### The Mizar article:

### On the Components of the Complement of a Special Polygonal Curve

**by****Andrzej Trybulec, and****Yatsuka Nakamura**

- Received January 21, 1999
Copyright (c) 1999 Association of Mizar Users

- MML identifier: SPRECT_4
- [ MML identifier index ]

environ vocabulary COMPTS_1, PRE_TOPC, SPPOL_2, EUCLID, TOPREAL1, BOOLE, FINSEQ_4, JORDAN3, JORDAN5C, FINSEQ_1, GROUP_2, RELAT_1, FUNCT_1, ARYTM_1, SPRECT_2, SEQM_3, GOBOARD5, PSCOMP_1, GOBOARD9, SPRECT_1, GOBOARD2, TREES_1, MATRIX_1, ARYTM_3, TOPS_1, MCART_1, TOPREAL4, SPPOL_1, FINSEQ_5, FINSEQ_6; notation TARSKI, XBOOLE_0, XREAL_0, REAL_1, NAT_1, FUNCT_1, FINSEQ_1, FINSEQ_4, BINARITH, MATRIX_1, STRUCT_0, PRE_TOPC, TOPS_1, COMPTS_1, EUCLID, TOPREAL1, TOPREAL4, FINSEQ_5, FINSEQ_6, GOBOARD1, GOBOARD2, SPPOL_1, SPPOL_2, GOBOARD5, GOBOARD9, JORDAN3, PSCOMP_1, JORDAN5C, SPRECT_1, SPRECT_2; constructors SPRECT_2, GOBOARD9, FINSEQ_4, JORDAN5C, TOPS_2, TOPMETR, BINARITH, COMPTS_1, TOPS_1, SPRECT_1, TOPREAL4, GOBOARD2, SPPOL_1, PSCOMP_1, JORDAN3, REAL_1, TOPREAL2, MATRIX_2, REALSET1, WAYBEL_3; clusters RELSET_1, SPRECT_1, SPPOL_2, GOBOARD2, FINSEQ_6, BORSUK_2, SPRECT_3, GOBOARD9, REVROT_1, ARYTM_3, PRE_TOPC, WAYBEL_3, MEMBERED; requirements REAL, NUMERALS, BOOLE, SUBSET, ARITHM; theorems TOPREAL1, JORDAN5C, JORDAN3, FINSEQ_3, SPPOL_2, TOPS_1, SPPOL_1, AXIOMS, TOPREAL3, FINSEQ_5, REAL_1, NAT_1, SPRECT_1, JORDAN5B, AMI_5, SPRECT_2, GOBOARD5, FINSEQ_4, GOBOARD7, GOBOARD6, GOBOARD9, ZFMISC_1, TOPREAL4, PSCOMP_1, JORDAN4, FINSEQ_6, FINSEQ_1, GROUP_5, TARSKI, COMPTS_1, EUCLID, SUBSET_1, JORDAN5D, SPRECT_3, REVROT_1, PARTFUN2, XBOOLE_0, XBOOLE_1, XCMPLX_1; begin reserve i,j,l for Nat; definition let T be being_T2 (non empty TopSpace); cluster compact -> closed Subset of T; coherence by COMPTS_1:16; end; theorem Th1: for f being S-Sequence_in_R2, Q being closed Subset of TOP-REAL 2 st L~f meets Q & not f/.1 in Q holds L~R_Cut(f,First_Point(L~f,f/.1,f/.len f,Q)) /\ Q = { First_Point(L~f,f/.1,f/.len f,Q) } proof let f be S-Sequence_in_R2, Q be closed Subset of TOP-REAL 2 such that A1: L~f meets Q and A2: not f/.1 in Q; set p1 = f/.1, p2 = f/.len f, fp = First_Point(L~f,p1,p2,Q); len f >= 1+1 by TOPREAL1:def 10; then A3: len f > 1 by NAT_1:38; A4: L~f /\ Q is closed by TOPS_1:35; A5: L~f is_an_arc_of p1,p2 by TOPREAL1:31; then A6: fp in L~f /\ Q by A1,A4,JORDAN5C:def 1; then A7: fp in Q by XBOOLE_0:def 3; A8: fp in L~f by A6,XBOOLE_0:def 3; then A9: 1<=Index(fp,f) & Index(fp,f)<=len f by JORDAN3:41; then A10: Index(fp,f) in dom f by FINSEQ_3:27; 1 in dom f by A3,FINSEQ_3:27; then fp <> f.1 by A2,A7,FINSEQ_4:def 4; then fp in L~R_Cut(f,fp) by A8,JORDAN5B:23; then A11: fp in L~R_Cut(f,fp) /\ Q by A7,XBOOLE_0:def 3; now assume not L~R_Cut(f,fp) /\ Q c= { fp }; then consider q being set such that A12: q in L~R_Cut(f,fp) /\ Q and A13: not q in { fp } by TARSKI:def 3; reconsider q as Point of TOP-REAL 2 by A12; A14: q <> fp by A13,TARSKI:def 1; A15: q in Q by A12,XBOOLE_0:def 3; A16: q in L~R_Cut(f,fp) by A12,XBOOLE_0:def 3; L~R_Cut(f,fp) c= L~f by A8,JORDAN3:76; then A17: LE fp, q, L~f, p1, p2 by A4,A15,A16,JORDAN5C:15; per cases; suppose fp = f.1; then A18: q in L~<*fp*> by A16,JORDAN3:def 5; len<*fp*> = 1 by FINSEQ_1:56; hence contradiction by A18,TOPREAL1:28; suppose A19: fp <> f.1; set m = mid(f,1,Index(fp,f)); len m = Index(fp,f)-'1+1 by A9,JORDAN4:20; then len m <> 0 by NAT_1:29; then A20: m is non empty by FINSEQ_1:25; q in L~(m^<*fp*>) by A16,A19,JORDAN3:def 5; then A21: q in L~m \/ LSeg(m/.len m,fp) by A20,SPPOL_2:19; A22: Index(fp,f) < len f by A8,JORDAN3:41; A23: fp in LSeg(f,Index(fp,f)) by A8,JORDAN3:42; now per cases by A21,XBOOLE_0:def 2; suppose A24: q in L~m; A25: now assume Index(fp,f) <= 1; then Index(fp,f) = 1 by A9,AXIOMS:21; then len m = 1 by A3,JORDAN4:27; hence contradiction by A24,TOPREAL1:28; end; then A26: LE q, f/.Index(fp,f), L~f, p1, p2 by A9,A24,SPRECT_3:33; f/.Index(fp,f) in LSeg(f/.Index(fp,f),fp) by TOPREAL1:6; then LE f/.Index(fp,f), fp, L~f, p1, p2 by A22,A23,A25,SPRECT_3:40; then LE q, fp, L~f, p1, p2 by A5,A26,JORDAN5C:13; hence contradiction by A5,A14,A17,JORDAN5C:12; suppose A27: q in LSeg(m/.len m,fp); len m in dom m by A20,FINSEQ_5:6; then m/.len m = m.len m by FINSEQ_4:def 4 .= f.Index(fp,f) by A9,JORDAN4:22 .= f/.Index(fp,f) by A10,FINSEQ_4:def 4; then LE q, fp, L~f, p1, p2 by A9,A22,A23,A27,SPRECT_3:40; hence contradiction by A5,A14,A17,JORDAN5C:12; end; hence contradiction; end; hence L~R_Cut(f,fp) /\ Q = { fp } by A11,ZFMISC_1:39; end; theorem for f being non empty FinSequence of TOP-REAL 2, p being Point of TOP-REAL 2 st f is_S-Seq & p = f/.len f holds L~L_Cut (f,p) = {} proof let f be non empty FinSequence of TOP-REAL 2, p be Point of TOP-REAL 2; assume that A1: f is_S-Seq and A2: p = f/.len f; len f >= 2 by A1,TOPREAL1:def 10; then len f >= 1 by AXIOMS:22; then len f in dom f by FINSEQ_3:27; then p = f.len f by A2,FINSEQ_4:def 4; then L_Cut (f,p) = <*p*> by A1,JORDAN5B:17; then len L_Cut (f,p) = 1 by FINSEQ_1:56; hence thesis by TOPREAL1:28; end; canceled; theorem Th4: for f being S-Sequence_in_R2, p being Point of TOP-REAL 2, j st 1 <=j & j < len f & p in L~mid(f,j,len f) holds LE f/.j, p, L~f, f/.1, f/.len f proof let f be S-Sequence_in_R2, p be Point of TOP-REAL 2, j such that A1: 1 <=j & j < len f and A2: p in L~mid(f,j,len f); A3: L~f is_an_arc_of f/.1,f/.len f by TOPREAL1:31; consider i such that A4: 1 <= i & i+1 <= len mid(f,j,len f) and A5: p in LSeg(mid(f,j,len f),i) by A2,SPPOL_2:13; A6: len mid(f,j,len f)=len f-'j+1 by A1,JORDAN4:20; then i <= len f -' j by A4,REAL_1:53; then A7: i + j <= len f by A1,SPRECT_3:7; j+1 <= j+i by A4,AXIOMS:24; then A8: j <= j+i-'1 by JORDAN3:12; 1+1 <= j+i by A1,A4,REAL_1:55; then A9: 1 <= j+i-'1 by JORDAN3:12; i<len f-'j+1 by A4,A6,NAT_1:38; then A10: p in LSeg(f,j+i-'1) by A1,A4,A5,JORDAN4:31; j+i >= i by NAT_1:29; then 1 <= j+i by A4,AXIOMS:22; then A11: j+i-'1+1 <= len f by A7,AMI_5:4; then A12: LE f/.(j+i-'1), p, L~f, f/.1, f/.len f by A9,A10,JORDAN5C:25; j+i-'1+1 >= j+i-'1 by NAT_1:29; then len f >= j+i-'1 by A11,AXIOMS:22; then LE f/.j, f/.(j+i-'1), L~f, f/.1, f/.len f by A1,A8,JORDAN5C:24; hence LE f/.j, p, L~f, f/.1, f/.len f by A3,A12,JORDAN5C:13; end; theorem Th5: for f being S-Sequence_in_R2, p,q being Point of TOP-REAL 2, j st 1 <=j & j < len f & p in LSeg(f,j) & q in LSeg(p,f/.(j+1)) holds LE p, q, L~f, f/.1, f/.len f proof let f be S-Sequence_in_R2, p,q be Point of TOP-REAL 2, j such that A1: 1 <=j & j < len f and A2: p in LSeg(f,j) and A3: q in LSeg(p,f/.(j+1)); A4: L~f is_an_arc_of f/.1, f/.len f by TOPREAL1:31; A5: j+1 <= len f by A1,NAT_1:38; then A6: LSeg(f,j) = LSeg(f/.j,f/.(j+1)) by A1,TOPREAL1:def 5; f/.(j+1) in LSeg(f,j) by A1,A5,TOPREAL1:27; then A7: LSeg(p,f/.(j+1)) c= LSeg(f,j) by A2,A6,TOPREAL1:12; then A8: q in LSeg(f,j) by A3; A9: LSeg(f,j) c= L~f by TOPREAL3:26; per cases; suppose p = q; hence thesis by A2,A4,A9,JORDAN5C:9; suppose A10: q <> p; for i, j being Nat st p in LSeg(f,i) & q in LSeg(f,j) & 1 <= i & i+1 <= len f & 1 <= j & j+1 <= len f holds i <= j & (i = j implies LE p,q,f/.i,f/.(i+1)) proof let i1, i2 be Nat such that A11: p in LSeg(f,i1) and A12: q in LSeg(f,i2) and A13: 1 <= i1 & i1+1 <= len f and A14: 1 <= i2 & i2+1 <= len f; A15: now assume i1 + 1 > j & j+1 > i1; then i1 <= j & j <= i1 by NAT_1:38; hence i1 = j by AXIOMS:21; end; A16: p in LSeg(f,i1) /\ LSeg(f,j) by A2,A11,XBOOLE_0:def 3; then LSeg(f,i1) meets LSeg(f,j) by XBOOLE_0:4; then A17: i1 + 1 >= j & j + 1 >= i1 by TOPREAL1:def 9; A18: now assume A19: j + 1 = i1; j+(1+1) = j+1+1 by XCMPLX_1:1; then p in {f/.(j+1)} by A1,A13,A16,A19,TOPREAL1:def 8; then p = f/.(j+1) by TARSKI:def 1; then q in {p} by A3,TOPREAL1:7; hence contradiction by A10,TARSKI:def 1; end; then i1 + 1 = j or i1 = j by A15,A17,AXIOMS:21; then A20: i1 <= j by NAT_1:29; A21: now assume i2 + 1 > j & j+1 > i2; then i2 <= j & j <= i2 by NAT_1:38; hence i2 = j by AXIOMS:21; end; A22: q in LSeg(f,i2) /\ LSeg(f,j) by A3,A7,A12,XBOOLE_0:def 3; then LSeg(f,i2) meets LSeg(f,j) by XBOOLE_0:4; then i2 + 1 >= j & j + 1 >= i2 by TOPREAL1:def 9; then A23: i2 + 1 = j or i2 = j or j + 1 = i2 by A21,AXIOMS:21; 1 <= j+1 by NAT_1:29; then A24: j+1 in dom f by A5,FINSEQ_3:27; A25: j in dom f by A1,FINSEQ_3:27; A26:now assume f/.(j+1)=f/.j; then j = j+1 by A24,A25,PARTFUN2:17; hence contradiction by REAL_1:69; end; A27: now assume A28: i2+1 = j; i2+(1+1) = i2+1+1 by XCMPLX_1:1; then q in {f/.j} by A5,A14,A22,A28,TOPREAL1:def 8; then q = f/.j by TARSKI:def 1; hence contradiction by A3,A6,A7,A10,A26,SPPOL_1:24; end; then i2 >= j by A23,NAT_1:29; hence i1 <= i2 by A20,AXIOMS:22; assume A29: i1 = i2; not p in LSeg(q,f/.(j+1)) by A3,A10,SPRECT_3:16; then not LE q,p,f/.j,f/.(j+1) by JORDAN3:65; then LT p,q,f/.j,f/.(j+1) by A2,A6,A12,A15,A17,A18,A26,A27,A29,AXIOMS:21, JORDAN3:63; hence LE p,q,f/.i1,f/.(i1+1) by A15,A17,A18,A27,A29,AXIOMS:21,JORDAN3:def 7 ; end; hence LE p, q, L~f, f/.1, f/.len f by A2,A8,A9,A10,JORDAN5C:30; end; theorem Th6: for f being S-Sequence_in_R2, Q being closed Subset of TOP-REAL 2 st L~f meets Q & not f/.len f in Q holds L~L_Cut(f,Last_Point(L~f,f/.1,f/.len f,Q)) /\ Q = { Last_Point(L~f,f/.1,f/.len f,Q) } proof let f be S-Sequence_in_R2, Q be closed Subset of TOP-REAL 2 such that A1: L~f meets Q and A2: not f/.len f in Q; set p1 = f/.1, p2 = f/.len f, lp = Last_Point(L~f,p1,p2,Q); len f >= 1+1 by TOPREAL1:def 10; then A3: len f > 1 by NAT_1:38; A4: L~f /\ Q is closed by TOPS_1:35; A5: L~f is_an_arc_of p1,p2 by TOPREAL1:31; then A6: lp in L~f /\ Q by A1,A4,JORDAN5C:def 2; then A7: lp in Q by XBOOLE_0:def 3; A8: lp in L~f by A6,XBOOLE_0:def 3; then A9: 1<=Index(lp,f) & Index(lp,f)<=len f by JORDAN3:41; len f in dom f by A3,FINSEQ_3:27; then lp <> f.len f by A2,A7,FINSEQ_4:def 4; then lp in L~L_Cut(f,lp) by A8,JORDAN5B:22; then A10: lp in L~L_Cut(f,lp) /\ Q by A7,XBOOLE_0:def 3; now assume not L~L_Cut(f,lp) /\ Q c= { lp }; then consider q being set such that A11: q in L~L_Cut(f,lp) /\ Q and A12: not q in { lp } by TARSKI:def 3; reconsider q as Point of TOP-REAL 2 by A11; A13: q <> lp by A12,TARSKI:def 1; A14: q in Q by A11,XBOOLE_0:def 3; A15: q in L~L_Cut(f,lp) by A11,XBOOLE_0:def 3; L~L_Cut(f,lp) c= L~f by A8,JORDAN3:77; then A16: LE q, lp, L~f, p1, p2 by A4,A14,A15,JORDAN5C:16; set m = mid(f,Index(lp,f)+1,len f); A17: Index(lp,f) < len f by A8,JORDAN3:41; then A18: Index(lp,f)+1 <= len f by NAT_1:38; A19: 1 <= Index(lp,f)+1 by NAT_1:29; then len m = len f -' (Index(lp,f)+1)+1 by A18,JORDAN4:20; then len m <> 0 by NAT_1:29; then A20: m is non empty by FINSEQ_1:25; per cases; suppose lp=f.(Index(lp,f)+1); then A21: q in L~m by A15,JORDAN3:def 4; A22: lp in LSeg(f,Index(lp,f)) by A8,JORDAN3:42; now assume Index(lp,f)+1 >= len f; then Index(lp,f)+1 = len f by A18,AXIOMS:21; then len m = 1 by A3,JORDAN4:27; hence contradiction by A21,TOPREAL1:28; end; then A23: LE f/.(Index(lp,f)+1), q, L~f, f/.1, f/.len f by A19,A21,Th4; f/.(Index(lp,f)+1) in LSeg(lp,f/.(Index(lp,f)+1)) by TOPREAL1:6; then LE lp, f/.(Index(lp,f)+1), L~f, p1, p2 by A9,A17,A22,Th5; then LE lp, q, L~f, p1, p2 by A5,A23,JORDAN5C:13; hence contradiction by A5,A13,A16,JORDAN5C:12; suppose lp<>f.(Index(lp,f)+1); then q in L~(<*lp*>^m) by A15,JORDAN3:def 4; then A24: q in L~m \/ LSeg(m/.1,lp) by A20,SPPOL_2:20; 1 <= Index(lp,f)+1 by NAT_1:29; then A25: Index(lp,f)+1 in dom f by A18,FINSEQ_3:27; A26: lp in LSeg(f,Index(lp,f)) by A8,JORDAN3:42; now per cases by A24,XBOOLE_0:def 2; suppose A27: q in L~m; now assume Index(lp,f)+1 >= len f; then Index(lp,f)+1 = len f by A18,AXIOMS:21; then len m = 1 by A3,JORDAN4:27; hence contradiction by A27,TOPREAL1:28; end; then A28: LE f/.(Index(lp,f)+1), q, L~f, f/.1, f/.len f by A19,A27,Th4; f/.(Index(lp,f)+1) in LSeg(lp,f/.(Index(lp,f)+1)) by TOPREAL1:6; then LE lp, f/.(Index(lp,f)+1), L~f, p1, p2 by A9,A17,A26,Th5; then LE lp, q, L~f, p1, p2 by A5,A28,JORDAN5C:13; hence contradiction by A5,A13,A16,JORDAN5C:12; suppose A29: q in LSeg(lp,m/.1); 1 in dom m by A20,FINSEQ_5:6; then m/.1 = m.1 by FINSEQ_4:def 4 .= f.(Index(lp,f)+1) by A3,A18,A19,JORDAN3:27 .= f/.(Index(lp,f)+1) by A25,FINSEQ_4:def 4; then LE lp, q, L~f, p1, p2 by A9,A17,A26,A29,Th5; hence contradiction by A5,A13,A16,JORDAN5C:12; end; hence contradiction; end; hence L~L_Cut(f,lp) /\ Q = { lp } by A10,ZFMISC_1:39; end; reserve q for Point of TOP-REAL 2; Lm1: for f being clockwise_oriented (non constant standard special_circular_sequence) st f/.1 = N-min L~f holds LeftComp f <> RightComp f proof let f be clockwise_oriented (non constant standard special_circular_sequence); set A = N-min L~f; assume that A1: f/.1 = A and A2: LeftComp f = RightComp f; A3: LeftComp SpStSeq L~f c= LeftComp f by A1,SPRECT_3:58; consider i such that A4: 1 <= i and A5: i < len GoB f and A6: A = (GoB f)*(i,width GoB f) by SPRECT_3:45; set w = width GoB f; A7: len f > 4 by GOBOARD7:36; then A8: 1+1 <= len f by AXIOMS:22; A9: w > 1 by GOBOARD7:35; then A10: w-'1+1 = w by AMI_5:4; A11: [i,w] in Indices GoB f by A4,A5,A9,GOBOARD7:10; A12: 1 <= i+1 & i+1 <= len GoB f by A5,NAT_1:29,38; then A13: [i+1,w] in Indices GoB f by A9,GOBOARD7:10; A14: 1 in dom f by FINSEQ_5:6; A15: i in dom GoB f by A4,A5,FINSEQ_3:27; then A16: f/.(1+1) = (GoB f)*(i+1,w) by A1,A6,SPRECT_3:46; then A17: right_cell(f,1) = cell(GoB f,i,w-'1) by A1,A6,A8,A10,A11,A13, GOBOARD5:29; set z2 = 1/2*((GoB f)*(i,w-'1)+(GoB f)*(i+1,w)), p1 = 1/2*((GoB f)*(i,w)+(GoB f)*(i+1,w)), p2 = 1/2*((GoB f)*(i,w-'1)+(GoB f)*(i,w)); A18: 1 <= w-'1 by A9,JORDAN3:12; then A19: z2 in Int cell(GoB f,i,w-'1) by A4,A10,A12,GOBOARD6:34; A20: Int right_cell(f,1) c= RightComp f by A8,GOBOARD9:28; A21: z2 in RightComp SpStSeq L~f proof A22: W-bound L~SpStSeq L~f = W-bound L~f & S-bound L~SpStSeq L~f = S-bound L~f & N-bound L~SpStSeq L~f = N-bound L~f & E-bound L~SpStSeq L~f = E-bound L~f by SPRECT_1:66,67,68,69; A23: 1 < i+1 by A4,NAT_1:38; A24: 1 <= len GoB f by A4,A5,AXIOMS:22; A25: w-'1 < w by A10,REAL_1:69; A26: z2 = 1/2*(A+(GoB f)*(i+1,w-'1)) by A4,A6,A10,A12,A18,GOBOARD7:11; then A27: z2`1 = 1/2*(A+(GoB f)*(i+1,w-'1))`1 by TOPREAL3:9 .= 1/2*(A`1+(GoB f)*(i+1,w-'1)`1) by TOPREAL3:7 .= 1/2*A`1+1/2*((GoB f)*(i+1,w-'1))`1 by XCMPLX_1:8; A28: z2`2 = 1/2*(A+(GoB f)*(i+1,w-'1))`2 by A26,TOPREAL3:9 .= 1/2*(A`2+(GoB f)*(i+1,w-'1)`2) by TOPREAL3:7 .= 1/2*(N-bound L~f+(GoB f)*(i+1,w-'1)`2) by PSCOMP_1:94 .= 1/2*N-bound L~f+1/2*((GoB f)*(i+1,w-'1))`2 by XCMPLX_1:8; A29: 1/2*(W-bound L~f) + 1/2*(W-bound L~f) = 1/2*((W-bound L~f)+(W-bound L~f)) by XCMPLX_1:8 .= 1/2*(2*(W-bound L~f)) by XCMPLX_1:11 .= (W-bound L~f) by XCMPLX_1:108; A30: 1/2*(S-bound L~f) + 1/2*(S-bound L~f) = 1/2*((S-bound L~f)+(S-bound L~f)) by XCMPLX_1:8 .= 1/2*(2*(S-bound L~f)) by XCMPLX_1:11 .= (S-bound L~f) by XCMPLX_1:108; A31: 1/2*(N-bound L~f) + 1/2*(N-bound L~f) = 1/2*((N-bound L~f)+(N-bound L~f)) by XCMPLX_1:8 .= 1/2*(2*(N-bound L~f)) by XCMPLX_1:11 .= N-bound L~f by XCMPLX_1:108; A32: 1/2*(E-bound L~f) + 1/2*(E-bound L~f) = 1/2*((E-bound L~f)+(E-bound L~f)) by XCMPLX_1:8 .= 1/2*(2*(E-bound L~f)) by XCMPLX_1:11 .= (E-bound L~f) by XCMPLX_1:108; N-min L~f in L~f by SPRECT_1:13; then A`1 >= W-bound L~f by PSCOMP_1:71; then A33: 1/2*A`1 >= 1/2*W-bound L~f by AXIOMS:25; A34: (GoB f)*(1,w-'1)`1 = (GoB f)*(1,1)`1 by A18,A24,A25,GOBOARD5:3; (GoB f)*(i+1,w-'1)`1 > (GoB f)*(1,w-'1)`1 by A12,A18,A23,A25,GOBOARD5:4; then (GoB f)*(i+1,w-'1)`1 > W-bound L~f by A34,JORDAN5D:39; then 1/2*((GoB f)*(i+1,w-'1))`1 > 1/2*W-bound L~f by REAL_1:70; then A35: W-bound L~SpStSeq L~f < z2`1 by A22,A27,A29,A33,REAL_1:67; A36: A`1 < (N-max L~f)`1 by SPRECT_2:55; N-max L~f in L~f by SPRECT_1:13; then (N-max L~f)`1 <= E-bound L~f by PSCOMP_1:71; then A`1 < E-bound L~f by A36,AXIOMS:22; then A37: 1/2*A`1 < 1/2*E-bound L~f by REAL_1:70; A38: (GoB f)*(len GoB f,w-'1)`1 = (GoB f)*(len GoB f,1)`1 by A18,A24,A25,GOBOARD5:3; (GoB f)*(i+1,w-'1)`1 <= (GoB f)*(len GoB f,w-'1)`1 by A12,A18,A25, SPRECT_3:25; then (GoB f)*(i+1,w-'1)`1 <= E-bound L~f by A38,JORDAN5D:41; then 1/2*((GoB f)*(i+1,w-'1))`1 <= 1/2*E-bound L~f by AXIOMS:25; then A39: z2`1 < E-bound L~SpStSeq L~f by A22,A27,A32,A37,REAL_1:67; N-bound L~f > S-bound L~f by SPRECT_1:34; then A40: 1/2*N-bound L~f > 1/2*S-bound L~f by REAL_1:70; A41: (GoB f)*(i+1,1)`2 = (GoB f)*(1,1)`2 by A9,A12,GOBOARD5:2; (GoB f)*(i+1,w-'1)`2 >= (GoB f)*(i+1,1)`2 by A12,A18,A25,SPRECT_3:24; then (GoB f)*(i+1,w-'1)`2 >= S-bound L~f by A41,JORDAN5D:40; then 1/2*((GoB f)*(i+1,w-'1))`2 >= 1/2*S-bound L~f by AXIOMS:25; then A42: S-bound L~SpStSeq L~f < z2`2 by A22,A28,A30,A40,REAL_1:67; A43: (GoB f)*(i+1,w)`2 = (GoB f)*(1,width GoB f)`2 by A9,A12,GOBOARD5:2; (GoB f)*(i+1,w-'1)`2 < (GoB f)*(i+1,w)`2 by A12,A18,A25,GOBOARD5:5; then (GoB f)*(i+1,w-'1)`2 < N-bound L~f by A43,JORDAN5D:42; then 1/2*((GoB f)*(i+1,w-'1))`2 < 1/2*N-bound L~f by REAL_1:70; then A44: z2`2 < N-bound L~SpStSeq L~f by A22,A28,A31,REAL_1:53; RightComp SpStSeq L~f = {q : W-bound L~SpStSeq L~f < q`1 & q`1 < E-bound L~SpStSeq L~f & S-bound L~SpStSeq L~f < q`2 & q`2 < N-bound L~SpStSeq L~f} by SPRECT_3:54; hence thesis by A35,A39,A42,A44; end; consider z1 being set such that A45: z1 in LeftComp SpStSeq L~f by XBOOLE_0:def 1; LeftComp SpStSeq L~f misses RightComp SpStSeq L~f by SPRECT_1:96; then A46: z1 <> z2 by A21,A45,XBOOLE_0:3; reconsider z1 as Point of TOP-REAL 2 by A45; consider P being Subset of TOP-REAL 2 such that A47: P is_S-P_arc_joining z1,z2 and A48: P c= RightComp f by A2,A3,A17,A19,A20,A45,A46,TOPREAL4:30; consider Red' being FinSequence of TOP-REAL 2 such that A49: Red' is_S-Seq and A50: P = L~Red' and A51: z1 = Red'/.1 and A52: z2 = Red'/.len Red' by A47,TOPREAL4:def 1; A53: L~SpStSeq L~f meets L~Red' by A21,A45,A49,A51,A52,SPRECT_3:50; A54: 2 in dom f by A8,FINSEQ_3:27; A55: len f in dom f by FINSEQ_5:6; A56: len f -' 1 >= 1 & len f -' 1 <= len f by A8,JORDAN3:7,12; then A57: len f -' 1 in dom f by FINSEQ_3:27; 1 <= len f by A55,FINSEQ_3:27; then A58: len f -' 1 + 1 = len f by AMI_5:4; then A59: len f -' 1 < len f by NAT_1:38; A60: <*NW-corner L~f*> is_in_the_area_of f by SPRECT_2:30; A61: w-'1 < width GoB f by A10,NAT_1:38; then Int cell(GoB f,i,w-'1) misses L~SpStSeq L~f by A4,A5,A18,SPRECT_3:72; then A62: not z2 in L~SpStSeq L~f by A19,XBOOLE_0:3; A63: LSeg(NW-corner L~f,NE-corner L~f) c= L~SpStSeq L~f by SPRECT_3:26; A64: f/.1 = f/.len f by FINSEQ_6:def 1; A65: NW-corner L~f in LSeg(NW-corner L~f,NE-corner L~f) by TOPREAL1:6; A66: N-min L~f in LSeg(NW-corner L~f,NE-corner L~f) by SPRECT_3:28; then A67: LSeg(NW-corner L~f,N-min L~f) c= LSeg(NW-corner L~f,NE-corner L~f ) by A65,TOPREAL1:12; A68: LSeg(NW-corner L~f,NE-corner L~f) is horizontal by SPRECT_3:29; then A69: LSeg(NW-corner L~f,N-min L~f) is horizontal by A67,SPRECT_1:11; 1 + 2 <= len f by A7,AXIOMS:22; then A70: LSeg(f,1) /\ LSeg(f,1+1) = {f/.(1+1)} by TOPREAL1:def 8; len f >= 2 + 1 by A7,AXIOMS:22; then len f -' 1 >= 1 + 1 by JORDAN3:12; then len f -' 1 -' 1 >= 1 by JORDAN3:12; then A71: len f -' 2 >= 1 by JORDAN3:8; A72: len f -' 2 + 1 = len f -' 1 -' 1 + 1 by JORDAN3:8 .= len f -' 1 by A56,AMI_5:4; len f >= 2 by A7,AXIOMS:22; then len f -' 2 + 2 = len f by AMI_5:4; then A73: LSeg(f,len f -' 1) /\ LSeg(f,len f -' 2) = {f/.(len f -' 1)} by A71,A72,TOPREAL1:def 8; A74: f/.2 in N-most L~f by A1,SPRECT_2:34; N-most L~f c= LSeg(NW-corner L~f,NE-corner L~f) by SPRECT_3:27; then A75: LSeg(f/.1, f/.2) c= LSeg(NW-corner L~f,NE-corner L~f) by A1,A66,A74,TOPREAL1:12; A76: ((GoB f)*(i,w-'1))`1 = ((GoB f)*(i,1))`1 by A4,A5,A18,A61,GOBOARD5:3 .= A`1 by A4,A5,A6,A9,GOBOARD5:3; A77: (GoB f)*(i+1,w)`2 = (GoB f)*(1,w)`2 by A9,A12,GOBOARD5:2 .= A`2 by A4,A5,A6,A9,GOBOARD5:2; then A78: (f/.2)`2 = N-bound L~f by A16,PSCOMP_1:94; A79: <*(GoB f)*(i+1,w)*> is_in_the_area_of f by A9,A12,SPRECT_3:66; <*(GoB f)*(i,w-'1)*> is_in_the_area_of f by A4,A5,A18,A61,SPRECT_3:66; then <*(GoB f)*(i,w-'1),(GoB f)*(i+1,w)*> is_in_the_area_of f by A79,SPRECT_3:59; then <*z2*> is_in_the_area_of f by SPRECT_3:67; then A80: <*z2*> is_in_the_area_of SpStSeq L~f by SPRECT_3:70; A81: (GoB f)*(i,w-'1) = f/.(len f -' 1) by A1,A6,A15,SPRECT_3:46; then LSeg(A,f/.(len f -' 1)) is vertical by A76,SPPOL_1:37; then A82: LSeg(A,f/.(len f -' 1)) /\ LSeg(NW-corner L~f,A) = { A } by A69,SPRECT_3:22; A83: (NW-corner L~f)`2 = A`2 by PSCOMP_1:95; A84: (NW-corner L~f)`1 <= A`1 by PSCOMP_1:97; A`1 <= (f/.2)`1 by A74,PSCOMP_1:98; then A in LSeg(NW-corner L~f,f/.2) by A16,A77,A83,A84,GOBOARD7:9; then A85: LSeg(A,f/.2) /\ LSeg(NW-corner L~f,A) = { A } by TOPREAL1:14; ((GoB f)*(i,w))`2 = ((GoB f)*(1,w))`2 by A4,A5,A9,GOBOARD5:2 .= ((GoB f)*(i+1,w))`2 by A9,A12,GOBOARD5:2; then ((GoB f)*(i,w-'1)+(GoB f)*(i,w))`2 = ((GoB f)*(i,w-'1))`2+((GoB f)*(i+1,w))`2 by TOPREAL3:7 .= ((GoB f)*(i,w-'1)+(GoB f)*(i+1,w))`2 by TOPREAL3:7; then p2`2 = 1/2*((GoB f)*(i,w-'1)+(GoB f)*(i+1,w)) `2 by TOPREAL3:9 .= z2`2 by TOPREAL3:9; then A86: LSeg(p2,z2) is horizontal by SPPOL_1:36; A87: f/.(len f -' 1) in LSeg(f/.(len f -' 1), f/.len f) by TOPREAL1:6; A88: (GoB f)*(i,w) = f/.len f by A1,A6,FINSEQ_6:def 1; p2 = 1/2*((GoB f)*(i,w-'1))+(1 - 1/2)*((GoB f)*(i,w)) by EUCLID:36; then A89: p2 in LSeg(f/.(len f -' 1), f/.len f) by A81,A88,SPPOL_1:22; then A90: LSeg(p2,f/.(len f -' 1)) c= LSeg(f/.(len f -' 1), f/.len f) by A87,TOPREAL1:12; LSeg(f/.(len f -' 1), f/.len f) = LSeg(f,len f -' 1) by A56,A58,TOPREAL1:def 5; then LSeg(f/.(len f -' 1), f/.len f) c= L~f by TOPREAL3:26; then A91: LSeg(p2,f/.(len f -' 1)) c= L~f by A90,XBOOLE_1:1; A92: LSeg(p2,f/.(len f -' 1)) c= LSeg(f,len f -' 1) by A56,A58,A90,TOPREAL1:def 5; A93: p2`1 = 1/2*((GoB f)*(i,w-'1)+A)`1 by A6,TOPREAL3:9 .= 1/2*(A`1+A`1) by A76,TOPREAL3:7 .= 1/2*(2*A`1) by XCMPLX_1:11 .= (N-min L~f)`1 by XCMPLX_1:108; then A94: LSeg(p2,f/.(len f -' 1)) is vertical by A76,A81,SPPOL_1:37; A95: p2 in LSeg(p2,f/.(len f -' 1)) by TOPREAL1:6; then A96: p2 in L~f by A91; A97: now assume p2 = A; then f/.(len f -' 1) = f/.len f by A1,A6,A64,A81,SPRECT_3:15; hence contradiction by A55,A57,A58,GOBOARD7:31; end; 1 + 1 + 1 <= len f by A7,AXIOMS:22; then A98: 1 + 1 <= len f -' 1 by JORDAN3:12; then A99: 1 <= len f -' 1 -' 1 by JORDAN3:12; 1 <= len f -' 1 by A98,AXIOMS:22; then A100: len f -' 1 -'1 + 1 = len f -' 1 by AMI_5:4; A101: len f >= 2 by A7,AXIOMS:22; A102: len f -' 1 -' 1 + 2 = len f -' 2 + 2 by JORDAN3:8 .= len f by A101,AMI_5:4; A103: for i,j st 1 <= i & i <= j & j < len f -' 1 holds L~mid(f,i,j) misses LSeg(p2,f/.(len f -' 1)) proof let l,j such that A104: 1 <= l and A105: l <= j and A106: j < len f -' 1; len f -' 1 > l by A105,A106,AXIOMS:22; then len f -' 1 > 1 by A104,AXIOMS:22; then A107: len f -' 1 < len f by JORDAN3:14; then A108: j < len f by A106,AXIOMS:22; assume L~mid(f,l,j) meets LSeg(p2,f/.(len f -' 1)); then L~mid(f,l,j) /\ LSeg(p2,f/.(len f -' 1)) <> {} by XBOOLE_0:def 7; then consider p being Point of TOP-REAL 2 such that A109: p in L~mid(f,l,j) /\ LSeg(p2,f/.(len f -' 1)) by SUBSET_1:10; A110: p in LSeg(p2,f/.(len f -' 1)) by A109,XBOOLE_0:def 3; p in L~mid(f,l,j) by A109,XBOOLE_0:def 3; then consider ii being Nat such that A111: 1 <= ii and A112: ii+1 <= len mid(f,l,j) and A113: p in LSeg(mid(f,l,j),ii) by SPPOL_2:13; set k = ii+l-'1; len mid(f,l,j) = j-'l+1 by A104,A105,A108,JORDAN4:20; then A114: ii<j-'l+1 by A112,NAT_1:38; per cases by A105,REAL_1:def 5; suppose l = j; then len mid(f,l,j) = 1 by A104,A108,JORDAN4:27; then L~mid(f,l,j) = {} by TOPREAL1:28; hence thesis by A113,SPPOL_2:17; suppose A115: l < j; then A116: LSeg(mid(f,l,j),ii)=LSeg(f,k) by A104,A108,A111,A114, JORDAN4:31; A117: ii + l >= ii + 1 by A104,AXIOMS:24; ii + 1 >= 1 + 1 by A111,AXIOMS:24; then A118: ii + l >= 1 + 1 by A117,AXIOMS:22; then A119: k >= 1 by JORDAN3:12; A120: ii + l >= 1 by A118,AXIOMS:22; A121: now assume k + 1 >= len f -' 1; then k >= len f -'1 -' 1 by SPRECT_3:6; then A122: ii + l >= len f -' 1 by A120,SPRECT_3:9; ii + l < j -' l + 1 + l by A114,REAL_1:53; then ii + l < j -' l + l + 1 by XCMPLX_1:1; then ii + l < j + 1 by A115,AMI_5:4; then len f -' 1 < j + 1 by A122,AXIOMS:22; hence contradiction by A106,NAT_1:38; end; A123: p in LSeg(f,k) /\ LSeg(f,len f -' 1) by A92,A110,A113,A116,XBOOLE_0:def 3; then LSeg(f,k) meets LSeg(f,len f -' 1) by XBOOLE_0:4; then k <= 1 by A107,A121,GOBOARD5:def 4; then A124: k = 1 by A119,AXIOMS:21; LSeg(f,1) /\ LSeg(f,len f -' 1) = { f.1 } by JORDAN4:54 .= {f/.1} by A14,FINSEQ_4:def 4; then p = f/.1 by A123,A124,TARSKI:def 1; hence contradiction by A1,A64,A89,A97,A110,SPRECT_3:16; end; A125: for j st 1 <= j & j < len f -' 1 holds L~mid(f,j,len f -' 1) /\ LSeg(p2,f/.(len f -' 1)) = {f/.(len f -' 1)} proof let j such that A126: 1 <= j and A127: j < len f -' 1; A128: f/.(len f -' 1) in LSeg(p2,f/.(len f -' 1)) by TOPREAL1:6; A129: 1 <= len f -' 1 by A126,A127,AXIOMS:22; A130: len f -' 1 -' 1 = len f -' 2 by JORDAN3:8; then A131: len f -' 2 < len f -' 1 by A129,JORDAN5B:1; j <= len f -' 2 by A127,A130,JORDAN3:12; then A132: LSeg(p2,f/.(len f -' 1)) misses L~mid(f,j,len f -' 2) by A103,A126,A131; len f -' 1 <= len f by GOBOARD9:2; then L~mid(f,j,len f -' 1) = LSeg(f,len f -' 2) \/ L~mid(f,j,len f -' 2) by A126,A127,A130,SPRECT_3:36; hence L~mid(f,j,len f -' 1) /\ LSeg(p2,f/.(len f -' 1)) = LSeg(f,len f -' 2) /\ LSeg(p2,f/.(len f -' 1)) by A132,XBOOLE_1:78 .= {f/.(len f -' 1)} by A73,A92,A128,SPRECT_3:2; end; A133: LSeg(NW-corner L~f,N-min L~f) misses LSeg(f/.(len f -' 1),p2) proof assume LSeg(NW-corner L~f,A) meets LSeg(f/.(len f -' 1),p2); then LSeg(p2,f/.(len f -' 1)) /\ LSeg(NW-corner L~f,A) <> {} by XBOOLE_0:def 7; then consider p being Point of TOP-REAL 2 such that A134: p in LSeg(p2,f/.(len f -' 1)) /\ LSeg(NW-corner L~f,A) by SUBSET_1:10; A135: p in LSeg(p2,f/.(len f -' 1)) by A134,XBOOLE_0:def 3; p in LSeg(NW-corner L~f,A) by A134,XBOOLE_0:def 3; then p in { A } by A1,A64,A82,A90,A135,XBOOLE_0:def 3; then p = A by TARSKI:def 1; hence contradiction by A1,A64,A89,A97,A135,SPRECT_3:16; end; p2`2 <> A`2 by A93,A97,TOPREAL3:11; then A136: LSeg(NW-corner L~f,N-min L~f) misses LSeg(z2,p2) by A69,A86,SPRECT_3:21; set G = GoB f; A137: LSeg(z2,p2) c= Int cell(G,i,w-'1) \/ { p2 } by A4,A5,A10,A18,A61, GOBOARD6:43; A138: Int cell(GoB f,i,w-'1) misses L~f by A5,A61,GOBOARD7:14; L~f /\ (Int cell(G,i,w-'1) \/ { p2 }) = L~f /\ Int cell(G,i,w-'1) \/ L~f /\ { p2 } by XBOOLE_1:23 .= {} \/ L~f /\ { p2 } by A138,XBOOLE_0:def 7 .= L~f /\ { p2 }; then A139: LSeg(z2,p2) /\ L~f c= L~f /\ { p2 } by A137,XBOOLE_1:26; L~f /\ { p2 } c= { p2 } by XBOOLE_1:17; then A140: LSeg(z2,p2) /\ L~f c= { p2 } by A139,XBOOLE_1:1; A141: for i,j st 1 <= i & i < j & j < len f holds L~mid(f,i,j) misses LSeg(z2,p2) proof let l,j such that A142: 1 <= l and A143: l < j and A144: j < len f; assume L~mid(f,l,j) meets LSeg(z2,p2); then L~mid(f,l,j) /\ LSeg(z2,p2) <> {} by XBOOLE_0:def 7; then consider p being Point of TOP-REAL 2 such that A145: p in L~mid(f,l,j) /\ LSeg(z2,p2) by SUBSET_1:10; A146: p in LSeg(z2,p2) by A145,XBOOLE_0:def 3; p in L~mid(f,l,j) by A145,XBOOLE_0:def 3; then consider ii being Nat such that A147: 1 <= ii and A148: ii+1 <= len mid(f,l,j) and A149: p in LSeg(mid(f,l,j),ii) by SPPOL_2:13; A150: len mid(f,l,j) = j-'l+1 by A142,A143,A144,JORDAN4:20; then ii<j-'l+1 by A148,NAT_1:38; then A151: p in LSeg(f,ii+l-'1) by A142,A143,A144,A147,A149,JORDAN4:31; then A152: p in LSeg(f,ii+l-'1) /\ LSeg(z2,p2) by A146,XBOOLE_0:def 3; set k = ii+l-'1; A153: ii + l >= ii + 1 by A142,AXIOMS:24; ii + 1 >= 1 + 1 by A147,AXIOMS:24; then ii + l >= 1 + 1 by A153,AXIOMS:22; then A154: k >= 1 by JORDAN3:12; len f >= 1 + 1 by A7,AXIOMS:22; then A155: len f -' 1 >= 1 by JORDAN3:12; LSeg(f,k) c= L~f by TOPREAL3:26; then LSeg(f,k) /\ LSeg(z2,p2) c= L~f /\ LSeg(z2,p2) by XBOOLE_1:26; then LSeg(f,k) /\ LSeg(z2,p2) c= { p2 } by A140,XBOOLE_1:1; then p = p2 by A152,TARSKI:def 1; then p2 in LSeg(f,k) /\ LSeg(f/.(len f -' 1), f/.len f) by A89,A151,XBOOLE_0:def 3; then A156: p2 in LSeg(f,k) /\ LSeg(f,len f -' 1) by A58,A155,TOPREAL1: def 5; then A157: LSeg(f,k) meets LSeg(f,len f -' 1) by XBOOLE_0:4; per cases by A59,A157,GOBOARD5:def 4; suppose k <= 1; then A158: k = 1 by A154,AXIOMS:21; LSeg(f,1) /\ LSeg(f,len f -' 1) = {f.1} by JORDAN4:54 .= {f/.1} by A14,FINSEQ_4:def 4; hence contradiction by A1,A97,A156,A158,TARSKI:def 1; suppose k+1 >= len f -' 1; then A159: k >= len f -' 1 -' 1 by SPRECT_3:6; ii+l >= l by NAT_1:29; then A160: ii+l >= 1 by A142,AXIOMS:22; ii <= j -' l by A148,A150,REAL_1:53; then ii + l <= j by A143,SPRECT_3:7; then ii+l < len f by A144,AXIOMS:22; then ii+l-'1 < len f -' 1 by A160,SPRECT_3:9; then ii+l-'1 <= len f -' 1 -' 1 by JORDAN3:12; then k = len f -' 1 -' 1 by A159,AXIOMS:21; then p2 in {f/.(len f -' 1)} by A99,A100,A102,A156,TOPREAL1:def 8; then p2 = f/.(len f -' 1) by TARSKI:def 1; hence contradiction by A6,A81,A97,SPRECT_3:15; end; f is_in_the_area_of f by SPRECT_2:25; then A161: <*p2*> is_in_the_area_of f by A91,A95,SPRECT_3:63; then A162: <*p2*> is_in_the_area_of SpStSeq L~f by SPRECT_3:70; <*p2,z2*> = <*p2*>^<*z2*> by FINSEQ_1:def 9; then <*p2,z2*> is_in_the_area_of SpStSeq L~f by A80,A162,SPRECT_2:28; then A163: L~SpStSeq L~f /\ LSeg(p2,z2) c= { p2 } by A62,SPRECT_3:64; w-'1 <= width(GoB f) by GOBOARD9:2; then ((GoB f)*(i,w-'1))`1 = ((GoB f)*(i,1))`1 by A4,A5,A18,GOBOARD5:3 .= ((GoB f)*(i,w))`1 by A4,A5,A9,GOBOARD5:3; then ((GoB f)*(i,w)+(GoB f)*(i+1,w))`1 = ((GoB f)*(i,w-'1))`1+((GoB f)*(i+1,w))`1 by TOPREAL3:7 .= ((GoB f)*(i,w-'1)+(GoB f)*(i+1,w))`1 by TOPREAL3:7; then p1`1 = 1/2*((GoB f)*(i,w-'1)+(GoB f)*(i+1,w))`1 by TOPREAL3:9 .= z2`1 by TOPREAL3:9; then A164: LSeg(p1,z2) is vertical by SPPOL_1:37; A165: f/.2 in LSeg(f/.1, f/.(1+1)) by TOPREAL1:6; p1 = 1/2*((GoB f)*(i,w))+(1-1/2)*((GoB f)*(i+1,w)) by EUCLID:36; then A166: p1 in LSeg(f/.1, f/.2) by A1,A6,A16,SPPOL_1:22; then A167: LSeg(p1,f/.2) c= LSeg(f/.1, f/.2) by A165,TOPREAL1:12; A168: LSeg(f/.1, f/.(1+1)) = LSeg(f,1) by A8,TOPREAL1:def 5; then LSeg(f/.1, f/.2) c= L~f by TOPREAL3:26; then A169: LSeg(p1,f/.2) c= L~f by A167,XBOOLE_1:1; A170: p1`2 = 1/2*(A+(GoB f)*(i+1,w))`2 by A6,TOPREAL3:9 .= 1/2*(A`2+A`2) by A77,TOPREAL3:7 .= 1/2*(2*A`2) by XCMPLX_1:11 .= (N-min L~f)`2 by XCMPLX_1:108 .= N-bound L~f by PSCOMP_1:94; A171: p1 in LSeg(p1,f/.2) by TOPREAL1:6; A172: now assume p1 = A; then f/.1 = f/.(1+1) by A1,A6,A16,SPRECT_3:15; hence contradiction by A14,A54,GOBOARD7:31; end; A173: for i,j st 2 < i & i <= j & j<= len f holds L~mid(f,i,j) misses LSeg(p1,f/.2) proof let l,j such that A174: 2 < l and A175: l <= j and A176: j<= len f; A177: 1 < l by A174,AXIOMS:22; assume L~mid(f,l,j) meets LSeg(p1,f/.2); then L~mid(f,l,j) /\ LSeg(p1,f/.2) <> {} by XBOOLE_0:def 7; then consider p being Point of TOP-REAL 2 such that A178: p in L~mid(f,l,j) /\ LSeg(p1,f/.2) by SUBSET_1:10; A179: p in LSeg(p1,f/.2) by A178,XBOOLE_0:def 3; p in L~mid(f,l,j) by A178,XBOOLE_0:def 3; then consider ii being Nat such that A180: 1 <= ii and A181: ii+1 <= len mid(f,l,j) and A182: p in LSeg(mid(f,l,j),ii) by SPPOL_2:13; set k = ii+l-'1; A183: len mid(f,l,j) = j-'l+1 by A175,A176,A177,JORDAN4:20; then A184: ii<j-'l+1 by A181,NAT_1:38; A185: ii + l > ii + 2 by A174,REAL_1:53; ii + 2 >= 1 + 2 by A180,AXIOMS:24; then ii + l > 1 + 2 by A185,AXIOMS:22; then A186: k > 1+1 by SPRECT_3:5; per cases by A175,REAL_1:def 5; suppose l = j; then len mid(f,l,j) = 1 by A176,A177,JORDAN4:27; then L~mid(f,l,j) = {} by TOPREAL1:28; hence thesis by A182,SPPOL_2:17; suppose l < j; then LSeg(mid(f,l,j),ii)=LSeg(f,k) by A176,A177,A180,A184,JORDAN4:31; then A187: p in LSeg(f,k) /\ LSeg(f,1) by A167,A168,A179,A182,XBOOLE_0:def 3; then LSeg(f,k) meets LSeg(f,1) by XBOOLE_0:4; then k+1 >= len f by A186,GOBOARD5:def 4; then A188: k >= len f -' 1 by SPRECT_3:6; ii <= j -' l by A181,A183,REAL_1:53; then ii + l <= j by A175,SPRECT_3:7; then ii+l <= len f by A176,AXIOMS:22; then ii+l-'1 <= len f -' 1 by JORDAN3:5; then A189: k = len f -' 1 by A188,AXIOMS:21; LSeg(f,1) /\ LSeg(f,len f -' 1) = { f.1 } by JORDAN4:54 .= {f/.1} by A14,FINSEQ_4:def 4; then p = f/.1 by A187,A189,TARSKI:def 1; hence contradiction by A1,A166,A172,A179,SPRECT_3:16; end; A190: for j st 2 < j & j <= len f holds L~mid(f,2,j) /\ LSeg(p1,f/.2) = {f/.2} proof let j such that A191: 2 < j and A192: j <= len f; 2+1 <= j by A191,NAT_1:38; then A193: LSeg(p1,f/.2) misses L~mid(f,2+1,j) by A173,A192; A194: f/.2 in LSeg(p1,f/.2) by TOPREAL1:6; L~mid(f,2,j) = LSeg(f,2) \/ L~mid(f,2+1,j) by A191,A192,SPRECT_3:35; hence L~mid(f,2,j) /\ LSeg(p1,f/.2) = LSeg(f,2) /\ LSeg(p1,f/.2) by A193,XBOOLE_1:78 .= {f/.2} by A70,A167,A168,A194,SPRECT_3:2; end; A195: LSeg(p1,f/.2) misses LSeg(NW-corner L~f,A) proof assume LSeg(p1,f/.2) meets LSeg(NW-corner L~f,A); then LSeg(p1,f/.2) /\ LSeg(NW-corner L~f,A) <> {} by XBOOLE_0:def 7; then consider p being Point of TOP-REAL 2 such that A196: p in LSeg(p1,f/.2) /\ LSeg(NW-corner L~f,A) by SUBSET_1:10; A197: p in LSeg(p1,f/.2) by A196,XBOOLE_0:def 3; p in LSeg(NW-corner L~f,A) by A196,XBOOLE_0:def 3; then p in { A } by A1,A85,A167,A197,XBOOLE_0:def 3; then p = A by TARSKI:def 1; hence contradiction by A1,A166,A172,A197,SPRECT_3:16; end; A198: LSeg(NW-corner L~f,N-min L~f) misses LSeg(z2,p1) proof assume LSeg(NW-corner L~f,N-min L~f) meets LSeg(z2,p1); then LSeg(NW-corner L~f,N-min L~f) /\ LSeg(z2,p1) <> {} by XBOOLE_0:def 7 ; then consider p being Point of TOP-REAL 2 such that A199: p in LSeg(NW-corner L~f,N-min L~f) /\ LSeg(z2,p1) by SUBSET_1:10; A200: p in LSeg(NW-corner L~f,N-min L~f) by A199,XBOOLE_0:def 3; A201: p in LSeg(z2,p1) by A199,XBOOLE_0:def 3; LSeg(NW-corner L~f,NE-corner L~f) /\ LSeg(z2,p1) = {p1} by A68,A75,A164,A166,SPRECT_3:23; then p in {p1} by A67,A200,A201,XBOOLE_0:def 3; then p1 in LSeg(NW-corner L~f,N-min L~f) by A200,TARSKI:def 1; then p1 in LSeg(NW-corner L~f,N-min L~f) /\ L~f by A169,A171,XBOOLE_0:def 3; then p1 in { N-min L~f } by PSCOMP_1:102; hence contradiction by A172,TARSKI:def 1; end; A202: for i,j st 1 < i & i < j & j<= len f holds L~mid(f,i,j) misses LSeg(z2,p1) proof let l,j such that A203: 1 < l and A204: l < j and A205: j<= len f; assume L~mid(f,l,j) meets LSeg(z2,p1); then L~mid(f,l,j) /\ LSeg(z2,p1) <> {} by XBOOLE_0:def 7; then consider p being Point of TOP-REAL 2 such that A206: p in L~mid(f,l,j) /\ LSeg(z2,p1) by SUBSET_1:10; A207: p in LSeg(z2,p1) by A206,XBOOLE_0:def 3; p in L~mid(f,l,j) by A206,XBOOLE_0:def 3; then consider ii being Nat such that A208: 1 <= ii and A209: ii+1 <= len mid(f,l,j) and A210: p in LSeg(mid(f,l,j),ii) by SPPOL_2:13; A211: len mid(f,l,j) = j-'l+1 by A203,A204,A205,JORDAN4:20; then ii<j-'l+1 by A209,NAT_1:38; then A212: p in LSeg(f,ii+l-'1) by A203,A204,A205,A208,A210,JORDAN4:31; then A213: p in LSeg(f,ii+l-'1) /\ LSeg(z2,p1) by A207,XBOOLE_0:def 3; set k = ii+l-'1, G = GoB f; A214: ii + l > ii + 1 by A203,REAL_1:53; ii + 1 >= 1 + 1 by A208,AXIOMS:24; then ii + l > 1 + 1 by A214,AXIOMS:22; then A215: k > 1 by SPRECT_3:5; A216: LSeg(z2,p1) c= Int cell(G,i,w-'1) \/ { p1 } by A4,A5,A10,A18,A61, GOBOARD6:44; A217: Int cell(GoB f,i,w-'1) misses L~f by A5,A61,GOBOARD7:14; L~f /\ (Int cell(G,i,w-'1) \/ { p1 }) = L~f /\ Int cell(G,i,w-'1) \/ L~f /\ { p1 } by XBOOLE_1:23 .= {} \/ L~f /\ { p1 } by A217,XBOOLE_0:def 7 .= L~f /\ { p1 }; then A218: LSeg(z2,p1) /\ L~f c= L~f /\ { p1 } by A216,XBOOLE_1:26; L~f /\ { p1 } c= { p1 } by XBOOLE_1:17; then A219: LSeg(z2,p1) /\ L~f c= { p1 } by A218,XBOOLE_1:1; A220: len f >= 2 by A7,AXIOMS:22; LSeg(f,k) c= L~f by TOPREAL3:26; then LSeg(f,k) /\ LSeg(z2,p1) c= L~f /\ LSeg(z2,p1) by XBOOLE_1:26; then LSeg(f,k) /\ LSeg(z2,p1) c= { p1 } by A219,XBOOLE_1:1; then p = p1 by A213,TARSKI:def 1; then p1 in LSeg(f,k) /\ LSeg(f/.1,f/.(1+1)) by A166,A212,XBOOLE_0:def 3; then A221: p1 in LSeg(f,k) /\ LSeg(f,1) by A220,TOPREAL1:def 5; then LSeg(f,k) meets LSeg(f,1) by XBOOLE_0:4; then A222: k <= 1+1 or k+1 >= len f by GOBOARD5:def 4; per cases by A222; suppose A223: k <= 2; k >= 1+1 by A215,NAT_1:38; then A224: k = 2 by A223,AXIOMS:21; 1 + 2 <= len f by A7,AXIOMS:22; then p1 in {f/.(1+1)} by A221,A224,TOPREAL1:def 8; then p1 = f/.(1+1) by TARSKI:def 1; hence contradiction by A6,A16,A172,SPRECT_3:15; suppose k+1 >= len f; then A225: k >= len f -' 1 by SPRECT_3:6; ii <= j -' l by A209,A211,REAL_1:53; then ii + l <= j by A204,SPRECT_3:7; then ii+l <= len f by A205,AXIOMS:22; then ii+l-'1 <= len f -' 1 by JORDAN3:5; then A226: k = len f -' 1 by A225,AXIOMS:21; LSeg(f,1) /\ LSeg(f,len f -' 1) = { f.1 } by JORDAN4:54 .= {f/.1} by A14,FINSEQ_4:def 4; hence contradiction by A1,A172,A221,A226,TARSKI:def 1; end; LSeg(f/.1,f/.2) c= L~SpStSeq L~f by A1,SPRECT_3:48; then A227: L~SpStSeq L~f /\ LSeg(p1,z2) = { p1 } by A62,A80,A166,SPRECT_3:65; A228: LSeg(p1,f/.2) is horizontal by A78,A170,SPPOL_1:36; A229: p1 in LSeg(p1,f/.2) by TOPREAL1:6; then A230: p1 in L~f by A169; f is_in_the_area_of f by SPRECT_2:25; then A231: <*p1*> is_in_the_area_of f by A169,A229,SPRECT_3:63; RightComp f misses L~f by SPRECT_3:42; then A232: L~f misses L~Red' by A48,A50,XBOOLE_1:63; reconsider Red' as S-Sequence_in_R2 by A49; len Red' in dom Red' by FINSEQ_5:6; then A233: z2 in L~Red' by A52,SPPOL_2:48; set u1 = Last_Point(L~Red',Red'/.1,Red'/.len Red',L~SpStSeq L~f), Red = L_Cut(Red',u1), u2 = First_Point(L~Red,Red/.1,Red/.len Red,LSeg(z2,p1)), u3 = First_Point(L~Red,Red/.1,Red/.len Red,LSeg(z2,p2)); NW-corner L~SpStSeq L~f = NW-corner L~f by SPRECT_1:70; then A234: u1 <> NW-corner L~f by A21,A45,A51,A52,SPRECT_3:55; A235: L~Red' is_an_arc_of z1,z2 by A51,A52,TOPREAL1:31; L~Red' /\ L~SpStSeq L~f is closed by TOPS_1:35; then A236: u1 in L~Red' /\ L~SpStSeq L~f by A51,A52,A53,A235,JORDAN5C:def 2; then A237: u1 in L~SpStSeq L~f by XBOOLE_0:def 3; A238: u1 in L~Red' by A236,XBOOLE_0:def 3; A239: now assume u1 in L~f; then u1 in L~f /\ L~Red' by A238,XBOOLE_0:def 3; hence contradiction by A232,XBOOLE_0:def 7; end; len Red' in dom Red' by FINSEQ_5:6; then u1 <> Red'.len Red' by A52,A62,A237,FINSEQ_4:def 4; then reconsider Red as S-Sequence_in_R2 by A238,JORDAN3:69; 1 in dom Red by FINSEQ_5:6; then A240: Red/.1 =Red.1 by FINSEQ_4:def 4 .= u1 by A238,JORDAN3:58; A241: L~SpStSeq L~f /\ L~Red = { u1 } by A52,A53,A62,Th6; len Red' in dom Red' by FINSEQ_5:6; then z2 = Red'.len Red' by A52,FINSEQ_4:def 4; then A242: z2 in L~Red by A62,A233,A237,A238,JORDAN5B:25; Red is_in_the_area_of SpStSeq L~f by A21,A45,A51,A52,SPRECT_3:71; then A243: Red is_in_the_area_of f by SPRECT_3:70; A244: L~Red c= L~Red' by A238,JORDAN3:77; then A245: L~f misses L~Red by A232,XBOOLE_1:63; z2 in LSeg(p1,z2) by TOPREAL1:6; then A246: LSeg(p1,z2) meets L~Red by A242,XBOOLE_0:3; z2 in LSeg(p2,z2) by TOPREAL1:6; then A247: LSeg(p2,z2) meets L~Red by A242,XBOOLE_0:3; A248: L~Red /\ LSeg(p1,z2) is closed by TOPS_1:35; A249: L~Red /\ LSeg(p2,z2) is closed by TOPS_1:35; L~Red is_an_arc_of Red/.1,Red/.len Red by TOPREAL1:31; then A250: u3 in LSeg(p2,z2) /\ L~Red by A247,A249,JORDAN5C:def 1; then A251: u3 in L~Red by XBOOLE_0:def 3; A252: u3 in LSeg(p2,z2) by A250,XBOOLE_0:def 3; A253: u3 <> p2 by A91,A95,A232,A244,A251,XBOOLE_0:3; p2 in LSeg(p2,z2) by TOPREAL1:6; then A254: LSeg(p2,u3) c= LSeg(p2,z2) by A252,TOPREAL1:12; then A255: LSeg(u3,p2) is horizontal by A86,SPRECT_1:11; A256: LSeg(NW-corner L~f,N-min L~f) misses LSeg(u3,p2) by A136,A254,XBOOLE_1:63 ; A257: for i,j st 1 <= i & i < j & j < len f holds L~mid(f,i,j) misses LSeg(u3,p2) proof let i,j; assume 1 <= i & i < j & j < len f; then L~mid(f,i,j) misses LSeg(z2,p2) by A141; hence thesis by A254,XBOOLE_1:63; end; A258: now assume A259: u3 = Red.1; 1 in dom Red by FINSEQ_5:6; then u3 in L~SpStSeq L~f by A237,A240,A259,FINSEQ_4:def 4; then u3 in LSeg(p2,z2) /\ L~SpStSeq L~f by A252,XBOOLE_0:def 3; hence contradiction by A163,A253,TARSKI:def 1; end; L~Red is_an_arc_of Red/.1,Red/.len Red by TOPREAL1:31; then A260: u2 in LSeg(p1,z2) /\ L~Red by A246,A248,JORDAN5C:def 1; then A261: u2 in L~Red by XBOOLE_0:def 3; A262: u2 in LSeg(p1,z2) by A260,XBOOLE_0:def 3; A263: u2 <> p1 by A169,A229,A232,A244,A261,XBOOLE_0:3; p1 in LSeg(p1,z2) by TOPREAL1:6; then A264: LSeg(p1,u2) c= LSeg(p1,z2) by A262,TOPREAL1:12; then A265: LSeg(p1,u2) is vertical by A164,SPRECT_1:12; A266: LSeg(NW-corner L~f,N-min L~f) misses LSeg(u2,p1) by A198,A264,XBOOLE_1:63 ; A267: for i,j st 1 < i & i < j & j<= len f holds L~mid(f,i,j) misses LSeg(u2,p1) proof let i,j; assume 1 < i & i < j & j<= len f; then L~mid(f,i,j) misses LSeg(z2,p1) by A202; hence thesis by A264,XBOOLE_1:63; end; A268: now assume A269: u2 = Red.1; 1 in dom Red by FINSEQ_5:6; then u2 in L~SpStSeq L~f by A237,A240,A269,FINSEQ_4:def 4; then u2 in { p1 } by A227,A262,XBOOLE_0:def 3; hence contradiction by A263,TARSKI:def 1; end; reconsider Red2=R_Cut(Red,u3) as S-Sequence_in_R2 by A251,A258,JORDAN3:70; A270: Rev Red2 is S-Sequence_in_R2 by SPPOL_2:47; A271: Red2/.1 = u1 by A240,A251,SPRECT_3:39; A272: len Red2 in dom Red2 by FINSEQ_5:6; A273: (Rev Red2)/.1 = Red2/.len Red2 by FINSEQ_5:68 .= Red2.len Red2 by A272,FINSEQ_4:def 4 .= u3 by A251,JORDAN3:59; then A274: ((Rev Red2)/.1)`2 = p2`2 & (Rev Red2)/.1 <> p2 by A86,A91,A95,A232,A244,A251,A252,SPRECT_3:19,XBOOLE_0:3; A275: L~Rev Red2 = L~Red2 by SPPOL_2:22; u3 in L~Red2 & u3 in LSeg(p2,u3) by A251,A258,JORDAN5B:23,TOPREAL1:6; then A276: u3 in L~Red2 /\ LSeg(p2,u3) by XBOOLE_0:def 3; now assume u1 in LSeg(p2,z2); then u1 in L~SpStSeq L~f /\ LSeg(p2,z2) by A237,XBOOLE_0:def 3; hence contradiction by A96,A163,A239,TARSKI:def 1; end; then LSeg(p2,z2) /\ L~Red2={ u3 } by A240,A247,Th1; then LSeg(p2,u3) /\ L~Red2 c={ u3 } by A254,XBOOLE_1:26; then LSeg(p2,u3) /\ L~Red2={ u3 } by A276,ZFMISC_1:39; then reconsider RB2 = <*p2*>^Rev Red2 as S-Sequence_in_R2 by A270,A273,A274,A275,SPRECT_3:31; A277: L~Red2 c= L~Red by A251,JORDAN3:76; LSeg(p2,f/.(len f -' 1)) misses L~Red by A91,A245,XBOOLE_1:63; then LSeg(p2,f/.(len f -' 1)) misses L~Red2 by A277,XBOOLE_1:63; then A278: LSeg(p2,f/.(len f -' 1)) /\ L~Red2 = {} by XBOOLE_0:def 7; 1 in dom Red2 by FINSEQ_5:6; then u1 in L~Red2 by A271,SPPOL_2:48; then u1 in L~SpStSeq L~f /\ L~Red2 & L~SpStSeq L~f /\ L~Red2 c= { u1 } by A237,A241,A277,XBOOLE_0:def 3,XBOOLE_1:26; then A279: L~SpStSeq L~f /\ L~Red2 = { u1 } by ZFMISC_1:39; reconsider Red1=R_Cut(Red,u2) as S-Sequence_in_R2 by A261,A268,JORDAN3:70; len Red1 in dom Red1 by FINSEQ_5:6; then A280: Red1/.len Red1 = Red1.len Red1 by FINSEQ_4:def 4 .= u2 by A261,JORDAN3:59; A281: Red1/.1 = u1 by A240,A261,SPRECT_3:39; A282: (Red1/.len Red1)`1 = p1`1 & Red1/.len Red1 <> p1 by A164,A169,A229,A232,A244,A261,A262,A280,SPRECT_3:20,XBOOLE_0:3; u2 in L~Red1 & u2 in LSeg(p1,u2) by A261,A268,JORDAN5B:23,TOPREAL1:6; then A283: u2 in L~Red1 /\ LSeg(p1,u2) by XBOOLE_0:def 3; now assume u1 in LSeg(p1,z2); then u1 in L~SpStSeq L~f /\ LSeg(p1,z2) by A237,XBOOLE_0:def 3; hence contradiction by A227,A230,A239,TARSKI:def 1; end; then LSeg(p1,z2) /\ L~Red1={ u2 } by A240,A246,Th1; then LSeg(p1,u2) /\ L~Red1 c={ u2 } by A264,XBOOLE_1:26; then LSeg(p1,u2) /\ L~Red1={ u2 } by A283,ZFMISC_1:39; then reconsider RB1 = Red1^<*p1*> as S-Sequence_in_R2 by A280,A282,SPRECT_2:65; A284: L~Red1 c= L~Red by A261,JORDAN3:76; LSeg(p1,f/.2) misses L~Red by A169,A245,XBOOLE_1:63; then LSeg(p1,f/.2) misses L~Red1 by A284,XBOOLE_1:63; then A285: LSeg(p1,f/.2) /\ L~Red1 = {} by XBOOLE_0:def 7; 1 in dom Red1 by FINSEQ_5:6; then A286: u1 in L~Red1 by A281,SPPOL_2:48; then u1 in L~SpStSeq L~f /\ L~Red1 & L~SpStSeq L~f /\ L~Red1 c= { u1 } by A237,A241,A284,XBOOLE_0:def 3,XBOOLE_1:26; then A287: L~SpStSeq L~f /\ L~Red1 = { u1 } by ZFMISC_1:39; len Rev Red2 = len Red2 by FINSEQ_5:def 3; then A288: RB2/.len RB2 = (Rev Red2)/.len Red2 by SPRECT_3:11 .= u1 by A271,FINSEQ_5:68; A289: RB2/.1 = p2 by FINSEQ_5:16; L~Rev Red2 = L~Red2 by SPPOL_2:22; then A290: L~RB2 = L~Red2 \/ LSeg(p2,u3) by A273,SPPOL_2:20; then A291: LSeg(p2,f/.(len f -' 1)) /\ L~RB2 = {} \/ LSeg(p2,f/.(len f -' 1)) /\ LSeg(u3,p2) by A278,XBOOLE_1:23 .={ p2 } by A94,A255,SPRECT_3:22; <*u3*> is_in_the_area_of f by A243,A251,SPRECT_3:63; then Red2 is_in_the_area_of f by A243,A251,SPRECT_3:69; then Rev Red2 is_in_the_area_of f by SPRECT_3:68; then A292: RB2 is_in_the_area_of f by A161,SPRECT_2:28; 1 in dom Red1 by FINSEQ_5:6; then A293: RB1/.1 = u1 by A281,GROUP_5:95; len <*p1*> = 1 by FINSEQ_1:56; then len RB1 = len Red1 + 1 by FINSEQ_1:35; then A294: RB1/.len RB1 = p1 by TOPREAL4:1; A295: L~RB1 = L~Red1 \/ LSeg(u2,p1) by A280,SPPOL_2:19; then A296: LSeg(p1,f/.2) /\ L~RB1 = {} \/ LSeg(p1,f/.2) /\ LSeg(u2,p1) by A285,XBOOLE_1:23 .={ p1 } by A228,A265,SPRECT_3:22; <*u2*> is_in_the_area_of f by A243,A261,SPRECT_3:63; then Red1 is_in_the_area_of f by A243,A261,SPRECT_3:69; then A297: RB1 is_in_the_area_of f by A231,SPRECT_2:28; A298: L~Red' is_an_arc_of z1,z2 by A47,A50,TOPREAL4:3; L~Red' /\ L~SpStSeq L~f is closed by TOPS_1:35; then u1 in L~Red' /\ L~SpStSeq L~f by A51,A52,A53,A298,JORDAN5C:def 2; then u1 in L~SpStSeq L~f by XBOOLE_0:def 3; then u1 in (LSeg(NW-corner L~f,NE-corner L~f) \/ LSeg(NE-corner L~f,SE-corner L~f)) \/ (LSeg(SE-corner L~f,SW-corner L~f) \/ LSeg(SW-corner L~f,NW-corner L~f)) by SPRECT_1:43; then A299: u1 in LSeg(NW-corner L~f,NE-corner L~f) \/ LSeg(NE-corner L~f, SE-corner L~f) or u1 in LSeg(SE-corner L~f,SW-corner L~f) \/ LSeg(SW-corner L~f,NW-corner L~f) by XBOOLE_0:def 2; N-most L~f = LSeg(NW-corner L~f, NE-corner L~f) /\ L~f by PSCOMP_1:def 39 ; then A300: N-most L~f c= LSeg(NW-corner L~f,NE-corner L~f) & A in N-most L~f by PSCOMP_1:101,XBOOLE_1:17; then LSeg(NW-corner L~f,NE-corner L~f) = LSeg(NW-corner L~f,A) \/ LSeg(A,NE-corner L~f) by TOPREAL1:11; then A301: u1 in LSeg(NW-corner L~f,NE-corner L~f) implies u1 in LSeg(NW-corner L~f,A) or u1 in LSeg(A,NE-corner L~f) by XBOOLE_0:def 2; per cases by A299,A301,XBOOLE_0:def 2; suppose A302: u1 in LSeg(NW-corner L~f,A); set i1 = (S-min L~f)..f, i2 = (E-min L~f)..f; A303: S-min L~f in rng f by SPRECT_2:45; then A304: i1 in dom f by FINSEQ_4:30; A305: E-min L~f in rng f by SPRECT_2:49; then A306: i2 in dom f by FINSEQ_4:30; (S-max L~f)..f < i1 & i2 <= (S-max L~f)..f by A1,SPRECT_2:76,77; then A307: i2 < i1 by AXIOMS:22; (W-min L~f)..f < len f & i1 <= (W-min L~f)..f by A1,SPRECT_2:78,80; then A308: i1 < len f by AXIOMS:22; (N-max L~f)..f > 1 by A1,SPRECT_2:73; then A309: (N-max L~f)..f >= 1+1 by NAT_1:38; (N-max L~f)..f <= (E-max L~f)..f by A1,SPRECT_2:74; then A310: 2 <= (E-max L~f)..f by A309,AXIOMS:22; (E-max L~f)..f < (E-min L~f)..f by A1,SPRECT_2:75; then A311: 2 < i2 by A310,AXIOMS:22; then A312: 2 < i1 by A307,AXIOMS:22; then 1 < i1 by AXIOMS:22; then reconsider M1 = mid(f,i1,len f) as S-Sequence_in_R2 by A308,JORDAN4:52; (NW-corner L~f)`2 = N-bound L~f & A`2 = N-bound L~f by PSCOMP_1:75,94; then A313: u1`2 = (NW-corner L~f)`2 by A302,GOBOARD7:6; u1 in LSeg(NW-corner L~f,u1) by TOPREAL1:6; then u1 in LSeg(NW-corner L~f,u1) /\ L~Red1 by A286,XBOOLE_0:def 3; then A314: { u1 } c= LSeg(NW-corner L~f,u1) /\ L~Red1 by ZFMISC_1:37; A315: NW-corner L~f in LSeg(NW-corner L~f,NE-corner L~f) by TOPREAL1:6; then LSeg(NW-corner L~f,A) c= LSeg(NW-corner L~f,NE-corner L~f) by A300,TOPREAL1:12; then LSeg(NW-corner L~f,u1) c= LSeg(NW-corner L~f,NE-corner L~f) by A302,A315,TOPREAL1:12; then LSeg(NW-corner L~f,u1) c= L~SpStSeq L~f by A63,XBOOLE_1:1; then LSeg(NW-corner L~f,u1) /\ L~Red1 c= { u1 } by A287,XBOOLE_1:26; then A316: LSeg(NW-corner L~f,u1) /\ L~Red1={ u1 } by A314,XBOOLE_0:def 10; NW-corner L~f in LSeg(NW-corner L~f,A) by TOPREAL1:6; then A317: LSeg(NW-corner L~f,u1) c= LSeg(NW-corner L~f,A) by A302,TOPREAL1: 12; then LSeg(NW-corner L~f,u1) misses LSeg(u2,p1) by A266,XBOOLE_1:63; then LSeg(NW-corner L~f,u1) /\ LSeg(u2,p1) = {} by XBOOLE_0:def 7; then LSeg(NW-corner L~f,u1) /\ L~RB1 = { u1 } \/ {} by A295,A316,XBOOLE_1:23 .= { u1 }; then reconsider M3 = <*NW-corner L~f*>^RB1 as S-Sequence_in_R2 by A234,A293,A313,SPRECT_3:31; A318: i2 <= len f by A306,FINSEQ_3:27; then reconsider M4 = mid(f,2,i2) as S-Sequence_in_R2 by A311,JORDAN4:52; A319: M3/.len M3 = p1 by A294,SPRECT_3:11; A320: M4/.1 = f/.2 by A54,A306,SPRECT_2:12; A321: L~M3 = LSeg(NW-corner L~f,RB1/.1) \/ L~RB1 by SPPOL_2:20; A322: L~M4 c= L~f by A54,A306,SPRECT_3:34; A323: now assume L~f meets LSeg(NW-corner L~f,u1); then consider u being set such that A324: u in L~f and A325: u in LSeg(NW-corner L~f,u1) by XBOOLE_0:3; reconsider u as Point of TOP-REAL 2 by A324; A326: u in LSeg(NW-corner L~f,A) /\ L~f by A317,A324,A325,XBOOLE_0:def 3; LSeg(NW-corner L~f,A) /\ L~f = {A} by PSCOMP_1:102; then u = A by A326,TARSKI:def 1; hence contradiction by A239,A302,A324,A325,SPRECT_3:16; end; then A327: L~M4 misses LSeg(NW-corner L~f,u1) by A322,XBOOLE_1:63; L~M4 misses L~Red by A245,A322,XBOOLE_1:63; then A328: L~M4 misses L~Red1 by A284,XBOOLE_1:63; L~M4 misses LSeg(u2,p1) by A267,A311,A318; then L~RB1 misses L~M4 by A295,A328,XBOOLE_1:70; then A329: L~M3 misses L~M4 by A293,A321,A327,XBOOLE_1:70; LSeg(p1,f/.2) misses LSeg(NW-corner L~f,u1) by A195,A317,XBOOLE_1:63 ; then LSeg(p1,f/.2) /\ LSeg(NW-corner L~f,u1) = {} by XBOOLE_0:def 7; then A330: LSeg(p1,f/.2) /\ L~M3 = {} \/ LSeg(p1,f/.2) /\ L~RB1 by A293,A321,XBOOLE_1:23 .= { p1 } by A296; LSeg(p1,f/.2) /\ L~M4 = { f/.2 } & (M3/.len M3)`2 = (M4/.1)`2 by A78,A170,A190,A294,A311,A318,A320, SPRECT_3:11; then reconsider M2 = M3^M4 as S-Sequence_in_R2 by A319,A320,A329,A330,SPRECT_3:38; A331: len M2 >= 2 & len M1 >= 2 by TOPREAL1:def 10; A332: M1 is_in_the_area_of f by A55,A304,SPRECT_2:27; A333: (M1/.1)`2 = (f/.i1)`2 by A55,A304,SPRECT_2:12 .= (S-min L~f)`2 by A303,FINSEQ_5:41 .= S-bound L~f by PSCOMP_1:114; (M1/.len M1)`2 = (f/.len f)`2 by A55,A304,SPRECT_2:13 .= (f/.1)`2 by FINSEQ_6:def 1 .= N-bound L~f by A1,PSCOMP_1:94; then A334: M1 is_a_v.c._for f by A332,A333,SPRECT_2:def 3; <*NW-corner L~f*>^RB1 is_in_the_area_of f & mid(f,2,i2) is_in_the_area_of f by A54,A60,A297,A306,SPRECT_2:27,28; then A335: M2 is_in_the_area_of f by SPRECT_2:28; M2 = <*NW-corner L~f*>^(RB1^mid(f,2,i2)) by FINSEQ_1:45; then A336: (M2/.1)`1 = (NW-corner L~f)`1 by FINSEQ_5:16 .= W-bound L~f by PSCOMP_1:74; (M2/.len M2)`1 = (mid(f,2,i2)/.len mid(f,2,i2))`1 by SPRECT_3:11 .= (f/.i2)`1 by A54,A306,SPRECT_2:13 .= (E-min L~f)`1 by A305,FINSEQ_5:41 .= E-bound L~f by PSCOMP_1:104; then M2 is_a_h.c._for f by A335,A336,SPRECT_2:def 2; then A337: L~M1 meets L~M2 by A331,A334,SPRECT_2:33; L~M2 = L~M3 \/ LSeg(M3/.len M3,M4/.1) \/ L~M4 & L~M1 misses L~M4 by A307,A308,A311,SPPOL_2:23,SPRECT_2:51; then L~M1 meets L~M3 \/ LSeg(M3/.len M3,M4/.1) & L~M1 misses LSeg(p1,f/.2) by A173,A308,A312,A337,XBOOLE_1:70; then A338: L~M1 meets L~M3 by A319,A320,XBOOLE_1:70; 1 < i1 by A312,AXIOMS:22; then A339: L~M1 misses LSeg(u2,p1) by A267,A308; L~M1 c= L~f by A55,A304,SPRECT_3:34; then L~M1 misses LSeg(NW-corner L~f,u1) & L~M3 = LSeg(NW-corner L~f,RB1/.1) \/ L~RB1 by A323,SPPOL_2:20,XBOOLE_1:63; then L~M1 meets L~RB1 by A293,A338,XBOOLE_1:70; then L~M1 meets L~Red1 by A295,A339,XBOOLE_1:70; then A340: L~M1 meets L~Red by A284,XBOOLE_1:63; L~M1 c= L~f by A55,A304,SPRECT_3:34; hence contradiction by A245,A340,XBOOLE_1:63; suppose that A341: u1 in LSeg(A,NE-corner L~f) and A342: N-min L~f = NW-corner L~f; set i1 = (S-max L~f)..f, i2 = (E-max L~f)..f; A343: S-max L~f in rng f by SPRECT_2:46; then A344: i1 in dom f by FINSEQ_4:30; A345: E-max L~f in rng f by SPRECT_2:50; then A346: i2 in dom f by FINSEQ_4:30; A347: (E-max L~f)..f < (E-min L~f)..f by A1,SPRECT_2:75; (E-min L~f)..f <= (S-max L~f)..f by A1,SPRECT_2:76; then A348: i2 < i1 by A347,AXIOMS:22; A349: (S-min L~f)..f <= (W-min L~f)..f by A1,SPRECT_2:78; (S-max L~f)..f < (S-min L~f)..f by A1,SPRECT_2:77; then (S-max L~f)..f < (W-min L~f)..f by A349,AXIOMS:22; then A350: (S-max L~f)..f + 1 <= (W-min L~f)..f by NAT_1:38; (W-min L~f)..f < len f by A1,SPRECT_2:80; then (S-max L~f)..f + 1 < len f by A350,AXIOMS:22; then A351: i1 < len f -' 1 by A58,AXIOMS:24; A352: (N-max L~f)..f > 1 by A1,SPRECT_2:73; (N-max L~f)..f <= (E-max L~f)..f by A1,SPRECT_2:74; then A353: 1 < i2 by A352,AXIOMS:22; then A354: 1 < i1 by A348,AXIOMS:22; then reconsider M4 = mid(f,i1,len f -' 1) as S-Sequence_in_R2 by A58,A351,JORDAN4:51; A355: M4/.len M4 = f/.(len f -' 1) by A57,A344,SPRECT_2:13; L~M4 c= L~f by A57,A344,SPRECT_3:34; then L~M4 misses L~Red by A245,XBOOLE_1:63; then A356: L~M4 misses L~Red2 by A277,XBOOLE_1:63; L~M4 misses LSeg(u3,p2) by A59,A257,A351,A354; then A357: L~RB2 misses L~M4 by A290,A356,XBOOLE_1:70; LSeg(M4/.len M4,p2) /\ L~M4={ M4/.len M4 } by A125,A351,A354,A355; then reconsider M1 = M4^RB2 as S-Sequence_in_R2 by A76,A81,A93,A289,A291,A355,A357,SPRECT_3:38; i1 < len f by A351,JORDAN3:7; then A358: i2 < len f by A348,AXIOMS:22; then i2+1 <= len f by NAT_1:38; then reconsider M2 = mid(f,1,i2) as S-Sequence_in_R2 by A353,JORDAN4:51; A359: len M2 >= 2 & len M1 >= 2 by TOPREAL1:def 10; mid(f,i1,len f -' 1) is_in_the_area_of f by A57,A344,SPRECT_2:27; then A360: M1 is_in_the_area_of f by A292,SPRECT_2:28; mid(f,i1,len f -' 1) is non empty by A57,A344,SPRECT_2:11; then 1 in dom mid(f,i1,len f -' 1) by FINSEQ_5:6; then A361: (M1/.1)`2 = (mid(f,i1,len f -' 1)/.1)`2 by GROUP_5:95 .= (f/.i1)`2 by A57,A344,SPRECT_2:12 .= (S-max L~f)`2 by A343,FINSEQ_5:41 .= S-bound L~f by PSCOMP_1:114; A362: (NE-corner L~f)`2 = N-bound L~f & A`2 = N-bound L~f by PSCOMP_1:77,94; (M1/.len M1)`2 = (RB2/.len RB2)`2 by SPRECT_3:11 .= N-bound L~f by A288,A341,A362,GOBOARD7:6; then A363: M1 is_a_v.c._for f by A360,A361,SPRECT_2:def 3; A364: M2 is_in_the_area_of f by A14,A346,SPRECT_2:27; A365: (M2/.1)`1 = (f/.1)`1 by A14,A346,SPRECT_2:12 .= W-bound L~f by A1,A342,PSCOMP_1:74; (M2/.len M2)`1 = (f/.i2)`1 by A14,A346,SPRECT_2:13 .= (E-max L~f)`1 by A345,FINSEQ_5:41 .= E-bound L~f by PSCOMP_1:104; then M2 is_a_h.c._for f by A364,A365,SPRECT_2:def 2; then A366: L~M1 meets L~M2 by A359,A363,SPRECT_2:33; A367: L~M1 = L~M4 \/ LSeg(M4/.len M4,p2) \/ L~RB2 by A289,SPPOL_2:23 .= L~M4 \/ (LSeg(M4/.len M4,p2) \/ L~RB2) by XBOOLE_1:4; A368: L~M2 misses LSeg(u3,p2) by A257,A353,A358; L~M2 misses L~M4 by A59,A348,A351,A353,SPRECT_2:51; then A369: L~M2 meets L~RB2 \/ LSeg(p2,M4/.len M4) by A366,A367,XBOOLE_1:70; i2 < len f -' 1 by A348,A351,AXIOMS:22; then L~M2 misses LSeg(p2,f/.(len f -' 1)) by A103,A353; then L~M2 meets L~RB2 by A355,A369,XBOOLE_1:70; then L~M2 meets L~Red2 by A290,A368,XBOOLE_1:70; then A370: L~M2 meets L~Red by A277,XBOOLE_1:63; L~M2 c= L~f by A14,A346,SPRECT_3:34; hence contradiction by A245,A370,XBOOLE_1:63; suppose that A371: u1 in LSeg(A,NE-corner L~f) and A372: N-min L~f <> NW-corner L~f; A373: W-max L~f <> N-min L~f by A372,PSCOMP_1:124; set i1 = (S-min L~f)..f, i2 = (E-max L~f)..f; A374: S-min L~f in rng f by SPRECT_2:45; then A375: i1 in dom f by FINSEQ_4:30; A376: E-max L~f in rng f by SPRECT_2:50; then A377: i2 in dom f by FINSEQ_4:30; (E-max L~f)..f < (E-min L~f)..f & (E-min L~f)..f <= (S-max L~f)..f by A1,SPRECT_2:75,76; then A378: (E-max L~f)..f < (S-max L~f)..f by AXIOMS:22; (S-max L~f)..f < (S-min L~f)..f by A1,SPRECT_2:77; then A379: i2 < i1 by A378,AXIOMS:22; (S-min L~f)..f <= (W-min L~f)..f & (W-min L~f)..f < (W-max L~f)..f by A1,A373,SPRECT_2:78,79; then (S-min L~f)..f < (W-max L~f)..f by AXIOMS:22; then (S-min L~f)..f + 1 <= (W-max L~f)..f & (W-max L~f)..f < len f by A1,NAT_1:38,SPRECT_2:81; then (S-min L~f)..f + 1 < len f by AXIOMS:22; then A380: i1 < len f -' 1 by A58,AXIOMS:24; (N-max L~f)..f > 1 & (N-max L~f)..f <= (E-max L~f)..f by A1,SPRECT_2:73,74; then A381: 1 < i2 by AXIOMS:22; then A382: 1 < i1 by A379,AXIOMS:22; then reconsider M4 = mid(f,i1,len f -' 1) as S-Sequence_in_R2 by A58,A380,JORDAN4:51; A383: M4/.len M4 = f/.(len f -' 1) by A57,A375,SPRECT_2:13; L~M4 c= L~f by A57,A375,SPRECT_3:34; then L~M4 misses L~Red by A245,XBOOLE_1:63; then A384: L~M4 misses L~Red2 by A277,XBOOLE_1:63; L~M4 misses LSeg(u3,p2) by A59,A257,A380,A382; then A385: L~RB2 misses L~M4 by A290,A384,XBOOLE_1:70; LSeg(M4/.len M4,p2) /\ L~M4={ M4/.len M4 } by A125,A380,A382,A383; then reconsider M1 = M4^RB2 as S-Sequence_in_R2 by A76,A81,A93,A289,A291,A383,A385,SPRECT_3:38; i1 < len f by A380,JORDAN3:7; then A386: i2 < len f by A379,AXIOMS:22; then i2+1 <= len f by NAT_1:38; then reconsider M3 = mid(f,1,i2) as S-Sequence_in_R2 by A381,JORDAN4:51; A387: M3/.1 = f/.1 by A14,A377,SPRECT_2:12; A388: (N-min L~f)`2 = (NW-corner L~f)`2 by PSCOMP_1:95; A389: M3/.1 = N-min L~f by A1,A14,A377,SPRECT_2:12; A390: LSeg(NW-corner L~f, N-min L~f) /\ L~f = {N-min L~f} by PSCOMP_1:102; L~M3 c= L~f by A14,A377,SPRECT_3:34; then A391: LSeg(NW-corner L~f,N-min L~f) /\ L~M3 c= { N-min L~f } by A390, XBOOLE_1:26; 1 in dom M3 by FINSEQ_5:6; then A392: N-min L~f in L~M3 by A389,SPPOL_2:48; N-min L~f in LSeg(NW-corner L~f,N-min L~f) by TOPREAL1:6; then N-min L~f in LSeg(NW-corner L~f,N-min L~f) /\ L~M3 by A392,XBOOLE_0:def 3; then {N-min L~f} c= LSeg(NW-corner L~f,N-min L~f) /\ L~M3 by ZFMISC_1:37; then LSeg(NW-corner L~f,N-min L~f) /\ L~M3={ N-min L~f } by A391,XBOOLE_0:def 10; then reconsider M2 = <*NW-corner L~f*>^M3 as S-Sequence_in_R2 by A372,A388,A389,SPRECT_3:31; A393: len M2 >= 2 & len M1 >= 2 by TOPREAL1:def 10; mid(f,i1,len f -' 1) is_in_the_area_of f by A57,A375,SPRECT_2:27; then A394: M1 is_in_the_area_of f by A292,SPRECT_2:28; mid(f,i1,len f -' 1) is non empty by A57,A375,SPRECT_2:11; then 1 in dom mid(f,i1,len f -' 1) by FINSEQ_5:6; then A395: (M1/.1)`2 = (mid(f,i1,len f -' 1)/.1)`2 by GROUP_5:95 .= (f/.i1)`2 by A57,A375,SPRECT_2:12 .= (S-min L~f)`2 by A374,FINSEQ_5:41 .= S-bound L~f by PSCOMP_1:114; A396: (NE-corner L~f)`2 = N-bound L~f & A`2 = N-bound L~f by PSCOMP_1:77,94; (M1/.len M1)`2 = u1`2 by A288,SPRECT_3:11 .= N-bound L~f by A371,A396,GOBOARD7:6; then A397: M1 is_a_v.c._for f by A394,A395,SPRECT_2:def 3; mid(f,1,i2) is_in_the_area_of f by A14,A377,SPRECT_2:27; then A398: M2 is_in_the_area_of f by A60,SPRECT_2:28; A399: (M2/.1)`1 = (NW-corner L~f)`1 by FINSEQ_5:16 .= W-bound L~f by PSCOMP_1:74; (M2/.len M2)`1 = (mid(f,1,i2)/.len mid(f,1,i2))`1 by SPRECT_3:11 .= (f/.i2)`1 by A14,A377,SPRECT_2:13 .= (E-max L~f)`1 by A376,FINSEQ_5:41 .= E-bound L~f by PSCOMP_1:104; then M2 is_a_h.c._for f by A398,A399,SPRECT_2:def 2; then A400: L~M1 meets L~M2 by A393,A397,SPRECT_2:33; A401: L~M1 = L~M4 \/ LSeg(M4/.len M4,p2) \/ L~RB2 by A289,SPPOL_2:23 .= L~M4 \/ (LSeg(M4/.len M4,p2) \/ L~RB2) by XBOOLE_1:4; A402: now L~M4 c= L~f by A57,A375,SPRECT_3:34; then A403: LSeg(NW-corner L~f,N-min L~f) /\ L~M4 c= { N-min L~f } by A390, XBOOLE_1:26; assume LSeg(NW-corner L~f,N-min L~f) meets L~M4; then LSeg(NW-corner L~f,N-min L~f) /\ L~M4 <> {} by XBOOLE_0:def 7; then LSeg(NW-corner L~f,N-min L~f) /\ L~M4 = { N-min L~f } by A403,ZFMISC_1:39; then N-min L~f in LSeg(NW-corner L~f,N-min L~f) /\ L~M4 by TARSKI:def 1; then N-min L~f in L~M4 by XBOOLE_0:def 3; hence contradiction by A1,A59,A380,A382,SPRECT_3:47; end; now NW-corner L~f in LSeg(NW-corner L~f,NE-corner L~f) & A in LSeg(NW-corner L~f,NE-corner L~f) by A300,TOPREAL1:6; then LSeg(NW-corner L~f,A) c= LSeg(NW-corner L~f,NE-corner L~f) by TOPREAL1:12; then LSeg(NW-corner L~f,N-min L~f) c= L~SpStSeq L~f by A63,XBOOLE_1:1; then A404: LSeg(NW-corner L~f,N-min L~f) /\ L~Red2 c= { u1 } by A279,XBOOLE_1: 26; assume LSeg(NW-corner L~f,N-min L~f) meets L~Red2; then LSeg(NW-corner L~f,N-min L~f) /\ L~Red2 <> {} by XBOOLE_0:def 7; then LSeg(NW-corner L~f,N-min L~f) /\ L~Red2 = { u1} by A404,ZFMISC_1:39; then u1 in LSeg(NW-corner L~f,N-min L~f) /\ L~Red2 by TARSKI:def 1; then u1 in LSeg(NW-corner L~f,N-min L~f) by XBOOLE_0:def 3; then A405: u1 in LSeg(NW-corner L~f,A) /\ LSeg(A,NE-corner L~f) by A371,XBOOLE_0:def 3; LSeg(NW-corner L~f,A) /\ LSeg(A,NE-corner L~f) = {A} by A300,TOPREAL1:14; then u1 = A by A405,TARSKI:def 1; hence contradiction by A239,SPRECT_1:13; end; then LSeg(NW-corner L~f,N-min L~f) misses L~RB2 by A256,A290,XBOOLE_1:70; then LSeg(NW-corner L~f,N-min L~f) misses LSeg(M4/.len M4,p2) \/ L~RB2 by A133,A383,XBOOLE_1:70; then LSeg(NW-corner L~f,N-min L~f) misses L~M1 & L~M2 = LSeg(NW-corner L~f,M3/.1) \/ L~M3 by A401,A402,SPPOL_2:20,XBOOLE_1:70; then A406: L~M3 meets L~M1 by A1,A387,A400,XBOOLE_1:70; L~M3 misses L~M4 by A59,A379,A380,A381,SPRECT_2:51; then A407: L~M3 meets L~RB2 \/ LSeg(p2,M4/.len M4) by A401,A406,XBOOLE_1:70; i2 < len f -' 1 by A379,A380,AXIOMS:22; then L~M3 misses LSeg(p2,f/.(len f -' 1)) by A103,A381; then L~M3 meets L~RB2 & L~M3 misses LSeg(u3,p2) by A257,A381,A383,A386,A407,XBOOLE_1:70; then L~M3 meets L~Red2 by A290,XBOOLE_1:70; then A408: L~M3 meets L~Red by A277,XBOOLE_1:63; L~M3 c= L~f by A14,A377,SPRECT_3:34; hence contradiction by A245,A408,XBOOLE_1:63; suppose that A409: u1 in LSeg(NE-corner L~f,SE-corner L~f) and A410: N-min L~f = W-max L~f; A411: N-min L~f = NW-corner L~f by A410,PSCOMP_1:124; set i1 = (S-max L~f)..f, i2 = (N-max L~f)..f, i3 = (W-min L~f)..f; A412: S-max L~f in rng f by SPRECT_2:46; then A413: i1 in dom f by FINSEQ_4:30; A414: N-max L~f in rng f by SPRECT_2:44; then A415: i2 in dom f by FINSEQ_4:30; W-min L~f in rng f by SPRECT_2:47; then A416: i3 in dom f by FINSEQ_4:30; A417: (S-max L~f)..f < (S-min L~f)..f by A1,SPRECT_2:77; (S-min L~f)..f <= (W-min L~f)..f by A1,SPRECT_2:78; then A418: i3 > i1 by A417,AXIOMS:22; A419: 1 < i2 by A1,SPRECT_2:73; A420: (N-max L~f)..f <= (E-max L~f)..f by A1,SPRECT_2:74; (E-max L~f)..f < (E-min L~f)..f by A1,SPRECT_2:75; then A421: (N-max L~f)..f < (E-min L~f)..f by A420,AXIOMS:22; (E-min L~f)..f <= (S-max L~f)..f by A1,SPRECT_2:76; then A422: i2 < i1 by A421,AXIOMS:22; i1 <= len f by A413,FINSEQ_3:27; then A423: mid(f,i2,i1) is S-Sequence_in_R2 by A419,A422,JORDAN4:52; A424: mid(f,i1,i2) = Rev mid(f,i2,i1) by JORDAN4:30; then reconsider M1 = mid(f,i1,i2) as S-Sequence_in_R2 by A423,SPPOL_2:47; A425: len M1 >= 2 & len RB2 >= 2 by TOPREAL1:def 10; A426: M1 is_in_the_area_of f by A413,A415,SPRECT_2:27; A427: (M1/.1)`2 = (f/.i1)`2 by A413,A415,SPRECT_2:12 .= (S-max L~f)`2 by A412,FINSEQ_5:41 .= S-bound L~f by PSCOMP_1:114; (M1/.len M1)`2 = (f/.i2)`2 by A413,A415,SPRECT_2:13 .= (N-max L~f)`2 by A414,FINSEQ_5:41 .= N-bound L~f by PSCOMP_1:94; then A428: M1 is_a_v.c._for f by A426,A427,SPRECT_2:def 3; A429: (RB2/.1)`1 = W-bound L~f by A93,A289,A411,PSCOMP_1:74; (NE-corner L~f)`1 = E-bound L~f & (SE-corner L~f)`1 = E-bound L~f by PSCOMP_1:76,78; then (RB2/.len RB2)`1 = E-bound L~f by A288,A409,GOBOARD7:5; then RB2 is_a_h.c._for f by A292,A429,SPRECT_2:def 2; then A430: L~M1 meets L~RB2 by A425,A428,SPRECT_2:33; i3 <= len f by A416,FINSEQ_3:27; then i1 < len f by A418,AXIOMS:22; then L~mid(f,i2,i1) misses LSeg(u3,p2) by A257,A419,A422; then L~M1 misses LSeg(u3,p2) by A424,SPPOL_2:22; then L~M1 meets L~Red2 by A290,A430,XBOOLE_1:70; then A431: L~M1 meets L~Red by A277,XBOOLE_1:63; L~M1 c= L~f by A413,A415,SPRECT_3:34; hence contradiction by A245,A431,XBOOLE_1:63; suppose that A432: u1 in LSeg(NE-corner L~f,SE-corner L~f) and A433: N-min L~f <> W-max L~f; set i1 = (S-max L~f)..f, i2 = (N-max L~f)..f, i3 = (W-min L~f)..f; A434: S-max L~f in rng f by SPRECT_2:46; then A435: i1 in dom f by FINSEQ_4:30; A436: N-max L~f in rng f by SPRECT_2:44; then A437: i2 in dom f by FINSEQ_4:30; A438: W-min L~f in rng f by SPRECT_2:47; then A439: i3 in dom f by FINSEQ_4:30; A440: (S-max L~f)..f < (S-min L~f)..f by A1,SPRECT_2:77; (S-min L~f)..f <= (W-min L~f)..f by A1,SPRECT_2:78; then A441: i3 > i1 by A440,AXIOMS:22; A442: 1 < i2 by A1,SPRECT_2:73; A443: (N-max L~f)..f <= (E-max L~f)..f by A1,SPRECT_2:74; (E-max L~f)..f < (E-min L~f)..f by A1,SPRECT_2:75; then A444: (N-max L~f)..f < (E-min L~f)..f by A443,AXIOMS:22; (E-min L~f)..f <= (S-max L~f)..f by A1,SPRECT_2:76; then A445: i2 < i1 by A444,AXIOMS:22; (W-max L~f)..f < len f by A1,SPRECT_2:81; then A446: (W-max L~f)..f <= len f -' 1 by JORDAN3:12; (W-min L~f)..f < (W-max L~f)..f by A1,A433,SPRECT_2:79; then A447: i3 < len f -' 1 by A446,AXIOMS:22; i1 <= len f by A435,FINSEQ_3:27; then A448: mid(f,i2,i1) is S-Sequence_in_R2 by A442,A445,JORDAN4:52; A449: mid(f,i1,i2) = Rev mid(f,i2,i1) by JORDAN4:30; then reconsider M1 = mid(f,i1,i2) as S-Sequence_in_R2 by A448,SPPOL_2:47; A450: 1 <= i3 by A439,FINSEQ_3:27; then reconsider M3 = mid(f,i3,len f -' 1) as S-Sequence_in_R2 by A58,A447,JORDAN4:51; A451: M3/.len M3 = f/.(len f -' 1) by A57,A439,SPRECT_2:13; L~M3 c= L~f by A57,A439,SPRECT_3:34; then L~M3 misses L~Red by A245,XBOOLE_1:63; then A452: L~M3 misses L~Red2 by A277,XBOOLE_1:63; L~M3 misses LSeg(u3,p2) by A59,A257,A447,A450; then A453: L~RB2 misses L~M3 by A290,A452,XBOOLE_1:70; LSeg(M3/.len M3,p2) /\ L~M3={ M3/.len M3 } by A125,A447,A450,A451; then reconsider M2 = M3^RB2 as S-Sequence_in_R2 by A76,A81,A93,A289,A291,A451,A453,SPRECT_3:38; A454: len M2 >= 2 & len M1 >= 2 by TOPREAL1:def 10; A455: M1 is_in_the_area_of f by A435,A437,SPRECT_2:27; A456: (M1/.1)`2 = (f/.i1)`2 by A435,A437,SPRECT_2:12 .= (S-max L~f)`2 by A434,FINSEQ_5:41 .= S-bound L~f by PSCOMP_1:114; (M1/.len M1)`2 = (f/.i2)`2 by A435,A437,SPRECT_2:13 .= (N-max L~f)`2 by A436,FINSEQ_5:41 .= N-bound L~f by PSCOMP_1:94; then A457: M1 is_a_v.c._for f by A455,A456,SPRECT_2:def 3; mid(f,i3,len f -' 1) is_in_the_area_of f by A57,A439,SPRECT_2:27; then A458: M2 is_in_the_area_of f by A292,SPRECT_2:28; mid(f,i3,len f -' 1) is non empty by A57,A439,SPRECT_2:11; then 1 in dom mid(f,i3,len f -' 1) by FINSEQ_5:6; then A459: (M2/.1)`1 = (mid(f,i3,len f -' 1)/.1)`1 by GROUP_5:95 .= (f/.i3)`1 by A57,A439,SPRECT_2:12 .= (W-min L~f)`1 by A438,FINSEQ_5:41 .= W-bound L~f by PSCOMP_1:84; A460: (NE-corner L~f)`1 = E-bound L~f & (SE-corner L~f)`1 = E-bound L~f by PSCOMP_1:76,78; (M2/.len M2)`1 = (RB2/.len RB2)`1 by SPRECT_3:11 .= E-bound L~f by A288,A432,A460,GOBOARD7:5; then M2 is_a_h.c._for f by A458,A459,SPRECT_2:def 2; then A461: L~M1 meets L~M2 by A454,A457,SPRECT_2:33; A462: L~M2 = L~M3 \/ LSeg(M3/.len M3,p2) \/ L~RB2 by A289,SPPOL_2:23 .= L~M3 \/ (LSeg(M3/.len M3,p2) \/ L~RB2) by XBOOLE_1:4; i3 < len f by A58,A447,NAT_1:38; then i1 < len f by A441,AXIOMS:22; then L~mid(f,i2,i1) misses LSeg(u3,p2) by A257,A442,A445; then A463: L~M1 misses LSeg(u3,p2) by A449,SPPOL_2:22; L~M1 misses L~M3 by A59,A441,A442,A445,A447,SPRECT_2:54; then A464: L~M1 meets L~RB2 \/ LSeg(p2,M3/.len M3) by A461,A462,XBOOLE_1:70; i1 < len f -' 1 by A441,A447,AXIOMS:22; then L~mid(f,i2,i1) misses LSeg(p2,f/.(len f -' 1)) by A103,A442,A445; then L~M1 misses LSeg(p2,f/.(len f -' 1)) by A449,SPPOL_2:22; then L~M1 meets L~RB2 by A451,A464,XBOOLE_1:70; then L~M1 meets L~Red2 by A290,A463,XBOOLE_1:70; then A465: L~M1 meets L~Red by A277,XBOOLE_1:63; L~M1 c= L~f by A435,A437,SPRECT_3:34; hence contradiction by A245,A465,XBOOLE_1:63; suppose A466: u1 in LSeg(SE-corner L~f,SW-corner L~f); set i1 = (E-min L~f)..f, i2 = (W-min L~f)..f; A467: E-min L~f in rng f by SPRECT_2:49; then A468: i1 in dom f by FINSEQ_4:30; A469: W-min L~f in rng f by SPRECT_2:47; then A470: i2 in dom f by FINSEQ_4:30; A471: (N-max L~f)..f > 1 by A1,SPRECT_2:73; (N-max L~f)..f <= (E-max L~f)..f by A1,SPRECT_2:74; then A472: (E-max L~f)..f > 1 by A471,AXIOMS:22; (E-max L~f)..f < (E-min L~f)..f by A1,SPRECT_2:75; then A473: 1 < i1 by A472,AXIOMS:22; A474: (E-min L~f)..f <= (S-max L~f)..f by A1,SPRECT_2:76; (S-max L~f)..f < (S-min L~f)..f by A1,SPRECT_2:77; then A475: (E-min L~f)..f < (S-min L~f)..f by A474,AXIOMS:22; (S-min L~f)..f <= (W-min L~f)..f by A1,SPRECT_2:78; then A476: i1 < i2 by A475,AXIOMS:22; A477: i2 <= len f by A470,FINSEQ_3:27; then A478: mid(f,i1,i2) is S-Sequence_in_R2 by A473,A476,JORDAN4:52; A479: mid(f,i2,i1) = Rev mid(f,i1,i2) by JORDAN4:30; then reconsider M2 = mid(f,i2,i1) as S-Sequence_in_R2 by A478,SPPOL_2:47; A480: len M2 >= 2 & len RB1 >= 2 by TOPREAL1:def 10; (SE-corner L~f)`2 = S-bound L~f & (SW-corner L~f)`2 = S-bound L~f by PSCOMP_1:73,79; then (RB1/.1)`2 = S-bound L~f by A293,A466,GOBOARD7:6; then A481: RB1 is_a_v.c._for f by A170,A294,A297,SPRECT_2:def 3; A482: M2 is_in_the_area_of f by A468,A470,SPRECT_2:27; A483: (M2/.len M2)`1 = (f/.i1)`1 by A468,A470,SPRECT_2:13 .= (E-min L~f)`1 by A467,FINSEQ_5:41 .= E-bound L~f by PSCOMP_1:104; (M2/.1)`1 = (f/.i2)`1 by A468,A470,SPRECT_2:12 .= (W-min L~f)`1 by A469,FINSEQ_5:41 .= W-bound L~f by PSCOMP_1:84; then M2 is_a_h.c._for f by A482,A483,SPRECT_2:def 2; then A484: L~RB1 meets L~M2 by A480,A481,SPRECT_2:33; L~mid(f,i1,i2) misses LSeg(u2,p1) by A267,A473,A476,A477; then L~M2 misses LSeg(u2,p1) by A479,SPPOL_2:22; then L~M2 meets L~Red1 by A295,A484,XBOOLE_1:70; then A485: L~M2 meets L~Red by A284,XBOOLE_1:63; L~M2 c= L~f by A468,A470,SPRECT_3:34; hence contradiction by A245,A485,XBOOLE_1:63; suppose A486: u1 in LSeg(SW-corner L~f,NW-corner L~f); set i1 = (S-min L~f)..f, i3 = (E-min L~f)..f; A487: S-min L~f in rng f by SPRECT_2:45; then A488: i1 in dom f by FINSEQ_4:30; A489: E-min L~f in rng f by SPRECT_2:49; then A490: i3 in dom f by FINSEQ_4:30; A491: (E-min L~f)..f <= (S-max L~f)..f by A1,SPRECT_2:76; (S-max L~f)..f < (S-min L~f)..f by A1,SPRECT_2:77; then A492: i3 < i1 by A491,AXIOMS:22; A493: (W-min L~f)..f < len f by A1,SPRECT_2:80; (S-min L~f)..f <= (W-min L~f)..f by A1,SPRECT_2:78; then A494: i1 < len f by A493,AXIOMS:22; A495: (N-max L~f)..f > 1 by A1,SPRECT_2:73; (N-max L~f)..f <= (E-max L~f)..f by A1,SPRECT_2:74; then (E-max L~f)..f > 1 by A495,AXIOMS:22; then A496: (E-max L~f)..f >= 1+1 by NAT_1:38; (E-max L~f)..f < (E-min L~f)..f by A1,SPRECT_2:75; then A497: 2 < i3 by A496,AXIOMS:22; then A498: 2 < i1 by A492,AXIOMS:22; then A499: 1 < i1 by AXIOMS:22; then reconsider M1 = mid(f,i1,len f) as S-Sequence_in_R2 by A494,JORDAN4:52; A500: i3 <= len f by A490,FINSEQ_3:27; then reconsider M3 = mid(f,2,i3) as S-Sequence_in_R2 by A497,JORDAN4:52; A501: M3/.1 = f/.2 by A54,A490,SPRECT_2:12; L~M3 c= L~f by A54,A490,SPRECT_3:34; then L~M3 misses L~Red by A245,XBOOLE_1:63; then A502: L~M3 misses L~Red1 by A284,XBOOLE_1:63; L~M3 misses LSeg(u2,p1) by A267,A497,A500; then A503: L~RB1 misses L~M3 by A295,A502,XBOOLE_1:70; LSeg(p1,M3/.1) /\ L~M3={ M3/.1 } by A190,A497,A500,A501; then reconsider M2 = RB1^M3 as S-Sequence_in_R2 by A78,A170,A294,A296,A501,A503,SPRECT_3:38; A504: len M2 >= 2 & len M1 >= 2 by TOPREAL1:def 10; A505: M1 is_in_the_area_of f by A55,A488,SPRECT_2:27; A506: (M1/.1)`2 = (f/.i1)`2 by A55,A488,SPRECT_2:12 .= (S-min L~f)`2 by A487,FINSEQ_5:41 .= S-bound L~f by PSCOMP_1:114; (M1/.len M1)`2 = (f/.len f)`2 by A55,A488,SPRECT_2:13 .= (f/.1)`2 by FINSEQ_6:def 1 .= N-bound L~f by A1,PSCOMP_1:94; then A507: M1 is_a_v.c._for f by A505,A506,SPRECT_2:def 3; mid(f,2,i3) is_in_the_area_of f by A54,A490,SPRECT_2:27; then A508: M2 is_in_the_area_of f by A297,SPRECT_2:28; A509: (NW-corner L~f)`1 = W-bound L~f & (SW-corner L~f)`1 = W-bound L~f by PSCOMP_1:72,74; 1 in dom RB1 by FINSEQ_5:6; then A510: (M2/.1)`1 = (RB1/.1)`1 by GROUP_5:95 .= W-bound L~f by A293,A486,A509,GOBOARD7:5; (M2/.len M2)`1 = (mid(f,2,i3)/.len mid(f,2,i3))`1 by SPRECT_3:11 .= (f/.i3)`1 by A54,A490,SPRECT_2:13 .= (E-min L~f)`1 by A489,FINSEQ_5:41 .= E-bound L~f by PSCOMP_1:104; then M2 is_a_h.c._for f by A508,A510,SPRECT_2:def 2; then A511: L~M1 meets L~M2 by A504,A507,SPRECT_2:33; A512: L~M2 = L~RB1 \/ LSeg(p1,M3/.1) \/ L~M3 by A294,SPPOL_2:23; A513: L~M1 misses LSeg(u2,p1) by A267,A494,A499; L~M1 misses L~M3 by A492,A494,A497,SPRECT_2:51; then A514: L~M1 meets L~RB1 \/ LSeg(p1,M3/.1) by A511,A512,XBOOLE_1:70; L~M1 misses LSeg(p1,f/.2) by A173,A494,A498; then L~M1 meets L~RB1 by A501,A514,XBOOLE_1:70; then L~M1 meets L~Red1 by A295,A513,XBOOLE_1:70; then A515: L~M1 meets L~Red by A284,XBOOLE_1:63; L~M1 c= L~f by A55,A488,SPRECT_3:34; hence contradiction by A245,A515,XBOOLE_1:63; end; Lm2: for f being non constant standard special_circular_sequence st f/.1 = N-min L~f holds LeftComp f <> RightComp f proof let f be non constant standard special_circular_sequence such that A1: f/.1 = N-min L~f; per cases by REVROT_1:38; suppose f is clockwise_oriented; hence LeftComp f <> RightComp f by A1,Lm1; suppose A2: Rev f is clockwise_oriented; A3: N-min L~Rev f = N-min L~f by SPPOL_2:22 .= Rev f/.len f by A1,FINSEQ_5:68 .= Rev f/.len Rev f by FINSEQ_5:def 3 .= Rev f/.1 by FINSEQ_6:def 1; LeftComp Rev f = RightComp f & RightComp Rev f = LeftComp f by GOBOARD9:26,27; hence LeftComp f <> RightComp f by A2,A3,Lm1; end; theorem for f being non constant standard special_circular_sequence holds LeftComp f <> RightComp f proof let f be non constant standard special_circular_sequence; set g = Rotate(f,N-min L~f); A1: N-min L~f in rng f by SPRECT_2:43; L~f = L~g by REVROT_1:33; then A2: g/.1 = N-min L~g by A1,FINSEQ_6:98; LeftComp g = LeftComp f & RightComp g = RightComp f by REVROT_1:36,37; hence LeftComp f <> RightComp f by A2,Lm2; end;

Back to top