### The Mizar article:

### Properties of Left and Right Components

**by****Artur Kornilowicz**

- Received May 5, 1999
Copyright (c) 1999 Association of Mizar Users

- MML identifier: GOBRD14
- [ MML identifier index ]

environ vocabulary SEQM_3, GOBOARD5, SPRECT_2, PRE_TOPC, EUCLID, COMPTS_1, SPPOL_1, GOBOARD1, METRIC_1, CONNSP_1, RELAT_2, RELAT_1, JORDAN2C, SUBSET_1, LATTICES, BOOLE, TOPREAL1, TARSKI, COMPLEX1, ABSVALUE, SETFAM_1, ARYTM_1, FINSEQ_1, SQUARE_1, MCART_1, TREES_1, CARD_3, FUNCOP_1, RCOMP_1, FUNCT_1, MATRIX_1, JORDAN8, GROUP_1, ARYTM_3, PSCOMP_1, INT_1, POWER, GOBOARD9, GOBOARD2, TOPS_1, JORDAN1, FINSEQ_6, SPRECT_1, ORDINAL2, FUNCT_5, SEQ_2, FINSEQ_4, ARYTM; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, RELAT_1, FUNCT_1, FUNCT_2, BINOP_1, ORDINAL1, XCMPLX_0, XREAL_0, REAL_1, SQUARE_1, NAT_1, ABSVALUE, INT_1, POWER, MATRIX_1, BINARITH, FUNCT_4, SEQ_4, STRUCT_0, GROUP_1, METRIC_1, LIMFUNC1, TBSP_1, FINSEQ_1, CARD_3, CARD_4, FINSEQ_4, FINSEQ_6, RCOMP_1, PRE_TOPC, TOPS_1, TOPS_2, CONNSP_1, COMPTS_1, EUCLID, TOPREAL1, GOBOARD1, GOBOARD2, GOBOARD5, GOBOARD9, PSCOMP_1, SPPOL_1, JGRAPH_1, SPRECT_1, SPRECT_2, JORDAN2C, JORDAN8; constructors BINARITH, CARD_4, COMPTS_1, CONNSP_1, FINSEQ_4, GOBOARD2, GOBOARD9, JORDAN1, JORDAN2C, JORDAN8, LIMFUNC1, POWER, PSCOMP_1, RCOMP_1, REAL_1, REALSET1, SPPOL_1, SPRECT_1, SPRECT_2, TBSP_1, TOPGRP_1, TOPREAL2, TOPREAL4, TOPS_1, TOPS_2, WAYBEL_3, TOPRNS_1, MEMBERED, PARTFUN1; clusters SUBSET_1, XREAL_0, EUCLID, GOBOARD2, GOBRD11, PRE_TOPC, PSCOMP_1, RELSET_1, SPRECT_1, SPRECT_3, TOPREAL6, TOPS_1, REVROT_1, INT_1, JORDAN2C, MEMBERED, FUNCT_2, SEQM_3, RELAT_1, ZFMISC_1; requirements REAL, NUMERALS, SUBSET, BOOLE, ARITHM; definitions TARSKI, TOPS_2, JORDAN2C, SPRECT_1, XBOOLE_0; theorems ABSVALUE, AXIOMS, BINARITH, CARD_3, FUNCT_4, COMPTS_1, CONNSP_1, CQC_THE1, EUCLID, FINSEQ_1, FINSEQ_3, FUNCT_1, FUNCT_2, GOBOARD5, GOBOARD6, GOBOARD7, GOBOARD9, GOBRD11, GOBRD12, HEINE, INT_1, JGRAPH_1, JORDAN1, JORDAN2C, JORDAN3, JORDAN8, METRIC_1, NAT_1, POWER, PRE_TOPC, PRE_FF, PSCOMP_1, RCOMP_1, REAL_1, REAL_2, RELAT_1, RLVECT_1, SEQ_4, SPRECT_1, SPRECT_3, SPRECT_4, SQUARE_1, SPPOL_2, SUBSET_1, TARSKI, TBSP_1, TDLAT_1, TOPREAL1, TOPREAL3, TOPREAL6, TOPS_1, TOPS_2, SPRECT_2, REVROT_1, FINSEQ_6, TOPREAL5, AMI_5, SETFAM_1, XBOOLE_0, XBOOLE_1, XREAL_0, XCMPLX_0, XCMPLX_1; begin :: Components reserve i, j, n for Nat, f for non constant standard special_circular_sequence, g for clockwise_oriented non constant standard special_circular_sequence, p, q for Point of TOP-REAL 2, P, Q, R for Subset of TOP-REAL 2, C for compact non vertical non horizontal Subset of TOP-REAL 2, G for Go-board; Lm1: Euclid 2 = MetrStruct(#REAL 2,Pitag_dist 2#) by EUCLID:def 7; theorem for T being TopSpace, A being Subset of T, B being Subset of T st B is_a_component_of A holds B is connected proof let T be TopSpace, A be Subset of T, B be Subset of T; assume B is_a_component_of A; then consider C being Subset of T|A such that A1: C = B and A2: C is_a_component_of T|A by CONNSP_1:def 6; C is connected by A2,CONNSP_1:def 5; hence thesis by A1,CONNSP_1:24; end; theorem for A being Subset of TOP-REAL n, B being Subset of TOP-REAL n st B is_inside_component_of A holds B is connected proof let A be Subset of TOP-REAL n, B be Subset of TOP-REAL n; assume B is_inside_component_of A; then consider C being Subset of (TOP-REAL n)|A` such that A1: C = B and A2: C is_a_component_of (TOP-REAL n)|A` and C is bounded Subset of Euclid n by JORDAN2C:17; C is connected by A2,CONNSP_1:def 5; hence thesis by A1,CONNSP_1:24; end; theorem Th3: for A being Subset of TOP-REAL n, B being Subset of TOP-REAL n st B is_outside_component_of A holds B is connected proof let A be Subset of TOP-REAL n, B be Subset of TOP-REAL n; assume B is_outside_component_of A; then consider C being Subset of (TOP-REAL n)|A` such that A1: C = B and A2: C is_a_component_of (TOP-REAL n)|A` and not C is bounded Subset of Euclid n by JORDAN2C:18; C is connected by A2,CONNSP_1:def 5; hence thesis by A1,CONNSP_1:24; end; theorem for A being Subset of TOP-REAL n, B being Subset of TOP-REAL n st B is_a_component_of A` holds A misses B proof let A be Subset of TOP-REAL n, B be Subset of TOP-REAL n; assume B is_a_component_of A`; then B c= A` by SPRECT_1:7; then A misses B by SUBSET_1:43; hence A /\ B = {} by XBOOLE_0:def 7; end; theorem P is_outside_component_of Q & R is_inside_component_of Q implies P misses R proof assume P is_outside_component_of Q; then A1: P c= UBD Q by JORDAN2C:27; assume R is_inside_component_of Q; then A2: R c= BDD Q by JORDAN2C:26; (BDD Q) misses (UBD Q) by JORDAN2C:28; then P misses BDD Q by A1,XBOOLE_1:63; hence thesis by A2,XBOOLE_1:63; end; theorem Th6: for A, B being Subset of TOP-REAL 2 st A is_outside_component_of L~f & B is_outside_component_of L~f holds A = B proof let A, B be Subset of TOP-REAL 2 such that A1: A is_outside_component_of L~f & B is_outside_component_of L~f; L~f is Bounded by JORDAN2C:73; then consider C being Subset of TOP-REAL 2 such that A2: C is_outside_component_of L~f and A3: C = UBD L~f by JORDAN2C:76; A4: A is_a_component_of (L~f)` & B is_a_component_of (L~f)` by A1,JORDAN2C:def 4; then A5: A <> {} & B <> {} by SPRECT_1:6; A6: C is_a_component_of (L~f)` by A2,JORDAN2C:def 4; A c= UBD L~f & B c= UBD L~f by A1,JORDAN2C:27; then A meets C & B meets C by A3,A5,XBOOLE_1:69; then A = C & B = C by A4,A6,GOBOARD9:3; hence thesis; end; theorem for p being Point of Euclid 2 st p = 0.REAL 2 & P is_outside_component_of L~ f ex r being Real st r > 0 & Ball(p,r)` c= P proof let p be Point of Euclid 2 such that A1: p = 0.REAL 2 and A2: P is_outside_component_of L~f; A3: L~f is Bounded by JORDAN2C:73; then consider D being Subset of Euclid 2 such that A4: D = L~f and A5: D is bounded by JORDAN2C:def 2; consider r being Real, c being Point of Euclid 2 such that A6: 0 < r and c in D and A7: for z being Point of Euclid 2 st z in D holds dist(c,z) <= r by A4,A5,TBSP_1:15; set rr = dist(p,c)+r+1; A8: D c= Ball(p,rr) proof let x be set; assume A9: x in D; then reconsider z = x as Point of Euclid 2; dist(c,z) <= r by A7,A9; then A10: dist(p,c) + dist(c,z) <= dist(p,c) + r by REAL_1:55; dist(p,z) <= dist(p,c) + dist(c,z) by METRIC_1:4; then A11: dist(p,z) <= dist(p,c) + r by A10,AXIOMS:22; dist(p,c) + r + 0 < dist(p,c) + r + 1 by REAL_1:53; then dist(p,z) < dist(p,c) + r + 1 by A11,AXIOMS:22; hence x in Ball(p,rr) by METRIC_1:12; end; take rr; A12: 0 <= dist(p,c) by METRIC_1:5; dist(p,c)+r+0 < rr by REAL_1:67; hence A13: 0 < rr by A6,A12,REAL_1:67; let x be set; assume A14: x in Ball(p,rr)`; then reconsider y = x as Point of Euclid 2; reconsider z = y as Point of TOP-REAL 2 by TOPREAL3:13; consider B being Subset of TOP-REAL 2 such that A15: B is_outside_component_of L~f and A16: B = UBD L~f by A3,JORDAN2C:76; A17: UBD L~f = union{W where W is Subset of TOP-REAL 2: W is_outside_component_of L~f} by JORDAN2C:def 6; A18: P = B by A2,A15,Th6; set L = (REAL 2) \ {a where a is Point of TOP-REAL 2: |.a.| < rr}; A19: the carrier of TOP-REAL 2 = REAL 2 by EUCLID:25; A20: L c= REAL 2 proof let l be set; assume l in L; hence l in REAL 2 by XBOOLE_0:def 4; end; rr = abs rr by A13,ABSVALUE:def 1; then for a being Point of TOP-REAL 2 holds a <> |[0,rr]| or |.a.| >= rr by TOPREAL6:31; then not |[0,rr]| in {a where a is Point of TOP-REAL 2: |.a.| < rr}; then reconsider L as non empty Subset of TOP-REAL 2 by A19,A20,XBOOLE_0:def 4; A21: z in REAL 2 by A19; A22: not x in Ball(p,rr) by A14,SUBSET_1:54; dist(p,y) = |.z.| by A1,TOPREAL6:33; then for k being Point of TOP-REAL 2 holds k <> z or |.k.| >= rr by A22,METRIC_1:12; then not x in {a where a is Point of TOP-REAL 2: |.a.| < rr}; then A23: x in L by A21,XBOOLE_0:def 4; A24: L is connected by JORDAN2C:61; L c= (L~f)` proof let l be set; assume A25: l in L; then A26: not l in {a where a is Point of TOP-REAL 2: |.a.| < rr} by XBOOLE_0 :def 4; reconsider j = l as Point of TOP-REAL 2 by A25; reconsider t = j as Point of Euclid 2 by TOPREAL3:13; A27: |.j.| >= rr by A26; now assume l in L~f; then dist(t,p) < rr by A4,A8,METRIC_1:12; hence contradiction by A1,A27,TOPREAL6:33; end; hence l in (L~f)` by A25,SUBSET_1:50; end; then consider M being Subset of TOP-REAL 2 such that A28: M is_a_component_of (L~f)` and A29: L c= M by A24,GOBOARD9:5; M is_outside_component_of L~f proof thus M is_a_component_of (L~f)` by A28; not L is Bounded proof thus for C being Subset of Euclid 2 holds C <> L or not C is bounded by JORDAN2C:70; end; hence not M is Bounded by A29,JORDAN2C:16; end; then M in {W where W is Subset of TOP-REAL 2: W is_outside_component_of L~ f}; hence x in P by A16,A17,A18,A23,A29,TARSKI:def 4; end; definition let C be closed Subset of TOP-REAL 2; cluster BDD C -> open; coherence proof set F = {B where B is Subset of TOP-REAL 2: B is_inside_component_of C}; F c= bool the carrier of TOP-REAL 2 proof let f be set; assume f in F; then ex B being Subset of TOP-REAL 2 st f = B & B is_inside_component_of C; hence thesis; end; then F is Subset-Family of TOP-REAL 2 by SETFAM_1:def 7; then reconsider F as Subset-Family of TOP-REAL 2; A1: BDD C = union F by JORDAN2C:def 5; F is open proof let P be Subset of TOP-REAL 2; assume P in F; then consider B being Subset of TOP-REAL 2 such that A2: P = B and A3: B is_inside_component_of C; B is_a_component_of C` by A3,JORDAN2C:def 3; then B is_a_component_of C`; hence P is open by A2,SPRECT_3:18; end; hence thesis by A1,TOPS_2:26; end; cluster UBD C -> open; coherence proof set F = {B where B is Subset of TOP-REAL 2: B is_outside_component_of C}; F c= bool the carrier of TOP-REAL 2 proof let f be set; assume f in F; then ex B being Subset of TOP-REAL 2 st f = B & B is_outside_component_of C; hence thesis; end; then F is Subset-Family of TOP-REAL 2 by SETFAM_1:def 7; then reconsider F as Subset-Family of TOP-REAL 2; A4: UBD C = union F by JORDAN2C:def 6; F is open proof let P be Subset of TOP-REAL 2; assume P in F; then consider B being Subset of TOP-REAL 2 such that A5: P = B and A6: B is_outside_component_of C; B is_a_component_of C` by A6,JORDAN2C:def 4; then B is_a_component_of C`; hence P is open by A5,SPRECT_3:18; end; hence thesis by A4,TOPS_2:26; end; end; definition let C be compact Subset of TOP-REAL 2; cluster UBD C -> connected; coherence proof C is Bounded by JORDAN2C:73; then ex B being Subset of TOP-REAL 2 st B is_outside_component_of C & B = UBD C by JORDAN2C:76; hence thesis by Th3; end; end; begin :: Go-boards theorem :: TOPREAL1:29 for f being FinSequence of TOP-REAL n st L~f <> {} holds 2 <= len f proof let f be FinSequence of TOP-REAL n; assume L~f <> {}; then len f <> 0 & len f <> 1 by TOPREAL1:28; then len f > 1 by CQC_THE1:2; then len f >= 1 + 1 by NAT_1:38; hence thesis; end; definition let n be Nat, a, b be Point of TOP-REAL n; func dist(a,b) -> Real means :Def1: ex p, q being Point of Euclid n st p = a & q = b & it = dist(p,q); existence proof reconsider p = a, q = b as Point of Euclid n by TOPREAL3:13; take dist(p,q); thus thesis; end; uniqueness; commutativity; end; theorem Th9: dist(p,q) = sqrt ((p`1-q`1)^2 + (p`2-q`2)^2) proof A1: ex a, b being Point of Euclid 2 st p = a & q = b & dist(a,b) = dist(p,q) by Def1; p = |[p`1, p`2]| & q = |[q`1, q`2]| by EUCLID:57; hence thesis by A1,GOBOARD6:9; end; theorem Th10: for p being Point of TOP-REAL n holds dist(p,p) = 0 proof let p be Point of TOP-REAL n; ex a, b being Point of Euclid n st a = p & b = p & dist(a,b) = dist(p,p) by Def1; hence thesis by METRIC_1:1; end; theorem for p, q, r being Point of TOP-REAL n holds dist(p,r) <= dist (p,q) + dist(q,r) proof let p, q, r be Point of TOP-REAL n; (ex a, b being Point of Euclid n st a = p & b = q & dist(a,b) = dist(p,q) ) & (ex a, b being Point of Euclid n st a = p & b = r & dist(a,b) = dist(p,r)) & ex a, b being Point of Euclid n st a = q & b = r & dist(a,b) = dist(q,r) by Def1; hence thesis by METRIC_1:4; end; theorem for x1, x2, y1, y2 being real number, a, b being Point of TOP-REAL 2 st x1 <= a`1 & a`1 <= x2 & y1 <= a`2 & a`2 <= y2 & x1 <= b`1 & b`1 <= x2 & y1 <= b`2 & b`2 <= y2 holds dist(a,b) <= (x2-x1) + (y2-y1) proof let x1, x2, y1, y2 be real number, a, b be Point of TOP-REAL 2 such that A1: x1 <= a`1 & a`1 <= x2 & y1 <= a`2 & a`2 <= y2 & x1 <= b`1 & b`1 <= x2 & y1 <= b`2 & b`2 <= y2; A2: dist(a,b) = sqrt ((a`1-b`1)^2 + (a`2-b`2)^2) by Th9; x1 is Real & x2 is Real & y1 is Real & y2 is Real by XREAL_0:def 1; then abs(a`1-b`1) <= x2 - x1 & abs(a`2-b`2) <= y2 - y1 by A1,JGRAPH_1:31; then A3: abs(a`1-b`1) + abs(a`2-b`2) <= (x2 - x1) + (y2 - y1) by REAL_1:55; (a`1-b`1)^2 >= 0 & (a`2-b`2)^2 >= 0 by SQUARE_1:72; then dist(a,b) <= sqrt(a`1-b`1)^2 + sqrt(a`2-b`2)^2 by A2,TOPREAL6:6; then dist(a,b) <= abs(a`1-b`1) + sqrt(a`2-b`2)^2 by SQUARE_1:91; then dist(a,b) <= abs(a`1-b`1) + abs(a`2-b`2) by SQUARE_1:91; hence thesis by A3,AXIOMS:22; end; theorem Th13: 1 <= i & i < len G & 1 <= j & j < width G implies cell(G,i,j) = product ((1,2) --> ([.G*(i,1)`1,G*(i+1,1)`1.],[.G*(1,j)`2,G*(1,j+1)`2.])) proof assume 1 <= i & i < len G & 1 <= j & j < width G; then A1: cell(G,i,j) = { |[r,s]| where r, s is Real: G*(i,1)`1 <= r & r <= G*(i+1,1)`1 & G*(1,j)`2 <= s & s <= G*(1,j+1)`2 } by GOBRD11:32; A2: [.G*(i,1)`1,G*(i+1,1)`1.] = {a where a is Real : G*(i,1)`1 <= a & a <= G*(i+1,1)`1 } & [.G*(1,j)`2,G*(1,j+1)`2.] = {b where b is Real : G*(1,j)`2 <= b & b <= G*(1,j+1)`2 } by RCOMP_1:def 1; set f = (1,2) --> ([.G*(i,1)`1,G*(i+1,1)`1.],[.G*(1,j)`2,G*(1,j+1)`2.]); A3: dom f = {1,2} by FUNCT_4:65; thus cell(G,i,j) c= product f proof let c be set; assume c in cell(G,i,j); then consider r, s being Real such that A4: c = |[r,s]| and A5: G*(i,1)`1 <= r & r <= G*(i+1,1)`1 & G*(1,j)`2 <= s & s <= G*(1,j+1)`2 by A1; A6: r in [.G*(i,1)`1,G*(i+1,1)`1.] & s in [.G*(1,j)`2,G*(1,j+1)`2.] by A2,A5; A7: dom <*r,s*> = {1,2} by FINSEQ_1:4,FINSEQ_3:29; A8: <*r,s*> = c by A4,EUCLID:def 16; for x being set st x in dom f holds <*r,s*>.x in f.x proof let x be set; assume x in dom f; then A9: x = 1 or x = 2 by A3,TARSKI:def 2; A10: |[r,s]| = <*r,s*> by EUCLID:def 16; A11: r = |[r,s]|`1 by EUCLID:56 .= <*r,s*>.1 by A10,EUCLID:def 14; s = |[r,s]|`2 by EUCLID:56 .= <*r,s*>.2 by A10,EUCLID:def 15; hence thesis by A6,A9,A11,FUNCT_4:66; end; hence thesis by A3,A7,A8,CARD_3:18; end; let c be set; assume c in product f; then consider g being Function such that A12: c = g and A13: dom g = dom f and A14: for x being set st x in dom f holds g.x in f.x by CARD_3:def 5; 1 in dom f & 2 in dom f by A3,TARSKI:def 2; then A15: g.1 in f.1 & g.2 in f.2 by A14; then g.1 in [.G*(i,1)`1,G*(i+1,1)`1.] by FUNCT_4:66; then consider a being Real such that A16: g.1 = a and A17: G*(i,1)`1 <= a & a <= G*(i+1,1)`1 by A2; g.2 in [.G*(1,j)`2,G*(1,j+1)`2.] by A15,FUNCT_4:66; then consider b being Real such that A18: g.2 = b and A19: G*(1,j)`2 <= b & b <= G*(1,j+1)`2 by A2; A20: dom <*a,b*> = {1,2} by FINSEQ_1:4,FINSEQ_3:29; A21: dom g = {1,2} by A13,FUNCT_4:65; for k being set st k in dom g holds g.k = <*a,b*>.k proof let k be set; assume k in dom g; then k = 1 or k = 2 by A21,TARSKI:def 2; hence thesis by A16,A18,FINSEQ_1:61; end; then g = <*a,b*> by A20,A21,FUNCT_1:9; then c = |[a,b]| by A12,EUCLID:def 16; hence thesis by A1,A17,A19; end; theorem 1 <= i & i < len G & 1 <= j & j < width G implies cell(G,i,j) is compact proof assume 1 <= i & i < len G & 1 <= j & j < width G; then cell(G,i,j) = product ((1,2) --> ([.G*(i,1)`1,G*(i+1,1)`1.], [.G*(1,j)`2,G*(1,j+1)`2.])) by Th13; hence thesis by TOPREAL6:87; end; theorem [i,j] in Indices G & [i+n,j] in Indices G implies dist(G*(i,j),G*(i+n,j)) = G*(i+n,j)`1 - G*(i,j)`1 proof assume A1: [i,j] in Indices G & [i+n,j] in Indices G; set x = G*(i,j), y = G*(i+n,j); per cases; suppose A2: n = 0; hence dist(G*(i,j),G*(i+n,j)) = 0 by Th10 .= G*(i+n,j)`1 - G*(i,j)`1 by A2,XCMPLX_1:14; suppose A3: n <> 0; A4: 1 <= i+n & i+n <= len G by A1,GOBOARD5:1; A5: 1 <= j & j <= width G & 1 <= i & i <= len G by A1,GOBOARD5:1; then A6: x`2 = G*(1,j)`2 by GOBOARD5:2 .= y`2 by A4,A5,GOBOARD5:2; 1 <= n by A3,RLVECT_1:99; then i < i+n by RLVECT_1:103; then x`1 < y`1 by A4,A5,GOBOARD5:4; then x`1 - x`1 < y`1 - x`1 by REAL_1:92; then A7: 0 < y`1 - x`1 by XCMPLX_1:14; thus dist(G*(i,j),G*(i+n,j)) = sqrt ((x`1-y`1)^2 + (x`2-y`2)^2) by Th9 .= sqrt ((x`1-y`1)^2 + 0) by A6,SQUARE_1:60,XCMPLX_1:14 .= abs(x`1-y`1) by SQUARE_1:91 .= abs(-(x`1-y`1)) by ABSVALUE:17 .= abs(y`1-x`1) by XCMPLX_1:143 .= G*(i+n,j)`1 - G*(i,j)`1 by A7,ABSVALUE:def 1; end; theorem [i,j] in Indices G & [i,j+n] in Indices G implies dist(G*(i,j),G*(i,j+n)) = G*(i,j+n)`2 - G*(i,j)`2 proof assume A1: [i,j] in Indices G & [i,j+n] in Indices G; set x = G*(i,j), y = G*(i,j+n); per cases; suppose A2: n = 0; hence dist(G*(i,j),G*(i,j+n)) = 0 by Th10 .= G*(i,j+n)`2 - G*(i,j)`2 by A2,XCMPLX_1:14; suppose A3: n <> 0; A4: 1 <= j+n & j+n <= width G by A1,GOBOARD5:1; A5: 1 <= j & j <= width G & 1 <= i & i <= len G by A1,GOBOARD5:1; then A6: x`1 = G*(i,1)`1 by GOBOARD5:3 .= y`1 by A4,A5,GOBOARD5:3; 1 <= n by A3,RLVECT_1:99; then j < j+n by RLVECT_1:103; then x`2 < y`2 by A4,A5,GOBOARD5:5; then x`2 - x`2 < y`2 - x`2 by REAL_1:92; then A7: 0 < y`2 - x`2 by XCMPLX_1:14; thus dist(G*(i,j),G*(i,j+n)) = sqrt ((x`1-y`1)^2 + (x`2-y`2)^2) by Th9 .= sqrt (0 + (x`2-y`2)^2) by A6,SQUARE_1:60,XCMPLX_1:14 .= abs(x`2-y`2) by SQUARE_1:91 .= abs(-(x`2-y`2)) by ABSVALUE:17 .= abs(y`2-x`2) by XCMPLX_1:143 .= G*(i,j+n)`2 - G*(i,j)`2 by A7,ABSVALUE:def 1; end; theorem 3 <= len Gauge(C,n)-'1 proof set G = Gauge(C,n); 4 <= len G by JORDAN8:13; then A1: 4 - 1 <= len G-1 by REAL_1:92; then 0 <= len G - 1 by AXIOMS:22; hence thesis by A1,BINARITH:def 3; end; theorem i <= j implies for a, b being Nat st 2 <= a & a <= len Gauge(C,i) - 1 & 2 <= b & b <= len Gauge(C,i) - 1 ex c, d being Nat st 2 <= c & c <= len Gauge(C,j) - 1 & 2 <= d & d <= len Gauge(C,j) - 1 & [c,d] in Indices Gauge(C,j) & Gauge(C,i)*(a,b) = Gauge(C,j)*(c,d) & c = 2 + 2|^(j-'i)*(a-'2) & d = 2 + 2|^(j-'i)*(b-'2) proof assume A1: i <= j; let a, b be Nat such that A2: 2 <= a and A3: a <= len Gauge(C,i) - 1 and A4: 2 <= b and A5: b <= len Gauge(C,i) - 1; set c = 2 + 2|^(j-'i)*(a-'2), d = 2 + 2|^(j-'i)*(b-'2); take c, d; 2|^i + 2 - 2 = 2|^i + (2 - 2) by XCMPLX_1:29 .= 2|^i + 0; then A6: 2|^i + 2 - 2 >= 0 by NAT_1:18; A7: 0 <= 2|^(j-'i) by NAT_1:18; A8: 0 <> 2|^i by HEINE:5; A9: 0 <> 2|^j by HEINE:5; A10: len Gauge(C,i) - 1 = 2|^i + 3 - 1 by JORDAN8:def 1 .= 2|^i + (3 - 1) by XCMPLX_1:29 .= 2|^i + 2; A11: 2|^i + 2 -' 2 = 2|^i + 2 - 2 by A6,BINARITH:def 3 .= 2|^i + (2 - 2) by XCMPLX_1:29 .= 2|^i + 0; A12: 2|^(j-'i)*(2|^i) = 2|^j / 2|^i*(2|^i) by A1,TOPREAL6:15 .= 2|^j by A8,XCMPLX_1:88; a -' 2 <= 2|^i + 2 -' 2 by A3,A10,JORDAN3:5; then A13: 2|^(j-'i)*(a-'2) <= 2|^j by A7,A11,A12,AXIOMS:25; b -' 2 <= 2|^i + 2 -' 2 by A5,A10,JORDAN3:5; then A14: 2|^(j-'i)*(b-'2) <= 2|^j by A7,A11,A12,AXIOMS:25; A15: len Gauge(C,j) - 1 = 2|^j + 3 - 1 by JORDAN8:def 1 .= 2|^j + (3 - 1) by XCMPLX_1:29 .= 2|^j + 2; 0 <= 2|^(j-'i)*(a-'2) & 0 <= 2|^(j-'i)*(b-'2) by NAT_1:18; then 2+0 <= 2+2|^(j-'i)*(a-'2) & 2+0 <= 2+2|^(j-'i)*(b-'2) by AXIOMS:24; hence A16: 2 <= c & c <= len Gauge(C,j) - 1 & 2 <= d & d <= len Gauge(C,j) - 1 by A13,A14,A15,AXIOMS:24; A17: width Gauge(C,j) = len Gauge(C,j) by JORDAN8:def 1; len Gauge(C,j) - 1 < len Gauge(C,j) - 0 by REAL_1:92; then 1 <= c & c <= len Gauge(C,j) & 1 <= d & d <= width Gauge(C,j) by A16,A17,AXIOMS:22; hence A18: [c,d] in Indices Gauge(C,j) by GOBOARD7:10; A19: width Gauge(C,i) = len Gauge(C,i) by JORDAN8:def 1; set n = N-bound C, e = E-bound C, s = S-bound C, w = W-bound C; len Gauge(C,i) - 1 < len Gauge(C,i) - 0 by REAL_1:92; then 1 <= a & a <= len Gauge(C,i) & 1 <= b & b <= width Gauge(C,i) by A2,A3,A4,A5,A19,AXIOMS:22; then [a,b] in Indices Gauge(C,i) by GOBOARD7:10; then A20: Gauge(C,i)*(a,b) = |[w+(e-w)/(2|^i)*(a-2), s+(n-s)/(2|^i)*(b-2)]| by JORDAN8:def 1; A21: Gauge(C,j)*(c,d) = |[w+(e-w)/(2|^j)*(c-2), s+(n-s)/(2|^j)*(d-2)]| by A18,JORDAN8:def 1; A22: 0 <= a-2 by A2,SQUARE_1:12; A23: 0 <= b-2 by A4,SQUARE_1:12; A24: (e-w)/(2|^j)*(c-2) = (e-w)/(2|^j)*(2 - 2 + 2|^(j-'i)*(a-'2)) by XCMPLX_1:29 .= (e-w)/(2|^j)*((2|^j/2|^i)*(a-'2)) by A1,TOPREAL6:15 .= (e-w)/(2|^j)*(2|^j/2|^i)*(a-'2) by XCMPLX_1:4 .= (e-w)/((2|^j)/(2|^j/2|^i))*(a-'2) by XCMPLX_1:82 .= (e-w)/(2|^i)*(a-'2) by A9,XCMPLX_1:52 .= (e-w)/(2|^i)*(a-2) by A22,BINARITH:def 3; A25: (n-s)/(2|^j)*(d-2) = (n-s)/(2|^j)*(2 - 2 + 2|^(j-'i)*(b-'2)) by XCMPLX_1:29 .= (n-s)/(2|^j)*((2|^j/2|^i)*(b-'2)) by A1,TOPREAL6:15 .= (n-s)/(2|^j)*(2|^j/2|^i)*(b-'2) by XCMPLX_1:4 .= (n-s)/((2|^j)/(2|^j/2|^i))*(b-'2) by XCMPLX_1:82 .= (n-s)/(2|^i)*(b-'2) by A9,XCMPLX_1:52 .= (n-s)/(2|^i)*(b-2) by A23,BINARITH:def 3; A26: Gauge(C,i)*(a,b)`1 = w+(e-w)/(2|^i)*(a-2) by A20,EUCLID:56 .= Gauge(C,j)*(c,d)`1 by A21,A24,EUCLID:56; Gauge(C,i)*(a,b)`2 = s+(n-s)/(2|^i)*(b-2) by A20,EUCLID:56 .= Gauge(C,j)*(c,d)`2 by A21,A25,EUCLID:56; hence Gauge(C,i)*(a,b) = Gauge(C,j)*(c,d) by A26,TOPREAL3:11; thus thesis; end; theorem Th19: [i,j] in Indices Gauge(C,n) & [i,j+1] in Indices Gauge(C,n) implies dist(Gauge(C,n)*(i,j),Gauge(C,n)*(i,j+1)) = (N-bound C - S-bound C)/2|^n proof set G = Gauge(C,n), a = N-bound C, e = E-bound C, s = S-bound C, w = W-bound C, p1 = G*(i,j), p2 = G*(i,j+1); assume that A1: [i,j] in Indices G and A2: [i,j+1] in Indices G; consider p, q being Point of Euclid 2 such that A3: p = p1 & q = p2 and A4: dist(p1,p2) = dist(p,q) by Def1; A5: p1 = |[w+(e-w)/(2|^n)*(i-2), s+(a-s)/(2|^n)*(j-2)]| by A1,JORDAN8:def 1; A6: p2 = |[w+(e-w)/(2|^n)*(i-2), s+(a-s)/(2|^n)*(j+1-2)]| by A2,JORDAN8:def 1; set x = (a-s)/(2|^n); A7: 2|^n > 0 by HEINE:5; a >= s by SPRECT_1:24; then a - s >= s - s by REAL_1:49; then A8: a - s >= 0 by XCMPLX_1:14; dist(p,q) = (Pitag_dist 2).(p,q) by Lm1,METRIC_1:def 1 .= sqrt ((p1`1 - p2`1)^2 + (p1`2 - p2`2)^2) by A3,TOPREAL3:12 .= sqrt ((w+(e-w)/(2|^n)*(i-2) - p2`1)^2 + (p1`2 - p2`2)^2) by A5,EUCLID:56 .= sqrt ((w+(e-w)/(2|^n)*(i-2) - (w+(e-w)/(2|^n)*(i-2)))^2 + (p1`2 - p2`2)^2) by A6,EUCLID:56 .= sqrt (0 + (p1`2 - p2`2)^2) by SQUARE_1:60,XCMPLX_1:14 .= abs(p1`2 - p2`2) by SQUARE_1:91 .= abs(s+x*(j-2) - p2`2) by A5,EUCLID:56 .= abs(s+x*(j-2) - (s+x*(j+1-2))) by A6,EUCLID:56 .= abs(s+x*(j-2) - s - x*(j+1-2)) by XCMPLX_1:36 .= abs(s-s+x*(j-2) - x*(j+1-2)) by XCMPLX_1:29 .= abs(0+x*(j-2) - x*(j+1-2)) by XCMPLX_1:14 .= abs(x*j-x*2 - x*(j+1-2)) by XCMPLX_1:40 .= abs(x*j-x*2 - (x*j+x*1-x*2)) by XCMPLX_1:43 .= abs(x*j-x*2 - (x*j+x*1) + x*2) by XCMPLX_1:37 .= abs(x*j-x*2 - x*j-x*1 + x*2) by XCMPLX_1:36 .= abs(x*j-(x*2 + x*j)-x + x*2) by XCMPLX_1:36 .= abs(x*j-x*j - x*2-x + x*2) by XCMPLX_1:36 .= abs(0 - x*2-x + x*2) by XCMPLX_1:14 .= abs(-x*2-x + x*2) by XCMPLX_1:150 .= abs(-x*2 + x*2 - x) by XCMPLX_1:29 .= abs(0 - x) by XCMPLX_0:def 6 .= abs(-x) by XCMPLX_1:150 .= abs(x) by ABSVALUE:17 .= abs(a-s)/abs(2|^n) by ABSVALUE:16 .= (a-s)/abs(2|^n) by A8,ABSVALUE:def 1; hence thesis by A4,A7,ABSVALUE:def 1; end; theorem Th20: [i,j] in Indices Gauge(C,n) & [i+1,j] in Indices Gauge(C,n) implies dist(Gauge(C,n)*(i,j),Gauge(C,n)*(i+1,j)) = (E-bound C - W-bound C)/2|^n proof set G = Gauge(C,n), a = N-bound C, e = E-bound C, s = S-bound C, w = W-bound C, p1 = G*(i,j), p2 = G*(i+1,j); assume that A1: [i,j] in Indices G and A2: [i+1,j] in Indices G; consider p, q being Point of Euclid 2 such that A3: p = p1 & q = p2 and A4: dist(p1,p2) = dist(p,q) by Def1; A5: p1 = |[w+(e-w)/(2|^n)*(i-2), s+(a-s)/(2|^n)*(j-2)]| by A1,JORDAN8:def 1; A6: p2 = |[w+(e-w)/(2|^n)*(i+1-2), s+(a-s)/(2|^n)*(j-2)]| by A2,JORDAN8:def 1; set x = (e-w)/(2|^n); A7: 2|^n > 0 by HEINE:5; e >= w by SPRECT_1:23; then e - w >= w - w by REAL_1:49; then A8: e - w >= 0 by XCMPLX_1:14; dist(p,q) = (Pitag_dist 2).(p,q) by Lm1,METRIC_1:def 1 .= sqrt ((p1`1 - p2`1)^2 + (p1`2 - p2`2)^2) by A3,TOPREAL3:12 .= sqrt ((p1`1 - p2`1)^2 + (s+(a-s)/(2|^n)*(j-2) - p2`2)^2) by A5,EUCLID:56 .= sqrt ((p1`1 - p2`1)^2 + (s+(a-s)/(2|^n)*(j-2) - (s+(a-s)/(2|^n)*(j-2)))^2) by A6,EUCLID:56 .= sqrt ((p1`1 - p2`1)^2 + 0) by SQUARE_1:60,XCMPLX_1:14 .= abs(p1`1 - p2`1) by SQUARE_1:91 .= abs(w+x*(i-2) - p2`1) by A5,EUCLID:56 .= abs(w+x*(i-2) - (w+x*(i+1-2))) by A6,EUCLID:56 .= abs(w+x*(i-2) - w - x*(i+1-2)) by XCMPLX_1:36 .= abs(w-w+x*(i-2) - x*(i+1-2)) by XCMPLX_1:29 .= abs(0+x*(i-2) - x*(i+1-2)) by XCMPLX_1:14 .= abs(x*i-x*2 - x*(i+1-2)) by XCMPLX_1:40 .= abs(x*i-x*2 - (x*i+x*1-x*2)) by XCMPLX_1:43 .= abs(x*i-x*2 - (x*i+x*1) + x*2) by XCMPLX_1:37 .= abs(x*i-x*2 - x*i-x*1 + x*2) by XCMPLX_1:36 .= abs(x*i-(x*2 + x*i)-x + x*2) by XCMPLX_1:36 .= abs(x*i-x*i - x*2-x + x*2) by XCMPLX_1:36 .= abs(0 - x*2-x + x*2) by XCMPLX_1:14 .= abs(-x*2-x + x*2) by XCMPLX_1:150 .= abs(-x*2 + x*2 - x) by XCMPLX_1:29 .= abs(0 - x) by XCMPLX_0:def 6 .= abs(-x) by XCMPLX_1:150 .= abs(x) by ABSVALUE:17 .= abs(e-w)/abs(2|^n) by ABSVALUE:16 .= (e-w)/abs(2|^n) by A8,ABSVALUE:def 1; hence thesis by A4,A7,ABSVALUE:def 1; end; theorem for r, t being real number holds r > 0 & t > 0 implies ex n being Nat st 1 < n & dist(Gauge(C,n)*(1,1),Gauge(C,n)*(1,2)) < r & dist(Gauge(C,n)*(1,1),Gauge(C,n)*(2,1)) < t proof let r, t be real number; assume that A1: r > 0 and A2: t > 0; set n = N-bound C, e = E-bound C, s = S-bound C, w = W-bound C; set a = abs([\ log(2,(n-s)/r) /]) + 1, b = abs([\ log(2,(e-w)/t) /]) + 1; take i = max(a,b)+1; abs([\ log(2,(n-s)/r) /]) >= 0 by ABSVALUE:5; then A3: a >= 0+1 by REAL_1:55; max(a,b) >= a by SQUARE_1:46; then max(a,b) >= 1 by A3,AXIOMS:22; then max(a,b)+1 >= 1+1 by REAL_1:55; hence 1 < i by AXIOMS:22; a <= max(a,b) & b <= max(a,b) by SQUARE_1:46; then A4: a+1 <= max(a,b)+1 & b+1 <= max(a,b)+1 by AXIOMS:24; a < a+1 & b < b+1 by REAL_1:69; then A5: i > a & i > b by A4,AXIOMS:22; A6: 2 to_power i > 0 by POWER:39; then A7: 2|^i > 0 by POWER:48; [\ log(2,(n-s)/r) /] > log(2,(n-s)/r) - 1 & [\ log(2,(e-w)/t) /] > log(2,(e-w)/t) - 1 by INT_1:def 4; then A8: [\ log(2,(n-s)/r) /] + 1 > log(2,(n-s)/r) - 1 + 1 & [\ log(2,(e-w)/t) /] + 1 > log(2,(e-w)/t) - 1 + 1 by REAL_1:53; [\ log(2,(n-s)/r) /] <= abs([\ log(2,(n-s)/r) /]) & [\ log(2,(e-w)/t) /] <= abs([\ log(2,(e-w)/t) /]) by ABSVALUE:11; then [\ log(2,(n-s)/r) /] + 1 <= a & [\ log(2,(e-w)/t) /] + 1 <= b by AXIOMS:24; then a > log(2,(n-s)/r) - 1 + 1 & b > log(2,(e-w)/t) - 1 + 1 by A8,AXIOMS:22; then a > log(2,(n-s)/r) & b > log(2,(e-w)/t) by XCMPLX_1:27; then i > log(2,(n-s)/r) & i > log(2,(e-w)/t) by A5,AXIOMS:22; then log(2,2 to_power i) > log(2,(n-s)/r) & log(2,2 to_power i) > log(2,(e-w)/t) by A6,POWER:def 3; then 2 to_power i > (n-s)/r & 2 to_power i > (e-w)/t by A6,PRE_FF:12; then 2|^i > (n-s)/r & 2|^i > (e-w)/t by POWER:48; then 2|^i * r > (n-s)/r * r & 2|^i * t > (e-w)/t * t by A1,A2,REAL_1:70; then 2|^i * r > n-s & 2|^i * t > e-w by A1,A2,XCMPLX_1:88; then 2|^i * r / 2|^i > (n-s) / 2|^i & 2|^i * t / 2|^i > (e-w) / 2|^i by A7,REAL_1:73; then A9: (n-s)/2|^i < r & (e-w)/2|^i < t by A7,XCMPLX_1:90; A10: len Gauge(C,i) = width Gauge(C,i) by JORDAN8:def 1; len Gauge(C,i) >= 4 by JORDAN8:13; then 1 <= len Gauge(C,i) & 1+1 <= width Gauge(C,i) by A10,AXIOMS:22; then [1,1] in Indices Gauge(C,i) & [1,1+1] in Indices Gauge(C,i) & [1+1,1] in Indices Gauge(C,i) by A10,GOBOARD7:10; hence thesis by A9,Th19,Th20; end; begin :: LeftComp and RightComp theorem Th22: for P being Subset of (TOP-REAL 2)|(L~f)` st P is_a_component_of (TOP-REAL 2)|(L~f)` holds P = RightComp f or P = LeftComp f proof let P be Subset of (TOP-REAL 2)|(L~f)`; assume that A1: P is_a_component_of (TOP-REAL 2)|(L~f)` and A2: P <> RightComp f; LeftComp f is_a_component_of (L~f)` by GOBOARD9:def 1; then consider L being Subset of (TOP-REAL 2)|(L~f)` such that A3: L = LeftComp f & L is_a_component_of (TOP-REAL 2)|(L~f)` by CONNSP_1:def 6; RightComp f is_a_component_of (L~f)` by GOBOARD9:def 2; then consider R being Subset of (TOP-REAL 2)|(L~f)` such that A4: R = RightComp f and A5: R is_a_component_of (TOP-REAL 2)|(L~f)` by CONNSP_1:def 6; A6: P misses R by A1,A2,A4,A5,CONNSP_1:37; P <> {}((TOP-REAL 2)|(L~f)`) by A1,CONNSP_1:34; then consider a being Point of (TOP-REAL 2)|(L~f)` such that A7: a in P by SUBSET_1:10; A8: the carrier of (TOP-REAL 2)|(L~f)` = (L~f)` by JORDAN1:1; (L~f)` = LeftComp f \/ RightComp f by GOBRD12:11; then a in LeftComp f or a in RightComp f by A8,XBOOLE_0:def 2; then P meets LeftComp f by A4,A6,A7,XBOOLE_0:3; hence P = LeftComp f by A1,A3,CONNSP_1:37; end; theorem for A1, A2 being Subset of TOP-REAL 2 st (L~f)` = A1 \/ A2 & A1 misses A2 & for C1, C2 being Subset of (TOP-REAL 2)|(L~f)` st C1 = A1 & C2 = A2 holds C1 is_a_component_of (TOP-REAL 2)|(L~f)` & C2 is_a_component_of (TOP-REAL 2)|(L~f)` holds A1 = RightComp f & A2 = LeftComp f or A1 = LeftComp f & A2 = RightComp f proof let A1, A2 be Subset of TOP-REAL 2 such that A1: (L~f)` = A1 \/ A2 and A2: A1 /\ A2 = {} and A3: for C1, C2 being Subset of (TOP-REAL 2)|(L~f)` st C1 = A1 & C2 = A2 holds C1 is_a_component_of (TOP-REAL 2)|(L~f)` & C2 is_a_component_of (TOP-REAL 2)|(L~f)`; A4: the carrier of (TOP-REAL 2)|(L~f)` = (L~f)` by JORDAN1:1; A1 c= A1 \/ A2 & A2 c= A1 \/ A2 by XBOOLE_1:7; then reconsider C1 = A1, C2 = A2 as Subset of (TOP-REAL 2)|(L~f)` by A1,A4; A5: C1 = A1 & C2 = A2; then A6: C1 is_a_component_of (TOP-REAL 2)|(L~f)` by A3; C2 is_a_component_of (TOP-REAL 2)|(L~f)` by A3,A5; then A7: (C1 = RightComp f or C1 = LeftComp f) & (C2 = RightComp f or C2 = LeftComp f) by A6,Th22; assume not thesis; hence contradiction by A2,A7; end; theorem Th24: LeftComp f misses RightComp f proof assume LeftComp f /\ RightComp f <> {}; then consider x being set such that A1: x in LeftComp f /\ RightComp f by XBOOLE_0:def 1; A2: LeftComp f meets RightComp f proof now take x; thus x in LeftComp f & x in RightComp f by A1,XBOOLE_0:def 3; end; hence thesis by XBOOLE_0:3; end; LeftComp f is_a_component_of (L~f)` & RightComp f is_a_component_of (L~f) ` by GOBOARD9:def 1,def 2; then LeftComp f = RightComp f by A2,GOBOARD9:3; hence thesis by SPRECT_4:7; end; theorem Th25: L~f \/ RightComp f \/ LeftComp f = the carrier of TOP-REAL 2 proof A1: (L~f)` = RightComp f \/ LeftComp f by GOBRD12:11; (L~f)` \/ L~f = [#]the carrier of TOP-REAL 2 by SUBSET_1:25 .= the carrier of TOP-REAL 2 by SUBSET_1:def 4; hence thesis by A1,XBOOLE_1:4; end; theorem Th26: p in L~f iff not p in LeftComp f & not p in RightComp f proof A1: (L~f)` = LeftComp f \/ RightComp f by GOBRD12:11; p in L~f iff not p in (L~f)` by SUBSET_1:50,54; hence thesis by A1,XBOOLE_0:def 2; end; theorem Th27: p in LeftComp f iff not p in L~f & not p in RightComp f proof A1: (L~f)` = LeftComp f \/ RightComp f by GOBRD12:11; LeftComp f misses RightComp f by Th24; then A2: LeftComp f /\ RightComp f = {} by XBOOLE_0:def 7; p in L~f iff not p in (L~f)` by SUBSET_1:50,54; hence thesis by A1,A2,XBOOLE_0:def 2,def 3; end; theorem p in RightComp f iff not p in L~f & not p in LeftComp f by Th26,Th27; theorem Th29: L~f = (Cl RightComp f) \ RightComp f proof thus L~f c= (Cl RightComp f) \ RightComp f proof let x be set; assume A1: x in L~f; then reconsider p = x as Point of TOP-REAL 2; consider i such that A2: 1 <= i & i+1 <= len f and A3: p in LSeg(f,i) by A1,SPPOL_2:13; for O being Subset of TOP-REAL 2 st O is open holds p in O implies RightComp f meets O proof let O be Subset of TOP-REAL 2 such that A4: O is open and A5: p in O; left_cell(f,i) /\ right_cell(f,i) = LSeg(f,i) by A2,GOBOARD5:32; then LSeg(f,i) c= right_cell(f,i) by XBOOLE_1:17; then A6: p in right_cell(f,i) by A3; f is_sequence_on GoB f by GOBOARD5:def 5; then consider i1, j1, i2, j2 being Nat such that A7: [i1,j1] in Indices GoB f & f/.i = GoB f*(i1,j1) & [i2,j2] in Indices GoB f & f/.(i+1) = GoB f*(i2,j2) and A8: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A2,JORDAN8:6; A9: 1 <= i1 & 1 <= j1 by A7,GOBOARD5:1; A10: i1 <= len GoB f & j1 <= width GoB f by A7,GOBOARD5:1; A11: i2 <= len GoB f & j2 <= width GoB f by A7,GOBOARD5:1; now per cases by A8; case A12: i1 = i2 & j1+1 = j2; set w = i1-'1; A13: w+1 = i1 by A9,AMI_5:4; then right_cell(f,i) = cell(GoB f,w+1,j1) by A2,A7,A12,GOBOARD5:28; hence p in Cl Int right_cell(f,i) by A6,A10,A13,GOBRD11:35; case A14: i1+1 = i2 & j1 = j2; set w = j1-'1; w+1 = j1 by A9,AMI_5:4; then A15: right_cell(f,i) = cell(GoB f,i1,w) by A2,A7,A14,GOBOARD5:29; w <= w+1 & w+1 <= width GoB f by A9,A10,AMI_5:4,NAT_1:29; then w <= width GoB f by AXIOMS:22; hence p in Cl Int right_cell(f,i) by A6,A10,A15,GOBRD11:35; case A16: i1 = i2+1 & j1 = j2; set w = j1-'1; A17: w+1 = j1 by A9,AMI_5:4; then right_cell(f,i) = cell(GoB f,i2,w+1) by A2,A7,A16,GOBOARD5:30; hence p in Cl Int right_cell(f,i) by A6,A10,A11,A17,GOBRD11:35; case A18: i1 = i2 & j1 = j2+1; set z = i1-'1; z+1 = i1 by A9,AMI_5:4; then A19: right_cell(f,i) = cell(GoB f,z,j2) by A2,A7,A18,GOBOARD5:31; z <= z+1 & z+1 <= len GoB f by A9,A10,AMI_5:4,NAT_1:29; then z <= len GoB f by AXIOMS:22; hence p in Cl Int right_cell(f,i) by A6,A11,A19,GOBRD11:35; end; then A20: O meets Int right_cell(f,i) by A4,A5,PRE_TOPC:def 13; Int right_cell(f,i) c= RightComp f by A2,GOBOARD9:28; hence RightComp f meets O by A20,XBOOLE_1:63; end; then A21: p in Cl RightComp f by PRE_TOPC:def 13; not x in RightComp f by A1,Th26; hence thesis by A21,XBOOLE_0:def 4; end; assume not (Cl RightComp f) \ RightComp f c= L~f; then consider q being set such that A22: q in (Cl RightComp f) \ RightComp f and A23: not q in L~f by TARSKI:def 3; reconsider q as Point of TOP-REAL 2 by A22; A24: q in Cl RightComp f & not q in RightComp f by A22,XBOOLE_0:def 4; then q in LeftComp f by A23,Th26; then LeftComp f meets RightComp f by A24,PRE_TOPC:def 13; hence contradiction by Th24; end; theorem Th30: L~f = (Cl LeftComp f) \ LeftComp f proof thus L~f c= (Cl LeftComp f) \ LeftComp f proof let x be set; assume A1: x in L~f; then reconsider p = x as Point of TOP-REAL 2; consider i such that A2: 1 <= i & i+1 <= len f and A3: p in LSeg(f,i) by A1,SPPOL_2:13; for O being Subset of TOP-REAL 2 st O is open holds p in O implies LeftComp f meets O proof let O be Subset of TOP-REAL 2 such that A4: O is open and A5: p in O; left_cell(f,i) /\ right_cell(f,i) = LSeg(f,i) by A2,GOBOARD5:32; then LSeg(f,i) c= left_cell(f,i) by XBOOLE_1:17; then A6: p in left_cell(f,i) by A3; f is_sequence_on GoB f by GOBOARD5:def 5; then consider i1, j1, i2, j2 being Nat such that A7: [i1,j1] in Indices GoB f & f/.i = GoB f*(i1,j1) & [i2,j2] in Indices GoB f & f/.(i+1) = GoB f*(i2,j2) and A8: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A2,JORDAN8:6; A9: 1 <= i1 & 1 <= j1 by A7,GOBOARD5:1; A10: i1 <= len GoB f & j1 <= width GoB f by A7,GOBOARD5:1; A11: i2 <= len GoB f & j2 <= width GoB f by A7,GOBOARD5:1; now per cases by A8; case A12: i1 = i2 & j1+1 = j2; set w = i1-'1; w+1 = i1 by A9,AMI_5:4; then A13: left_cell(f,i) = cell(GoB f,w,j1) by A2,A7,A12,GOBOARD5:28; w <= w+1 & w+1 <= len GoB f by A9,A10,AMI_5:4,NAT_1:29; then w <= len GoB f by AXIOMS:22; hence p in Cl Int left_cell(f,i) by A6,A10,A13,GOBRD11:35; case A14: i1+1 = i2 & j1 = j2; set w = j1-'1; w+1 = j1 by A9,AMI_5:4; then A15: left_cell(f,i) = cell(GoB f,i1,w+1) by A2,A7,A14,GOBOARD5:29; w+1 <= width GoB f by A9,A10,AMI_5:4; hence p in Cl Int left_cell(f,i) by A6,A10,A15,GOBRD11:35; case A16: i1 = i2+1 & j1 = j2; set w = j1-'1; w+1 = j1 by A9,AMI_5:4; then A17: left_cell(f,i) = cell(GoB f,i2,w) by A2,A7,A16,GOBOARD5:30; w <= w+1 & w+1 <= width GoB f by A9,A10,AMI_5:4,NAT_1:29; then w <= width GoB f by AXIOMS:22; hence p in Cl Int left_cell(f,i) by A6,A11,A17,GOBRD11:35; case A18: i1 = i2 & j1 = j2+1; set z = i1-'1; z+1 = i1 by A9,AMI_5:4; then A19: left_cell(f,i) = cell(GoB f,z+1,j2) by A2,A7,A18,GOBOARD5:31; z+1 <= len GoB f by A9,A10,AMI_5:4; hence p in Cl Int left_cell(f,i) by A6,A11,A19,GOBRD11:35; end; then A20: O meets Int left_cell(f,i) by A4,A5,PRE_TOPC:def 13; Int left_cell(f,i) c= LeftComp f by A2,GOBOARD9:24; hence LeftComp f meets O by A20,XBOOLE_1:63; end; then A21: p in Cl LeftComp f by PRE_TOPC:def 13; not x in LeftComp f by A1,Th26; hence thesis by A21,XBOOLE_0:def 4; end; assume not (Cl LeftComp f) \ LeftComp f c= L~f; then consider q being set such that A22: q in (Cl LeftComp f) \ LeftComp f and A23: not q in L~f by TARSKI:def 3; reconsider q as Point of TOP-REAL 2 by A22; A24: q in Cl LeftComp f & not q in LeftComp f by A22,XBOOLE_0:def 4; then q in RightComp f by A23,Th26; then RightComp f meets LeftComp f by A24,PRE_TOPC:def 13; hence contradiction by Th24; end; theorem Th31: Cl RightComp f = (RightComp f) \/ L~f proof thus Cl RightComp f c= RightComp f \/ L~f proof let x be set; assume A1: x in Cl RightComp f; now assume A2: not x in RightComp f; now assume x in LeftComp f; then LeftComp f meets RightComp f by A1,TOPS_1:39; hence contradiction by Th24; end; hence x in L~f by A1,A2,Th26; end; hence thesis by XBOOLE_0:def 2; end; L~f c= (Cl RightComp f) \ RightComp f & (Cl RightComp f) \ RightComp f c= Cl RightComp f by Th29,XBOOLE_1:36 ; then RightComp f c= Cl RightComp f & L~f c= Cl RightComp f by PRE_TOPC:48,XBOOLE_1:1; hence thesis by XBOOLE_1:8; end; theorem Cl LeftComp f = (LeftComp f) \/ L~f proof thus Cl LeftComp f c= LeftComp f \/ L~f proof let x be set; assume A1: x in Cl LeftComp f; now assume A2: not x in LeftComp f; now assume x in RightComp f; then LeftComp f meets RightComp f by A1,TOPS_1:39; hence contradiction by Th24; end; hence x in L~f by A1,A2,Th26; end; hence thesis by XBOOLE_0:def 2; end; L~f c= (Cl LeftComp f) \ LeftComp f & (Cl LeftComp f) \ LeftComp f c= Cl LeftComp f by Th30,XBOOLE_1:36; then LeftComp f c= Cl LeftComp f & L~f c= Cl LeftComp f by PRE_TOPC:48,XBOOLE_1:1; hence thesis by XBOOLE_1:8; end; definition let f be non constant standard special_circular_sequence; cluster L~f -> Jordan; coherence proof thus (L~f)` <> {}; take A1 = RightComp f, A2 = LeftComp f; thus (L~f)` = A1 \/ A2 by GOBRD12:11; A1 misses A2 & L~f = (Cl A1) \ A1 by Th24,Th29; hence thesis by Th30,GOBOARD9:def 1,def 2; end; end; reserve f for clockwise_oriented non constant standard special_circular_sequence; theorem Th33: p in RightComp f implies W-bound L~f < p`1 proof set g = Rotate(f,N-min L~f); A1: L~f = L~g by REVROT_1:33; N-min L~f in rng f by SPRECT_2:43; then A2: g/.1 = N-min L~g by A1,FINSEQ_6:98; reconsider u = p as Point of Euclid 2 by TOPREAL3:13; assume p in RightComp f; then p in RightComp g by REVROT_1:37; then u in Int RightComp g by TOPS_1:55; then consider r being real number such that A3: r > 0 and A4: Ball(u,r) c= RightComp g by GOBOARD6:8; reconsider r as Real by XREAL_0:def 1; A5: W-bound L~SpStSeq L~g = W-bound L~g by SPRECT_1:66; A6: W-bound L~SpStSeq L~g = inf(proj1.:L~SpStSeq L~g) by SPRECT_1:48; A7: proj1.:L~SpStSeq L~g is bounded_below by SPRECT_1:46; set x = |[p`1-r/2,N-bound L~SpStSeq L~g]|; reconsider k = |[p`1-r/2,p`2]| as Point of Euclid 2 by TOPREAL3:13; A8: r/2 > 0 by A3,REAL_2:127; dist(u,k) = (Pitag_dist 2).(u,k) by Lm1,METRIC_1:def 1 .= sqrt ((p`1 - |[p`1-r/2,p`2]|`1)^2 + (p`2 - |[p`1-r/2,p`2]|`2)^2) by TOPREAL3:12 .= sqrt ((p`1 - (p`1-r/2))^2 + (p`2 - |[p`1-r/2,p`2]|`2)^2) by EUCLID:56 .= sqrt ((p`1 - (p`1-r/2))^2 + (p`2 - p`2)^2) by EUCLID:56 .= sqrt ((p`1 - (p`1-r/2))^2 + 0) by SQUARE_1:60,XCMPLX_1:14 .= sqrt ((p`1 - p`1 + r/2)^2 + 0) by XCMPLX_1:37 .= sqrt ((0 + r/2)^2) by XCMPLX_1:14 .= r/2 by A8,SQUARE_1:89; then dist(u,k) < r/1 by A3,REAL_2:200; then A9: k in Ball(u,r) by METRIC_1:12; RightComp g misses LeftComp g by Th24; then A10: not |[p`1-r/2,p`2]| in LeftComp g by A4,A9,XBOOLE_0:3; then |[p`1-r/2,p`2]|`1 <= E-bound L~g by A2,JORDAN2C:119; then p`1-r/2 <= E-bound L~g by EUCLID:56; then A11: x`1 <= E-bound L~g by EUCLID:56; |[p`1-r/2,p`2]|`1 >= W-bound L~g by A2,A10,JORDAN2C:118; then p`1-r/2 >= W-bound L~g by EUCLID:56; then A12: x`1 >= W-bound L~g by EUCLID:56; x`2 = N-bound L~SpStSeq L~g by EUCLID:56; then A13: x`2 = N-bound L~g by SPRECT_1:68; LSeg(NW-corner L~g, NE-corner L~g) = { q : q`1 <= E-bound L~g & q`1 >= W-bound L~g & q`2 = N-bound L~g} by SPRECT_1:27; then A14: x in LSeg(NW-corner L~g,NE-corner L~g) by A11,A12,A13; A15: LSeg(NW-corner L~g,NE-corner L~g) c= L~SpStSeq L~g by SPRECT_3:26; proj1.x = x`1 by PSCOMP_1:def 28 .= p`1-r/2 by EUCLID:56; then p`1-r/2 in proj1.:L~SpStSeq L~g by A14,A15,FUNCT_2:43; then A16: inf(proj1.:L~SpStSeq L~g) <= p`1-r/2 by A7,SEQ_4:def 5; p`1-r/2 < p`1-0 by A8,REAL_1:92; hence thesis by A1,A5,A6,A16,AXIOMS:22 ; end; theorem Th34: p in RightComp f implies E-bound L~f > p`1 proof set g = Rotate(f,N-min L~f); A1: L~f = L~g by REVROT_1:33; N-min L~f in rng f by SPRECT_2:43; then A2: g/.1 = N-min L~g by A1,FINSEQ_6:98; reconsider u = p as Point of Euclid 2 by TOPREAL3:13; assume p in RightComp f; then p in RightComp g by REVROT_1:37; then u in Int RightComp g by TOPS_1:55; then consider r being real number such that A3: r > 0 and A4: Ball(u,r) c= RightComp g by GOBOARD6:8; reconsider r as Real by XREAL_0:def 1; A5: E-bound L~SpStSeq L~g = E-bound L~g by SPRECT_1:69; A6: E-bound L~SpStSeq L~g = sup(proj1.:L~SpStSeq L~g) by SPRECT_1:51; A7: proj1.:L~SpStSeq L~g is bounded_above by SPRECT_1:47; set x = |[p`1+r/2,N-bound L~SpStSeq L~g]|; reconsider k = |[p`1+r/2,p`2]| as Point of Euclid 2 by TOPREAL3:13; A8: r/2 > 0 by A3,REAL_2:127; dist(u,k) = (Pitag_dist 2).(u,k) by Lm1,METRIC_1:def 1 .= sqrt ((p`1 - |[p`1+r/2,p`2]|`1)^2 + (p`2 - |[p`1+r/2,p`2]|`2)^2) by TOPREAL3:12 .= sqrt ((p`1 - (p`1+r/2))^2 + (p`2 - |[p`1+r/2,p`2]|`2)^2) by EUCLID:56 .= sqrt ((p`1 - (p`1+r/2))^2 + (p`2 - p`2)^2) by EUCLID:56 .= sqrt ((p`1 - (p`1+r/2))^2 + 0) by SQUARE_1:60,XCMPLX_1:14 .= sqrt ((p`1 - p`1 - r/2)^2 + 0) by XCMPLX_1:36 .= sqrt ((0 - r/2)^2) by XCMPLX_1:14 .= sqrt ((-r/2)^2) by XCMPLX_1:150 .= sqrt ((r/2)^2) by SQUARE_1:61 .= r/2 by A8,SQUARE_1:89; then dist(u,k) < r/1 by A3,REAL_2:200; then A9: k in Ball(u,r) by METRIC_1:12; RightComp g misses LeftComp g by Th24; then A10: not |[p`1+r/2,p`2]| in LeftComp g by A4,A9,XBOOLE_0:3; then |[p`1+r/2,p`2]|`1 <= E-bound L~g by A2,JORDAN2C:119; then p`1+r/2 <= E-bound L~g by EUCLID:56; then A11: x`1 <= E-bound L~g by EUCLID:56; |[p`1+r/2,p`2]|`1 >= W-bound L~g by A2,A10,JORDAN2C:118; then p`1+r/2 >= W-bound L~g by EUCLID:56; then A12: x`1 >= W-bound L~g by EUCLID:56; x`2 = N-bound L~SpStSeq L~g by EUCLID:56; then A13: x`2 = N-bound L~g by SPRECT_1:68; LSeg(NW-corner L~g, NE-corner L~g) = { q : q`1 <= E-bound L~g & q`1 >= W-bound L~g & q`2 = N-bound L~g} by SPRECT_1:27; then A14: x in LSeg(NW-corner L~g,NE-corner L~g) by A11,A12,A13; A15: LSeg(NW-corner L~g,NE-corner L~g) c= L~SpStSeq L~g by SPRECT_3:26; proj1.x = x`1 by PSCOMP_1:def 28 .= p`1+r/2 by EUCLID:56; then p`1+r/2 in proj1.:L~SpStSeq L~g by A14,A15,FUNCT_2:43; then A16: sup(proj1.:L~SpStSeq L~g) >= p`1+r/2 by A7,SEQ_4:def 4; p`1+r/2 > p`1+0 by A8,REAL_1:53; hence thesis by A1,A5,A6,A16,AXIOMS:22; end; theorem Th35: p in RightComp f implies N-bound L~f > p`2 proof set g = Rotate(f,N-min L~f); A1: L~f = L~g by REVROT_1:33; N-min L~f in rng f by SPRECT_2:43; then A2: g/.1 = N-min L~g by A1,FINSEQ_6:98; reconsider u = p as Point of Euclid 2 by TOPREAL3:13; assume p in RightComp f; then p in RightComp g by REVROT_1:37; then u in Int RightComp g by TOPS_1:55; then consider r being real number such that A3: r > 0 and A4: Ball(u,r) c= RightComp g by GOBOARD6:8; reconsider r as Real by XREAL_0:def 1; A5: N-bound L~SpStSeq L~g = N-bound L~g by SPRECT_1:68; A6: N-bound L~SpStSeq L~g = sup(proj2.:L~SpStSeq L~g) by SPRECT_1:50; A7: proj2.:L~SpStSeq L~g is bounded_above by SPRECT_1:47; set x = |[E-bound L~SpStSeq L~g,p`2+r/2]|; reconsider k = |[p`1,p`2+r/2]| as Point of Euclid 2 by TOPREAL3:13; A8: r/2 > 0 by A3,REAL_2:127; dist(u,k) = (Pitag_dist 2).(u,k) by Lm1,METRIC_1:def 1 .= sqrt ((p`1 - |[p`1,p`2+r/2]|`1)^2 + (p`2 - |[p`1,p`2+r/2]|`2)^2) by TOPREAL3:12 .= sqrt ((p`1 - p`1)^2 + (p`2 - |[p`1,p`2+r/2]|`2)^2) by EUCLID:56 .= sqrt (0 + (p`2 - |[p`1,p`2+r/2]|`2)^2) by SQUARE_1:60,XCMPLX_1:14 .= sqrt ((p`2 - (p`2+r/2))^2) by EUCLID:56 .= sqrt ((p`2 - p`2 - r/2)^2) by XCMPLX_1:36 .= sqrt ((0 - r/2)^2) by XCMPLX_1:14 .= sqrt ((-r/2)^2) by XCMPLX_1:150 .= sqrt ((r/2)^2) by SQUARE_1:61 .= r/2 by A8,SQUARE_1:89; then dist(u,k) < r/1 by A3,REAL_2:200; then A9: k in Ball(u,r) by METRIC_1:12; RightComp g misses LeftComp g by Th24; then A10: not |[p`1,p`2+r/2]| in LeftComp g by A4,A9,XBOOLE_0:3; then |[p`1,p`2+r/2]|`2 <= N-bound L~g by A2,JORDAN2C:121; then p`2+r/2 <= N-bound L~g by EUCLID:56; then A11: x`2 <= N-bound L~g by EUCLID:56; |[p`1,p`2+r/2]|`2 >= S-bound L~g by A2,A10,JORDAN2C:120; then p`2+r/2 >= S-bound L~g by EUCLID:56; then A12: x`2 >= S-bound L~g by EUCLID:56; x`1 = E-bound L~SpStSeq L~g by EUCLID:56; then A13: x`1 = E-bound L~g by SPRECT_1:69; LSeg(SE-corner L~g, NE-corner L~g) = { q : q`1 = E-bound L~g & q`2 <= N-bound L~g & q`2 >= S-bound L~g } by SPRECT_1:25; then A14: x in LSeg(SE-corner L~g,NE-corner L~g) by A11,A12,A13; A15: LSeg(SE-corner L~g,NE-corner L~g) c= L~SpStSeq L~g by TOPREAL6:43; proj2.x = x`2 by PSCOMP_1:def 29 .= p`2+r/2 by EUCLID:56; then p`2+r/2 in proj2.:L~SpStSeq L~g by A14,A15,FUNCT_2:43; then A16: sup(proj2.:L~SpStSeq L~g) >= p`2+r/2 by A7,SEQ_4:def 4; p`2+r/2 > p`2+0 by A8,REAL_1:53; hence thesis by A1,A5,A6,A16,AXIOMS:22; end; theorem Th36: p in RightComp f implies S-bound L~f < p`2 proof set g = Rotate(f,N-min L~f); A1: L~f = L~g by REVROT_1:33; N-min L~f in rng f by SPRECT_2:43; then A2: g/.1 = N-min L~g by A1,FINSEQ_6:98; reconsider u = p as Point of Euclid 2 by TOPREAL3:13; assume p in RightComp f; then p in RightComp g by REVROT_1:37; then u in Int RightComp g by TOPS_1:55; then consider r being real number such that A3: r > 0 and A4: Ball(u,r) c= RightComp g by GOBOARD6:8; reconsider r as Real by XREAL_0:def 1; A5: S-bound L~SpStSeq L~g = S-bound L~g by SPRECT_1:67; A6: S-bound L~SpStSeq L~g = inf(proj2.:L~SpStSeq L~g) by SPRECT_1:49; A7: proj2.:L~SpStSeq L~g is bounded_below by SPRECT_1:46; set x = |[E-bound L~SpStSeq L~g,p`2-r/2]|; reconsider k = |[p`1,p`2-r/2]| as Point of Euclid 2 by TOPREAL3:13; A8: r/2 > 0 by A3,REAL_2:127; dist(u,k) = (Pitag_dist 2).(u,k) by Lm1,METRIC_1:def 1 .= sqrt ((p`1 - |[p`1,p`2-r/2]|`1)^2 + (p`2 - |[p`1,p`2-r/2]|`2)^2) by TOPREAL3:12 .= sqrt ((p`1 - p`1)^2 + (p`2 - |[p`1,p`2-r/2]|`2)^2) by EUCLID:56 .= sqrt (0 + (p`2 - |[p`1,p`2-r/2]|`2)^2) by SQUARE_1:60,XCMPLX_1:14 .= sqrt ((p`2 - (p`2-r/2))^2) by EUCLID:56 .= sqrt ((p`2 - p`2 + r/2)^2) by XCMPLX_1:37 .= sqrt ((0 + r/2)^2) by XCMPLX_1:14 .= r/2 by A8,SQUARE_1:89; then dist(u,k) < r/1 by A3,REAL_2:200; then A9: k in Ball(u,r) by METRIC_1:12; RightComp g misses LeftComp g by Th24; then A10: not |[p`1,p`2-r/2]| in LeftComp g by A4,A9,XBOOLE_0:3; then |[p`1,p`2-r/2]|`2 <= N-bound L~g by A2,JORDAN2C:121; then p`2-r/2 <= N-bound L~g by EUCLID:56; then A11: x`2 <= N-bound L~g by EUCLID:56; |[p`1,p`2-r/2]|`2 >= S-bound L~g by A2,A10,JORDAN2C:120; then p`2-r/2 >= S-bound L~g by EUCLID:56; then A12: x`2 >= S-bound L~g by EUCLID:56; x`1 = E-bound L~SpStSeq L~g by EUCLID:56; then A13: x`1 = E-bound L~g by SPRECT_1:69; LSeg(SE-corner L~g, NE-corner L~g) = { q : q`1 = E-bound L~g & q`2 <= N-bound L~g & q`2 >= S-bound L~g } by SPRECT_1:25; then A14: x in LSeg(SE-corner L~g,NE-corner L~g) by A11,A12,A13; A15: LSeg(SE-corner L~g,NE-corner L~g) c= L~SpStSeq L~g by TOPREAL6:43; proj2.x = x`2 by PSCOMP_1:def 29 .= p`2-r/2 by EUCLID:56; then p`2-r/2 in proj2.:L~SpStSeq L~g by A14,A15,FUNCT_2:43; then A16: inf(proj2.:L~SpStSeq L~g) <= p`2-r/2 by A7,SEQ_4:def 5; p`2-r/2 < p`2-0 by A8,REAL_1:92; hence thesis by A1,A5,A6,A16,AXIOMS:22 ; end; theorem Th37: p in RightComp f & q in LeftComp f implies LSeg(p,q) meets L~f proof assume A1: p in RightComp f & q in LeftComp f; assume LSeg(p,q) misses L~f; then LSeg(p,q) c= (L~f)` by TDLAT_1:2; then LSeg(p,q) c= (L~f)`; then LSeg(p,q) c= the carrier of (TOP-REAL 2)|(L~f)` by JORDAN1:1; then reconsider A = LSeg(p,q) as Subset of (TOP-REAL 2)|(L~f)` ; A2: A is connected by CONNSP_1:24; RightComp f is_a_component_of (L~f)` by GOBOARD9:def 2; then consider R being Subset of (TOP-REAL 2)|(L~f)` such that A3: R = RightComp f and A4: R is_a_component_of (TOP-REAL 2)|(L~f)` by CONNSP_1:def 6; LeftComp f is_a_component_of (L~f)` by GOBOARD9:def 1; then consider L being Subset of (TOP-REAL 2)|(L~f)` such that A5: L = LeftComp f and A6: L is_a_component_of (TOP-REAL 2)|(L~f)` by CONNSP_1:def 6; p in A & q in A by TOPREAL1:6; then R meets A & L meets A by A1,A3,A5,XBOOLE_0:3; then RightComp f = LeftComp f by A2,A3,A4,A5,A6,JORDAN2C:100; hence contradiction by SPRECT_4:7; end; theorem Th38: Cl RightComp SpStSeq C = product((1,2)-->([.W-bound L~SpStSeq C,E-bound L~SpStSeq C.], [.S-bound L~SpStSeq C,N-bound L~SpStSeq C.])) proof set g = (1,2)-->([.W-bound L~SpStSeq C,E-bound L~SpStSeq C.], [.S-bound L~SpStSeq C,N-bound L~SpStSeq C.]); A1: Cl RightComp SpStSeq C = (RightComp SpStSeq C) \/ L~SpStSeq C by Th31; A2: dom g = {1,2} by FUNCT_4:65; hereby let a be set; assume A3: a in Cl RightComp SpStSeq C; then consider r, s being Real such that A4: a = <*r,s*> by EUCLID:55; reconsider b = a as Point of TOP-REAL 2 by A3; reconsider h = a as FinSequence by A4; A5: dom h = {1,2} by A4,FINSEQ_1:4,FINSEQ_3:29; for x being set st x in {1,2} holds h.x in g.x proof let x be set such that A6: x in {1,2}; per cases by A6,TARSKI:def 2; suppose A7: x = 1; then A8: g.x = [.W-bound L~SpStSeq C,E-bound L~SpStSeq C.] by FUNCT_4:66; A9: h.x = r by A4,A7,FINSEQ_1:61; A10: b`1 = |[r,s]|`1 by A4,EUCLID:def 16 .= r by EUCLID:56; now per cases by A1,A3,XBOOLE_0:def 2; case a in RightComp SpStSeq C; then W-bound L~SpStSeq C < b`1 & E-bound L~SpStSeq C > b`1 by Th33,Th34; hence h.x in g.x by A8,A9,A10,TOPREAL5:1; case a in L~SpStSeq C; then W-bound L~SpStSeq C <= b`1 & b`1 <= E-bound L~SpStSeq C by PSCOMP_1:71; hence h.x in g.x by A8,A9,A10,TOPREAL5:1; end; hence h.x in g.x; suppose A11: x = 2; then A12: g.x = [.S-bound L~SpStSeq C,N-bound L~SpStSeq C.] by FUNCT_4:66; A13: h.x = s by A4,A11,FINSEQ_1:61; A14: b`2 = |[r,s]|`2 by A4,EUCLID:def 16 .= s by EUCLID:56; now per cases by A1,A3,XBOOLE_0:def 2; case a in RightComp SpStSeq C; then S-bound L~SpStSeq C < b`2 & N-bound L~SpStSeq C > b`2 by Th35,Th36; hence h.x in g.x by A12,A13,A14,TOPREAL5:1; case a in L~SpStSeq C; then S-bound L~SpStSeq C <= b`2 & b`2 <= N-bound L~SpStSeq C by PSCOMP_1:71; hence h.x in g.x by A12,A13,A14,TOPREAL5:1; end; hence h.x in g.x; end; hence a in product g by A2,A5,CARD_3:18; end; let a be set; assume a in product g; then consider h being Function such that A15: a = h and A16: dom h = dom g and A17: for x being set st x in dom g holds h.x in g.x by CARD_3:def 5; 1 in dom g & 2 in dom g by A2,TARSKI:def 2; then A18: h.1 in g.1 & h.2 in g.2 by A17; then A19: h.1 in [.W-bound L~SpStSeq C,E-bound L~SpStSeq C.] by FUNCT_4:66; [.W-bound L~SpStSeq C,E-bound L~SpStSeq C.] = {r where r is Real : W-bound L~SpStSeq C <= r & r <= E-bound L~SpStSeq C} by RCOMP_1:def 1; then consider r being Real such that A20: h.1 = r and A21: W-bound L~SpStSeq C <= r & r <= E-bound L~SpStSeq C by A19; A22: h.2 in [.S-bound L~SpStSeq C,N-bound L~SpStSeq C.] by A18,FUNCT_4:66; [.S-bound L~SpStSeq C,N-bound L~SpStSeq C.] = {s where s is Real: S-bound L~SpStSeq C <= s & s <= N-bound L~SpStSeq C} by RCOMP_1:def 1; then consider s being Real such that A23: h.2 = s and A24: S-bound L~SpStSeq C <= s & s <= N-bound L~SpStSeq C by A22; A25: dom <*r,s*> = {1,2} by FINSEQ_1:4,FINSEQ_3:29; A26: dom h = {1,2} by A16,FUNCT_4:65; for k being set st k in dom h holds h.k = <*r,s*>.k proof let k be set; assume k in dom h; then k = 1 or k = 2 by A26,TARSKI:def 2; hence thesis by A20,A23,FINSEQ_1:61; end; then a = <*r,s*> by A15,A25,A26,FUNCT_1:9; then A27: a = |[r,s]| by EUCLID:def 16; assume not a in Cl RightComp SpStSeq C; then not a in RightComp SpStSeq C & not a in L~SpStSeq C by A1,XBOOLE_0:def 2 ; then A28: a in LeftComp SpStSeq C by A27,Th26; LeftComp SpStSeq C = {q: not(W-bound L~SpStSeq C <= q`1 & q`1 <= E-bound L~SpStSeq C & S-bound L~SpStSeq C <= q`2 & q`2 <= N-bound L~SpStSeq C)} by SPRECT_3:54; then ex q st q = a & not(W-bound L~SpStSeq C <= q`1 & q`1 <= E-bound L~SpStSeq C & S-bound L~SpStSeq C <= q`2 & q`2 <= N-bound L~SpStSeq C) by A28; hence contradiction by A21,A24,A27,EUCLID:56; end; theorem Th39: proj1.:(Cl RightComp f) = proj1.:(L~f) proof set g = Rotate(f,N-min L~f); A1: L~f = L~g by REVROT_1:33; N-min L~f in rng f by SPRECT_2:43; then A2: g/.1 = N-min L~g by A1,FINSEQ_6:98; L~f = (Cl RightComp f) \ RightComp f by Th29; then A3: L~f c= Cl RightComp f by XBOOLE_1:36; thus proj1.:(Cl RightComp f) c= proj1.:(L~f) proof let a be set; assume a in proj1.:(Cl RightComp f); then consider x being set such that A4: x in the carrier of TOP-REAL 2 and A5: x in Cl RightComp f and A6: a = proj1.x by FUNCT_2:115; A7: Cl RightComp f = (RightComp f) \/ L~f by Th31; per cases by A5,A7,XBOOLE_0:def 2; suppose A8: x in RightComp f; reconsider x as Point of TOP-REAL 2 by A4; set b = |[x`1,N-bound L~f + 1]|; b`2 = N-bound L~f + 1 & N-bound L~f + 1 > N-bound L~f + 0 by EUCLID:56,REAL_1:53; then b in LeftComp g by A1,A2,JORDAN2C:121; then b in LeftComp f by REVROT_1:36; then LSeg(x,b) meets L~f by A8,Th37; then consider c being set such that A9: c in LSeg(x,b) & c in L~f by XBOOLE_0:3; reconsider c as Point of TOP-REAL 2 by A9; A10: b`1 = x`1 by EUCLID:56; proj1.c = c`1 by PSCOMP_1:def 28 .= x`1 by A9,A10,GOBOARD7:5 .= a by A6,PSCOMP_1:def 28; hence a in proj1.:(L~f) by A9,FUNCT_2:43; suppose x in L~f; hence a in proj1.:(L~f) by A6,FUNCT_2:43; end; thus thesis by A3,RELAT_1:156; end; theorem Th40: proj2.:(Cl RightComp f) = proj2.:(L~f) proof set g = Rotate(f,N-min L~f); A1: L~f = L~g by REVROT_1:33; N-min L~f in rng f by SPRECT_2:43; then A2: g/.1 = N-min L~g by A1,FINSEQ_6:98; L~f = (Cl RightComp f) \ RightComp f by Th29; then A3: L~f c= Cl RightComp f by XBOOLE_1:36; thus proj2.:(Cl RightComp f) c= proj2.:(L~f) proof let a be set; assume a in proj2.:(Cl RightComp f); then consider x being set such that A4: x in the carrier of TOP-REAL 2 and A5: x in Cl RightComp f and A6: a = proj2.x by FUNCT_2:115; A7: Cl RightComp f = (RightComp f) \/ L~f by Th31; per cases by A5,A7,XBOOLE_0:def 2; suppose A8: x in RightComp f; reconsider x as Point of TOP-REAL 2 by A4; set b = |[E-bound L~f + 1,x`2]|; b`1 = E-bound L~f + 1 & E-bound L~f + 1 > E-bound L~f + 0 by EUCLID:56,REAL_1:53; then b in LeftComp g by A1,A2,JORDAN2C:119; then b in LeftComp f by REVROT_1:36; then LSeg(x,b) meets L~f by A8,Th37; then consider c be set such that A9: c in LSeg(x,b) & c in L~f by XBOOLE_0:3; reconsider c as Point of TOP-REAL 2 by A9; A10: b`2 = x`2 by EUCLID:56; proj2.c = c`2 by PSCOMP_1:def 29 .= x`2 by A9,A10,GOBOARD7:6 .= a by A6,PSCOMP_1:def 29; hence a in proj2.:(L~f) by A9,FUNCT_2:43; suppose x in L~f; hence a in proj2.:(L~f) by A6,FUNCT_2:43; end; thus thesis by A3,RELAT_1:156; end; theorem Th41: RightComp g c= RightComp SpStSeq L~g proof let a be set; assume A1: a in RightComp g; then reconsider p = a as Point of TOP-REAL 2; p`1 > W-bound L~g & p`1 < E-bound L~g & p`2 > S-bound L~g & p`2 < N-bound L~g by A1,Th33,Th34,Th35,Th36; then A2: p`1 > W-bound L~SpStSeq L~g & p`1 < E-bound L~SpStSeq L~g & p`2 > S-bound L~SpStSeq L~g & p`2 < N-bound L~SpStSeq L~g by SPRECT_1:66,67,68,69; RightComp SpStSeq L~g = {q : W-bound L~SpStSeq L~g < q`1 & q`1 < E-bound L~SpStSeq L~g & S-bound L~SpStSeq L~g < q`2 & q`2 < N-bound L~SpStSeq L~g} by SPRECT_3:54; hence a in RightComp SpStSeq L~g by A2; end; theorem Th42: Cl RightComp g is compact proof Cl RightComp SpStSeq L~g = product((1,2)-->([.W-bound L~SpStSeq L~g,E-bound L~SpStSeq L~g.], [.S-bound L~SpStSeq L~g,N-bound L~SpStSeq L~g.])) by Th38; then A1: Cl RightComp SpStSeq L~g is compact by TOPREAL6:87; RightComp g c= RightComp SpStSeq L~g by Th41; then Cl RightComp g c= Cl RightComp SpStSeq L~g by PRE_TOPC:49; hence thesis by A1,COMPTS_1:18; end; theorem Th43: LeftComp g is non Bounded proof A1: L~g \/ RightComp g \/ LeftComp g = the carrier of TOP-REAL 2 by Th25; A2: RightComp g c= Cl RightComp g by PRE_TOPC:48; Cl RightComp g is compact by Th42; then A3: L~g is Bounded & Cl RightComp g is Bounded by JORDAN2C:73; then RightComp g is Bounded by A2,JORDAN2C:16; then L~g \/ RightComp g is Bounded by A3,TOPREAL6:76; hence thesis by A1,TOPREAL6:74; end; theorem Th44: LeftComp g is_outside_component_of L~g proof thus LeftComp g is_a_component_of (L~g)` by GOBOARD9:def 1; thus not LeftComp g is Bounded by Th43; end; theorem RightComp g is_inside_component_of L~g proof thus RightComp g is_a_component_of (L~g)` by GOBOARD9:def 2; A1: RightComp g c= Cl RightComp g by PRE_TOPC:48; Cl RightComp g is compact by Th42; then Cl RightComp g is Bounded by JORDAN2C:73; hence RightComp g is Bounded by A1,JORDAN2C:16; end; theorem Th46: UBD L~g = LeftComp g proof L~g is Bounded by JORDAN2C:73; then A1: ex B being Subset of TOP-REAL 2 st B is_outside_component_of L~g & B = UBD L~g by JORDAN2C:76; LeftComp g is_outside_component_of L~g by Th44; hence thesis by A1,Th6; end; theorem Th47: BDD L~g = RightComp g proof UBD L~g = LeftComp g & (L~g)` = LeftComp g \/ RightComp g & (L~g)` = (BDD L~g) \/ (UBD L~g) & LeftComp g misses RightComp g & (BDD L~g) misses (UBD L~g) by Th24,Th46,GOBRD12:11,JORDAN2C:28,31; hence thesis by XBOOLE_1:71; end; theorem P is_outside_component_of L~g implies P = LeftComp g proof L~g is Bounded by JORDAN2C:73; then consider B being Subset of TOP-REAL 2 such that A1: B is_outside_component_of L~g & B = UBD L~g by JORDAN2C:76; assume that A2: P is_outside_component_of L~g; P = B by A1,A2,Th6; hence thesis by A1,Th46; end; theorem Th49: P is_inside_component_of L~g implies P meets RightComp g proof assume that A1: P is_inside_component_of L~g; A2: BDD L~g = RightComp g by Th47; A3: P c= BDD L~g by A1,JORDAN2C:26; P is_a_component_of (L~g)` by A1,JORDAN2C:def 3; then P <> {} by SPRECT_1:6; hence thesis by A2,A3,XBOOLE_1:67; end; theorem P is_inside_component_of L~g implies P = BDD L~g proof assume that A1: P is_inside_component_of L~g; A2: P is_a_component_of (L~g)` by A1,JORDAN2C:def 3; A3: P meets RightComp g by A1,Th49; A4: RightComp g = BDD L~g by Th47; L~g is Bounded by JORDAN2C:73; then BDD L~g is_inside_component_of L~g by JORDAN2C:116; then BDD L~g is_a_component_of (L~g)` by JORDAN2C:def 3; hence thesis by A2,A3,A4,GOBOARD9:3; end; theorem W-bound L~g = W-bound RightComp g proof set A = (Cl RightComp g) \ RightComp g; A1: L~g = A by Th29; then reconsider A as non empty Subset of TOP-REAL 2; A2: proj1.:(Cl RightComp g) = proj1.:(L~g) by Th39; Cl RightComp g is compact by Th42; then RightComp g c= Cl RightComp g & Cl RightComp g is Bounded by JORDAN2C:73,PRE_TOPC:48; then A3: RightComp g is Bounded by JORDAN2C:16; W-bound A = inf (proj1.:A) & W-bound Cl RightComp g = inf (proj1.:(Cl RightComp g)) by SPRECT_1:48; hence thesis by A1,A2,A3,TOPREAL6:94; end; theorem E-bound L~g = E-bound RightComp g proof set A = (Cl RightComp g) \ RightComp g; A1: L~g = A by Th29; then reconsider A as non empty Subset of TOP-REAL 2; A2: proj1.:(Cl RightComp g) = proj1.:(L~g) by Th39; Cl RightComp g is compact by Th42; then RightComp g c= Cl RightComp g & Cl RightComp g is Bounded by JORDAN2C:73,PRE_TOPC:48; then A3: RightComp g is Bounded by JORDAN2C:16; E-bound A = sup (proj1.:A) & E-bound Cl RightComp g = sup (proj1.:(Cl RightComp g)) by SPRECT_1:51; hence thesis by A1,A2,A3,TOPREAL6:95; end; theorem N-bound L~g = N-bound RightComp g proof set A = (Cl RightComp g) \ RightComp g; A1: L~g = A by Th29; then reconsider A as non empty Subset of TOP-REAL 2; A2: proj2.:(Cl RightComp g) = proj2.:(L~g) by Th40; Cl RightComp g is compact by Th42; then RightComp g c= Cl RightComp g & Cl RightComp g is Bounded by JORDAN2C:73,PRE_TOPC:48; then A3: RightComp g is Bounded by JORDAN2C:16; N-bound A = sup (proj2.:A) & N-bound Cl RightComp g = sup (proj2.:(Cl RightComp g)) by SPRECT_1:50; hence thesis by A1,A2,A3,TOPREAL6:96; end; theorem S-bound L~g = S-bound RightComp g proof set A = (Cl RightComp g) \ RightComp g; A1: L~g = A by Th29; then reconsider A as non empty Subset of TOP-REAL 2; A2: proj2.:(Cl RightComp g) = proj2.:(L~g) by Th40; Cl RightComp g is compact by Th42; then RightComp g c= Cl RightComp g & Cl RightComp g is Bounded by JORDAN2C:73,PRE_TOPC:48; then A3: RightComp g is Bounded by JORDAN2C:16; S-bound A = inf (proj2.:A) & S-bound Cl RightComp g = inf (proj2.:(Cl RightComp g)) by SPRECT_1:49; hence thesis by A1,A2,A3,TOPREAL6:97; end;

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