def 5;
A154: godo/.1 = (go^'pion1)/.1 by FINSEQ_6:155
.= Wmin by A61,FINSEQ_6:155;
A155: len US >= 2 by A18,XXREAL_0:2;
A156: godo/.2 = (go^'pion1)/.2 by A89,FINSEQ_6:159
.= US/.2 by A35,A77,FINSEQ_6:159
.= (US^'LS)/.2 by A155,FINSEQ_6:159
.= Rotate(Cage(C,n),Wmin)/.2 by JORDAN1E:11;
A157: L~go \/ L~co is compact by COMPTS_1:10;
Wmin in L~go \/ L~co by A63,A78,XBOOLE_0:def 3;
then
A158: W-min (L~go \/ L~co) = Wmin by A147,A148,A157,JORDAN1J:21,XBOOLE_1:8;
A159: (W-min (L~go \/ L~co))`1 = W-bound (L~go \/ L~co) by EUCLID:52;
A160: Wmin`1 = Wbo by EUCLID:52;
A161: Gij`1 <= Gik`1 by A1,A2,A3,A4,A5,SPRECT_3:13;
then W-bound LSeg(Gik,Gij) = Gij`1 by SPRECT_1:54;
then
A162: W-bound L~pion1 = Gij`1 by A83,SPPOL_2:21;
Gij`1 >= Wbo by A9,A146,PSCOMP_1:24;
then Gij`1 > Wbo by A76,XXREAL_0:1;
then W-min (L~go\/L~co\/L~pion1) = W-min (L~go \/ L~co) by A157,A158,A159
,A160,A162,JORDAN1J:33;
then
A163: W-min L~godo = Wmin by A143,A158,XBOOLE_1:4;
A164: rng godo c= L~godo by A89,A91,SPPOL_2:18,XXREAL_0:2;
2 in dom godo by A92,FINSEQ_3:25;
then
A165: godo/.2 in rng godo by PARTFUN2:2;
godo/.2 in W-most L~Cage(C,n) by A156,JORDAN1I:25;
then (godo/.2)`1 = (W-min L~godo)`1 by A163,PSCOMP_1:31
.= W-bound L~godo by EUCLID:52;
then godo/.2 in W-most L~godo by A164,A165,SPRECT_2:12;
then Rotate(godo,W-min L~godo)/.2 in W-most L~godo by A154,A163,FINSEQ_6:89;
then reconsider godo as clockwise_oriented non constant standard
special_circular_sequence by JORDAN1I:25;
len US in dom US by FINSEQ_5:6;
then
A166: US.len US = US/.len US by PARTFUN1:def 6
.= Emax by JORDAN1F:7;
A167: east_halfline E-max C misses L~go
proof
assume east_halfline E-max C meets L~go;
then consider p be object such that
A168: p in east_halfline E-max C and
A169: p in L~go by XBOOLE_0:3;
reconsider p as Point of TOP-REAL 2 by A168;
p in L~US by A48,A169;
then p in east_halfline E-max C /\ L~Cage(C,n) by A145,A168,XBOOLE_0:def 4;
then
A170: p`1 = Ebo by JORDAN1A:83,PSCOMP_1:50;
then
A171: p = Emax by A48,A169,JORDAN1J:46;
then Emax = Gik by A10,A166,A169,JORDAN1J:43;
then Gik`1 = Ga*(len Ga,k)`1 by A3,A14,A170,A171,JORDAN1A:71;
hence contradiction by A3,A17,A32,JORDAN1G:7;
end;
now
assume east_halfline E-max C meets L~godo;
then
A172: east_halfline E-max C meets (L~go \/ L~pion1) or east_halfline E-max
C meets L~co by A143,XBOOLE_1:70;
per cases by A172,XBOOLE_1:70;
suppose
east_halfline E-max C meets L~go;
hence contradiction by A167;
end;
suppose
east_halfline E-max C meets L~pion1;
then consider p be object such that
A173: p in east_halfline E-max C and
A174: p in L~pion1 by XBOOLE_0:3;
reconsider p as Point of TOP-REAL 2 by A173;
A175: p`2 = (E-max C)`2 by A173,TOPREAL1:def 11;
k+1 <= len Ga by A3,NAT_1:13;
then k+1-1 <= len Ga-1 by XREAL_1:9;
then
A176: k <= len Ga-'1 by XREAL_0:def 2;
len Ga-'1 <= len Ga by NAT_D:35;
then
A177: Gik`1 <= Ga*(len Ga-'1,1)`1 by A4,A5,A11,A14,A21,A176,JORDAN1A:18;
p`1 <= Gik`1 by A83,A150,A161,A174,TOPREAL1:3;
then p`1 <= Ga*(len Ga-'1,1)`1 by A177,XXREAL_0:2;
then p`1 <= E-bound C by A21,JORDAN8:12;
then
A178: p`1 <= (E-max C)`1 by EUCLID:52;
p`1 >= (E-max C)`1 by A173,TOPREAL1:def 11;
then p`1 = (E-max C)`1 by A178,XXREAL_0:1;
then p = E-max C by A175,TOPREAL3:6;
hence contradiction by A8,A83,A134,A150,A174,XBOOLE_0:3;
end;
suppose
east_halfline E-max C meets L~co;
then consider p be object such that
A179: p in east_halfline E-max C and
A180: p in L~co by XBOOLE_0:3;
reconsider p as Point of TOP-REAL 2 by A179;
A181: p in LSeg(co,Index(p,co)) by A180,JORDAN3:9;
consider t be Nat such that
A182: t in dom LS and
A183: LS.t = Gij by A39,FINSEQ_2:10;
1 <= t by A182,FINSEQ_3:25;
then
A184: 1 < t by A34,A183,XXREAL_0:1;
t <= len LS by A182,FINSEQ_3:25;
then Index(Gij,LS)+1 = t by A183,A184,JORDAN3:12;
then
A185: len L_Cut(LS,Gij) = len LS-Index(Gij,LS) by A9,A183,JORDAN3:26;
Index(p,co) < len co by A180,JORDAN3:8;
then Index(p,co) < len LS-'Index(Gij,LS) by A185,XREAL_0:def 2;
then Index(p,co)+1 <= len LS-'Index(Gij,LS) by NAT_1:13;
then
A186: Index(p,co) <= len LS-'Index(Gij,LS)-1 by XREAL_1:19;
A187: co = mid(LS,Gij..LS,len LS) by A39,JORDAN1J:37;
p in L~LS by A55,A180;
then p in east_halfline E-max C /\ L~Cage(C,n) by A146,A179,
XBOOLE_0:def 4;
then
A188: p`1 = Ebo by JORDAN1A:83,PSCOMP_1:50;
A189: Index(Gij,LS)+1 = Gij..LS by A34,A39,JORDAN1J:56;
0+Index(Gij,LS) < len LS by A9,JORDAN3:8;
then len LS-Index(Gij,LS) > 0 by XREAL_1:20;
then Index(p,co) <= len LS-Index(Gij,LS)-1 by A186,XREAL_0:def 2;
then Index(p,co) <= len LS-Gij..LS by A189;
then Index(p,co) <= len LS-'Gij..LS by XREAL_0:def 2;
then
A190: Index(p,co) < len LS-'(Gij..LS)+1 by NAT_1:13;
A191: 1<=Index(p,co) by A180,JORDAN3:8;
A192: Gij..LS<=len LS by A39,FINSEQ_4:21;
Gij..LS <> len LS by A31,A39,FINSEQ_4:19;
then
A193: Gij..LS < len LS by A192,XXREAL_0:1;
A194: 1+1 <= len LS by A25,XXREAL_0:2;
then
A195: 2 in dom LS by FINSEQ_3:25;
set tt = Index(p,co)+(Gij..LS)-'1;
set RC = Rotate(Cage(C,n),Emax);
A196: E-max C in right_cell(RC,1) by JORDAN1I:7;
A197: GoB RC = GoB Cage(C,n) by REVROT_1:28
.= Ga by JORDAN1H:44;
A198: L~RC = L~Cage(C,n) by REVROT_1:33;
consider jj2 be Nat such that
A199: 1 <= jj2 and
A200: jj2 <= width Ga and
A201: Emax = Ga*(len Ga,jj2) by JORDAN1D:25;
A202: len Ga >= 4 by JORDAN8:10;
then len Ga >= 1 by XXREAL_0:2;
then
A203: [len Ga,jj2] in Indices Ga by A199,A200,MATRIX_0:30;
A204: len RC = len Cage(C,n) by FINSEQ_6:179;
LS = RC-:Wmin by JORDAN1G:18;
then
A205: LSeg(LS,1) = LSeg(RC,1) by A194,SPPOL_2:9;
A206: Emax in rng Cage(C,n) by SPRECT_2:46;
RC is_sequence_on Ga by A140,REVROT_1:34;
then consider ii,jj be Nat such that
A207: [ii,jj+1] in Indices Ga and
A208: [ii,jj] in Indices Ga and
A209: RC/.1 = Ga*(ii,jj+1) and
A210: RC/.(1+1) = Ga*(ii,jj) by A87,A198,A204,A206,FINSEQ_6:92,JORDAN1I:23;
A211: jj+1+1 <> jj;
A212: 1 <= jj by A208,MATRIX_0:32;
RC/.1 = E-max L~RC by A198,A206,FINSEQ_6:92;
then
A213: ii = len Ga by A198,A207,A209,A201,A203,GOBOARD1:5;
then ii-1 >= 4-1 by A202,XREAL_1:9;
then
A214: ii-1 >= 1 by XXREAL_0:2;
then
A215: 1 <= ii-'1 by XREAL_0:def 2;
A216: jj <= width Ga by A208,MATRIX_0:32;
then
A217: Ga*(len Ga,jj)`1 = Ebo by A11,A212,JORDAN1A:71;
A218: jj+1 <= width Ga by A207,MATRIX_0:32;
ii+1 <> ii;
then
A219: right_cell(RC,1) = cell(Ga,ii-'1,jj) by A87,A204,A197,A207,A208,A209,A210
,A211,GOBOARD5:def 6;
A220: ii <= len Ga by A208,MATRIX_0:32;
A221: 1 <= ii by A208,MATRIX_0:32;
A222: ii <= len Ga by A207,MATRIX_0:32;
A223: 1 <= jj+1 by A207,MATRIX_0:32;
then
A224: Ebo = Ga*(len Ga,jj+1)`1 by A11,A218,JORDAN1A:71;
A225: 1 <= ii by A207,MATRIX_0:32;
then
A226: ii-'1+1 = ii by XREAL_1:235;
then
A227: ii-'1 < len Ga by A222,NAT_1:13;
then
A228: Ga*(ii-'1,jj+1)`2 = Ga*(1,jj+1)`2 by A223,A218,A215,GOBOARD5:1
.= Ga*(ii,jj+1)`2 by A225,A222,A223,A218,GOBOARD5:1;
A229: (E-max C)`2 = p`2 by A179,TOPREAL1:def 11;
then
A230: p`2 <= Ga*(ii-'1,jj+1)`2 by A196,A222,A218,A212,A219,A226,A214,JORDAN9:17
;
A231: Ga*(ii-'1,jj)`2 = Ga*(1,jj)`2 by A212,A216,A215,A227,GOBOARD5:1
.= Ga*(ii,jj)`2 by A221,A220,A212,A216,GOBOARD5:1;
Ga*(ii-'1,jj)`2 <= p`2 by A229,A196,A222,A218,A212,A219,A226,A214,
JORDAN9:17;
then p in LSeg(RC/.1,RC/.(1+1)) by A188,A209,A210,A213,A230,A231,A228
,A217,A224,GOBOARD7:7;
then
A232: p in LSeg(LS,1) by A87,A205,A204,TOPREAL1:def 3;
1<=Gij..LS by A39,FINSEQ_4:21;
then
A233: LSeg(mid(LS,Gij..LS,len LS),Index(p,co)) = LSeg(LS,Index(p,co)+(
Gij..LS)-'1) by A193,A191,A190,JORDAN4:19;
1<=Index(Gij,LS) by A9,JORDAN3:8;
then
A234: 1+1 <= Gij..LS by A189,XREAL_1:7;
then Index(p,co)+Gij..LS >= 1+1+1 by A191,XREAL_1:7;
then Index(p,co)+Gij..LS-1 >= 1+1+1-1 by XREAL_1:9;
then
A235: tt >= 1+1 by XREAL_0:def 2;
now
per cases by A235,XXREAL_0:1;
suppose
tt > 1+1;
then LSeg(LS,1) misses LSeg(LS,tt) by TOPREAL1:def 7;
hence contradiction by A232,A181,A187,A233,XBOOLE_0:3;
end;
suppose
A236: tt = 1+1;
then 1+1 = Index(p,co)+(Gij..LS)-1 by XREAL_0:def 2;
then 1+1+1 = Index(p,co)+(Gij..LS);
then
A237: Gij..LS = 2 by A191,A234,JORDAN1E:6;
LSeg(LS,1) /\ LSeg(LS,tt) = {LS/.2} by A25,A236,TOPREAL1:def 6;
then p in {LS/.2} by A232,A181,A187,A233,XBOOLE_0:def 4;
then
A238: p = LS/.2 by TARSKI:def 1;
then
A239: p in rng LS by A195,PARTFUN2:2;
p..LS = 2 by A195,A238,FINSEQ_5:41;
then p = Gij by A39,A237,A239,FINSEQ_5:9;
then Gij`1 = Ebo by A238,JORDAN1G:32;
then Gij`1 = Ga*(len Ga,j)`1 by A1,A11,A13,JORDAN1A:71;
hence contradiction by A2,A3,A16,A69,JORDAN1G:7;
end;
end;
hence contradiction;
end;
end;
then east_halfline E-max C c= (L~godo)` by SUBSET_1:23;
then consider W be Subset of TOP-REAL 2 such that
A240: W is_a_component_of (L~godo)` and
A241: east_halfline E-max C c= W by GOBOARD9:3;
W is not bounded by A241,JORDAN2C:121,RLTOPSP1:42;
then W is_outside_component_of L~godo by A240,JORDAN2C:def 3;
then W c= UBD L~godo by JORDAN2C:23;
then
A242: east_halfline E-max C c= UBD L~godo by A241;
E-max C in east_halfline E-max C by TOPREAL1:38;
then E-max C in UBD L~godo by A242;
then E-max C in LeftComp godo by GOBRD14:36;
then LA meets L~godo by A132,A133,A134,A142,A153,JORDAN1J:36;
then
A243: LA meets (L~go \/ L~pion1) or LA meets L~co by A143,XBOOLE_1:70;
A244: LA c= C by JORDAN6:61;
per cases by A243,XBOOLE_1:70;
suppose
LA meets L~go;
then LA meets L~Cage(C,n) by A48,A145,XBOOLE_1:1,63;
hence contradiction by A244,JORDAN10:5,XBOOLE_1:63;
end;
suppose
LA meets L~pion1;
hence contradiction by A8,A83,A150;
end;
suppose
LA meets L~co;
then LA meets L~Cage(C,n) by A55,A146,XBOOLE_1:1,63;
hence contradiction by A244,JORDAN10:5,XBOOLE_1:63;
end;
end;
theorem Th29:
for C be Simple_closed_curve for i,j,k be Nat st 1 <
j & j <= k & k < len Gauge(C,n) & 1 <= i & i <= width Gauge(C,n) & LSeg(Gauge(C
,n)*(j,i),Gauge(C,n)*(k,i)) /\ L~Upper_Seq(C,n) = {Gauge(C,n)*(k,i)} & LSeg(
Gauge(C,n)*(j,i),Gauge(C,n)*(k,i)) /\ L~Lower_Seq(C,n) = {Gauge(C,n)*(j,i)}
holds LSeg(Gauge(C,n)*(j,i),Gauge(C,n)*(k,i)) meets Upper_Arc C
proof
let C be Simple_closed_curve;
let i,j,k be Nat;
set Ga = Gauge(C,n);
set US = Upper_Seq(C,n);
set LS = Lower_Seq(C,n);
set UA = Upper_Arc C;
set Wmin = W-min L~Cage(C,n);
set Emax = E-max L~Cage(C,n);
set Wbo = W-bound L~Cage(C,n);
set Ebo = E-bound L~Cage(C,n);
set Gij = Ga*(j,i);
set Gik = Ga*(k,i);
assume that
A1: 1 < j and
A2: j <= k and
A3: k < len Ga and
A4: 1 <= i and
A5: i <= width Ga and
A6: LSeg(Gij,Gik) /\ L~US = {Gik} and
A7: LSeg(Gij,Gik) /\ L~LS = {Gij} and
A8: LSeg(Gij,Gik) misses UA;
Gij in {Gij} by TARSKI:def 1;
then
A9: Gij in L~LS by A7,XBOOLE_0:def 4;
Gik in {Gik} by TARSKI:def 1;
then
A10: Gik in L~US by A6,XBOOLE_0:def 4;
A11: len Ga = width Ga by JORDAN8:def 1;
A12: j <> k by A1,A3,A4,A5,A9,A10,Th27;
A13: j <= width Ga by A2,A3,A11,XXREAL_0:2;
A14: 1 <= k by A1,A2,XXREAL_0:2;
A15: k <= width Ga by A3,JORDAN8:def 1;
A16: [j,i] in Indices Ga by A1,A4,A5,A11,A13,MATRIX_0:30;
A17: [k,i] in Indices Ga by A3,A4,A5,A14,MATRIX_0:30;
set go = R_Cut(US,Gik);
set co = L_Cut(LS,Gij);
A18: len US >= 3 by JORDAN1E:15;
then len US >= 1 by XXREAL_0:2;
then 1 in dom US by FINSEQ_3:25;
then
A19: US.1 = US/.1 by PARTFUN1:def 6
.= Wmin by JORDAN1F:5;
A20: Wmin`1 = Wbo by EUCLID:52
.= Ga*(1,k)`1 by A3,A14,JORDAN1A:73;
len Ga >= 4 by JORDAN8:10;
then
A21: len Ga >= 1 by XXREAL_0:2;
then
A22: [1,k] in Indices Ga by A14,A15,MATRIX_0:30;
then
A23: Gik <> US.1 by A1,A2,A17,A19,A20,JORDAN1G:7;
then reconsider go as being_S-Seq FinSequence of TOP-REAL 2 by A10,JORDAN3:35
;
A24: [1,j] in Indices Ga by A1,A13,A21,MATRIX_0:30;
A25: len LS >= 1+2 by JORDAN1E:15;
then
A26: len LS >= 1 by XXREAL_0:2;
then
A27: 1 in dom LS by FINSEQ_3:25;
len LS in dom LS by A26,FINSEQ_3:25;
then
A28: LS.len LS = LS/.len LS by PARTFUN1:def 6
.= Wmin by JORDAN1F:8;
A29: Wmin`1 = Wbo by EUCLID:52
.= Ga*(1,k)`1 by A3,A14,JORDAN1A:73;
A30: [j,i] in Indices Ga by A1,A4,A5,A11,A13,MATRIX_0:30;
then
A31: Gij <> LS.len LS by A1,A22,A28,A29,JORDAN1G:7;
then reconsider co as being_S-Seq FinSequence of TOP-REAL 2 by A9,JORDAN3:34;
A32: [len Ga,k] in Indices Ga by A14,A15,A21,MATRIX_0:30;
A33: LS.1 = LS/.1 by A27,PARTFUN1:def 6
.= Emax by JORDAN1F:6;
Emax`1 = Ebo by EUCLID:52
.= Ga*(len Ga,k)`1 by A3,A14,JORDAN1A:71;
then
A34: Gij <> LS.1 by A2,A3,A30,A32,A33,JORDAN1G:7;
A35: len go >= 1+1 by TOPREAL1:def 8;
A36: Gik in rng US by A4,A5,A10,A11,A14,A15,JORDAN1G:4,JORDAN1J:40;
then
A37: go is_sequence_on Ga by JORDAN1G:4,JORDAN1J:38;
A38: len co >= 1+1 by TOPREAL1:def 8;
A39: Gij in rng LS by A1,A4,A5,A9,A11,A13,JORDAN1G:5,JORDAN1J:40;
then
A40: co is_sequence_on Ga by JORDAN1G:5,JORDAN1J:39;
reconsider go as non constant s.c.c. being_S-Seq FinSequence of TOP-REAL 2
by A35,A37,JGRAPH_1:12,JORDAN8:5;
reconsider co as non constant s.c.c. being_S-Seq FinSequence of TOP-REAL 2
by A38,A40,JGRAPH_1:12,JORDAN8:5;
A41: len go > 1 by A35,NAT_1:13;
then
A42: len go in dom go by FINSEQ_3:25;
then
A43: go/.len go = go.len go by PARTFUN1:def 6
.= Gik by A10,JORDAN3:24;
len co >= 1 by A38,XXREAL_0:2;
then 1 in dom co by FINSEQ_3:25;
then
A44: co/.1 = co.1 by PARTFUN1:def 6
.= Gij by A9,JORDAN3:23;
reconsider m = len go - 1 as Nat by A42,FINSEQ_3:26;
A45: m+1 = len go;
then
A46: len go-'1 = m by NAT_D:34;
A47: LSeg(go,m) c= L~go by TOPREAL3:19;
A48: L~go c= L~US by A10,JORDAN3:41;
then LSeg(go,m) c= L~US by A47;
then
A49: LSeg(go,m) /\ LSeg(Gik,Gij) c= {Gik} by A6,XBOOLE_1:26;
m >= 1 by A35,XREAL_1:19;
then
A50: LSeg(go,m) = LSeg(go/.m,Gik) by A43,A45,TOPREAL1:def 3;
{Gik} c= LSeg(go,m) /\ LSeg(Gik,Gij)
proof
let x be object;
A51: Gik in LSeg(Gik,Gij) by RLTOPSP1:68;
assume x in {Gik};
then
A52: x = Gik by TARSKI:def 1;
Gik in LSeg(go,m) by A50,RLTOPSP1:68;
hence thesis by A52,A51,XBOOLE_0:def 4;
end;
then
A53: LSeg(go,m) /\ LSeg(Gik,Gij) = {Gik} by A49;
A54: LSeg(co,1) c= L~co by TOPREAL3:19;
A55: L~co c= L~LS by A9,JORDAN3:42;
then LSeg(co,1) c= L~LS by A54;
then
A56: LSeg(co,1) /\ LSeg(Gik,Gij) c= {Gij} by A7,XBOOLE_1:26;
A57: LSeg(co,1) = LSeg(Gij,co/.(1+1)) by A38,A44,TOPREAL1:def 3;
{Gij} c= LSeg(co,1) /\ LSeg(Gik,Gij)
proof
let x be object;
A58: Gij in LSeg(Gik,Gij) by RLTOPSP1:68;
assume x in {Gij};
then
A59: x = Gij by TARSKI:def 1;
Gij in LSeg(co,1) by A57,RLTOPSP1:68;
hence thesis by A59,A58,XBOOLE_0:def 4;
end;
then
A60: LSeg(Gik,Gij) /\ LSeg(co,1) = {Gij} by A56;
A61: go/.1 = US/.1 by A10,SPRECT_3:22
.= Wmin by JORDAN1F:5;
then
A62: go/.1 = LS/.len LS by JORDAN1F:8
.= co/.len co by A9,JORDAN1J:35;
A63: rng go c= L~go by A35,SPPOL_2:18;
A64: rng co c= L~co by A38,SPPOL_2:18;
A65: {go/.1} c= L~go /\ L~co
proof
let x be object;
assume x in {go/.1};
then
A66: x = go/.1 by TARSKI:def 1;
then
A67: x in rng go by FINSEQ_6:42;
x in rng co by A62,A66,FINSEQ_6:168;
hence thesis by A63,A64,A67,XBOOLE_0:def 4;
end;
A68: LS.1 = LS/.1 by A27,PARTFUN1:def 6
.= Emax by JORDAN1F:6;
A69: [len Ga,j] in Indices Ga by A1,A13,A21,MATRIX_0:30;
L~go /\ L~co c= {go/.1}
proof
let x be object;
assume
A70: x in L~go /\ L~co;
then
A71: x in L~co by XBOOLE_0:def 4;
A72: now
assume x = Emax;
then
A73: Emax = Gij by A9,A68,A71,JORDAN1E:7;
Ga*(len Ga,j)`1 = Ebo by A1,A11,A13,JORDAN1A:71;
then Emax`1 <> Ebo by A2,A3,A16,A69,A73,JORDAN1G:7;
hence contradiction by EUCLID:52;
end;
x in L~go by A70,XBOOLE_0:def 4;
then x in L~US /\ L~LS by A48,A55,A71,XBOOLE_0:def 4;
then x in {Wmin,Emax} by JORDAN1E:16;
then x = Wmin or x = Emax by TARSKI:def 2;
hence thesis by A61,A72,TARSKI:def 1;
end;
then
A74: L~go /\ L~co = {go/.1} by A65;
set W2 = go/.2;
A75: 2 in dom go by A35,FINSEQ_3:25;
A76: now
assume Gij`1 = Wbo;
then Ga*(1,j)`1 = Ga*(j,i)`1 by A1,A11,A13,JORDAN1A:73;
hence contradiction by A1,A16,A24,JORDAN1G:7;
end;
go = mid(US,1,Gik..US) by A36,JORDAN1G:49
.= US|(Gik..US) by A36,FINSEQ_4:21,FINSEQ_6:116;
then
A77: W2 = US/.2 by A75,FINSEQ_4:70;
A78: Wmin in rng go by A61,FINSEQ_6:42;
set pion = <*Gik,Gij*>;
A79: now
let n be Nat;
assume n in dom pion;
then n in {1,2} by FINSEQ_1:2,89;
then n = 1 or n = 2 by TARSKI:def 2;
hence ex j,i be Nat st [j,i] in Indices Ga & pion/.n = Ga*(j,i)
by A16,A17,FINSEQ_4:17;
end;
A80: Gik <> Gij by A12,A16,A17,GOBOARD1:5;
Gik`2 = Ga*(1,i)`2 by A3,A4,A5,A14,GOBOARD5:1
.= Gij`2 by A1,A4,A5,A11,A13,GOBOARD5:1;
then LSeg(Gik,Gij) is horizontal by SPPOL_1:15;
then pion is being_S-Seq by A80,JORDAN1B:8;
then consider pion1 be FinSequence of TOP-REAL 2 such that
A81: pion1 is_sequence_on Ga and
A82: pion1 is being_S-Seq and
A83: L~pion = L~pion1 and
A84: pion/.1 = pion1/.1 and
A85: pion/.len pion = pion1/.len pion1 and
A86: len pion <= len pion1 by A79,GOBOARD3:2;
reconsider pion1 as being_S-Seq FinSequence of TOP-REAL 2 by A82;
set godo = go^'pion1^'co;
A87: 1+1 <= len Cage(C,n) by GOBOARD7:34,XXREAL_0:2;
A88: 1+1 <= len Rotate(Cage(C,n),Wmin) by GOBOARD7:34,XXREAL_0:2;
len (go^'pion1) >= len go by TOPREAL8:7;
then
A89: len (go^'pion1) >= 1+1 by A35,XXREAL_0:2;
then
A90: len (go^'pion1) > 1+0 by NAT_1:13;
A91: len godo >= len (go^'pion1) by TOPREAL8:7;
then
A92: 1+1 <= len godo by A89,XXREAL_0:2;
A93: US is_sequence_on Ga by JORDAN1G:4;
A94: go/.len go = pion1/.1 by A43,A84,FINSEQ_4:17;
then
A95: go^'pion1 is_sequence_on Ga by A37,A81,TOPREAL8:12;
A96: (go^'pion1)/.len (go^'pion1) = pion/.len pion by A85,FINSEQ_6:156
.= pion/.2 by FINSEQ_1:44
.= co/.1 by A44,FINSEQ_4:17;
then
A97: godo is_sequence_on Ga by A40,A95,TOPREAL8:12;
LSeg(pion1,1) c= L~<*Gik,Gij*> by A83,TOPREAL3:19;
then LSeg(pion1,1) c= LSeg(Gik,Gij) by SPPOL_2:21;
then
A98: LSeg(go,len go-'1) /\ LSeg(pion1,1) c= {Gik} by A46,A53,XBOOLE_1:27;
A99: len pion1 >= 1+1 by A86,FINSEQ_1:44;
{Gik} c= LSeg(go,m) /\ LSeg(pion1,1)
proof
let x be object;
assume x in {Gik};
then
A100: x = Gik by TARSKI:def 1;
A101: Gik in LSeg(go,m) by A50,RLTOPSP1:68;
Gik in LSeg(pion1,1) by A43,A94,A99,TOPREAL1:21;
hence thesis by A100,A101,XBOOLE_0:def 4;
end;
then LSeg(go,len go-'1) /\ LSeg(pion1,1) = { go/.len go } by A43,A46,A98;
then
A102: go^'pion1 is unfolded by A94,TOPREAL8:34;
len pion1 >= 2+0 by A86,FINSEQ_1:44;
then
A103: len pion1-2 >= 0 by XREAL_1:19;
len (go^'pion1)+1-1 = len go+len pion1-1 by FINSEQ_6:139;
then len (go^'pion1)-1 = len go + (len pion1-2)
.= len go + (len pion1-'2) by A103,XREAL_0:def 2;
then
A104: len (go^'pion1)-'1 = len go + (len pion1-'2) by XREAL_0:def 2;
A105: len pion1-1 >= 1 by A99,XREAL_1:19;
then
A106: len pion1-'1 = len pion1-1 by XREAL_0:def 2;
A107: len pion1-'2+1 = len pion1-2+1 by A103,XREAL_0:def 2
.= len pion1-'1 by A105,XREAL_0:def 2;
len pion1-1+1 <= len pion1;
then
A108: len pion1-'1 < len pion1 by A106,NAT_1:13;
LSeg(pion1,len pion1-'1) c= L~<*Gik,Gij*> by A83,TOPREAL3:19;
then LSeg(pion1,len pion1-'1) c= LSeg(Gik,Gij) by SPPOL_2:21;
then
A109: LSeg(pion1,len pion1-'1) /\ LSeg(co,1) c= {Gij} by A60,XBOOLE_1:27;
{Gij} c= LSeg(pion1,len pion1-'1) /\ LSeg(co,1)
proof
let x be object;
assume x in {Gij};
then
A110: x = Gij by TARSKI:def 1;
pion1/.(len pion1-'1+1) = pion/.2 by A85,A106,FINSEQ_1:44
.= Gij by FINSEQ_4:17;
then
A111: Gij in LSeg(pion1,len pion1-'1) by A105,A106,TOPREAL1:21;
Gij in LSeg(co,1) by A57,RLTOPSP1:68;
hence thesis by A110,A111,XBOOLE_0:def 4;
end;
then LSeg(pion1,len pion1-'1) /\ LSeg(co,1) = {Gij} by A109;
then
A112: LSeg(go^'pion1,len go+(len pion1-'2)) /\ LSeg(co,1) = {(go^'pion1)/.
len (go^'pion1)} by A44,A94,A96,A107,A108,TOPREAL8:31;
A113: (go^'pion1) is non trivial by A89,NAT_D:60;
A114: rng pion1 c= L~pion1 by A99,SPPOL_2:18;
A115: {pion1/.1} c= L~go /\ L~pion1
proof
let x be object;
assume x in {pion1/.1};
then
A116: x = pion1/.1 by TARSKI:def 1;
then
A117: x in rng pion1 by FINSEQ_6:42;
x in rng go by A94,A116,FINSEQ_6:168;
hence thesis by A63,A114,A117,XBOOLE_0:def 4;
end;
L~go /\ L~pion1 c= {pion1/.1}
proof
let x be object;
assume
A118: x in L~go /\ L~pion1;
then
A119: x in L~pion1 by XBOOLE_0:def 4;
x in L~go by A118,XBOOLE_0:def 4;
then x in L~pion1 /\ L~US by A48,A119,XBOOLE_0:def 4;
hence thesis by A6,A43,A83,A94,SPPOL_2:21;
end;
then
A120: L~go /\ L~pion1 = {pion1/.1} by A115;
then
A121: (go^'pion1) is s.n.c. by A94,JORDAN1J:54;
rng go /\ rng pion1 c= {pion1/.1} by A63,A114,A120,XBOOLE_1:27;
then
A122: go^'pion1 is one-to-one by JORDAN1J:55;
A123: pion/.len pion = pion/.2 by FINSEQ_1:44
.= co/.1 by A44,FINSEQ_4:17;
A124: {pion1/.len pion1} c= L~co /\ L~pion1
proof
let x be object;
assume x in {pion1/.len pion1};
then
A125: x = pion1/.len pion1 by TARSKI:def 1;
then
A126: x in rng pion1 by FINSEQ_6:168;
x in rng co by A85,A123,A125,FINSEQ_6:42;
hence thesis by A64,A114,A126,XBOOLE_0:def 4;
end;
L~co /\ L~pion1 c= {pion1/.len pion1}
proof
let x be object;
assume
A127: x in L~co /\ L~pion1;
then
A128: x in L~pion1 by XBOOLE_0:def 4;
x in L~co by A127,XBOOLE_0:def 4;
then x in L~pion1 /\ L~LS by A55,A128,XBOOLE_0:def 4;
hence thesis by A7,A44,A83,A85,A123,SPPOL_2:21;
end;
then
A129: L~co /\ L~pion1 = {pion1/.len pion1} by A124;
A130: L~(go^'pion1) /\ L~co = (L~go \/ L~pion1) /\ L~co by A94,TOPREAL8:35
.= {go/.1} \/ {co/.1} by A74,A85,A123,A129,XBOOLE_1:23
.= {(go^'pion1)/.1} \/ {co/.1} by FINSEQ_6:155
.= {(go^'pion1)/.1,co/.1} by ENUMSET1:1;
co/.len co = (go^'pion1)/.1 by A62,FINSEQ_6:155;
then reconsider
godo as non constant standard special_circular_sequence by A92,A96,A97,A102
,A104,A112,A113,A121,A122,A130,JORDAN8:4,5,TOPREAL8:11,33,34;
A131: UA is_an_arc_of W-min C,E-max C by JORDAN6:def 8;
then
A132: UA is connected by JORDAN6:10;
A133: W-min C in UA by A131,TOPREAL1:1;
A134: E-max C in UA by A131,TOPREAL1:1;
set ff = Rotate(Cage(C,n),Wmin);
Wmin in rng Cage(C,n) by SPRECT_2:43;
then
A135: ff/.1 = Wmin by FINSEQ_6:92;
A136: L~ff = L~Cage(C,n) by REVROT_1:33;
then (W-max L~ff)..ff > 1 by A135,SPRECT_5:22;
then (N-min L~ff)..ff > 1 by A135,A136,SPRECT_5:23,XXREAL_0:2;
then (N-max L~ff)..ff > 1 by A135,A136,SPRECT_5:24,XXREAL_0:2;
then
A137: Emax..ff > 1 by A135,A136,SPRECT_5:25,XXREAL_0:2;
A138: now
assume
A139: Gik..US <= 1;
Gik..US >= 1 by A36,FINSEQ_4:21;
then Gik..US = 1 by A139,XXREAL_0:1;
then Gik = US/.1 by A36,FINSEQ_5:38;
hence contradiction by A19,A23,JORDAN1F:5;
end;
A140: Cage(C,n) is_sequence_on Ga by JORDAN9:def 1;
then
A141: ff is_sequence_on Ga by REVROT_1:34;
A142: right_cell(godo,1,Ga)\L~godo c= RightComp godo by A92,A97,JORDAN9:27;
A143: L~godo = L~(go^'pion1) \/ L~co by A96,TOPREAL8:35
.= L~go \/ L~pion1 \/ L~co by A94,TOPREAL8:35;
A144: L~Cage(C,n) = L~US \/ L~LS by JORDAN1E:13;
then
A145: L~US c= L~Cage(C,n) by XBOOLE_1:7;
A146: L~LS c= L~Cage(C,n) by A144,XBOOLE_1:7;
A147: L~go c= L~Cage(C,n) by A48,A145;
A148: L~co c= L~Cage(C,n) by A55,A146;
A149: W-min C in C by SPRECT_1:13;
A150: L~pion = LSeg(Gik,Gij) by SPPOL_2:21;
A151: now
assume W-min C in L~godo;
then
A152: W-min C in L~go \/ L~pion1 or W-min C in L~co by A143,XBOOLE_0:def 3;
per cases by A152,XBOOLE_0:def 3;
suppose
W-min C in L~go;
then C meets L~Cage(C,n) by A147,A149,XBOOLE_0:3;
hence contradiction by JORDAN10:5;
end;
suppose
W-min C in L~pion1;
hence contradiction by A8,A83,A133,A150,XBOOLE_0:3;
end;
suppose
W-min C in L~co;
then C meets L~Cage(C,n) by A148,A149,XBOOLE_0:3;
hence contradiction by JORDAN10:5;
end;
end;
right_cell(Rotate(Cage(C,n),Wmin),1) = right_cell(ff,1,GoB ff) by A88,
JORDAN1H:23
.= right_cell(ff,1,GoB Cage(C,n)) by REVROT_1:28
.= right_cell(ff,1,Ga) by JORDAN1H:44
.= right_cell(ff-:Emax,1,Ga) by A137,A141,JORDAN1J:53
.= right_cell(US,1,Ga) by JORDAN1E:def 1
.= right_cell(R_Cut(US,Gik),1,Ga) by A36,A93,A138,JORDAN1J:52
.= right_cell(go^'pion1,1,Ga) by A41,A95,JORDAN1J:51
.= right_cell(godo,1,Ga) by A90,A97,JORDAN1J:51;
then W-min C in right_cell(godo,1,Ga) by JORDAN1I:6;
then
A153: W-min C in right_cell(godo,1,Ga)\L~godo by A151,XBOOLE_0:def 5;
A154: godo/.1 = (go^'pion1)/.1 by FINSEQ_6:155
.= Wmin by A61,FINSEQ_6:155;
A155: len US >= 2 by A18,XXREAL_0:2;
A156: godo/.2 = (go^'pion1)/.2 by A89,FINSEQ_6:159
.= US/.2 by A35,A77,FINSEQ_6:159
.= (US^'LS)/.2 by A155,FINSEQ_6:159
.= Rotate(Cage(C,n),Wmin)/.2 by JORDAN1E:11;
A157: L~go \/ L~co is compact by COMPTS_1:10;
Wmin in L~go \/ L~co by A63,A78,XBOOLE_0:def 3;
then
A158: W-min (L~go \/ L~co) = Wmin by A147,A148,A157,JORDAN1J:21,XBOOLE_1:8;
A159: (W-min (L~go \/ L~co))`1 = W-bound (L~go \/ L~co) by EUCLID:52;
A160: Wmin`1 = Wbo by EUCLID:52;
A161: Gij`1 <= Gik`1 by A1,A2,A3,A4,A5,SPRECT_3:13;
then W-bound LSeg(Gik,Gij) = Gij`1 by SPRECT_1:54;
then
A162: W-bound L~pion1 = Gij`1 by A83,SPPOL_2:21;
Gij`1 >= Wbo by A9,A146,PSCOMP_1:24;
then Gij`1 > Wbo by A76,XXREAL_0:1;
then W-min (L~go\/L~co\/L~pion1) = W-min (L~go \/ L~co) by A157,A158,A159
,A160,A162,JORDAN1J:33;
then
A163: W-min L~godo = Wmin by A143,A158,XBOOLE_1:4;
A164: rng godo c= L~godo by A89,A91,SPPOL_2:18,XXREAL_0:2;
2 in dom godo by A92,FINSEQ_3:25;
then
A165: godo/.2 in rng godo by PARTFUN2:2;
godo/.2 in W-most L~Cage(C,n) by A156,JORDAN1I:25;
then (godo/.2)`1 = (W-min L~godo)`1 by A163,PSCOMP_1:31
.= W-bound L~godo by EUCLID:52;
then godo/.2 in W-most L~godo by A164,A165,SPRECT_2:12;
then Rotate(godo,W-min L~godo)/.2 in W-most L~godo by A154,A163,FINSEQ_6:89;
then reconsider godo as clockwise_oriented non constant standard
special_circular_sequence by JORDAN1I:25;
len US in dom US by FINSEQ_5:6;
then
A166: US.len US = US/.len US by PARTFUN1:def 6
.= Emax by JORDAN1F:7;
A167: east_halfline E-max C misses L~go
proof
assume east_halfline E-max C meets L~go;
then consider p be object such that
A168: p in east_halfline E-max C and
A169: p in L~go by XBOOLE_0:3;
reconsider p as Point of TOP-REAL 2 by A168;
p in L~US by A48,A169;
then p in east_halfline E-max C /\ L~Cage(C,n) by A145,A168,XBOOLE_0:def 4;
then
A170: p`1 = Ebo by JORDAN1A:83,PSCOMP_1:50;
then
A171: p = Emax by A48,A169,JORDAN1J:46;
then Emax = Gik by A10,A166,A169,JORDAN1J:43;
then Gik`1 = Ga*(len Ga,k)`1 by A3,A14,A170,A171,JORDAN1A:71;
hence contradiction by A3,A17,A32,JORDAN1G:7;
end;
now
assume east_halfline E-max C meets L~godo;
then
A172: east_halfline E-max C meets (L~go \/ L~pion1) or east_halfline E-max
C meets L~co by A143,XBOOLE_1:70;
per cases by A172,XBOOLE_1:70;
suppose
east_halfline E-max C meets L~go;
hence contradiction by A167;
end;
suppose
east_halfline E-max C meets L~pion1;
then consider p be object such that
A173: p in east_halfline E-max C and
A174: p in L~pion1 by XBOOLE_0:3;
reconsider p as Point of TOP-REAL 2 by A173;
A175: p`2 = (E-max C)`2 by A173,TOPREAL1:def 11;
k+1 <= len Ga by A3,NAT_1:13;
then k+1-1 <= len Ga-1 by XREAL_1:9;
then
A176: k <= len Ga-'1 by XREAL_0:def 2;
len Ga-'1 <= len Ga by NAT_D:35;
then
A177: Gik`1 <= Ga*(len Ga-'1,1)`1 by A4,A5,A11,A14,A21,A176,JORDAN1A:18;
p`1 <= Gik`1 by A83,A150,A161,A174,TOPREAL1:3;
then p`1 <= Ga*(len Ga-'1,1)`1 by A177,XXREAL_0:2;
then p`1 <= E-bound C by A21,JORDAN8:12;
then
A178: p`1 <= (E-max C)`1 by EUCLID:52;
p`1 >= (E-max C)`1 by A173,TOPREAL1:def 11;
then p`1 = (E-max C)`1 by A178,XXREAL_0:1;
then p = E-max C by A175,TOPREAL3:6;
hence contradiction by A8,A83,A134,A150,A174,XBOOLE_0:3;
end;
suppose
east_halfline E-max C meets L~co;
then consider p be object such that
A179: p in east_halfline E-max C and
A180: p in L~co by XBOOLE_0:3;
reconsider p as Point of TOP-REAL 2 by A179;
A181: p in LSeg(co,Index(p,co)) by A180,JORDAN3:9;
consider t be Nat such that
A182: t in dom LS and
A183: LS.t = Gij by A39,FINSEQ_2:10;
1 <= t by A182,FINSEQ_3:25;
then
A184: 1 < t by A34,A183,XXREAL_0:1;
t <= len LS by A182,FINSEQ_3:25;
then Index(Gij,LS)+1 = t by A183,A184,JORDAN3:12;
then
A185: len L_Cut(LS,Gij) = len LS-Index(Gij,LS) by A9,A183,JORDAN3:26;
Index(p,co) < len co by A180,JORDAN3:8;
then Index(p,co) < len LS-'Index(Gij,LS) by A185,XREAL_0:def 2;
then Index(p,co)+1 <= len LS-'Index(Gij,LS) by NAT_1:13;
then
A186: Index(p,co) <= len LS-'Index(Gij,LS)-1 by XREAL_1:19;
A187: co = mid(LS,Gij..LS,len LS) by A39,JORDAN1J:37;
p in L~LS by A55,A180;
then p in east_halfline E-max C /\ L~Cage(C,n) by A146,A179,
XBOOLE_0:def 4;
then
A188: p`1 = Ebo by JORDAN1A:83,PSCOMP_1:50;
A189: Index(Gij,LS)+1 = Gij..LS by A34,A39,JORDAN1J:56;
0+Index(Gij,LS) < len LS by A9,JORDAN3:8;
then len LS-Index(Gij,LS) > 0 by XREAL_1:20;
then Index(p,co) <= len LS-Index(Gij,LS)-1 by A186,XREAL_0:def 2;
then Index(p,co) <= len LS-Gij..LS by A189;
then Index(p,co) <= len LS-'Gij..LS by XREAL_0:def 2;
then
A190: Index(p,co) < len LS-'(Gij..LS)+1 by NAT_1:13;
A191: 1<=Index(p,co) by A180,JORDAN3:8;
A192: Gij..LS<=len LS by A39,FINSEQ_4:21;
Gij..LS <> len LS by A31,A39,FINSEQ_4:19;
then
A193: Gij..LS < len LS by A192,XXREAL_0:1;
A194: 1+1 <= len LS by A25,XXREAL_0:2;
then
A195: 2 in dom LS by FINSEQ_3:25;
set tt = Index(p,co)+(Gij..LS)-'1;
set RC = Rotate(Cage(C,n),Emax);
A196: E-max C in right_cell(RC,1) by JORDAN1I:7;
A197: GoB RC = GoB Cage(C,n) by REVROT_1:28
.= Ga by JORDAN1H:44;
A198: L~RC = L~Cage(C,n) by REVROT_1:33;
consider jj2 be Nat such that
A199: 1 <= jj2 and
A200: jj2 <= width Ga and
A201: Emax = Ga*(len Ga,jj2) by JORDAN1D:25;
A202: len Ga >= 4 by JORDAN8:10;
then len Ga >= 1 by XXREAL_0:2;
then
A203: [len Ga,jj2] in Indices Ga by A199,A200,MATRIX_0:30;
A204: len RC = len Cage(C,n) by FINSEQ_6:179;
LS = RC-:Wmin by JORDAN1G:18;
then
A205: LSeg(LS,1) = LSeg(RC,1) by A194,SPPOL_2:9;
A206: Emax in rng Cage(C,n) by SPRECT_2:46;
RC is_sequence_on Ga by A140,REVROT_1:34;
then consider ii,jj be Nat such that
A207: [ii,jj+1] in Indices Ga and
A208: [ii,jj] in Indices Ga and
A209: RC/.1 = Ga*(ii,jj+1) and
A210: RC/.(1+1) = Ga*(ii,jj) by A87,A198,A204,A206,FINSEQ_6:92,JORDAN1I:23;
A211: jj+1+1 <> jj;
A212: 1 <= jj by A208,MATRIX_0:32;
RC/.1 = E-max L~RC by A198,A206,FINSEQ_6:92;
then
A213: ii = len Ga by A198,A207,A209,A201,A203,GOBOARD1:5;
then ii-1 >= 4-1 by A202,XREAL_1:9;
then
A214: ii-1 >= 1 by XXREAL_0:2;
then
A215: 1 <= ii-'1 by XREAL_0:def 2;
A216: jj <= width Ga by A208,MATRIX_0:32;
then
A217: Ga*(len Ga,jj)`1 = Ebo by A11,A212,JORDAN1A:71;
A218: jj+1 <= width Ga by A207,MATRIX_0:32;
ii+1 <> ii;
then
A219: right_cell(RC,1) = cell(Ga,ii-'1,jj) by A87,A204,A197,A207,A208,A209,A210
,A211,GOBOARD5:def 6;
A220: ii <= len Ga by A208,MATRIX_0:32;
A221: 1 <= ii by A208,MATRIX_0:32;
A222: ii <= len Ga by A207,MATRIX_0:32;
A223: 1 <= jj+1 by A207,MATRIX_0:32;
then
A224: Ebo = Ga*(len Ga,jj+1)`1 by A11,A218,JORDAN1A:71;
A225: 1 <= ii by A207,MATRIX_0:32;
then
A226: ii-'1+1 = ii by XREAL_1:235;
then
A227: ii-'1 < len Ga by A222,NAT_1:13;
then
A228: Ga*(ii-'1,jj+1)`2 = Ga*(1,jj+1)`2 by A223,A218,A215,GOBOARD5:1
.= Ga*(ii,jj+1)`2 by A225,A222,A223,A218,GOBOARD5:1;
A229: (E-max C)`2 = p`2 by A179,TOPREAL1:def 11;
then
A230: p`2 <= Ga*(ii-'1,jj+1)`2 by A196,A222,A218,A212,A219,A226,A214,JORDAN9:17
;
A231: Ga*(ii-'1,jj)`2 = Ga*(1,jj)`2 by A212,A216,A215,A227,GOBOARD5:1
.= Ga*(ii,jj)`2 by A221,A220,A212,A216,GOBOARD5:1;
Ga*(ii-'1,jj)`2 <= p`2 by A229,A196,A222,A218,A212,A219,A226,A214,
JORDAN9:17;
then p in LSeg(RC/.1,RC/.(1+1)) by A188,A209,A210,A213,A230,A231,A228
,A217,A224,GOBOARD7:7;
then
A232: p in LSeg(LS,1) by A87,A205,A204,TOPREAL1:def 3;
1<=Gij..LS by A39,FINSEQ_4:21;
then
A233: LSeg(mid(LS,Gij..LS,len LS),Index(p,co)) = LSeg(LS,Index(p,co)+(
Gij..LS)-'1) by A193,A191,A190,JORDAN4:19;
1<=Index(Gij,LS) by A9,JORDAN3:8;
then
A234: 1+1 <= Gij..LS by A189,XREAL_1:7;
then Index(p,co)+Gij..LS >= 1+1+1 by A191,XREAL_1:7;
then Index(p,co)+Gij..LS-1 >= 1+1+1-1 by XREAL_1:9;
then
A235: tt >= 1+1 by XREAL_0:def 2;
now
per cases by A235,XXREAL_0:1;
suppose
tt > 1+1;
then LSeg(LS,1) misses LSeg(LS,tt) by TOPREAL1:def 7;
hence contradiction by A232,A181,A187,A233,XBOOLE_0:3;
end;
suppose
A236: tt = 1+1;
then 1+1 = Index(p,co)+(Gij..LS)-1 by XREAL_0:def 2;
then 1+1+1 = Index(p,co)+(Gij..LS);
then
A237: Gij..LS = 2 by A191,A234,JORDAN1E:6;
LSeg(LS,1) /\ LSeg(LS,tt) = {LS/.2} by A25,A236,TOPREAL1:def 6;
then p in {LS/.2} by A232,A181,A187,A233,XBOOLE_0:def 4;
then
A238: p = LS/.2 by TARSKI:def 1;
then
A239: p in rng LS by A195,PARTFUN2:2;
p..LS = 2 by A195,A238,FINSEQ_5:41;
then p = Gij by A39,A237,A239,FINSEQ_5:9;
then Gij`1 = Ebo by A238,JORDAN1G:32;
then Gij`1 = Ga*(len Ga,j)`1 by A1,A11,A13,JORDAN1A:71;
hence contradiction by A2,A3,A16,A69,JORDAN1G:7;
end;
end;
hence contradiction;
end;
end;
then east_halfline E-max C c= (L~godo)` by SUBSET_1:23;
then consider W be Subset of TOP-REAL 2 such that
A240: W is_a_component_of (L~godo)` and
A241: east_halfline E-max C c= W by GOBOARD9:3;
W is not bounded by A241,JORDAN2C:121,RLTOPSP1:42;
then W is_outside_component_of L~godo by A240,JORDAN2C:def 3;
then W c= UBD L~godo by JORDAN2C:23;
then
A242: east_halfline E-max C c= UBD L~godo by A241;
E-max C in east_halfline E-max C by TOPREAL1:38;
then E-max C in UBD L~godo by A242;
then E-max C in LeftComp godo by GOBRD14:36;
then UA meets L~godo by A132,A133,A134,A142,A153,JORDAN1J:36;
then
A243: UA meets (L~go \/ L~pion1) or UA meets L~co by A143,XBOOLE_1:70;
A244: UA c= C by JORDAN6:61;
per cases by A243,XBOOLE_1:70;
suppose
UA meets L~go;
then UA meets L~Cage(C,n) by A48,A145,XBOOLE_1:1,63;
hence contradiction by A244,JORDAN10:5,XBOOLE_1:63;
end;
suppose
UA meets L~pion1;
hence contradiction by A8,A83,A150;
end;
suppose
UA meets L~co;
then UA meets L~Cage(C,n) by A55,A146,XBOOLE_1:1,63;
hence contradiction by A244,JORDAN10:5,XBOOLE_1:63;
end;
end;
theorem Th30:
for C be Simple_closed_curve for i,j,k be Nat st 1 <
j & j <= k & k < len Gauge(C,n) & 1 <= i & i <= width Gauge(C,n) & Gauge(C,n)*(
k,i) in L~Upper_Seq(C,n) & Gauge(C,n)*(j,i) in L~Lower_Seq(C,n) holds LSeg(
Gauge(C,n)*(j,i),Gauge(C,n)*(k,i)) meets Lower_Arc C
proof
let C be Simple_closed_curve;
let i,j,k be Nat;
assume that
A1: 1 < j and
A2: j <= k and
A3: k < len Gauge(C,n) and
A4: 1 <= i and
A5: i <= width Gauge(C,n) and
A6: Gauge(C,n)*(k,i) in L~Upper_Seq(C,n) and
A7: Gauge(C,n)*(j,i) in L~Lower_Seq(C,n);
consider j1,k1 be Nat such that
A8: j <= j1 and
A9: j1 <= k1 and
A10: k1 <= k and
A11: LSeg(Gauge(C,n)*(j1,i),Gauge(C,n)*(k1,i)) /\ L~Lower_Seq(C,n) = {
Gauge(C,n)*(j1,i)} and
A12: LSeg(Gauge(C,n)*(j1,i),Gauge(C,n)*(k1,i)) /\ L~Upper_Seq(C,n) = {
Gauge(C,n)*(k1,i)} by A1,A2,A3,A4,A5,A6,A7,Th14;
A13: k1 < len Gauge(C,n) by A3,A10,XXREAL_0:2;
1 < j1 by A1,A8,XXREAL_0:2;
then
LSeg(Gauge(C,n)*(j1,i),Gauge(C,n)*(k1,i)) meets Lower_Arc C by A4,A5,A9,A11
,A12,A13,Th28;
hence thesis by A1,A3,A4,A5,A8,A9,A10,Th6,XBOOLE_1:63;
end;
theorem Th31:
for C be Simple_closed_curve for i,j,k be Nat st 1 <
j & j <= k & k < len Gauge(C,n) & 1 <= i & i <= width Gauge(C,n) & Gauge(C,n)*(
k,i) in L~Upper_Seq(C,n) & Gauge(C,n)*(j,i) in L~Lower_Seq(C,n) holds LSeg(
Gauge(C,n)*(j,i),Gauge(C,n)*(k,i)) meets Upper_Arc C
proof
let C be Simple_closed_curve;
let i,j,k be Nat;
assume that
A1: 1 < j and
A2: j <= k and
A3: k < len Gauge(C,n) and
A4: 1 <= i and
A5: i <= width Gauge(C,n) and
A6: Gauge(C,n)*(k,i) in L~Upper_Seq(C,n) and
A7: Gauge(C,n)*(j,i) in L~Lower_Seq(C,n);
consider j1,k1 be Nat such that
A8: j <= j1 and
A9: j1 <= k1 and
A10: k1 <= k and
A11: LSeg(Gauge(C,n)*(j1,i),Gauge(C,n)*(k1,i)) /\ L~Lower_Seq(C,n) = {
Gauge(C,n)*(j1,i)} and
A12: LSeg(Gauge(C,n)*(j1,i),Gauge(C,n)*(k1,i)) /\ L~Upper_Seq(C,n) = {
Gauge(C,n)*(k1,i)} by A1,A2,A3,A4,A5,A6,A7,Th14;
A13: k1 < len Gauge(C,n) by A3,A10,XXREAL_0:2;
1 < j1 by A1,A8,XXREAL_0:2;
then
LSeg(Gauge(C,n)*(j1,i),Gauge(C,n)*(k1,i)) meets Upper_Arc C by A4,A5,A9,A11
,A12,A13,Th29;
hence thesis by A1,A3,A4,A5,A8,A9,A10,Th6,XBOOLE_1:63;
end;
theorem Th32:
for C be Simple_closed_curve for i,j,k be Nat st 1 <
j & j <= k & k < len Gauge(C,n) & 1 <= i & i <= width Gauge(C,n) & n > 0 &
Gauge(C,n)*(k,i) in Upper_Arc L~Cage(C,n) & Gauge(C,n)*(j,i) in Lower_Arc L~
Cage(C,n) holds LSeg(Gauge(C,n)*(j,i),Gauge(C,n)*(k,i)) meets Lower_Arc C
proof
let C be Simple_closed_curve;
let i,j,k be Nat;
assume that
A1: 1 < j and
A2: j <= k and
A3: k < len Gauge(C,n) and
A4: 1 <= i and
A5: i <= width Gauge(C,n) and
A6: n > 0 and
A7: Gauge(C,n)*(k,i) in Upper_Arc L~Cage(C,n) and
A8: Gauge(C,n)*(j,i) in Lower_Arc L~Cage(C,n);
A9: L~Lower_Seq(C,n) = Lower_Arc L~Cage(C,n) by A6,JORDAN1G:56;
L~Upper_Seq(C,n) = Upper_Arc L~Cage(C,n) by A6,JORDAN1G:55;
hence thesis by A1,A2,A3,A4,A5,A7,A8,A9,Th30;
end;
theorem Th33:
for C be Simple_closed_curve for i,j,k be Nat st 1 <
j & j <= k & k < len Gauge(C,n) & 1 <= i & i <= width Gauge(C,n) & n > 0 &
Gauge(C,n)*(k,i) in Upper_Arc L~Cage(C,n) & Gauge(C,n)*(j,i) in Lower_Arc L~
Cage(C,n) holds LSeg(Gauge(C,n)*(j,i),Gauge(C,n)*(k,i)) meets Upper_Arc C
proof
let C be Simple_closed_curve;
let i,j,k be Nat;
assume that
A1: 1 < j and
A2: j <= k and
A3: k < len Gauge(C,n) and
A4: 1 <= i and
A5: i <= width Gauge(C,n) and
A6: n > 0 and
A7: Gauge(C,n)*(k,i) in Upper_Arc L~Cage(C,n) and
A8: Gauge(C,n)*(j,i) in Lower_Arc L~Cage(C,n);
A9: L~Lower_Seq(C,n) = Lower_Arc L~Cage(C,n) by A6,JORDAN1G:56;
L~Upper_Seq(C,n) = Upper_Arc L~Cage(C,n) by A6,JORDAN1G:55;
hence thesis by A1,A2,A3,A4,A5,A7,A8,A9,Th31;
end;
theorem
for C be Simple_closed_curve for j,k be Nat holds 1 < j & j
<= k & k < len Gauge(C,n+1) & Gauge(C,n+1)*(k,Center Gauge(C,n+1)) in Upper_Arc
L~Cage(C,n+1) & Gauge(C,n+1)*(j,Center Gauge(C,n+1)) in Lower_Arc L~Cage(C,n+1)
implies LSeg(Gauge(C,n+1)*(j,Center Gauge(C,n+1)), Gauge(C,n+1)*(k,Center Gauge
(C,n+1))) meets Lower_Arc C
proof
let C be Simple_closed_curve;
let j,k be Nat;
assume that
A1: 1 < j and
A2: j <= k and
A3: k < len Gauge(C,n+1) and
A4: Gauge(C,n+1)*(k,Center Gauge(C,n+1)) in Upper_Arc L~Cage(C,n+1) and
A5: Gauge(C,n+1)*(j,Center Gauge(C,n+1)) in Lower_Arc L~Cage(C,n+1);
A6: len Gauge(C,n+1) >= 4 by JORDAN8:10;
then len Gauge(C,n+1) >= 3 by XXREAL_0:2;
then Center Gauge(C,n+1) < len Gauge(C,n+1) by JORDAN1B:15;
then
A7: Center Gauge(C,n+1) < width Gauge(C,n+1) by JORDAN8:def 1;
len Gauge(C,n+1) >= 2 by A6,XXREAL_0:2;
then 1 < Center Gauge(C,n+1) by JORDAN1B:14;
hence thesis by A1,A2,A3,A4,A5,A7,Th32;
end;
theorem
for C be Simple_closed_curve for j,k be Nat holds 1 < j & j
<= k & k < len Gauge(C,n+1) & Gauge(C,n+1)*(k,Center Gauge(C,n+1)) in Upper_Arc
L~Cage(C,n+1) & Gauge(C,n+1)*(j,Center Gauge(C,n+1)) in Lower_Arc L~Cage(C,n+1)
implies LSeg(Gauge(C,n+1)*(j,Center Gauge(C,n+1)), Gauge(C,n+1)*(k,Center Gauge
(C,n+1))) meets Upper_Arc C
proof
let C be Simple_closed_curve;
let j,k be Nat;
assume that
A1: 1 < j and
A2: j <= k and
A3: k < len Gauge(C,n+1) and
A4: Gauge(C,n+1)*(k,Center Gauge(C,n+1)) in Upper_Arc L~Cage(C,n+1) and
A5: Gauge(C,n+1)*(j,Center Gauge(C,n+1)) in Lower_Arc L~Cage(C,n+1);
A6: len Gauge(C,n+1) >= 4 by JORDAN8:10;
then len Gauge(C,n+1) >= 3 by XXREAL_0:2;
then Center Gauge(C,n+1) < len Gauge(C,n+1) by JORDAN1B:15;
then
A7: Center Gauge(C,n+1) < width Gauge(C,n+1) by JORDAN8:def 1;
len Gauge(C,n+1) >= 2 by A6,XXREAL_0:2;
then 1 < Center Gauge(C,n+1) by JORDAN1B:14;
hence thesis by A1,A2,A3,A4,A5,A7,Th33;
end;
theorem Th36:
for C be Simple_closed_curve for i,j,k be Nat st 1 <
j & j <= k & k < len Gauge(C,n) & 1 <= i & i <= width Gauge(C,n) & LSeg(Gauge(C
,n)*(j,i),Gauge(C,n)*(k,i)) /\ L~Upper_Seq(C,n) = {Gauge(C,n)*(j,i)} & LSeg(
Gauge(C,n)*(j,i),Gauge(C,n)*(k,i)) /\ L~Lower_Seq(C,n) = {Gauge(C,n)*(k,i)}
holds LSeg(Gauge(C,n)*(j,i),Gauge(C,n)*(k,i)) meets Lower_Arc C
proof
let C be Simple_closed_curve;
let i,j,k be Nat;
set Ga = Gauge(C,n);
set US = Upper_Seq(C,n);
set LS = Lower_Seq(C,n);
set LA = Lower_Arc C;
set Wmin = W-min L~Cage(C,n);
set Emax = E-max L~Cage(C,n);
set Wbo = W-bound L~Cage(C,n);
set Ebo = E-bound L~Cage(C,n);
set Gij = Ga*(j,i);
set Gik = Ga*(k,i);
assume that
A1: 1 < j and
A2: j <= k and
A3: k < len Ga and
A4: 1 <= i and
A5: i <= width Ga and
A6: LSeg(Gij,Gik) /\ L~US = {Gij} and
A7: LSeg(Gij,Gik) /\ L~LS = {Gik} and
A8: LSeg(Gij,Gik) misses LA;
Gik in {Gik} by TARSKI:def 1;
then
A9: Gik in L~LS by A7,XBOOLE_0:def 4;
Gij in {Gij} by TARSKI:def 1;
then
A10: Gij in L~US by A6,XBOOLE_0:def 4;
A11: len Ga = width Ga by JORDAN8:def 1;
A12: j <> k by A1,A3,A4,A5,A9,A10,Th27;
A13: j < width Ga by A2,A3,A11,XXREAL_0:2;
A14: 1 < k by A1,A2,XXREAL_0:2;
A15: k < width Ga by A3,JORDAN8:def 1;
A16: [j,i] in Indices Ga by A1,A4,A5,A11,A13,MATRIX_0:30;
A17: [k,i] in Indices Ga by A3,A4,A5,A14,MATRIX_0:30;
set go = R_Cut(US,Gij);
set co = L_Cut(LS,Gik);
A18: len US >= 3 by JORDAN1E:15;
then len US >= 1 by XXREAL_0:2;
then 1 in dom US by FINSEQ_3:25;
then
A19: US.1 = US/.1 by PARTFUN1:def 6
.= Wmin by JORDAN1F:5;
A20: Wmin`1 = Wbo by EUCLID:52
.= Ga*(1,i)`1 by A4,A5,A11,JORDAN1A:73;
len Ga >= 4 by JORDAN8:10;
then
A21: len Ga >= 1 by XXREAL_0:2;
then
A22: [1,k] in Indices Ga by A14,A15,MATRIX_0:30;
A23: [1,i] in Indices Ga by A4,A5,A21,MATRIX_0:30;
then
A24: Gij <> US.1 by A1,A16,A19,A20,JORDAN1G:7;
then reconsider go as being_S-Seq FinSequence of TOP-REAL 2 by A10,JORDAN3:35
;
A25: len LS >= 1+2 by JORDAN1E:15;
then
A26: len LS >= 1 by XXREAL_0:2;
then
A27: 1 in dom LS by FINSEQ_3:25;
len LS in dom LS by A26,FINSEQ_3:25;
then
A28: LS.len LS = LS/.len LS by PARTFUN1:def 6
.= Wmin by JORDAN1F:8;
Wmin`1 = Wbo by EUCLID:52
.= Ga*(1,i)`1 by A4,A5,A11,JORDAN1A:73;
then
A29: Gik <> LS.len LS by A1,A2,A17,A23,A28,JORDAN1G:7;
then reconsider co as being_S-Seq FinSequence of TOP-REAL 2 by A9,JORDAN3:34;
A30: [len Ga,k] in Indices Ga by A14,A15,A21,MATRIX_0:30;
A31: LS.1 = LS/.1 by A27,PARTFUN1:def 6
.= Emax by JORDAN1F:6;
Emax`1 = Ebo by EUCLID:52
.= Ga*(len Ga,k)`1 by A3,A14,JORDAN1A:71;
then
A32: Gik <> LS.1 by A3,A17,A30,A31,JORDAN1G:7;
A33: len go >= 1+1 by TOPREAL1:def 8;
A34: Gij in rng US by A1,A4,A5,A10,A11,A13,JORDAN1G:4,JORDAN1J:40;
then
A35: go is_sequence_on Ga by JORDAN1G:4,JORDAN1J:38;
A36: len co >= 1+1 by TOPREAL1:def 8;
A37: Gik in rng LS by A4,A5,A9,A11,A14,A15,JORDAN1G:5,JORDAN1J:40;
then
A38: co is_sequence_on Ga by JORDAN1G:5,JORDAN1J:39;
reconsider go as non constant s.c.c. being_S-Seq FinSequence of TOP-REAL 2
by A33,A35,JGRAPH_1:12,JORDAN8:5;
reconsider co as non constant s.c.c. being_S-Seq FinSequence of TOP-REAL 2
by A36,A38,JGRAPH_1:12,JORDAN8:5;
A39: len go > 1 by A33,NAT_1:13;
then
A40: len go in dom go by FINSEQ_3:25;
then
A41: go/.len go = go.len go by PARTFUN1:def 6
.= Gij by A10,JORDAN3:24;
len co >= 1 by A36,XXREAL_0:2;
then 1 in dom co by FINSEQ_3:25;
then
A42: co/.1 = co.1 by PARTFUN1:def 6
.= Gik by A9,JORDAN3:23;
reconsider m = len go - 1 as Nat by A40,FINSEQ_3:26;
A43: m+1 = len go;
then
A44: len go-'1 = m by NAT_D:34;
A45: LSeg(go,m) c= L~go by TOPREAL3:19;
A46: L~go c= L~US by A10,JORDAN3:41;
then LSeg(go,m) c= L~US by A45;
then
A47: LSeg(go,m) /\ LSeg(Gik,Gij) c= {Gij} by A6,XBOOLE_1:26;
m >= 1 by A33,XREAL_1:19;
then
A48: LSeg(go,m) = LSeg(go/.m,Gij) by A41,A43,TOPREAL1:def 3;
{Gij} c= LSeg(go,m) /\ LSeg(Gik,Gij)
proof
let x be object;
A49: Gij in LSeg(Gik,Gij) by RLTOPSP1:68;
assume x in {Gij};
then
A50: x = Gij by TARSKI:def 1;
Gij in LSeg(go,m) by A48,RLTOPSP1:68;
hence thesis by A50,A49,XBOOLE_0:def 4;
end;
then
A51: LSeg(go,m) /\ LSeg(Gik,Gij) = {Gij} by A47;
A52: LSeg(co,1) c= L~co by TOPREAL3:19;
A53: L~co c= L~LS by A9,JORDAN3:42;
then LSeg(co,1) c= L~LS by A52;
then
A54: LSeg(co,1) /\ LSeg(Gik,Gij) c= {Gik} by A7,XBOOLE_1:26;
A55: LSeg(co,1) = LSeg(Gik,co/.(1+1)) by A36,A42,TOPREAL1:def 3;
{Gik} c= LSeg(co,1) /\ LSeg(Gik,Gij)
proof
let x be object;
A56: Gik in LSeg(Gik,Gij) by RLTOPSP1:68;
assume x in {Gik};
then
A57: x = Gik by TARSKI:def 1;
Gik in LSeg(co,1) by A55,RLTOPSP1:68;
hence thesis by A57,A56,XBOOLE_0:def 4;
end;
then
A58: LSeg(Gik,Gij) /\ LSeg(co,1) = {Gik} by A54;
A59: go/.1 = US/.1 by A10,SPRECT_3:22
.= Wmin by JORDAN1F:5;
then
A60: go/.1 = LS/.len LS by JORDAN1F:8
.= co/.len co by A9,JORDAN1J:35;
A61: rng go c= L~go by A33,SPPOL_2:18;
A62: rng co c= L~co by A36,SPPOL_2:18;
A63: {go/.1} c= L~go /\ L~co
proof
let x be object;
assume x in {go/.1};
then
A64: x = go/.1 by TARSKI:def 1;
then
A65: x in rng go by FINSEQ_6:42;
x in rng co by A60,A64,FINSEQ_6:168;
hence thesis by A61,A62,A65,XBOOLE_0:def 4;
end;
A66: LS.1 = LS/.1 by A27,PARTFUN1:def 6
.= Emax by JORDAN1F:6;
A67: [len Ga,j] in Indices Ga by A1,A13,A21,MATRIX_0:30;
L~go /\ L~co c= {go/.1}
proof
let x be object;
assume
A68: x in L~go /\ L~co;
then
A69: x in L~co by XBOOLE_0:def 4;
A70: now
assume x = Emax;
then
A71: Emax = Gik by A9,A66,A69,JORDAN1E:7;
Ga*(len Ga,j)`1 = Ebo by A1,A11,A13,JORDAN1A:71;
then Emax`1 <> Ebo by A3,A17,A67,A71,JORDAN1G:7;
hence contradiction by EUCLID:52;
end;
x in L~go by A68,XBOOLE_0:def 4;
then x in L~US /\ L~LS by A46,A53,A69,XBOOLE_0:def 4;
then x in {Wmin,Emax} by JORDAN1E:16;
then x = Wmin or x = Emax by TARSKI:def 2;
hence thesis by A59,A70,TARSKI:def 1;
end;
then
A72: L~go /\ L~co = {go/.1} by A63;
set W2 = go/.2;
A73: 2 in dom go by A33,FINSEQ_3:25;
A74: now
assume Gij`1 = Wbo;
then Ga*(1,k)`1 = Ga*(j,i)`1 by A3,A14,JORDAN1A:73;
hence contradiction by A1,A16,A22,JORDAN1G:7;
end;
go = mid(US,1,Gij..US) by A34,JORDAN1G:49
.= US|(Gij..US) by A34,FINSEQ_4:21,FINSEQ_6:116;
then
A75: W2 = US/.2 by A73,FINSEQ_4:70;
A76: Wmin in rng go by A59,FINSEQ_6:42;
set pion = <*Gij,Gik*>;
A77: now
let n be Nat;
assume n in dom pion;
then n in {1,2} by FINSEQ_1:2,89;
then n = 1 or n = 2 by TARSKI:def 2;
hence ex j,i be Nat st [j,i] in Indices Ga & pion/.n = Ga*(j,i)
by A16,A17,FINSEQ_4:17;
end;
A78: Gik <> Gij by A12,A16,A17,GOBOARD1:5;
Gik`2 = Ga*(1,i)`2 by A3,A4,A5,A14,GOBOARD5:1
.= Gij`2 by A1,A4,A5,A11,A13,GOBOARD5:1;
then LSeg(Gik,Gij) is horizontal by SPPOL_1:15;
then pion is being_S-Seq by A78,JORDAN1B:8;
then consider pion1 be FinSequence of TOP-REAL 2 such that
A79: pion1 is_sequence_on Ga and
A80: pion1 is being_S-Seq and
A81: L~pion = L~pion1 and
A82: pion/.1 = pion1/.1 and
A83: pion/.len pion = pion1/.len pion1 and
A84: len pion <= len pion1 by A77,GOBOARD3:2;
reconsider pion1 as being_S-Seq FinSequence of TOP-REAL 2 by A80;
set godo = go^'pion1^'co;
A85: 1+1 <= len Cage(C,n) by GOBOARD7:34,XXREAL_0:2;
A86: 1+1 <= len Rotate(Cage(C,n),Wmin) by GOBOARD7:34,XXREAL_0:2;
len (go^'pion1) >= len go by TOPREAL8:7;
then
A87: len (go^'pion1) >= 1+1 by A33,XXREAL_0:2;
then
A88: len (go^'pion1) > 1+0 by NAT_1:13;
A89: len godo >= len (go^'pion1) by TOPREAL8:7;
then
A90: 1+1 <= len godo by A87,XXREAL_0:2;
A91: US is_sequence_on Ga by JORDAN1G:4;
A92: go/.len go = pion1/.1 by A41,A82,FINSEQ_4:17;
then
A93: go^'pion1 is_sequence_on Ga by A35,A79,TOPREAL8:12;
A94: (go^'pion1)/.len (go^'pion1) = pion/.len pion by A83,FINSEQ_6:156
.= pion/.2 by FINSEQ_1:44
.= co/.1 by A42,FINSEQ_4:17;
then
A95: godo is_sequence_on Ga by A38,A93,TOPREAL8:12;
LSeg(pion1,1) c= L~<*Gij,Gik*> by A81,TOPREAL3:19;
then LSeg(pion1,1) c= LSeg(Gij,Gik) by SPPOL_2:21;
then
A96: LSeg(go,len go-'1) /\ LSeg(pion1,1) c= {Gij} by A44,A51,XBOOLE_1:27;
A97: len pion1 >= 1+1 by A84,FINSEQ_1:44;
{Gij} c= LSeg(go,m) /\ LSeg(pion1,1)
proof
let x be object;
assume x in {Gij};
then
A98: x = Gij by TARSKI:def 1;
A99: Gij in LSeg(go,m) by A48,RLTOPSP1:68;
Gij in LSeg(pion1,1) by A41,A92,A97,TOPREAL1:21;
hence thesis by A98,A99,XBOOLE_0:def 4;
end;
then LSeg(go,len go-'1) /\ LSeg(pion1,1) = { go/.len go } by A41,A44,A96;
then
A100: go^'pion1 is unfolded by A92,TOPREAL8:34;
len pion1 >= 2+0 by A84,FINSEQ_1:44;
then
A101: len pion1-2 >= 0 by XREAL_1:19;
len (go^'pion1)+1-1 = len go+len pion1-1 by FINSEQ_6:139;
then len (go^'pion1)-1 = len go + (len pion1-2)
.= len go + (len pion1-'2) by A101,XREAL_0:def 2;
then
A102: len (go^'pion1)-'1 = len go + (len pion1-'2) by XREAL_0:def 2;
A103: len pion1-1 >= 1 by A97,XREAL_1:19;
then
A104: len pion1-'1 = len pion1-1 by XREAL_0:def 2;
A105: len pion1-'2+1 = len pion1-2+1 by A101,XREAL_0:def 2
.= len pion1-'1 by A103,XREAL_0:def 2;
len pion1-1+1 <= len pion1;
then
A106: len pion1-'1 < len pion1 by A104,NAT_1:13;
LSeg(pion1,len pion1-'1) c= L~<*Gij,Gik*> by A81,TOPREAL3:19;
then LSeg(pion1,len pion1-'1) c= LSeg(Gij,Gik) by SPPOL_2:21;
then
A107: LSeg(pion1,len pion1-'1) /\ LSeg(co,1) c= {Gik} by A58,XBOOLE_1:27;
{Gik} c= LSeg(pion1,len pion1-'1) /\ LSeg(co,1)
proof
let x be object;
assume x in {Gik};
then
A108: x = Gik by TARSKI:def 1;
pion1/.(len pion1-'1+1) = pion/.2 by A83,A104,FINSEQ_1:44
.= Gik by FINSEQ_4:17;
then
A109: Gik in LSeg(pion1,len pion1-'1) by A103,A104,TOPREAL1:21;
Gik in LSeg(co,1) by A55,RLTOPSP1:68;
hence thesis by A108,A109,XBOOLE_0:def 4;
end;
then LSeg(pion1,len pion1-'1) /\ LSeg(co,1) = {Gik} by A107;
then
A110: LSeg(go^'pion1,len go+(len pion1-'2)) /\ LSeg(co,1) = {(go^'pion1)/.
len (go^'pion1)} by A42,A92,A94,A105,A106,TOPREAL8:31;
A111: (go^'pion1) is non trivial by A87,NAT_D:60;
A112: rng pion1 c= L~pion1 by A97,SPPOL_2:18;
A113: {pion1/.1} c= L~go /\ L~pion1
proof
let x be object;
assume x in {pion1/.1};
then
A114: x = pion1/.1 by TARSKI:def 1;
then
A115: x in rng pion1 by FINSEQ_6:42;
x in rng go by A92,A114,FINSEQ_6:168;
hence thesis by A61,A112,A115,XBOOLE_0:def 4;
end;
L~go /\ L~pion1 c= {pion1/.1}
proof
let x be object;
assume
A116: x in L~go /\ L~pion1;
then
A117: x in L~pion1 by XBOOLE_0:def 4;
x in L~go by A116,XBOOLE_0:def 4;
then x in L~pion1 /\ L~US by A46,A117,XBOOLE_0:def 4;
hence thesis by A6,A41,A81,A92,SPPOL_2:21;
end;
then
A118: L~go /\ L~pion1 = {pion1/.1} by A113;
then
A119: (go^'pion1) is s.n.c. by A92,JORDAN1J:54;
rng go /\ rng pion1 c= {pion1/.1} by A61,A112,A118,XBOOLE_1:27;
then
A120: go^'pion1 is one-to-one by JORDAN1J:55;
A121: pion/.len pion = pion/.2 by FINSEQ_1:44
.= co/.1 by A42,FINSEQ_4:17;
A122: {pion1/.len pion1} c= L~co /\ L~pion1
proof
let x be object;
assume x in {pion1/.len pion1};
then
A123: x = pion1/.len pion1 by TARSKI:def 1;
then
A124: x in rng pion1 by FINSEQ_6:168;
x in rng co by A83,A121,A123,FINSEQ_6:42;
hence thesis by A62,A112,A124,XBOOLE_0:def 4;
end;
L~co /\ L~pion1 c= {pion1/.len pion1}
proof
let x be object;
assume
A125: x in L~co /\ L~pion1;
then
A126: x in L~pion1 by XBOOLE_0:def 4;
x in L~co by A125,XBOOLE_0:def 4;
then x in L~pion1 /\ L~LS by A53,A126,XBOOLE_0:def 4;
hence thesis by A7,A42,A81,A83,A121,SPPOL_2:21;
end;
then
A127: L~co /\ L~pion1 = {pion1/.len pion1} by A122;
A128: L~(go^'pion1) /\ L~co = (L~go \/ L~pion1) /\ L~co by A92,TOPREAL8:35
.= {go/.1} \/ {co/.1} by A72,A83,A121,A127,XBOOLE_1:23
.= {(go^'pion1)/.1} \/ {co/.1} by FINSEQ_6:155
.= {(go^'pion1)/.1,co/.1} by ENUMSET1:1;
co/.len co = (go^'pion1)/.1 by A60,FINSEQ_6:155;
then reconsider
godo as non constant standard special_circular_sequence by A90,A94,A95,A100
,A102,A110,A111,A119,A120,A128,JORDAN8:4,5,TOPREAL8:11,33,34;
A129: LA is_an_arc_of E-max C,W-min C by JORDAN6:def 9;
then
A130: LA is connected by JORDAN6:10;
A131: W-min C in LA by A129,TOPREAL1:1;
A132: E-max C in LA by A129,TOPREAL1:1;
set ff = Rotate(Cage(C,n),Wmin);
Wmin in rng Cage(C,n) by SPRECT_2:43;
then
A133: ff/.1 = Wmin by FINSEQ_6:92;
A134: L~ff = L~Cage(C,n) by REVROT_1:33;
then (W-max L~ff)..ff > 1 by A133,SPRECT_5:22;
then (N-min L~ff)..ff > 1 by A133,A134,SPRECT_5:23,XXREAL_0:2;
then (N-max L~ff)..ff > 1 by A133,A134,SPRECT_5:24,XXREAL_0:2;
then
A135: Emax..ff > 1 by A133,A134,SPRECT_5:25,XXREAL_0:2;
A136: now
assume
A137: Gij..US <= 1;
Gij..US >= 1 by A34,FINSEQ_4:21;
then Gij..US = 1 by A137,XXREAL_0:1;
then Gij = US/.1 by A34,FINSEQ_5:38;
hence contradiction by A19,A24,JORDAN1F:5;
end;
A138: Cage(C,n) is_sequence_on Ga by JORDAN9:def 1;
then
A139: ff is_sequence_on Ga by REVROT_1:34;
A140: right_cell(godo,1,Ga)\L~godo c= RightComp godo by A90,A95,JORDAN9:27;
A141: L~godo = L~(go^'pion1) \/ L~co by A94,TOPREAL8:35
.= L~go \/ L~pion1 \/ L~co by A92,TOPREAL8:35;
A142: L~Cage(C,n) = L~US \/ L~LS by JORDAN1E:13;
then
A143: L~US c= L~Cage(C,n) by XBOOLE_1:7;
A144: L~LS c= L~Cage(C,n) by A142,XBOOLE_1:7;
A145: L~go c= L~Cage(C,n) by A46,A143;
A146: L~co c= L~Cage(C,n) by A53,A144;
A147: W-min C in C by SPRECT_1:13;
A148: L~pion = LSeg(Gik,Gij) by SPPOL_2:21;
A149: now
assume W-min C in L~godo;
then
A150: W-min C in L~go \/ L~pion1 or W-min C in L~co by A141,XBOOLE_0:def 3;
per cases by A150,XBOOLE_0:def 3;
suppose
W-min C in L~go;
then C meets L~Cage(C,n) by A145,A147,XBOOLE_0:3;
hence contradiction by JORDAN10:5;
end;
suppose
W-min C in L~pion1;
hence contradiction by A8,A81,A131,A148,XBOOLE_0:3;
end;
suppose
W-min C in L~co;
then C meets L~Cage(C,n) by A146,A147,XBOOLE_0:3;
hence contradiction by JORDAN10:5;
end;
end;
right_cell(Rotate(Cage(C,n),Wmin),1) = right_cell(ff,1,GoB ff) by A86,
JORDAN1H:23
.= right_cell(ff,1,GoB Cage(C,n)) by REVROT_1:28
.= right_cell(ff,1,Ga) by JORDAN1H:44
.= right_cell(ff-:Emax,1,Ga) by A135,A139,JORDAN1J:53
.= right_cell(US,1,Ga) by JORDAN1E:def 1
.= right_cell(R_Cut(US,Gij),1,Ga) by A34,A91,A136,JORDAN1J:52
.= right_cell(go^'pion1,1,Ga) by A39,A93,JORDAN1J:51
.= right_cell(godo,1,Ga) by A88,A95,JORDAN1J:51;
then W-min C in right_cell(godo,1,Ga) by JORDAN1I:6;
then
A151: W-min C in right_cell(godo,1,Ga)\L~godo by A149,XBOOLE_0:def 5;
A152: godo/.1 = (go^'pion1)/.1 by FINSEQ_6:155
.= Wmin by A59,FINSEQ_6:155;
A153: len US >= 2 by A18,XXREAL_0:2;
A154: godo/.2 = (go^'pion1)/.2 by A87,FINSEQ_6:159
.= US/.2 by A33,A75,FINSEQ_6:159
.= (US^'LS)/.2 by A153,FINSEQ_6:159
.= Rotate(Cage(C,n),Wmin)/.2 by JORDAN1E:11;
A155: L~go \/ L~co is compact by COMPTS_1:10;
Wmin in L~go \/ L~co by A61,A76,XBOOLE_0:def 3;
then
A156: W-min (L~go \/ L~co) = Wmin by A145,A146,A155,JORDAN1J:21,XBOOLE_1:8;
A157: (W-min (L~go \/ L~co))`1 = W-bound (L~go \/ L~co) by EUCLID:52;
A158: Wmin`1 = Wbo by EUCLID:52;
A159: Gij`1 <= Gik`1 by A1,A2,A3,A4,A5,SPRECT_3:13;
then W-bound LSeg(Gik,Gij) = Gij`1 by SPRECT_1:54;
then
A160: W-bound L~pion1 = Gij`1 by A81,SPPOL_2:21;
Gij`1 >= Wbo by A10,A143,PSCOMP_1:24;
then Gij`1 > Wbo by A74,XXREAL_0:1;
then W-min (L~go\/L~co\/L~pion1) = W-min (L~go \/ L~co) by A155,A156,A157
,A158,A160,JORDAN1J:33;
then
A161: W-min L~godo = Wmin by A141,A156,XBOOLE_1:4;
A162: rng godo c= L~godo by A87,A89,SPPOL_2:18,XXREAL_0:2;
2 in dom godo by A90,FINSEQ_3:25;
then
A163: godo/.2 in rng godo by PARTFUN2:2;
godo/.2 in W-most L~Cage(C,n) by A154,JORDAN1I:25;
then (godo/.2)`1 = (W-min L~godo)`1 by A161,PSCOMP_1:31
.= W-bound L~godo by EUCLID:52;
then godo/.2 in W-most L~godo by A162,A163,SPRECT_2:12;
then Rotate(godo,W-min L~godo)/.2 in W-most L~godo by A152,A161,FINSEQ_6:89;
then reconsider godo as clockwise_oriented non constant standard
special_circular_sequence by JORDAN1I:25;
len US in dom US by FINSEQ_5:6;
then
A164: US.len US = US/.len US by PARTFUN1:def 6
.= Emax by JORDAN1F:7;
A165: east_halfline E-max C misses L~go
proof
assume east_halfline E-max C meets L~go;
then consider p be object such that
A166: p in east_halfline E-max C and
A167: p in L~go by XBOOLE_0:3;
reconsider p as Point of TOP-REAL 2 by A166;
p in L~US by A46,A167;
then p in east_halfline E-max C /\ L~Cage(C,n) by A143,A166,XBOOLE_0:def 4;
then
A168: p`1 = Ebo by JORDAN1A:83,PSCOMP_1:50;
then
A169: p = Emax by A46,A167,JORDAN1J:46;
then Emax = Gij by A10,A164,A167,JORDAN1J:43;
then Gij`1 = Ga*(len Ga,k)`1 by A3,A14,A168,A169,JORDAN1A:71;
hence contradiction by A2,A3,A16,A30,JORDAN1G:7;
end;
now
assume east_halfline E-max C meets L~godo;
then
A170: east_halfline E-max C meets (L~go \/ L~pion1) or east_halfline E-max
C meets L~co by A141,XBOOLE_1:70;
per cases by A170,XBOOLE_1:70;
suppose
east_halfline E-max C meets L~go;
hence contradiction by A165;
end;
suppose
east_halfline E-max C meets L~pion1;
then consider p be object such that
A171: p in east_halfline E-max C and
A172: p in L~pion1 by XBOOLE_0:3;
reconsider p as Point of TOP-REAL 2 by A171;
A173: p`2 = (E-max C)`2 by A171,TOPREAL1:def 11;
k+1 <= len Ga by A3,NAT_1:13;
then k+1-1 <= len Ga-1 by XREAL_1:9;
then
A174: k <= len Ga-'1 by XREAL_0:def 2;
len Ga-'1 <= len Ga by NAT_D:35;
then
A175: Gik`1 <= Ga*(len Ga-'1,1)`1 by A4,A5,A11,A14,A21,A174,JORDAN1A:18;
p`1 <= Gik`1 by A81,A148,A159,A172,TOPREAL1:3;
then p`1 <= Ga*(len Ga-'1,1)`1 by A175,XXREAL_0:2;
then p`1 <= E-bound C by A21,JORDAN8:12;
then
A176: p`1 <= (E-max C)`1 by EUCLID:52;
p`1 >= (E-max C)`1 by A171,TOPREAL1:def 11;
then p`1 = (E-max C)`1 by A176,XXREAL_0:1;
then p = E-max C by A173,TOPREAL3:6;
hence contradiction by A8,A81,A132,A148,A172,XBOOLE_0:3;
end;
suppose
east_halfline E-max C meets L~co;
then consider p be object such that
A177: p in east_halfline E-max C and
A178: p in L~co by XBOOLE_0:3;
reconsider p as Point of TOP-REAL 2 by A177;
A179: p in LSeg(co,Index(p,co)) by A178,JORDAN3:9;
consider t be Nat such that
A180: t in dom LS and
A181: LS.t = Gik by A37,FINSEQ_2:10;
1 <= t by A180,FINSEQ_3:25;
then
A182: 1 < t by A32,A181,XXREAL_0:1;
t <= len LS by A180,FINSEQ_3:25;
then Index(Gik,LS)+1 = t by A181,A182,JORDAN3:12;
then
A183: len L_Cut(LS,Gik) = len LS-Index(Gik,LS) by A9,A181,JORDAN3:26;
Index(p,co) < len co by A178,JORDAN3:8;
then Index(p,co) < len LS-'Index(Gik,LS) by A183,XREAL_0:def 2;
then Index(p,co)+1 <= len LS-'Index(Gik,LS) by NAT_1:13;
then
A184: Index(p,co) <= len LS-'Index(Gik,LS)-1 by XREAL_1:19;
A185: co = mid(LS,Gik..LS,len LS) by A37,JORDAN1J:37;
p in L~LS by A53,A178;
then p in east_halfline E-max C /\ L~Cage(C,n) by A144,A177,
XBOOLE_0:def 4;
then
A186: p`1 = Ebo by JORDAN1A:83,PSCOMP_1:50;
A187: Index(Gik,LS)+1 = Gik..LS by A32,A37,JORDAN1J:56;
0+Index(Gik,LS) < len LS by A9,JORDAN3:8;
then len LS-Index(Gik,LS) > 0 by XREAL_1:20;
then Index(p,co) <= len LS-Index(Gik,LS)-1 by A184,XREAL_0:def 2;
then Index(p,co) <= len LS-Gik..LS by A187;
then Index(p,co) <= len LS-'Gik..LS by XREAL_0:def 2;
then
A188: Index(p,co) < len LS-'(Gik..LS)+1 by NAT_1:13;
A189: 1<=Index(p,co) by A178,JORDAN3:8;
A190: Gik..LS<=len LS by A37,FINSEQ_4:21;
Gik..LS <> len LS by A29,A37,FINSEQ_4:19;
then
A191: Gik..LS < len LS by A190,XXREAL_0:1;
A192: 1+1 <= len LS by A25,XXREAL_0:2;
then
A193: 2 in dom LS by FINSEQ_3:25;
set tt = Index(p,co)+(Gik..LS)-'1;
set RC = Rotate(Cage(C,n),Emax);
A194: E-max C in right_cell(RC,1) by JORDAN1I:7;
A195: GoB RC = GoB Cage(C,n) by REVROT_1:28
.= Ga by JORDAN1H:44;
A196: L~RC = L~Cage(C,n) by REVROT_1:33;
consider g2 be Nat such that
A197: 1 <= g2 and
A198: g2 <= width Ga and
A199: Emax = Ga*(len Ga,g2) by JORDAN1D:25;
A200: len Ga >= 4 by JORDAN8:10;
then len Ga >= 1 by XXREAL_0:2;
then
A201: [len Ga,g2] in Indices Ga by A197,A198,MATRIX_0:30;
A202: len RC = len Cage(C,n) by FINSEQ_6:179;
LS = RC-:Wmin by JORDAN1G:18;
then
A203: LSeg(LS,1) = LSeg(RC,1) by A192,SPPOL_2:9;
A204: Emax in rng Cage(C,n) by SPRECT_2:46;
RC is_sequence_on Ga by A138,REVROT_1:34;
then consider ii,g be Nat such that
A205: [ii,g+1] in Indices Ga and
A206: [ii,g] in Indices Ga and
A207: RC/.1 = Ga*(ii,g+1) and
A208: RC/.(1+1) = Ga*(ii,g) by A85,A196,A202,A204,FINSEQ_6:92,JORDAN1I:23;
A209: g+1+1 <> g;
A210: 1 <= g by A206,MATRIX_0:32;
RC/.1 = E-max L~RC by A196,A204,FINSEQ_6:92;
then
A211: ii = len Ga by A196,A205,A207,A199,A201,GOBOARD1:5;
then ii-1 >= 4-1 by A200,XREAL_1:9;
then
A212: ii-1 >= 1 by XXREAL_0:2;
then
A213: 1 <= ii-'1 by XREAL_0:def 2;
A214: g <= width Ga by A206,MATRIX_0:32;
then
A215: Ga*(len Ga,g)`1 = Ebo by A11,A210,JORDAN1A:71;
A216: g+1 <= width Ga by A205,MATRIX_0:32;
ii+1 <> ii;
then
A217: right_cell(RC,1) = cell(Ga,ii-'1,g) by A85,A202,A195,A205,A206,A207,A208
,A209,GOBOARD5:def 6;
A218: ii <= len Ga by A206,MATRIX_0:32;
A219: 1 <= ii by A206,MATRIX_0:32;
A220: ii <= len Ga by A205,MATRIX_0:32;
A221: 1 <= g+1 by A205,MATRIX_0:32;
then
A222: Ebo = Ga*(len Ga,g+1)`1 by A11,A216,JORDAN1A:71;
A223: 1 <= ii by A205,MATRIX_0:32;
then
A224: ii-'1+1 = ii by XREAL_1:235;
then
A225: ii-'1 < len Ga by A220,NAT_1:13;
then
A226: Ga*(ii-'1,g+1)`2 = Ga*(1,g+1)`2 by A221,A216,A213,GOBOARD5:1
.= Ga*(ii,g+1)`2 by A223,A220,A221,A216,GOBOARD5:1;
A227: (E-max C)`2 = p`2 by A177,TOPREAL1:def 11;
then
A228: p`2 <= Ga*(ii-'1,g+1)`2 by A194,A220,A216,A210,A217,A224,A212,JORDAN9:17;
A229: Ga*(ii-'1,g)`2 = Ga*(1,g)`2 by A210,A214,A213,A225,GOBOARD5:1
.= Ga*(ii,g)`2 by A219,A218,A210,A214,GOBOARD5:1;
Ga*(ii-'1,g)`2 <= p`2 by A227,A194,A220,A216,A210,A217,A224,A212,
JORDAN9:17;
then p in LSeg(RC/.1,RC/.(1+1)) by A186,A207,A208,A211,A228,A229,A226
,A215,A222,GOBOARD7:7;
then
A230: p in LSeg(LS,1) by A85,A203,A202,TOPREAL1:def 3;
1<=Gik..LS by A37,FINSEQ_4:21;
then
A231: LSeg(mid(LS,Gik..LS,len LS),Index(p,co)) = LSeg(LS,Index(p,co)+(
Gik..LS)-'1) by A191,A189,A188,JORDAN4:19;
1<=Index(Gik,LS) by A9,JORDAN3:8;
then
A232: 1+1 <= Gik..LS by A187,XREAL_1:7;
then Index(p,co)+Gik..LS >= 1+1+1 by A189,XREAL_1:7;
then Index(p,co)+Gik..LS-1 >= 1+1+1-1 by XREAL_1:9;
then
A233: tt >= 1+1 by XREAL_0:def 2;
now
per cases by A233,XXREAL_0:1;
suppose
tt > 1+1;
then LSeg(LS,1) misses LSeg(LS,tt) by TOPREAL1:def 7;
hence contradiction by A230,A179,A185,A231,XBOOLE_0:3;
end;
suppose
A234: tt = 1+1;
then 1+1 = Index(p,co)+(Gik..LS)-1 by XREAL_0:def 2;
then 1+1+1 = Index(p,co)+(Gik..LS);
then
A235: Gik..LS = 2 by A189,A232,JORDAN1E:6;
LSeg(LS,1) /\ LSeg(LS,tt) = {LS/.2} by A25,A234,TOPREAL1:def 6;
then p in {LS/.2} by A230,A179,A185,A231,XBOOLE_0:def 4;
then
A236: p = LS/.2 by TARSKI:def 1;
then
A237: p in rng LS by A193,PARTFUN2:2;
p..LS = 2 by A193,A236,FINSEQ_5:41;
then p = Gik by A37,A235,A237,FINSEQ_5:9;
then Gik`1 = Ebo by A236,JORDAN1G:32;
then Gik`1 = Ga*(len Ga,j)`1 by A1,A11,A13,JORDAN1A:71;
hence contradiction by A3,A17,A67,JORDAN1G:7;
end;
end;
hence contradiction;
end;
end;
then east_halfline E-max C c= (L~godo)` by SUBSET_1:23;
then consider W be Subset of TOP-REAL 2 such that
A238: W is_a_component_of (L~godo)` and
A239: east_halfline E-max C c= W by GOBOARD9:3;
W is not bounded by A239,JORDAN2C:121,RLTOPSP1:42;
then W is_outside_component_of L~godo by A238,JORDAN2C:def 3;
then W c= UBD L~godo by JORDAN2C:23;
then
A240: east_halfline E-max C c= UBD L~godo by A239;
E-max C in east_halfline E-max C by TOPREAL1:38;
then E-max C in UBD L~godo by A240;
then E-max C in LeftComp godo by GOBRD14:36;
then LA meets L~godo by A130,A131,A132,A140,A151,JORDAN1J:36;
then
A241: LA meets (L~go \/ L~pion1) or LA meets L~co by A141,XBOOLE_1:70;
A242: LA c= C by JORDAN6:61;
per cases by A241,XBOOLE_1:70;
suppose
LA meets L~go;
then LA meets L~Cage(C,n) by A46,A143,XBOOLE_1:1,63;
hence contradiction by A242,JORDAN10:5,XBOOLE_1:63;
end;
suppose
LA meets L~pion1;
hence contradiction by A8,A81,A148;
end;
suppose
LA meets L~co;
then LA meets L~Cage(C,n) by A53,A144,XBOOLE_1:1,63;
hence contradiction by A242,JORDAN10:5,XBOOLE_1:63;
end;
end;
theorem Th37:
for C be Simple_closed_curve for i,j,k be Nat st 1 <
j & j <= k & k < len Gauge(C,n) & 1 <= i & i <= width Gauge(C,n) & LSeg(Gauge(C
,n)*(j,i),Gauge(C,n)*(k,i)) /\ L~Upper_Seq(C,n) = {Gauge(C,n)*(j,i)} & LSeg(
Gauge(C,n)*(j,i),Gauge(C,n)*(k,i)) /\ L~Lower_Seq(C,n) = {Gauge(C,n)*(k,i)}
holds LSeg(Gauge(C,n)*(j,i),Gauge(C,n)*(k,i)) meets Upper_Arc C
proof
let C be Simple_closed_curve;
let i,j,k be Nat;
set Ga = Gauge(C,n);
set US = Upper_Seq(C,n);
set LS = Lower_Seq(C,n);
set UA = Upper_Arc C;
set Wmin = W-min L~Cage(C,n);
set Emax = E-max L~Cage(C,n);
set Wbo = W-bound L~Cage(C,n);
set Ebo = E-bound L~Cage(C,n);
set Gij = Ga*(j,i);
set Gik = Ga*(k,i);
assume that
A1: 1 < j and
A2: j <= k and
A3: k < len Ga and
A4: 1 <= i and
A5: i <= width Ga and
A6: LSeg(Gij,Gik) /\ L~US = {Gij} and
A7: LSeg(Gij,Gik) /\ L~LS = {Gik} and
A8: LSeg(Gij,Gik) misses UA;
Gik in {Gik} by TARSKI:def 1;
then
A9: Gik in L~LS by A7,XBOOLE_0:def 4;
Gij in {Gij} by TARSKI:def 1;
then
A10: Gij in L~US by A6,XBOOLE_0:def 4;
A11: len Ga = width Ga by JORDAN8:def 1;
A12: j <> k by A1,A3,A4,A5,A9,A10,Th27;
A13: j < width Ga by A2,A3,A11,XXREAL_0:2;
A14: 1 < k by A1,A2,XXREAL_0:2;
A15: k < width Ga by A3,JORDAN8:def 1;
A16: [j,i] in Indices Ga by A1,A4,A5,A11,A13,MATRIX_0:30;
A17: [k,i] in Indices Ga by A3,A4,A5,A14,MATRIX_0:30;
set go = R_Cut(US,Gij);
set co = L_Cut(LS,Gik);
A18: len US >= 3 by JORDAN1E:15;
then len US >= 1 by XXREAL_0:2;
then 1 in dom US by FINSEQ_3:25;
then
A19: US.1 = US/.1 by PARTFUN1:def 6
.= Wmin by JORDAN1F:5;
A20: Wmin`1 = Wbo by EUCLID:52
.= Ga*(1,i)`1 by A4,A5,A11,JORDAN1A:73;
len Ga >= 4 by JORDAN8:10;
then
A21: len Ga >= 1 by XXREAL_0:2;
then
A22: [1,k] in Indices Ga by A14,A15,MATRIX_0:30;
A23: [1,i] in Indices Ga by A4,A5,A21,MATRIX_0:30;
then
A24: Gij <> US.1 by A1,A16,A19,A20,JORDAN1G:7;
then reconsider go as being_S-Seq FinSequence of TOP-REAL 2 by A10,JORDAN3:35
;
A25: len LS >= 1+2 by JORDAN1E:15;
then
A26: len LS >= 1 by XXREAL_0:2;
then
A27: 1 in dom LS by FINSEQ_3:25;
len LS in dom LS by A26,FINSEQ_3:25;
then
A28: LS.len LS = LS/.len LS by PARTFUN1:def 6
.= Wmin by JORDAN1F:8;
Wmin`1 = Wbo by EUCLID:52
.= Ga*(1,i)`1 by A4,A5,A11,JORDAN1A:73;
then
A29: Gik <> LS.len LS by A1,A2,A17,A23,A28,JORDAN1G:7;
then reconsider co as being_S-Seq FinSequence of TOP-REAL 2 by A9,JORDAN3:34;
A30: [len Ga,k] in Indices Ga by A14,A15,A21,MATRIX_0:30;
A31: LS.1 = LS/.1 by A27,PARTFUN1:def 6
.= Emax by JORDAN1F:6;
Emax`1 = Ebo by EUCLID:52
.= Ga*(len Ga,k)`1 by A3,A14,JORDAN1A:71;
then
A32: Gik <> LS.1 by A3,A17,A30,A31,JORDAN1G:7;
A33: len go >= 1+1 by TOPREAL1:def 8;
A34: Gij in rng US by A1,A4,A5,A10,A11,A13,JORDAN1G:4,JORDAN1J:40;
then
A35: go is_sequence_on Ga by JORDAN1G:4,JORDAN1J:38;
A36: len co >= 1+1 by TOPREAL1:def 8;
A37: Gik in rng LS by A4,A5,A9,A11,A14,A15,JORDAN1G:5,JORDAN1J:40;
then
A38: co is_sequence_on Ga by JORDAN1G:5,JORDAN1J:39;
reconsider go as non constant s.c.c. being_S-Seq FinSequence of TOP-REAL 2
by A33,A35,JGRAPH_1:12,JORDAN8:5;
reconsider co as non constant s.c.c. being_S-Seq FinSequence of TOP-REAL 2
by A36,A38,JGRAPH_1:12,JORDAN8:5;
A39: len go > 1 by A33,NAT_1:13;
then
A40: len go in dom go by FINSEQ_3:25;
then
A41: go/.len go = go.len go by PARTFUN1:def 6
.= Gij by A10,JORDAN3:24;
len co >= 1 by A36,XXREAL_0:2;
then 1 in dom co by FINSEQ_3:25;
then
A42: co/.1 = co.1 by PARTFUN1:def 6
.= Gik by A9,JORDAN3:23;
reconsider m = len go - 1 as Nat by A40,FINSEQ_3:26;
A43: m+1 = len go;
then
A44: len go-'1 = m by NAT_D:34;
A45: LSeg(go,m) c= L~go by TOPREAL3:19;
A46: L~go c= L~US by A10,JORDAN3:41;
then LSeg(go,m) c= L~US by A45;
then
A47: LSeg(go,m) /\ LSeg(Gik,Gij) c= {Gij} by A6,XBOOLE_1:26;
m >= 1 by A33,XREAL_1:19;
then
A48: LSeg(go,m) = LSeg(go/.m,Gij) by A41,A43,TOPREAL1:def 3;
{Gij} c= LSeg(go,m) /\ LSeg(Gik,Gij)
proof
let x be object;
A49: Gij in LSeg(Gik,Gij) by RLTOPSP1:68;
assume x in {Gij};
then
A50: x = Gij by TARSKI:def 1;
Gij in LSeg(go,m) by A48,RLTOPSP1:68;
hence thesis by A50,A49,XBOOLE_0:def 4;
end;
then
A51: LSeg(go,m) /\ LSeg(Gik,Gij) = {Gij} by A47;
A52: LSeg(co,1) c= L~co by TOPREAL3:19;
A53: L~co c= L~LS by A9,JORDAN3:42;
then LSeg(co,1) c= L~LS by A52;
then
A54: LSeg(co,1) /\ LSeg(Gik,Gij) c= {Gik} by A7,XBOOLE_1:26;
A55: LSeg(co,1) = LSeg(Gik,co/.(1+1)) by A36,A42,TOPREAL1:def 3;
{Gik} c= LSeg(co,1) /\ LSeg(Gik,Gij)
proof
let x be object;
A56: Gik in LSeg(Gik,Gij) by RLTOPSP1:68;
assume x in {Gik};
then
A57: x = Gik by TARSKI:def 1;
Gik in LSeg(co,1) by A55,RLTOPSP1:68;
hence thesis by A57,A56,XBOOLE_0:def 4;
end;
then
A58: LSeg(Gik,Gij) /\ LSeg(co,1) = {Gik} by A54;
A59: go/.1 = US/.1 by A10,SPRECT_3:22
.= Wmin by JORDAN1F:5;
then
A60: go/.1 = LS/.len LS by JORDAN1F:8
.= co/.len co by A9,JORDAN1J:35;
A61: rng go c= L~go by A33,SPPOL_2:18;
A62: rng co c= L~co by A36,SPPOL_2:18;
A63: {go/.1} c= L~go /\ L~co
proof
let x be object;
assume x in {go/.1};
then
A64: x = go/.1 by TARSKI:def 1;
then
A65: x in rng go by FINSEQ_6:42;
x in rng co by A60,A64,FINSEQ_6:168;
hence thesis by A61,A62,A65,XBOOLE_0:def 4;
end;
A66: LS.1 = LS/.1 by A27,PARTFUN1:def 6
.= Emax by JORDAN1F:6;
A67: [len Ga,j] in Indices Ga by A1,A13,A21,MATRIX_0:30;
L~go /\ L~co c= {go/.1}
proof
let x be object;
assume
A68: x in L~go /\ L~co;
then
A69: x in L~co by XBOOLE_0:def 4;
A70: now
assume x = Emax;
then
A71: Emax = Gik by A9,A66,A69,JORDAN1E:7;
Ga*(len Ga,j)`1 = Ebo by A1,A11,A13,JORDAN1A:71;
then Emax`1 <> Ebo by A3,A17,A67,A71,JORDAN1G:7;
hence contradiction by EUCLID:52;
end;
x in L~go by A68,XBOOLE_0:def 4;
then x in L~US /\ L~LS by A46,A53,A69,XBOOLE_0:def 4;
then x in {Wmin,Emax} by JORDAN1E:16;
then x = Wmin or x = Emax by TARSKI:def 2;
hence thesis by A59,A70,TARSKI:def 1;
end;
then
A72: L~go /\ L~co = {go/.1} by A63;
set W2 = go/.2;
A73: 2 in dom go by A33,FINSEQ_3:25;
A74: now
assume Gij`1 = Wbo;
then Ga*(1,k)`1 = Ga*(j,i)`1 by A3,A14,JORDAN1A:73;
hence contradiction by A1,A16,A22,JORDAN1G:7;
end;
go = mid(US,1,Gij..US) by A34,JORDAN1G:49
.= US|(Gij..US) by A34,FINSEQ_4:21,FINSEQ_6:116;
then
A75: W2 = US/.2 by A73,FINSEQ_4:70;
A76: Wmin in rng go by A59,FINSEQ_6:42;
set pion = <*Gij,Gik*>;
A77: now
let n be Nat;
assume n in dom pion;
then n in {1,2} by FINSEQ_1:2,89;
then n = 1 or n = 2 by TARSKI:def 2;
hence ex j,i be Nat st [j,i] in Indices Ga & pion/.n = Ga*(j,i)
by A16,A17,FINSEQ_4:17;
end;
A78: Gik <> Gij by A12,A16,A17,GOBOARD1:5;
Gik`2 = Ga*(1,i)`2 by A3,A4,A5,A14,GOBOARD5:1
.= Gij`2 by A1,A4,A5,A11,A13,GOBOARD5:1;
then LSeg(Gik,Gij) is horizontal by SPPOL_1:15;
then pion is being_S-Seq by A78,JORDAN1B:8;
then consider pion1 be FinSequence of TOP-REAL 2 such that
A79: pion1 is_sequence_on Ga and
A80: pion1 is being_S-Seq and
A81: L~pion = L~pion1 and
A82: pion/.1 = pion1/.1 and
A83: pion/.len pion = pion1/.len pion1 and
A84: len pion <= len pion1 by A77,GOBOARD3:2;
reconsider pion1 as being_S-Seq FinSequence of TOP-REAL 2 by A80;
set godo = go^'pion1^'co;
A85: 1+1 <= len Cage(C,n) by GOBOARD7:34,XXREAL_0:2;
A86: 1+1 <= len Rotate(Cage(C,n),Wmin) by GOBOARD7:34,XXREAL_0:2;
len (go^'pion1) >= len go by TOPREAL8:7;
then
A87: len (go^'pion1) >= 1+1 by A33,XXREAL_0:2;
then
A88: len (go^'pion1) > 1+0 by NAT_1:13;
A89: len godo >= len (go^'pion1) by TOPREAL8:7;
then
A90: 1+1 <= len godo by A87,XXREAL_0:2;
A91: US is_sequence_on Ga by JORDAN1G:4;
A92: go/.len go = pion1/.1 by A41,A82,FINSEQ_4:17;
then
A93: go^'pion1 is_sequence_on Ga by A35,A79,TOPREAL8:12;
A94: (go^'pion1)/.len (go^'pion1) = pion/.len pion by A83,FINSEQ_6:156
.= pion/.2 by FINSEQ_1:44
.= co/.1 by A42,FINSEQ_4:17;
then
A95: godo is_sequence_on Ga by A38,A93,TOPREAL8:12;
LSeg(pion1,1) c= L~<*Gij,Gik*> by A81,TOPREAL3:19;
then LSeg(pion1,1) c= LSeg(Gij,Gik) by SPPOL_2:21;
then
A96: LSeg(go,len go-'1) /\ LSeg(pion1,1) c= {Gij} by A44,A51,XBOOLE_1:27;
A97: len pion1 >= 1+1 by A84,FINSEQ_1:44;
{Gij} c= LSeg(go,m) /\ LSeg(pion1,1)
proof
let x be object;
assume x in {Gij};
then
A98: x = Gij by TARSKI:def 1;
A99: Gij in LSeg(go,m) by A48,RLTOPSP1:68;
Gij in LSeg(pion1,1) by A41,A92,A97,TOPREAL1:21;
hence thesis by A98,A99,XBOOLE_0:def 4;
end;
then LSeg(go,len go-'1) /\ LSeg(pion1,1) = { go/.len go } by A41,A44,A96;
then
A100: go^'pion1 is unfolded by A92,TOPREAL8:34;
len pion1 >= 2+0 by A84,FINSEQ_1:44;
then
A101: len pion1-2 >= 0 by XREAL_1:19;
len (go^'pion1)+1-1 = len go+len pion1-1 by FINSEQ_6:139;
then len (go^'pion1)-1 = len go + (len pion1-2)
.= len go + (len pion1-'2) by A101,XREAL_0:def 2;
then
A102: len (go^'pion1)-'1 = len go + (len pion1-'2) by XREAL_0:def 2;
A103: len pion1-1 >= 1 by A97,XREAL_1:19;
then
A104: len pion1-'1 = len pion1-1 by XREAL_0:def 2;
A105: len pion1-'2+1 = len pion1-2+1 by A101,XREAL_0:def 2
.= len pion1-'1 by A103,XREAL_0:def 2;
len pion1-1+1 <= len pion1;
then
A106: len pion1-'1 < len pion1 by A104,NAT_1:13;
LSeg(pion1,len pion1-'1) c= L~<*Gij,Gik*> by A81,TOPREAL3:19;
then LSeg(pion1,len pion1-'1) c= LSeg(Gij,Gik) by SPPOL_2:21;
then
A107: LSeg(pion1,len pion1-'1) /\ LSeg(co,1) c= {Gik} by A58,XBOOLE_1:27;
{Gik} c= LSeg(pion1,len pion1-'1) /\ LSeg(co,1)
proof
let x be object;
assume x in {Gik};
then
A108: x = Gik by TARSKI:def 1;
pion1/.(len pion1-'1+1) = pion/.2 by A83,A104,FINSEQ_1:44
.= Gik by FINSEQ_4:17;
then
A109: Gik in LSeg(pion1,len pion1-'1) by A103,A104,TOPREAL1:21;
Gik in LSeg(co,1) by A55,RLTOPSP1:68;
hence thesis by A108,A109,XBOOLE_0:def 4;
end;
then LSeg(pion1,len pion1-'1) /\ LSeg(co,1) = {Gik} by A107;
then
A110: LSeg(go^'pion1,len go+(len pion1-'2)) /\ LSeg(co,1) = {(go^'pion1)/.
len (go^'pion1)} by A42,A92,A94,A105,A106,TOPREAL8:31;
A111: (go^'pion1) is non trivial by A87,NAT_D:60;
A112: rng pion1 c= L~pion1 by A97,SPPOL_2:18;
A113: {pion1/.1} c= L~go /\ L~pion1
proof
let x be object;
assume x in {pion1/.1};
then
A114: x = pion1/.1 by TARSKI:def 1;
then
A115: x in rng pion1 by FINSEQ_6:42;
x in rng go by A92,A114,FINSEQ_6:168;
hence thesis by A61,A112,A115,XBOOLE_0:def 4;
end;
L~go /\ L~pion1 c= {pion1/.1}
proof
let x be object;
assume
A116: x in L~go /\ L~pion1;
then
A117: x in L~pion1 by XBOOLE_0:def 4;
x in L~go by A116,XBOOLE_0:def 4;
then x in L~pion1 /\ L~US by A46,A117,XBOOLE_0:def 4;
hence thesis by A6,A41,A81,A92,SPPOL_2:21;
end;
then
A118: L~go /\ L~pion1 = {pion1/.1} by A113;
then
A119: (go^'pion1) is s.n.c. by A92,JORDAN1J:54;
rng go /\ rng pion1 c= {pion1/.1} by A61,A112,A118,XBOOLE_1:27;
then
A120: go^'pion1 is one-to-one by JORDAN1J:55;
A121: pion/.len pion = pion/.2 by FINSEQ_1:44
.= co/.1 by A42,FINSEQ_4:17;
A122: {pion1/.len pion1} c= L~co /\ L~pion1
proof
let x be object;
assume x in {pion1/.len pion1};
then
A123: x = pion1/.len pion1 by TARSKI:def 1;
then
A124: x in rng pion1 by FINSEQ_6:168;
x in rng co by A83,A121,A123,FINSEQ_6:42;
hence thesis by A62,A112,A124,XBOOLE_0:def 4;
end;
L~co /\ L~pion1 c= {pion1/.len pion1}
proof
let x be object;
assume
A125: x in L~co /\ L~pion1;
then
A126: x in L~pion1 by XBOOLE_0:def 4;
x in L~co by A125,XBOOLE_0:def 4;
then x in L~pion1 /\ L~LS by A53,A126,XBOOLE_0:def 4;
hence thesis by A7,A42,A81,A83,A121,SPPOL_2:21;
end;
then
A127: L~co /\ L~pion1 = {pion1/.len pion1} by A122;
A128: L~(go^'pion1) /\ L~co = (L~go \/ L~pion1) /\ L~co by A92,TOPREAL8:35
.= {go/.1} \/ {co/.1} by A72,A83,A121,A127,XBOOLE_1:23
.= {(go^'pion1)/.1} \/ {co/.1} by FINSEQ_6:155
.= {(go^'pion1)/.1,co/.1} by ENUMSET1:1;
co/.len co = (go^'pion1)/.1 by A60,FINSEQ_6:155;
then reconsider
godo as non constant standard special_circular_sequence by A90,A94,A95,A100
,A102,A110,A111,A119,A120,A128,JORDAN8:4,5,TOPREAL8:11,33,34;
A129: UA is_an_arc_of W-min C,E-max C by JORDAN6:def 8;
then
A130: UA is connected by JORDAN6:10;
A131: W-min C in UA by A129,TOPREAL1:1;
A132: E-max C in UA by A129,TOPREAL1:1;
set ff = Rotate(Cage(C,n),Wmin);
Wmin in rng Cage(C,n) by SPRECT_2:43;
then
A133: ff/.1 = Wmin by FINSEQ_6:92;
A134: L~ff = L~Cage(C,n) by REVROT_1:33;
then (W-max L~ff)..ff > 1 by A133,SPRECT_5:22;
then (N-min L~ff)..ff > 1 by A133,A134,SPRECT_5:23,XXREAL_0:2;
then (N-max L~ff)..ff > 1 by A133,A134,SPRECT_5:24,XXREAL_0:2;
then
A135: Emax..ff > 1 by A133,A134,SPRECT_5:25,XXREAL_0:2;
A136: now
assume
A137: Gij..US <= 1;
Gij..US >= 1 by A34,FINSEQ_4:21;
then Gij..US = 1 by A137,XXREAL_0:1;
then Gij = US/.1 by A34,FINSEQ_5:38;
hence contradiction by A19,A24,JORDAN1F:5;
end;
A138: Cage(C,n) is_sequence_on Ga by JORDAN9:def 1;
then
A139: ff is_sequence_on Ga by REVROT_1:34;
A140: right_cell(godo,1,Ga)\L~godo c= RightComp godo by A90,A95,JORDAN9:27;
A141: L~godo = L~(go^'pion1) \/ L~co by A94,TOPREAL8:35
.= L~go \/ L~pion1 \/ L~co by A92,TOPREAL8:35;
A142: L~Cage(C,n) = L~US \/ L~LS by JORDAN1E:13;
then
A143: L~US c= L~Cage(C,n) by XBOOLE_1:7;
A144: L~LS c= L~Cage(C,n) by A142,XBOOLE_1:7;
A145: L~go c= L~Cage(C,n) by A46,A143;
A146: L~co c= L~Cage(C,n) by A53,A144;
A147: W-min C in C by SPRECT_1:13;
A148: L~pion = LSeg(Gik,Gij) by SPPOL_2:21;
A149: now
assume W-min C in L~godo;
then
A150: W-min C in L~go \/ L~pion1 or W-min C in L~co by A141,XBOOLE_0:def 3;
per cases by A150,XBOOLE_0:def 3;
suppose
W-min C in L~go;
then C meets L~Cage(C,n) by A145,A147,XBOOLE_0:3;
hence contradiction by JORDAN10:5;
end;
suppose
W-min C in L~pion1;
hence contradiction by A8,A81,A131,A148,XBOOLE_0:3;
end;
suppose
W-min C in L~co;
then C meets L~Cage(C,n) by A146,A147,XBOOLE_0:3;
hence contradiction by JORDAN10:5;
end;
end;
right_cell(Rotate(Cage(C,n),Wmin),1) = right_cell(ff,1,GoB ff) by A86,
JORDAN1H:23
.= right_cell(ff,1,GoB Cage(C,n)) by REVROT_1:28
.= right_cell(ff,1,Ga) by JORDAN1H:44
.= right_cell(ff-:Emax,1,Ga) by A135,A139,JORDAN1J:53
.= right_cell(US,1,Ga) by JORDAN1E:def 1
.= right_cell(R_Cut(US,Gij),1,Ga) by A34,A91,A136,JORDAN1J:52
.= right_cell(go^'pion1,1,Ga) by A39,A93,JORDAN1J:51
.= right_cell(godo,1,Ga) by A88,A95,JORDAN1J:51;
then W-min C in right_cell(godo,1,Ga) by JORDAN1I:6;
then
A151: W-min C in right_cell(godo,1,Ga)\L~godo by A149,XBOOLE_0:def 5;
A152: godo/.1 = (go^'pion1)/.1 by FINSEQ_6:155
.= Wmin by A59,FINSEQ_6:155;
A153: len US >= 2 by A18,XXREAL_0:2;
A154: godo/.2 = (go^'pion1)/.2 by A87,FINSEQ_6:159
.= US/.2 by A33,A75,FINSEQ_6:159
.= (US^'LS)/.2 by A153,FINSEQ_6:159
.= Rotate(Cage(C,n),Wmin)/.2 by JORDAN1E:11;
A155: L~go \/ L~co is compact by COMPTS_1:10;
Wmin in L~go \/ L~co by A61,A76,XBOOLE_0:def 3;
then
A156: W-min (L~go \/ L~co) = Wmin by A145,A146,A155,JORDAN1J:21,XBOOLE_1:8;
A157: (W-min (L~go \/ L~co))`1 = W-bound (L~go \/ L~co) by EUCLID:52;
A158: Wmin`1 = Wbo by EUCLID:52;
A159: Gij`1 <= Gik`1 by A1,A2,A3,A4,A5,SPRECT_3:13;
then W-bound LSeg(Gik,Gij) = Gij`1 by SPRECT_1:54;
then
A160: W-bound L~pion1 = Gij`1 by A81,SPPOL_2:21;
Gij`1 >= Wbo by A10,A143,PSCOMP_1:24;
then Gij`1 > Wbo by A74,XXREAL_0:1;
then W-min (L~go\/L~co\/L~pion1) = W-min (L~go \/ L~co) by A155,A156,A157
,A158,A160,JORDAN1J:33;
then
A161: W-min L~godo = Wmin by A141,A156,XBOOLE_1:4;
A162: rng godo c= L~godo by A87,A89,SPPOL_2:18,XXREAL_0:2;
2 in dom godo by A90,FINSEQ_3:25;
then
A163: godo/.2 in rng godo by PARTFUN2:2;
godo/.2 in W-most L~Cage(C,n) by A154,JORDAN1I:25;
then (godo/.2)`1 = (W-min L~godo)`1 by A161,PSCOMP_1:31
.= W-bound L~godo by EUCLID:52;
then godo/.2 in W-most L~godo by A162,A163,SPRECT_2:12;
then Rotate(godo,W-min L~godo)/.2 in W-most L~godo by A152,A161,FINSEQ_6:89;
then reconsider godo as clockwise_oriented non constant standard
special_circular_sequence by JORDAN1I:25;
len US in dom US by FINSEQ_5:6;
then
A164: US.len US = US/.len US by PARTFUN1:def 6
.= Emax by JORDAN1F:7;
A165: east_halfline E-max C misses L~go
proof
assume east_halfline E-max C meets L~go;
then consider p be object such that
A166: p in east_halfline E-max C and
A167: p in L~go by XBOOLE_0:3;
reconsider p as Point of TOP-REAL 2 by A166;
p in L~US by A46,A167;
then p in east_halfline E-max C /\ L~Cage(C,n) by A143,A166,XBOOLE_0:def 4;
then
A168: p`1 = Ebo by JORDAN1A:83,PSCOMP_1:50;
then
A169: p = Emax by A46,A167,JORDAN1J:46;
then Emax = Gij by A10,A164,A167,JORDAN1J:43;
then Gij`1 = Ga*(len Ga,k)`1 by A3,A14,A168,A169,JORDAN1A:71;
hence contradiction by A2,A3,A16,A30,JORDAN1G:7;
end;
now
assume east_halfline E-max C meets L~godo;
then
A170: east_halfline E-max C meets (L~go \/ L~pion1) or east_halfline E-max
C meets L~co by A141,XBOOLE_1:70;
per cases by A170,XBOOLE_1:70;
suppose
east_halfline E-max C meets L~go;
hence contradiction by A165;
end;
suppose
east_halfline E-max C meets L~pion1;
then consider p be object such that
A171: p in east_halfline E-max C and
A172: p in L~pion1 by XBOOLE_0:3;
reconsider p as Point of TOP-REAL 2 by A171;
A173: p`2 = (E-max C)`2 by A171,TOPREAL1:def 11;
k+1 <= len Ga by A3,NAT_1:13;
then k+1-1 <= len Ga-1 by XREAL_1:9;
then
A174: k <= len Ga-'1 by XREAL_0:def 2;
len Ga-'1 <= len Ga by NAT_D:35;
then
A175: Gik`1 <= Ga*(len Ga-'1,1)`1 by A4,A5,A11,A14,A21,A174,JORDAN1A:18;
p`1 <= Gik`1 by A81,A148,A159,A172,TOPREAL1:3;
then p`1 <= Ga*(len Ga-'1,1)`1 by A175,XXREAL_0:2;
then p`1 <= E-bound C by A21,JORDAN8:12;
then
A176: p`1 <= (E-max C)`1 by EUCLID:52;
p`1 >= (E-max C)`1 by A171,TOPREAL1:def 11;
then p`1 = (E-max C)`1 by A176,XXREAL_0:1;
then p = E-max C by A173,TOPREAL3:6;
hence contradiction by A8,A81,A132,A148,A172,XBOOLE_0:3;
end;
suppose
east_halfline E-max C meets L~co;
then consider p be object such that
A177: p in east_halfline E-max C and
A178: p in L~co by XBOOLE_0:3;
reconsider p as Point of TOP-REAL 2 by A177;
A179: p in LSeg(co,Index(p,co)) by A178,JORDAN3:9;
consider t be Nat such that
A180: t in dom LS and
A181: LS.t = Gik by A37,FINSEQ_2:10;
1 <= t by A180,FINSEQ_3:25;
then
A182: 1 < t by A32,A181,XXREAL_0:1;
t <= len LS by A180,FINSEQ_3:25;
then Index(Gik,LS)+1 = t by A181,A182,JORDAN3:12;
then
A183: len L_Cut(LS,Gik) = len LS-Index(Gik,LS) by A9,A181,JORDAN3:26;
Index(p,co) < len co by A178,JORDAN3:8;
then Index(p,co) < len LS-'Index(Gik,LS) by A183,XREAL_0:def 2;
then Index(p,co)+1 <= len LS-'Index(Gik,LS) by NAT_1:13;
then
A184: Index(p,co) <= len LS-'Index(Gik,LS)-1 by XREAL_1:19;
A185: co = mid(LS,Gik..LS,len LS) by A37,JORDAN1J:37;
p in L~LS by A53,A178;
then p in east_halfline E-max C /\ L~Cage(C,n) by A144,A177,
XBOOLE_0:def 4;
then
A186: p`1 = Ebo by JORDAN1A:83,PSCOMP_1:50;
A187: Index(Gik,LS)+1 = Gik..LS by A32,A37,JORDAN1J:56;
0+Index(Gik,LS) < len LS by A9,JORDAN3:8;
then len LS-Index(Gik,LS) > 0 by XREAL_1:20;
then Index(p,co) <= len LS-Index(Gik,LS)-1 by A184,XREAL_0:def 2;
then Index(p,co) <= len LS-Gik..LS by A187;
then Index(p,co) <= len LS-'Gik..LS by XREAL_0:def 2;
then
A188: Index(p,co) < len LS-'(Gik..LS)+1 by NAT_1:13;
A189: 1<=Index(p,co) by A178,JORDAN3:8;
A190: Gik..LS<=len LS by A37,FINSEQ_4:21;
Gik..LS <> len LS by A29,A37,FINSEQ_4:19;
then
A191: Gik..LS < len LS by A190,XXREAL_0:1;
A192: 1+1 <= len LS by A25,XXREAL_0:2;
then
A193: 2 in dom LS by FINSEQ_3:25;
set tt = Index(p,co)+(Gik..LS)-'1;
set RC = Rotate(Cage(C,n),Emax);
A194: E-max C in right_cell(RC,1) by JORDAN1I:7;
A195: GoB RC = GoB Cage(C,n) by REVROT_1:28
.= Ga by JORDAN1H:44;
A196: L~RC = L~Cage(C,n) by REVROT_1:33;
consider g2 be Nat such that
A197: 1 <= g2 and
A198: g2 <= width Ga and
A199: Emax = Ga*(len Ga,g2) by JORDAN1D:25;
A200: len Ga >= 4 by JORDAN8:10;
then len Ga >= 1 by XXREAL_0:2;
then
A201: [len Ga,g2] in Indices Ga by A197,A198,MATRIX_0:30;
A202: len RC = len Cage(C,n) by FINSEQ_6:179;
LS = RC-:Wmin by JORDAN1G:18;
then
A203: LSeg(LS,1) = LSeg(RC,1) by A192,SPPOL_2:9;
A204: Emax in rng Cage(C,n) by SPRECT_2:46;
RC is_sequence_on Ga by A138,REVROT_1:34;
then consider ii,g be Nat such that
A205: [ii,g+1] in Indices Ga and
A206: [ii,g] in Indices Ga and
A207: RC/.1 = Ga*(ii,g+1) and
A208: RC/.(1+1) = Ga*(ii,g) by A85,A196,A202,A204,FINSEQ_6:92,JORDAN1I:23;
A209: g+1+1 <> g;
A210: 1 <= g by A206,MATRIX_0:32;
RC/.1 = E-max L~RC by A196,A204,FINSEQ_6:92;
then
A211: ii = len Ga by A196,A205,A207,A199,A201,GOBOARD1:5;
then ii-1 >= 4-1 by A200,XREAL_1:9;
then
A212: ii-1 >= 1 by XXREAL_0:2;
then
A213: 1 <= ii-'1 by XREAL_0:def 2;
A214: g <= width Ga by A206,MATRIX_0:32;
then
A215: Ga*(len Ga,g)`1 = Ebo by A11,A210,JORDAN1A:71;
A216: g+1 <= width Ga by A205,MATRIX_0:32;
ii+1 <> ii;
then
A217: right_cell(RC,1) = cell(Ga,ii-'1,g) by A85,A202,A195,A205,A206,A207,A208
,A209,GOBOARD5:def 6;
A218: ii <= len Ga by A206,MATRIX_0:32;
A219: 1 <= ii by A206,MATRIX_0:32;
A220: ii <= len Ga by A205,MATRIX_0:32;
A221: 1 <= g+1 by A205,MATRIX_0:32;
then
A222: Ebo = Ga*(len Ga,g+1)`1 by A11,A216,JORDAN1A:71;
A223: 1 <= ii by A205,MATRIX_0:32;
then
A224: ii-'1+1 = ii by XREAL_1:235;
then
A225: ii-'1 < len Ga by A220,NAT_1:13;
then
A226: Ga*(ii-'1,g+1)`2 = Ga*(1,g+1)`2 by A221,A216,A213,GOBOARD5:1
.= Ga*(ii,g+1)`2 by A223,A220,A221,A216,GOBOARD5:1;
A227: (E-max C)`2 = p`2 by A177,TOPREAL1:def 11;
then
A228: p`2 <= Ga*(ii-'1,g+1)`2 by A194,A220,A216,A210,A217,A224,A212,JORDAN9:17;
A229: Ga*(ii-'1,g)`2 = Ga*(1,g)`2 by A210,A214,A213,A225,GOBOARD5:1
.= Ga*(ii,g)`2 by A219,A218,A210,A214,GOBOARD5:1;
Ga*(ii-'1,g)`2 <= p`2 by A227,A194,A220,A216,A210,A217,A224,A212,
JORDAN9:17;
then p in LSeg(RC/.1,RC/.(1+1)) by A186,A207,A208,A211,A228,A229,A226
,A215,A222,GOBOARD7:7;
then
A230: p in LSeg(LS,1) by A85,A203,A202,TOPREAL1:def 3;
1<=Gik..LS by A37,FINSEQ_4:21;
then
A231: LSeg(mid(LS,Gik..LS,len LS),Index(p,co)) = LSeg(LS,Index(p,co)+(
Gik..LS)-'1) by A191,A189,A188,JORDAN4:19;
1<=Index(Gik,LS) by A9,JORDAN3:8;
then
A232: 1+1 <= Gik..LS by A187,XREAL_1:7;
then Index(p,co)+Gik..LS >= 1+1+1 by A189,XREAL_1:7;
then Index(p,co)+Gik..LS-1 >= 1+1+1-1 by XREAL_1:9;
then
A233: tt >= 1+1 by XREAL_0:def 2;
now
per cases by A233,XXREAL_0:1;
suppose
tt > 1+1;
then LSeg(LS,1) misses LSeg(LS,tt) by TOPREAL1:def 7;
hence contradiction by A230,A179,A185,A231,XBOOLE_0:3;
end;
suppose
A234: tt = 1+1;
then 1+1 = Index(p,co)+(Gik..LS)-1 by XREAL_0:def 2;
then 1+1+1 = Index(p,co)+(Gik..LS);
then
A235: Gik..LS = 2 by A189,A232,JORDAN1E:6;
LSeg(LS,1) /\ LSeg(LS,tt) = {LS/.2} by A25,A234,TOPREAL1:def 6;
then p in {LS/.2} by A230,A179,A185,A231,XBOOLE_0:def 4;
then
A236: p = LS/.2 by TARSKI:def 1;
then
A237: p in rng LS by A193,PARTFUN2:2;
p..LS = 2 by A193,A236,FINSEQ_5:41;
then p = Gik by A37,A235,A237,FINSEQ_5:9;
then Gik`1 = Ebo by A236,JORDAN1G:32;
then Gik`1 = Ga*(len Ga,j)`1 by A1,A11,A13,JORDAN1A:71;
hence contradiction by A3,A17,A67,JORDAN1G:7;
end;
end;
hence contradiction;
end;
end;
then east_halfline E-max C c= (L~godo)` by SUBSET_1:23;
then consider W be Subset of TOP-REAL 2 such that
A238: W is_a_component_of (L~godo)` and
A239: east_halfline E-max C c= W by GOBOARD9:3;
W is not bounded by A239,JORDAN2C:121,RLTOPSP1:42;
then W is_outside_component_of L~godo by A238,JORDAN2C:def 3;
then W c= UBD L~godo by JORDAN2C:23;
then
A240: east_halfline E-max C c= UBD L~godo by A239;
E-max C in east_halfline E-max C by TOPREAL1:38;
then E-max C in UBD L~godo by A240;
then E-max C in LeftComp godo by GOBRD14:36;
then UA meets L~godo by A130,A131,A132,A140,A151,JORDAN1J:36;
then
A241: UA meets (L~go \/ L~pion1) or UA meets L~co by A141,XBOOLE_1:70;
A242: UA c= C by JORDAN6:61;
per cases by A241,XBOOLE_1:70;
suppose
UA meets L~go;
then UA meets L~Cage(C,n) by A46,A143,XBOOLE_1:1,63;
hence contradiction by A242,JORDAN10:5,XBOOLE_1:63;
end;
suppose
UA meets L~pion1;
hence contradiction by A8,A81,A148;
end;
suppose
UA meets L~co;
then UA meets L~Cage(C,n) by A53,A144,XBOOLE_1:1,63;
hence contradiction by A242,JORDAN10:5,XBOOLE_1:63;
end;
end;
theorem Th38:
for C be Simple_closed_curve for i,j,k be Nat st 1 <
j & j <= k & k < len Gauge(C,n) & 1 <= i & i <= width Gauge(C,n) & Gauge(C,n)*(
j,i) in L~Upper_Seq(C,n) & Gauge(C,n)*(k,i) in L~Lower_Seq(C,n) holds LSeg(
Gauge(C,n)*(j,i),Gauge(C,n)*(k,i)) meets Lower_Arc C
proof
let C be Simple_closed_curve;
let i,j,k be Nat;
assume that
A1: 1 < j and
A2: j <= k and
A3: k < len Gauge(C,n) and
A4: 1 <= i and
A5: i <= width Gauge(C,n) and
A6: Gauge(C,n)*(j,i) in L~Upper_Seq(C,n) and
A7: Gauge(C,n)*(k,i) in L~Lower_Seq(C,n);
consider j1,k1 be Nat such that
A8: j <= j1 and
A9: j1 <= k1 and
A10: k1 <= k and
A11: LSeg(Gauge(C,n)*(j1,i),Gauge(C,n)*(k1,i)) /\ L~Upper_Seq(C,n) = {
Gauge(C,n)*(j1,i)} and
A12: LSeg(Gauge(C,n)*(j1,i),Gauge(C,n)*(k1,i)) /\ L~Lower_Seq(C,n) = {
Gauge(C,n)*(k1,i)} by A1,A2,A3,A4,A5,A6,A7,Th20;
A13: k1 < len Gauge(C,n) by A3,A10,XXREAL_0:2;
1 < j1 by A1,A8,XXREAL_0:2;
then
LSeg(Gauge(C,n)*(j1,i),Gauge(C,n)*(k1,i)) meets Lower_Arc C by A4,A5,A9,A11
,A12,A13,Th36;
hence thesis by A1,A3,A4,A5,A8,A9,A10,Th6,XBOOLE_1:63;
end;
theorem Th39:
for C be Simple_closed_curve for i,j,k be Nat st 1 <
j & j <= k & k < len Gauge(C,n) & 1 <= i & i <= width Gauge(C,n) & Gauge(C,n)*(
j,i) in L~Upper_Seq(C,n) & Gauge(C,n)*(k,i) in L~Lower_Seq(C,n) holds LSeg(
Gauge(C,n)*(j,i),Gauge(C,n)*(k,i)) meets Upper_Arc C
proof
let C be Simple_closed_curve;
let i,j,k be Nat;
assume that
A1: 1 < j and
A2: j <= k and
A3: k < len Gauge(C,n) and
A4: 1 <= i and
A5: i <= width Gauge(C,n) and
A6: Gauge(C,n)*(j,i) in L~Upper_Seq(C,n) and
A7: Gauge(C,n)*(k,i) in L~Lower_Seq(C,n);
consider j1,k1 be Nat such that
A8: j <= j1 and
A9: j1 <= k1 and
A10: k1 <= k and
A11: LSeg(Gauge(C,n)*(j1,i),Gauge(C,n)*(k1,i)) /\ L~Upper_Seq(C,n) = {
Gauge(C,n)*(j1,i)} and
A12: LSeg(Gauge(C,n)*(j1,i),Gauge(C,n)*(k1,i)) /\ L~Lower_Seq(C,n) = {
Gauge(C,n)*(k1,i)} by A1,A2,A3,A4,A5,A6,A7,Th20;
A13: k1 < len Gauge(C,n) by A3,A10,XXREAL_0:2;
1 < j1 by A1,A8,XXREAL_0:2;
then
LSeg(Gauge(C,n)*(j1,i),Gauge(C,n)*(k1,i)) meets Upper_Arc C by A4,A5,A9,A11
,A12,A13,Th37;
hence thesis by A1,A3,A4,A5,A8,A9,A10,Th6,XBOOLE_1:63;
end;
theorem Th40:
for C be Simple_closed_curve for i,j,k be Nat st 1 <
j & j <= k & k < len Gauge(C,n) & 1 <= i & i <= width Gauge(C,n) & n > 0 &
Gauge(C,n)*(j,i) in Upper_Arc L~Cage(C,n) & Gauge(C,n)*(k,i) in Lower_Arc L~
Cage(C,n) holds LSeg(Gauge(C,n)*(j,i),Gauge(C,n)*(k,i)) meets Lower_Arc C
proof
let C be Simple_closed_curve;
let i,j,k be Nat;
assume that
A1: 1 < j and
A2: j <= k and
A3: k < len Gauge(C,n) and
A4: 1 <= i and
A5: i <= width Gauge(C,n) and
A6: n > 0 and
A7: Gauge(C,n)*(j,i) in Upper_Arc L~Cage(C,n) and
A8: Gauge(C,n)*(k,i) in Lower_Arc L~Cage(C,n);
A9: L~Lower_Seq(C,n) = Lower_Arc L~Cage(C,n) by A6,JORDAN1G:56;
L~Upper_Seq(C,n) = Upper_Arc L~Cage(C,n) by A6,JORDAN1G:55;
hence thesis by A1,A2,A3,A4,A5,A7,A8,A9,Th38;
end;
theorem Th41:
for C be Simple_closed_curve for i,j,k be Nat st 1 <
j & j <= k & k < len Gauge(C,n) & 1 <= i & i <= width Gauge(C,n) & n > 0 &
Gauge(C,n)*(j,i) in Upper_Arc L~Cage(C,n) & Gauge(C,n)*(k,i) in Lower_Arc L~
Cage(C,n) holds LSeg(Gauge(C,n)*(j,i),Gauge(C,n)*(k,i)) meets Upper_Arc C
proof
let C be Simple_closed_curve;
let i,j,k be Nat;
assume that
A1: 1 < j and
A2: j <= k and
A3: k < len Gauge(C,n) and
A4: 1 <= i and
A5: i <= width Gauge(C,n) and
A6: n > 0 and
A7: Gauge(C,n)*(j,i) in Upper_Arc L~Cage(C,n) and
A8: Gauge(C,n)*(k,i) in Lower_Arc L~Cage(C,n);
A9: L~Lower_Seq(C,n) = Lower_Arc L~Cage(C,n) by A6,JORDAN1G:56;
L~Upper_Seq(C,n) = Upper_Arc L~Cage(C,n) by A6,JORDAN1G:55;
hence thesis by A1,A2,A3,A4,A5,A7,A8,A9,Th39;
end;
theorem
for C be Simple_closed_curve for j,k be Nat holds 1 < j & j
<= k & k < len Gauge(C,n+1) & Gauge(C,n+1)*(j,Center Gauge(C,n+1)) in Upper_Arc
L~Cage(C,n+1) & Gauge(C,n+1)*(k,Center Gauge(C,n+1)) in Lower_Arc L~Cage(C,n+1)
implies LSeg(Gauge(C,n+1)*(j,Center Gauge(C,n+1)), Gauge(C,n+1)*(k,Center Gauge
(C,n+1))) meets Lower_Arc C
proof
let C be Simple_closed_curve;
let j,k be Nat;
assume that
A1: 1 < j and
A2: j <= k and
A3: k < len Gauge(C,n+1) and
A4: Gauge(C,n+1)*(j,Center Gauge(C,n+1)) in Upper_Arc L~Cage(C,n+1) and
A5: Gauge(C,n+1)*(k,Center Gauge(C,n+1)) in Lower_Arc L~Cage(C,n+1);
A6: len Gauge(C,n+1) >= 4 by JORDAN8:10;
then len Gauge(C,n+1) >= 3 by XXREAL_0:2;
then Center Gauge(C,n+1) < len Gauge(C,n+1) by JORDAN1B:15;
then
A7: Center Gauge(C,n+1) < width Gauge(C,n+1) by JORDAN8:def 1;
len Gauge(C,n+1) >= 2 by A6,XXREAL_0:2;
then 1 < Center Gauge(C,n+1) by JORDAN1B:14;
hence thesis by A1,A2,A3,A4,A5,A7,Th40;
end;
theorem
for C be Simple_closed_curve for j,k be Nat holds 1 < j & j
<= k & k < len Gauge(C,n+1) & Gauge(C,n+1)*(j,Center Gauge(C,n+1)) in Upper_Arc
L~Cage(C,n+1) & Gauge(C,n+1)*(k,Center Gauge(C,n+1)) in Lower_Arc L~Cage(C,n+1)
implies LSeg(Gauge(C,n+1)*(j,Center Gauge(C,n+1)), Gauge(C,n+1)*(k,Center Gauge
(C,n+1))) meets Upper_Arc C
proof
let C be Simple_closed_curve;
let j,k be Nat;
assume that
A1: 1 < j and
A2: j <= k and
A3: k < len Gauge(C,n+1) and
A4: Gauge(C,n+1)*(j,Center Gauge(C,n+1)) in Upper_Arc L~Cage(C,n+1) and
A5: Gauge(C,n+1)*(k,Center Gauge(C,n+1)) in Lower_Arc L~Cage(C,n+1);
A6: len Gauge(C,n+1) >= 4 by JORDAN8:10;
then len Gauge(C,n+1) >= 3 by XXREAL_0:2;
then Center Gauge(C,n+1) < len Gauge(C,n+1) by JORDAN1B:15;
then
A7: Center Gauge(C,n+1) < width Gauge(C,n+1) by JORDAN8:def 1;
len Gauge(C,n+1) >= 2 by A6,XXREAL_0:2;
then 1 < Center Gauge(C,n+1) by JORDAN1B:14;
hence thesis by A1,A2,A3,A4,A5,A7,Th41;
end;
theorem Th44:
for C be Simple_closed_curve for i1,i2,j,k be Nat st
1 < i1 & i1 <= i2 & i2 < len Gauge(C,n) & 1 <= j & j <= k & k <= width Gauge(C,
n) & (LSeg(Gauge(C,n)*(i1,j),Gauge(C,n)*(i1,k)) \/ LSeg(Gauge(C,n)*(i1,k),Gauge
(C,n)*(i2,k))) /\ L~Upper_Seq(C,n) = {Gauge(C,n)*(i2,k)} & (LSeg(Gauge(C,n)*(i1
,j),Gauge(C,n)*(i1,k)) \/ LSeg(Gauge(C,n)*(i1,k),Gauge(C,n)*(i2,k))) /\ L~
Lower_Seq(C,n) = {Gauge(C,n)*(i1,j)} holds (LSeg(Gauge(C,n)*(i1,j),Gauge(C,n)*(
i1,k)) \/ LSeg(Gauge(C,n)*(i1,k),Gauge(C,n)*(i2,k))) meets Upper_Arc C
proof
let C be Simple_closed_curve;
let i1,i2,j,k be Nat;
set G = Gauge(C,n);
set pio = LSeg(G*(i1,j),G*(i1,k));
set poz = LSeg(G*(i1,k),G*(i2,k));
set US = Upper_Seq(C,n);
set LS = Lower_Seq(C,n);
assume that
A1: 1 < i1 and
A2: i1 <= i2 and
A3: i2 < len G and
A4: 1 <= j and
A5: j <= k and
A6: k <= width G and
A7: (pio \/ poz) /\ L~US = {G*(i2,k)} and
A8: (pio \/ poz) /\ L~LS = {G*(i1,j)} and
A9: (pio \/ poz) misses Upper_Arc C;
set Gij = G*(i1,j);
A10: j <= width G by A5,A6,XXREAL_0:2;
A11: i1 < len G by A2,A3,XXREAL_0:2;
then
A12: [i1,j] in Indices G by A1,A4,A10,MATRIX_0:30;
set Gi1k = G*(i1,k);
set Gik = G*(i2,k);
A13: L~<*Gik,Gi1k,Gij*> = poz \/ pio by TOPREAL3:16;
len G >= 4 by JORDAN8:10;
then
A14: len G >= 1 by XXREAL_0:2;
then
A15: [len G,j] in Indices G by A4,A10,MATRIX_0:30;
A16: 1 <= k by A4,A5,XXREAL_0:2;
then
A17: [1,k] in Indices G by A6,A14,MATRIX_0:30;
A18: 1 < i2 by A1,A2,XXREAL_0:2;
then
A19: [i2,k] in Indices G by A3,A6,A16,MATRIX_0:30;
A20: Gi1k`2 = G*(1,k)`2 by A1,A6,A11,A16,GOBOARD5:1
.= Gik`2 by A3,A6,A18,A16,GOBOARD5:1;
Gi1k`1 = G*(i1,1)`1 by A1,A6,A11,A16,GOBOARD5:2
.= Gij`1 by A1,A4,A11,A10,GOBOARD5:2;
then
A21: Gi1k = |[Gij`1,Gik`2]| by A20,EUCLID:53;
A22: [len G,k] in Indices G by A6,A16,A14,MATRIX_0:30;
A23: [i1,j] in Indices G by A1,A4,A11,A10,MATRIX_0:30;
set Wbo = W-bound L~Cage(C,n);
set Wmin = W-min L~Cage(C,n);
A24: len G = width G by JORDAN8:def 1;
set Ebo = E-bound L~Cage(C,n);
set Emax = E-max L~Cage(C,n);
A25: len LS >= 1+2 by JORDAN1E:15;
then
A26: len LS >= 1 by XXREAL_0:2;
then
A27: 1 in dom LS by FINSEQ_3:25;
then
A28: LS.1 = LS/.1 by PARTFUN1:def 6
.= Emax by JORDAN1F:6;
len LS in dom LS by A26,FINSEQ_3:25;
then
A29: LS.len LS = LS/.len LS by PARTFUN1:def 6
.= Wmin by JORDAN1F:8;
set co = L_Cut(LS,Gij);
Gij in {Gij} by TARSKI:def 1;
then
A30: Gij in L~LS by A8,XBOOLE_0:def 4;
Wmin`1 = Wbo by EUCLID:52
.= G*(1,k)`1 by A6,A16,A24,JORDAN1A:73;
then
A31: Gij <> LS.len LS by A1,A17,A29,A12,JORDAN1G:7;
then reconsider co as being_S-Seq FinSequence of TOP-REAL 2 by A30,JORDAN3:34
;
A32: Gij in rng LS by A1,A4,A11,A30,A10,JORDAN1G:5,JORDAN1J:40;
then
A33: co is_sequence_on G by JORDAN1G:5,JORDAN1J:39;
Emax`1 = Ebo by EUCLID:52
.= G*(len G,k)`1 by A6,A16,A24,JORDAN1A:71;
then
A34: Gij <> LS.1 by A2,A3,A12,A22,A28,JORDAN1G:7;
A35: len co >= 1+1 by TOPREAL1:def 8;
then reconsider
co as non constant s.c.c. being_S-Seq FinSequence of TOP-REAL 2
by A33,JGRAPH_1:12,JORDAN8:5;
A36: L~co c= L~LS by A30,JORDAN3:42;
A37: [1,j] in Indices G by A4,A10,A14,MATRIX_0:30;
A38: now
assume Gij`1 = Wbo;
then G*(1,j)`1 = G*(i1,j)`1 by A4,A10,A24,JORDAN1A:73;
hence contradiction by A1,A23,A37,JORDAN1G:7;
end;
set pion = <*Gik,Gi1k,Gij*>;
A39: Gi1k in poz by RLTOPSP1:68;
set UA = Upper_Arc C;
A40: Gi1k in pio by RLTOPSP1:68;
set go = R_Cut(US,Gik);
A41: len US >= 3 by JORDAN1E:15;
then len US >= 1 by XXREAL_0:2;
then 1 in dom US by FINSEQ_3:25;
then
A42: US.1 = US/.1 by PARTFUN1:def 6
.= Wmin by JORDAN1F:5;
A43: [i1,k] in Indices G by A1,A6,A11,A16,MATRIX_0:30;
A44: now
let n be Nat;
assume n in dom pion;
then n in {1,2,3} by FINSEQ_1:89,FINSEQ_3:1;
then n = 1 or n = 2 or n = 3 by ENUMSET1:def 1;
hence
ex i,j be Nat st [i,j] in Indices G & pion/.n = G*(i,j) by A23
,A19,A43,FINSEQ_4:18;
end;
Gik in {Gik} by TARSKI:def 1;
then
A45: Gik in L~US by A7,XBOOLE_0:def 4;
Wmin`1 = Wbo by EUCLID:52
.= G*(1,k)`1 by A6,A16,A24,JORDAN1A:73;
then
A46: Gik <> US.1 by A1,A2,A19,A42,A17,JORDAN1G:7;
then reconsider go as being_S-Seq FinSequence of TOP-REAL 2 by A45,JORDAN3:35
;
A47: Gik in rng US by A3,A6,A18,A45,A16,JORDAN1G:4,JORDAN1J:40;
then
A48: go is_sequence_on G by JORDAN1G:4,JORDAN1J:38;
len co >= 1 by A35,XXREAL_0:2;
then 1 in dom co by FINSEQ_3:25;
then
A49: co/.1 = co.1 by PARTFUN1:def 6
.= Gij by A30,JORDAN3:23;
then
A50: LSeg(co,1) = LSeg(Gij,co/.(1+1)) by A35,TOPREAL1:def 3;
A51: {Gij} c= LSeg(co,1) /\ L~<*Gik,Gi1k,Gij*>
proof
let x be object;
assume x in {Gij};
then
A52: x = Gij by TARSKI:def 1;
Gij in LSeg(Gi1k,Gij) by RLTOPSP1:68;
then Gij in LSeg(Gik,Gi1k) \/ LSeg(Gi1k,Gij) by XBOOLE_0:def 3;
then
A53: Gij in L~<*Gik,Gi1k,Gij*> by SPRECT_1:8;
Gij in LSeg(co,1) by A50,RLTOPSP1:68;
hence thesis by A52,A53,XBOOLE_0:def 4;
end;
LSeg(co,1) c= L~co by TOPREAL3:19;
then LSeg(co,1) c= L~LS by A36;
then LSeg(co,1) /\ L~<*Gik,Gi1k,Gij*> c= {Gij} by A8,A13,XBOOLE_1:26;
then
A54: L~<*Gik,Gi1k,Gij*> /\ LSeg(co,1) = {Gij} by A51;
A55: rng co c= L~co by A35,SPPOL_2:18;
A56: len go >= 1+1 by TOPREAL1:def 8;
then reconsider
go as non constant s.c.c. being_S-Seq FinSequence of TOP-REAL 2
by A48,JGRAPH_1:12,JORDAN8:5;
A57: L~go c= L~US by A45,JORDAN3:41;
A58: len go > 1 by A56,NAT_1:13;
then
A59: len go in dom go by FINSEQ_3:25;
then
A60: go/.len go = go.len go by PARTFUN1:def 6
.= Gik by A45,JORDAN3:24;
reconsider m = len go - 1 as Nat by A59,FINSEQ_3:26;
A61: m+1 = len go;
then
A62: len go-'1 = m by NAT_D:34;
m >= 1 by A56,XREAL_1:19;
then
A63: LSeg(go,m) = LSeg(go/.m,Gik) by A60,A61,TOPREAL1:def 3;
A64: {Gik} c= LSeg(go,m) /\ L~<*Gik,Gi1k,Gij*>
proof
let x be object;
assume x in {Gik};
then
A65: x = Gik by TARSKI:def 1;
Gik in LSeg(Gik,Gi1k) by RLTOPSP1:68;
then Gik in LSeg(Gik,Gi1k) \/ LSeg(Gi1k,Gij) by XBOOLE_0:def 3;
then
A66: Gik in L~<*Gik,Gi1k,Gij*> by SPRECT_1:8;
Gik in LSeg(go,m) by A63,RLTOPSP1:68;
hence thesis by A65,A66,XBOOLE_0:def 4;
end;
LSeg(go,m) c= L~go by TOPREAL3:19;
then LSeg(go,m) c= L~US by A57;
then LSeg(go,m) /\ L~<*Gik,Gi1k,Gij*> c= {Gik} by A7,A13,XBOOLE_1:26;
then
A67: LSeg(go,m) /\ L~<*Gik,Gi1k,Gij*> = {Gik} by A64;
A68: go/.1 = US/.1 by A45,SPRECT_3:22
.= Wmin by JORDAN1F:5;
A69: LS.1 = LS/.1 by A27,PARTFUN1:def 6
.= Emax by JORDAN1F:6;
A70: L~go /\ L~co c= {go/.1}
proof
let x be object;
assume
A71: x in L~go /\ L~co;
then
A72: x in L~co by XBOOLE_0:def 4;
A73: now
assume x = Emax;
then
A74: Emax = Gij by A30,A69,A72,JORDAN1E:7;
G*(len G,j)`1 = Ebo by A4,A10,A24,JORDAN1A:71;
then Emax`1 <> Ebo by A2,A3,A23,A15,A74,JORDAN1G:7;
hence contradiction by EUCLID:52;
end;
x in L~go by A71,XBOOLE_0:def 4;
then x in L~US /\ L~LS by A57,A36,A72,XBOOLE_0:def 4;
then x in {Wmin,Emax} by JORDAN1E:16;
then x = Wmin or x = Emax by TARSKI:def 2;
hence thesis by A68,A73,TARSKI:def 1;
end;
set W2 = go/.2;
A75: 2 in dom go by A56,FINSEQ_3:25;
go = mid(US,1,Gik..US) by A47,JORDAN1G:49
.= US|(Gik..US) by A47,FINSEQ_4:21,FINSEQ_6:116;
then
A76: W2 = US/.2 by A75,FINSEQ_4:70;
A77: rng go c= L~go by A56,SPPOL_2:18;
A78: go/.1 = LS/.len LS by A68,JORDAN1F:8
.= co/.len co by A30,JORDAN1J:35;
{go/.1} c= L~go /\ L~co
proof
let x be object;
assume x in {go/.1};
then
A79: x = go/.1 by TARSKI:def 1;
then
A80: x in rng go by FINSEQ_6:42;
x in rng co by A78,A79,FINSEQ_6:168;
hence thesis by A77,A55,A80,XBOOLE_0:def 4;
end;
then
A81: L~go /\ L~co = {go/.1} by A70;
now
per cases;
suppose
Gij`1 <> Gik`1 & Gij`2 <> Gik`2;
then pion is being_S-Seq by A21,TOPREAL3:35;
then consider pion1 be FinSequence of TOP-REAL 2 such that
A82: pion1 is_sequence_on G and
A83: pion1 is being_S-Seq and
A84: L~pion = L~pion1 and
A85: pion/.1 = pion1/.1 and
A86: pion/.len pion = pion1/.len pion1 and
A87: len pion <= len pion1 by A44,GOBOARD3:2;
reconsider pion1 as being_S-Seq FinSequence of TOP-REAL 2 by A83;
A88: (go^'pion1)/.len (go^'pion1) = pion/.len pion by A86,FINSEQ_6:156
.= pion/.3 by FINSEQ_1:45
.= co/.1 by A49,FINSEQ_4:18;
A89: go/.len go = pion1/.1 by A60,A85,FINSEQ_4:18;
A90: L~go /\ L~pion1 c= {pion1/.1}
proof
let x be object;
assume
A91: x in L~go /\ L~pion1;
then
A92: x in L~pion1 by XBOOLE_0:def 4;
x in L~go by A91,XBOOLE_0:def 4;
hence thesis by A7,A13,A60,A57,A84,A89,A92,XBOOLE_0:def 4;
end;
len pion1 >= 2+1 by A87,FINSEQ_1:45;
then
A93: len pion1 > 1+1 by NAT_1:13;
then
A94: rng pion1 c= L~pion1 by SPPOL_2:18;
{pion1/.1} c= L~go /\ L~pion1
proof
let x be object;
assume x in {pion1/.1};
then
A95: x = pion1/.1 by TARSKI:def 1;
then
A96: x in rng pion1 by FINSEQ_6:42;
x in rng go by A89,A95,FINSEQ_6:168;
hence thesis by A77,A94,A96,XBOOLE_0:def 4;
end;
then
A97: L~go /\ L~pion1 = {pion1/.1} by A90;
then
A98: (go^'pion1) is s.n.c. by A89,JORDAN1J:54;
A99: pion/.len pion = pion/.3 by FINSEQ_1:45
.= co/.1 by A49,FINSEQ_4:18;
A100: {pion1/.len pion1} c= L~co /\ L~pion1
proof
let x be object;
assume x in {pion1/.len pion1};
then
A101: x = pion1/.len pion1 by TARSKI:def 1;
then
A102: x in rng pion1 by FINSEQ_6:168;
x in rng co by A86,A99,A101,FINSEQ_6:42;
hence thesis by A55,A94,A102,XBOOLE_0:def 4;
end;
L~co /\ L~pion1 c= {pion1/.len pion1}
proof
let x be object;
assume
A103: x in L~co /\ L~pion1;
then
A104: x in L~pion1 by XBOOLE_0:def 4;
x in L~co by A103,XBOOLE_0:def 4;
hence thesis by A8,A13,A49,A36,A84,A86,A99,A104,XBOOLE_0:def 4;
end;
then
A105: L~co /\ L~pion1 = {pion1/.len pion1} by A100;
A106: L~(go^'pion1) /\ L~co = (L~go \/ L~pion1) /\ L~co by A89,TOPREAL8:35
.= {go/.1} \/ {co/.1} by A81,A86,A99,A105,XBOOLE_1:23
.= {(go^'pion1)/.1} \/ {co/.1} by FINSEQ_6:155
.= {(go^'pion1)/.1,co/.1} by ENUMSET1:1;
A107: UA is_an_arc_of W-min C,E-max C by JORDAN6:def 8;
then
A108: UA is connected by JORDAN6:10;
set godo = go^'pion1^'co;
A109: co/.len co = (go^'pion1)/.1 by A78,FINSEQ_6:155;
A110: go^'pion1 is_sequence_on G by A48,A82,A89,TOPREAL8:12;
then
A111: godo is_sequence_on G by A33,A88,TOPREAL8:12;
A112: len pion1-1 >= 1 by A93,XREAL_1:19;
then
A113: len pion1-'1 = len pion1-1 by XREAL_0:def 2;
A114: {Gij} c= LSeg(pion1,len pion1-'1) /\ LSeg(co,1)
proof
let x be object;
assume x in {Gij};
then
A115: x = Gij by TARSKI:def 1;
pion1/.(len pion1-'1+1) = pion/.3 by A86,A113,FINSEQ_1:45
.= Gij by FINSEQ_4:18;
then
A116: Gij in LSeg(pion1,len pion1-'1) by A112,A113,TOPREAL1:21;
Gij in LSeg(co,1) by A50,RLTOPSP1:68;
hence thesis by A115,A116,XBOOLE_0:def 4;
end;
LSeg(pion1,len pion1-'1) c= L~pion by A84,TOPREAL3:19;
then LSeg(pion1,len pion1-'1) /\ LSeg(co,1) c= {Gij} by A54,XBOOLE_1:27;
then
A117: LSeg(pion1,len pion1-'1) /\ LSeg(co,1) = {Gij} by A114;
len pion1-1+1 <= len pion1;
then
A118: len pion1-'1 < len pion1 by A113,NAT_1:13;
len pion1 >= 2+1 by A87,FINSEQ_1:45;
then
A119: len pion1-2 >= 0 by XREAL_1:19;
then len pion1-'2+1 = len pion1-2+1 by XREAL_0:def 2
.= len pion1-'1 by A112,XREAL_0:def 2;
then
A120: LSeg(go^'pion1,len go+(len pion1-'2)) /\ LSeg(co,1) = {(go^'pion1)
/.len (go^'pion1)} by A49,A89,A88,A118,A117,TOPREAL8:31;
rng go /\ rng pion1 c= {pion1/.1} by A77,A94,A97,XBOOLE_1:27;
then
A121: go^'pion1 is one-to-one by JORDAN1J:55;
len (go^'pion1)+1-1 = len go+len pion1-1 by FINSEQ_6:139;
then len (go^'pion1)-1 = len go + (len pion1-2)
.= len go + (len pion1-'2) by A119,XREAL_0:def 2;
then
A122: len (go^'pion1)-'1 = len go + (len pion1-'2) by XREAL_0:def 2;
A123: L~Cage(C,n) = L~US \/ L~LS by JORDAN1E:13;
then
A124: L~US c= L~Cage(C,n) by XBOOLE_1:7;
then
A125: L~go c=L~Cage(C,n) by A57;
A126: {Gik} c= LSeg(go,m) /\ LSeg(pion1,1)
proof
let x be object;
assume x in {Gik};
then
A127: x = Gik by TARSKI:def 1;
A128: Gik in LSeg(go,m) by A63,RLTOPSP1:68;
Gik in LSeg(pion1,1) by A60,A89,A93,TOPREAL1:21;
hence thesis by A127,A128,XBOOLE_0:def 4;
end;
LSeg(pion1,1) c= L~pion by A84,TOPREAL3:19;
then LSeg(go,len go-'1) /\ LSeg(pion1,1) c={Gik} by A62,A67,XBOOLE_1:27;
then LSeg(go,len go-'1) /\ LSeg(pion1,1) = { go/.len go } by A60,A62,A126
;
then
A129: go^'pion1 is unfolded by A89,TOPREAL8:34;
len (go^'pion1) >= len go by TOPREAL8:7;
then
A130: len (go^'pion1) >= 1+1 by A56,XXREAL_0:2;
then
A131: len (go^'pion1) > 1+0 by NAT_1:13;
A132: now
assume
A133: Gik..US <= 1;
Gik..US >= 1 by A47,FINSEQ_4:21;
then Gik..US = 1 by A133,XXREAL_0:1;
then Gik = US/.1 by A47,FINSEQ_5:38;
hence contradiction by A42,A46,JORDAN1F:5;
end;
A134: US is_sequence_on G by JORDAN1G:4;
A135: Wmin`1 = Wbo by EUCLID:52;
set ff = Rotate(Cage(C,n),Wmin);
A136: 1+1 <= len Cage(C,n) by GOBOARD7:34,XXREAL_0:2;
A137: len godo >= len (go^'pion1) by TOPREAL8:7;
then
A138: 1+1 <= len godo by A130,XXREAL_0:2;
(go^'pion1) is non trivial by A130,NAT_D:60;
then reconsider
godo as non constant standard special_circular_sequence by A138,A88,A111
,A129,A122,A120,A98,A121,A106,A109,JORDAN8:4,5,TOPREAL8:11,33,34;
A139: L~godo = L~(go^'pion1) \/ L~co by A88,TOPREAL8:35
.= L~go \/ L~pion1 \/ L~co by A89,TOPREAL8:35;
A140: right_cell(godo,1,G)\L~godo c= RightComp godo by A138,A111,JORDAN9:27;
2 in dom godo by A138,FINSEQ_3:25;
then
A141: godo/.2 in rng godo by PARTFUN2:2;
A142: W-min C in UA by A107,TOPREAL1:1;
Wmin in rng Cage(C,n) by SPRECT_2:43;
then
A143: ff/.1 = Wmin by FINSEQ_6:92;
A144: L~ff = L~Cage(C,n) by REVROT_1:33;
then (W-max L~ff)..ff > 1 by A143,SPRECT_5:22;
then (N-min L~ff)..ff > 1 by A143,A144,SPRECT_5:23,XXREAL_0:2;
then (N-max L~ff)..ff > 1 by A143,A144,SPRECT_5:24,XXREAL_0:2;
then
A145: Emax..ff > 1 by A143,A144,SPRECT_5:25,XXREAL_0:2;
A146: Cage(C,n) is_sequence_on G by JORDAN9:def 1;
then
A147: ff is_sequence_on G by REVROT_1:34;
A148: Gi1k`1 = G*(i1,1)`1 by A1,A6,A11,A16,GOBOARD5:2
.= Gij`1 by A1,A4,A11,A10,GOBOARD5:2;
then
A149: W-bound pio = Gij`1 by SPRECT_1:54;
A150: L~LS c= L~Cage(C,n) by A123,XBOOLE_1:7;
then
A151: L~co c=L~Cage(C,n) by A36;
A152: W-min C in C by SPRECT_1:13;
A153: now
assume W-min C in L~godo;
then
A154: W-min C in L~go \/ L~pion1 or W-min C in L~co by A139,XBOOLE_0:def 3;
per cases by A154,XBOOLE_0:def 3;
suppose
W-min C in L~go;
then C meets L~Cage(C,n) by A125,A152,XBOOLE_0:3;
hence contradiction by JORDAN10:5;
end;
suppose
W-min C in L~pion1;
hence contradiction by A9,A13,A84,A142,XBOOLE_0:3;
end;
suppose
W-min C in L~co;
then C meets L~Cage(C,n) by A151,A152,XBOOLE_0:3;
hence contradiction by JORDAN10:5;
end;
end;
A155: len US >= 2 by A41,XXREAL_0:2;
A156: L~go \/ L~co is compact by COMPTS_1:10;
1+1 <= len Rotate(Cage(C,n),Wmin) by GOBOARD7:34,XXREAL_0:2;
then right_cell(Rotate(Cage(C,n),Wmin),1) = right_cell(ff,1,GoB ff) by
JORDAN1H:23
.= right_cell(ff,1,GoB Cage(C,n)) by REVROT_1:28
.= right_cell(ff,1,G) by JORDAN1H:44
.= right_cell(ff-:Emax,1,G) by A145,A147,JORDAN1J:53
.= right_cell(US,1,G) by JORDAN1E:def 1
.= right_cell(R_Cut(US,Gik),1,G) by A47,A134,A132,JORDAN1J:52
.= right_cell(go^'pion1,1,G) by A58,A110,JORDAN1J:51
.= right_cell(godo,1,G) by A131,A111,JORDAN1J:51;
then W-min C in right_cell(godo,1,G) by JORDAN1I:6;
then
A157: W-min C in right_cell(godo,1,G)\L~godo by A153,XBOOLE_0:def 5;
A158: rng godo c= L~godo by A130,A137,SPPOL_2:18,XXREAL_0:2;
A159: godo/.1 = (go^'pion1)/.1 by FINSEQ_6:155
.= Wmin by A68,FINSEQ_6:155;
A160: Gi1k`1 <= Gik`1 by A1,A2,A3,A6,A16,JORDAN1A:18;
then
A161: W-bound poz = Gi1k`1 by SPRECT_1:54;
W-bound (poz \/ pio) = min(W-bound poz, W-bound pio) by SPRECT_1:47
.= Gij`1 by A148,A161,A149;
then
A162: W-bound L~pion1 = Gij`1 by A84,TOPREAL3:16;
A163: UA c= C by JORDAN6:61;
Gij`1 >= Wbo by A30,A150,PSCOMP_1:24;
then
A164: Gij`1 > Wbo by A38,XXREAL_0:1;
A165: E-max C in UA by A107,TOPREAL1:1;
Wmin in rng go by A68,FINSEQ_6:42;
then Wmin in L~go \/ L~co by A77,XBOOLE_0:def 3;
then
A166: W-min (L~go \/ L~co) = Wmin by A125,A151,A156,JORDAN1J:21,XBOOLE_1:8;
(W-min (L~go \/ L~co))`1 = W-bound (L~go \/ L~co) by EUCLID:52;
then W-min (L~go\/L~co\/L~pion1) = W-min (L~go \/ L~co) by A162,A156,A166
,A135,A164,JORDAN1J:33;
then
A167: W-min L~godo = Wmin by A139,A166,XBOOLE_1:4;
godo/.2 = (go^'pion1)/.2 by A130,FINSEQ_6:159
.= US/.2 by A56,A76,FINSEQ_6:159
.= (US^'LS)/.2 by A155,FINSEQ_6:159
.= Rotate(Cage(C,n),Wmin)/.2 by JORDAN1E:11;
then godo/.2 in W-most L~Cage(C,n) by JORDAN1I:25;
then (godo/.2)`1 = (W-min L~godo)`1 by A167,PSCOMP_1:31
.= W-bound L~godo by EUCLID:52;
then godo/.2 in W-most L~godo by A158,A141,SPRECT_2:12;
then Rotate(godo,W-min L~godo)/.2 in W-most L~godo by A159,A167,
FINSEQ_6:89;
then reconsider godo as clockwise_oriented non constant standard
special_circular_sequence by JORDAN1I:25;
len US in dom US by FINSEQ_5:6;
then
A168: US.len US = US/.len US by PARTFUN1:def 6
.= Emax by JORDAN1F:7;
A169: east_halfline E-max C misses L~go
proof
assume east_halfline E-max C meets L~go;
then consider p be object such that
A170: p in east_halfline E-max C and
A171: p in L~go by XBOOLE_0:3;
reconsider p as Point of TOP-REAL 2 by A170;
p in L~US by A57,A171;
then p in east_halfline E-max C /\ L~Cage(C,n) by A124,A170,
XBOOLE_0:def 4;
then
A172: p`1 = Ebo by JORDAN1A:83,PSCOMP_1:50;
then
A173: p = Emax by A57,A171,JORDAN1J:46;
then Emax = Gik by A45,A168,A171,JORDAN1J:43;
then Gik`1 = G*(len G,k)`1 by A6,A16,A24,A172,A173,JORDAN1A:71;
hence contradiction by A3,A19,A22,JORDAN1G:7;
end;
now
assume east_halfline E-max C meets L~godo;
then
A174: east_halfline E-max C meets (L~go \/ L~pion1) or east_halfline
E-max C meets L~co by A139,XBOOLE_1:70;
per cases by A174,XBOOLE_1:70;
suppose
east_halfline E-max C meets L~go;
hence contradiction by A169;
end;
suppose
east_halfline E-max C meets L~pion1;
then consider p be object such that
A175: p in east_halfline E-max C and
A176: p in L~pion1 by XBOOLE_0:3;
reconsider p as Point of TOP-REAL 2 by A175;
A177: p`2 = (E-max C)`2 by A175,TOPREAL1:def 11;
A178: now
per cases by A13,A84,A176,XBOOLE_0:def 3;
suppose
p in poz;
hence p`1 <= Gik`1 by A160,TOPREAL1:3;
end;
suppose
p in pio;
hence p`1 <= Gik`1 by A148,A160,GOBOARD7:5;
end;
end;
i2+1 <= len G by A3,NAT_1:13;
then i2+1-1 <= len G-1 by XREAL_1:9;
then
A179: i2 <= len G-'1 by XREAL_0:def 2;
len G-'1 <= len G by NAT_D:35;
then Gik`1 <= G*(len G-'1,1)`1 by A6,A18,A16,A24,A14,A179,JORDAN1A:18
;
then p`1 <= G*(len G-'1,1)`1 by A178,XXREAL_0:2;
then p`1 <= E-bound C by A14,JORDAN8:12;
then
A180: p`1 <= (E-max C)`1 by EUCLID:52;
p`1 >= (E-max C)`1 by A175,TOPREAL1:def 11;
then p`1 = (E-max C)`1 by A180,XXREAL_0:1;
then p = E-max C by A177,TOPREAL3:6;
hence contradiction by A9,A13,A84,A165,A176,XBOOLE_0:3;
end;
suppose
east_halfline E-max C meets L~co;
then consider p be object such that
A181: p in east_halfline E-max C and
A182: p in L~co by XBOOLE_0:3;
reconsider p as Point of TOP-REAL 2 by A181;
A183: p in LSeg(co,Index(p,co)) by A182,JORDAN3:9;
consider t be Nat such that
A184: t in dom LS and
A185: LS.t = Gij by A32,FINSEQ_2:10;
1 <= t by A184,FINSEQ_3:25;
then
A186: 1 < t by A34,A185,XXREAL_0:1;
t <= len LS by A184,FINSEQ_3:25;
then Index(Gij,LS)+1 = t by A185,A186,JORDAN3:12;
then
A187: len L_Cut(LS,Gij) = len LS-Index(Gij,LS) by A30,A185,JORDAN3:26;
Index(p,co) < len co by A182,JORDAN3:8;
then Index(p,co) < len LS-'Index(Gij,LS) by A187,XREAL_0:def 2;
then Index(p,co)+1 <= len LS-'Index(Gij,LS) by NAT_1:13;
then
A188: Index(p,co) <= len LS-'Index(Gij,LS)-1 by XREAL_1:19;
A189: co = mid(LS,Gij..LS,len LS) by A32,JORDAN1J:37;
p in L~LS by A36,A182;
then p in east_halfline E-max C /\ L~Cage(C,n) by A150,A181,
XBOOLE_0:def 4;
then
A190: p`1 = Ebo by JORDAN1A:83,PSCOMP_1:50;
A191: Index(Gij,LS)+1 = Gij..LS by A34,A32,JORDAN1J:56;
0+Index(Gij,LS) < len LS by A30,JORDAN3:8;
then len LS-Index(Gij,LS) > 0 by XREAL_1:20;
then Index(p,co) <= len LS-Index(Gij,LS)-1 by A188,XREAL_0:def 2;
then Index(p,co) <= len LS-Gij..LS by A191;
then Index(p,co) <= len LS-'Gij..LS by XREAL_0:def 2;
then
A192: Index(p,co) < len LS-'(Gij..LS)+1 by NAT_1:13;
A193: 1<=Index(p,co) by A182,JORDAN3:8;
A194: Gij..LS<=len LS by A32,FINSEQ_4:21;
Gij..LS <> len LS by A31,A32,FINSEQ_4:19;
then
A195: Gij..LS < len LS by A194,XXREAL_0:1;
A196: 1+1 <= len LS by A25,XXREAL_0:2;
then
A197: 2 in dom LS by FINSEQ_3:25;
set tt = Index(p,co)+(Gij..LS)-'1;
set RC = Rotate(Cage(C,n),Emax);
A198: E-max C in right_cell(RC,1) by JORDAN1I:7;
A199: GoB RC = GoB Cage(C,n) by REVROT_1:28
.= G by JORDAN1H:44;
A200: L~RC = L~Cage(C,n) by REVROT_1:33;
consider jj2 be Nat such that
A201: 1 <= jj2 and
A202: jj2 <= width G and
A203: Emax = G*(len G,jj2) by JORDAN1D:25;
A204: len G >= 4 by JORDAN8:10;
then len G >= 1 by XXREAL_0:2;
then
A205: [len G,jj2] in Indices G by A201,A202,MATRIX_0:30;
A206: len RC = len Cage(C,n) by FINSEQ_6:179;
LS = RC-:Wmin by JORDAN1G:18;
then
A207: LSeg(LS,1) = LSeg(RC,1) by A196,SPPOL_2:9;
A208: Emax in rng Cage(C,n) by SPRECT_2:46;
RC is_sequence_on G by A146,REVROT_1:34;
then consider ii,jj be Nat such that
A209: [ii,jj+1] in Indices G and
A210: [ii,jj] in Indices G and
A211: RC/.1 = G*(ii,jj+1) and
A212: RC/.(1+1) = G*(ii,jj) by A136,A200,A206,A208,FINSEQ_6:92,JORDAN1I:23;
A213: jj+1+1 <> jj;
A214: 1 <= jj by A210,MATRIX_0:32;
RC/.1 = E-max L~RC by A200,A208,FINSEQ_6:92;
then
A215: ii = len G by A200,A209,A211,A203,A205,GOBOARD1:5;
then ii-1 >= 4-1 by A204,XREAL_1:9;
then
A216: ii-1 >= 1 by XXREAL_0:2;
then
A217: 1 <= ii-'1 by XREAL_0:def 2;
A218: jj <= width G by A210,MATRIX_0:32;
then
A219: G*(len G,jj)`1 = Ebo by A24,A214,JORDAN1A:71;
A220: jj+1 <= width G by A209,MATRIX_0:32;
ii+1 <> ii;
then
A221: right_cell(RC,1) = cell(G,ii-'1,jj) by A136,A206,A199,A209,A210,A211
,A212,A213,GOBOARD5:def 6;
A222: ii <= len G by A210,MATRIX_0:32;
A223: 1 <= ii by A210,MATRIX_0:32;
A224: ii <= len G by A209,MATRIX_0:32;
A225: 1 <= jj+1 by A209,MATRIX_0:32;
then
A226: Ebo = G*(len G,jj+1)`1 by A24,A220,JORDAN1A:71;
A227: 1 <= ii by A209,MATRIX_0:32;
then
A228: ii-'1+1 = ii by XREAL_1:235;
then
A229: ii-'1 < len G by A224,NAT_1:13;
then
A230: G*(ii-'1,jj+1)`2 = G*(1,jj+1)`2 by A225,A220,A217,GOBOARD5:1
.= G*(ii,jj+1)`2 by A227,A224,A225,A220,GOBOARD5:1;
A231: (E-max C)`2 = p`2 by A181,TOPREAL1:def 11;
then
A232: p`2 <= G*(ii-'1,jj+1)`2 by A198,A224,A220,A214,A221,A228,A216,
JORDAN9:17;
A233: G*(ii-'1,jj)`2 = G*(1,jj)`2 by A214,A218,A217,A229,GOBOARD5:1
.= G*(ii,jj)`2 by A223,A222,A214,A218,GOBOARD5:1;
G*(ii-'1,jj)`2 <= p`2 by A231,A198,A224,A220,A214,A221,A228,A216,
JORDAN9:17;
then p in LSeg(RC/.1,RC/.(1+1)) by A190,A211,A212,A215,A232,A233,A230
,A219,A226,GOBOARD7:7;
then
A234: p in LSeg(LS,1) by A136,A207,A206,TOPREAL1:def 3;
1<=Gij..LS by A32,FINSEQ_4:21;
then
A235: LSeg(mid(LS,Gij..LS,len LS),Index(p,co)) = LSeg(LS,Index(p,co)
+(Gij..LS)-'1) by A195,A193,A192,JORDAN4:19;
1<=Index(Gij,LS) by A30,JORDAN3:8;
then
A236: 1+1 <= Gij..LS by A191,XREAL_1:7;
then Index(p,co)+Gij..LS >= 1+1+1 by A193,XREAL_1:7;
then Index(p,co)+Gij..LS-1 >= 1+1+1-1 by XREAL_1:9;
then
A237: tt >= 1+1 by XREAL_0:def 2;
now
per cases by A237,XXREAL_0:1;
suppose
tt > 1+1;
then LSeg(LS,1) misses LSeg(LS,tt) by TOPREAL1:def 7;
hence contradiction by A234,A183,A189,A235,XBOOLE_0:3;
end;
suppose
A238: tt = 1+1;
then 1+1 = Index(p,co)+(Gij..LS)-1 by XREAL_0:def 2;
then 1+1+1 = Index(p,co)+(Gij..LS);
then
A239: Gij..LS = 2 by A193,A236,JORDAN1E:6;
LSeg(LS,1) /\ LSeg(LS,tt) = {LS/.2} by A25,A238,TOPREAL1:def 6;
then p in {LS/.2} by A234,A183,A189,A235,XBOOLE_0:def 4;
then
A240: p = LS/.2 by TARSKI:def 1;
then
A241: p in rng LS by A197,PARTFUN2:2;
p..LS = 2 by A197,A240,FINSEQ_5:41;
then p = Gij by A32,A239,A241,FINSEQ_5:9;
then Gij`1 = Ebo by A240,JORDAN1G:32;
then Gij`1 = G*(len G,j)`1 by A4,A10,A24,JORDAN1A:71;
hence contradiction by A2,A3,A23,A15,JORDAN1G:7;
end;
end;
hence contradiction;
end;
end;
then east_halfline E-max C c= (L~godo)` by SUBSET_1:23;
then consider W be Subset of TOP-REAL 2 such that
A242: W is_a_component_of (L~godo)` and
A243: east_halfline E-max C c= W by GOBOARD9:3;
W is not bounded by A243,JORDAN2C:121,RLTOPSP1:42;
then W is_outside_component_of L~godo by A242,JORDAN2C:def 3;
then W c= UBD L~godo by JORDAN2C:23;
then
A244: east_halfline E-max C c= UBD L~godo by A243;
E-max C in east_halfline E-max C by TOPREAL1:38;
then E-max C in UBD L~godo by A244;
then E-max C in LeftComp godo by GOBRD14:36;
then UA meets L~godo by A108,A142,A165,A140,A157,JORDAN1J:36;
then
A245: UA meets (L~go \/ L~pion1) or UA meets L~co by A139,XBOOLE_1:70;
now
per cases by A245,XBOOLE_1:70;
suppose
UA meets L~go;
then UA meets L~Cage(C,n) by A57,A124,XBOOLE_1:1,63;
hence contradiction by A163,JORDAN10:5,XBOOLE_1:63;
end;
suppose
UA meets L~pion1;
hence contradiction by A9,A13,A84;
end;
suppose
UA meets L~co;
then UA meets L~Cage(C,n) by A36,A150,XBOOLE_1:1,63;
hence contradiction by A163,JORDAN10:5,XBOOLE_1:63;
end;
end;
hence contradiction;
end;
suppose
Gij`1 = Gik`1;
then
A246: i1 = i2 by A23,A19,JORDAN1G:7;
then poz = {Gi1k} by RLTOPSP1:70;
then poz c= pio by A40,ZFMISC_1:31;
then pio \/ poz = pio by XBOOLE_1:12;
hence contradiction by A1,A3,A4,A5,A6,A7,A8,A9,A246,JORDAN1J:59;
end;
suppose
Gij`2 = Gik`2;
then
A247: j = k by A23,A19,JORDAN1G:6;
then pio = {Gi1k} by RLTOPSP1:70;
then pio c= poz by A39,ZFMISC_1:31;
then pio \/ poz = poz by XBOOLE_1:12;
hence contradiction by A1,A2,A3,A4,A6,A7,A8,A9,A247,Th29;
end;
end;
hence contradiction;
end;
theorem Th45:
for C be Simple_closed_curve for i1,i2,j,k be Nat st
1 < i1 & i1 <= i2 & i2 < len Gauge(C,n) & 1 <= j & j <= k & k <= width Gauge(C,
n) & (LSeg(Gauge(C,n)*(i1,j),Gauge(C,n)*(i1,k)) \/ LSeg(Gauge(C,n)*(i1,k),Gauge
(C,n)*(i2,k))) /\ L~Upper_Seq(C,n) = {Gauge(C,n)*(i2,k)} & (LSeg(Gauge(C,n)*(i1
,j),Gauge(C,n)*(i1,k)) \/ LSeg(Gauge(C,n)*(i1,k),Gauge(C,n)*(i2,k))) /\ L~
Lower_Seq(C,n) = {Gauge(C,n)*(i1,j)} holds (LSeg(Gauge(C,n)*(i1,j),Gauge(C,n)*(
i1,k)) \/ LSeg(Gauge(C,n)*(i1,k),Gauge(C,n)*(i2,k))) meets Lower_Arc C
proof
let C be Simple_closed_curve;
let i1,i2,j,k be Nat;
set G = Gauge(C,n);
set pio = LSeg(G*(i1,j),G*(i1,k));
set poz = LSeg(G*(i1,k),G*(i2,k));
set US = Upper_Seq(C,n);
set LS = Lower_Seq(C,n);
assume that
A1: 1 < i1 and
A2: i1 <= i2 and
A3: i2 < len G and
A4: 1 <= j and
A5: j <= k and
A6: k <= width G and
A7: (pio \/ poz) /\ L~US = {G*(i2,k)} and
A8: (pio \/ poz) /\ L~LS = {G*(i1,j)} and
A9: (pio \/ poz) misses Lower_Arc C;
set Gij = G*(i1,j);
A10: j <= width G by A5,A6,XXREAL_0:2;
A11: i1 < len G by A2,A3,XXREAL_0:2;
then
A12: [i1,j] in Indices G by A1,A4,A10,MATRIX_0:30;
set Gi1k = G*(i1,k);
set Gik = G*(i2,k);
A13: L~<*Gik,Gi1k,Gij*> = poz \/ pio by TOPREAL3:16;
len G >= 4 by JORDAN8:10;
then
A14: len G >= 1 by XXREAL_0:2;
then
A15: [len G,j] in Indices G by A4,A10,MATRIX_0:30;
A16: 1 <= k by A4,A5,XXREAL_0:2;
then
A17: [1,k] in Indices G by A6,A14,MATRIX_0:30;
A18: 1 < i2 by A1,A2,XXREAL_0:2;
then
A19: [i2,k] in Indices G by A3,A6,A16,MATRIX_0:30;
A20: Gi1k`2 = G*(1,k)`2 by A1,A6,A11,A16,GOBOARD5:1
.= Gik`2 by A3,A6,A18,A16,GOBOARD5:1;
Gi1k`1 = G*(i1,1)`1 by A1,A6,A11,A16,GOBOARD5:2
.= Gij`1 by A1,A4,A11,A10,GOBOARD5:2;
then
A21: Gi1k = |[Gij`1,Gik`2]| by A20,EUCLID:53;
A22: [len G,k] in Indices G by A6,A16,A14,MATRIX_0:30;
A23: [i1,j] in Indices G by A1,A4,A11,A10,MATRIX_0:30;
set Wbo = W-bound L~Cage(C,n);
set Wmin = W-min L~Cage(C,n);
A24: len G = width G by JORDAN8:def 1;
set Ebo = E-bound L~Cage(C,n);
set Emax = E-max L~Cage(C,n);
A25: len LS >= 1+2 by JORDAN1E:15;
then
A26: len LS >= 1 by XXREAL_0:2;
then
A27: 1 in dom LS by FINSEQ_3:25;
then
A28: LS.1 = LS/.1 by PARTFUN1:def 6
.= Emax by JORDAN1F:6;
len LS in dom LS by A26,FINSEQ_3:25;
then
A29: LS.len LS = LS/.len LS by PARTFUN1:def 6
.= Wmin by JORDAN1F:8;
set co = L_Cut(LS,Gij);
Gij in {Gij} by TARSKI:def 1;
then
A30: Gij in L~LS by A8,XBOOLE_0:def 4;
Wmin`1 = Wbo by EUCLID:52
.= G*(1,k)`1 by A6,A16,A24,JORDAN1A:73;
then
A31: Gij <> LS.len LS by A1,A17,A29,A12,JORDAN1G:7;
then reconsider co as being_S-Seq FinSequence of TOP-REAL 2 by A30,JORDAN3:34
;
A32: Gij in rng LS by A1,A4,A11,A30,A10,JORDAN1G:5,JORDAN1J:40;
then
A33: co is_sequence_on G by JORDAN1G:5,JORDAN1J:39;
Emax`1 = Ebo by EUCLID:52
.= G*(len G,k)`1 by A6,A16,A24,JORDAN1A:71;
then
A34: Gij <> LS.1 by A2,A3,A12,A22,A28,JORDAN1G:7;
A35: len co >= 1+1 by TOPREAL1:def 8;
then reconsider
co as non constant s.c.c. being_S-Seq FinSequence of TOP-REAL 2
by A33,JGRAPH_1:12,JORDAN8:5;
A36: L~co c= L~LS by A30,JORDAN3:42;
A37: [1,j] in Indices G by A4,A10,A14,MATRIX_0:30;
A38: now
assume Gij`1 = Wbo;
then G*(1,j)`1 = G*(i1,j)`1 by A4,A10,A24,JORDAN1A:73;
hence contradiction by A1,A23,A37,JORDAN1G:7;
end;
set pion = <*Gik,Gi1k,Gij*>;
A39: Gi1k in poz by RLTOPSP1:68;
set LA = Lower_Arc C;
A40: Gi1k in pio by RLTOPSP1:68;
set go = R_Cut(US,Gik);
A41: len US >= 3 by JORDAN1E:15;
then len US >= 1 by XXREAL_0:2;
then 1 in dom US by FINSEQ_3:25;
then
A42: US.1 = US/.1 by PARTFUN1:def 6
.= Wmin by JORDAN1F:5;
A43: [i1,k] in Indices G by A1,A6,A11,A16,MATRIX_0:30;
A44: now
let n be Nat;
assume n in dom pion;
then n in {1,2,3} by FINSEQ_1:89,FINSEQ_3:1;
then n = 1 or n = 2 or n = 3 by ENUMSET1:def 1;
hence
ex i,j be Nat st [i,j] in Indices G & pion/.n = G*(i,j) by A23
,A19,A43,FINSEQ_4:18;
end;
Gik in {Gik} by TARSKI:def 1;
then
A45: Gik in L~US by A7,XBOOLE_0:def 4;
Wmin`1 = Wbo by EUCLID:52
.= G*(1,k)`1 by A6,A16,A24,JORDAN1A:73;
then
A46: Gik <> US.1 by A1,A2,A19,A42,A17,JORDAN1G:7;
then reconsider go as being_S-Seq FinSequence of TOP-REAL 2 by A45,JORDAN3:35
;
A47: Gik in rng US by A3,A6,A18,A45,A16,JORDAN1G:4,JORDAN1J:40;
then
A48: go is_sequence_on G by JORDAN1G:4,JORDAN1J:38;
len co >= 1 by A35,XXREAL_0:2;
then 1 in dom co by FINSEQ_3:25;
then
A49: co/.1 = co.1 by PARTFUN1:def 6
.= Gij by A30,JORDAN3:23;
then
A50: LSeg(co,1) = LSeg(Gij,co/.(1+1)) by A35,TOPREAL1:def 3;
A51: {Gij} c= LSeg(co,1) /\ L~<*Gik,Gi1k,Gij*>
proof
let x be object;
assume x in {Gij};
then
A52: x = Gij by TARSKI:def 1;
Gij in LSeg(Gi1k,Gij) by RLTOPSP1:68;
then Gij in LSeg(Gik,Gi1k) \/ LSeg(Gi1k,Gij) by XBOOLE_0:def 3;
then
A53: Gij in L~<*Gik,Gi1k,Gij*> by SPRECT_1:8;
Gij in LSeg(co,1) by A50,RLTOPSP1:68;
hence thesis by A52,A53,XBOOLE_0:def 4;
end;
LSeg(co,1) c= L~co by TOPREAL3:19;
then LSeg(co,1) c= L~LS by A36;
then LSeg(co,1) /\ L~<*Gik,Gi1k,Gij*> c= {Gij} by A8,A13,XBOOLE_1:26;
then
A54: L~<*Gik,Gi1k,Gij*> /\ LSeg(co,1) = {Gij} by A51;
A55: rng co c= L~co by A35,SPPOL_2:18;
A56: len go >= 1+1 by TOPREAL1:def 8;
then reconsider
go as non constant s.c.c. being_S-Seq FinSequence of TOP-REAL 2
by A48,JGRAPH_1:12,JORDAN8:5;
A57: L~go c= L~US by A45,JORDAN3:41;
A58: len go > 1 by A56,NAT_1:13;
then
A59: len go in dom go by FINSEQ_3:25;
then
A60: go/.len go = go.len go by PARTFUN1:def 6
.= Gik by A45,JORDAN3:24;
reconsider m = len go - 1 as Nat by A59,FINSEQ_3:26;
A61: m+1 = len go;
then
A62: len go-'1 = m by NAT_D:34;
m >= 1 by A56,XREAL_1:19;
then
A63: LSeg(go,m) = LSeg(go/.m,Gik) by A60,A61,TOPREAL1:def 3;
A64: {Gik} c= LSeg(go,m) /\ L~<*Gik,Gi1k,Gij*>
proof
let x be object;
assume x in {Gik};
then
A65: x = Gik by TARSKI:def 1;
Gik in LSeg(Gik,Gi1k) by RLTOPSP1:68;
then Gik in LSeg(Gik,Gi1k) \/ LSeg(Gi1k,Gij) by XBOOLE_0:def 3;
then
A66: Gik in L~<*Gik,Gi1k,Gij*> by SPRECT_1:8;
Gik in LSeg(go,m) by A63,RLTOPSP1:68;
hence thesis by A65,A66,XBOOLE_0:def 4;
end;
LSeg(go,m) c= L~go by TOPREAL3:19;
then LSeg(go,m) c= L~US by A57;
then LSeg(go,m) /\ L~<*Gik,Gi1k,Gij*> c= {Gik} by A7,A13,XBOOLE_1:26;
then
A67: LSeg(go,m) /\ L~<*Gik,Gi1k,Gij*> = {Gik} by A64;
A68: go/.1 = US/.1 by A45,SPRECT_3:22
.= Wmin by JORDAN1F:5;
then
A69: Wmin in rng go by FINSEQ_6:42;
A70: LS.1 = LS/.1 by A27,PARTFUN1:def 6
.= Emax by JORDAN1F:6;
A71: L~go /\ L~co c= {go/.1}
proof
let x be object;
assume
A72: x in L~go /\ L~co;
then
A73: x in L~co by XBOOLE_0:def 4;
A74: now
assume x = Emax;
then
A75: Emax = Gij by A30,A70,A73,JORDAN1E:7;
G*(len G,j)`1 = Ebo by A4,A10,A24,JORDAN1A:71;
then Emax`1 <> Ebo by A2,A3,A23,A15,A75,JORDAN1G:7;
hence contradiction by EUCLID:52;
end;
x in L~go by A72,XBOOLE_0:def 4;
then x in L~US /\ L~LS by A57,A36,A73,XBOOLE_0:def 4;
then x in {Wmin,Emax} by JORDAN1E:16;
then x = Wmin or x = Emax by TARSKI:def 2;
hence thesis by A68,A74,TARSKI:def 1;
end;
set W2 = go/.2;
A76: 2 in dom go by A56,FINSEQ_3:25;
go = mid(US,1,Gik..US) by A47,JORDAN1G:49
.= US|(Gik..US) by A47,FINSEQ_4:21,FINSEQ_6:116;
then
A77: W2 = US/.2 by A76,FINSEQ_4:70;
A78: rng go c= L~go by A56,SPPOL_2:18;
A79: go/.1 = LS/.len LS by A68,JORDAN1F:8
.= co/.len co by A30,JORDAN1J:35;
{go/.1} c= L~go /\ L~co
proof
let x be object;
assume x in {go/.1};
then
A80: x = go/.1 by TARSKI:def 1;
then
A81: x in rng go by FINSEQ_6:42;
x in rng co by A79,A80,FINSEQ_6:168;
hence thesis by A78,A55,A81,XBOOLE_0:def 4;
end;
then
A82: L~go /\ L~co = {go/.1} by A71;
now
per cases;
suppose
Gij`1 <> Gik`1 & Gij`2 <> Gik`2;
then pion is being_S-Seq by A21,TOPREAL3:35;
then consider pion1 be FinSequence of TOP-REAL 2 such that
A83: pion1 is_sequence_on G and
A84: pion1 is being_S-Seq and
A85: L~pion = L~pion1 and
A86: pion/.1 = pion1/.1 and
A87: pion/.len pion = pion1/.len pion1 and
A88: len pion <= len pion1 by A44,GOBOARD3:2;
reconsider pion1 as being_S-Seq FinSequence of TOP-REAL 2 by A84;
A89: (go^'pion1)/.len (go^'pion1) = pion/.len pion by A87,FINSEQ_6:156
.= pion/.3 by FINSEQ_1:45
.= co/.1 by A49,FINSEQ_4:18;
A90: go/.len go = pion1/.1 by A60,A86,FINSEQ_4:18;
A91: L~go /\ L~pion1 c= {pion1/.1}
proof
let x be object;
assume
A92: x in L~go /\ L~pion1;
then
A93: x in L~pion1 by XBOOLE_0:def 4;
x in L~go by A92,XBOOLE_0:def 4;
hence thesis by A7,A13,A60,A57,A85,A90,A93,XBOOLE_0:def 4;
end;
len pion1 >= 2+1 by A88,FINSEQ_1:45;
then
A94: len pion1 > 1+1 by NAT_1:13;
then
A95: rng pion1 c= L~pion1 by SPPOL_2:18;
{pion1/.1} c= L~go /\ L~pion1
proof
let x be object;
assume x in {pion1/.1};
then
A96: x = pion1/.1 by TARSKI:def 1;
then
A97: x in rng pion1 by FINSEQ_6:42;
x in rng go by A90,A96,FINSEQ_6:168;
hence thesis by A78,A95,A97,XBOOLE_0:def 4;
end;
then
A98: L~go /\ L~pion1 = {pion1/.1} by A91;
then
A99: (go^'pion1) is s.n.c. by A90,JORDAN1J:54;
A100: pion/.len pion = pion/.3 by FINSEQ_1:45
.= co/.1 by A49,FINSEQ_4:18;
A101: {pion1/.len pion1} c= L~co /\ L~pion1
proof
let x be object;
assume x in {pion1/.len pion1};
then
A102: x = pion1/.len pion1 by TARSKI:def 1;
then
A103: x in rng pion1 by FINSEQ_6:168;
x in rng co by A87,A100,A102,FINSEQ_6:42;
hence thesis by A55,A95,A103,XBOOLE_0:def 4;
end;
L~co /\ L~pion1 c= {pion1/.len pion1}
proof
let x be object;
assume
A104: x in L~co /\ L~pion1;
then
A105: x in L~pion1 by XBOOLE_0:def 4;
x in L~co by A104,XBOOLE_0:def 4;
hence thesis by A8,A13,A49,A36,A85,A87,A100,A105,XBOOLE_0:def 4;
end;
then
A106: L~co /\ L~pion1 = {pion1/.len pion1} by A101;
A107: L~(go^'pion1) /\ L~co = (L~go \/ L~pion1) /\ L~co by A90,TOPREAL8:35
.= {go/.1} \/ {co/.1} by A82,A87,A100,A106,XBOOLE_1:23
.= {(go^'pion1)/.1} \/ {co/.1} by FINSEQ_6:155
.= {(go^'pion1)/.1,co/.1} by ENUMSET1:1;
A108: LA is_an_arc_of E-max C,W-min C by JORDAN6:def 9;
then
A109: LA is connected by JORDAN6:10;
set godo = go^'pion1^'co;
A110: co/.len co = (go^'pion1)/.1 by A79,FINSEQ_6:155;
A111: go^'pion1 is_sequence_on G by A48,A83,A90,TOPREAL8:12;
then
A112: godo is_sequence_on G by A33,A89,TOPREAL8:12;
A113: len pion1-1 >= 1 by A94,XREAL_1:19;
then
A114: len pion1-'1 = len pion1-1 by XREAL_0:def 2;
A115: {Gij} c= LSeg(pion1,len pion1-'1) /\ LSeg(co,1)
proof
let x be object;
assume x in {Gij};
then
A116: x = Gij by TARSKI:def 1;
pion1/.(len pion1-'1+1) = pion/.3 by A87,A114,FINSEQ_1:45
.= Gij by FINSEQ_4:18;
then
A117: Gij in LSeg(pion1,len pion1-'1) by A113,A114,TOPREAL1:21;
Gij in LSeg(co,1) by A50,RLTOPSP1:68;
hence thesis by A116,A117,XBOOLE_0:def 4;
end;
LSeg(pion1,len pion1-'1) c= L~pion by A85,TOPREAL3:19;
then LSeg(pion1,len pion1-'1) /\ LSeg(co,1) c= {Gij} by A54,XBOOLE_1:27;
then
A118: LSeg(pion1,len pion1-'1) /\ LSeg(co,1) = {Gij} by A115;
len pion1-1+1 <= len pion1;
then
A119: len pion1-'1 < len pion1 by A114,NAT_1:13;
len pion1 >= 2+1 by A88,FINSEQ_1:45;
then
A120: len pion1-2 >= 0 by XREAL_1:19;
then len pion1-'2+1 = len pion1-2+1 by XREAL_0:def 2
.= len pion1-'1 by A113,XREAL_0:def 2;
then
A121: LSeg(go^'pion1,len go+(len pion1-'2)) /\ LSeg(co,1) = {(go^'pion1)
/.len (go^'pion1)} by A49,A90,A89,A119,A118,TOPREAL8:31;
rng go /\ rng pion1 c= {pion1/.1} by A78,A95,A98,XBOOLE_1:27;
then
A122: go^'pion1 is one-to-one by JORDAN1J:55;
len (go^'pion1)+1-1 = len go+len pion1-1 by FINSEQ_6:139;
then len (go^'pion1)-1 = len go + (len pion1-2)
.= len go + (len pion1-'2) by A120,XREAL_0:def 2;
then
A123: len (go^'pion1)-'1 = len go + (len pion1-'2) by XREAL_0:def 2;
A124: L~Cage(C,n) = L~US \/ L~LS by JORDAN1E:13;
then
A125: L~US c= L~Cage(C,n) by XBOOLE_1:7;
then
A126: L~go c=L~Cage(C,n) by A57;
A127: {Gik} c= LSeg(go,m) /\ LSeg(pion1,1)
proof
let x be object;
assume x in {Gik};
then
A128: x = Gik by TARSKI:def 1;
A129: Gik in LSeg(go,m) by A63,RLTOPSP1:68;
Gik in LSeg(pion1,1) by A60,A90,A94,TOPREAL1:21;
hence thesis by A128,A129,XBOOLE_0:def 4;
end;
LSeg(pion1,1) c= L~pion by A85,TOPREAL3:19;
then LSeg(go,len go-'1) /\ LSeg(pion1,1) c={Gik} by A62,A67,XBOOLE_1:27;
then LSeg(go,len go-'1) /\ LSeg(pion1,1) = { go/.len go } by A60,A62,A127
;
then
A130: go^'pion1 is unfolded by A90,TOPREAL8:34;
len (go^'pion1) >= len go by TOPREAL8:7;
then
A131: len (go^'pion1) >= 1+1 by A56,XXREAL_0:2;
then
A132: len (go^'pion1) > 1+0 by NAT_1:13;
A133: now
assume
A134: Gik..US <= 1;
Gik..US >= 1 by A47,FINSEQ_4:21;
then Gik..US = 1 by A134,XXREAL_0:1;
then Gik = US/.1 by A47,FINSEQ_5:38;
hence contradiction by A42,A46,JORDAN1F:5;
end;
A135: US is_sequence_on G by JORDAN1G:4;
A136: Wmin`1 = Wbo by EUCLID:52;
set ff = Rotate(Cage(C,n),Wmin);
A137: 1+1 <= len Cage(C,n) by GOBOARD7:34,XXREAL_0:2;
A138: len godo >= len (go^'pion1) by TOPREAL8:7;
then
A139: 1+1 <= len godo by A131,XXREAL_0:2;
(go^'pion1) is non trivial by A131,NAT_D:60;
then reconsider
godo as non constant standard special_circular_sequence by A139,A89,A112
,A130,A123,A121,A99,A122,A107,A110,JORDAN8:4,5,TOPREAL8:11,33,34;
A140: L~godo = L~(go^'pion1) \/ L~co by A89,TOPREAL8:35
.= L~go \/ L~pion1 \/ L~co by A90,TOPREAL8:35;
A141: right_cell(godo,1,G)\L~godo c= RightComp godo by A139,A112,JORDAN9:27;
2 in dom godo by A139,FINSEQ_3:25;
then
A142: godo/.2 in rng godo by PARTFUN2:2;
A143: W-min C in LA by A108,TOPREAL1:1;
Wmin in rng Cage(C,n) by SPRECT_2:43;
then
A144: ff/.1 = Wmin by FINSEQ_6:92;
A145: L~ff = L~Cage(C,n) by REVROT_1:33;
then (W-max L~ff)..ff > 1 by A144,SPRECT_5:22;
then (N-min L~ff)..ff > 1 by A144,A145,SPRECT_5:23,XXREAL_0:2;
then (N-max L~ff)..ff > 1 by A144,A145,SPRECT_5:24,XXREAL_0:2;
then
A146: Emax..ff > 1 by A144,A145,SPRECT_5:25,XXREAL_0:2;
A147: Cage(C,n) is_sequence_on G by JORDAN9:def 1;
then
A148: ff is_sequence_on G by REVROT_1:34;
A149: Gi1k`1 = G*(i1,1)`1 by A1,A6,A11,A16,GOBOARD5:2
.= Gij`1 by A1,A4,A11,A10,GOBOARD5:2;
then
A150: W-bound pio = Gij`1 by SPRECT_1:54;
A151: L~LS c= L~Cage(C,n) by A124,XBOOLE_1:7;
then
A152: L~co c=L~Cage(C,n) by A36;
A153: W-min C in C by SPRECT_1:13;
A154: now
assume W-min C in L~godo;
then
A155: W-min C in L~go \/ L~pion1 or W-min C in L~co by A140,XBOOLE_0:def 3;
per cases by A155,XBOOLE_0:def 3;
suppose
W-min C in L~go;
then C meets L~Cage(C,n) by A126,A153,XBOOLE_0:3;
hence contradiction by JORDAN10:5;
end;
suppose
W-min C in L~pion1;
hence contradiction by A9,A13,A85,A143,XBOOLE_0:3;
end;
suppose
W-min C in L~co;
then C meets L~Cage(C,n) by A152,A153,XBOOLE_0:3;
hence contradiction by JORDAN10:5;
end;
end;
A156: len US >= 2 by A41,XXREAL_0:2;
A157: L~go \/ L~co is compact by COMPTS_1:10;
1+1 <= len Rotate(Cage(C,n),Wmin) by GOBOARD7:34,XXREAL_0:2;
then right_cell(Rotate(Cage(C,n),Wmin),1) = right_cell(ff,1,GoB ff) by
JORDAN1H:23
.= right_cell(ff,1,GoB Cage(C,n)) by REVROT_1:28
.= right_cell(ff,1,G) by JORDAN1H:44
.= right_cell(ff-:Emax,1,G) by A146,A148,JORDAN1J:53
.= right_cell(US,1,G) by JORDAN1E:def 1
.= right_cell(R_Cut(US,Gik),1,G) by A47,A135,A133,JORDAN1J:52
.= right_cell(go^'pion1,1,G) by A58,A111,JORDAN1J:51
.= right_cell(godo,1,G) by A132,A112,JORDAN1J:51;
then W-min C in right_cell(godo,1,G) by JORDAN1I:6;
then
A158: W-min C in right_cell(godo,1,G)\L~godo by A154,XBOOLE_0:def 5;
A159: rng godo c= L~godo by A131,A138,SPPOL_2:18,XXREAL_0:2;
A160: godo/.1 = (go^'pion1)/.1 by FINSEQ_6:155
.= Wmin by A68,FINSEQ_6:155;
A161: Gi1k`1 <= Gik`1 by A1,A2,A3,A6,A16,JORDAN1A:18;
then
A162: W-bound poz = Gi1k`1 by SPRECT_1:54;
W-bound (poz \/ pio) = min(W-bound poz, W-bound pio) by SPRECT_1:47
.= Gij`1 by A149,A162,A150;
then
A163: W-bound L~pion1 = Gij`1 by A85,TOPREAL3:16;
A164: LA c= C by JORDAN6:61;
Gij`1 >= Wbo by A30,A151,PSCOMP_1:24;
then
A165: Gij`1 > Wbo by A38,XXREAL_0:1;
A166: E-max C in LA by A108,TOPREAL1:1;
Wmin in L~go \/ L~co by A78,A69,XBOOLE_0:def 3;
then
A167: W-min (L~go \/ L~co) = Wmin by A126,A152,A157,JORDAN1J:21,XBOOLE_1:8;
(W-min (L~go \/ L~co))`1 = W-bound (L~go \/ L~co) by EUCLID:52;
then W-min (L~go\/L~co\/L~pion1) = W-min (L~go \/ L~co) by A163,A157,A167
,A136,A165,JORDAN1J:33;
then
A168: W-min L~godo = Wmin by A140,A167,XBOOLE_1:4;
godo/.2 = (go^'pion1)/.2 by A131,FINSEQ_6:159
.= US/.2 by A56,A77,FINSEQ_6:159
.= (US^'LS)/.2 by A156,FINSEQ_6:159
.= Rotate(Cage(C,n),Wmin)/.2 by JORDAN1E:11;
then godo/.2 in W-most L~Cage(C,n) by JORDAN1I:25;
then (godo/.2)`1 = (W-min L~godo)`1 by A168,PSCOMP_1:31
.= W-bound L~godo by EUCLID:52;
then godo/.2 in W-most L~godo by A159,A142,SPRECT_2:12;
then Rotate(godo,W-min L~godo)/.2 in W-most L~godo by A160,A168,
FINSEQ_6:89;
then reconsider godo as clockwise_oriented non constant standard
special_circular_sequence by JORDAN1I:25;
len US in dom US by FINSEQ_5:6;
then
A169: US.len US = US/.len US by PARTFUN1:def 6
.= Emax by JORDAN1F:7;
A170: east_halfline E-max C misses L~go
proof
assume east_halfline E-max C meets L~go;
then consider p be object such that
A171: p in east_halfline E-max C and
A172: p in L~go by XBOOLE_0:3;
reconsider p as Point of TOP-REAL 2 by A171;
p in L~US by A57,A172;
then p in east_halfline E-max C /\ L~Cage(C,n) by A125,A171,
XBOOLE_0:def 4;
then
A173: p`1 = Ebo by JORDAN1A:83,PSCOMP_1:50;
then
A174: p = Emax by A57,A172,JORDAN1J:46;
then Emax = Gik by A45,A169,A172,JORDAN1J:43;
then Gik`1 = G*(len G,k)`1 by A6,A16,A24,A173,A174,JORDAN1A:71;
hence contradiction by A3,A19,A22,JORDAN1G:7;
end;
now
assume east_halfline E-max C meets L~godo;
then
A175: east_halfline E-max C meets (L~go \/ L~pion1) or east_halfline
E-max C meets L~co by A140,XBOOLE_1:70;
per cases by A175,XBOOLE_1:70;
suppose
east_halfline E-max C meets L~go;
hence contradiction by A170;
end;
suppose
east_halfline E-max C meets L~pion1;
then consider p be object such that
A176: p in east_halfline E-max C and
A177: p in L~pion1 by XBOOLE_0:3;
reconsider p as Point of TOP-REAL 2 by A176;
A178: p`2 = (E-max C)`2 by A176,TOPREAL1:def 11;
A179: now
per cases by A13,A85,A177,XBOOLE_0:def 3;
suppose
p in poz;
hence p`1 <= Gik`1 by A161,TOPREAL1:3;
end;
suppose
p in pio;
hence p`1 <= Gik`1 by A149,A161,GOBOARD7:5;
end;
end;
i2+1 <= len G by A3,NAT_1:13;
then i2+1-1 <= len G-1 by XREAL_1:9;
then
A180: i2 <= len G-'1 by XREAL_0:def 2;
len G-'1 <= len G by NAT_D:35;
then Gik`1 <= G*(len G-'1,1)`1 by A6,A18,A16,A24,A14,A180,JORDAN1A:18
;
then p`1 <= G*(len G-'1,1)`1 by A179,XXREAL_0:2;
then p`1 <= E-bound C by A14,JORDAN8:12;
then
A181: p`1 <= (E-max C)`1 by EUCLID:52;
p`1 >= (E-max C)`1 by A176,TOPREAL1:def 11;
then p`1 = (E-max C)`1 by A181,XXREAL_0:1;
then p = E-max C by A178,TOPREAL3:6;
hence contradiction by A9,A13,A85,A166,A177,XBOOLE_0:3;
end;
suppose
east_halfline E-max C meets L~co;
then consider p be object such that
A182: p in east_halfline E-max C and
A183: p in L~co by XBOOLE_0:3;
reconsider p as Point of TOP-REAL 2 by A182;
A184: p in LSeg(co,Index(p,co)) by A183,JORDAN3:9;
consider t be Nat such that
A185: t in dom LS and
A186: LS.t = Gij by A32,FINSEQ_2:10;
1 <= t by A185,FINSEQ_3:25;
then
A187: 1 < t by A34,A186,XXREAL_0:1;
t <= len LS by A185,FINSEQ_3:25;
then Index(Gij,LS)+1 = t by A186,A187,JORDAN3:12;
then
A188: len L_Cut(LS,Gij) = len LS-Index(Gij,LS) by A30,A186,JORDAN3:26;
Index(p,co) < len co by A183,JORDAN3:8;
then Index(p,co) < len LS-'Index(Gij,LS) by A188,XREAL_0:def 2;
then Index(p,co)+1 <= len LS-'Index(Gij,LS) by NAT_1:13;
then
A189: Index(p,co) <= len LS-'Index(Gij,LS)-1 by XREAL_1:19;
A190: co = mid(LS,Gij..LS,len LS) by A32,JORDAN1J:37;
p in L~LS by A36,A183;
then p in east_halfline E-max C /\ L~Cage(C,n) by A151,A182,
XBOOLE_0:def 4;
then
A191: p`1 = Ebo by JORDAN1A:83,PSCOMP_1:50;
A192: Index(Gij,LS)+1 = Gij..LS by A34,A32,JORDAN1J:56;
0+Index(Gij,LS) < len LS by A30,JORDAN3:8;
then len LS-Index(Gij,LS) > 0 by XREAL_1:20;
then Index(p,co) <= len LS-Index(Gij,LS)-1 by A189,XREAL_0:def 2;
then Index(p,co) <= len LS-Gij..LS by A192;
then Index(p,co) <= len LS-'Gij..LS by XREAL_0:def 2;
then
A193: Index(p,co) < len LS-'(Gij..LS)+1 by NAT_1:13;
A194: 1<=Index(p,co) by A183,JORDAN3:8;
A195: Gij..LS<=len LS by A32,FINSEQ_4:21;
Gij..LS <> len LS by A31,A32,FINSEQ_4:19;
then
A196: Gij..LS < len LS by A195,XXREAL_0:1;
A197: 1+1 <= len LS by A25,XXREAL_0:2;
then
A198: 2 in dom LS by FINSEQ_3:25;
set tt = Index(p,co)+(Gij..LS)-'1;
set RC = Rotate(Cage(C,n),Emax);
A199: E-max C in right_cell(RC,1) by JORDAN1I:7;
A200: GoB RC = GoB Cage(C,n) by REVROT_1:28
.= G by JORDAN1H:44;
A201: L~RC = L~Cage(C,n) by REVROT_1:33;
consider jj2 be Nat such that
A202: 1 <= jj2 and
A203: jj2 <= width G and
A204: Emax = G*(len G,jj2) by JORDAN1D:25;
A205: len G >= 4 by JORDAN8:10;
then len G >= 1 by XXREAL_0:2;
then
A206: [len G,jj2] in Indices G by A202,A203,MATRIX_0:30;
A207: len RC = len Cage(C,n) by FINSEQ_6:179;
LS = RC-:Wmin by JORDAN1G:18;
then
A208: LSeg(LS,1) = LSeg(RC,1) by A197,SPPOL_2:9;
A209: Emax in rng Cage(C,n) by SPRECT_2:46;
RC is_sequence_on G by A147,REVROT_1:34;
then consider ii,jj be Nat such that
A210: [ii,jj+1] in Indices G and
A211: [ii,jj] in Indices G and
A212: RC/.1 = G*(ii,jj+1) and
A213: RC/.(1+1) = G*(ii,jj) by A137,A201,A207,A209,FINSEQ_6:92,JORDAN1I:23;
A214: jj+1+1 <> jj;
A215: 1 <= jj by A211,MATRIX_0:32;
RC/.1 = E-max L~RC by A201,A209,FINSEQ_6:92;
then
A216: ii = len G by A201,A210,A212,A204,A206,GOBOARD1:5;
then ii-1 >= 4-1 by A205,XREAL_1:9;
then
A217: ii-1 >= 1 by XXREAL_0:2;
then
A218: 1 <= ii-'1 by XREAL_0:def 2;
A219: jj <= width G by A211,MATRIX_0:32;
then
A220: G*(len G,jj)`1 = Ebo by A24,A215,JORDAN1A:71;
A221: jj+1 <= width G by A210,MATRIX_0:32;
ii+1 <> ii;
then
A222: right_cell(RC,1) = cell(G,ii-'1,jj) by A137,A207,A200,A210,A211,A212
,A213,A214,GOBOARD5:def 6;
A223: ii <= len G by A211,MATRIX_0:32;
A224: 1 <= ii by A211,MATRIX_0:32;
A225: ii <= len G by A210,MATRIX_0:32;
A226: 1 <= jj+1 by A210,MATRIX_0:32;
then
A227: Ebo = G*(len G,jj+1)`1 by A24,A221,JORDAN1A:71;
A228: 1 <= ii by A210,MATRIX_0:32;
then
A229: ii-'1+1 = ii by XREAL_1:235;
then
A230: ii-'1 < len G by A225,NAT_1:13;
then
A231: G*(ii-'1,jj+1)`2 = G*(1,jj+1)`2 by A226,A221,A218,GOBOARD5:1
.= G*(ii,jj+1)`2 by A228,A225,A226,A221,GOBOARD5:1;
A232: (E-max C)`2 = p`2 by A182,TOPREAL1:def 11;
then
A233: p`2 <= G*(ii-'1,jj+1)`2 by A199,A225,A221,A215,A222,A229,A217,
JORDAN9:17;
A234: G*(ii-'1,jj)`2 = G*(1,jj)`2 by A215,A219,A218,A230,GOBOARD5:1
.= G*(ii,jj)`2 by A224,A223,A215,A219,GOBOARD5:1;
G*(ii-'1,jj)`2 <= p`2 by A232,A199,A225,A221,A215,A222,A229,A217,
JORDAN9:17;
then p in LSeg(RC/.1,RC/.(1+1)) by A191,A212,A213,A216,A233,A234,A231
,A220,A227,GOBOARD7:7;
then
A235: p in LSeg(LS,1) by A137,A208,A207,TOPREAL1:def 3;
1<=Gij..LS by A32,FINSEQ_4:21;
then
A236: LSeg(mid(LS,Gij..LS,len LS),Index(p,co)) = LSeg(LS,Index(p,co)
+(Gij..LS)-'1) by A196,A194,A193,JORDAN4:19;
1<=Index(Gij,LS) by A30,JORDAN3:8;
then
A237: 1+1 <= Gij..LS by A192,XREAL_1:7;
then Index(p,co)+Gij..LS >= 1+1+1 by A194,XREAL_1:7;
then Index(p,co)+Gij..LS-1 >= 1+1+1-1 by XREAL_1:9;
then
A238: tt >= 1+1 by XREAL_0:def 2;
now
per cases by A238,XXREAL_0:1;
suppose
tt > 1+1;
then LSeg(LS,1) misses LSeg(LS,tt) by TOPREAL1:def 7;
hence contradiction by A235,A184,A190,A236,XBOOLE_0:3;
end;
suppose
A239: tt = 1+1;
then 1+1 = Index(p,co)+(Gij..LS)-1 by XREAL_0:def 2;
then 1+1+1 = Index(p,co)+(Gij..LS);
then
A240: Gij..LS = 2 by A194,A237,JORDAN1E:6;
LSeg(LS,1) /\ LSeg(LS,tt) = {LS/.2} by A25,A239,TOPREAL1:def 6;
then p in {LS/.2} by A235,A184,A190,A236,XBOOLE_0:def 4;
then
A241: p = LS/.2 by TARSKI:def 1;
then
A242: p in rng LS by A198,PARTFUN2:2;
p..LS = 2 by A198,A241,FINSEQ_5:41;
then p = Gij by A32,A240,A242,FINSEQ_5:9;
then Gij`1 = Ebo by A241,JORDAN1G:32;
then Gij`1 = G*(len G,j)`1 by A4,A10,A24,JORDAN1A:71;
hence contradiction by A2,A3,A23,A15,JORDAN1G:7;
end;
end;
hence contradiction;
end;
end;
then east_halfline E-max C c= (L~godo)` by SUBSET_1:23;
then consider W be Subset of TOP-REAL 2 such that
A243: W is_a_component_of (L~godo)` and
A244: east_halfline E-max C c= W by GOBOARD9:3;
W is not bounded by A244,JORDAN2C:121,RLTOPSP1:42;
then W is_outside_component_of L~godo by A243,JORDAN2C:def 3;
then W c= UBD L~godo by JORDAN2C:23;
then
A245: east_halfline E-max C c= UBD L~godo by A244;
E-max C in east_halfline E-max C by TOPREAL1:38;
then E-max C in UBD L~godo by A245;
then E-max C in LeftComp godo by GOBRD14:36;
then LA meets L~godo by A109,A143,A166,A141,A158,JORDAN1J:36;
then
A246: LA meets (L~go \/ L~pion1) or LA meets L~co by A140,XBOOLE_1:70;
now
per cases by A246,XBOOLE_1:70;
suppose
LA meets L~go;
then LA meets L~Cage(C,n) by A57,A125,XBOOLE_1:1,63;
hence contradiction by A164,JORDAN10:5,XBOOLE_1:63;
end;
suppose
LA meets L~pion1;
hence contradiction by A9,A13,A85;
end;
suppose
LA meets L~co;
then LA meets L~Cage(C,n) by A36,A151,XBOOLE_1:1,63;
hence contradiction by A164,JORDAN10:5,XBOOLE_1:63;
end;
end;
hence contradiction;
end;
suppose
Gij`1 = Gik`1;
then
A247: i1 = i2 by A23,A19,JORDAN1G:7;
then poz = {Gi1k} by RLTOPSP1:70;
then poz c= pio by A40,ZFMISC_1:31;
then pio \/ poz = pio by XBOOLE_1:12;
hence contradiction by A1,A3,A4,A5,A6,A7,A8,A9,A247,JORDAN1J:58;
end;
suppose
Gij`2 = Gik`2;
then
A248: j = k by A23,A19,JORDAN1G:6;
then pio = {Gi1k} by RLTOPSP1:70;
then pio c= poz by A39,ZFMISC_1:31;
then pio \/ poz = poz by XBOOLE_1:12;
hence contradiction by A1,A2,A3,A4,A6,A7,A8,A9,A248,Th28;
end;
end;
hence contradiction;
end;
theorem Th46:
for C be Simple_closed_curve for i1,i2,j,k be Nat st
1 < i2 & i2 <= i1 & i1 < len Gauge(C,n) & 1 <= j & j <= k & k <= width Gauge(C,
n) & (LSeg(Gauge(C,n)*(i1,j),Gauge(C,n)*(i1,k)) \/ LSeg(Gauge(C,n)*(i1,k),Gauge
(C,n)*(i2,k))) /\ L~Upper_Seq(C,n) = {Gauge(C,n)*(i2,k)} & (LSeg(Gauge(C,n)*(i1
,j),Gauge(C,n)*(i1,k)) \/ LSeg(Gauge(C,n)*(i1,k),Gauge(C,n)*(i2,k))) /\ L~
Lower_Seq(C,n) = {Gauge(C,n)*(i1,j)} holds (LSeg(Gauge(C,n)*(i1,j),Gauge(C,n)*(
i1,k)) \/ LSeg(Gauge(C,n)*(i1,k),Gauge(C,n)*(i2,k))) meets Upper_Arc C
proof
let C be Simple_closed_curve;
let i1,i2,j,k be Nat;
set G = Gauge(C,n);
set pio = LSeg(G*(i1,j),G*(i1,k));
set poz = LSeg(G*(i1,k),G*(i2,k));
set US = Upper_Seq(C,n);
set LS = Lower_Seq(C,n);
assume that
A1: 1 < i2 and
A2: i2 <= i1 and
A3: i1 < len G and
A4: 1 <= j and
A5: j <= k and
A6: k <= width G and
A7: (pio \/ poz) /\ L~US = {G*(i2,k)} and
A8: (pio \/ poz) /\ L~LS = {G*(i1,j)} and
A9: (pio \/ poz) misses Upper_Arc C;
set Gi1k = G*(i1,k);
set Gik = G*(i2,k);
A10: 1 <= k by A4,A5,XXREAL_0:2;
A11: i2 < len G by A2,A3,XXREAL_0:2;
then
A12: [i2,k] in Indices G by A1,A6,A10,MATRIX_0:30;
set Wmin = W-min L~Cage(C,n);
set Wbo = W-bound L~Cage(C,n);
A13: len G = width G by JORDAN8:def 1;
set go = R_Cut(US,Gik);
A14: len US >= 3 by JORDAN1E:15;
then len US >= 1 by XXREAL_0:2;
then 1 in dom US by FINSEQ_3:25;
then
A15: US.1 = US/.1 by PARTFUN1:def 6
.= Wmin by JORDAN1F:5;
set Gij = G*(i1,j);
set co = L_Cut(LS,Gij);
Gij in {Gij} by TARSKI:def 1;
then
A16: Gij in L~LS by A8,XBOOLE_0:def 4;
A17: 1 < i1 by A1,A2,XXREAL_0:2;
then
A18: Gi1k`2 = G*(1,k)`2 by A3,A6,A10,GOBOARD5:1
.= Gik`2 by A1,A6,A11,A10,GOBOARD5:1;
A19: j <= width G by A5,A6,XXREAL_0:2;
then
A20: [i1,j] in Indices G by A3,A4,A17,MATRIX_0:30;
len G >= 4 by JORDAN8:10;
then
A21: len G >= 1 by XXREAL_0:2;
then
A22: [len G,j] in Indices G by A4,A19,MATRIX_0:30;
A23: [1,k] in Indices G by A6,A10,A21,MATRIX_0:30;
A24: now
assume Gik`1 = Wbo;
then G*(1,k)`1 = G*(i2,k)`1 by A6,A10,A13,JORDAN1A:73;
hence contradiction by A1,A12,A23,JORDAN1G:7;
end;
A25: [i1,j] in Indices G by A3,A4,A17,A19,MATRIX_0:30;
set pion = <*Gik,Gi1k,Gij*>;
A26: Gi1k in poz by RLTOPSP1:68;
set UA = Upper_Arc C;
A27: Gi1k in pio by RLTOPSP1:68;
A28: [i1,k] in Indices G by A3,A6,A17,A10,MATRIX_0:30;
A29: now
let n be Nat;
assume n in dom pion;
then n in {1,2,3} by FINSEQ_1:89,FINSEQ_3:1;
then n = 1 or n = 2 or n = 3 by ENUMSET1:def 1;
hence
ex i,j be Nat st [i,j] in Indices G & pion/.n = G*(i,j) by A25
,A12,A28,FINSEQ_4:18;
end;
Gik in {Gik} by TARSKI:def 1;
then
A30: Gik in L~US by A7,XBOOLE_0:def 4;
set Emax = E-max L~Cage(C,n);
A31: len LS >= 1+2 by JORDAN1E:15;
then
A32: len LS >= 1 by XXREAL_0:2;
then
A33: 1 in dom LS by FINSEQ_3:25;
then
A34: LS.1 = LS/.1 by PARTFUN1:def 6
.= Emax by JORDAN1F:6;
len LS in dom LS by A32,FINSEQ_3:25;
then
A35: LS.len LS = LS/.len LS by PARTFUN1:def 6
.= Wmin by JORDAN1F:8;
set Ebo = E-bound L~Cage(C,n);
A36: L~<*Gik,Gi1k,Gij*> = poz \/ pio by TOPREAL3:16;
Wmin`1 = Wbo by EUCLID:52
.= G*(1,k)`1 by A6,A10,A13,JORDAN1A:73;
then
A37: Gik <> US.1 by A1,A12,A15,A23,JORDAN1G:7;
then reconsider go as being_S-Seq FinSequence of TOP-REAL 2 by A30,JORDAN3:35
;
A38: Gik in rng US by A1,A6,A11,A30,A10,JORDAN1G:4,JORDAN1J:40;
then
A39: go is_sequence_on G by JORDAN1G:4,JORDAN1J:38;
Gi1k`1 = G*(i1,1)`1 by A3,A6,A17,A10,GOBOARD5:2
.= Gij`1 by A3,A4,A17,A19,GOBOARD5:2;
then
A40: Gi1k = |[Gij`1,Gik`2]| by A18,EUCLID:53;
A41: [len G,k] in Indices G by A6,A10,A21,MATRIX_0:30;
A42: len go >= 1+1 by TOPREAL1:def 8;
Wmin`1 = Wbo by EUCLID:52
.= G*(1,k)`1 by A6,A10,A13,JORDAN1A:73;
then
A43: Gij <> LS.len LS by A1,A2,A23,A35,A20,JORDAN1G:7;
then reconsider co as being_S-Seq FinSequence of TOP-REAL 2 by A16,JORDAN3:34
;
A44: Gij in rng LS by A3,A4,A17,A16,A19,JORDAN1G:5,JORDAN1J:40;
then
A45: co is_sequence_on G by JORDAN1G:5,JORDAN1J:39;
Emax`1 = Ebo by EUCLID:52
.= G*(len G,k)`1 by A6,A10,A13,JORDAN1A:71;
then
A46: Gij <> LS.1 by A3,A20,A41,A34,JORDAN1G:7;
A47: len co >= 1+1 by TOPREAL1:def 8;
then reconsider
co as non constant s.c.c. being_S-Seq FinSequence of TOP-REAL 2
by A45,JGRAPH_1:12,JORDAN8:5;
A48: L~co c= L~LS by A16,JORDAN3:42;
len co >= 1 by A47,XXREAL_0:2;
then 1 in dom co by FINSEQ_3:25;
then
A49: co/.1 = co.1 by PARTFUN1:def 6
.= Gij by A16,JORDAN3:23;
then
A50: LSeg(co,1) = LSeg(Gij,co/.(1+1)) by A47,TOPREAL1:def 3;
A51: {Gij} c= LSeg(co,1) /\ L~<*Gik,Gi1k,Gij*>
proof
let x be object;
assume x in {Gij};
then
A52: x = Gij by TARSKI:def 1;
Gij in LSeg(Gi1k,Gij) by RLTOPSP1:68;
then Gij in LSeg(Gik,Gi1k) \/ LSeg(Gi1k,Gij) by XBOOLE_0:def 3;
then
A53: Gij in L~<*Gik,Gi1k,Gij*> by SPRECT_1:8;
Gij in LSeg(co,1) by A50,RLTOPSP1:68;
hence thesis by A52,A53,XBOOLE_0:def 4;
end;
LSeg(co,1) c= L~co by TOPREAL3:19;
then LSeg(co,1) c= L~LS by A48;
then LSeg(co,1) /\ L~<*Gik,Gi1k,Gij*> c= {Gij} by A8,A36,XBOOLE_1:26;
then
A54: L~<*Gik,Gi1k,Gij*> /\ LSeg(co,1) = {Gij} by A51;
A55: rng co c= L~co by A47,SPPOL_2:18;
reconsider go as non constant s.c.c. being_S-Seq FinSequence of TOP-REAL 2
by A42,A39,JGRAPH_1:12,JORDAN8:5;
A56: L~go c= L~US by A30,JORDAN3:41;
A57: len go > 1 by A42,NAT_1:13;
then
A58: len go in dom go by FINSEQ_3:25;
then
A59: go/.len go = go.len go by PARTFUN1:def 6
.= Gik by A30,JORDAN3:24;
reconsider m = len go - 1 as Nat by A58,FINSEQ_3:26;
A60: m+1 = len go;
then
A61: len go-'1 = m by NAT_D:34;
m >= 1 by A42,XREAL_1:19;
then
A62: LSeg(go,m) = LSeg(go/.m,Gik) by A59,A60,TOPREAL1:def 3;
A63: {Gik} c= LSeg(go,m) /\ L~<*Gik,Gi1k,Gij*>
proof
let x be object;
assume x in {Gik};
then
A64: x = Gik by TARSKI:def 1;
Gik in LSeg(Gik,Gi1k) by RLTOPSP1:68;
then Gik in LSeg(Gik,Gi1k) \/ LSeg(Gi1k,Gij) by XBOOLE_0:def 3;
then
A65: Gik in L~<*Gik,Gi1k,Gij*> by SPRECT_1:8;
Gik in LSeg(go,m) by A62,RLTOPSP1:68;
hence thesis by A64,A65,XBOOLE_0:def 4;
end;
LSeg(go,m) c= L~go by TOPREAL3:19;
then LSeg(go,m) c= L~US by A56;
then LSeg(go,m) /\ L~<*Gik,Gi1k,Gij*> c= {Gik} by A7,A36,XBOOLE_1:26;
then
A66: LSeg(go,m) /\ L~<*Gik,Gi1k,Gij*> = {Gik} by A63;
A67: go/.1 = US/.1 by A30,SPRECT_3:22
.= Wmin by JORDAN1F:5;
then
A68: Wmin in rng go by FINSEQ_6:42;
A69: LS.1 = LS/.1 by A33,PARTFUN1:def 6
.= Emax by JORDAN1F:6;
A70: L~go /\ L~co c= {go/.1}
proof
let x be object;
assume
A71: x in L~go /\ L~co;
then
A72: x in L~co by XBOOLE_0:def 4;
A73: now
assume x = Emax;
then
A74: Emax = Gij by A16,A69,A72,JORDAN1E:7;
G*(len G,j)`1 = Ebo by A4,A19,A13,JORDAN1A:71;
then Emax`1 <> Ebo by A3,A25,A22,A74,JORDAN1G:7;
hence contradiction by EUCLID:52;
end;
x in L~go by A71,XBOOLE_0:def 4;
then x in L~US /\ L~LS by A56,A48,A72,XBOOLE_0:def 4;
then x in {Wmin,Emax} by JORDAN1E:16;
then x = Wmin or x = Emax by TARSKI:def 2;
hence thesis by A67,A73,TARSKI:def 1;
end;
set W2 = go/.2;
A75: 2 in dom go by A42,FINSEQ_3:25;
go = mid(US,1,Gik..US) by A38,JORDAN1G:49
.= US|(Gik..US) by A38,FINSEQ_4:21,FINSEQ_6:116;
then
A76: W2 = US/.2 by A75,FINSEQ_4:70;
A77: rng go c= L~go by A42,SPPOL_2:18;
A78: go/.1 = LS/.len LS by A67,JORDAN1F:8
.= co/.len co by A16,JORDAN1J:35;
{go/.1} c= L~go /\ L~co
proof
let x be object;
assume x in {go/.1};
then
A79: x = go/.1 by TARSKI:def 1;
then
A80: x in rng go by FINSEQ_6:42;
x in rng co by A78,A79,FINSEQ_6:168;
hence thesis by A77,A55,A80,XBOOLE_0:def 4;
end;
then
A81: L~go /\ L~co = {go/.1} by A70;
now
per cases;
suppose
Gij`1 <> Gik`1 & Gij`2 <> Gik`2;
then pion is being_S-Seq by A40,TOPREAL3:35;
then consider pion1 be FinSequence of TOP-REAL 2 such that
A82: pion1 is_sequence_on G and
A83: pion1 is being_S-Seq and
A84: L~pion = L~pion1 and
A85: pion/.1 = pion1/.1 and
A86: pion/.len pion = pion1/.len pion1 and
A87: len pion <= len pion1 by A29,GOBOARD3:2;
reconsider pion1 as being_S-Seq FinSequence of TOP-REAL 2 by A83;
A88: (go^'pion1)/.len (go^'pion1) = pion/.len pion by A86,FINSEQ_6:156
.= pion/.3 by FINSEQ_1:45
.= co/.1 by A49,FINSEQ_4:18;
A89: go/.len go = pion1/.1 by A59,A85,FINSEQ_4:18;
A90: L~go /\ L~pion1 c= {pion1/.1}
proof
let x be object;
assume
A91: x in L~go /\ L~pion1;
then
A92: x in L~pion1 by XBOOLE_0:def 4;
x in L~go by A91,XBOOLE_0:def 4;
hence thesis by A7,A36,A59,A56,A84,A89,A92,XBOOLE_0:def 4;
end;
len pion1 >= 2+1 by A87,FINSEQ_1:45;
then
A93: len pion1 > 1+1 by NAT_1:13;
then
A94: rng pion1 c= L~pion1 by SPPOL_2:18;
{pion1/.1} c= L~go /\ L~pion1
proof
let x be object;
assume x in {pion1/.1};
then
A95: x = pion1/.1 by TARSKI:def 1;
then
A96: x in rng pion1 by FINSEQ_6:42;
x in rng go by A89,A95,FINSEQ_6:168;
hence thesis by A77,A94,A96,XBOOLE_0:def 4;
end;
then
A97: L~go /\ L~pion1 = {pion1/.1} by A90;
then
A98: (go^'pion1) is s.n.c. by A89,JORDAN1J:54;
A99: {Gik} c= LSeg(go,m) /\ LSeg(pion1,1)
proof
let x be object;
assume x in {Gik};
then
A100: x = Gik by TARSKI:def 1;
A101: Gik in LSeg(go,m) by A62,RLTOPSP1:68;
Gik in LSeg(pion1,1) by A59,A89,A93,TOPREAL1:21;
hence thesis by A100,A101,XBOOLE_0:def 4;
end;
LSeg(pion1,1) c= L~pion by A84,TOPREAL3:19;
then LSeg(go,len go-'1) /\ LSeg(pion1,1) c={Gik} by A61,A66,XBOOLE_1:27;
then LSeg(go,len go-'1) /\ LSeg(pion1,1) = { go/.len go } by A59,A61,A99;
then
A102: go^'pion1 is unfolded by A89,TOPREAL8:34;
len (go^'pion1) >= len go by TOPREAL8:7;
then
A103: len (go^'pion1) >= 1+1 by A42,XXREAL_0:2;
then
A104: len (go^'pion1) > 1+0 by NAT_1:13;
A105: pion/.len pion = pion/.3 by FINSEQ_1:45
.= co/.1 by A49,FINSEQ_4:18;
A106: {pion1/.len pion1} c= L~co /\ L~pion1
proof
let x be object;
assume x in {pion1/.len pion1};
then
A107: x = pion1/.len pion1 by TARSKI:def 1;
then
A108: x in rng pion1 by FINSEQ_6:168;
x in rng co by A86,A105,A107,FINSEQ_6:42;
hence thesis by A55,A94,A108,XBOOLE_0:def 4;
end;
L~co /\ L~pion1 c= {pion1/.len pion1}
proof
let x be object;
assume
A109: x in L~co /\ L~pion1;
then
A110: x in L~pion1 by XBOOLE_0:def 4;
x in L~co by A109,XBOOLE_0:def 4;
hence thesis by A8,A36,A49,A48,A84,A86,A105,A110,XBOOLE_0:def 4;
end;
then
A111: L~co /\ L~pion1 = {pion1/.len pion1} by A106;
A112: L~(go^'pion1) /\ L~co = (L~go \/ L~pion1) /\ L~co by A89,TOPREAL8:35
.= {go/.1} \/ {co/.1} by A81,A86,A105,A111,XBOOLE_1:23
.= {(go^'pion1)/.1} \/ {co/.1} by FINSEQ_6:155
.= {(go^'pion1)/.1,co/.1} by ENUMSET1:1;
A113: UA is_an_arc_of W-min C,E-max C by JORDAN6:def 8;
then
A114: UA is connected by JORDAN6:10;
set godo = go^'pion1^'co;
A115: co/.len co = (go^'pion1)/.1 by A78,FINSEQ_6:155;
A116: go^'pion1 is_sequence_on G by A39,A82,A89,TOPREAL8:12;
then
A117: godo is_sequence_on G by A45,A88,TOPREAL8:12;
A118: len pion1-1 >= 1 by A93,XREAL_1:19;
then
A119: len pion1-'1 = len pion1-1 by XREAL_0:def 2;
A120: {Gij} c= LSeg(pion1,len pion1-'1) /\ LSeg(co,1)
proof
let x be object;
assume x in {Gij};
then
A121: x = Gij by TARSKI:def 1;
pion1/.(len pion1-'1+1) = pion/.3 by A86,A119,FINSEQ_1:45
.= Gij by FINSEQ_4:18;
then
A122: Gij in LSeg(pion1,len pion1-'1) by A118,A119,TOPREAL1:21;
Gij in LSeg(co,1) by A50,RLTOPSP1:68;
hence thesis by A121,A122,XBOOLE_0:def 4;
end;
LSeg(pion1,len pion1-'1) c= L~pion by A84,TOPREAL3:19;
then LSeg(pion1,len pion1-'1) /\ LSeg(co,1) c= {Gij} by A54,XBOOLE_1:27;
then
A123: LSeg(pion1,len pion1-'1) /\ LSeg(co,1) = {Gij} by A120;
len pion1-1+1 <= len pion1;
then
A124: len pion1-'1 < len pion1 by A119,NAT_1:13;
len pion1 >= 2+1 by A87,FINSEQ_1:45;
then
A125: len pion1-2 >= 0 by XREAL_1:19;
then len pion1-'2+1 = len pion1-2+1 by XREAL_0:def 2
.= len pion1-'1 by A118,XREAL_0:def 2;
then
A126: LSeg(go^'pion1,len go+(len pion1-'2)) /\ LSeg(co,1) = {(go^'pion1)
/.len (go^'pion1)} by A49,A89,A88,A124,A123,TOPREAL8:31;
rng go /\ rng pion1 c= {pion1/.1} by A77,A94,A97,XBOOLE_1:27;
then
A127: go^'pion1 is one-to-one by JORDAN1J:55;
len (go^'pion1)+1-1 = len go+len pion1-1 by FINSEQ_6:139;
then len (go^'pion1)-1 = len go + (len pion1-2)
.= len go + (len pion1-'2) by A125,XREAL_0:def 2;
then
A128: len (go^'pion1)-'1 = len go + (len pion1-'2) by XREAL_0:def 2;
A129: L~Cage(C,n) = L~US \/ L~LS by JORDAN1E:13;
then
A130: L~US c= L~Cage(C,n) by XBOOLE_1:7;
then
A131: L~go c=L~Cage(C,n) by A56;
A132: len godo >= len (go^'pion1) by TOPREAL8:7;
then
A133: 1+1 <= len godo by A103,XXREAL_0:2;
(go^'pion1) is non trivial by A103,NAT_D:60;
then reconsider
godo as non constant standard special_circular_sequence by A133,A88,A117
,A102,A128,A126,A98,A127,A112,A115,JORDAN8:4,5,TOPREAL8:11,33,34;
A134: L~godo = L~(go^'pion1) \/ L~co by A88,TOPREAL8:35
.= L~go \/ L~pion1 \/ L~co by A89,TOPREAL8:35;
A135: now
assume
A136: Gik..US <= 1;
Gik..US >= 1 by A38,FINSEQ_4:21;
then Gik..US = 1 by A136,XXREAL_0:1;
then Gik = US/.1 by A38,FINSEQ_5:38;
hence contradiction by A15,A37,JORDAN1F:5;
end;
A137: US is_sequence_on G by JORDAN1G:4;
A138: Gik`1 <= Gi1k`1 by A1,A2,A3,A6,A10,JORDAN1A:18;
then
A139: W-bound poz = Gik`1 by SPRECT_1:54;
A140: Gi1k`1 = G*(i1,1)`1 by A3,A6,A17,A10,GOBOARD5:2
.= Gij`1 by A3,A4,A17,A19,GOBOARD5:2;
then
A141: W-bound pio = Gij`1 by SPRECT_1:54;
W-bound (poz \/ pio) = min(W-bound poz, W-bound pio) by SPRECT_1:47
.= Gik`1 by A140,A138,A139,A141,XXREAL_0:def 9;
then
A142: W-bound L~pion1 = Gik`1 by A84,TOPREAL3:16;
A143: UA c= C by JORDAN6:61;
Gik`1 >= Wbo by A30,A130,PSCOMP_1:24;
then
A144: Gik`1 > Wbo by A24,XXREAL_0:1;
A145: len US >= 2 by A14,XXREAL_0:2;
A146: L~go \/ L~co is compact by COMPTS_1:10;
A147: L~LS c= L~Cage(C,n) by A129,XBOOLE_1:7;
then
A148: L~co c=L~Cage(C,n) by A48;
A149: right_cell(godo,1,G)\L~godo c= RightComp godo by A133,A117,JORDAN9:27;
2 in dom godo by A133,FINSEQ_3:25;
then
A150: godo/.2 in rng godo by PARTFUN2:2;
A151: rng godo c= L~godo by A103,A132,SPPOL_2:18,XXREAL_0:2;
A152: godo/.1 = (go^'pion1)/.1 by FINSEQ_6:155
.= Wmin by A67,FINSEQ_6:155;
A153: W-min C in UA by A113,TOPREAL1:1;
A154: W-min C in C by SPRECT_1:13;
A155: now
assume W-min C in L~godo;
then
A156: W-min C in L~go \/ L~pion1 or W-min C in L~co by A134,XBOOLE_0:def 3;
per cases by A156,XBOOLE_0:def 3;
suppose
W-min C in L~go;
then C meets L~Cage(C,n) by A131,A154,XBOOLE_0:3;
hence contradiction by JORDAN10:5;
end;
suppose
W-min C in L~pion1;
hence contradiction by A9,A36,A84,A153,XBOOLE_0:3;
end;
suppose
W-min C in L~co;
then C meets L~Cage(C,n) by A148,A154,XBOOLE_0:3;
hence contradiction by JORDAN10:5;
end;
end;
A157: Wmin`1 = Wbo by EUCLID:52;
set ff = Rotate(Cage(C,n),Wmin);
A158: 1+1 <= len Cage(C,n) by GOBOARD7:34,XXREAL_0:2;
Wmin in rng Cage(C,n) by SPRECT_2:43;
then
A159: ff/.1 = Wmin by FINSEQ_6:92;
A160: L~ff = L~Cage(C,n) by REVROT_1:33;
then (W-max L~ff)..ff > 1 by A159,SPRECT_5:22;
then (N-min L~ff)..ff > 1 by A159,A160,SPRECT_5:23,XXREAL_0:2;
then (N-max L~ff)..ff > 1 by A159,A160,SPRECT_5:24,XXREAL_0:2;
then
A161: Emax..ff > 1 by A159,A160,SPRECT_5:25,XXREAL_0:2;
A162: Cage(C,n) is_sequence_on G by JORDAN9:def 1;
then
A163: ff is_sequence_on G by REVROT_1:34;
1+1 <= len Rotate(Cage(C,n),Wmin) by GOBOARD7:34,XXREAL_0:2;
then right_cell(Rotate(Cage(C,n),Wmin),1) = right_cell(ff,1,GoB ff) by
JORDAN1H:23
.= right_cell(ff,1,GoB Cage(C,n)) by REVROT_1:28
.= right_cell(ff,1,G) by JORDAN1H:44
.= right_cell(ff-:Emax,1,G) by A161,A163,JORDAN1J:53
.= right_cell(US,1,G) by JORDAN1E:def 1
.= right_cell(R_Cut(US,Gik),1,G) by A38,A137,A135,JORDAN1J:52
.= right_cell(go^'pion1,1,G) by A57,A116,JORDAN1J:51
.= right_cell(godo,1,G) by A104,A117,JORDAN1J:51;
then W-min C in right_cell(godo,1,G) by JORDAN1I:6;
then
A164: W-min C in right_cell(godo,1,G)\L~godo by A155,XBOOLE_0:def 5;
A165: E-max C in UA by A113,TOPREAL1:1;
Wmin in L~go \/ L~co by A77,A68,XBOOLE_0:def 3;
then
A166: W-min (L~go \/ L~co) = Wmin by A131,A148,A146,JORDAN1J:21,XBOOLE_1:8;
(W-min (L~go \/ L~co))`1 = W-bound (L~go \/ L~co) by EUCLID:52;
then W-min (L~go\/L~co\/L~pion1) = W-min (L~go \/ L~co) by A142,A146,A166
,A157,A144,JORDAN1J:33;
then
A167: W-min L~godo = Wmin by A134,A166,XBOOLE_1:4;
godo/.2 = (go^'pion1)/.2 by A103,FINSEQ_6:159
.= US/.2 by A42,A76,FINSEQ_6:159
.= (US^'LS)/.2 by A145,FINSEQ_6:159
.= Rotate(Cage(C,n),Wmin)/.2 by JORDAN1E:11;
then godo/.2 in W-most L~Cage(C,n) by JORDAN1I:25;
then (godo/.2)`1 = (W-min L~godo)`1 by A167,PSCOMP_1:31
.= W-bound L~godo by EUCLID:52;
then godo/.2 in W-most L~godo by A151,A150,SPRECT_2:12;
then Rotate(godo,W-min L~godo)/.2 in W-most L~godo by A152,A167,
FINSEQ_6:89;
then reconsider godo as clockwise_oriented non constant standard
special_circular_sequence by JORDAN1I:25;
len US in dom US by FINSEQ_5:6;
then
A168: US.len US = US/.len US by PARTFUN1:def 6
.= Emax by JORDAN1F:7;
A169: east_halfline E-max C misses L~go
proof
assume east_halfline E-max C meets L~go;
then consider p be object such that
A170: p in east_halfline E-max C and
A171: p in L~go by XBOOLE_0:3;
reconsider p as Point of TOP-REAL 2 by A170;
p in L~US by A56,A171;
then p in east_halfline E-max C /\ L~Cage(C,n) by A130,A170,
XBOOLE_0:def 4;
then
A172: p`1 = Ebo by JORDAN1A:83,PSCOMP_1:50;
then
A173: p = Emax by A56,A171,JORDAN1J:46;
then Emax = Gik by A30,A168,A171,JORDAN1J:43;
then Gik`1 = G*(len G,k)`1 by A6,A10,A13,A172,A173,JORDAN1A:71;
hence contradiction by A2,A3,A12,A41,JORDAN1G:7;
end;
now
assume east_halfline E-max C meets L~godo;
then
A174: east_halfline E-max C meets (L~go \/ L~pion1) or east_halfline
E-max C meets L~co by A134,XBOOLE_1:70;
per cases by A174,XBOOLE_1:70;
suppose
east_halfline E-max C meets L~go;
hence contradiction by A169;
end;
suppose
east_halfline E-max C meets L~pion1;
then consider p be object such that
A175: p in east_halfline E-max C and
A176: p in L~pion1 by XBOOLE_0:3;
reconsider p as Point of TOP-REAL 2 by A175;
A177: p`2 = (E-max C)`2 by A175,TOPREAL1:def 11;
A178: now
per cases by A36,A84,A176,XBOOLE_0:def 3;
suppose
p in poz;
hence p`1 <= Gi1k`1 by A138,TOPREAL1:3;
end;
suppose
p in pio;
hence p`1 <= Gi1k`1 by A140,GOBOARD7:5;
end;
end;
i1+1 <= len G by A3,NAT_1:13;
then i1+1-1 <= len G-1 by XREAL_1:9;
then
A179: i1 <= len G-'1 by XREAL_0:def 2;
len G-'1 <= len G by NAT_D:35;
then Gi1k`1 <= G*(len G-'1,1)`1 by A6,A17,A10,A13,A21,A179,
JORDAN1A:18;
then p`1 <= G*(len G-'1,1)`1 by A178,XXREAL_0:2;
then p`1 <= E-bound C by A21,JORDAN8:12;
then
A180: p`1 <= (E-max C)`1 by EUCLID:52;
p`1 >= (E-max C)`1 by A175,TOPREAL1:def 11;
then p`1 = (E-max C)`1 by A180,XXREAL_0:1;
then p = E-max C by A177,TOPREAL3:6;
hence contradiction by A9,A36,A84,A165,A176,XBOOLE_0:3;
end;
suppose
east_halfline E-max C meets L~co;
then consider p be object such that
A181: p in east_halfline E-max C and
A182: p in L~co by XBOOLE_0:3;
reconsider p as Point of TOP-REAL 2 by A181;
A183: p in LSeg(co,Index(p,co)) by A182,JORDAN3:9;
consider t be Nat such that
A184: t in dom LS and
A185: LS.t = Gij by A44,FINSEQ_2:10;
1 <= t by A184,FINSEQ_3:25;
then
A186: 1 < t by A46,A185,XXREAL_0:1;
t <= len LS by A184,FINSEQ_3:25;
then Index(Gij,LS)+1 = t by A185,A186,JORDAN3:12;
then
A187: len L_Cut(LS,Gij) = len LS-Index(Gij,LS) by A16,A185,JORDAN3:26;
Index(p,co) < len co by A182,JORDAN3:8;
then Index(p,co) < len LS-'Index(Gij,LS) by A187,XREAL_0:def 2;
then Index(p,co)+1 <= len LS-'Index(Gij,LS) by NAT_1:13;
then
A188: Index(p,co) <= len LS-'Index(Gij,LS)-1 by XREAL_1:19;
A189: co = mid(LS,Gij..LS,len LS) by A44,JORDAN1J:37;
p in L~LS by A48,A182;
then p in east_halfline E-max C /\ L~Cage(C,n) by A147,A181,
XBOOLE_0:def 4;
then
A190: p`1 = Ebo by JORDAN1A:83,PSCOMP_1:50;
A191: Index(Gij,LS)+1 = Gij..LS by A46,A44,JORDAN1J:56;
0+Index(Gij,LS) < len LS by A16,JORDAN3:8;
then len LS-Index(Gij,LS) > 0 by XREAL_1:20;
then Index(p,co) <= len LS-Index(Gij,LS)-1 by A188,XREAL_0:def 2;
then Index(p,co) <= len LS-Gij..LS by A191;
then Index(p,co) <= len LS-'Gij..LS by XREAL_0:def 2;
then
A192: Index(p,co) < len LS-'(Gij..LS)+1 by NAT_1:13;
A193: 1<=Index(p,co) by A182,JORDAN3:8;
A194: Gij..LS<=len LS by A44,FINSEQ_4:21;
Gij..LS <> len LS by A43,A44,FINSEQ_4:19;
then
A195: Gij..LS < len LS by A194,XXREAL_0:1;
A196: 1+1 <= len LS by A31,XXREAL_0:2;
then
A197: 2 in dom LS by FINSEQ_3:25;
set tt = Index(p,co)+(Gij..LS)-'1;
set RC = Rotate(Cage(C,n),Emax);
A198: E-max C in right_cell(RC,1) by JORDAN1I:7;
A199: GoB RC = GoB Cage(C,n) by REVROT_1:28
.= G by JORDAN1H:44;
A200: L~RC = L~Cage(C,n) by REVROT_1:33;
consider jj2 be Nat such that
A201: 1 <= jj2 and
A202: jj2 <= width G and
A203: Emax = G*(len G,jj2) by JORDAN1D:25;
A204: len G >= 4 by JORDAN8:10;
then len G >= 1 by XXREAL_0:2;
then
A205: [len G,jj2] in Indices G by A201,A202,MATRIX_0:30;
A206: len RC = len Cage(C,n) by FINSEQ_6:179;
LS = RC-:Wmin by JORDAN1G:18;
then
A207: LSeg(LS,1) = LSeg(RC,1) by A196,SPPOL_2:9;
A208: Emax in rng Cage(C,n) by SPRECT_2:46;
RC is_sequence_on G by A162,REVROT_1:34;
then consider ii,jj be Nat such that
A209: [ii,jj+1] in Indices G and
A210: [ii,jj] in Indices G and
A211: RC/.1 = G*(ii,jj+1) and
A212: RC/.(1+1) = G*(ii,jj) by A158,A200,A206,A208,FINSEQ_6:92,JORDAN1I:23;
A213: jj+1+1 <> jj;
A214: 1 <= jj by A210,MATRIX_0:32;
RC/.1 = E-max L~RC by A200,A208,FINSEQ_6:92;
then
A215: ii = len G by A200,A209,A211,A203,A205,GOBOARD1:5;
then ii-1 >= 4-1 by A204,XREAL_1:9;
then
A216: ii-1 >= 1 by XXREAL_0:2;
then
A217: 1 <= ii-'1 by XREAL_0:def 2;
A218: jj <= width G by A210,MATRIX_0:32;
then
A219: G*(len G,jj)`1 = Ebo by A13,A214,JORDAN1A:71;
A220: jj+1 <= width G by A209,MATRIX_0:32;
ii+1 <> ii;
then
A221: right_cell(RC,1) = cell(G,ii-'1,jj) by A158,A206,A199,A209,A210,A211
,A212,A213,GOBOARD5:def 6;
A222: ii <= len G by A210,MATRIX_0:32;
A223: 1 <= ii by A210,MATRIX_0:32;
A224: ii <= len G by A209,MATRIX_0:32;
A225: 1 <= jj+1 by A209,MATRIX_0:32;
then
A226: Ebo = G*(len G,jj+1)`1 by A13,A220,JORDAN1A:71;
A227: 1 <= ii by A209,MATRIX_0:32;
then
A228: ii-'1+1 = ii by XREAL_1:235;
then
A229: ii-'1 < len G by A224,NAT_1:13;
then
A230: G*(ii-'1,jj+1)`2 = G*(1,jj+1)`2 by A225,A220,A217,GOBOARD5:1
.= G*(ii,jj+1)`2 by A227,A224,A225,A220,GOBOARD5:1;
A231: (E-max C)`2 = p`2 by A181,TOPREAL1:def 11;
then
A232: p`2 <= G*(ii-'1,jj+1)`2 by A198,A224,A220,A214,A221,A228,A216,
JORDAN9:17;
A233: G*(ii-'1,jj)`2 = G*(1,jj)`2 by A214,A218,A217,A229,GOBOARD5:1
.= G*(ii,jj)`2 by A223,A222,A214,A218,GOBOARD5:1;
G*(ii-'1,jj)`2 <= p`2 by A231,A198,A224,A220,A214,A221,A228,A216,
JORDAN9:17;
then p in LSeg(RC/.1,RC/.(1+1)) by A190,A211,A212,A215,A232,A233,A230
,A219,A226,GOBOARD7:7;
then
A234: p in LSeg(LS,1) by A158,A207,A206,TOPREAL1:def 3;
1<=Gij..LS by A44,FINSEQ_4:21;
then
A235: LSeg(mid(LS,Gij..LS,len LS),Index(p,co)) = LSeg(LS,Index(p,co)
+(Gij..LS)-'1) by A195,A193,A192,JORDAN4:19;
1<=Index(Gij,LS) by A16,JORDAN3:8;
then
A236: 1+1 <= Gij..LS by A191,XREAL_1:7;
then Index(p,co)+Gij..LS >= 1+1+1 by A193,XREAL_1:7;
then Index(p,co)+Gij..LS-1 >= 1+1+1-1 by XREAL_1:9;
then
A237: tt >= 1+1 by XREAL_0:def 2;
now
per cases by A237,XXREAL_0:1;
suppose
tt > 1+1;
then LSeg(LS,1) misses LSeg(LS,tt) by TOPREAL1:def 7;
hence contradiction by A234,A183,A189,A235,XBOOLE_0:3;
end;
suppose
A238: tt = 1+1;
then 1+1 = Index(p,co)+(Gij..LS)-1 by XREAL_0:def 2;
then 1+1+1 = Index(p,co)+(Gij..LS);
then
A239: Gij..LS = 2 by A193,A236,JORDAN1E:6;
LSeg(LS,1) /\ LSeg(LS,tt) = {LS/.2} by A31,A238,TOPREAL1:def 6;
then p in {LS/.2} by A234,A183,A189,A235,XBOOLE_0:def 4;
then
A240: p = LS/.2 by TARSKI:def 1;
then
A241: p in rng LS by A197,PARTFUN2:2;
p..LS = 2 by A197,A240,FINSEQ_5:41;
then p = Gij by A44,A239,A241,FINSEQ_5:9;
then Gij`1 = Ebo by A240,JORDAN1G:32;
then Gij`1 = G*(len G,j)`1 by A4,A19,A13,JORDAN1A:71;
hence contradiction by A3,A25,A22,JORDAN1G:7;
end;
end;
hence contradiction;
end;
end;
then east_halfline E-max C c= (L~godo)` by SUBSET_1:23;
then consider W be Subset of TOP-REAL 2 such that
A242: W is_a_component_of (L~godo)` and
A243: east_halfline E-max C c= W by GOBOARD9:3;
W is not bounded by A243,JORDAN2C:121,RLTOPSP1:42;
then W is_outside_component_of L~godo by A242,JORDAN2C:def 3;
then W c= UBD L~godo by JORDAN2C:23;
then
A244: east_halfline E-max C c= UBD L~godo by A243;
E-max C in east_halfline E-max C by TOPREAL1:38;
then E-max C in UBD L~godo by A244;
then E-max C in LeftComp godo by GOBRD14:36;
then UA meets L~godo by A114,A153,A165,A149,A164,JORDAN1J:36;
then
A245: UA meets (L~go \/ L~pion1) or UA meets L~co by A134,XBOOLE_1:70;
now
per cases by A245,XBOOLE_1:70;
suppose
UA meets L~go;
then UA meets L~Cage(C,n) by A56,A130,XBOOLE_1:1,63;
hence contradiction by A143,JORDAN10:5,XBOOLE_1:63;
end;
suppose
UA meets L~pion1;
hence contradiction by A9,A36,A84;
end;
suppose
UA meets L~co;
then UA meets L~Cage(C,n) by A48,A147,XBOOLE_1:1,63;
hence contradiction by A143,JORDAN10:5,XBOOLE_1:63;
end;
end;
hence contradiction;
end;
suppose
Gij`1 = Gik`1;
then
A246: i1 = i2 by A25,A12,JORDAN1G:7;
then poz = {Gi1k} by RLTOPSP1:70;
then poz c= pio by A27,ZFMISC_1:31;
then pio \/ poz = pio by XBOOLE_1:12;
hence contradiction by A1,A3,A4,A5,A6,A7,A8,A9,A246,JORDAN1J:59;
end;
suppose
Gij`2 = Gik`2;
then
A247: j = k by A25,A12,JORDAN1G:6;
then pio = {Gi1k} by RLTOPSP1:70;
then pio c= poz by A26,ZFMISC_1:31;
then pio \/ poz = poz by XBOOLE_1:12;
hence contradiction by A1,A2,A3,A4,A6,A7,A8,A9,A247,Th37;
end;
end;
hence contradiction;
end;
theorem Th47:
for C be Simple_closed_curve for i1,i2,j,k be Nat st
1 < i2 & i2 <= i1 & i1 < len Gauge(C,n) & 1 <= j & j <= k & k <= width Gauge(C,
n) & (LSeg(Gauge(C,n)*(i1,j),Gauge(C,n)*(i1,k)) \/ LSeg(Gauge(C,n)*(i1,k),Gauge
(C,n)*(i2,k))) /\ L~Upper_Seq(C,n) = {Gauge(C,n)*(i2,k)} & (LSeg(Gauge(C,n)*(i1
,j),Gauge(C,n)*(i1,k)) \/ LSeg(Gauge(C,n)*(i1,k),Gauge(C,n)*(i2,k))) /\ L~
Lower_Seq(C,n) = {Gauge(C,n)*(i1,j)} holds (LSeg(Gauge(C,n)*(i1,j),Gauge(C,n)*(
i1,k)) \/ LSeg(Gauge(C,n)*(i1,k),Gauge(C,n)*(i2,k))) meets Lower_Arc C
proof
let C be Simple_closed_curve;
let i1,i2,j,k be Nat;
set G = Gauge(C,n);
set pio = LSeg(G*(i1,j),G*(i1,k));
set poz = LSeg(G*(i1,k),G*(i2,k));
set US = Upper_Seq(C,n);
set LS = Lower_Seq(C,n);
assume that
A1: 1 < i2 and
A2: i2 <= i1 and
A3: i1 < len G and
A4: 1 <= j and
A5: j <= k and
A6: k <= width G and
A7: (pio \/ poz) /\ L~US = {G*(i2,k)} and
A8: (pio \/ poz) /\ L~LS = {G*(i1,j)} and
A9: (pio \/ poz) misses Lower_Arc C;
set Gi1k = G*(i1,k);
set Gik = G*(i2,k);
A10: 1 <= k by A4,A5,XXREAL_0:2;
A11: i2 < len G by A2,A3,XXREAL_0:2;
then
A12: [i2,k] in Indices G by A1,A6,A10,MATRIX_0:30;
set Wmin = W-min L~Cage(C,n);
set Wbo = W-bound L~Cage(C,n);
A13: len G = width G by JORDAN8:def 1;
set go = R_Cut(US,Gik);
A14: len US >= 3 by JORDAN1E:15;
then len US >= 1 by XXREAL_0:2;
then 1 in dom US by FINSEQ_3:25;
then
A15: US.1 = US/.1 by PARTFUN1:def 6
.= Wmin by JORDAN1F:5;
set Gij = G*(i1,j);
set co = L_Cut(LS,Gij);
Gij in {Gij} by TARSKI:def 1;
then
A16: Gij in L~LS by A8,XBOOLE_0:def 4;
A17: 1 < i1 by A1,A2,XXREAL_0:2;
then
A18: Gi1k`2 = G*(1,k)`2 by A3,A6,A10,GOBOARD5:1
.= Gik`2 by A1,A6,A11,A10,GOBOARD5:1;
A19: j <= width G by A5,A6,XXREAL_0:2;
then
A20: [i1,j] in Indices G by A3,A4,A17,MATRIX_0:30;
len G >= 4 by JORDAN8:10;
then
A21: len G >= 1 by XXREAL_0:2;
then
A22: [len G,j] in Indices G by A4,A19,MATRIX_0:30;
A23: [1,k] in Indices G by A6,A10,A21,MATRIX_0:30;
A24: now
assume Gik`1 = Wbo;
then G*(1,k)`1 = G*(i2,k)`1 by A6,A10,A13,JORDAN1A:73;
hence contradiction by A1,A12,A23,JORDAN1G:7;
end;
A25: [i1,j] in Indices G by A3,A4,A17,A19,MATRIX_0:30;
set pion = <*Gik,Gi1k,Gij*>;
A26: Gi1k in poz by RLTOPSP1:68;
set LA = Lower_Arc C;
A27: Gi1k in pio by RLTOPSP1:68;
A28: [i1,k] in Indices G by A3,A6,A17,A10,MATRIX_0:30;
A29: now
let n be Nat;
assume n in dom pion;
then n in {1,2,3} by FINSEQ_1:89,FINSEQ_3:1;
then n = 1 or n = 2 or n = 3 by ENUMSET1:def 1;
hence
ex i,j be Nat st [i,j] in Indices G & pion/.n = G*(i,j) by A25
,A12,A28,FINSEQ_4:18;
end;
Gik in {Gik} by TARSKI:def 1;
then
A30: Gik in L~US by A7,XBOOLE_0:def 4;
set Emax = E-max L~Cage(C,n);
A31: len LS >= 1+2 by JORDAN1E:15;
then
A32: len LS >= 1 by XXREAL_0:2;
then
A33: 1 in dom LS by FINSEQ_3:25;
then
A34: LS.1 = LS/.1 by PARTFUN1:def 6
.= Emax by JORDAN1F:6;
len LS in dom LS by A32,FINSEQ_3:25;
then
A35: LS.len LS = LS/.len LS by PARTFUN1:def 6
.= Wmin by JORDAN1F:8;
set Ebo = E-bound L~Cage(C,n);
A36: L~<*Gik,Gi1k,Gij*> = poz \/ pio by TOPREAL3:16;
Wmin`1 = Wbo by EUCLID:52
.= G*(1,k)`1 by A6,A10,A13,JORDAN1A:73;
then
A37: Gik <> US.1 by A1,A12,A15,A23,JORDAN1G:7;
then reconsider go as being_S-Seq FinSequence of TOP-REAL 2 by A30,JORDAN3:35
;
A38: Gik in rng US by A1,A6,A11,A30,A10,JORDAN1G:4,JORDAN1J:40;
then
A39: go is_sequence_on G by JORDAN1G:4,JORDAN1J:38;
Gi1k`1 = G*(i1,1)`1 by A3,A6,A17,A10,GOBOARD5:2
.= Gij`1 by A3,A4,A17,A19,GOBOARD5:2;
then
A40: Gi1k = |[Gij`1,Gik`2]| by A18,EUCLID:53;
A41: [len G,k] in Indices G by A6,A10,A21,MATRIX_0:30;
A42: len go >= 1+1 by TOPREAL1:def 8;
Wmin`1 = Wbo by EUCLID:52
.= G*(1,k)`1 by A6,A10,A13,JORDAN1A:73;
then
A43: Gij <> LS.len LS by A1,A2,A23,A35,A20,JORDAN1G:7;
then reconsider co as being_S-Seq FinSequence of TOP-REAL 2 by A16,JORDAN3:34
;
A44: Gij in rng LS by A3,A4,A17,A16,A19,JORDAN1G:5,JORDAN1J:40;
then
A45: co is_sequence_on G by JORDAN1G:5,JORDAN1J:39;
Emax`1 = Ebo by EUCLID:52
.= G*(len G,k)`1 by A6,A10,A13,JORDAN1A:71;
then
A46: Gij <> LS.1 by A3,A20,A41,A34,JORDAN1G:7;
A47: len co >= 1+1 by TOPREAL1:def 8;
then reconsider
co as non constant s.c.c. being_S-Seq FinSequence of TOP-REAL 2
by A45,JGRAPH_1:12,JORDAN8:5;
A48: L~co c= L~LS by A16,JORDAN3:42;
len co >= 1 by A47,XXREAL_0:2;
then 1 in dom co by FINSEQ_3:25;
then
A49: co/.1 = co.1 by PARTFUN1:def 6
.= Gij by A16,JORDAN3:23;
then
A50: LSeg(co,1) = LSeg(Gij,co/.(1+1)) by A47,TOPREAL1:def 3;
A51: {Gij} c= LSeg(co,1) /\ L~<*Gik,Gi1k,Gij*>
proof
let x be object;
assume x in {Gij};
then
A52: x = Gij by TARSKI:def 1;
Gij in LSeg(Gi1k,Gij) by RLTOPSP1:68;
then Gij in LSeg(Gik,Gi1k) \/ LSeg(Gi1k,Gij) by XBOOLE_0:def 3;
then
A53: Gij in L~<*Gik,Gi1k,Gij*> by SPRECT_1:8;
Gij in LSeg(co,1) by A50,RLTOPSP1:68;
hence thesis by A52,A53,XBOOLE_0:def 4;
end;
LSeg(co,1) c= L~co by TOPREAL3:19;
then LSeg(co,1) c= L~LS by A48;
then LSeg(co,1) /\ L~<*Gik,Gi1k,Gij*> c= {Gij} by A8,A36,XBOOLE_1:26;
then
A54: L~<*Gik,Gi1k,Gij*> /\ LSeg(co,1) = {Gij} by A51;
A55: rng co c= L~co by A47,SPPOL_2:18;
reconsider go as non constant s.c.c. being_S-Seq FinSequence of TOP-REAL 2
by A42,A39,JGRAPH_1:12,JORDAN8:5;
A56: L~go c= L~US by A30,JORDAN3:41;
A57: len go > 1 by A42,NAT_1:13;
then
A58: len go in dom go by FINSEQ_3:25;
then
A59: go/.len go = go.len go by PARTFUN1:def 6
.= Gik by A30,JORDAN3:24;
reconsider m = len go - 1 as Nat by A58,FINSEQ_3:26;
A60: m+1 = len go;
then
A61: len go-'1 = m by NAT_D:34;
m >= 1 by A42,XREAL_1:19;
then
A62: LSeg(go,m) = LSeg(go/.m,Gik) by A59,A60,TOPREAL1:def 3;
A63: {Gik} c= LSeg(go,m) /\ L~<*Gik,Gi1k,Gij*>
proof
let x be object;
assume x in {Gik};
then
A64: x = Gik by TARSKI:def 1;
Gik in LSeg(Gik,Gi1k) by RLTOPSP1:68;
then Gik in LSeg(Gik,Gi1k) \/ LSeg(Gi1k,Gij) by XBOOLE_0:def 3;
then
A65: Gik in L~<*Gik,Gi1k,Gij*> by SPRECT_1:8;
Gik in LSeg(go,m) by A62,RLTOPSP1:68;
hence thesis by A64,A65,XBOOLE_0:def 4;
end;
LSeg(go,m) c= L~go by TOPREAL3:19;
then LSeg(go,m) c= L~US by A56;
then LSeg(go,m) /\ L~<*Gik,Gi1k,Gij*> c= {Gik} by A7,A36,XBOOLE_1:26;
then
A66: LSeg(go,m) /\ L~<*Gik,Gi1k,Gij*> = {Gik} by A63;
A67: go/.1 = US/.1 by A30,SPRECT_3:22
.= Wmin by JORDAN1F:5;
then
A68: Wmin in rng go by FINSEQ_6:42;
A69: LS.1 = LS/.1 by A33,PARTFUN1:def 6
.= Emax by JORDAN1F:6;
A70: L~go /\ L~co c= {go/.1}
proof
let x be object;
assume
A71: x in L~go /\ L~co;
then
A72: x in L~co by XBOOLE_0:def 4;
A73: now
assume x = Emax;
then
A74: Emax = Gij by A16,A69,A72,JORDAN1E:7;
G*(len G,j)`1 = Ebo by A4,A19,A13,JORDAN1A:71;
then Emax`1 <> Ebo by A3,A25,A22,A74,JORDAN1G:7;
hence contradiction by EUCLID:52;
end;
x in L~go by A71,XBOOLE_0:def 4;
then x in L~US /\ L~LS by A56,A48,A72,XBOOLE_0:def 4;
then x in {Wmin,Emax} by JORDAN1E:16;
then x = Wmin or x = Emax by TARSKI:def 2;
hence thesis by A67,A73,TARSKI:def 1;
end;
set W2 = go/.2;
A75: 2 in dom go by A42,FINSEQ_3:25;
go = mid(US,1,Gik..US) by A38,JORDAN1G:49
.= US|(Gik..US) by A38,FINSEQ_4:21,FINSEQ_6:116;
then
A76: W2 = US/.2 by A75,FINSEQ_4:70;
A77: rng go c= L~go by A42,SPPOL_2:18;
A78: go/.1 = LS/.len LS by A67,JORDAN1F:8
.= co/.len co by A16,JORDAN1J:35;
{go/.1} c= L~go /\ L~co
proof
let x be object;
assume x in {go/.1};
then
A79: x = go/.1 by TARSKI:def 1;
then
A80: x in rng go by FINSEQ_6:42;
x in rng co by A78,A79,FINSEQ_6:168;
hence thesis by A77,A55,A80,XBOOLE_0:def 4;
end;
then
A81: L~go /\ L~co = {go/.1} by A70;
now
per cases;
suppose
Gij`1 <> Gik`1 & Gij`2 <> Gik`2;
then pion is being_S-Seq by A40,TOPREAL3:35;
then consider pion1 be FinSequence of TOP-REAL 2 such that
A82: pion1 is_sequence_on G and
A83: pion1 is being_S-Seq and
A84: L~pion = L~pion1 and
A85: pion/.1 = pion1/.1 and
A86: pion/.len pion = pion1/.len pion1 and
A87: len pion <= len pion1 by A29,GOBOARD3:2;
reconsider pion1 as being_S-Seq FinSequence of TOP-REAL 2 by A83;
A88: (go^'pion1)/.len (go^'pion1) = pion/.len pion by A86,FINSEQ_6:156
.= pion/.3 by FINSEQ_1:45
.= co/.1 by A49,FINSEQ_4:18;
A89: go/.len go = pion1/.1 by A59,A85,FINSEQ_4:18;
A90: L~go /\ L~pion1 c= {pion1/.1}
proof
let x be object;
assume
A91: x in L~go /\ L~pion1;
then
A92: x in L~pion1 by XBOOLE_0:def 4;
x in L~go by A91,XBOOLE_0:def 4;
hence thesis by A7,A36,A59,A56,A84,A89,A92,XBOOLE_0:def 4;
end;
len pion1 >= 2+1 by A87,FINSEQ_1:45;
then
A93: len pion1 > 1+1 by NAT_1:13;
then
A94: rng pion1 c= L~pion1 by SPPOL_2:18;
{pion1/.1} c= L~go /\ L~pion1
proof
let x be object;
assume x in {pion1/.1};
then
A95: x = pion1/.1 by TARSKI:def 1;
then
A96: x in rng pion1 by FINSEQ_6:42;
x in rng go by A89,A95,FINSEQ_6:168;
hence thesis by A77,A94,A96,XBOOLE_0:def 4;
end;
then
A97: L~go /\ L~pion1 = {pion1/.1} by A90;
then
A98: (go^'pion1) is s.n.c. by A89,JORDAN1J:54;
A99: {Gik} c= LSeg(go,m) /\ LSeg(pion1,1)
proof
let x be object;
assume x in {Gik};
then
A100: x = Gik by TARSKI:def 1;
A101: Gik in LSeg(go,m) by A62,RLTOPSP1:68;
Gik in LSeg(pion1,1) by A59,A89,A93,TOPREAL1:21;
hence thesis by A100,A101,XBOOLE_0:def 4;
end;
LSeg(pion1,1) c= L~pion by A84,TOPREAL3:19;
then LSeg(go,len go-'1) /\ LSeg(pion1,1) c={Gik} by A61,A66,XBOOLE_1:27;
then LSeg(go,len go-'1) /\ LSeg(pion1,1) = { go/.len go } by A59,A61,A99;
then
A102: go^'pion1 is unfolded by A89,TOPREAL8:34;
len (go^'pion1) >= len go by TOPREAL8:7;
then
A103: len (go^'pion1) >= 1+1 by A42,XXREAL_0:2;
then
A104: len (go^'pion1) > 1+0 by NAT_1:13;
A105: pion/.len pion = pion/.3 by FINSEQ_1:45
.= co/.1 by A49,FINSEQ_4:18;
A106: {pion1/.len pion1} c= L~co /\ L~pion1
proof
let x be object;
assume x in {pion1/.len pion1};
then
A107: x = pion1/.len pion1 by TARSKI:def 1;
then
A108: x in rng pion1 by FINSEQ_6:168;
x in rng co by A86,A105,A107,FINSEQ_6:42;
hence thesis by A55,A94,A108,XBOOLE_0:def 4;
end;
L~co /\ L~pion1 c= {pion1/.len pion1}
proof
let x be object;
assume
A109: x in L~co /\ L~pion1;
then
A110: x in L~pion1 by XBOOLE_0:def 4;
x in L~co by A109,XBOOLE_0:def 4;
hence thesis by A8,A36,A49,A48,A84,A86,A105,A110,XBOOLE_0:def 4;
end;
then
A111: L~co /\ L~pion1 = {pion1/.len pion1} by A106;
A112: L~(go^'pion1) /\ L~co = (L~go \/ L~pion1) /\ L~co by A89,TOPREAL8:35
.= {go/.1} \/ {co/.1} by A81,A86,A105,A111,XBOOLE_1:23
.= {(go^'pion1)/.1} \/ {co/.1} by FINSEQ_6:155
.= {(go^'pion1)/.1,co/.1} by ENUMSET1:1;
A113: LA is_an_arc_of E-max C,W-min C by JORDAN6:def 9;
then
A114: LA is connected by JORDAN6:10;
set godo = go^'pion1^'co;
A115: co/.len co = (go^'pion1)/.1 by A78,FINSEQ_6:155;
A116: go^'pion1 is_sequence_on G by A39,A82,A89,TOPREAL8:12;
then
A117: godo is_sequence_on G by A45,A88,TOPREAL8:12;
A118: len pion1-1 >= 1 by A93,XREAL_1:19;
then
A119: len pion1-'1 = len pion1-1 by XREAL_0:def 2;
A120: {Gij} c= LSeg(pion1,len pion1-'1) /\ LSeg(co,1)
proof
let x be object;
assume x in {Gij};
then
A121: x = Gij by TARSKI:def 1;
pion1/.(len pion1-'1+1) = pion/.3 by A86,A119,FINSEQ_1:45
.= Gij by FINSEQ_4:18;
then
A122: Gij in LSeg(pion1,len pion1-'1) by A118,A119,TOPREAL1:21;
Gij in LSeg(co,1) by A50,RLTOPSP1:68;
hence thesis by A121,A122,XBOOLE_0:def 4;
end;
LSeg(pion1,len pion1-'1) c= L~pion by A84,TOPREAL3:19;
then LSeg(pion1,len pion1-'1) /\ LSeg(co,1) c= {Gij} by A54,XBOOLE_1:27;
then
A123: LSeg(pion1,len pion1-'1) /\ LSeg(co,1) = {Gij} by A120;
len pion1-1+1 <= len pion1;
then
A124: len pion1-'1 < len pion1 by A119,NAT_1:13;
len pion1 >= 2+1 by A87,FINSEQ_1:45;
then
A125: len pion1-2 >= 0 by XREAL_1:19;
then len pion1-'2+1 = len pion1-2+1 by XREAL_0:def 2
.= len pion1-'1 by A118,XREAL_0:def 2;
then
A126: LSeg(go^'pion1,len go+(len pion1-'2)) /\ LSeg(co,1) = {(go^'pion1)
/.len (go^'pion1)} by A49,A89,A88,A124,A123,TOPREAL8:31;
rng go /\ rng pion1 c= {pion1/.1} by A77,A94,A97,XBOOLE_1:27;
then
A127: go^'pion1 is one-to-one by JORDAN1J:55;
len (go^'pion1)+1-1 = len go+len pion1-1 by FINSEQ_6:139;
then len (go^'pion1)-1 = len go + (len pion1-2)
.= len go + (len pion1-'2) by A125,XREAL_0:def 2;
then
A128: len (go^'pion1)-'1 = len go + (len pion1-'2) by XREAL_0:def 2;
A129: L~Cage(C,n) = L~US \/ L~LS by JORDAN1E:13;
then
A130: L~US c= L~Cage(C,n) by XBOOLE_1:7;
then
A131: L~go c=L~Cage(C,n) by A56;
A132: len godo >= len (go^'pion1) by TOPREAL8:7;
then
A133: 1+1 <= len godo by A103,XXREAL_0:2;
(go^'pion1) is non trivial by A103,NAT_D:60;
then reconsider
godo as non constant standard special_circular_sequence by A133,A88,A117
,A102,A128,A126,A98,A127,A112,A115,JORDAN8:4,5,TOPREAL8:11,33,34;
A134: L~godo = L~(go^'pion1) \/ L~co by A88,TOPREAL8:35
.= L~go \/ L~pion1 \/ L~co by A89,TOPREAL8:35;
A135: now
assume
A136: Gik..US <= 1;
Gik..US >= 1 by A38,FINSEQ_4:21;
then Gik..US = 1 by A136,XXREAL_0:1;
then Gik = US/.1 by A38,FINSEQ_5:38;
hence contradiction by A15,A37,JORDAN1F:5;
end;
A137: US is_sequence_on G by JORDAN1G:4;
A138: Gik`1 <= Gi1k`1 by A1,A2,A3,A6,A10,JORDAN1A:18;
then
A139: W-bound poz = Gik`1 by SPRECT_1:54;
A140: Gi1k`1 = G*(i1,1)`1 by A3,A6,A17,A10,GOBOARD5:2
.= Gij`1 by A3,A4,A17,A19,GOBOARD5:2;
then
A141: W-bound pio = Gij`1 by SPRECT_1:54;
W-bound (poz \/ pio) = min(W-bound poz, W-bound pio) by SPRECT_1:47
.= Gik`1 by A140,A138,A139,A141,XXREAL_0:def 9;
then
A142: W-bound L~pion1 = Gik`1 by A84,TOPREAL3:16;
A143: LA c= C by JORDAN6:61;
Gik`1 >= Wbo by A30,A130,PSCOMP_1:24;
then
A144: Gik`1 > Wbo by A24,XXREAL_0:1;
A145: len US >= 2 by A14,XXREAL_0:2;
A146: L~go \/ L~co is compact by COMPTS_1:10;
A147: L~LS c= L~Cage(C,n) by A129,XBOOLE_1:7;
then
A148: L~co c=L~Cage(C,n) by A48;
A149: right_cell(godo,1,G)\L~godo c= RightComp godo by A133,A117,JORDAN9:27;
2 in dom godo by A133,FINSEQ_3:25;
then
A150: godo/.2 in rng godo by PARTFUN2:2;
A151: rng godo c= L~godo by A103,A132,SPPOL_2:18,XXREAL_0:2;
A152: godo/.1 = (go^'pion1)/.1 by FINSEQ_6:155
.= Wmin by A67,FINSEQ_6:155;
A153: W-min C in LA by A113,TOPREAL1:1;
A154: W-min C in C by SPRECT_1:13;
A155: now
assume W-min C in L~godo;
then
A156: W-min C in L~go \/ L~pion1 or W-min C in L~co by A134,XBOOLE_0:def 3;
per cases by A156,XBOOLE_0:def 3;
suppose
W-min C in L~go;
then C meets L~Cage(C,n) by A131,A154,XBOOLE_0:3;
hence contradiction by JORDAN10:5;
end;
suppose
W-min C in L~pion1;
hence contradiction by A9,A36,A84,A153,XBOOLE_0:3;
end;
suppose
W-min C in L~co;
then C meets L~Cage(C,n) by A148,A154,XBOOLE_0:3;
hence contradiction by JORDAN10:5;
end;
end;
A157: Wmin`1 = Wbo by EUCLID:52;
set ff = Rotate(Cage(C,n),Wmin);
A158: 1+1 <= len Cage(C,n) by GOBOARD7:34,XXREAL_0:2;
Wmin in rng Cage(C,n) by SPRECT_2:43;
then
A159: ff/.1 = Wmin by FINSEQ_6:92;
A160: L~ff = L~Cage(C,n) by REVROT_1:33;
then (W-max L~ff)..ff > 1 by A159,SPRECT_5:22;
then (N-min L~ff)..ff > 1 by A159,A160,SPRECT_5:23,XXREAL_0:2;
then (N-max L~ff)..ff > 1 by A159,A160,SPRECT_5:24,XXREAL_0:2;
then
A161: Emax..ff > 1 by A159,A160,SPRECT_5:25,XXREAL_0:2;
A162: Cage(C,n) is_sequence_on G by JORDAN9:def 1;
then
A163: ff is_sequence_on G by REVROT_1:34;
1+1 <= len Rotate(Cage(C,n),Wmin) by GOBOARD7:34,XXREAL_0:2;
then right_cell(Rotate(Cage(C,n),Wmin),1) = right_cell(ff,1,GoB ff) by
JORDAN1H:23
.= right_cell(ff,1,GoB Cage(C,n)) by REVROT_1:28
.= right_cell(ff,1,G) by JORDAN1H:44
.= right_cell(ff-:Emax,1,G) by A161,A163,JORDAN1J:53
.= right_cell(US,1,G) by JORDAN1E:def 1
.= right_cell(R_Cut(US,Gik),1,G) by A38,A137,A135,JORDAN1J:52
.= right_cell(go^'pion1,1,G) by A57,A116,JORDAN1J:51
.= right_cell(godo,1,G) by A104,A117,JORDAN1J:51;
then W-min C in right_cell(godo,1,G) by JORDAN1I:6;
then
A164: W-min C in right_cell(godo,1,G)\L~godo by A155,XBOOLE_0:def 5;
A165: E-max C in LA by A113,TOPREAL1:1;
Wmin in L~go \/ L~co by A77,A68,XBOOLE_0:def 3;
then
A166: W-min (L~go \/ L~co) = Wmin by A131,A148,A146,JORDAN1J:21,XBOOLE_1:8;
(W-min (L~go \/ L~co))`1 = W-bound (L~go \/ L~co) by EUCLID:52;
then W-min (L~go\/L~co\/L~pion1) = W-min (L~go \/ L~co) by A142,A146,A166
,A157,A144,JORDAN1J:33;
then
A167: W-min L~godo = Wmin by A134,A166,XBOOLE_1:4;
godo/.2 = (go^'pion1)/.2 by A103,FINSEQ_6:159
.= US/.2 by A42,A76,FINSEQ_6:159
.= (US^'LS)/.2 by A145,FINSEQ_6:159
.= Rotate(Cage(C,n),Wmin)/.2 by JORDAN1E:11;
then godo/.2 in W-most L~Cage(C,n) by JORDAN1I:25;
then (godo/.2)`1 = (W-min L~godo)`1 by A167,PSCOMP_1:31
.= W-bound L~godo by EUCLID:52;
then godo/.2 in W-most L~godo by A151,A150,SPRECT_2:12;
then Rotate(godo,W-min L~godo)/.2 in W-most L~godo by A152,A167,
FINSEQ_6:89;
then reconsider godo as clockwise_oriented non constant standard
special_circular_sequence by JORDAN1I:25;
len US in dom US by FINSEQ_5:6;
then
A168: US.len US = US/.len US by PARTFUN1:def 6
.= Emax by JORDAN1F:7;
A169: east_halfline E-max C misses L~go
proof
assume east_halfline E-max C meets L~go;
then consider p be object such that
A170: p in east_halfline E-max C and
A171: p in L~go by XBOOLE_0:3;
reconsider p as Point of TOP-REAL 2 by A170;
p in L~US by A56,A171;
then p in east_halfline E-max C /\ L~Cage(C,n) by A130,A170,
XBOOLE_0:def 4;
then
A172: p`1 = Ebo by JORDAN1A:83,PSCOMP_1:50;
then
A173: p = Emax by A56,A171,JORDAN1J:46;
then Emax = Gik by A30,A168,A171,JORDAN1J:43;
then Gik`1 = G*(len G,k)`1 by A6,A10,A13,A172,A173,JORDAN1A:71;
hence contradiction by A2,A3,A12,A41,JORDAN1G:7;
end;
now
assume east_halfline E-max C meets L~godo;
then
A174: east_halfline E-max C meets (L~go \/ L~pion1) or east_halfline
E-max C meets L~co by A134,XBOOLE_1:70;
per cases by A174,XBOOLE_1:70;
suppose
east_halfline E-max C meets L~go;
hence contradiction by A169;
end;
suppose
east_halfline E-max C meets L~pion1;
then consider p be object such that
A175: p in east_halfline E-max C and
A176: p in L~pion1 by XBOOLE_0:3;
reconsider p as Point of TOP-REAL 2 by A175;
A177: p`2 = (E-max C)`2 by A175,TOPREAL1:def 11;
A178: now
per cases by A36,A84,A176,XBOOLE_0:def 3;
suppose
p in poz;
hence p`1 <= Gi1k`1 by A138,TOPREAL1:3;
end;
suppose
p in pio;
hence p`1 <= Gi1k`1 by A140,GOBOARD7:5;
end;
end;
i1+1 <= len G by A3,NAT_1:13;
then i1+1-1 <= len G-1 by XREAL_1:9;
then
A179: i1 <= len G-'1 by XREAL_0:def 2;
len G-'1 <= len G by NAT_D:35;
then Gi1k`1 <= G*(len G-'1,1)`1 by A6,A17,A10,A13,A21,A179,
JORDAN1A:18;
then p`1 <= G*(len G-'1,1)`1 by A178,XXREAL_0:2;
then p`1 <= E-bound C by A21,JORDAN8:12;
then
A180: p`1 <= (E-max C)`1 by EUCLID:52;
p`1 >= (E-max C)`1 by A175,TOPREAL1:def 11;
then p`1 = (E-max C)`1 by A180,XXREAL_0:1;
then p = E-max C by A177,TOPREAL3:6;
hence contradiction by A9,A36,A84,A165,A176,XBOOLE_0:3;
end;
suppose
east_halfline E-max C meets L~co;
then consider p be object such that
A181: p in east_halfline E-max C and
A182: p in L~co by XBOOLE_0:3;
reconsider p as Point of TOP-REAL 2 by A181;
A183: p in LSeg(co,Index(p,co)) by A182,JORDAN3:9;
consider t be Nat such that
A184: t in dom LS and
A185: LS.t = Gij by A44,FINSEQ_2:10;
1 <= t by A184,FINSEQ_3:25;
then
A186: 1 < t by A46,A185,XXREAL_0:1;
t <= len LS by A184,FINSEQ_3:25;
then Index(Gij,LS)+1 = t by A185,A186,JORDAN3:12;
then
A187: len L_Cut(LS,Gij) = len LS-Index(Gij,LS) by A16,A185,JORDAN3:26;
Index(p,co) < len co by A182,JORDAN3:8;
then Index(p,co) < len LS-'Index(Gij,LS) by A187,XREAL_0:def 2;
then Index(p,co)+1 <= len LS-'Index(Gij,LS) by NAT_1:13;
then
A188: Index(p,co) <= len LS-'Index(Gij,LS)-1 by XREAL_1:19;
A189: co = mid(LS,Gij..LS,len LS) by A44,JORDAN1J:37;
p in L~LS by A48,A182;
then p in east_halfline E-max C /\ L~Cage(C,n) by A147,A181,
XBOOLE_0:def 4;
then
A190: p`1 = Ebo by JORDAN1A:83,PSCOMP_1:50;
A191: Index(Gij,LS)+1 = Gij..LS by A46,A44,JORDAN1J:56;
0+Index(Gij,LS) < len LS by A16,JORDAN3:8;
then len LS-Index(Gij,LS) > 0 by XREAL_1:20;
then Index(p,co) <= len LS-Index(Gij,LS)-1 by A188,XREAL_0:def 2;
then Index(p,co) <= len LS-Gij..LS by A191;
then Index(p,co) <= len LS-'Gij..LS by XREAL_0:def 2;
then
A192: Index(p,co) < len LS-'(Gij..LS)+1 by NAT_1:13;
A193: 1<=Index(p,co) by A182,JORDAN3:8;
A194: Gij..LS<=len LS by A44,FINSEQ_4:21;
Gij..LS <> len LS by A43,A44,FINSEQ_4:19;
then
A195: Gij..LS < len LS by A194,XXREAL_0:1;
A196: 1+1 <= len LS by A31,XXREAL_0:2;
then
A197: 2 in dom LS by FINSEQ_3:25;
set tt = Index(p,co)+(Gij..LS)-'1;
set RC = Rotate(Cage(C,n),Emax);
A198: E-max C in right_cell(RC,1) by JORDAN1I:7;
A199: GoB RC = GoB Cage(C,n) by REVROT_1:28
.= G by JORDAN1H:44;
A200: L~RC = L~Cage(C,n) by REVROT_1:33;
consider jj2 be Nat such that
A201: 1 <= jj2 and
A202: jj2 <= width G and
A203: Emax = G*(len G,jj2) by JORDAN1D:25;
A204: len G >= 4 by JORDAN8:10;
then len G >= 1 by XXREAL_0:2;
then
A205: [len G,jj2] in Indices G by A201,A202,MATRIX_0:30;
A206: len RC = len Cage(C,n) by FINSEQ_6:179;
LS = RC-:Wmin by JORDAN1G:18;
then
A207: LSeg(LS,1) = LSeg(RC,1) by A196,SPPOL_2:9;
A208: Emax in rng Cage(C,n) by SPRECT_2:46;
RC is_sequence_on G by A162,REVROT_1:34;
then consider ii,jj be Nat such that
A209: [ii,jj+1] in Indices G and
A210: [ii,jj] in Indices G and
A211: RC/.1 = G*(ii,jj+1) and
A212: RC/.(1+1) = G*(ii,jj) by A158,A200,A206,A208,FINSEQ_6:92,JORDAN1I:23;
A213: jj+1+1 <> jj;
A214: 1 <= jj by A210,MATRIX_0:32;
RC/.1 = E-max L~RC by A200,A208,FINSEQ_6:92;
then
A215: ii = len G by A200,A209,A211,A203,A205,GOBOARD1:5;
then ii-1 >= 4-1 by A204,XREAL_1:9;
then
A216: ii-1 >= 1 by XXREAL_0:2;
then
A217: 1 <= ii-'1 by XREAL_0:def 2;
A218: jj <= width G by A210,MATRIX_0:32;
then
A219: G*(len G,jj)`1 = Ebo by A13,A214,JORDAN1A:71;
A220: jj+1 <= width G by A209,MATRIX_0:32;
ii+1 <> ii;
then
A221: right_cell(RC,1) = cell(G,ii-'1,jj) by A158,A206,A199,A209,A210,A211
,A212,A213,GOBOARD5:def 6;
A222: ii <= len G by A210,MATRIX_0:32;
A223: 1 <= ii by A210,MATRIX_0:32;
A224: ii <= len G by A209,MATRIX_0:32;
A225: 1 <= jj+1 by A209,MATRIX_0:32;
then
A226: Ebo = G*(len G,jj+1)`1 by A13,A220,JORDAN1A:71;
A227: 1 <= ii by A209,MATRIX_0:32;
then
A228: ii-'1+1 = ii by XREAL_1:235;
then
A229: ii-'1 < len G by A224,NAT_1:13;
then
A230: G*(ii-'1,jj+1)`2 = G*(1,jj+1)`2 by A225,A220,A217,GOBOARD5:1
.= G*(ii,jj+1)`2 by A227,A224,A225,A220,GOBOARD5:1;
A231: (E-max C)`2 = p`2 by A181,TOPREAL1:def 11;
then
A232: p`2 <= G*(ii-'1,jj+1)`2 by A198,A224,A220,A214,A221,A228,A216,
JORDAN9:17;
A233: G*(ii-'1,jj)`2 = G*(1,jj)`2 by A214,A218,A217,A229,GOBOARD5:1
.= G*(ii,jj)`2 by A223,A222,A214,A218,GOBOARD5:1;
G*(ii-'1,jj)`2 <= p`2 by A231,A198,A224,A220,A214,A221,A228,A216,
JORDAN9:17;
then p in LSeg(RC/.1,RC/.(1+1)) by A190,A211,A212,A215,A232,A233,A230
,A219,A226,GOBOARD7:7;
then
A234: p in LSeg(LS,1) by A158,A207,A206,TOPREAL1:def 3;
1<=Gij..LS by A44,FINSEQ_4:21;
then
A235: LSeg(mid(LS,Gij..LS,len LS),Index(p,co)) = LSeg(LS,Index(p,co)
+(Gij..LS)-'1) by A195,A193,A192,JORDAN4:19;
1<=Index(Gij,LS) by A16,JORDAN3:8;
then
A236: 1+1 <= Gij..LS by A191,XREAL_1:7;
then Index(p,co)+Gij..LS >= 1+1+1 by A193,XREAL_1:7;
then Index(p,co)+Gij..LS-1 >= 1+1+1-1 by XREAL_1:9;
then
A237: tt >= 1+1 by XREAL_0:def 2;
now
per cases by A237,XXREAL_0:1;
suppose
tt > 1+1;
then LSeg(LS,1) misses LSeg(LS,tt) by TOPREAL1:def 7;
hence contradiction by A234,A183,A189,A235,XBOOLE_0:3;
end;
suppose
A238: tt = 1+1;
then 1+1 = Index(p,co)+(Gij..LS)-1 by XREAL_0:def 2;
then 1+1+1 = Index(p,co)+(Gij..LS);
then
A239: Gij..LS = 2 by A193,A236,JORDAN1E:6;
LSeg(LS,1) /\ LSeg(LS,tt) = {LS/.2} by A31,A238,TOPREAL1:def 6;
then p in {LS/.2} by A234,A183,A189,A235,XBOOLE_0:def 4;
then
A240: p = LS/.2 by TARSKI:def 1;
then
A241: p in rng LS by A197,PARTFUN2:2;
p..LS = 2 by A197,A240,FINSEQ_5:41;
then p = Gij by A44,A239,A241,FINSEQ_5:9;
then Gij`1 = Ebo by A240,JORDAN1G:32;
then Gij`1 = G*(len G,j)`1 by A4,A19,A13,JORDAN1A:71;
hence contradiction by A3,A25,A22,JORDAN1G:7;
end;
end;
hence contradiction;
end;
end;
then east_halfline E-max C c= (L~godo)` by SUBSET_1:23;
then consider W be Subset of TOP-REAL 2 such that
A242: W is_a_component_of (L~godo)` and
A243: east_halfline E-max C c= W by GOBOARD9:3;
W is not bounded by A243,JORDAN2C:121,RLTOPSP1:42;
then W is_outside_component_of L~godo by A242,JORDAN2C:def 3;
then W c= UBD L~godo by JORDAN2C:23;
then
A244: east_halfline E-max C c= UBD L~godo by A243;
E-max C in east_halfline E-max C by TOPREAL1:38;
then E-max C in UBD L~godo by A244;
then E-max C in LeftComp godo by GOBRD14:36;
then LA meets L~godo by A114,A153,A165,A149,A164,JORDAN1J:36;
then
A245: LA meets (L~go \/ L~pion1) or LA meets L~co by A134,XBOOLE_1:70;
now
per cases by A245,XBOOLE_1:70;
suppose
LA meets L~go;
then LA meets L~Cage(C,n) by A56,A130,XBOOLE_1:1,63;
hence contradiction by A143,JORDAN10:5,XBOOLE_1:63;
end;
suppose
LA meets L~pion1;
hence contradiction by A9,A36,A84;
end;
suppose
LA meets L~co;
then LA meets L~Cage(C,n) by A48,A147,XBOOLE_1:1,63;
hence contradiction by A143,JORDAN10:5,XBOOLE_1:63;
end;
end;
hence contradiction;
end;
suppose
Gij`1 = Gik`1;
then
A246: i1 = i2 by A25,A12,JORDAN1G:7;
then poz = {Gi1k} by RLTOPSP1:70;
then poz c= pio by A27,ZFMISC_1:31;
then pio \/ poz = pio by XBOOLE_1:12;
hence contradiction by A1,A3,A4,A5,A6,A7,A8,A9,A246,JORDAN1J:58;
end;
suppose
Gij`2 = Gik`2;
then
A247: j = k by A25,A12,JORDAN1G:6;
then pio = {Gi1k} by RLTOPSP1:70;
then pio c= poz by A26,ZFMISC_1:31;
then pio \/ poz = poz by XBOOLE_1:12;
hence contradiction by A1,A2,A3,A4,A6,A7,A8,A9,A247,Th36;
end;
end;
hence contradiction;
end;
theorem Th48:
for C be Simple_closed_curve for i1,i2,j,k be Nat
holds 1 < i1 & i1 < len Gauge(C,n+1) & 1 < i2 & i2 < len Gauge(C,n+1) & 1 <= j
& j <= k & k <= width Gauge(C,n+1) & Gauge(C,n+1)*(i1,k) in Upper_Arc L~Cage(C,
n+1) & Gauge(C,n+1)*(i2,j) in Lower_Arc L~Cage(C,n+1) implies LSeg(Gauge(C,n+1)
*(i2,j),Gauge(C,n+1)*(i2,k)) \/ LSeg(Gauge(C,n+1)*(i2,k),Gauge(C,n+1)*(i1,k))
meets Upper_Arc C
proof
let C be Simple_closed_curve;
let i1,i2,j,k be Nat;
set G=Gauge(C,n+1);
assume that
A1: 1 < i1 and
A2: i1 < len G and
A3: 1 < i2 and
A4: i2 < len G and
A5: 1 <= j and
A6: j <= k and
A7: k <= width G and
A8: G*(i1,k) in Upper_Arc L~Cage(C,n+1) and
A9: G*(i2,j) in Lower_Arc L~Cage(C,n+1);
A10: 1 <= k by A5,A6,XXREAL_0:2;
then
A11: [i2,k] in Indices G by A3,A4,A7,MATRIX_0:30;
A12: [i1,k] in Indices G by A1,A2,A7,A10,MATRIX_0:30;
G*(i2,k)`2 = G*(1,k)`2 by A3,A4,A7,A10,GOBOARD5:1
.= G*(i1,k)`2 by A1,A2,A7,A10,GOBOARD5:1;
then
A13: LSeg(G*(i2,k),G*(i1,k)) is horizontal by SPPOL_1:15;
A14: Lower_Arc L~Cage(C,n+1) = L~Lower_Seq(C,n+1) by JORDAN1G:56;
A15: j <= width G by A6,A7,XXREAL_0:2;
then
A16: [i2,j] in Indices G by A3,A4,A5,MATRIX_0:30;
G*(i2,j)`1 = G*(i2,1)`1 by A3,A4,A5,A15,GOBOARD5:2
.= G*(i2,k)`1 by A3,A4,A7,A10,GOBOARD5:2;
then
A17: LSeg(G*(i2,j),G*(i2,k)) is vertical by SPPOL_1:16;
A18: Upper_Arc L~Cage(C,n+1) = L~Upper_Seq(C,n+1) by JORDAN1G:55;
A19: [i2,k] in Indices G by A3,A4,A7,A10,MATRIX_0:30;
now
per cases;
suppose
A20: LSeg(G*(i2,j),G*(i2,k)) meets Upper_Arc L~Cage(C,n+1);
set X = LSeg(G*(i2,j),G*(i2,k)) /\ L~Upper_Seq(C,n+1);
ex x be object st x in LSeg(G*(i2,j),G*(i2,k)) & x in L~Upper_Seq(C,n+
1) by A18,A20,XBOOLE_0:3;
then reconsider X1=X as non empty compact Subset of TOP-REAL 2 by
XBOOLE_0:def 4;
consider pp be object such that
A21: pp in S-most X1 by XBOOLE_0:def 1;
reconsider pp as Point of TOP-REAL 2 by A21;
A22: pp in X by A21,XBOOLE_0:def 4;
then
A23: pp in L~Upper_Seq(C,n+1) by XBOOLE_0:def 4;
A24: pp in LSeg(G*(i2,j),G*(i2,k)) by A22,XBOOLE_0:def 4;
consider m be Nat such that
A25: j <= m and
A26: m <= k and
A27: G*(i2,m)`2 = lower_bound(proj2.:(LSeg(G*(i2,j),G*(i2,k)) /\ L~Upper_Seq
(C,n+1))) by A6,A18,A16,A19,A20,JORDAN1F:1,JORDAN1G:4;
A28: m <= width G by A7,A26,XXREAL_0:2;
1 <= m by A5,A25,XXREAL_0:2;
then
A29: G*(i2,m)`1 = G*(i2,1)`1 by A3,A4,A28,GOBOARD5:2;
then
A30: |[G*(i2,1)`1,lower_bound(proj2.:X)]| = G*(i2,m) by A27,EUCLID:53;
then
G*(i2,j)`1 = |[G*(i2,1)`1,lower_bound(proj2.:X)]|`1
by A3,A4,A5,A15,A29,GOBOARD5:2;
then
A31: pp`1 = |[G*(i2,1)`1,lower_bound(proj2.:X)]|`1 by A17,A24,SPPOL_1:41;
|[G*(i2,1)`1,lower_bound(proj2.:X)]|`2 = S-bound X by A27,A30,SPRECT_1:44
.= (S-min X)`2 by EUCLID:52
.= pp`2 by A21,PSCOMP_1:55;
then G*(i2,m) in Upper_Arc L~Cage(C,n+1) by A18,A30,A23,A31,TOPREAL3:6;
then LSeg(G*(i2,j),G*(i2,m)) meets Upper_Arc C by A3,A4,A5,A9,A25,A28
,Th24;
then LSeg(G*(i2,j),G*(i2,k)) meets Upper_Arc C by A3,A4,A5,A7,A25,A26,Th5
,XBOOLE_1:63;
hence thesis by XBOOLE_1:70;
end;
suppose
A32: LSeg(G*(i2,k),G*(i1,k)) meets Lower_Arc L~Cage(C,n+1) & i2 <= i1;
set X = LSeg(G*(i2,k),G*(i1,k)) /\ L~Lower_Seq(C,n+1);
ex x be object st x in LSeg(G*(i2,k),G*(i1,k)) & x in L~Lower_Seq(C,n+
1) by A14,A32,XBOOLE_0:3;
then reconsider X1=X as non empty compact Subset of TOP-REAL 2 by
XBOOLE_0:def 4;
consider pp be object such that
A33: pp in E-most X1 by XBOOLE_0:def 1;
reconsider pp as Point of TOP-REAL 2 by A33;
A34: pp in X by A33,XBOOLE_0:def 4;
then
A35: pp in L~Lower_Seq(C,n+1) by XBOOLE_0:def 4;
A36: pp in LSeg(G*(i2,k),G*(i1,k)) by A34,XBOOLE_0:def 4;
consider m be Nat such that
A37: i2 <= m and
A38: m <= i1 and
A39: G*(m,k)`1 = upper_bound(proj1.:(LSeg(G*(i2,k),G*(i1,k)) /\ L~Lower_Seq(
C,n+1))) by A14,A11,A12,A32,JORDAN1F:4,JORDAN1G:5;
A40: 1 < m by A3,A37,XXREAL_0:2;
m < len G by A2,A38,XXREAL_0:2;
then
A41: G*(m,k)`2 = G*(1,k)`2 by A7,A10,A40,GOBOARD5:1;
then
A42: |[upper_bound(proj1.:X),G*(1,k)`2]| = G*(m,k) by A39,EUCLID:53;
then G*(i2,k)`2 = |[upper_bound(proj1.:X),G*(1,k)`2]|`2
by A3,A4,A7,A10,A41,GOBOARD5:1;
then
A43: pp`2 = |[upper_bound(proj1.:X),G*(1,k)`2]|`2 by A13,A36,SPPOL_1:40;
|[upper_bound(proj1.:X),G*(1,k)`2]|`1 = E-bound X by A39,A42,SPRECT_1:46
.= (E-min X)`1 by EUCLID:52
.= pp`1 by A33,PSCOMP_1:47;
then G*(m,k) in Lower_Arc L~Cage(C,n+1) by A14,A42,A35,A43,TOPREAL3:6;
then LSeg(G*(m,k),G*(i1,k)) meets Upper_Arc C by A2,A7,A8,A10,A38,A40
,Th33;
then
LSeg(G*(i2,k),G*(i1,k)) meets Upper_Arc C by A2,A3,A7,A10,A37,A38,Th6,
XBOOLE_1:63;
hence thesis by XBOOLE_1:70;
end;
suppose
A44: LSeg(G*(i2,k),G*(i1,k)) meets Lower_Arc L~Cage(C,n+1) & i1 < i2;
set X = LSeg(G*(i1,k),G*(i2,k)) /\ L~Lower_Seq(C,n+1);
ex x be object st x in LSeg(G*(i1,k),G*(i2,k)) & x in L~Lower_Seq(C,n+
1) by A14,A44,XBOOLE_0:3;
then reconsider X1=X as non empty compact Subset of TOP-REAL 2 by
XBOOLE_0:def 4;
consider pp be object such that
A45: pp in W-most X1 by XBOOLE_0:def 1;
reconsider pp as Point of TOP-REAL 2 by A45;
A46: pp in X by A45,XBOOLE_0:def 4;
then
A47: pp in L~Lower_Seq(C,n+1) by XBOOLE_0:def 4;
A48: pp in LSeg(G*(i1,k),G*(i2,k)) by A46,XBOOLE_0:def 4;
consider m be Nat such that
A49: i1 <= m and
A50: m <= i2 and
A51: G*(m,k)`1 = lower_bound(proj1.:(LSeg(G*(i1,k),G*(i2,k)) /\ L~Lower_Seq(
C,n+1))) by A14,A11,A12,A44,JORDAN1F:3,JORDAN1G:5;
A52: m < len G by A4,A50,XXREAL_0:2;
1 < m by A1,A49,XXREAL_0:2;
then
A53: G*(m,k)`2 = G*(1,k)`2 by A7,A10,A52,GOBOARD5:1;
then
A54: |[lower_bound(proj1.:X),G*(1,k)`2]| = G*(m,k) by A51,EUCLID:53;
then G*(i1,k)`2 = |[lower_bound(proj1.:X),G*(1,k)`2]|`2
by A1,A2,A7,A10,A53,GOBOARD5:1;
then
A55: pp`2 = |[lower_bound(proj1.:X),G*(1,k)`2]|`2 by A13,A48,SPPOL_1:40;
|[lower_bound(proj1.:X),G*(1,k)`2]|`1 = W-bound X by A51,A54,SPRECT_1:43
.= (W-min X)`1 by EUCLID:52
.= pp`1 by A45,PSCOMP_1:31;
then G*(m,k) in Lower_Arc L~Cage(C,n+1) by A14,A54,A47,A55,TOPREAL3:6;
then LSeg(G*(i1,k),G*(m,k)) meets Upper_Arc C by A1,A7,A8,A10,A49,A52
,Th41;
then
LSeg(G*(i1,k),G*(i2,k)) meets Upper_Arc C by A1,A4,A7,A10,A49,A50,Th6,
XBOOLE_1:63;
hence thesis by XBOOLE_1:70;
end;
suppose
A56: LSeg(G*(i2,j),G*(i2,k)) misses Upper_Arc L~Cage(C,n+1) & LSeg(
Gauge(C,n+1)*(i2,k),Gauge(C,n+1)*(i1,k)) misses Lower_Arc L~Cage(C,n+1);
consider j1 be Nat such that
A57: j <= j1 and
A58: j1 <= k and
A59: LSeg(G*(i2,j1),G*(i2,k)) /\ L~Lower_Seq(C,n+1) = {G*(i2,j1)} by A3,A4,A5
,A6,A7,A9,A14,Th9;
G*(i2,j1) in LSeg(G*(i2,j1),G*(i2,k)) /\ L~Lower_Seq(C,n+1) by A59,
TARSKI:def 1;
then
A60: G*(i2,j1) in L~Lower_Seq(C,n+1) by XBOOLE_0:def 4;
A61: 1 <= j1 by A5,A57,XXREAL_0:2;
now
per cases;
suppose
A62: i2 <= i1;
A63: LSeg(G*(i2,j1),G*(i2,k)) c= LSeg(G*(i2,j),G*(i2,k)) by A3,A4,A5,A7
,A57,A58,Th5;
consider i3 be Nat such that
A64: i2 <= i3 and
A65: i3 <= i1 and
A66: LSeg(G*(i2,k),G*(i3,k)) /\ L~Upper_Seq(C,n+1) = {G*(i3,k)}
by A2,A3,A7,A8,A18,A10,A62,Th13;
A67: LSeg(G*(i2,k),G*(i3,k)) c= LSeg(G*(i2,k),G*(i1,k)) by A2,A3,A7,A10
,A64,A65,Th6;
then
A68: LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3,k)) c= LSeg(G*
(i2,j),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i1,k)) by A63,XBOOLE_1:13;
G*(i3,k) in LSeg(G*(i2,k),G*(i3,k)) /\ L~Upper_Seq(C,n+1) by A66,
TARSKI:def 1;
then
A69: G*(i3,k) in L~Upper_Seq(C,n+1) by XBOOLE_0:def 4;
A70: (LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3,k))) /\ L~
Upper_Seq(C,n+1) = {G*(i3,k)}
proof
thus (LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3,k))) /\ L~
Upper_Seq(C,n+1) c= {G*(i3,k)}
proof
let x be object;
assume
A71: x in (LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3,
k))) /\ L~Upper_Seq(C,n+1);
then x in LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3,k))
by XBOOLE_0:def 4;
then
A72: x in LSeg(G*(i2,j1),G*(i2,k)) or x in LSeg(G*(i2,k),G*(i3,
k)) by XBOOLE_0:def 3;
x in L~Upper_Seq(C,n+1) by A71,XBOOLE_0:def 4;
hence thesis by A18,A56,A66,A63,A72,XBOOLE_0:def 4;
end;
let x be object;
G*(i3,k) in LSeg(G*(i2,k),G*(i3,k)) by RLTOPSP1:68;
then
A73: G*(i3,k) in LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3,
k)) by XBOOLE_0:def 3;
assume x in {G*(i3,k)};
then x = G*(i3,k) by TARSKI:def 1;
hence thesis by A69,A73,XBOOLE_0:def 4;
end;
A74: (LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3,k))) /\ L~
Lower_Seq(C,n+1) = {G*(i2,j1)}
proof
thus (LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3,k))) /\ L~
Lower_Seq(C,n+1) c= {G*(i2,j1)}
proof
let x be object;
assume
A75: x in (LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3,
k))) /\ L~Lower_Seq(C,n+1);
then x in LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3,k))
by XBOOLE_0:def 4;
then
A76: x in LSeg(G*(i2,j1),G*(i2,k)) or x in LSeg(G*(i2,k),G*(i3,
k)) by XBOOLE_0:def 3;
x in L~Lower_Seq(C,n+1) by A75,XBOOLE_0:def 4;
hence thesis by A14,A56,A59,A67,A76,XBOOLE_0:def 4;
end;
let x be object;
G*(i2,j1) in LSeg(G*(i2,j1),G*(i2,k)) by RLTOPSP1:68;
then
A77: G*(i2,j1) in LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3
,k)) by XBOOLE_0:def 3;
assume x in {G*(i2,j1)};
then x = G*(i2,j1) by TARSKI:def 1;
hence thesis by A60,A77,XBOOLE_0:def 4;
end;
i3 < len G by A2,A65,XXREAL_0:2;
hence thesis by A3,A7,A58,A61,A64,A68,A70,A74,Th44,XBOOLE_1:63;
end;
suppose
A78: i1 < i2;
A79: LSeg(G*(i2,j1),G*(i2,k)) c= LSeg(G*(i2,j),G*(i2,k)) by A3,A4,A5,A7
,A57,A58,Th5;
consider i3 be Nat such that
A80: i1 <= i3 and
A81: i3 <= i2 and
A82: LSeg(G*(i3,k),G*(i2,k)) /\ L~Upper_Seq(C,n+1) = {G*(i3,k)}
by A1,A4,A7,A8,A18,A10,A78,Th18;
A83: LSeg(G*(i2,k),G*(i3,k)) c= LSeg(G*(i2,k),G*(i1,k)) by A1,A4,A7,A10
,A80,A81,Th6;
then
A84: LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3,k)) c= LSeg(G*
(i2,j),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i1,k)) by A79,XBOOLE_1:13;
G*(i3,k) in LSeg(G*(i2,k),G*(i3,k)) /\ L~Upper_Seq(C,n+1) by A82,
TARSKI:def 1;
then
A85: G*(i3,k) in L~Upper_Seq(C,n+1) by XBOOLE_0:def 4;
A86: (LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3,k))) /\ L~
Upper_Seq(C,n+1) = {G*(i3,k)}
proof
thus (LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3,k))) /\ L~
Upper_Seq(C,n+1) c= {G*(i3,k)}
proof
let x be object;
assume
A87: x in (LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3,
k))) /\ L~Upper_Seq(C,n+1);
then x in LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3,k))
by XBOOLE_0:def 4;
then
A88: x in LSeg(G*(i2,j1),G*(i2,k)) or x in LSeg(G*(i2,k),G*(i3,
k)) by XBOOLE_0:def 3;
x in L~Upper_Seq(C,n+1) by A87,XBOOLE_0:def 4;
hence thesis by A18,A56,A82,A79,A88,XBOOLE_0:def 4;
end;
let x be object;
G*(i3,k) in LSeg(G*(i2,k),G*(i3,k)) by RLTOPSP1:68;
then
A89: G*(i3,k) in LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3,
k)) by XBOOLE_0:def 3;
assume x in {G*(i3,k)};
then x = G*(i3,k) by TARSKI:def 1;
hence thesis by A85,A89,XBOOLE_0:def 4;
end;
A90: (LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3,k))) /\ L~
Lower_Seq(C,n+1) = {G*(i2,j1)}
proof
thus (LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3,k))) /\ L~
Lower_Seq(C,n+1) c= {G*(i2,j1)}
proof
let x be object;
assume
A91: x in (LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3,
k))) /\ L~Lower_Seq(C,n+1);
then x in LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3,k))
by XBOOLE_0:def 4;
then
A92: x in LSeg(G*(i2,j1),G*(i2,k)) or x in LSeg(G*(i2,k),G*(i3,
k)) by XBOOLE_0:def 3;
x in L~Lower_Seq(C,n+1) by A91,XBOOLE_0:def 4;
hence thesis by A14,A56,A59,A83,A92,XBOOLE_0:def 4;
end;
let x be object;
G*(i2,j1) in LSeg(G*(i2,j1),G*(i2,k)) by RLTOPSP1:68;
then
A93: G*(i2,j1) in LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3
,k)) by XBOOLE_0:def 3;
assume x in {G*(i2,j1)};
then x = G*(i2,j1) by TARSKI:def 1;
hence thesis by A60,A93,XBOOLE_0:def 4;
end;
1 < i3 by A1,A80,XXREAL_0:2;
hence thesis by A4,A7,A58,A61,A81,A84,A86,A90,Th46,XBOOLE_1:63;
end;
end;
hence thesis;
end;
end;
hence thesis;
end;
theorem Th49:
for C be Simple_closed_curve for i1,i2,j,k be Nat
holds 1 < i1 & i1 < len Gauge(C,n+1) & 1 < i2 & i2 < len Gauge(C,n+1) & 1 <= j
& j <= k & k <= width Gauge(C,n+1) & Gauge(C,n+1)*(i1,k) in Upper_Arc L~Cage(C,
n+1) & Gauge(C,n+1)*(i2,j) in Lower_Arc L~Cage(C,n+1) implies LSeg(Gauge(C,n+1)
*(i2,j),Gauge(C,n+1)*(i2,k)) \/ LSeg(Gauge(C,n+1)*(i2,k),Gauge(C,n+1)*(i1,k))
meets Lower_Arc C
proof
let C be Simple_closed_curve;
let i1,i2,j,k be Nat;
set G=Gauge(C,n+1);
assume that
A1: 1 < i1 and
A2: i1 < len G and
A3: 1 < i2 and
A4: i2 < len G and
A5: 1 <= j and
A6: j <= k and
A7: k <= width G and
A8: G*(i1,k) in Upper_Arc L~Cage(C,n+1) and
A9: G*(i2,j) in Lower_Arc L~Cage(C,n+1);
A10: 1 <= k by A5,A6,XXREAL_0:2;
then
A11: [i2,k] in Indices G by A3,A4,A7,MATRIX_0:30;
A12: [i1,k] in Indices G by A1,A2,A7,A10,MATRIX_0:30;
G*(i2,k)`2 = G*(1,k)`2 by A3,A4,A7,A10,GOBOARD5:1
.= G*(i1,k)`2 by A1,A2,A7,A10,GOBOARD5:1;
then
A13: LSeg(G*(i2,k),G*(i1,k)) is horizontal by SPPOL_1:15;
A14: Lower_Arc L~Cage(C,n+1) = L~Lower_Seq(C,n+1) by JORDAN1G:56;
A15: j <= width G by A6,A7,XXREAL_0:2;
then
A16: [i2,j] in Indices G by A3,A4,A5,MATRIX_0:30;
G*(i2,j)`1 = G*(i2,1)`1 by A3,A4,A5,A15,GOBOARD5:2
.= G*(i2,k)`1 by A3,A4,A7,A10,GOBOARD5:2;
then
A17: LSeg(G*(i2,j),G*(i2,k)) is vertical by SPPOL_1:16;
A18: Upper_Arc L~Cage(C,n+1) = L~Upper_Seq(C,n+1) by JORDAN1G:55;
A19: [i2,k] in Indices G by A3,A4,A7,A10,MATRIX_0:30;
now
per cases;
suppose
A20: LSeg(G*(i2,j),G*(i2,k)) meets Upper_Arc L~Cage(C,n+1);
set X = LSeg(G*(i2,j),G*(i2,k)) /\ L~Upper_Seq(C,n+1);
ex x be object st x in LSeg(G*(i2,j),G*(i2,k)) & x in L~Upper_Seq(C,n+
1) by A18,A20,XBOOLE_0:3;
then reconsider X1=X as non empty compact Subset of TOP-REAL 2 by
XBOOLE_0:def 4;
consider pp be object such that
A21: pp in S-most X1 by XBOOLE_0:def 1;
reconsider pp as Point of TOP-REAL 2 by A21;
A22: pp in X by A21,XBOOLE_0:def 4;
then
A23: pp in L~Upper_Seq(C,n+1) by XBOOLE_0:def 4;
A24: pp in LSeg(G*(i2,j),G*(i2,k)) by A22,XBOOLE_0:def 4;
consider m be Nat such that
A25: j <= m and
A26: m <= k and
A27: G*(i2,m)`2 = lower_bound(proj2.:(LSeg(G*(i2,j),G*(i2,k)) /\ L~Upper_Seq
(C,n+1))) by A6,A18,A16,A19,A20,JORDAN1F:1,JORDAN1G:4;
A28: m <= width G by A7,A26,XXREAL_0:2;
1 <= m by A5,A25,XXREAL_0:2;
then
A29: G*(i2,m)`1 = G*(i2,1)`1 by A3,A4,A28,GOBOARD5:2;
then
A30: |[G*(i2,1)`1,lower_bound(proj2.:X)]| = G*(i2,m) by A27,EUCLID:53;
then
G*(i2,j)`1 = |[G*(i2,1)`1,lower_bound(proj2.:X)]|`1
by A3,A4,A5,A15,A29,GOBOARD5:2;
then
A31: pp`1 = |[G*(i2,1)`1,lower_bound(proj2.:X)]|`1 by A17,A24,SPPOL_1:41;
|[G*(i2,1)`1,lower_bound(proj2.:X)]|`2 = S-bound X by A27,A30,SPRECT_1:44
.= (S-min X)`2 by EUCLID:52
.= pp`2 by A21,PSCOMP_1:55;
then G*(i2,m) in Upper_Arc L~Cage(C,n+1) by A18,A30,A23,A31,TOPREAL3:6;
then LSeg(G*(i2,j),G*(i2,m)) meets Lower_Arc C by A3,A4,A5,A9,A25,A28
,Th23;
then LSeg(G*(i2,j),G*(i2,k)) meets Lower_Arc C by A3,A4,A5,A7,A25,A26,Th5
,XBOOLE_1:63;
hence thesis by XBOOLE_1:70;
end;
suppose
A32: LSeg(G*(i2,k),G*(i1,k)) meets Lower_Arc L~Cage(C,n+1) & i2 <= i1;
set X = LSeg(G*(i2,k),G*(i1,k)) /\ L~Lower_Seq(C,n+1);
ex x be object st x in LSeg(G*(i2,k),G*(i1,k)) & x in L~Lower_Seq(C,n+
1) by A14,A32,XBOOLE_0:3;
then reconsider X1=X as non empty compact Subset of TOP-REAL 2 by
XBOOLE_0:def 4;
consider pp be object such that
A33: pp in E-most X1 by XBOOLE_0:def 1;
reconsider pp as Point of TOP-REAL 2 by A33;
A34: pp in X by A33,XBOOLE_0:def 4;
then
A35: pp in L~Lower_Seq(C,n+1) by XBOOLE_0:def 4;
A36: pp in LSeg(G*(i2,k),G*(i1,k)) by A34,XBOOLE_0:def 4;
consider m be Nat such that
A37: i2 <= m and
A38: m <= i1 and
A39: G*(m,k)`1 = upper_bound(proj1.:(LSeg(G*(i2,k),G*(i1,k)) /\ L~Lower_Seq(
C,n+1))) by A14,A11,A12,A32,JORDAN1F:4,JORDAN1G:5;
A40: 1 < m by A3,A37,XXREAL_0:2;
m < len G by A2,A38,XXREAL_0:2;
then
A41: G*(m,k)`2 = G*(1,k)`2 by A7,A10,A40,GOBOARD5:1;
then
A42: |[upper_bound(proj1.:X),G*(1,k)`2]| = G*(m,k) by A39,EUCLID:53;
then G*(i2,k)`2 = |[upper_bound(proj1.:X),G*(1,k)`2]|`2
by A3,A4,A7,A10,A41,GOBOARD5:1;
then
A43: pp`2 = |[upper_bound(proj1.:X),G*(1,k)`2]|`2 by A13,A36,SPPOL_1:40;
|[upper_bound(proj1.:X),G*(1,k)`2]|`1 = E-bound X by A39,A42,SPRECT_1:46
.= (E-min X)`1 by EUCLID:52
.= pp`1 by A33,PSCOMP_1:47;
then G*(m,k) in Lower_Arc L~Cage(C,n+1) by A14,A42,A35,A43,TOPREAL3:6;
then LSeg(G*(m,k),G*(i1,k)) meets Lower_Arc C by A2,A7,A8,A10,A38,A40
,Th32;
then
LSeg(G*(i2,k),G*(i1,k)) meets Lower_Arc C by A2,A3,A7,A10,A37,A38,Th6,
XBOOLE_1:63;
hence thesis by XBOOLE_1:70;
end;
suppose
A44: LSeg(G*(i2,k),G*(i1,k)) meets Lower_Arc L~Cage(C,n+1) & i1 < i2;
set X = LSeg(G*(i1,k),G*(i2,k)) /\ L~Lower_Seq(C,n+1);
ex x be object st x in LSeg(G*(i1,k),G*(i2,k)) & x in L~Lower_Seq(C,n+
1) by A14,A44,XBOOLE_0:3;
then reconsider X1=X as non empty compact Subset of TOP-REAL 2 by
XBOOLE_0:def 4;
consider pp be object such that
A45: pp in W-most X1 by XBOOLE_0:def 1;
reconsider pp as Point of TOP-REAL 2 by A45;
A46: pp in X by A45,XBOOLE_0:def 4;
then
A47: pp in L~Lower_Seq(C,n+1) by XBOOLE_0:def 4;
A48: pp in LSeg(G*(i1,k),G*(i2,k)) by A46,XBOOLE_0:def 4;
consider m be Nat such that
A49: i1 <= m and
A50: m <= i2 and
A51: G*(m,k)`1 = lower_bound(proj1.:(LSeg(G*(i1,k),G*(i2,k)) /\ L~Lower_Seq(
C,n+1))) by A14,A11,A12,A44,JORDAN1F:3,JORDAN1G:5;
A52: m < len G by A4,A50,XXREAL_0:2;
1 < m by A1,A49,XXREAL_0:2;
then
A53: G*(m,k)`2 = G*(1,k)`2 by A7,A10,A52,GOBOARD5:1;
then
A54: |[lower_bound(proj1.:X),G*(1,k)`2]| = G*(m,k) by A51,EUCLID:53;
then G*(i1,k)`2 = |[lower_bound(proj1.:X),G*(1,k)`2]|`2
by A1,A2,A7,A10,A53,GOBOARD5:1;
then
A55: pp`2 = |[lower_bound(proj1.:X),G*(1,k)`2]|`2 by A13,A48,SPPOL_1:40;
|[lower_bound(proj1.:X),G*(1,k)`2]|`1 = W-bound X by A51,A54,SPRECT_1:43
.= (W-min X)`1 by EUCLID:52
.= pp`1 by A45,PSCOMP_1:31;
then G*(m,k) in Lower_Arc L~Cage(C,n+1) by A14,A54,A47,A55,TOPREAL3:6;
then LSeg(G*(i1,k),G*(m,k)) meets Lower_Arc C by A1,A7,A8,A10,A49,A52
,Th40;
then
LSeg(G*(i1,k),G*(i2,k)) meets Lower_Arc C by A1,A4,A7,A10,A49,A50,Th6,
XBOOLE_1:63;
hence thesis by XBOOLE_1:70;
end;
suppose
A56: LSeg(G*(i2,j),G*(i2,k)) misses Upper_Arc L~Cage(C,n+1) & LSeg(
Gauge(C,n+1)*(i2,k),Gauge(C,n+1)*(i1,k)) misses Lower_Arc L~Cage(C,n+1);
consider j1 be Nat such that
A57: j <= j1 and
A58: j1 <= k and
A59: LSeg(G*(i2,j1),G*(i2,k)) /\ L~Lower_Seq(C,n+1) = {G*(i2,j1)} by A3,A4,A5
,A6,A7,A9,A14,Th9;
G*(i2,j1) in LSeg(G*(i2,j1),G*(i2,k)) /\ L~Lower_Seq(C,n+1) by A59,
TARSKI:def 1;
then
A60: G*(i2,j1) in L~Lower_Seq(C,n+1) by XBOOLE_0:def 4;
A61: 1 <= j1 by A5,A57,XXREAL_0:2;
now
per cases;
suppose
A62: i2 <= i1;
A63: LSeg(G*(i2,j1),G*(i2,k)) c= LSeg(G*(i2,j),G*(i2,k)) by A3,A4,A5,A7
,A57,A58,Th5;
consider i3 be Nat such that
A64: i2 <= i3 and
A65: i3 <= i1 and
A66: LSeg(G*(i2,k),G*(i3,k)) /\ L~Upper_Seq(C,n+1) = {G*(i3,k)}
by A2,A3,A7,A8,A18,A10,A62,Th13;
A67: LSeg(G*(i2,k),G*(i3,k)) c= LSeg(G*(i2,k),G*(i1,k)) by A2,A3,A7,A10
,A64,A65,Th6;
then
A68: LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3,k)) c= LSeg(G*
(i2,j),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i1,k)) by A63,XBOOLE_1:13;
G*(i3,k) in LSeg(G*(i2,k),G*(i3,k)) /\ L~Upper_Seq(C,n+1) by A66,
TARSKI:def 1;
then
A69: G*(i3,k) in L~Upper_Seq(C,n+1) by XBOOLE_0:def 4;
A70: (LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3,k))) /\ L~
Upper_Seq(C,n+1) = {G*(i3,k)}
proof
thus (LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3,k))) /\ L~
Upper_Seq(C,n+1) c= {G*(i3,k)}
proof
let x be object;
assume
A71: x in (LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3,
k))) /\ L~Upper_Seq(C,n+1);
then x in LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3,k))
by XBOOLE_0:def 4;
then
A72: x in LSeg(G*(i2,j1),G*(i2,k)) or x in LSeg(G*(i2,k),G*(i3,
k)) by XBOOLE_0:def 3;
x in L~Upper_Seq(C,n+1) by A71,XBOOLE_0:def 4;
hence thesis by A18,A56,A66,A63,A72,XBOOLE_0:def 4;
end;
let x be object;
G*(i3,k) in LSeg(G*(i2,k),G*(i3,k)) by RLTOPSP1:68;
then
A73: G*(i3,k) in LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3,
k)) by XBOOLE_0:def 3;
assume x in {G*(i3,k)};
then x = G*(i3,k) by TARSKI:def 1;
hence thesis by A69,A73,XBOOLE_0:def 4;
end;
A74: (LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3,k))) /\ L~
Lower_Seq(C,n+1) = {G*(i2,j1)}
proof
thus (LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3,k))) /\ L~
Lower_Seq(C,n+1) c= {G*(i2,j1)}
proof
let x be object;
assume
A75: x in (LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3,
k))) /\ L~Lower_Seq(C,n+1);
then x in LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3,k))
by XBOOLE_0:def 4;
then
A76: x in LSeg(G*(i2,j1),G*(i2,k)) or x in LSeg(G*(i2,k),G*(i3,
k)) by XBOOLE_0:def 3;
x in L~Lower_Seq(C,n+1) by A75,XBOOLE_0:def 4;
hence thesis by A14,A56,A59,A67,A76,XBOOLE_0:def 4;
end;
let x be object;
G*(i2,j1) in LSeg(G*(i2,j1),G*(i2,k)) by RLTOPSP1:68;
then
A77: G*(i2,j1) in LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3
,k)) by XBOOLE_0:def 3;
assume x in {G*(i2,j1)};
then x = G*(i2,j1) by TARSKI:def 1;
hence thesis by A60,A77,XBOOLE_0:def 4;
end;
i3 < len G by A2,A65,XXREAL_0:2;
hence thesis by A3,A7,A58,A61,A64,A68,A70,A74,Th45,XBOOLE_1:63;
end;
suppose
A78: i1 < i2;
A79: LSeg(G*(i2,j1),G*(i2,k)) c= LSeg(G*(i2,j),G*(i2,k)) by A3,A4,A5,A7
,A57,A58,Th5;
consider i3 be Nat such that
A80: i1 <= i3 and
A81: i3 <= i2 and
A82: LSeg(G*(i3,k),G*(i2,k)) /\ L~Upper_Seq(C,n+1) = {G*(i3,k)}
by A1,A4,A7,A8,A18,A10,A78,Th18;
A83: LSeg(G*(i2,k),G*(i3,k)) c= LSeg(G*(i2,k),G*(i1,k)) by A1,A4,A7,A10
,A80,A81,Th6;
then
A84: LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3,k)) c= LSeg(G*
(i2,j),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i1,k)) by A79,XBOOLE_1:13;
G*(i3,k) in LSeg(G*(i2,k),G*(i3,k)) /\ L~Upper_Seq(C,n+1) by A82,
TARSKI:def 1;
then
A85: G*(i3,k) in L~Upper_Seq(C,n+1) by XBOOLE_0:def 4;
A86: (LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3,k))) /\ L~
Upper_Seq(C,n+1) = {G*(i3,k)}
proof
thus (LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3,k))) /\ L~
Upper_Seq(C,n+1) c= {G*(i3,k)}
proof
let x be object;
assume
A87: x in (LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3,
k))) /\ L~Upper_Seq(C,n+1);
then x in LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3,k))
by XBOOLE_0:def 4;
then
A88: x in LSeg(G*(i2,j1),G*(i2,k)) or x in LSeg(G*(i2,k),G*(i3,
k)) by XBOOLE_0:def 3;
x in L~Upper_Seq(C,n+1) by A87,XBOOLE_0:def 4;
hence thesis by A18,A56,A82,A79,A88,XBOOLE_0:def 4;
end;
let x be object;
G*(i3,k) in LSeg(G*(i2,k),G*(i3,k)) by RLTOPSP1:68;
then
A89: G*(i3,k) in LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3,
k)) by XBOOLE_0:def 3;
assume x in {G*(i3,k)};
then x = G*(i3,k) by TARSKI:def 1;
hence thesis by A85,A89,XBOOLE_0:def 4;
end;
A90: (LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3,k))) /\ L~
Lower_Seq(C,n+1) = {G*(i2,j1)}
proof
thus (LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3,k))) /\ L~
Lower_Seq(C,n+1) c= {G*(i2,j1)}
proof
let x be object;
assume
A91: x in (LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3,
k))) /\ L~Lower_Seq(C,n+1);
then x in LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3,k))
by XBOOLE_0:def 4;
then
A92: x in LSeg(G*(i2,j1),G*(i2,k)) or x in LSeg(G*(i2,k),G*(i3,
k)) by XBOOLE_0:def 3;
x in L~Lower_Seq(C,n+1) by A91,XBOOLE_0:def 4;
hence thesis by A14,A56,A59,A83,A92,XBOOLE_0:def 4;
end;
let x be object;
G*(i2,j1) in LSeg(G*(i2,j1),G*(i2,k)) by RLTOPSP1:68;
then
A93: G*(i2,j1) in LSeg(G*(i2,j1),G*(i2,k)) \/ LSeg(G*(i2,k),G*(i3
,k)) by XBOOLE_0:def 3;
assume x in {G*(i2,j1)};
then x = G*(i2,j1) by TARSKI:def 1;
hence thesis by A60,A93,XBOOLE_0:def 4;
end;
1 < i3 by A1,A80,XXREAL_0:2;
hence thesis by A4,A7,A58,A61,A81,A84,A86,A90,Th47,XBOOLE_1:63;
end;
end;
hence thesis;
end;
end;
hence thesis;
end;
theorem
for C be Simple_closed_curve for i,j,k be Nat holds 1 < i &
i < len Gauge(C,n+1) & 1 <= j & j <= k & k <= width Gauge(C,n+1) & Gauge(C,n+1)
*(i,k) in Upper_Arc L~Cage(C,n+1) & Gauge(C,n+1)*(Center Gauge(C,n+1),j) in
Lower_Arc L~Cage(C,n+1) implies LSeg(Gauge(C,n+1)*(Center Gauge(C,n+1),j),
Gauge(C,n+1)*(Center Gauge(C,n+1),k)) \/ LSeg(Gauge(C,n+1)*(Center Gauge(C,n+1)
,k),Gauge(C,n+1)*(i,k)) meets Upper_Arc C
proof
let C be Simple_closed_curve;
let i,j,k be Nat;
assume that
A1: 1 < i and
A2: i < len Gauge(C,n+1) and
A3: 1 <= j and
A4: j <= k and
A5: k <= width Gauge(C,n+1) and
A6: Gauge(C,n+1)*(i,k) in Upper_Arc L~Cage(C,n+1) and
A7: Gauge(C,n+1)*(Center Gauge(C,n+1),j) in Lower_Arc L~Cage(C,n+1);
A8: len Gauge(C,n+1) >= 4 by JORDAN8:10;
then len Gauge(C,n+1) >= 3 by XXREAL_0:2;
then
A9: Center Gauge(C,n+1) < len Gauge(C,n+1) by JORDAN1B:15;
len Gauge(C,n+1) >= 2 by A8,XXREAL_0:2;
then 1 < Center Gauge(C,n+1) by JORDAN1B:14;
hence thesis by A1,A2,A3,A4,A5,A6,A7,A9,Th48;
end;
theorem
for C be Simple_closed_curve for i,j,k be Nat holds 1 < i &
i < len Gauge(C,n+1) & 1 <= j & j <= k & k <= width Gauge(C,n+1) & Gauge(C,n+1)
*(i,k) in Upper_Arc L~Cage(C,n+1) & Gauge(C,n+1)*(Center Gauge(C,n+1),j) in
Lower_Arc L~Cage(C,n+1) implies LSeg(Gauge(C,n+1)*(Center Gauge(C,n+1),j),
Gauge(C,n+1)*(Center Gauge(C,n+1),k)) \/ LSeg(Gauge(C,n+1)*(Center Gauge(C,n+1)
,k),Gauge(C,n+1)*(i,k)) meets Lower_Arc C
proof
let C be Simple_closed_curve;
let i,j,k be Nat;
assume that
A1: 1 < i and
A2: i < len Gauge(C,n+1) and
A3: 1 <= j and
A4: j <= k and
A5: k <= width Gauge(C,n+1) and
A6: Gauge(C,n+1)*(i,k) in Upper_Arc L~Cage(C,n+1) and
A7: Gauge(C,n+1)*(Center Gauge(C,n+1),j) in Lower_Arc L~Cage(C,n+1);
A8: len Gauge(C,n+1) >= 4 by JORDAN8:10;
then len Gauge(C,n+1) >= 3 by XXREAL_0:2;
then
A9: Center Gauge(C,n+1) < len Gauge(C,n+1) by JORDAN1B:15;
len Gauge(C,n+1) >= 2 by A8,XXREAL_0:2;
then 1 < Center Gauge(C,n+1) by JORDAN1B:14;
hence thesis by A1,A2,A3,A4,A5,A6,A7,A9,Th49;
end;