### The Mizar article:

### Graph Theoretical Properties of Arcs in the Plane and Fashoda Meet Theorem

**by****Yatsuka Nakamura**

- Received August 21, 1998
Copyright (c) 1998 Association of Mizar Users

- MML identifier: JGRAPH_1
- [ MML identifier index ]

environ vocabulary GRAPH_1, FINSEQ_1, ORDERS_1, SQUARE_1, ABSVALUE, ARYTM_1, GRAPH_4, FUNCT_1, FINSEQ_4, BOOLE, FUNCT_3, RELAT_1, FINSET_1, CARD_1, EUCLID, TOPREAL1, GOBOARD5, PRE_TOPC, TARSKI, MCART_1, GOBOARD1, GOBOARD4, COMPLEX1, METRIC_1, RLVECT_1, RVSUM_1, PCOMPS_1, COMPTS_1, WEIERSTR, NAT_1, ARYTM_3, BORSUK_1, TOPS_2, ORDINAL2, RCOMP_1, TBSP_1, SUBSET_1, PSCOMP_1, LATTICES, JGRAPH_1, ARYTM; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, ORDINAL1, NUMBERS, XREAL_0, REAL_1, RELAT_1, FUNCT_1, FUNCT_2, FINSEQ_1, FINSEQ_2, FINSET_1, CARD_1, NAT_1, GRAPH_1, STRUCT_0, BINARITH, GRAPH_2, FINSEQ_4, GRAPH_4, TOPREAL1, GOBOARD5, PRE_TOPC, GOBOARD1, GOBOARD4, TOPMETR, PCOMPS_1, COMPTS_1, WEIERSTR, TOPS_2, METRIC_1, TBSP_1, RCOMP_1, ABSVALUE, SQUARE_1, PSCOMP_1, RVSUM_1, TOPRNS_1, EUCLID, FUNCT_3; constructors REAL_1, BINARITH, GRAPH_2, GRAPH_4, GOBOARD5, GOBOARD4, WEIERSTR, TOPS_2, TBSP_1, RCOMP_1, ABSVALUE, SQUARE_1, PSCOMP_1, FUNCT_3, FINSEQ_4, TOPRNS_1, GOBOARD1, JORDAN7; clusters STRUCT_0, RELSET_1, GRAPH_4, EUCLID, PRE_TOPC, BORSUK_1, XREAL_0, FINSEQ_1, METRIC_1, ARYTM_3, MEMBERED; requirements REAL, NUMERALS, BOOLE, SUBSET, ARITHM; definitions TARSKI, XBOOLE_0; theorems AXIOMS, TARSKI, FUNCT_1, FINSEQ_1, NAT_1, JORDAN3, ZFMISC_1, REAL_1, REAL_2, SPPOL_1, FINSEQ_4, CARD_1, FINSET_1, GRAPH_1, GRAPH_2, BINARITH, FUNCT_3, RELAT_1, GRAPH_4, FINSEQ_3, TOPREAL1, GOBOARD1, GOBOARD5, GOBOARD4, TOPREAL3, SQUARE_1, EUCLID, JORDAN7, UNIFORM1, WEIERSTR, JORDAN5A, TOPS_2, PRE_TOPC, TBSP_1, HEINE, COMPTS_1, TOPMETR, PSCOMP_1, TOPREAL5, BORSUK_1, RCOMP_1, ABSVALUE, METRIC_1, RVSUM_1, FINSEQ_2, FUNCT_2, GROUP_4, SCMFSA_7, JORDAN6, XBOOLE_0, XBOOLE_1, TOPRNS_1, JORDAN1, XCMPLX_0, XCMPLX_1; schemes FINSEQ_1; begin :: GRAPHS BY CARTESIAN PRODUCT reserve x,y for set; reserve G for Graph; reserve vs,vs' for FinSequence of the Vertices of G; reserve IT for oriented Chain of G; reserve n,m,k,i,j for Nat; reserve r,r1,r2 for Real; canceled; theorem Th2: sqrt(r1^2+r2^2)<=abs(r1)+abs(r2) proof A1: r1^2>=0 by SQUARE_1:72; r2^2>=0 by SQUARE_1:72; then r1^2+r2^2>=0+0 by A1,REAL_1:55; then A2:(sqrt(r1^2+r2^2))^2=r1^2+r2^2 by SQUARE_1:def 4; A3:(abs(r1)+abs(r2))^2=(abs(r1))^2+2*(abs(r1))*(abs(r2))+(abs(r2))^2 by SQUARE_1:63; (abs(r1))^2=r1^2 by SQUARE_1:62; then A4:(abs(r1)+abs(r2))^2 - (sqrt(r1^2+r2^2))^2= r1^2+2*(abs(r1))*(abs(r2))+r2^2-(r1^2+r2^2) by A2,A3,SQUARE_1:62 .=2*(abs(r1))*(abs(r2))+r1^2+r2^2-r1^2-r2^2 by XCMPLX_1:36 .=2*(abs(r1))*(abs(r2))+r2^2+r1^2-r1^2-r2^2 by XCMPLX_1:1 .=2*(abs(r1))*(abs(r2))+r2^2-r2^2 by XCMPLX_1:26 .=2*(abs(r1))*(abs(r2)) by XCMPLX_1:26; A5:abs(r1)>=0 by ABSVALUE:5; then A6: 2*abs(r1)>=0 by SQUARE_1:19; A7:abs(r2)>=0 by ABSVALUE:5; then 2*abs(r1)*abs(r2)>=0 by A6,SQUARE_1:19; then A8:(abs(r1)+abs(r2))^2>= (sqrt(r1^2+r2^2))^2 by A4,REAL_2:105; (sqrt(r1^2+r2^2))^2>=0 by SQUARE_1:72; then A9:sqrt((abs(r1)+abs(r2))^2)>=sqrt((sqrt(r1^2+r2^2))^2) by A8,SQUARE_1: 94; abs(r1)+abs(r2)>=0+0 by A5,A7,REAL_1:55; hence thesis by A2,A9,SQUARE_1:89; end; theorem Th3: abs(r1)<=sqrt(r1^2+r2^2) & abs(r2)<=sqrt(r1^2+r2^2) proof A1: r1^2>=0 by SQUARE_1:72; A2:r2^2>=0 by SQUARE_1:72; then r1^2+0<= r1^2+r2^2 by AXIOMS:24; then sqrt(r1^2)<= sqrt(r1^2+r2^2) by A1,SQUARE_1:94; hence abs(r1)<= sqrt(r1^2+r2^2) by SQUARE_1:91; 0+r2^2<= r1^2+r2^2 by A1,REAL_1:55; then sqrt(r2^2)<= sqrt(r1^2+r2^2) by A2,SQUARE_1:94; hence thesis by SQUARE_1:91; end; theorem Th4: for vs st IT is Simple & vs is_oriented_vertex_seq_of IT holds for n,m st 1<=n & n<m & m<=len vs & vs.n=vs.m holds n=1 & m=len vs proof let vs;assume A1:IT is Simple & vs is_oriented_vertex_seq_of IT; then consider vs' such that A2: vs' is_oriented_vertex_seq_of IT & (for n,m st 1<=n & n<m & m<=len vs' & vs'.n=vs'.m holds n=1 & m=len vs') by GRAPH_4:def 7; A3:len vs = len IT + 1 & for n st 1<=n & n<=len IT holds IT.n orientedly_joins vs/.n, vs/.(n+1) by A1,GRAPH_4:def 5; per cases; suppose IT<> {}; then vs=vs' by A1,A2,GRAPH_4:11; hence thesis by A2; suppose IT= {}; then len IT=0 by FINSEQ_1:25; hence thesis by A3,AXIOMS:22; end; definition let X be set; func PGraph(X) -> MultiGraphStruct equals :Def1: MultiGraphStruct(#X,[:X,X:],pr1(X,X),pr2(X,X)#); coherence; end; theorem Th5:for X being non empty set holds PGraph(X) is Graph proof let X be non empty set; PGraph(X)=MultiGraphStruct(#X,[:X,X:],pr1(X,X),pr2(X,X)#) by Def1; hence thesis by GRAPH_1:def 1; end; theorem Th6:for X being set holds the Vertices of PGraph(X)=X proof let X be set; PGraph(X)=MultiGraphStruct(#X,[:X,X:],pr1(X,X),pr2(X,X)#) by Def1; hence thesis; end; definition let f be FinSequence; func PairF(f) -> FinSequence means :Def2: len it=len f-'1 & for i being Nat st 1<=i & i<len f holds it.i=[f.i,f.(i+1)]; existence proof deffunc F(Nat)=[f.$1,f.($1+1)]; ex p being FinSequence st len p = len f-'1 & for k being Nat st k in Seg (len f-'1) holds p.k=F(k) from SeqLambda; then consider p being FinSequence such that A1: len p = len f-'1 & for k being Nat st k in Seg (len f-'1) holds p.k=[f.k,f.(k+1)]; for i being Nat st 1<=i & i<len f holds p.i=[f.i,f.(i+1)] proof let i be Nat;assume A2:1<=i & i<len f; then len f>1 by AXIOMS:22; then A3:len f-1=len f-'1 by SCMFSA_7:3; i+1<=len f by A2,NAT_1:38; then i+1-1<=len f-1 by REAL_1:49; then i<=len f-'1 by A3,XCMPLX_1:26; then i in Seg (len f-'1) by A2,FINSEQ_1:3; hence thesis by A1; end; hence thesis by A1; end; uniqueness proof let g1,g2 be FinSequence; assume A4:(len g1=len f-'1 & for i being Nat st 1<=i & i<len f holds g1.i=[f.i,f.(i+1)])& (len g2=len f-'1 & for i being Nat st 1<=i & i<len f holds g2.i=[f.i,f.(i+1)]); per cases; suppose A5:len f>=1; for j being Nat st 1<=j & j<=len g1 holds g1.j=g2.j proof let j be Nat;assume A6:1<=j & j<=len g1; len f<len f+1 by NAT_1:38; then len f-1<len f by REAL_1:84; then len g1<len f by A4,A5,SCMFSA_7:3; then A7:j<len f by A6,AXIOMS:22; then g1.j=[f.j,f.(j+1)] by A4,A6; hence thesis by A4,A6,A7; end; hence thesis by A4,FINSEQ_1:18; suppose len f<1; then len f+1<=1 by NAT_1:38; then len f+1-1<=1-1 by REAL_1:49; then len f<=0 by XCMPLX_1:26; then A8:len f=0 by NAT_1:18; 0-1<0; then A9:len g1=0 & len g2=0 by A4,A8,BINARITH:def 3; then g1= {} by FINSEQ_1:25; hence thesis by A9,FINSEQ_1:25; end; end; reserve X for non empty set; definition let X be non empty set; cluster PGraph(X) -> Graph-like; coherence by Th5; end; theorem Th7:for f being FinSequence of X holds f is FinSequence of the Vertices of PGraph(X) proof let f be FinSequence of X; PGraph(X)=MultiGraphStruct(#X,[:X,X:],pr1(X,X),pr2(X,X)#) by Def1; hence thesis; end; theorem Th8:for f being FinSequence of X holds PairF(f) is FinSequence of the Edges of PGraph(X) proof let f be FinSequence of X; A1:rng PairF(f) c= [:X,X:] proof let y be set;assume y in rng PairF(f); then consider x being set such that A2:x in dom PairF(f) & y=(PairF(f)).x by FUNCT_1:def 5; A3:x in Seg len PairF(f) by A2,FINSEQ_1:def 3; reconsider n=x as Nat by A2; A4:1<=n & n<=len PairF(f) by A3,FINSEQ_1:3; A5:len PairF(f)=len f-'1 by Def2; then 1<=len f-'1 by A4,AXIOMS:22; then A6:len f-'1=len f-1 by JORDAN3:1; len f-'1<len f-'1+1 by NAT_1:38; then len f-'1<len f by A6,XCMPLX_1:27; then A7:n<len f by A4,A5,AXIOMS:22; then A8:(PairF(f)).n =[f.n,f.(n+1)] by A4,Def2; A9:rng f c= X by FINSEQ_1:def 4; n in dom f by A4,A7,FINSEQ_3:27; then A10: f.n in rng f by FUNCT_1:def 5; A11:1<n+1 by A4,NAT_1:38; n+1<=len f by A7,NAT_1:38; then n+1 in dom f by A11,FINSEQ_3:27; then f.(n+1) in rng f by FUNCT_1:def 5; hence y in [:X,X:] by A2,A8,A9,A10,ZFMISC_1:def 2; end; PGraph(X)=MultiGraphStruct(#X,[:X,X:],pr1(X,X),pr2(X,X)#) by Def1; hence thesis by A1,FINSEQ_1:def 4; end; definition let X be non empty set,f be FinSequence of X; redefine func PairF(f) -> FinSequence of the Edges of PGraph(X); coherence by Th8; end; theorem Th9:for n being Nat,f being FinSequence of X st 1 <= n & n <= len PairF(f) holds (PairF(f)).n in the Edges of PGraph(X) proof let n be Nat,f be FinSequence of X; assume A1:1 <= n & n <= len PairF(f); A2:len PairF(f)=len f-'1 by Def2; then 1<=len f-'1 by A1,AXIOMS:22; then A3:len f-'1=len f-1 by JORDAN3:1; len f-'1<len f-'1+1 by NAT_1:38; then len f-'1<len f by A3,XCMPLX_1:27; then A4:n<len f by A1,A2,AXIOMS:22; then A5:(PairF(f)).n =[f.n,f.(n+1)] by A1,Def2; A6:rng f c= X by FINSEQ_1:def 4; n in dom f by A1,A4,FINSEQ_3:27; then A7:f.n in rng f by FUNCT_1:def 5; A8:1<n+1 by A1,NAT_1:38; n+1<=len f by A4,NAT_1:38; then n+1 in dom f by A8,FINSEQ_3:27; then A9:f.(n+1) in rng f by FUNCT_1:def 5; PGraph(X)=MultiGraphStruct(#X,[:X,X:],pr1(X,X),pr2(X,X)#) by Def1; hence thesis by A5,A6,A7,A9,ZFMISC_1:def 2; end; theorem Th10:for f being FinSequence of X holds PairF(f) is oriented Chain of PGraph(X) proof let f be FinSequence of X; A1: now per cases; case len f>=1; then len f-1+1=len f-'1+1 by SCMFSA_7:3; then len f=len f-'1+1 by XCMPLX_1:27; then A2:len f = len PairF(f) + 1 by Def2; A3:for n st 1 <= n & n <= len f holds f.n in the Vertices of PGraph(X) proof let n;assume A4:1 <= n & n <= len f; A5:rng f c= X by FINSEQ_1:def 4; n in dom f by A4,FINSEQ_3:27; then A6: f.n in rng f by FUNCT_1:def 5; PGraph(X)=MultiGraphStruct(#X,[:X,X:],pr1(X,X),pr2(X,X)#) by Def1; hence thesis by A5,A6; end; for n st 1 <= n & n <= len PairF(f) ex x', y' being Element of the Vertices of PGraph(X) st x' = f.n & y' = f.(n+1) & (PairF(f)).n joins x', y' proof let n;assume A7:1 <= n & n <= len PairF(f); A8:len PairF(f)=len f-'1 by Def2; then 1<=len f-'1 by A7,AXIOMS:22; then A9:len f-'1=len f-1 by JORDAN3:1; len f-'1<len f-'1+1 by NAT_1:38; then len f-'1<len f by A9,XCMPLX_1:27; then A10:n<len f by A7,A8,AXIOMS:22; A11:rng f c= X by FINSEQ_1:def 4; n in dom f by A7,A10,FINSEQ_3:27; then A12: f.n in rng f by FUNCT_1:def 5; A13:1<n+1 by A7,NAT_1:38; n+1<=len f by A10,NAT_1:38; then n+1 in dom f by A13,FINSEQ_3:27; then A14: f.(n+1) in rng f by FUNCT_1:def 5; A15:PGraph(X)=MultiGraphStruct(#X,[:X,X:],pr1(X,X),pr2(X,X)#) by Def1; then reconsider a=f.n,b=f.(n+1) as Element of the Vertices of PGraph(X ) by A11,A12,A14; A16:(pr1(X,X)).([f.n,f.(n+1)])=f.n by A11,A12,A14,FUNCT_3:def 5; (pr2(X,X)).([f.n,f.(n+1)])=f.(n+1) by A11,A12,A14,FUNCT_3:def 6; then ((the Source of PGraph(X)).((PairF(f)).n) = a & (the Target of PGraph(X)).((PairF(f)).n) = b) by A7,A10,A15,A16,Def2; then (PairF(f)).n joins a, b by GRAPH_1:def 9; hence thesis; end; then (for n st 1 <= n & n <= len PairF(f) holds (PairF(f)).n in the Edges of PGraph(X)) & ex p being FinSequence st len p = len PairF(f) + 1 & (for n st 1 <= n & n <= len p holds p.n in the Vertices of PGraph(X)) & for n st 1 <= n & n <= len PairF(f) ex x', y' being Element of the Vertices of PGraph(X) st x' = p.n & y' = p.(n+1) & (PairF(f)).n joins x', y' by A2,A3,Th9; hence PairF(f) is Chain of PGraph(X) by GRAPH_1:def 11; case len f<1; then len f+1<=1 by NAT_1:38; then len f+1-1<=1-1 by REAL_1:49; then len f<=0 by XCMPLX_1:26; then A17:len f=0 by NAT_1:18; 0-1<0; then len f-'1=0 by A17,BINARITH:def 3; then len PairF(f)=0 by Def2; then PairF(f)= {} by FINSEQ_1:25; hence PairF(f) is oriented Chain of PGraph(X) by GRAPH_1:14; end; for n st 1 <= n & n < len PairF(f) holds (the Source of PGraph(X)).((PairF(f)).(n+1)) = (the Target of PGraph(X)).((PairF(f)).n) proof let n;assume A18:1 <= n & n < len PairF(f); then A19:n+1<=len PairF(f) by NAT_1:38; A20:len PairF(f)=len f-'1 by Def2; then 1<=len f-'1 by A18,AXIOMS:22; then A21:len f-'1=len f-1 by JORDAN3:1; len f-'1<len f-'1+1 by NAT_1:38; then A22:len f-'1<len f by A21,XCMPLX_1:27; then A23:n<len f by A18,A20,AXIOMS:22; A24:n+1<len f by A19,A20,A22,AXIOMS:22; A25:rng f c= X by FINSEQ_1:def 4; n in dom f by A18,A23,FINSEQ_3:27; then A26: f.n in rng f by FUNCT_1:def 5; A27:1<n+1 by A18,NAT_1:38; n+1<=len f by A23,NAT_1:38; then n+1 in dom f by A27,FINSEQ_3:27; then A28: f.(n+1) in rng f by FUNCT_1:def 5; A29:PGraph(X)=MultiGraphStruct(#X,[:X,X:],pr1(X,X),pr2(X,X)#) by Def1; then reconsider a=f.n,b=f.(n+1) as Element of the Vertices of PGraph(X) by A25,A26,A28; A30:(pr1(X,X)).([f.n,f.(n+1)])=f.n by A25,A26,A28,FUNCT_3:def 5; (pr2(X,X)).([f.n,f.(n+1)])=f.(n+1) by A25,A26,A28,FUNCT_3:def 6; then A31:((the Source of PGraph(X)).((PairF(f)).n) = a & (the Target of PGraph(X)).((PairF(f)).n) = b) by A18,A23,A29,A30,Def2; A32:rng f c= X by FINSEQ_1:def 4; n+1 in dom f by A24,A27,FINSEQ_3:27; then A33: f.(n+1) in rng f by FUNCT_1:def 5; A34:1<n+1+1 by A27,NAT_1:38; n+1+1<=len f by A24,NAT_1:38; then n+1+1 in dom f by A34,FINSEQ_3:27; then A35: f.(n+1+1) in rng f by FUNCT_1:def 5; A36:PGraph(X)=MultiGraphStruct(#X,[:X,X:],pr1(X,X),pr2(X,X)#) by Def1; (pr1(X,X)).([f.(n+1),f.((n+1)+1)])=f.(n+1) by A32,A33,A35,FUNCT_3:def 5; hence thesis by A24,A27,A31,A36,Def2; end; hence thesis by A1,GRAPH_1:def 12; end; definition let X be non empty set,f be FinSequence of X; redefine func PairF(f) -> oriented Chain of PGraph(X); coherence by Th10; end; theorem Th11:for f being FinSequence of X, f1 being FinSequence of the Vertices of PGraph(X) st len f>=1 & f=f1 holds f1 is_oriented_vertex_seq_of PairF(f) proof let f be FinSequence of X, f1 be FinSequence of the Vertices of PGraph(X); assume A1:len f>=1 & f=f1; then len f-'1=len f-1 by SCMFSA_7:3; then len f-1+1=len PairF(f)+1 by Def2; then A2:len f1=len PairF(f)+1 by A1,XCMPLX_1:27; for n st 1<=n & n<=len PairF(f) holds ((PairF(f)).n) orientedly_joins f1/.n, f1/.(n+1) proof let n;assume A3: 1<=n & n<=len PairF(f); A4:len PairF(f)=len f-'1 by Def2; then 1<=len f-'1 by A3,AXIOMS:22; then A5:len f-'1=len f-1 by JORDAN3:1; len f-'1<len f-'1+1 by NAT_1:38; then len f-'1<len f by A5,XCMPLX_1:27; then A6:n<len f by A3,A4,AXIOMS:22; A7:rng f c= X by FINSEQ_1:def 4; A8:n in dom f by A3,A6,FINSEQ_3:27; then A9: f.n in rng f by FUNCT_1:def 5; A10:1<n+1 by A3,NAT_1:38; n+1<=len f by A6,NAT_1:38; then A11:n+1 in dom f by A10,FINSEQ_3:27; then A12: f.(n+1) in rng f by FUNCT_1:def 5; A13:PGraph(X)=MultiGraphStruct(#X,[:X,X:],pr1(X,X),pr2(X,X)#) by Def1; A14:f1/.n=f1.n by A1,A8,FINSEQ_4:def 4; A15:f1/.(n+1)=f1.(n+1) by A1,A11,FINSEQ_4:def 4; A16:(pr1(X,X)).([f.n,f.(n+1)])=f.n by A7,A9,A12,FUNCT_3:def 5; (pr2(X,X)).([f.n,f.(n+1)])=f.(n+1) by A7,A9,A12,FUNCT_3:def 6; then (the Source of PGraph(X)).((PairF(f)).n) = f1/.n & (the Target of PGraph(X)).((PairF(f)).n) = f1/.(n+1) by A1,A3,A6,A13,A14,A15,A16,Def2; hence thesis by GRAPH_4:def 1; end; hence thesis by A2,GRAPH_4:def 5; end; begin :: SHORTCUTS OF FINITE SEQUENCES IN PLANE definition let X be non empty set,f,g be FinSequence of X; pred g is_Shortcut_of f means :Def3: f.1=g.1 & f.len f=g.len g & ex fc being FinSubsequence of PairF(f),fvs being FinSubsequence of f, sc being oriented simple Chain of PGraph(X), g1 being FinSequence of the Vertices of PGraph(X) st Seq fc = sc & Seq fvs = g & g1=g & g1 is_oriented_vertex_seq_of sc; end; theorem Th12:for f,g being FinSequence of X st g is_Shortcut_of f holds 1<=len g & len g <= len f proof let f,g be FinSequence of X;assume g is_Shortcut_of f; then consider fc being FinSubsequence of PairF(f),fvs being FinSubsequence of f, sc being oriented simple Chain of PGraph(X), g1 being FinSequence of the Vertices of PGraph(X) such that A1:Seq fc = sc & Seq fvs = g & g1=g & g1 is_oriented_vertex_seq_of sc by Def3; A2:len g1 = len sc + 1 by A1,GRAPH_4:def 5; fvs c= f by GRAPH_2:def 5; then A3:dom fvs c= dom f by RELAT_1:25; A4:g=fvs*Sgm(dom fvs) by A1,FINSEQ_1:def 14; rng Sgm(dom fvs) c= dom fvs by FINSEQ_1:71; then A5:dom g=dom (Sgm(dom fvs)) by A4,RELAT_1:46; A6:Seg len f is finite set; then dom f is finite set by FINSEQ_1:def 3; then reconsider dfvs=dom fvs as finite set by A3,FINSET_1:13; reconsider df=dom f as finite set by A6,FINSEQ_1:def 3; A7:dom fvs c=Seg len f by A3,FINSEQ_1:def 3; A8:len g=len Sgm(dom fvs) by A5,FINSEQ_3:31 .=card dfvs by A7,FINSEQ_3:44; card df=card (Seg len f) by FINSEQ_1:def 3 .=len f by FINSEQ_1:78; hence thesis by A1,A2,A3,A8,CARD_1:80,NAT_1:37; end; theorem Th13:for f being FinSequence of X st len f>=1 holds ex g being FinSequence of X st g is_Shortcut_of f proof let f be FinSequence of X;assume A1: len f>=1; reconsider f1=f as FinSequence of the Vertices of PGraph(X) by Th7; f1 is_oriented_vertex_seq_of PairF(f) by A1,Th11; then consider fc being FinSubsequence of PairF(f), fvs being FinSubsequence of f1, sc being oriented simple Chain of PGraph(X), vs1 being FinSequence of the Vertices of PGraph(X) such that A2: Seq fc = sc & Seq fvs = vs1 & vs1 is_oriented_vertex_seq_of sc & f1.1 = vs1.1 & f1.len f1 = vs1.len vs1 by GRAPH_4:23; A3:PGraph(X)=MultiGraphStruct(#X,[:X,X:],pr1(X,X),pr2(X,X)#) by Def1; then reconsider g1=vs1 as FinSequence of X; g1 is_Shortcut_of f by A2,A3,Def3; hence thesis; end; theorem Th14:for f,g being FinSequence of X st g is_Shortcut_of f holds rng PairF(g) c= rng PairF(f) proof let f,g be FinSequence of X;assume g is_Shortcut_of f; then consider fc being FinSubsequence of PairF(f), fvs being FinSubsequence of f, sc being oriented simple Chain of PGraph(X), g1 being FinSequence of the Vertices of PGraph(X) such that A1:Seq fc = sc & Seq fvs = g & g1=g & g1 is_oriented_vertex_seq_of sc by Def3; let y be set;assume y in rng PairF(g); then consider x being set such that A2: x in dom PairF(g) & y=(PairF(g)).x by FUNCT_1:def 5; A3:x in Seg len PairF(g) by A2,FINSEQ_1:def 3; reconsider n=x as Nat by A2; A4:1<=n & n<=len PairF(g) by A3,FINSEQ_1:3; A5:len PairF(g)=len g-'1 by Def2; 1<=len PairF(g) by A4,AXIOMS:22; then A6:len g-'1=len g-1 by A5,JORDAN3:1; len g<len g+1 by NAT_1:38; then len g-1<len g by REAL_1:84; then A7:n<len g by A4,A5,A6,AXIOMS:22; then A8:(PairF(g)).x=[g.n,g.(n+1)] by A4,Def2; A9:n+1<=len g by A7,NAT_1:38; A10:fc* Sgm(dom fc)=sc by A1,FINSEQ_1:def 14; rng Sgm(dom fc) = dom fc by FINSEQ_1:71; then A11:rng fc = rng sc by A10,RELAT_1:47; fc c= PairF(f) by GRAPH_2:def 5; then A12:rng sc c= rng PairF(f) by A11,RELAT_1:25; n+1<=len sc+1 by A1,A9,GRAPH_4:def 5; then A13:n<=len sc by REAL_1:53; then A14: (sc.n) orientedly_joins g1/.n, g1/.(n+1) by A1,A4,GRAPH_4:def 5; n in dom sc by A4,A13,FINSEQ_3:27; then sc.n in rng sc by FUNCT_1:def 5; then consider z being set such that A15: z in dom PairF(f) & (PairF(f)).z=sc.n by A12,FUNCT_1:def 5; A16:z in Seg len PairF(f) by A15,FINSEQ_1:def 3; reconsider m=z as Nat by A15; A17: 1<=m & m<=len PairF(f) by A16,FINSEQ_1:3; then A18:1<=m & m<=len f -'1 by Def2; then 1<=len f-'1 by AXIOMS:22; then A19:len f-'1=len f-1 by JORDAN3:1; len f<len f+1 by NAT_1:38; then len f-1<len f by REAL_1:84; then A20:1<=m & m<len f by A18,A19,AXIOMS:22; then A21:[f.m,f.(m+1)]=sc.n by A15,Def2; m in dom f by A20,FINSEQ_3:27; then A22:f.m in rng f by FUNCT_1:def 5; A23:1<m+1 by A17,NAT_1:38; m+1<=len f by A20,NAT_1:38; then m+1 in dom f by A23,FINSEQ_3:27; then A24:f.(m+1) in rng f by FUNCT_1:def 5; A25: rng f c= X by FINSEQ_1:def 4; A26:PGraph(X)=MultiGraphStruct(#X,[:X,X:],pr1(X,X),pr2(X,X)#) by Def1; then A27:(the Source of PGraph(X)).(sc.n)= f.m by A21,A22,A24,A25,FUNCT_3: def 5; (the Target of PGraph(X)).(sc.n)= f.(m+1) by A21,A22,A24,A25,A26,FUNCT_3: def 6; then A28:g1/.n=f.m & g1/.(n+1)=f.(m+1) by A14,A27,GRAPH_4:def 1; 1<=n+1 & n+1<=len g1 by A1,A4,A7,NAT_1:38; then g.n=f.m & g.(n+1)=f.(m+1) by A1,A4,A7,A28,FINSEQ_4:24; hence thesis by A2,A8,A15,A21,FUNCT_1:def 5; end; theorem Th15:for f,g being FinSequence of X st f.1<>f.len f & g is_Shortcut_of f holds g is one-to-one & rng PairF(g) c= rng PairF(f) & g.1=f.1 & g.len g=f.len f proof let f,g be FinSequence of X; assume A1:f.1<>f.len f & g is_Shortcut_of f; then A2: f.1=g.1 & f.len f=g.len g & ex fc being FinSubsequence of PairF(f),fvs being FinSubsequence of f, sc being oriented simple Chain of PGraph(X), g1 being FinSequence of the Vertices of PGraph(X) st Seq fc = sc & Seq fvs = g & g1=g & g1 is_oriented_vertex_seq_of sc by Def3; consider fc being FinSubsequence of PairF(f), fvs being FinSubsequence of f, sc being oriented simple Chain of PGraph(X), g1 being FinSequence of the Vertices of PGraph(X) such that A3:Seq fc = sc & Seq fvs = g & g1=g & g1 is_oriented_vertex_seq_of sc by A1,Def3; sc is Simple by GRAPH_4:20; then consider vs being FinSequence of the Vertices of PGraph(X) such that A4: vs is_oriented_vertex_seq_of sc & (for n,m being Nat st 1<=n & n<m & m<=len vs & vs.n=vs.m holds n=1 & m=len vs) by GRAPH_4:def 7; A5:len g1 = len sc + 1 & for n st 1<=n & n<=len sc holds sc.n orientedly_joins g1/.n,g1/.(n+1) by A3,GRAPH_4:def 5; then 1<=len g1 by NAT_1:29; then 1<len g1 by A1,A2,A3,REAL_1:def 5; then len sc>=1 by A5,NAT_1:38; then len sc>0 by AXIOMS:22; then sc<> {} by FINSEQ_1:25; then A6:g1=vs by A3,A4,GRAPH_4:11; for x,y being set st x in dom g & y in dom g & g.x=g.y holds x=y proof let x,y be set;assume A7: x in dom g & y in dom g & g.x=g.y; then A8:x in Seg len g by FINSEQ_1:def 3; reconsider i1=x as Nat by A7; A9:y in Seg len g by A7,FINSEQ_1:def 3; reconsider i2=y as Nat by A7; A10:1<=i1 & i1<=len g by A8,FINSEQ_1:3; A11:1<=i2 & i2<=len g by A9,FINSEQ_1:3; now assume A12:x<>y; per cases; suppose i1<=i2; then i1<i2 by A12,REAL_1:def 5; then i1=1 & i2=len g by A3,A4,A6,A7,A10,A11; hence contradiction by A1,A2,A7; suppose i1>i2; then i2=1 & i1=len g by A3,A4,A6,A7,A10,A11; hence contradiction by A1,A2,A7; end; hence x=y; end; hence g is one-to-one by FUNCT_1:def 8; thus rng PairF(g) c= rng PairF(f) by A1,Th14; thus thesis by A1,Def3; end; definition let n; let IT be FinSequence of TOP-REAL n; attr IT is nodic means :Def4: for i,j st LSeg(IT,i) meets LSeg(IT,j) holds LSeg(IT,i) /\ LSeg(IT,j)={IT.i} &(IT.i=IT.j or IT.i=IT.(j+1)) or LSeg(IT,i) /\ LSeg(IT,j)={IT.(i+1)} &(IT.(i+1)=IT.j or IT.(i+1)=IT.(j+1)) or LSeg(IT,i)=LSeg(IT,j); end; theorem for f being FinSequence of TOP-REAL 2 st f is s.n.c. holds f is s.c.c. proof let f be FinSequence of TOP-REAL 2; assume f is s.n.c.; then for i,j st i+1 < j & (i > 1 & j < len f or j+1 < len f) holds LSeg(f,i) misses LSeg(f,j) by TOPREAL1:def 9; hence thesis by GOBOARD5:def 4; end; theorem Th17:for f being FinSequence of TOP-REAL 2 st f is s.c.c. & LSeg(f,1) misses LSeg(f,len f -'1) holds f is s.n.c. proof let f be FinSequence of TOP-REAL 2; assume A1:f is s.c.c. & LSeg(f,1) misses LSeg(f,len f -'1); for i,j st i+1 < j holds LSeg(f,i) misses LSeg(f,j) proof let i,j;assume A2:i+1 < j; per cases; suppose len f<>0; then len f>0 by NAT_1:19; then A3: len f>=0+1 by NAT_1:38; now per cases; case A4:1<=i & j+1<=len f; then A5:j<len f by NAT_1:38; now per cases; case A6:i=1 & j+1=len f; then j=len f-1 by XCMPLX_1:26; then LSeg(f,i) misses LSeg(f,j) by A1,A3,A6,SCMFSA_7:3; hence LSeg(f,i) /\ LSeg(f,j) = {} by XBOOLE_0:def 7; case not(i=1 & j+1=len f); then i>1 or j+1<len f by A4,REAL_1:def 5; then LSeg(f,i) misses LSeg(f,j) by A1,A2,A5,GOBOARD5:def 4; hence LSeg(f,i) /\ LSeg(f,j) = {} by XBOOLE_0:def 7; end; hence LSeg(f,i) /\ LSeg(f,j) = {}; case A7:not (1<=i & j+1<=len f); now per cases by A7; case 1>i; then LSeg(f,i)={} by TOPREAL1:def 5; hence LSeg(f,i) /\ LSeg(f,j) = {}; case j+1>len f; then LSeg(f,j)={} by TOPREAL1:def 5; hence LSeg(f,i) /\ LSeg(f,j) = {}; end; hence LSeg(f,i) /\ LSeg(f,j) = {}; end; hence thesis by XBOOLE_0:def 7; suppose A8:len f=0; now per cases; case i>=1; then i>len f by A8,AXIOMS:22; then i+1>len f by NAT_1:38; then LSeg(f,i)={} by TOPREAL1:def 5; hence LSeg(f,i) /\ LSeg(f,j) = {}; case i<1; then LSeg(f,i)={} by TOPREAL1:def 5; hence LSeg(f,i) /\ LSeg(f,j) = {}; end; hence thesis by XBOOLE_0:def 7; end; hence thesis by TOPREAL1:def 9; end; theorem Th18: for f being FinSequence of TOP-REAL 2 st f is nodic & PairF(f) is Simple holds f is s.c.c. proof let f be FinSequence of TOP-REAL 2; assume A1:f is nodic & PairF(f) is Simple; reconsider f1=f as FinSequence of the Vertices of PGraph(the carrier of TOP-REAL 2) by Th6; per cases; suppose len f>=1; then A2:f1 is_oriented_vertex_seq_of PairF(f) by Th11; for i,j st i+1 < j & (i > 1 & j < len f or j+1 < len f) holds LSeg(f,i) misses LSeg(f,j) proof let i,j;assume A3:i+1 < j & (i > 1 & j < len f or j+1 < len f); per cases; suppose A4:i>=1; A5:i<j by A3,NAT_1:38; A6:i+1<j+1 by A3,NAT_1:38; A7:1<=j by A4,A5,AXIOMS:22; A8:j<len f by A3,NAT_1:38; then A9: i+1<len f by A3,AXIOMS:22; A10:i<j+1 by A5,NAT_1:38; A11: j+1<=len f by A3,NAT_1:38; then A12:i<len f by A10,AXIOMS:22; A13:1<i+1 by A4,NAT_1:38; A14:1<j+1 by A7,NAT_1:38; now assume A15:LSeg(f,i) meets LSeg(f,j); now per cases by A1,A15,Def4; case A16:LSeg(f,i) /\ LSeg(f,j)={f.i} &(f.i=f.j or f.i=f.(j+1))& LSeg(f,i)<>LSeg(f,j); now per cases by A16; case f.i=f.j; hence contradiction by A1,A2,A4,A5,A8,Th4; case f.i=f.(j+1); hence contradiction by A1,A2,A3,A4,A10,A11,Th4; end; hence contradiction; case A17:LSeg(f,i) /\ LSeg(f,j)={f.(i+1)} &(f.(i+1)=f.j or f.(i+1)=f.(j+1))&LSeg(f,i)<>LSeg(f,j); now per cases by A17; case f.(i+1)=f.j; hence contradiction by A1,A2,A3,A8,A13,Th4; case f.(i+1)=f.(j+1); hence contradiction by A1,A2,A6,A11,A13,Th4; end; hence contradiction; case LSeg(f,i)=LSeg(f,j); then LSeg(f/.i,f/.(i+1))=LSeg(f,j) by A4,A9,TOPREAL1:def 5; then A18: LSeg(f/.i,f/.(i+1))=LSeg(f/.j,f/.(j+1)) by A7,A11,TOPREAL1:def 5; A19:f/.i=f.i by A4,A12,FINSEQ_4:24; A20:f/.(i+1)=f.(i+1) by A9,A13,FINSEQ_4:24; A21:f/.j=f.j by A7,A8,FINSEQ_4:24; A22:f/.(j+1)=f.(j+1) by A11,A14,FINSEQ_4:24; now per cases by A18,A19,A20,A21,A22,SPPOL_1:25; case f.i=f.j & f.(i+1)=f.(j+1); hence contradiction by A1,A2,A6,A11,A13,Th4; case f.i=f.(j+1) & f.(i+1)=f.j; hence contradiction by A1,A2,A3,A8,A13,Th4; end; hence contradiction; end; hence contradiction; end; hence thesis; suppose i<1; then LSeg(f,i)={} by TOPREAL1:def 5; then LSeg(f,i) /\ LSeg(f,j) = {}; hence thesis by XBOOLE_0:def 7; end; hence thesis by GOBOARD5:def 4; suppose A23:len f<1; for i,j st i+1 < j & (i > 1 & j < len f or j+1 < len f) holds LSeg(f,i) misses LSeg(f,j) proof let i,j;assume i+1 < j & (i > 1 & j < len f or j+1 < len f); per cases; suppose i>=1; then i>len f by A23,AXIOMS:22; then i+1>len f by NAT_1:38; then LSeg(f,i)={} by TOPREAL1:def 5; then LSeg(f,i) /\ LSeg(f,j) = {}; hence thesis by XBOOLE_0:def 7; suppose i<1; then LSeg(f,i)={} by TOPREAL1:def 5; then LSeg(f,i) /\ LSeg(f,j) = {}; hence thesis by XBOOLE_0:def 7; end; hence thesis by GOBOARD5:def 4; end; theorem Th19:for f being FinSequence of TOP-REAL 2 st f is nodic & PairF(f) is Simple & f.1<>f.len f holds f is s.n.c. proof let f be FinSequence of TOP-REAL 2; assume A1:f is nodic & PairF(f) is Simple & f.1<>f.len f; then A2:f is s.c.c. by Th18; reconsider f1=f as FinSequence of the Vertices of PGraph(the carrier of TOP-REAL 2) by Th6; A3:len f-'1<=len f by JORDAN3:7; per cases; suppose A4:len f-'1>2; then A5:len f-'1>1 by AXIOMS:22; then A6:1<len f by JORDAN3:7; len f>=1 by A5,JORDAN3:7; then A7:f1 is_oriented_vertex_seq_of PairF(f) by Th11; A8:1+1<len f by A4,JORDAN3:7; A9:len f-'1=len f-1 by A5,JORDAN3:1; then A10:(len f-'1)+1=len f by XCMPLX_1:27; now assume A11: LSeg(f,1) meets LSeg(f,len f-'1); now per cases by A1,A11,Def4; case LSeg(f,1) /\ LSeg(f,(len f-'1))={f.1} &(f.1=f.(len f-'1) or f.1=f.((len f-'1)+1)); hence contradiction by A1,A3,A5,A7,A9,Th4,XCMPLX_1:27; case A12:LSeg(f,1) /\ LSeg(f,(len f-'1))={f.(1+1)} &(f.(1+1)=f.(len f-'1) or f.(1+1)=f.((len f-'1)+1)); now per cases by A12; case f.(1+1)=f.(len f-'1); hence contradiction by A1,A3,A4,A7,Th4; case f.(1+1)=f.((len f-'1)+1); hence contradiction by A1,A7,A8,A10,Th4; end; hence contradiction; case A13:LSeg(f,1)=LSeg(f,(len f-'1)); A14:1+1<(len f-'1)+1 by A5,REAL_1:53; A15:(len f-'1)<len f by A10,NAT_1:38; A16:1<(len f-'1)+1 by A6,A9,XCMPLX_1:27; A17: (len f-'1)+1<=len f by A9,XCMPLX_1:27; A18:1<len f by A5,JORDAN3:7; LSeg(f/.1,f/.(1+1))=LSeg(f,(len f-'1)) by A8,A13,TOPREAL1:def 5; then A19: LSeg(f/.1,f/.(1+1))=LSeg(f/.(len f-'1),f/.((len f-'1)+1)) by A5,A17,TOPREAL1:def 5; A20:f/.1=f.1 by A18,FINSEQ_4:24; A21:f/.(1+1)=f.(1+1) by A8,FINSEQ_4:24; A22:f/.(len f-'1)=f.(len f-'1) by A5,A15,FINSEQ_4:24; A23:f/.((len f-'1)+1)=f.((len f-'1)+1) by A16,A17,FINSEQ_4:24; now per cases by A19,A20,A21,A22,A23,SPPOL_1:25; case f.1=f.(len f-'1) & f.(1+1)=f.((len f-'1)+1); hence contradiction by A1,A7,A14,A17,Th4; case f.1=f.((len f-'1)+1) & f.(1+1)=f.(len f-'1); hence contradiction by A1,A9,XCMPLX_1:27; end; hence contradiction; end; hence contradiction; end; hence thesis by A2,Th17; suppose A24:len f-'1<=2; for i,j st i+1 < j holds LSeg(f,i) misses LSeg(f,j) proof let i,j;assume A25:i+1 < j; per cases; suppose A26:1<=i & j+1<=len f; i+1+1<j+1 by A25,REAL_1:53; then A27:i+1+1<len f by A26,AXIOMS:22; 1<i+1 by A26,NAT_1:38; then 1+1<i+1+1 by REAL_1:67; then 1+1+1<=i+1+1 by NAT_1:38; then 1+1+1<len f by A27,AXIOMS:22; then A28: 1+1+1-1<len f-1 by REAL_1:54; then 1<len f-1 by AXIOMS:22; hence thesis by A24,A28,JORDAN3:1; suppose A29:1>i or j+1>len f; now per cases by A29; case 1>i; then LSeg(f,i)={} by TOPREAL1:def 5; hence LSeg(f,i) /\ LSeg(f,j) = {}; case j+1>len f; then LSeg(f,j)={} by TOPREAL1:def 5; hence LSeg(f,i) /\ LSeg(f,j) = {}; end; hence thesis by XBOOLE_0:def 7; end; hence thesis by TOPREAL1:def 9; end; theorem Th20:for p1,p2,p3 being Point of TOP-REAL n st (ex x being set st x<>p2 & x in LSeg(p1,p2)/\ LSeg(p2,p3)) holds p1 in LSeg(p2,p3) or p3 in LSeg(p1,p2) proof let p1,p2,p3 be Point of TOP-REAL n; given x being set such that A1:x<>p2 & x in LSeg(p1,p2)/\ LSeg(p2,p3); A2:x in LSeg(p1,p2) & x in LSeg(p2,p3) by A1,XBOOLE_0:def 3; reconsider p=x as Point of TOP-REAL n by A1; p in { (1-r1)*p1 + r1*p2 : 0 <= r1 & r1 <= 1 } by A2,TOPREAL1:def 4; then consider r1 such that A3:p=(1-r1)*p1 + r1*p2 &( 0 <= r1 & r1 <= 1 ); p in { (1-r2)*p2 + r2*p3 : 0 <= r2 & r2 <= 1 } by A2,TOPREAL1:def 4; then consider r2 such that A4:p=(1-r2)*p2 + r2*p3 &( 0 <= r2 & r2 <= 1 ); per cases; suppose A5:r1>=1-r2; now per cases; case A6:r2<>0; r2*p3=(1-r1)*p1 + r1*p2-(1-r2)*p2 by A3,A4,EUCLID:52; then r2*p3=(1-r1)*p1 + (r1*p2-(1-r2)*p2) by EUCLID:49; then r2"*(r2*p3)=r2"*((1-r1)*p1 + (r1-(1-r2))*p2) by EUCLID:54; then (r2"*r2)*p3=r2"*((1-r1)*p1 + (r1-(1-r2))*p2) by EUCLID:34; then 1*p3=r2"*((1-r1)*p1 + (r1-(1-r2))*p2) by A6,XCMPLX_0:def 7; then A7: p3=r2"*((1-r1)*p1 + (r1-(1-r2))*p2) by EUCLID:33 .=r2"*((1-r1)*p1) +r2"*((r1-(1-r2))*p2) by EUCLID:36 .=(r2"*(1-r1))*p1 +r2"*((r1-(1-r2))*p2) by EUCLID:34 .=(r2"*(1-r1))*p1 +(r2"*(r1-(1-r2)))*p2 by EUCLID:34; r2"*(1-r1) +r2"*(r1-(1-r2))=r2"*(1-r1+(r1-(1-r2))) by XCMPLX_1:8 .=r2"*(1-r1+r1-(1-r2)) by XCMPLX_1:29 .=r2"*(1-(1-r2)) by XCMPLX_1:27 .=r2"*(1-1+r2) by XCMPLX_1:37 .=1 by A6,XCMPLX_0:def 7; then A8:r2"*(1-r1)=1-r2"*(r1-(1-r2)) by XCMPLX_1:26; A9:(r1-(1-r2))>=0 by A5,SQUARE_1:12; A10:r2">0 by A4,A6,REAL_1:72; then A11:0<=r2"*(r1-(1-r2)) by A9,REAL_2:121; r1<=1+0 by A3; then r1-1<=0 by REAL_1:86; then r1-1+r2<=0+r2 by AXIOMS:24; then (r1-(1-r2))<=r2 by XCMPLX_1:37; then r2"*(r1-(1-r2))<=r2"*r2 by A10,AXIOMS:25; then r2"*(r1-(1-r2))<=1 by A6,XCMPLX_0:def 7; then p3 in {(1-r)*p1+r*p2:0<=r & r<=1} by A7,A8,A11; hence p1 in LSeg(p2,p3) or p3 in LSeg(p1,p2) by TOPREAL1:def 4; case r2=0; then p=1*p2+0.REAL n by A4,EUCLID:33; then p=p2+0.REAL n by EUCLID:33; hence p1 in LSeg(p2,p3) or p3 in LSeg(p1,p2) by A1,EUCLID:31; end; hence thesis; suppose A12:r1<1-r2; set s1=1-r2; set s2=1-r1; A13:1-s1=1-1+r2 by XCMPLX_1:37 .=r2; A14:1-s2=1-1+r1 by XCMPLX_1:37 .=r1; A15: 0+s1<=1-s1+s1 by A4,A13,AXIOMS:24; 1-s2+s2<=1+s2 by A3,A14,AXIOMS:24; then 1<=1+s2 by XCMPLX_1:27; then A16: 1-1<=s2 by REAL_1:86; now per cases; case A17:s2<>0; s2*p1=(1-s1)*p3 + s1*p2-(1-s2)*p2 by A3,A4,A13,A14,EUCLID:52 .=(1-s1)*p3 + (s1*p2-(1-s2)*p2) by EUCLID:49 .=(1-s1)*p3 + (s1-(1-s2))*p2 by EUCLID:54; then (s2"*s2)*p1=s2"*((1-s1)*p3 + (s1-(1-s2))*p2) by EUCLID:34; then 1*p1=s2"*((1-s1)*p3 + (s1-(1-s2))*p2) by A17,XCMPLX_0:def 7; then p1=s2"*((1-s1)*p3 + (s1-(1-s2))*p2) by EUCLID:33 .=s2"*((1-s1)*p3) +s2"*((s1-(1-s2))*p2) by EUCLID:36 .=(s2"*(1-s1))*p3 +s2"*((s1-(1-s2))*p2) by EUCLID:34; then A18:p1=(s2"*(1-s1))*p3 +(s2"*(s1-(1-s2)))*p2 by EUCLID:34; s2"*(1-s1) +s2"*(s1-(1-s2))=s2"*(1-s1+(s1-(1-s2))) by XCMPLX_1:8 .=s2"*(1-s1+s1-(1-s2)) by XCMPLX_1:29 .=s2"*(1-(1-s2)) by XCMPLX_1:27 .=s2"*(1-1+s2) by XCMPLX_1:37 .=1 by A17,XCMPLX_0:def 7; then A19:s2"*(1-s1)=1-s2"*(s1-(1-s2)) by XCMPLX_1:26; A20:(s1-(1-s2))>=0 by A12,A14,SQUARE_1:12; A21:s2">0 by A16,A17,REAL_1:72; then A22:0<=s2"*(s1-(1-s2)) by A20,REAL_2:121; s1<=1+0 by A15,XCMPLX_1:27; then s1-1<=0 by REAL_1:86; then s1-1+s2<=0+s2 by AXIOMS:24; then (s1-(1-s2))<=s2 by XCMPLX_1:37; then s2"*(s1-(1-s2))<=s2"*s2 by A21,AXIOMS:25; then s2"*(s1-(1-s2))<=1 by A17,XCMPLX_0:def 7; then p1 in {(1-r)*p3+r*p2:0<=r & r<=1} by A18,A19,A22; hence p1 in LSeg(p2,p3) or p3 in LSeg(p1,p2) by TOPREAL1:def 4; case s2=0; then p=1*p2+0.REAL n by A3,A14,EUCLID:33; then p=p2+0.REAL n by EUCLID:33; hence p1 in LSeg(p2,p3) or p3 in LSeg(p1,p2) by A1,EUCLID:31; end; hence thesis; end; theorem Th21:for f being FinSequence of TOP-REAL 2 st f is s.n.c. & LSeg(f,1) /\ LSeg(f,1+1) c= {f/.(1+1)} & LSeg(f,len f-'2) /\ LSeg(f,len f-'1) c= {f/.(len f-'1)} holds f is unfolded proof let f be FinSequence of TOP-REAL 2; assume A1: f is s.n.c. & LSeg(f,1) /\ LSeg(f,1+1) c= {f/.(1+1)} & LSeg(f,len f-'2) /\ LSeg(f,len f-'1) c= {f/.(len f-'1)}; for i st 1 <= i & i + 2 <= len f holds LSeg(f,i) /\ LSeg(f,i+1) = {f/.(i+1)} proof let i;assume A2:1 <= i & i + 2 <= len f; A3:i+1+1=i+(1+1) by XCMPLX_1:1 .=i+2; A4: 1<i+1 by A2,NAT_1:38; A5:i+1<len f by A2,A3,NAT_1:38; then A6:i<len f by NAT_1:38; A7:1<i+1+1 by A4,NAT_1:38; A8:LSeg(f,i)=LSeg(f/.i,f/.(i+1)) by A2,A5,TOPREAL1:def 5; A9:LSeg(f,i+1)=LSeg(f/.(i+1),f/.(i+1+1)) by A2,A3,A4,TOPREAL1:def 5; A10:f/.(i+1) in LSeg(f/.i,f/.(i+1)) by TOPREAL1:6; f/.(i+1) in LSeg(f/.(i+1),f/.(i+1+1)) by TOPREAL1:6; then f/.(i+1) in LSeg(f,i) /\ LSeg(f,i+1) by A8,A9,A10,XBOOLE_0:def 3; then A11:{f/.(i+1)} c= LSeg(f,i) /\ LSeg(f,i+1) by ZFMISC_1:37; per cases; suppose i=1; hence thesis by A1,A11,XBOOLE_0:def 10; suppose A12:i<>1; now per cases; case A13:i+2=len f; then A14:i=len f-2 by XCMPLX_1:26; then A15:len f-2=len f-'2 by A2,JORDAN3:1; A16:i+1=len f-1 by A3,A13,XCMPLX_1:26; then len f-1=len f-'1 by A4,JORDAN3:1; hence LSeg(f,i) /\ LSeg(f,i+1) = {f/.(i+1)} by A1,A11,A14,A15,A16,XBOOLE_0:def 10; case i+2<>len f; then i+2<len f by A2,REAL_1:def 5; then A17:i+1+1+1<=len f by A3,NAT_1:38; 1<i by A2,A12,REAL_1:def 5; then 1+1<=i by NAT_1:38; then A18: 1+1-1<=i-1 by REAL_1:49; now assume A19:LSeg(f,i) /\ LSeg(f,i+1)<>{f/.(i+1)}; A20:f/.(i+1) in LSeg(f,i+1) by A9,TOPREAL1:6; f/.(i+1) in LSeg(f,i) by A8,TOPREAL1:6; then f/.(i+1) in LSeg(f,i) /\ LSeg(f,i+1) by A20,XBOOLE_0:def 3; then {f/.(i+1)} c= LSeg(f,i) /\ LSeg(f,i+1) by ZFMISC_1:37; then not LSeg(f,i) /\ LSeg(f,i+1) c= {f/.(i+1)} by A19,XBOOLE_0:def 10; then consider x being set such that A21: x in LSeg(f,i) /\ LSeg(f,i+1) & not x in {f/.(i+1)} by TARSKI:def 3; A22:x<>f/.(i+1) by A21,TARSKI:def 1; A23:LSeg(f,i+1+1)=LSeg(f/.(i+1+1),f/.(i+1+1+1)) by A7,A17,TOPREAL1:def 5; now per cases by A8,A9,A21,A22,Th20; case A24: f/.i in LSeg(f/.(i+1),f/.(i+1+1)); A25:i-'1=i-1 by A2,SCMFSA_7:3; then A26:i-'1+1=i by XCMPLX_1:27; then i-'1+1<i+1 by NAT_1:38; then LSeg(f,i-'1) misses LSeg(f,i+1) by A1,TOPREAL1:def 9; then A27:LSeg(f,i-'1)/\ LSeg(f,i+1)={} by XBOOLE_0:def 7; LSeg(f,i-'1)=LSeg(f/.(i-'1),f/.(i-'1+1)) by A6,A18,A25,A26,TOPREAL1:def 5; then f/.i in LSeg(f,i-'1) by A26,TOPREAL1:6; hence contradiction by A9,A24,A27,XBOOLE_0:def 3; case A28: f/.(i+1+1) in LSeg(f/.i,f/.(i+1)); i+1<i+1+1 by NAT_1:38; then LSeg(f,i) misses LSeg(f,i+1+1) by A1,TOPREAL1:def 9; then A29:LSeg(f,i)/\ LSeg(f,i+1+1)={} by XBOOLE_0:def 7; f/.(i+1+1) in LSeg(f,i+1+1) by A23,TOPREAL1:6; hence contradiction by A8,A28,A29,XBOOLE_0:def 3; end; hence contradiction; end; hence LSeg(f,i) /\ LSeg(f,i+1) = {f/.(i+1)}; end; hence thesis; end; hence thesis by TOPREAL1:def 8; end; theorem Th22:for f being FinSequence of X holds PairF(f) is Simple & f.1<>f.len f implies f is one-to-one & len f<>1 proof let f be FinSequence of X; thus PairF(f) is Simple & f.1<>f.len f implies f is one-to-one & len f<>1 proof assume A1:PairF(f) is Simple & f.1<>f.len f; then consider vs being FinSequence of the Vertices of PGraph(X) such that A2:vs is_oriented_vertex_seq_of PairF(f) & (for n,m st 1<=n & n<m & m<=len vs & vs.n=vs.m holds n=1 & m=len vs) by GRAPH_4:def 7; reconsider f1=f as FinSequence of the Vertices of PGraph(X) by Th7; now per cases; case A3:len f>=1; now per cases by A3,REAL_1:def 5; case A4: len f>1; A5: f1 is_oriented_vertex_seq_of PairF(f) by A3,Th11; then len f1=len PairF(f)+1 by GRAPH_4:def 5; then 1-1<len PairF(f)+1-1 by A4,REAL_1:54; then 1-1<len PairF(f) by XCMPLX_1:26; then PairF(f) <> {} by FINSEQ_1:25; then A6:vs = f1 by A2,A5,GRAPH_4:11; for x,y st x in dom f & y in dom f & f.x=f.y holds x=y proof let x,y;assume A7:x in dom f & y in dom f & f.x=f.y; A8:dom f=Seg len f by FINSEQ_1:def 3; reconsider i=x,j=y as Nat by A7; A9:1<=i & i<=len f by A7,A8,FINSEQ_1:3; A10:1<=j & j<=len f by A7,A8,FINSEQ_1:3; now assume A11:i<>j; now per cases by A11,REAL_1:def 5; case i<j; then i=1 & j=len f by A2,A6,A7,A9,A10; hence contradiction by A1,A7; case j<i; then j=1 & i=len f by A2,A6,A7,A9,A10; hence contradiction by A1,A7; end; hence contradiction; end; hence thesis; end; hence f is one-to-one by FUNCT_1:def 8; case len f=1; hence f is one-to-one by A1; end; hence f is one-to-one; case len f<1; then len f+1<=1 by NAT_1:38; then len f+1-1<=1-1 by REAL_1:49; then len f<=1-1 by XCMPLX_1:26; then len f=0 by NAT_1:18; then for x,y st x in dom f & y in dom f & f.x=f.y holds x=y by FINSEQ_1:4, def 3; hence f is one-to-one by FUNCT_1:def 8; end; hence thesis by A1; end; end; theorem for f being FinSequence of X holds f is one-to-one & len f>1 implies PairF(f) is Simple & f.1<>f.len f proof let f be FinSequence of X; assume A1:f is one-to-one & len f>1; then A2:1 in dom f by FINSEQ_3:27; A3: len f in dom f by A1,FINSEQ_3:27; reconsider f1=f as FinSequence of the Vertices of PGraph(X) by Th7; A4: f1 is_oriented_vertex_seq_of PairF(f) by A1,Th11; for i,j st 1<=i & i<j & j<=len f1 & f1.i=f1.j holds i=1 & j=len f1 proof let i,j;assume A5:1<=i & i<j & j<=len f1 & f1.i=f1.j; then A6:i<len f by AXIOMS:22; 1<j by A5,AXIOMS:22; then A7: j in Seg len f by A5,FINSEQ_1:3; i in Seg len f by A5,A6,FINSEQ_1:3; then i in dom f & j in dom f by A7,FINSEQ_1:def 3; hence thesis by A1,A5,FUNCT_1:def 8; end; hence thesis by A1,A2,A3,A4,FUNCT_1:def 8,GRAPH_4:def 7; end; theorem for f being FinSequence of TOP-REAL 2 st f is nodic & PairF(f) is Simple & f.1<>f.len f holds f is unfolded proof let f be FinSequence of TOP-REAL 2; assume A1:f is nodic & PairF(f) is Simple & f.1<>f.len f; then A2:f is s.n.c. by Th19; per cases; suppose A3:2<len f; then A4:2+1<=len f by NAT_1:38; A5:1+1<=len f by A3; A6:LSeg(f,1)=LSeg(f/.1,f/.(1+1)) by A3,TOPREAL1:def 5; A7:LSeg(f,2)=LSeg(f/.2,f/.(2+1)) by A4,TOPREAL1:def 5; A8:f is one-to-one by A1,Th22; A9:1<len f by A3,AXIOMS:22; then A10:1 in dom f by FINSEQ_3:27; A11:2 in dom f by A3,FINSEQ_3:27; A12:f.1=f/.1 by A9,FINSEQ_4:24; A13:2+1 in dom f by A4,FINSEQ_3:27; A14:f.2=f/.2 by A3,FINSEQ_4:24; then A15:f.2 in LSeg(f,1) by A5,TOPREAL1:27; A16:now assume A17:LSeg(f,1)=LSeg(f,2); now per cases by A6,A7,A17,SPPOL_1:25; case A18:f/.1=f/.2 & f/.(1+1)=f/.(2+1); A19:f.1=f/.1 by A9,FINSEQ_4:24; f.2=f/.2 by A3,FINSEQ_4:24; hence contradiction by A8,A10,A11,A18,A19,FUNCT_1:def 8; case A20:f/.1=f/.(2+1) & f/.(1+1)=f/.2; f.(2+1)=f/.(2+1) by A4,FINSEQ_4:24; hence contradiction by A8,A10,A12,A13,A20,FUNCT_1:def 8; end; hence contradiction; end; f.2 in LSeg(f,2) by A4,A14,TOPREAL1:27; then LSeg(f,1)/\ LSeg(f,2)<>{} by A15,XBOOLE_0:def 3; then LSeg(f,1) meets LSeg(f,2) by XBOOLE_0:def 7; then LSeg(f,1) /\ LSeg(f,2)={f.1} &(f.1=f.2 or f.1=f.(2+1)) or LSeg(f,1) /\ LSeg(f,2)={f.(1+1)} &(f.(1+1)=f.2 or f.(1+1)=f.(2+1)) by A1,A16,Def4; then A21:LSeg(f,1) /\ LSeg(f,1+1) c= {f/.(1+1)} by A3,A8,A10,A13,FINSEQ_4:24,FUNCT_1:def 8; A22:1+1-1<=len f-1 by A3,REAL_1:49; A23:2+1-2<=len f-2 by A4,REAL_1:49; A24:len f-1=len f-'1 by A9,SCMFSA_7:3; then A25:len f-'1+1=len f by XCMPLX_1:27; A26:len f-'2=len f-2 by A3,SCMFSA_7:3; then A27:len f-'2+1=len f-(1+1-1) by XCMPLX_1:37; len f<len f+1 by NAT_1:38; then A28:len f-1<len f by REAL_1:84; then len f-1-1<len f-1 by REAL_1:54; then A29:len f-(1+1)<len f-1 by XCMPLX_1:36; then A30:len f-2<len f by A28,AXIOMS:22; A31:len f-'2<len f by A26,A28,A29,AXIOMS:22; then A32:(len f-'2)+1<=len f by NAT_1:38; then A33:LSeg(f,(len f-'2))=LSeg(f/.(len f-'2),f/.((len f-'2)+1)) by A23,A26,TOPREAL1:def 5; A34:LSeg(f,(len f-'1))=LSeg(f/.(len f-'1),f/.((len f-'1)+1)) by A22,A24,A25,TOPREAL1:def 5; A35:f.(len f-'2)=f/.(len f-'2) by A23,A26,A31,FINSEQ_4:24; A36:f.(len f-'2)=f/.(len f-'2) by A23,A26,A30,FINSEQ_4:24; A37:now assume A38:LSeg(f,(len f-'2))=LSeg(f,(len f-'1)); A39:(len f-'2) in dom f by A23,A26,A31,FINSEQ_3:27; A40:(len f-'1) in dom f by A22,A24,A28,FINSEQ_3:27; now per cases by A33,A34,A38,SPPOL_1:25; case A41:f/.(len f-'2)=f/.(len f-'1) & f/.((len f-'2)+1)=f/.((len f-'1)+1); f.(len f-'1)=f/.(len f-'1) by A22,A24,A28,FINSEQ_4:24; hence contradiction by A8,A24,A26,A29,A35,A39,A40,A41,FUNCT_1:def 8; case A42:f/.(len f-'2)=f/.((len f-'1)+1) & f/.((len f-'2)+1)=f/.(len f-'1); (len f-'1)+1 in Seg len f by A9,A25,FINSEQ_1:3; then A43:(len f-'1)+1 in dom f by FINSEQ_1:def 3; f.((len f-'1)+1)=f/.((len f-'1)+1) by A9,A25,FINSEQ_4:24; hence contradiction by A8,A25,A26,A28,A29,A36,A39,A42,A43,FUNCT_1:def 8; end; hence contradiction; end; A44:f.(len f-'1)=f/.(len f-'1) by A22,A24,A28,FINSEQ_4:24; then A45:f.(len f-'1) in LSeg(f,(len f-'2)) by A23,A24,A26,A27,A32,TOPREAL1:27; f.(len f-'1) in LSeg(f,(len f-'1)) by A22,A24,A25,A44,TOPREAL1:27; then LSeg(f,(len f-'2))/\ LSeg(f,(len f-'1))<>{} by A45,XBOOLE_0:def 3; then LSeg(f,(len f-'2)) meets LSeg(f,(len f-'1)) by XBOOLE_0:def 7; then A46:LSeg(f,(len f-'2)) /\ LSeg(f,(len f-'1))={f.(len f-'2)} &(f.(len f-'2)=f.(len f-'1) or f.(len f-'2)=f.((len f-'1)+1)) or LSeg(f,(len f-'2)) /\ LSeg(f,(len f-'1))={f.((len f-'2)+1)} &(f.((len f-'2)+1)=f.(len f-'1) or f.((len f-'2)+1)=f.((len f-'1)+1)) by A1,A37,Def4; A47:len f-'2 in dom f by A23,A26,A31,FINSEQ_3:27; len f-'1+1 in dom f by A9,A25,FINSEQ_3:27; then LSeg(f,len f-'2) /\ LSeg(f,len f-'1) c= {f/.(len f-'1)} by A8,A22,A24,A25,A26,A27,A28,A29,A46,A47,FINSEQ_4:24,FUNCT_1:def 8; hence thesis by A2,A21,Th21; suppose A48:len f<=2; for i st 1 <= i & i + 2 <= len f holds LSeg(f,i) /\ LSeg(f,i+1) = {f/.(i+1)} proof let i;assume A49:1 <= i & i + 2 <= len f; then i+2<=2 by A48,AXIOMS:22; then i<=2-2 by REAL_1:84; hence thesis by A49,AXIOMS:22; end; hence thesis by TOPREAL1:def 8; end; theorem Th25: for f,g being FinSequence of TOP-REAL 2,i st g is_Shortcut_of f & 1 <= i & i+1 <= len g holds ex k1 being Nat st 1<=k1 & k1+1<=len f & f/.k1=g/.i & f/.(k1+1)=g/.(i+1) & f.k1=g.i & f.(k1+1)=g.(i+1) proof let f,g be FinSequence of TOP-REAL 2,i;assume that A1:g is_Shortcut_of f and A2:1 <= i & i+1 <= len g; A3:i<=len g by A2,NAT_1:38; A4:1<=i+1 by A2,NAT_1:38; A5: rng PairF(g) c= rng PairF(f) by A1,Th14; A6:i<len g by A2,NAT_1:38; then A7:(PairF(g)).i=[g.i,g.(i+1)] by A2,Def2; A8:len PairF(g)=len g -'1 by Def2; A9:i<=len g-1 by A2,REAL_1:84; then 1<=len g-1 by A2,AXIOMS:22; then len g-'1=len g-1 by JORDAN3:1; then i in dom (PairF(g)) by A2,A8,A9,FINSEQ_3:27; then [ g.i,g.(i+1) ] in rng PairF(g) by A7,FUNCT_1:def 5; then consider x such that A10:x in dom PairF(f) & (PairF(f)).x=[g.i,g.(i+1)] by A5,FUNCT_1:def 5; A11:x in Seg len PairF(f) by A10,FINSEQ_1:def 3; reconsider k=x as Nat by A10; A12:1<=k & k<=len PairF(f) by A11,FINSEQ_1:3; A13:len PairF(f)=len f-'1 by Def2; A14:1<len g by A2,A6,AXIOMS:22; len g<=len f by A1,Th12; then 1<=len f by A14,AXIOMS:22; then len f-'1=len f-1 by SCMFSA_7:3; then k+1<=len f-1+1 by A12,A13,AXIOMS:24; then A15:k+1<=len f by XCMPLX_1:27; then A16:k<len f by NAT_1:38; then [ g.i,g.(i+1)]=[ f.k,f.(k+1) ] by A10,A12,Def2; then A17:g.i=f.k & g.(i+1)=f.(k+1) by ZFMISC_1:33; A18:1<k+1 by A12,NAT_1:38; A19:g/.i=g.i by A2,A3,FINSEQ_4:24; A20:g/.(i+1)=g.(i+1) by A2,A4,FINSEQ_4:24; A21:f/.k=f.k by A12,A16,FINSEQ_4:24; f/.(k+1)=f.(k+1) by A15,A18,FINSEQ_4:24; hence thesis by A12,A15,A17,A19,A20,A21; end; theorem Th26: for f,g being FinSequence of TOP-REAL 2 st g is_Shortcut_of f holds rng g c= rng f proof let f,g be FinSequence of TOP-REAL 2; assume A1: g is_Shortcut_of f; let x;assume x in rng g; then consider z being set such that A2:z in dom g & x=g.z by FUNCT_1:def 5; A3:z in Seg len g by A2,FINSEQ_1:def 3; reconsider i=z as Nat by A2; A4:1<=i & i<=len g by A3,FINSEQ_1:3; per cases; suppose i<len g; then i+1<=len g by NAT_1:38; then consider k1 being Nat such that A5:1<=k1 & k1+1<=len f & f/.k1=g/.i & f/.(k1+1)=g/.(i+1) & f.k1=g.i & f.(k1+1)=g.(i+1) by A1,A4,Th25; k1<len f by A5,NAT_1:38; then k1 in dom f by A5,FINSEQ_3:27; hence x in rng f by A2,A5,FUNCT_1:def 5; suppose i>=len g; then A6:i=len g by A4,AXIOMS:21; now per cases; case A7:1<i; then 1-1<i-1 by REAL_1:54; then 0<i-'1 by A7,SCMFSA_7:3; then A8:0+1<=i-'1 by NAT_1:38; i-'1=i-1 by A7,SCMFSA_7:3; then A9:i-'1+1=len g by A6,XCMPLX_1:27; then consider k1 being Nat such that A10:1<=k1 & k1+1<=len f & f/.k1=g/.(i-'1) & f/.(k1+1)=g/.(i-'1+1) & f.k1=g.(i-'1) & f.(k1+1)=g.(i-'1+1) by A1,A8,Th25; 1<k1+1 by A10,NAT_1:38; then k1+1 in dom f by A10,FINSEQ_3:27; hence x in rng f by A2,A6,A9,A10,FUNCT_1:def 5; case 1>=i; then A11:i=1 by A4,AXIOMS:21; A12:f.1=g.1 by A1,Def3; len g<=len f by A1,Th12; then 1 in dom f by A6,A11,FINSEQ_3:27; hence x in rng f by A2,A11,A12,FUNCT_1:def 5; end; hence thesis; end; theorem Th27: for f,g being FinSequence of TOP-REAL 2 st g is_Shortcut_of f holds L~g c= L~f proof let f,g be FinSequence of TOP-REAL 2; assume A1: g is_Shortcut_of f; let x be set;assume x in L~g; then x in union { LSeg(g,i) : 1 <= i & i+1 <= len g } by TOPREAL1:def 6; then consider y such that A2:x in y & y in { LSeg(g,i) : 1 <= i & i+1 <= len g } by TARSKI:def 4; consider i such that A3: y=LSeg(g,i) &(1<=i & i+1<=len g) by A2; A4:x in LSeg(g/.i,g/.(i+1)) by A2,A3,TOPREAL1:def 5; consider k1 being Nat such that A5: 1<=k1 & k1+1<=len f & f/.k1=g/.i & f/.(k1+1)=g/.(i+1) & f.k1=g.i & f.(k1+1)=g.(i+1) by A1,A3,Th25; A6:x in LSeg(f,k1) by A4,A5,TOPREAL1:def 5; LSeg(f,k1) in {LSeg(f,k):1<=k & k+1<=len f} by A5; then x in union { LSeg(f,k) : 1 <= k & k+1 <= len f } by A6,TARSKI:def 4; hence thesis by TOPREAL1:def 6; end; theorem Th28: for f,g being FinSequence of TOP-REAL 2 st f is special & g is_Shortcut_of f holds g is special proof let f,g be FinSequence of TOP-REAL 2;assume A1:f is special & g is_Shortcut_of f; for i st 1 <= i & i+1 <= len g holds (g/.i)`1 = (g/.(i+1))`1 or (g/.i)`2 = (g/.(i+1))`2 proof let i;assume A2:1 <= i & i+1 <= len g; then A3:i<=len g by NAT_1:38; A4:1<=i+1 by A2,NAT_1:38; A5: rng PairF(g) c= rng PairF(f) by A1,Th14; A6:i<len g by A2,NAT_1:38; then A7:(PairF(g)).i=[g.i,g.(i+1)] by A2,Def2; A8:len PairF(g)=len g -'1 by Def2; A9:i<=len g-1 by A2,REAL_1:84; then 1<=len g-1 by A2,AXIOMS:22; then len g-'1=len g-1 by JORDAN3:1; then i in dom PairF(g) by A2,A8,A9,FINSEQ_3:27; then [ (g.i),(g.(i+1)) ] in rng PairF(g) by A7,FUNCT_1:def 5; then consider x such that A10:x in dom PairF(f) & (PairF(f)).x=[g.i,g.(i+1)] by A5,FUNCT_1:def 5; A11:x in Seg len PairF(f) by A10,FINSEQ_1:def 3; reconsider k=x as Nat by A10; A12:1<=k & k<=len PairF(f) by A11,FINSEQ_1:3; A13:len PairF(f)=len f-'1 by Def2; A14:1<len g by A2,A6,AXIOMS:22; len g<=len f by A1,Th12; then 1<=len f by A14,AXIOMS:22; then len f-'1=len f-1 by SCMFSA_7:3; then k+1<=len f-1+1 by A12,A13,AXIOMS:24; then A15:k+1<=len f by XCMPLX_1:27; then A16:k<len f by NAT_1:38; then (PairF(f)).k=[f.k,f.(k+1)] by A12,Def2; then A17:g.i=f.k & g.(i+1)=f.(k+1) by A10,ZFMISC_1:33; A18:1<k+1 by A12,NAT_1:38; A19:g/.i=g.i by A2,A3,FINSEQ_4:24; A20:g/.(i+1)=g.(i+1) by A2,A4,FINSEQ_4:24; A21:f/.k=f.k by A12,A16,FINSEQ_4:24; f/.(k+1)=f.(k+1) by A15,A18,FINSEQ_4:24; hence thesis by A1,A12,A15,A17,A19,A20,A21,TOPREAL1:def 7; end; hence thesis by TOPREAL1:def 7; end; theorem Th29:for f being FinSequence of TOP-REAL 2 st f is special & 2<=len f & f.1 <> f.len f holds ex g being FinSequence of TOP-REAL 2 st 2<=len g & g is special & g is one-to-one & L~g c= L~f & f.1=g.1 & f.len f=g.len g & rng g c= rng f proof let f be FinSequence of TOP-REAL 2;assume A1:f is special & 2<=len f & f.1 <> f.len f; then 1<=len f by AXIOMS:22; then consider g being FinSequence of TOP-REAL 2 such that A2:g is_Shortcut_of f by Th13; A3:g is one-to-one & rng PairF(g) c= rng PairF(f) & g.1=f.1 & g.len g=f.len f by A1,A2,Th15; A4:g is special by A1,A2,Th28; A5:L~g c= L~f by A2,Th27; A6: rng g c= rng f by A2,Th26; 1<=len g by A2,Th12; then 1<len g by A1,A3,REAL_1:def 5; then 1+1<=len g by NAT_1:38; hence thesis by A3,A4,A5,A6; end; :: Goboard Theorem for general special sequences theorem Th30:for f1,f2 being FinSequence of TOP-REAL 2 st f1 is special & f2 is special & 2<=len f1 & 2<=len f2 & f1.1<>f1.len f1 & f2.1<>f2.len f2 & X_axis(f1) lies_between (X_axis(f1)).1, (X_axis(f1)).(len f1) & X_axis(f2) lies_between (X_axis(f1)).1, (X_axis(f1)).(len f1) & Y_axis(f1) lies_between (Y_axis(f2)).1, (Y_axis(f2)).(len f2) & Y_axis(f2) lies_between (Y_axis(f2)).1, (Y_axis(f2)).(len f2) holds L~f1 meets L~f2 proof let f1,f2 be FinSequence of TOP-REAL 2; assume A1:f1 is special & f2 is special & 2<=len f1 & 2<=len f2 & f1.1<>f1.len f1 & f2.1<>f2.len f2 & X_axis(f1) lies_between (X_axis(f1)).1, (X_axis(f1)).(len f1) & X_axis(f2) lies_between (X_axis(f1)).1, (X_axis(f1)).(len f1) & Y_axis(f1) lies_between (Y_axis(f2)).1, (Y_axis(f2)).(len f2) & Y_axis(f2) lies_between (Y_axis(f2)).1, (Y_axis(f2)).(len f2); then consider g1 being FinSequence of TOP-REAL 2 such that A2: 2<=len g1 & g1 is special & g1 is one-to-one & L~g1 c= L~f1 & f1.1=g1.1 & f1.len f1=g1.len g1 & rng g1 c= rng f1 by Th29; consider g2 being FinSequence of TOP-REAL 2 such that A3: 2<=len g2 & g2 is special & g2 is one-to-one & L~g2 c= L~f2 & f2.1=g2.1 & f2.len f2=g2.len g2 & rng g2 c= rng f2 by A1,Th29; for i st i in dom X_axis(g1) holds (X_axis(g1)).1 <= X_axis(g1).i & X_axis(g1).i <= (X_axis(g1)).(len g1) proof let i;assume A4:i in dom X_axis(g1); A5:len (X_axis(f1)) = len f1 & for j st j in dom X_axis(f1) holds (X_axis(f1)).j = (f1/.j)`1 by GOBOARD1:def 3; A6:len (X_axis(g1)) = len g1 & for j st j in dom X_axis(g1) holds (X_axis(g1)).j = (g1/.j)`1 by GOBOARD1:def 3; A7:1<=len g1 by A2,AXIOMS:22; then A8:1 in dom X_axis(g1) by A6,FINSEQ_3:27; A9:g1/.1=g1.1 by A7,FINSEQ_4:24; A10:1<=len f1 by A1,AXIOMS:22; then A11:g1/.1=f1/.1 by A2,A9,FINSEQ_4:24; g1/.len g1=g1.(len g1) by A7,FINSEQ_4:24; then A12:g1/.len g1=f1/.len f1 by A2,A10,FINSEQ_4:24; A13:1 in dom X_axis(f1) by A5,A10,FINSEQ_3:27; i in Seg len g1 by A4,A6,FINSEQ_1:def 3; then A14:i in dom g1 by FINSEQ_1:def 3; then g1.i in rng g1 by FUNCT_1:def 5; then consider y being set such that A15:y in dom f1 & g1.i=f1.y by A2,FUNCT_1:def 5; reconsider j=y as Nat by A15; f1.j=f1/.j by A15,FINSEQ_4:def 4; then A16:j in dom f1 & g1/.i=f1/.j by A14,A15,FINSEQ_4:def 4; j in Seg len f1 by A15,FINSEQ_1:def 3; then A17:j in dom (X_axis(f1)) by A5,FINSEQ_1:def 3; then A18:(X_axis(f1)).1 <= X_axis(f1).j & X_axis(f1).j <= (X_axis(f1)).(len f1) by A1,GOBOARD4:def 1; A19:X_axis(f1).j =(f1/.j)`1 by A17,GOBOARD1:def 3; len f1 in Seg len f1 by A10,FINSEQ_1:3; then len f1 in dom (X_axis(f1)) by A5,FINSEQ_1:def 3; then A20:(g1/.1)`1<=(g1/.i)`1 & (g1/.i)`1<=(g1/.len g1)`1 by A11,A12,A13,A16,A18,A19,GOBOARD1:def 3; A21:X_axis(g1).i = (g1/.i)`1 by A4,GOBOARD1:def 3; len g1 in Seg len g1 by A7,FINSEQ_1:3; then len g1 in dom (X_axis(g1)) by A6,FINSEQ_1:def 3; hence thesis by A8,A20,A21,GOBOARD1:def 3; end; then A22:X_axis(g1) lies_between (X_axis(g1)).1, (X_axis(g1)).(len g1) by GOBOARD4:def 1; for i st i in dom X_axis(g2) holds (X_axis(g1)).1 <= X_axis(g2).i & X_axis(g2).i <= (X_axis(g1)).(len g1) proof let i;assume A23:i in dom X_axis(g2); A24:len (X_axis(f2)) = len f2 & for j st j in dom X_axis(f2) holds (X_axis(f2)).j = (f2/.j)`1 by GOBOARD1:def 3; A25:len (X_axis(f1)) = len f1 & for j st j in dom X_axis(f1) holds (X_axis(f1)).j = (f1/.j)`1 by GOBOARD1:def 3; A26:len (X_axis(g2)) = len g2 & for j st j in dom X_axis(g2) holds (X_axis(g2)).j = (g2/.j)`1 by GOBOARD1:def 3; A27:len (X_axis(g1)) = len g1 & for j st j in dom X_axis(g1) holds (X_axis(g1)).j = (g1/.j)`1 by GOBOARD1:def 3; A28:1<=len g1 by A2,AXIOMS:22; then 1 in Seg len g1 by FINSEQ_1:3; then A29:1 in dom (X_axis(g1)) by A27,FINSEQ_1:def 3; A30:g1/.1=g1.1 by A28,FINSEQ_4:24; A31:1<=len f1 by A1,AXIOMS:22; then A32:g1/.1=f1/.1 by A2,A30,FINSEQ_4:24; g1/.len g1=g1.(len g1) by A28,FINSEQ_4:24; then A33:g1/.len g1=f1/.len f1 by A2,A31,FINSEQ_4:24; 1 in Seg len f1 by A31,FINSEQ_1:3; then A34:1 in dom (X_axis(f1)) by A25,FINSEQ_1:def 3; i in Seg len g2 by A23,A26,FINSEQ_1:def 3; then A35:i in dom g2 by FINSEQ_1:def 3; then g2.i in rng g2 by FUNCT_1:def 5; then consider y being set such that A36:y in dom f2 & g2.i=f2.y by A3,FUNCT_1:def 5; reconsider j=y as Nat by A36; f2.j=f2/.j by A36,FINSEQ_4:def 4; then A37:j in dom f2 & g2/.i=f2/.j by A35,A36,FINSEQ_4:def 4; j in Seg len f2 by A36,FINSEQ_1:def 3; then A38:j in dom (X_axis(f2)) by A24,FINSEQ_1:def 3; then A39:(X_axis(f1)).1 <= X_axis(f2).j & X_axis(f2).j <= (X_axis(f1)).(len f1) by A1,GOBOARD4:def 1; A40:X_axis(f2).j =(f2/.j)`1 by A38,GOBOARD1:def 3; len f1 in Seg len f1 by A31,FINSEQ_1:3; then len f1 in dom (X_axis(f1)) by A25,FINSEQ_1:def 3; then A41:(g1/.1)`1<=(g2/.i)`1 & (g2/.i)`1<=(g1/.len g1)`1 by A32,A33,A34,A37,A39,A40,GOBOARD1:def 3; A42:X_axis(g2).i =(g2/.i)`1 by A23,GOBOARD1:def 3; len g1 in Seg len g1 by A28,FINSEQ_1:3; then len g1 in dom (X_axis(g1)) by A27,FINSEQ_1:def 3; hence thesis by A29,A41,A42,GOBOARD1:def 3; end; then A43:X_axis(g2) lies_between (X_axis(g1)).1, (X_axis(g1)).(len g1) by GOBOARD4:def 1; for i st i in dom Y_axis(g1) holds (Y_axis(g2)).1 <= Y_axis(g1).i & Y_axis(g1).i <= (Y_axis(g2)).(len g2) proof let i;assume A44:i in dom Y_axis(g1); A45:len (Y_axis(f1)) = len f1 & for j st j in dom Y_axis(f1) holds (Y_axis(f1)).j = (f1/.j)`2 by GOBOARD1:def 4; A46:len (Y_axis(f2)) = len f2 & for j st j in dom Y_axis(f2) holds (Y_axis(f2)).j = (f2/.j)`2 by GOBOARD1:def 4; A47:len (Y_axis(g1)) = len g1 & for j st j in dom Y_axis(g1) holds (Y_axis(g1)).j = (g1/.j)`2 by GOBOARD1:def 4; A48:len (Y_axis(g2)) = len g2 & for j st j in dom Y_axis(g2) holds (Y_axis(g2)).j = (g2/.j)`2 by GOBOARD1:def 4; A49:1<=len g2 by A3,AXIOMS:22; then 1 in Seg len g2 by FINSEQ_1:3; then A50:1 in dom (Y_axis(g2)) by A48,FINSEQ_1:def 3; A51:g2/.1=g2.1 by A49,FINSEQ_4:24; A52:1<=len f2 by A1,AXIOMS:22; then A53:g2/.1=f2/.1 by A3,A51,FINSEQ_4:24; g2/.len g2=g2.(len g2) by A49,FINSEQ_4:24; then A54:g2/.len g2=f2/.len f2 by A3,A52,FINSEQ_4:24; 1 in Seg len f2 by A52,FINSEQ_1:3; then A55:1 in dom (Y_axis(f2)) by A46,FINSEQ_1:def 3; i in Seg len g1 by A44,A47,FINSEQ_1:def 3; then A56:i in dom g1 by FINSEQ_1:def 3; then g1.i in rng g1 by FUNCT_1:def 5; then consider y being set such that A57:y in dom f1 & g1.i=f1.y by A2,FUNCT_1:def 5; reconsider j=y as Nat by A57; f1.j=f1/.j by A57,FINSEQ_4:def 4; then A58:j in dom f1 & g1/.i=f1/.j by A56,A57,FINSEQ_4:def 4; j in Seg len f1 by A57,FINSEQ_1:def 3; then A59:j in dom (Y_axis(f1)) by A45,FINSEQ_1:def 3; then A60:(Y_axis(f2)).1 <= Y_axis(f1).j & Y_axis(f1).j <= (Y_axis(f2)).(len f2) by A1,GOBOARD4:def 1; A61:Y_axis(f1).j =(f1/.j)`2 by A59,GOBOARD1:def 4; len f2 in Seg len f2 by A52,FINSEQ_1:3; then len f2 in dom (Y_axis(f2)) by A46,FINSEQ_1:def 3; then A62:(g2/.1)`2<=(g1/.i)`2 & (g1/.i)`2<=(g2/.len g2)`2 by A53,A54,A55,A58,A60,A61,GOBOARD1:def 4; A63:Y_axis(g1).i = (g1/.i)`2 by A44,GOBOARD1:def 4; len g2 in Seg len g2 by A49,FINSEQ_1:3; then len g2 in dom (Y_axis(g2)) by A48,FINSEQ_1:def 3; hence thesis by A50,A62,A63,GOBOARD1:def 4; end; then A64:Y_axis(g1) lies_between (Y_axis(g2)).1, (Y_axis(g2)).(len g2) by GOBOARD4:def 1; for i st i in dom Y_axis(g2) holds (Y_axis(g2)).1 <= Y_axis(g2).i & Y_axis(g2).i <= (Y_axis(g2)).(len g2) proof let i;assume A65:i in dom Y_axis(g2); A66:len (Y_axis(f2)) = len f2 & for j st j in dom Y_axis(f2) holds (Y_axis(f2)).j = (f2/.j)`2 by GOBOARD1:def 4; A67:len (Y_axis(g2)) = len g2 & for j st j in dom Y_axis(g2) holds (Y_axis(g2)).j = (g2/.j)`2 by GOBOARD1:def 4; A68:1<=len g2 by A3,AXIOMS:22; then 1 in Seg len g2 by FINSEQ_1:3; then A69:1 in dom (Y_axis(g2)) by A67,FINSEQ_1:def 3; A70:g2/.1=g2.1 by A68,FINSEQ_4:24; A71:1<=len f2 by A1,AXIOMS:22; then A72:g2/.1=f2/.1 by A3,A70,FINSEQ_4:24; g2/.len g2=g2.(len g2) by A68,FINSEQ_4:24; then A73:g2/.len g2=f2/.len f2 by A3,A71,FINSEQ_4:24; 1 in Seg len f2 by A71,FINSEQ_1:3; then A74:1 in dom (Y_axis(f2)) by A66,FINSEQ_1:def 3; i in Seg len g2 by A65,A67,FINSEQ_1:def 3; then A75:i in dom g2 by FINSEQ_1:def 3; then g2.i in rng g2 by FUNCT_1:def 5; then consider y being set such that A76:y in dom f2 & g2.i=f2.y by A3,FUNCT_1:def 5; reconsider j=y as Nat by A76; f2.j=f2/.j by A76,FINSEQ_4:def 4; then A77:j in dom f2 & g2/.i=f2/.j by A75,A76,FINSEQ_4:def 4; j in Seg len f2 by A76,FINSEQ_1:def 3; then A78:j in dom (Y_axis(f2)) by A66,FINSEQ_1:def 3; then A79:(Y_axis(f2)).1 <= Y_axis(f2).j & Y_axis(f2).j <= (Y_axis(f2)).(len f2) by A1,GOBOARD4:def 1; A80:Y_axis(f2).j =(f2/.j)`2 by A78,GOBOARD1:def 4; len f2 in Seg len f2 by A71,FINSEQ_1:3; then len f2 in dom (Y_axis(f2)) by A66,FINSEQ_1:def 3; then A81:(g2/.1)`2<=(g2/.i)`2 & (g2/.i)`2<=(g2/.len g2)`2 by A72,A73,A74,A77,A79,A80,GOBOARD1:def 4; A82:Y_axis(g2).i =(g2/.i)`2 by A65,GOBOARD1:def 4; len g2 in Seg len (Y_axis(g2)) by A67,A68,FINSEQ_1:3; then len g2 in dom (Y_axis(g2)) by FINSEQ_1:def 3; hence thesis by A69,A81,A82,GOBOARD1:def 4; end; then A83:Y_axis(g2) lies_between (Y_axis(g2)).1, (Y_axis(g2)).(len g2) by GOBOARD4:def 1; A84:for k being Nat st k in dom g1 & k+1 in dom g1 holds g1/.k <> g1/.(k+1) proof let k be Nat;assume A85:k in dom g1 & k+1 in dom g1; k<k+1 by NAT_1:38; then A86:g1.k<>g1.(k+1) by A2,A85,FUNCT_1:def 8; g1.k=g1/.k by A85,FINSEQ_4:def 4; hence thesis by A85,A86,FINSEQ_4:def 4; end; for k being Nat st k in dom g2 & k+1 in dom g2 holds g2/.k <> g2/.(k+1) proof let k be Nat;assume A87:k in dom g2 & k+1 in dom g2; k<k+1 by NAT_1:38; then A88:g2.k<>g2.(k+1) by A3,A87,FUNCT_1:def 8; g2.k=g2/.k by A87,FINSEQ_4:def 4; hence thesis by A87,A88,FINSEQ_4:def 4; end; then L~g1 meets L~g2 by A2,A3,A22,A43,A64,A83,A84,GOBOARD4:4; then A89: L~g1 /\ L~g2 <> {} by XBOOLE_0:def 7; L~g1 /\ L~g2 c= L~f1 /\ L~f2 by A2,A3,XBOOLE_1:27; then L~f1 /\ L~f2 <> {} by A89,XBOOLE_1:3; hence thesis by XBOOLE_0:def 7; end; begin :: NORM OF POINTS IN TOPREAL n theorem Th31: for a,b,r1,r2 being Real st a<=r1 & r1<=b & a<=r2 & r2<=b holds abs(r1-r2)<=b-a proof let a,b,r1,r2 be Real; assume A1:a<=r1 & r1<=b & a<=r2 & r2<=b; per cases; suppose r1-r2>=0; then A2:abs(r1-r2)=r1-r2 by ABSVALUE:def 1; A3:r1-r2<=b-r2 by A1,REAL_1:49; b-r2<=b-a by A1,REAL_2:106; hence thesis by A2,A3,AXIOMS:22; suppose r1-r2<0; then A4:abs(r1-r2)=-(r1-r2) by ABSVALUE:def 1 .=r2-r1 by XCMPLX_1:143; A5:r2-r1<=b-r1 by A1,REAL_1:49; b-r1<=b-a by A1,REAL_2:106; hence thesis by A4,A5,AXIOMS:22; end; definition let n;let p be Point of TOP-REAL n; redefine func |. p .| means :Def5: for w being Element of REAL n st p=w holds it=|.w.|; compatibility proof let r be Real; thus r = |. p .| implies for w being Element of REAL n st p=w holds r=|.w.| by TOPRNS_1:def 6; reconsider y = p as Element of REAL n by EUCLID:25; assume for w being Element of REAL n st p=w holds r=|.w.|; then r=|.y.|; hence r = |. p .| by TOPRNS_1:def 6; end; end; reserve p,p1,p2 for Point of TOP-REAL n; canceled 13; theorem Th45: for x1,x2 being Point of Euclid n st x1=p1 & x2=p2 holds |.p1 - p2.| = dist(x1,x2) proof let x1,x2 be Point of Euclid n; assume A1:x1=p1 & x2=p2; Euclid n= MetrStruct(#REAL n,Pitag_dist n#) by EUCLID:def 7; then reconsider x1'=x1,x2'=x2 as Element of REAL n; p1-p2=x1'-x2' by A1,EUCLID:def 13; then |.p1-p2.|=|.x1'-x2'.| by Def5; then A2:(Pitag_dist n).(x1',x2')=|.p1-p2.| by EUCLID:def 6; Euclid n = MetrStruct(#REAL n,Pitag_dist n#) by EUCLID:def 7; hence thesis by A2,METRIC_1:def 1; end; theorem Th46: for p being Point of TOP-REAL 2 holds (|.p.|)^2 = (p`1)^2+(p`2)^2 proof let p be Point of TOP-REAL 2; reconsider w=p as Element of REAL 2 by EUCLID:25; A1:|.p.|=|.w.| by Def5; A2:|.w.| = sqrt Sum sqr w by EUCLID:def 5; 0 <= Sum sqr w by RVSUM_1:116; then A3: (|.p.|)^2=Sum sqr w by A1,A2,SQUARE_1:def 4; A4: p`1=w.1 by EUCLID:def 14; A5: p`2=w.2 by EUCLID:def 15; A6:len sqr w =2 by FINSEQ_2:109; A7:(sqr w).1=(p`1)^2 by A4,RVSUM_1:78; (sqr w).2=(p`2)^2 by A5,RVSUM_1:78; then sqr w=<*(p`1)^2,(p`2)^2*> by A6,A7,FINSEQ_1:61; hence thesis by A3,RVSUM_1:107; end; theorem Th47: for p being Point of TOP-REAL 2 holds |.p.| = sqrt((p`1)^2+(p`2)^2) proof let p be Point of TOP-REAL 2; A1:(|.p.|)^2= (p`1)^2+(p`2)^2 by Th46; |.p.| >=0 by TOPRNS_1:26; hence thesis by A1,SQUARE_1:89; end; theorem Th48: for p being Point of TOP-REAL 2 holds |.p.|<=abs(p`1)+abs(p`2) proof let p be Point of TOP-REAL 2; A1:(|.p.|)^2= (p`1)^2+(p`2)^2 by Th46; |.p.| >=0 by TOPRNS_1:26; then sqrt((p`1)^2+(p`2)^2)=|.p.| by A1,SQUARE_1:89; hence thesis by Th2; end; theorem Th49: for p1,p2 being Point of TOP-REAL 2 holds |.p1-p2.|<=abs(p1`1-p2`1)+abs(p1`2-p2`2) proof let p1,p2 be Point of TOP-REAL 2; p1`1-p2`1=(p1-p2)`1 & p1`2-p2`2=(p1-p2)`2 by TOPREAL3:8; hence thesis by Th48; end; theorem Th50: for p being Point of TOP-REAL 2 holds abs(p`1)<=|.p.| & abs(p`2)<=|.p.| proof let p be Point of TOP-REAL 2; |.p.| = sqrt((p`1)^2+(p`2)^2) by Th47; hence thesis by Th3; end; theorem Th51: for p1,p2 being Point of TOP-REAL 2 holds abs(p1`1-p2`1)<=|.p1-p2.| & abs(p1`2-p2`2)<=|.p1-p2.| proof let p1,p2 be Point of TOP-REAL 2; p1`1-p2`1=(p1-p2)`1 & p1`2-p2`2=(p1-p2)`2 by TOPREAL3:8; hence thesis by Th50; end; theorem Th52: p in LSeg(p1,p2) implies ex r st 0<=r & r<=1 & p=(1-r)*p1+r*p2 proof assume p in LSeg(p1,p2); then p in { (1-r)*p1 + r*p2 : 0 <= r & r <= 1 } by TOPREAL1:def 4; then consider r1 such that A1: p=(1-r1)*p1 + r1*p2 &( 0 <= r1 & r1 <= 1); thus thesis by A1; end; theorem Th53: p in LSeg(p1,p2) implies |.p-p1.|<=|.p1-p2.| & |.p-p2.|<=|.p1-p2.| proof assume A1:p in LSeg(p1,p2); then consider r such that A2: 0<=r & r<=1 & p=(1-r)*p1+r*p2 by Th52; p-p1= (1-r)*p1+r*p2+-p1 by A2,EUCLID:45 .= (1-r)*p1+-p1+r*p2 by EUCLID:30 .=(1-r)*p1-p1+r*p2 by EUCLID:45 .=(1-r)*p1-1*p1+r*p2 by EUCLID:33 .=(1-r-1)*p1+r*p2 by EUCLID:54 .=(1+-r-1)*p1+r*p2 by XCMPLX_0:def 8 .=(-r)*p1+r*p2 by XCMPLX_1:26 .=r*p2+-r*p1 by EUCLID:44 .=r*p2+r*(-p1) by EUCLID:44 .=r*(p2+(-p1)) by EUCLID:36 .=r*(p2-p1) by EUCLID:45; then |.p-p1.|=abs(r)*|.p2-p1.| by TOPRNS_1:8 .=abs(r)*|.p1-p2.| by TOPRNS_1:28 .=r*|.p1-p2.| by A2,ABSVALUE:def 1; then A3:|.p1-p2.|- |.p-p1.|=1*|.p1-p2.|-r*|.p1-p2.|.=(1-r)*|.p1-p2.| by XCMPLX_1:40; A4:0<=1-r by A2,SQUARE_1:12; |.p1-p2.|>=0 by TOPRNS_1:26; then A5:|.p1-p2.|- |.p-p1.|>=0 by A3,A4,SQUARE_1:19; consider r such that A6: 0<=r & r<=1 & p=(1-r)*p2+r*p1 by A1,Th52; p-p2= (1-r)*p2+r*p1+-p2 by A6,EUCLID:45 .= (1-r)*p2+-p2+r*p1 by EUCLID:30 .=(1-r)*p2-p2+r*p1 by EUCLID:45 .=(1-r)*p2-1*p2+r*p1 by EUCLID:33 .=(1-r-1)*p2+r*p1 by EUCLID:54 .=(1+-r-1)*p2+r*p1 by XCMPLX_0:def 8 .=(-r)*p2+r*p1 by XCMPLX_1:26 .=r*p1+-r*p2 by EUCLID:44 .=r*p1+r*(-p2) by EUCLID:44 .=r*(p1+(-p2)) by EUCLID:36 .=r*(p1-p2) by EUCLID:45; then |.p-p2.|=abs(r)*|.p1-p2.| by TOPRNS_1:8 .=r*|.p1-p2.| by A6,ABSVALUE:def 1; then A7:|.p1-p2.|- |.p-p2.|=1*|.p1-p2.|-r*|.p1-p2.|.=(1-r)*|.p1-p2.| by XCMPLX_1:40; A8:0<=1-r by A6,SQUARE_1:12; |.p1-p2.|>=0 by TOPRNS_1:26; then |.p1-p2.|- |.p-p2.|>=0 by A7,A8,SQUARE_1:19; hence thesis by A5,REAL_2:105; end; begin :: EXTENDED GOBOARD THEOREM AND FASHODA MEET THEOREM reserve M for non empty MetrSpace; theorem Th54: for P,Q being Subset of TopSpaceMetr(M) st P <> {} & P is compact & Q <> {} & Q is compact holds min_dist_min(P,Q)>=0 proof let P,Q be Subset of TopSpaceMetr(M); assume P <> {} & P is compact & Q <> {} & Q is compact; then consider x1,x2 being Point of M such that A1:x1 in P & x2 in Q & dist(x1,x2) = min_dist_min(P,Q) by WEIERSTR:36; thus thesis by A1,METRIC_1:5; end; theorem Th55: for P,Q being Subset of TopSpaceMetr(M) st P <> {} & P is compact & Q <> {} & Q is compact holds P misses Q iff min_dist_min(P,Q)>0 proof let P,Q be Subset of TopSpaceMetr(M); assume A1:P <> {} & P is compact & Q <> {} & Q is compact; then consider x1,x2 being Point of M such that A2:x1 in P & x2 in Q & dist(x1,x2) = min_dist_min(P,Q) by WEIERSTR:36; A3: the distance of M is discerning by METRIC_1:def 8; A4: now assume not min_dist_min(P,Q)>0; then dist(x1,x2)=0 by A1,A2,Th54; then (the distance of M).(x1,x2)=0 by METRIC_1:def 1; then x1=x2 by A3,METRIC_1:def 4; hence P /\ Q <> {} by A2,XBOOLE_0:def 3; end; now assume A5:P /\ Q <>{}; consider p being Element of P /\ Q; A6:p in P & p in Q by A5,XBOOLE_0:def 3; then reconsider p'=p as Element of TopSpaceMetr(M); reconsider q=p' as Point of M by TOPMETR:16; the distance of M is Reflexive by METRIC_1:def 7; then (the distance of M).(q,q)=0 by METRIC_1:def 3; then dist(q,q)=0 by METRIC_1:def 1; hence not min_dist_min(P,Q)>0 by A1,A6,WEIERSTR:40; end; hence thesis by A4,XBOOLE_0:def 7; end; :: Approximation of finite sequence by special finite sequence theorem Th56: for f being FinSequence of TOP-REAL 2,a,c,d being Real st 1<=len f & X_axis(f) lies_between (X_axis(f)).1, (X_axis(f)).(len f) & Y_axis(f) lies_between c, d & a>0 & (for i st 1<=i & i+1<=len f holds |. f/.i-f/.(i+1) .|<a) holds ex g being FinSequence of TOP-REAL 2 st g is special & g.1=f.1 & g.len g=f.len f & len g>=len f & X_axis(g) lies_between (X_axis(f)).1, (X_axis(f)).(len f) & Y_axis(g) lies_between c, d & (for j st j in dom g holds ex k st k in dom f & |. g/.j - f/.k .|<a)& (for j st 1<=j & j+1<=len g holds |. g/.j - g/.(j+1) .|<a) proof let f be FinSequence of TOP-REAL 2,a,c,d be Real; assume A1:1<=len f & X_axis(f) lies_between (X_axis(f)).1, (X_axis(f)).(len f) & Y_axis(f) lies_between c, d & a>0 & (for i st 1<=i & i+1<=len f holds |. f/.i-f/.(i+1) .|<a); defpred P[set,set] means (for j st $1=2*j or $1=2*j -'1 holds ($1 =2*j implies $2=|[(f/.j)`1,(f/.(j+1))`2]|) &($1 =2*j-'1 implies $2=f/.j)); A2: for k st k in Seg (2*(len f)-'1) ex x st P[k,x] proof let k;assume A3: k in Seg (2*(len f)-'1); then A4:1<=k & k<=(2*(len f)-'1) by FINSEQ_1:3; per cases by GROUP_4:100; suppose A5:k mod 2=0; consider i such that A6: k=2*i+ (k mod 2) & k mod 2<2 by NAT_1:def 2; for j st k=2*j or k=2*j -'1 holds (k =2*j implies |[(f/.i)`1,(f/.(i+1))`2]| =|[(f/.j)`1,(f/.(j+1))`2]|) &(k =2*j-'1 implies |[(f/.i)`1,(f/.(i+1))`2]|=f/.j) proof let j;assume A7:k=2*j or k=2*j -'1; per cases by A7; suppose k=2*j; then i=2*j/2 by A5,A6,XCMPLX_1:90; then A8: i=j by XCMPLX_1:90; now assume A9:k=2*j-'1; then k=2*j-1 by A4,JORDAN3:1; then A10:k=2*j-2*1+2-1 by XCMPLX_1:27 .=2*(j-1)+(1+1)-1 by XCMPLX_1:40 .=2*(j-1)+1+1-1 by XCMPLX_1:1 .=2*(j-1)+1 by XCMPLX_1:26; now assume A11:j=0; 0-1<0; then 0-'1=0 by BINARITH:def 3; hence contradiction by A3,A9,A11,FINSEQ_1:3; end; then j>0 by NAT_1:19; then j>=0+1 by NAT_1:38; then k= 2*(j-'1)+1 & 1<=2 by A10,SCMFSA_7:3; hence contradiction by A5,NAT_1:def 2; end; hence thesis by A8; suppose A12:k=2*j-'1; then k=2*j-1 by A4,JORDAN3:1; then A13:k=2*j-2*1+2-1 by XCMPLX_1:27 .=2*(j-1)+(1+1)-1 by XCMPLX_1: 40 .=2*(j-1)+1+1-1 by XCMPLX_1:1 .=2*(j-1)+1 by XCMPLX_1:26; now assume A14:j=0; 0-1<0; then 0-'1=0 by BINARITH:def 3; hence contradiction by A3,A12,A14,FINSEQ_1:3; end; then j>0 by NAT_1:19; then j>=0+1 by NAT_1:38; then k= 2*(j-'1)+1 & 1<=2 by A13,SCMFSA_7:3; hence thesis by A5,NAT_1:def 2; end; hence thesis; suppose A15:k mod 2=1; consider i such that A16: k=2*i+ (k mod 2) & k mod 2<2 by NAT_1:def 2; for j st k=2*j or k=2*j -'1 holds (k =2*j implies f/.(i+1)=|[(f/.j)`1,(f/.(j+1))`2]|) &(k =2*j-'1 implies f/.(i+1)=f/.j) proof let j;assume A17:k=2*j or k=2*j -'1; per cases by A17; suppose k=2*j-'1; then A18:k=2*j-1 by A4,JORDAN3:1; A19:now assume k=2*j-'1; then k=2*j-1 by A4,JORDAN3:1; then k=2*j-2*1+2-1 by XCMPLX_1:27 .=2*(j-1)+(1+1)-1 by XCMPLX_1:40 .=2*(j-1)+1+1-1 by XCMPLX_1:1 .=2*(j-1)+1 by XCMPLX_1:26; then 2*(j-1)=2*i+1-1 by A15,A16,XCMPLX_1:26; then 2*(j-1)/2=2*i/2 by XCMPLX_1:26; then (j-1)=2*i/2 by XCMPLX_1:90; then j-1+1=i+1 by XCMPLX_1:90; hence f/.(i+1)=f/.j by XCMPLX_1:27; end; now assume k=2*j; then k-k=1 by A18,XCMPLX_1:18; hence contradiction by XCMPLX_1:14; end; hence thesis by A19; suppose k=2*j; then 2*j-2*i=1 by A15,A16,XCMPLX_1:26; then A20:2*(j-i)=1 by XCMPLX_1:40; then j-i>=0 by SQUARE_1:24; then A21:j-i=j-'i by BINARITH:def 3; j-i=0 or j-i>0 by A20,SQUARE_1:24; then j-i>=0+1 by A20,A21,NAT_1:38; then 2*(j-i)>=2*1 by AXIOMS:25; hence thesis by A20; end; hence thesis; end; A22:for k being Nat,y1,y2 being set st k in Seg (2*(len f)-'1) & P[k,y1] & P[k,y2] holds y1=y2 proof let k be Nat,y1,y2 be set; assume A23: k in Seg (2*(len f)-'1) & (for j st k=2*j or k=2*j -'1 holds (k =2*j implies y1=|[(f/.j)`1,(f/.(j+1))`2]|) &(k =2*j-'1 implies y1=f/.j))& (for j st k=2*j or k=2*j -'1 holds (k =2*j implies y2=|[(f/.j)`1,(f/.(j+1))`2]|) &(k =2*j-'1 implies y2=f/.j)); per cases by GROUP_4:100; suppose A24:k mod 2=0; consider i such that A25: k=2*i+ (k mod 2) & k mod 2<2 by NAT_1:def 2; y1=|[(f/.i)`1,(f/.(i+1))`2]| by A23,A24,A25; hence thesis by A23,A24,A25; suppose A26:k mod 2=1; consider i such that A27: k=2*i+ (k mod 2) & k mod 2<2 by NAT_1:def 2; k=2*i+2*1-(1+1)+1 by A26,A27,XCMPLX_1:26 .=2*(i+1)-(1+1)+1 by XCMPLX_1:8 .=2*(i+1)+1-(1+1) by XCMPLX_1:29 .=2*(i+1)+1-1-1 by XCMPLX_1:36; then A28:k=2*(i+1)-1 by XCMPLX_1:26; 1<=1+i by NAT_1:37; then 2*1<=2*(i+1) by AXIOMS:25; then 1<=2*(i+1) by AXIOMS:22; then A29:2*(i+1)-'1= 2*(i+1)-1 by SCMFSA_7:3; then y1=f/.(i+1) by A23,A28; hence thesis by A23,A28,A29; end; ex p being FinSequence st dom p = Seg (2*(len f)-'1) & for k st k in Seg (2*(len f)-'1) holds P[k,p.k] from SeqEx(A22,A2); then consider p being FinSequence such that A30: dom p = Seg (2*(len f)-'1) & for k st k in Seg (2*(len f)-'1) holds (for j st k=2*j or k=2*j -'1 holds (k =2*j implies p.k=|[(f/.j)`1,(f/.(j+1))`2]|) &(k =2*j-'1 implies p.k=f/.j)); A31:len p=(2*(len f)-'1) by A30,FINSEQ_1:def 3; rng p c= the carrier of TOP-REAL 2 proof let y;assume y in rng p; then consider x such that A32:x in dom p & y=p.x by FUNCT_1:def 5; A33:x in Seg len p by A32,FINSEQ_1:def 3; reconsider i=x as Nat by A32; A34:1 <= i & i <= len p by A33,FINSEQ_1:3; per cases by GROUP_4:100; suppose A35:i mod 2=0; consider j such that A36: i=2*j+ (i mod 2) & i mod 2<2 by NAT_1:def 2; p.i=|[(f/.j)`1,(f/.(j+1))`2]| by A30,A32,A35,A36; hence thesis by A32; suppose A37:i mod 2=1; consider j such that A38: i=2*j+ (i mod 2) & i mod 2<2 by NAT_1:def 2; i=2*j+1+1-1 by A37,A38,XCMPLX_1:26 .=2*j+(1+1)-1 by XCMPLX_1:1 .=2*j+2*1-1; then A39:i=2*(j+1)-1 by XCMPLX_1:8; then 2*(j+1)-'1 =2*(j+1)-1 by A34,JORDAN3:1; then p.i=f/.(j+1) by A30,A32,A39; hence thesis by A32; end; then reconsider g1=p as FinSequence of TOP-REAL 2 by FINSEQ_1:def 4; for i st 1 <= i & i+1 <= len g1 holds (g1/.i)`1 = (g1/.(i+1))`1 or (g1/.i)`2 = (g1/.(i+1))`2 proof let i;assume A40:1 <= i & i+1 <= len g1; then A41:i<len g1 by NAT_1:38; A42:1<i+1 by A40,NAT_1:38; then A43:i+1 in Seg len g1 by A40,FINSEQ_1:3; A44: i in Seg (2*(len f)-'1) by A31,A40,A41,FINSEQ_1:3; A45:g1.i=g1/.i by A40,A41,FINSEQ_4:24; A46:g1.(i+1)=g1/.(i+1) by A40,A42,FINSEQ_4:24; per cases by GROUP_4:100; suppose A47:i mod 2=0; consider j such that A48: i=2*j+ (i mod 2) & i mod 2<2 by NAT_1:def 2; A49:g1.i=|[(f/.j)`1,(f/.(j+1))`2]| by A30,A44,A47,A48; i+1=2*j+1+1-1 by A47,A48,XCMPLX_1:26 .=2*j+(1+1)-1 by XCMPLX_1:1 .=2*j+2*1-1; then A50:i+1=2*(j+1)-1 by XCMPLX_1:8; 1<=1+i by NAT_1:29; then 2*(j+1)-'1 =2*(j+1)-1 by A50,JORDAN3:1; then g1.(i+1)=f/.(j+1) by A30,A31,A43,A50; hence thesis by A45,A46,A49,EUCLID:56; suppose A51:i mod 2=1; consider j such that A52: i=2*j+ (i mod 2) & i mod 2<2 by NAT_1:def 2; i=2*j+1+1-1 by A51,A52,XCMPLX_1:26 .=2*j+(1+1)-1 by XCMPLX_1:1 .=2*j+2*1-1; then A53:i=2*(j+1)-1 by XCMPLX_1:8; then 2*(j+1)-'1 =2*(j+1)-1 by A40,JORDAN3:1; then A54: g1.i=f/.(j+1) by A30,A44,A53; i+1=2*j+(1+1) by A51,A52,XCMPLX_1:1 .=2*j+2*1 .=2*(j+1) by XCMPLX_1:8; then g1.(i+1)=|[(f/.(j+1))`1,(f/.(j+1+1))`2]| by A30,A31,A43; hence thesis by A45,A46,A54,EUCLID:56; end; then A55:g1 is special by TOPREAL1:def 7; A56:f.len f=f/.len f by A1,FINSEQ_4:24; A57: 2*(len f)>=2*1 by A1,AXIOMS:25; then 2*(len f)>=1 by AXIOMS:22; then A58:2*(len f)-'1= 2*(len f)-1 by SCMFSA_7:3; A59:2*(len f)-1>= 1+1-1 by A57,REAL_1:49; then A60: 1 in Seg (2*(len f)-'1) by A58,FINSEQ_1:3; 2*1-'1=1+1-'1 .=1 by BINARITH:39; then p.1=f/.1 by A30,A60; then A61:g1.1=f.1 by A1,FINSEQ_4:24; (2*(len f)-'1) in Seg (2*(len f)-'1) by A58,A59,FINSEQ_1:3; then A62:g1.len g1=f.len f by A30,A31,A56; len f + len f>=len f+1 by A1,AXIOMS:24; then 2*(len f)>=len f +1 by XCMPLX_1:11; then 2*(len f)-1>=len f +1-1 by REAL_1:49; then A63:len g1>=len f by A31,A58,XCMPLX_1:26; for i st i in dom (X_axis(g1)) holds (X_axis(f)).1 <= (X_axis(g1)).i & (X_axis(g1)).i <= (X_axis(f)).(len f) proof let i;assume A64:i in dom (X_axis(g1)); then A65: i in Seg len (X_axis(g1)) by FINSEQ_1:def 3; then i in Seg len g1 by GOBOARD1:def 3; then A66: 1<=i & i<=len g1 by FINSEQ_1:3; A67:i in Seg len g1 by A65,GOBOARD1:def 3; A68:(X_axis(g1)).i = (g1/.i)`1 by A64,GOBOARD1:def 3; A69:len (X_axis(f))=len f by GOBOARD1:def 3; A70:g1/.i=(g1.i) by A66,FINSEQ_4:24; per cases by GROUP_4:100; suppose A71:i mod 2=0; consider j such that A72: i=2*j+ (i mod 2) & i mod 2<2 by NAT_1:def 2; g1.i=|[(f/.j)`1,(f/.(j+1))`2]| by A30,A31,A67,A71,A72; then A73:(g1/.i)`1=(f/.j)`1 by A70,EUCLID:56; j<>0 by A65,A71,A72,FINSEQ_1:3; then j>0 by NAT_1:19; then A74: j>=0+1 by NAT_1:38; 2*j+1<=2*(len f)-1+1 by A31,A58,A66,A71,A72,AXIOMS:24; then 2*j+1<=2*(len f) by XCMPLX_1:27; then 2*j<2*(len f) by NAT_1:38; then 2*j/2<2*(len f)/2 by REAL_1:73; then j<2*(len f)/2 by XCMPLX_1:90; then j<len f by XCMPLX_1:90; then j in Seg len f by A74,FINSEQ_1:3; then A75:j in dom (X_axis(f)) by A69,FINSEQ_1:def 3; then (X_axis(f)).j = (f/.j)`1 by GOBOARD1:def 3; hence thesis by A1,A68,A73,A75,GOBOARD4:def 1; suppose A76:i mod 2=1; consider j such that A77: i=2*j+ (i mod 2) & i mod 2<2 by NAT_1:def 2; i=2*j+1+1-1 by A76,A77,XCMPLX_1:26; then i=2*j+(1+1)-1 by XCMPLX_1:1; then i=2*j+2*1-1; then A78:i=2*(j+1)-1 by XCMPLX_1:8; then 2*(j+1)-'1 =2*(j+1)-1 by A66,JORDAN3:1; then A79:(g1/.i)`1=(f/.(j+1))`1 by A30,A31,A67,A70,A78; A80:1<=j+1 by NAT_1:29; 2*j+1<=2*(len f)-1+1 by A31,A58,A66,A76,A77,NAT_1:38; then 2*j+1<=2*(len f) by XCMPLX_1:27; then 2*j<2*(len f) by NAT_1:38; then 2*j/2<2*(len f)/2 by REAL_1:73; then j<2*(len f)/2 by XCMPLX_1:90; then j<len f by XCMPLX_1:90; then j+1<=len f by NAT_1:38; then j+1 in Seg len f by A80,FINSEQ_1:3; then A81:j+1 in dom (X_axis(f)) by A69,FINSEQ_1:def 3; then (X_axis(f)).(j+1) = (f/.(j+1))`1 by GOBOARD1:def 3; hence thesis by A1,A68,A79,A81,GOBOARD4:def 1; end; then A82:X_axis(g1) lies_between (X_axis(f)).1, (X_axis(f)).(len f) by GOBOARD4:def 1; for i st i in dom (Y_axis(g1)) holds c <= (Y_axis(g1)).i & (Y_axis(g1)).i <= d proof let i;assume A83:i in dom (Y_axis(g1)); then A84: i in Seg len (Y_axis(g1)) by FINSEQ_1:def 3; then i in Seg len g1 by GOBOARD1:def 4; then A85: 1<=i & i<=len g1 by FINSEQ_1:3; A86:i in Seg len g1 by A84,GOBOARD1:def 4; A87:(Y_axis(g1)).i = (g1/.i)`2 by A83,GOBOARD1:def 4; A88:len (Y_axis(f))=len f by GOBOARD1:def 4; A89:g1/.i=(g1.i) by A85,FINSEQ_4:24; per cases by GROUP_4:100; suppose A90:i mod 2=0; consider j such that A91: i=2*j+ (i mod 2) & i mod 2<2 by NAT_1:def 2; g1.i=|[(f/.j)`1,(f/.(j+1))`2]| by A30,A31,A86,A90,A91; then A92:(g1/.i)`2=(f/.(j+1))`2 by A89,EUCLID:56; A93:1<=j+1 by NAT_1:29; 2*j+1<=2*(len f)-1+1 by A31,A58,A85,A90,A91,AXIOMS:24; then 2*j+1<=2*(len f) by XCMPLX_1:27; then 2*j<2*(len f) by NAT_1:38; then 2*j/2<2*(len f)/2 by REAL_1:73; then j<2*(len f)/2 by XCMPLX_1:90; then j<len f by XCMPLX_1:90; then j+1<=len f by NAT_1:38; then j+1 in Seg len f by A93,FINSEQ_1:3; then A94:j+1 in dom (Y_axis(f)) by A88,FINSEQ_1:def 3; then (Y_axis(f)).(j+1) = (f/.(j+1))`2 by GOBOARD1:def 4; hence thesis by A1,A87,A92,A94,GOBOARD4:def 1; suppose A95:i mod 2=1; consider j such that A96: i=2*j+ (i mod 2) & i mod 2<2 by NAT_1:def 2; i=2*j+1+1-1 by A95,A96,XCMPLX_1:26; then i=2*j+(1+1)-1 by XCMPLX_1:1; then i=2*j+2*1-1; then A97:i=2*(j+1)-1 by XCMPLX_1:8; then 2*(j+1)-'1 =2*(j+1)-1 by A85,JORDAN3:1; then A98:(g1/.i)`2=(f/.(j+1))`2 by A30,A31,A86,A89,A97; A99:1<=j+1 by NAT_1:29; 2*j+1<=2*(len f)-1+1 by A31,A58,A85,A95,A96,NAT_1:38; then 2*j+1<=2*(len f) by XCMPLX_1:27; then 2*j<2*(len f) by NAT_1:38; then 2*j/2<2*(len f)/2 by REAL_1:73; then j<2*(len f)/2 by XCMPLX_1:90; then j<len f by XCMPLX_1:90; then j+1<=len f by NAT_1:38; then j+1 in Seg len f by A99,FINSEQ_1:3; then A100:j+1 in dom (Y_axis(f)) by A88,FINSEQ_1:def 3; then (Y_axis(f)).(j+1) = (f/.(j+1))`2 by GOBOARD1:def 4; hence thesis by A1,A87,A98,A100,GOBOARD4:def 1; end; then A101:Y_axis(g1) lies_between c, d by GOBOARD4:def 1; A102:for i st i in dom g1 holds ex k st k in dom f & |. g1/.i - f/.k .| < a proof let i;assume A103:i in dom g1; then A104: i in Seg len g1 by FINSEQ_1:def 3; then A105:1<=i & i<=len g1 by FINSEQ_1:3; then A106:g1.i=g1/.i by FINSEQ_4:24; per cases by GROUP_4:100; suppose A107:i mod 2=0; consider j such that A108: i=2*j+ (i mod 2) & i mod 2<2 by NAT_1:def 2; A109:g1.i=|[(f/.j)`1,(f/.(j+1))`2]| by A30,A103,A107,A108; then A110:(g1/.i)`1=(f/.j)`1 by A106,EUCLID:56; A111:(g1/.i)`2=(f/.(j+1))`2 by A106,A109,EUCLID:56; A112:g1/.i-f/.j=|[(g1/.i)`1-(f/.j)`1,(g1/.i)`2-( (f/.j))`2]| by EUCLID:65 .=|[0,(g1/.i)`2-(f/.j)`2]| by A110,XCMPLX_1:14; then (g1/.i-f/.j)`2=(g1/.i)`2-(f/.j)`2 by EUCLID:56; then |.g1/.i-f/.j.|= sqrt(((g1/.i-f/.j)`1)^2+((g1/.i)`2-(f/.j)`2)^2) by Th47 .= sqrt(0+(((f/.(j+1))`2-(f/.j)`2))^2) by A111,A112,EUCLID:56,SQUARE_1 :60 .= sqrt((((f/.(j+1))`2-(f/.j)`2))^2); then |.g1/.i-f/.j.|=abs((f/.(j+1))`2-(f/.j)`2) by SQUARE_1:91 .=abs((f/.j)`2 -(f/.(j+1))`2) by UNIFORM1:13; then A113:|.g1/.i-f/.j.| <=|.f/.j-f/.(j+1).| by Th51; j<>0 by A104,A107,A108,FINSEQ_1:3; then j>0 by NAT_1:19; then A114: j>=0+1 by NAT_1:38; 2*j+1<=2*(len f)-1+1 by A31,A58,A105,A107,A108,AXIOMS:24; then 2*j+1<=2*(len f) by XCMPLX_1:27; then 2*j<2*(len f) by NAT_1:38; then 2*j/2<2*(len f)/2 by REAL_1:73; then j<2*(len f)/2 by XCMPLX_1:90; then A115: j<len f by XCMPLX_1:90; then j+1<=len f by NAT_1:38; then |. f/.j-f/.(j+1) .|<a by A1,A114; then A116: |. g1/.i - f/.j .|<a by A113,AXIOMS:22; j in dom f by A114,A115,FINSEQ_3:27; hence thesis by A116; suppose A117:i mod 2=1; consider j such that A118: i=2*j+ (i mod 2) & i mod 2<2 by NAT_1:def 2; i=2*j+1+1-1 by A117,A118,XCMPLX_1:26 .=2*j+(1+1)-1 by XCMPLX_1:1 .=2*j+2*1-1; then A119:i=2*(j+1)-1 by XCMPLX_1:8; then 2*(j+1)-'1 =2*(j+1)-1 by A105,JORDAN3:1; then g1.i=f/.(j+1) by A30,A103,A119; then A120:|.g1/.i-f/.(j+1).|=|.0.REAL 2.| by A106,EUCLID:46 .=0 by TOPRNS_1:24; 2*j+1+1<=2*(len f)-1+1 by A31,A58,A105,A117,A118,AXIOMS:24; then 2*j+1+1<=2*(len f) by XCMPLX_1:27; then 2*j+1<2*(len f) by NAT_1:38; then 2*j<2*(len f) by NAT_1:38; then 2*j/2<2*(len f)/2 by REAL_1:73; then j<2*(len f)/2 by XCMPLX_1:90; then j<len f by XCMPLX_1:90; then A121: j+1<=len f by NAT_1:38; 1<=j+1 by NAT_1:29; then j+1 in dom f by A121,FINSEQ_3:27; hence thesis by A1,A120; end; for i st 1<=i & i+1<=len g1 holds |. g1/.i - g1/.(i+1) .|<a proof let i;assume A122:1<=i & i+1<=len g1; then i<=len g1 by NAT_1:38; then A123:i in Seg len g1 by A122,FINSEQ_1:3; A124:1<=i & i<=len g1 by A122,NAT_1:38; A125:1<=i+1 by NAT_1:29; then A126: i+1 in Seg (2*(len f)-'1) by A31,A122,FINSEQ_1:3; A127:g1.(i+1)=g1/.(i+1) by A122,A125,FINSEQ_4:24; A128:g1.i=g1/.i by A124,FINSEQ_4:24; per cases by GROUP_4:100; suppose A129:i mod 2=0; consider j such that A130: i=2*j+ (i mod 2) & i mod 2<2 by NAT_1:def 2; A131:g1.i=|[(f/.j)`1,(f/.(j+1))`2]| by A30,A31,A123,A129,A130; i+1=2*j+1+1-1 by A129,A130,XCMPLX_1:26 .=2*j+(1+1)-1 by XCMPLX_1:1 .=2*j+2*1-1; then A132:i+1=2*(j+1)-1 by XCMPLX_1:8; 1<=1+i by NAT_1:29; then 2*(j+1)-'1 =2*(j+1)-1 by A132,JORDAN3:1; then A133:g1.(i+1)=f/.(j+1) by A30,A126,A132; A134:(g1/.i)`1=(f/.j)`1 by A128,A131,EUCLID:56; A135:(g1/.i)`2=(f/.(j+1))`2 by A128,A131,EUCLID:56; A136:(g1/.(i+1))`1=(f/.(j+1))`1 by A122,A125,A133,FINSEQ_4:24; A137:(g1/.(i+1))`2=(f/.(j+1))`2 by A122,A125,A133,FINSEQ_4:24; A138:g1/.i-g1/.(i+1) =|[(g1/.i)`1-(g1/.(i+1))`1,(g1/.i)`2-(g1/.(i+1))`2]| by EUCLID:65 .=|[(f/.j)`1-(f/.(j+1))`1,0]| by A134,A135,A136,A137,XCMPLX_1:14; then A139:(g1/.i-g1/.(i+1))`1=(f/.j)`1-(f/.(j+1))`1 by EUCLID:56; |.g1/.i-g1/.(i+1).|= sqrt((((g1/.i-g1/.(i+1))`1))^2+(((g1/.i-g1/.(i+1))`2))^2) by Th47 .= sqrt((((f/.j)`1-(f/.(j+1))`1))^2+0) by A138,A139,EUCLID:56,SQUARE_1 :60 .= sqrt((((f/.j)`1-(f/.(j+1))`1))^2); then |.g1/.i-g1/.(i+1).|=abs((f/.j)`1-(f/.(j+1))`1) by SQUARE_1:91; then A140:|.g1/.i-g1/.(i+1).| <=|.f/.j-f/.(j+1).| by Th51; j<>0 by A122,A129,A130; then j>0 by NAT_1:19; then A141: j>=0+1 by NAT_1:38; 2*j+1<=2*(len f)-1+1 by A31,A58,A122,A129,A130,NAT_1:38; then 2*j+1<=2*(len f) by XCMPLX_1:27; then 2*j<2*(len f) by NAT_1:38; then 2*j/2<2*(len f)/2 by REAL_1:73; then j<2*(len f)/2 by XCMPLX_1:90; then j<len f by XCMPLX_1:90; then j+1<=len f by NAT_1:38; then |. f/.j-f/.(j+1) .|<a by A1,A141; hence thesis by A140,AXIOMS:22; suppose A142:i mod 2=1; consider j such that A143: i=2*j+ (i mod 2) & i mod 2<2 by NAT_1:def 2; i=2*j+1+1-1 by A142,A143,XCMPLX_1:26; then i=2*j+(1+1)-1 by XCMPLX_1:1; then i=2*j+2*1-1; then A144:i=2*(j+1)-1 by XCMPLX_1:8; then A145: 2*(j+1)-'1 =2*(j+1)-1 by A122,JORDAN3:1; i+1=2*j+(1+1) by A142,A143,XCMPLX_1:1 .=2*j+2*1; then A146:i+1=2*(j+1) by XCMPLX_1:8; then A147:g1/.(i+1)=|[(f/.(j+1))`1,(f/.(j+1+1))`2]| by A30,A126,A127; then A148:(g1/.(i+1))`1=(f/.(j+1))`1 by EUCLID:56; A149:(g1/.(i+1))`2=(f/.(j+1+1))`2 by A147,EUCLID:56; A150:(g1/.i)`1=(f/.(j+1))`1 by A30,A31,A123,A128,A144,A145; A151:(g1/.i)`2=(f/.(j+1))`2 by A30,A31,A123,A128,A144,A145; A152:g1/.i-g1/.(i+1) =|[(g1/.i)`1-(g1/.(i+1))`1,(g1/.i)`2-(g1/.(i+1))`2]| by EUCLID:65 .=|[0,(f/.(j+1))`2-(f/.(j+1+1))`2]| by A148,A149,A150,A151,XCMPLX_1:14; then A153:(g1/.i-g1/.(i+1))`1=0 by EUCLID:56; |.g1/.i-g1/.(i+1).|= sqrt((((g1/.i-g1/.(i+1))`1))^2+(((g1/.i- g1/.(i+1))`2))^2) by Th47 .= sqrt((((f/.(j+1))`2-(f/.(j+1+1))`2))^2) by A152,A153,EUCLID:56, SQUARE_1:60; then |.g1/.i-g1/.(i+1).|=abs((f/.(j+1))`2-(f/.(j+1+1))`2) by SQUARE_1:91; then A154:|.g1/.i-g1/.(i+1).| <=|.f/.(j+1)-f/.(j+1+1).| by Th51; A155: j+1>=1 by NAT_1:29; 2*(j+1)+1<=2*(len f)-1+1 by A31,A58,A122,A146,AXIOMS:24; then 2*(j+1)+1<=2*(len f) by XCMPLX_1:27; then 2*(j+1)<2*(len f) by NAT_1:38; then 2*(j+1)/2<2*(len f)/2 by REAL_1:73; then (j+1)<2*(len f)/2 by XCMPLX_1:90; then (j+1)<len f by XCMPLX_1:90; then (j+1)+1<=len f by NAT_1:38; then |. f/.(j+1)-f/.(j+1+1) .|<a by A1,A155; hence thesis by A154,AXIOMS:22; end; hence thesis by A55,A61,A62,A63,A82,A101,A102; end; theorem Th57: for f being FinSequence of TOP-REAL 2,a,c,d being Real st 1<=len f & Y_axis(f) lies_between (Y_axis(f)).1, (Y_axis(f)).(len f) & X_axis(f) lies_between c, d & a>0 & (for i st 1<=i & i+1<=len f holds |. (f/.i)-(f/.(i+1)) .|<a) holds ex g being FinSequence of TOP-REAL 2 st g is special & g.1=f.1 & g.len g=f.len f & len g>=len f & Y_axis(g) lies_between (Y_axis(f)).1, (Y_axis(f)).(len f) & X_axis(g) lies_between c, d & (for j st j in dom g holds ex k st k in dom f & |. g/.j - f/.k .| < a)& (for j st 1<=j & j+1<=len g holds |. g/.j - g/.(j+1) .|<a) proof let f be FinSequence of TOP-REAL 2,a,c,d be Real; assume A1:1<=len f & Y_axis(f) lies_between (Y_axis(f)).1, (Y_axis(f)).(len f) & X_axis(f) lies_between c, d & a>0 & (for i st 1<=i & i+1<=len f holds |. f/.i-f/.(i+1) .|<a); defpred P[set,set] means (for j st $1=2*j or $1=2*j -'1 holds ($1 =2*j implies $2=|[(f/.j)`1,(f/.(j+1))`2]|) &($1 =2*j-'1 implies $2=f/.j)); A2: for k st k in Seg (2*(len f)-'1) ex x st P[k,x] proof let k;assume A3: k in Seg (2*(len f)-'1); then A4:1<=k & k<=2*(len f)-'1 by FINSEQ_1:3; per cases by GROUP_4:100; suppose A5:k mod 2=0; consider i such that A6: k=2*i+ (k mod 2) & k mod 2<2 by NAT_1:def 2; for j st k=2*j or k=2*j -'1 holds (k =2*j implies |[(f/.i)`1,(f/.(i+1))`2]| =|[(f/.j)`1,(f/.(j+1))`2]|) &(k =2*j-'1 implies |[(f/.i)`1,(f/.(i+1))`2]|=f/.j) proof let j;assume A7:k=2*j or k=2*j -'1; per cases by A7; suppose k=2*j; then i=2*j/2 by A5,A6,XCMPLX_1:90; then A8: i=j by XCMPLX_1:90; now assume A9:k=2*j-'1; then k=2*j-1 by A4,JORDAN3:1; then A10:k=2*j-2*1+2-1 by XCMPLX_1:27 .=2*(j-1)+(1+1)-1 by XCMPLX_1:40 .=2*(j-1)+1+1-1 by XCMPLX_1:1 .=2*(j-1)+1 by XCMPLX_1:26; now assume A11:j=0; 0-1<0; then 0-'1=0 by BINARITH:def 3; hence contradiction by A3,A9,A11,FINSEQ_1:3; end; then j>0 by NAT_1:19; then j>=0+1 by NAT_1:38; then k= 2*(j-'1)+1 & 1<=2 by A10,SCMFSA_7:3; hence contradiction by A5,NAT_1:def 2; end; hence thesis by A8; suppose A12: k=2*j-'1; then k=2*j-1 by A4,JORDAN3:1; then A13:k=2*j-2*1+2-1 by XCMPLX_1:27 .=2*(j-1)+(1+1)-1 by XCMPLX_1: 40 .=2*(j-1)+1+1-1 by XCMPLX_1:1 .=2*(j-1)+1 by XCMPLX_1:26; now assume A14:j=0; 0-1<0; then 0-'1=0 by BINARITH:def 3; hence contradiction by A3,A12,A14,FINSEQ_1:3; end; then j>0 by NAT_1:19; then j>=0+1 by NAT_1:38; then j-'1=j-1 by SCMFSA_7:3; hence thesis by A5,A13,NAT_1:def 2; end; hence thesis; suppose A15:k mod 2=1; consider i such that A16: k=2*i+ (k mod 2) & k mod 2<2 by NAT_1:def 2; for j st k=2*j or k=2*j -'1 holds (k =2*j implies f/.(i+1)=|[(f/.j)`1,(f/.(j+1))`2]|) &(k =2*j-'1 implies f/.(i+1)=f/.j) proof let j;assume A17:k=2*j or k=2*j -'1; per cases by A17; suppose k=2*j-'1; then A18:k=2*j-1 by A4,JORDAN3:1; A19:now assume k=2*j-'1; then k=2*j-1 by A4,JORDAN3:1; then k=2*j-2*1+2-1 by XCMPLX_1:27 .=2*(j-1)+(1+1)-1 by XCMPLX_1:40 .=2*(j-1)+1+1-1 by XCMPLX_1:1 .=2*(j-1)+1 by XCMPLX_1:26; then 2*(j-1)=2*i+1-1 by A15,A16,XCMPLX_1:26; then 2*(j-1)/2=2*i/2 by XCMPLX_1:26; then (j-1)=2*i/2 by XCMPLX_1:90; then (j-1)=i by XCMPLX_1:90; hence f/.(i+1)=f/.j by XCMPLX_1:27; end; now assume k=2*j; then k-k=1 by A18,XCMPLX_1:18; hence contradiction by XCMPLX_1:14; end; hence thesis by A19; suppose k=2*j; then 2*j-2*i=1 by A15,A16,XCMPLX_1:26; then A20:2*(j-i)=1 by XCMPLX_1:40; then j-i>=0 by SQUARE_1:24; then A21:j-i=j-'i by BINARITH:def 3; j-i=0 or j-i>0 by A20,SQUARE_1:24; then j-i>=0+1 by A20,A21,NAT_1:38; then 2*(j-i)>=2*1 by AXIOMS:25; hence thesis by A20; end; hence thesis; end; A22:for k being Nat,y1,y2 being set st k in Seg (2*(len f)-'1) & P[k,y1] & P[k,y2] holds y1=y2 proof let k be Nat,y1,y2 be set; assume A23: k in Seg (2*(len f)-'1) & (for j st k=2*j or k=2*j -'1 holds (k =2*j implies y1=|[(f/.j)`1,(f/.(j+1))`2]|) &(k =2*j-'1 implies y1=f/.j))& (for j st k=2*j or k=2*j -'1 holds (k =2*j implies y2=|[(f/.j)`1,(f/.(j+1))`2]|) &(k =2*j-'1 implies y2=f/.j)); per cases by GROUP_4:100; suppose A24:k mod 2=0; consider i such that A25: k=2*i+ (k mod 2) & k mod 2<2 by NAT_1:def 2; y1=|[(f/.i)`1,(f/.(i+1))`2]| by A23,A24,A25; hence thesis by A23,A24,A25; suppose A26:k mod 2=1; consider i such that A27: k=2*i+ (k mod 2) & k mod 2<2 by NAT_1:def 2; A28: k=2*i+2*1-(1+1)+1 by A26,A27,XCMPLX_1:26 .=2*(i+1)-(1+1)+1 by XCMPLX_1:8 .=2*(i+1)+1-(1+1) by XCMPLX_1:29 .=2*(i+1)+1-1-1 by XCMPLX_1:36 .=2*(i+1)-1 by XCMPLX_1:26; 1<=1+i by NAT_1:37; then 2*1<=2*(i+1) by AXIOMS:25; then 1<=2*(i+1) by AXIOMS:22; then A29:2*(i+1)-'1= 2*(i+1)-1 by SCMFSA_7:3; then y1=f/.(i+1) by A23,A28; hence thesis by A23,A28,A29; end; ex p being FinSequence st dom p = Seg (2*(len f)-'1) & for k st k in Seg (2*(len f)-'1) holds P[k,p.k] from SeqEx(A22,A2); then consider p being FinSequence such that A30: dom p = Seg (2*(len f)-'1) & for k st k in Seg (2*(len f)-'1) holds (for j st k=2*j or k=2*j -'1 holds (k =2*j implies p.k=|[(f/.j)`1,(f/.(j+1))`2]|) &(k =2*j-'1 implies p.k=f/.j)); A31:len p=(2*(len f)-'1) by A30,FINSEQ_1:def 3; rng p c= the carrier of TOP-REAL 2 proof let y;assume y in rng p; then consider x such that A32:x in dom p & y=p.x by FUNCT_1:def 5; A33:x in Seg len p by A32,FINSEQ_1:def 3; reconsider i=x as Nat by A32; A34:1 <= i & i <= len p by A33,FINSEQ_1:3; per cases by GROUP_4:100; suppose A35:i mod 2=0; consider j such that A36: i=2*j+ (i mod 2) & i mod 2<2 by NAT_1:def 2; p.i=|[(f/.j)`1,(f/.(j+1))`2]| by A30,A32,A35,A36; hence thesis by A32; suppose A37:i mod 2=1; consider j such that A38: i=2*j+ (i mod 2) & i mod 2<2 by NAT_1:def 2; A39: i=2*j+1+1-1 by A37,A38,XCMPLX_1:26 .=2*j+(1+1)-1 by XCMPLX_1:1 .=2*j+2*1-1 .=2*(j+1)-1 by XCMPLX_1:8; then 2*(j+1)-1=2*(j+1)-'1 by A34,JORDAN3:1; then p.i=f/.(j+1) by A30,A32,A39; hence thesis by A32; end; then reconsider g1=p as FinSequence of TOP-REAL 2 by FINSEQ_1:def 4; for i st 1 <= i & i+1 <= len g1 holds (g1/.i)`1 = (g1/.(i+1))`1 or (g1/.i)`2 = (g1/.(i+1))`2 proof let i;assume A40:1 <= i & i+1 <= len g1; then A41:i<len g1 by NAT_1:38; A42:1<i+1 by A40,NAT_1:38; then A43:i+1 in Seg len g1 by A40,FINSEQ_1:3; A44: i in Seg (2*(len f)-'1) by A31,A40,A41,FINSEQ_1:3; A45:g1.i=g1/.i by A40,A41,FINSEQ_4:24; A46:g1/.(i+1)=g1.(i+1) by A40,A42,FINSEQ_4:24; per cases by GROUP_4:100; suppose A47:i mod 2=0; consider j such that A48: i=2*j+ (i mod 2) & i mod 2<2 by NAT_1:def 2; A49:g1/.i=|[(f/.j)`1,(f/.(j+1))`2]| by A30,A44,A45,A47,A48; A50: i+1=2*j+1+1-1 by A47,A48,XCMPLX_1:26 .=2*j+(1+1)-1 by XCMPLX_1:1 .=2*j+2*1-1 .=2*(j+1)-1 by XCMPLX_1:8; 1<=1+i by NAT_1:29; then 2*(j+1)-'1 =2*(j+1)-1 by A50,JORDAN3:1; then g1/.(i+1)=f/.(j+1) by A30,A31,A43,A46,A50; hence thesis by A49,EUCLID:56; suppose A51:i mod 2=1; consider j such that A52: i=2*j+ (i mod 2) & i mod 2<2 by NAT_1:def 2; i=2*j+1+1-1 by A51,A52,XCMPLX_1:26; then i=2*j+(1+1)-1 by XCMPLX_1:1; then i=2*j+2*1-1; then A53:i=2*(j+1)-1 by XCMPLX_1:8; then 2*(j+1)-'1 =2*(j+1)-1 by A40,JORDAN3:1; then A54:g1/.i=f/.(j+1) by A30,A44,A45,A53; i+1=2*j+(1+1) by A51,A52,XCMPLX_1:1 .=2*j+2*1 .=2*(j+1) by XCMPLX_1:8; then g1/.(i+1)=|[(f/.(j+1))`1,(f/.(j+1+1))`2]| by A30,A31,A43,A46; hence thesis by A54,EUCLID:56; end; then A55:g1 is special by TOPREAL1:def 7; A56: 2*(len f)>=2*1 by A1,AXIOMS:25; then 2*(len f)>=1 by AXIOMS:22; then A57:2*(len f)-'1= 2*(len f)-1 by SCMFSA_7:3; A58:2*(len f)-1>= 1+1-1 by A56,REAL_1:49; then A59: 1 in Seg (2*(len f)-'1) by A57,FINSEQ_1:3; 2*1-'1=1+1-'1 .=1 by BINARITH:39; then p.1=f/.1 by A30,A59; then A60:g1.1=f.1 by A1,FINSEQ_4:24; (2*(len f)-'1) in Seg (2*(len f)-'1) by A57,A58,FINSEQ_1:3; then p.(2*(len f)-'1)=f/.len f by A30; then A61:g1.len g1=f.len f by A1,A31,FINSEQ_4:24; len f + len f>=len f+1 by A1,AXIOMS:24; then 2*(len f)>=len f +1 by XCMPLX_1:11; then 2*(len f)-1>=len f +1-1 by REAL_1:49; then A62:len g1>=len f by A31,A57,XCMPLX_1:26; for i st i in dom (Y_axis(g1)) holds (Y_axis(f)).1 <= (Y_axis(g1)).i & (Y_axis(g1)).i <= (Y_axis(f)).(len f) proof let i;assume A63:i in dom (Y_axis(g1)); then A64: i in Seg len (Y_axis(g1)) by FINSEQ_1:def 3; then i in Seg len g1 by GOBOARD1:def 4; then A65: 1<=i & i<=len g1 by FINSEQ_1:3; A66:i in Seg len g1 by A64,GOBOARD1:def 4; A67:(Y_axis(g1)).i = (g1/.i)`2 by A63,GOBOARD1:def 4; A68:len (Y_axis(f))=len f by GOBOARD1:def 4; A69:g1/.i=(g1.i) by A65,FINSEQ_4:24; per cases by GROUP_4:100; suppose A70:i mod 2=0; consider j such that A71: i=2*j+ (i mod 2) & (i mod 2)<2 by NAT_1:def 2; g1/.i=|[(f/.j)`1,(f/.(j+1))`2]| by A30,A31,A66,A69,A70,A71; then A72:(g1/.i)`2=(f/.(j+1))`2 by EUCLID:56; A73:1<=j+1 by NAT_1:29; 2*j+1<=2*(len f)-1+1 by A31,A57,A65,A70,A71,AXIOMS:24; then 2*j+1<=2*(len f) by XCMPLX_1:27; then 2*j<2*(len f) by NAT_1:38; then 2*j/2<2*(len f)/2 by REAL_1:73; then j<2*(len f)/2 by XCMPLX_1:90; then j<len f by XCMPLX_1:90; then j+1<=len f by NAT_1:38; then j+1 in Seg len f by A73,FINSEQ_1:3; then A74:j+1 in dom (Y_axis(f)) by A68,FINSEQ_1:def 3; then (Y_axis(f)).(j+1) = (f/.(j+1))`2 by GOBOARD1:def 4; hence thesis by A1,A67,A72,A74,GOBOARD4:def 1; suppose A75:i mod 2=1; consider j such that A76: i=2*j+ (i mod 2) & (i mod 2)<2 by NAT_1:def 2; i=2*j+1+1-1 by A75,A76,XCMPLX_1:26 .=2*j+(1+1)-1 by XCMPLX_1:1 .=2*j+2*1-1; then A77:i=2*(j+1)-1 by XCMPLX_1:8; then 2*(j+1)-'1 =2*(j+1)-1 by A65,JORDAN3:1; then A78:(g1/.i)`2=(f/.(j+1))`2 by A30,A31,A66,A69,A77; A79:1<=j+1 by NAT_1:29; 2*j+1<=2*(len f)-1+1 by A31,A57,A65,A75,A76,NAT_1:38; then 2*j+1<=2*(len f) by XCMPLX_1:27; then 2*j<2*(len f) by NAT_1:38; then 2*j/2<2*(len f)/2 by REAL_1:73; then j<2*(len f)/2 by XCMPLX_1:90; then j<len f by XCMPLX_1:90; then j+1<=len f by NAT_1:38; then j+1 in Seg len f by A79,FINSEQ_1:3; then A80:j+1 in dom (Y_axis(f)) by A68,FINSEQ_1:def 3; then (Y_axis(f)).(j+1) = (f/.(j+1))`2 by GOBOARD1:def 4; hence thesis by A1,A67,A78,A80,GOBOARD4:def 1; end; then A81:Y_axis(g1) lies_between (Y_axis(f)).1, (Y_axis(f)).(len f) by GOBOARD4:def 1; for i st i in dom (X_axis(g1)) holds c <= (X_axis(g1)).i & (X_axis(g1)).i <= d proof let i;assume A82:i in dom (X_axis(g1)); then A83: i in Seg len (X_axis(g1)) by FINSEQ_1:def 3; then A84:i in Seg len g1 by GOBOARD1:def 3; then A85: 1<=i & i<=len g1 by FINSEQ_1:3; A86:(X_axis(g1)).i = (g1/.i)`1 by A82,GOBOARD1:def 3; A87:len (X_axis(f))=len f by GOBOARD1:def 3; A88:g1/.i=(g1.i) by A85,FINSEQ_4:24; per cases by GROUP_4:100; suppose A89:i mod 2=0; consider j such that A90: i=2*j+ (i mod 2) & (i mod 2)<2 by NAT_1:def 2; g1/.i=|[(f/.j)`1,(f/.(j+1))`2]| by A30,A31,A84,A88,A89,A90; then A91:(g1/.i)`1=(f/.j)`1 by EUCLID:56; j<>0 by A83,A89,A90,FINSEQ_1:3; then j>0 by NAT_1:19; then A92: j>=0+1 by NAT_1:38; 2*j+1<=2*(len f)-1+1 by A31,A57,A85,A89,A90,AXIOMS:24; then 2*j+1<=2*(len f) by XCMPLX_1:27; then 2*j<2*(len f) by NAT_1:38; then 2*j/2<2*(len f)/2 by REAL_1:73; then j<2*(len f)/2 by XCMPLX_1:90; then j<len f by XCMPLX_1:90; then j in Seg len f by A92,FINSEQ_1:3; then A93:j in dom (X_axis(f)) by A87,FINSEQ_1:def 3; then (X_axis(f)).j = (f/.j)`1 by GOBOARD1:def 3; hence thesis by A1,A86,A91,A93,GOBOARD4:def 1; suppose A94:i mod 2=1; consider j such that A95: i=2*j+ (i mod 2) & (i mod 2)<2 by NAT_1:def 2; i=2*j+1+1-1 by A94,A95,XCMPLX_1:26; then i=2*j+(1+1)-1 by XCMPLX_1:1; then i=2*j+2*1-1; then A96:i=2*(j+1)-1 by XCMPLX_1:8; then 2*(j+1)-'1 =2*(j+1)-1 by A85,JORDAN3:1; then A97:(g1/.i)`1=(f/.(j+1))`1 by A30,A31,A84,A88,A96; A98:1<=j+1 by NAT_1:29; 2*j+1<=2*(len f)-1+1 by A31,A57,A85,A94,A95,NAT_1:38; then 2*j+1<=2*(len f) by XCMPLX_1:27; then 2*j<2*(len f) by NAT_1:38; then 2*j/2<2*(len f)/2 by REAL_1:73; then j<2*(len f)/2 by XCMPLX_1:90; then j<len f by XCMPLX_1:90; then j+1<=len f by NAT_1:38; then j+1 in Seg len f by A98,FINSEQ_1:3; then A99:j+1 in dom (X_axis(f)) by A87,FINSEQ_1:def 3; then (X_axis(f)).(j+1) = (f/.(j+1))`1 by GOBOARD1:def 3; hence thesis by A1,A86,A97,A99,GOBOARD4:def 1; end; then A100:X_axis(g1) lies_between c, d by GOBOARD4:def 1; A101:for i st i in dom g1 holds ex k st k in dom f & |. g1/.i - f/.k .| < a proof let i;assume A102: i in dom g1; then A103:1<=i & i<=len g1 by FINSEQ_3:27; then A104:g1.i=g1/.i by FINSEQ_4:24; per cases by GROUP_4:100; suppose A105:i mod 2=0; consider j such that A106: i=2*j+ (i mod 2) & (i mod 2)<2 by NAT_1:def 2; A107:g1/.i=|[(f/.j)`1,(f/.(j+1))`2]| by A30,A102,A104,A105,A106; then A108:(g1/.i)`1=(f/.j)`1 by EUCLID:56; A109:g1/.i-f/.j =|[(g1/.i)`1-(f/.j)`1,(g1/.i)`2-(f/.j)`2]| by EUCLID:65 .=|[0,(g1/.i)`2-(f/.j)`2]| by A108,XCMPLX_1:14; then A110:(g1/.i-f/.j)`1=0 by EUCLID:56; (g1/.i-f/.j)`2=(g1/.i)`2-(f/.j)`2 by A109,EUCLID:56; then |.g1/.i-f/.j.|= sqrt((((g1/.i-f/.j)`1))^2+(((g1/.i)`2-(f/.j)`2))^2) by Th47 .= sqrt((((f/.(j+1))`2-(f/.j)`2))^2) by A107,A110,EUCLID:56,SQUARE_1: 60; then |.g1/.i-f/.j.|=abs((f/.(j+1))`2-(f/.j)`2) by SQUARE_1:91 .=abs((f/.j)`2 -(f/.(j+1))`2) by UNIFORM1:13; then A111:|.g1/.i-f/.j.| <=|.f/.j-f/.(j+1).| by Th51; j<>0 by A102,A105,A106,FINSEQ_3:27; then j>0 by NAT_1:19; then A112: j>=0+1 by NAT_1:38; 2*j+1<=2*(len f)-1+1 by A31,A57,A103,A105,A106,AXIOMS:24; then 2*j+1<=2*(len f) by XCMPLX_1:27; then 2*j<2*(len f) by NAT_1:38; then 2*j/2<2*(len f)/2 by REAL_1:73; then j<2*(len f)/2 by XCMPLX_1:90; then A113: j<len f by XCMPLX_1:90; then j+1<=len f by NAT_1:38; then |. f/.j-f/.(j+1) .|<a by A1,A112; then A114: |. g1/.i - f/.j .|<a by A111,AXIOMS:22; j in dom f by A112,A113,FINSEQ_3:27; hence thesis by A114; suppose A115:i mod 2=1; consider j such that A116: i=2*j+ (i mod 2) & (i mod 2)<2 by NAT_1:def 2; i=2*j+1+1-1 by A115,A116,XCMPLX_1:26; then i=2*j+(1+1)-1 by XCMPLX_1:1; then i=2*j+2*1-1; then A117:i=2*(j+1)-1 by XCMPLX_1:8; then 2*(j+1)-'1 =2*(j+1)-1 by A103,JORDAN3:1; then g1/.i=f/.(j+1) by A30,A102,A104,A117; then A118:|.g1/.i-f/.(j+1).|=|.0.REAL 2.| by EUCLID:46 .=0 by TOPRNS_1: 24; 2*j+1+1<=2*(len f)-1+1 by A31,A57,A103,A115,A116,AXIOMS:24; then 2*j+1+1<=2*(len f) by XCMPLX_1:27; then 2*j+1<2*(len f) by NAT_1:38; then 2*j<2*(len f) by NAT_1:38; then 2*j/2<2*(len f)/2 by REAL_1:73; then j<2*(len f)/2 by XCMPLX_1:90; then j<len f by XCMPLX_1:90; then A119: j+1<=len f by NAT_1:38; 1<=j+1 by NAT_1:29; then j+1 in dom f by A119,FINSEQ_3:27; hence thesis by A1,A118; end; for i st 1<=i & i+1<=len g1 holds |. g1/.i - g1/.(i+1) .|<a proof let i;assume A120:1<=i & i+1<=len g1; then i<=len g1 by NAT_1:38; then A121:i in Seg len g1 by A120,FINSEQ_1:3; A122:1<=i & i<=len g1 by A120,NAT_1:38; A123:1<=i+1 by NAT_1:29; then A124: i+1 in Seg (2*(len f)-'1) by A31,A120,FINSEQ_1:3; A125:g1.(i+1)=g1/.(i+1) by A120,A123,FINSEQ_4:24; A126:g1.i=g1/.i by A122,FINSEQ_4:24; per cases by GROUP_4:100; suppose A127:i mod 2=0; consider j such that A128: i=2*j+ (i mod 2) & (i mod 2)<2 by NAT_1:def 2; A129:g1/.i=|[(f/.j)`1,(f/.(j+1))`2]| by A30,A31,A121,A126,A127,A128; i+1=2*j+1+1-1 by A127,A128,XCMPLX_1:26 .=2*j+(1+1)-1 by XCMPLX_1:1 .=2*j+2*1-1; then A130:i+1=2*(j+1)-1 by XCMPLX_1:8; 1<=1+i by NAT_1:29; then 2*(j+1)-'1 =2*(j+1)-1 by A130,JORDAN3:1; then A131:g1/.(i+1)=f/.(j+1) by A30,A124,A125,A130; A132:(g1/.i)`1=(f/.j)`1 by A129,EUCLID:56; A133:(g1/.i)`2=(f/.(j+1))`2 by A129,EUCLID:56; A134:g1/.i-g1/.(i+1) =|[(g1/.i)`1-(g1/.(i+1))`1,(g1/.i)`2-(g1/.(i+1))`2]| by EUCLID:65 .=|[(f/.j)`1-(f/.(j+1))`1,0]| by A131,A132,A133,XCMPLX_1:14; then A135:(g1/.i-g1/.(i+1))`1=(f/.j)`1-(f/.(j+1))`1 by EUCLID:56; (g1/.i-g1/.(i+1))`2=0 by A134,EUCLID:56; then |.g1/.i-g1/.(i+1).|= sqrt((((f/.j)`1-(f/.(j+1))`1))^2+0) by A135, Th47,SQUARE_1:60 .= sqrt((((f/.j)`1-(f/.(j+1))`1))^2); then |.g1/.i-g1/.(i+1).|=abs((f/.j)`1-(f/.(j+1))`1) by SQUARE_1:91; then A136:|.g1/.i-g1/.(i+1).|<=|.f/.j-f/.(j+1).| by Th51; j<>0 by A120,A127,A128; then j>0 by NAT_1:19; then A137: j>=0+1 by NAT_1:38; 2*j+1<=2*(len f)-1+1 by A31,A57,A120,A127,A128,NAT_1:38; then 2*j+1<=2*(len f) by XCMPLX_1:27; then 2*j<2*(len f) by NAT_1:38; then 2*j/2<2*(len f)/2 by REAL_1:73; then j<2*(len f)/2 by XCMPLX_1:90; then j<len f by XCMPLX_1:90; then j+1<=len f by NAT_1:38; then |. f/.j-f/.(j+1) .|<a by A1,A137; hence thesis by A136,AXIOMS:22; suppose A138:i mod 2=1; consider j such that A139: i=2*j+ (i mod 2) & i mod 2<2 by NAT_1:def 2; i=2*j+1+1-1 by A138,A139,XCMPLX_1:26 .=2*j+(1+1)-1 by XCMPLX_1:1 .=2*j+2*1-1; then A140:i=2*(j+1)-1 by XCMPLX_1:8; then 2*(j+1)-'1 =2*(j+1)-1 by A120,JORDAN3:1; then A141:g1/.i=f/.(j+1) by A30,A31,A121,A126,A140; i+1=2*j+(1+1) by A138,A139,XCMPLX_1:1 .=2*j+2*1; then A142:i+1=2*(j+1) by XCMPLX_1:8; then A143:g1/.(i+1)=|[(f/.(j+1))`1,(f/.(j+1+1))`2]| by A30,A124,A125; then A144:(g1/.(i+1))`1=(f/.(j+1))`1 by EUCLID:56; A145:(g1/.(i+1))`2=(f/.(j+1+1))`2 by A143,EUCLID:56; A146:g1/.i-g1/.(i+1) =|[(g1/.i)`1-(g1/.(i+1))`1,(g1/.i)`2-(g1/.(i+1))`2]| by EUCLID:65 .=|[0,(f/.(j+1))`2-(f/.(j+1+1))`2]| by A141,A144,A145,XCMPLX_1:14; then A147:(g1/.i-g1/.(i+1))`1=0 by EUCLID:56; (g1/.i-g1/.(i+1))`2=(f/.(j+1))`2-(f/.(j+1+1))`2 by A146,EUCLID:56; then |.g1/.i-g1/.(i+1).|= sqrt(0+(((f/.(j+1))`2-(f/.(j+1+1))`2))^2) by A147,Th47,SQUARE_1:60 .= sqrt((((f/.(j+1))`2-(f/.(j+1+1))`2))^2); then |.g1/.i-g1/.(i+1).|=abs((f/.(j+1))`2-(f/.(j+1+1))`2) by SQUARE_1:91; then A148:|.g1/.i-g1/.(i+1).| <=|.f/.(j+1)-f/.(j+1+1).| by Th51; A149: j+1>=1 by NAT_1:29; 2*(j+1)+1<=2*(len f)-1+1 by A31,A57,A120,A142,AXIOMS:24; then 2*(j+1)+1<=2*(len f) by XCMPLX_1:27; then 2*(j+1)<2*(len f) by NAT_1:38; then 2*(j+1)/2<2*(len f)/2 by REAL_1:73; then (j+1)<2*(len f)/2 by XCMPLX_1:90; then (j+1)<len f by XCMPLX_1:90; then (j+1)+1<=len f by NAT_1:38; then |. f/.(j+1)-f/.(j+1+1) .|<a by A1,A149; hence thesis by A148,AXIOMS:22; end; hence thesis by A55,A60,A61,A62,A81,A100,A101; end; canceled; theorem Th59: for f being FinSequence of TOP-REAL 2 st 1 <=len f holds len X_axis(f)=len f & X_axis(f).1=(f/.1)`1 & X_axis(f).len f=(f/.len f)`1 proof let f be FinSequence of TOP-REAL 2; assume A1:1 <=len f; then A2: 1 in Seg len f by FINSEQ_1:3; A3: len (X_axis(f)) = len f & for k st k in dom (X_axis(f)) holds (X_axis(f)).k = (f/.k)`1 by GOBOARD1:def 3; then A4: 1 in dom (X_axis(f)) by A2,FINSEQ_1:def 3; len f in Seg len f by A1,FINSEQ_1:3; then len f in dom (X_axis(f)) by A3,FINSEQ_1:def 3; hence thesis by A4,GOBOARD1:def 3; end; theorem Th60: for f being FinSequence of TOP-REAL 2 st 1 <=len f holds len Y_axis(f)=len f & Y_axis(f).1=(f/.1)`2 & Y_axis(f).len f=(f/.len f)`2 proof let f be FinSequence of TOP-REAL 2; assume A1:1 <=len f; then A2: 1 in Seg len f by FINSEQ_1:3; A3: len (Y_axis(f)) = len f & for k st k in dom (Y_axis(f)) holds (Y_axis(f)).k = (f/.k)`2 by GOBOARD1:def 4; then A4: 1 in dom (Y_axis(f)) by A2,FINSEQ_1:def 3; len f in Seg len f by A1,FINSEQ_1:3; then len f in dom (Y_axis(f)) by A3,FINSEQ_1:def 3; hence thesis by A4,GOBOARD1:def 4; end; theorem Th61: for f being FinSequence of TOP-REAL 2 st i in dom f holds X_axis(f).i=(f/.i)`1 & Y_axis(f).i=(f/.i)`2 proof let f be FinSequence of TOP-REAL 2; assume A1: i in dom f; len (X_axis(f)) = len f by GOBOARD1:def 3; then i in dom (X_axis(f)) by A1,FINSEQ_3:31; hence (X_axis(f)).i = (f/.i)`1 by GOBOARD1:def 3; len (Y_axis(f)) = len f by GOBOARD1:def 4; then i in dom Y_axis f by A1,FINSEQ_3:31; hence thesis by GOBOARD1:def 4; end; :: Goboard Theorem in continuous case theorem Th62:for P,Q being non empty Subset of TOP-REAL 2, p1,p2,q1,q2 being Point of TOP-REAL 2 st P is_an_arc_of p1,p2 & Q is_an_arc_of q1,q2 & (for p being Point of TOP-REAL 2 st p in P holds p1`1<=p`1 & p`1<= p2`1) & (for p being Point of TOP-REAL 2 st p in Q holds p1`1<=p`1 & p`1<= p2`1) & (for p being Point of TOP-REAL 2 st p in P holds q1`2<=p`2 & p`2<= q2`2) & (for p being Point of TOP-REAL 2 st p in Q holds q1`2<=p`2 & p`2<= q2`2) holds P meets Q proof let P,Q be non empty Subset of TOP-REAL 2, p1,p2,q1,q2 be Point of TOP-REAL 2; assume that A1:P is_an_arc_of p1,p2 & Q is_an_arc_of q1,q2 and A2:(for p being Point of TOP-REAL 2 st p in P holds p1`1<=p`1 & p`1<= p2`1) and A3:(for p being Point of TOP-REAL 2 st p in Q holds p1`1<=p`1 & p`1<= p2`1) and A4:(for p being Point of TOP-REAL 2 st p in P holds q1`2<=p`2 & p`2<= q2`2) and A5:(for p being Point of TOP-REAL 2 st p in Q holds q1`2<=p`2 & p`2<= q2`2); A6:p1<>p2 by A1,JORDAN6:49; A7:q1<>q2 by A1,JORDAN6:49; consider f being map of I[01], (TOP-REAL 2)|P such that A8:f is_homeomorphism & f.0 = p1 & f.1 = p2 by A1,TOPREAL1:def 2; consider g being map of I[01], (TOP-REAL 2)|Q such that A9:g is_homeomorphism & g.0 = q1 & g.1 = q2 by A1,TOPREAL1:def 2; consider f1 being map of I[01],TOP-REAL 2 such that A10: f=f1 & f1 is continuous & f1 is one-to-one by A8,JORDAN7:15; consider g1 being map of I[01],TOP-REAL 2 such that A11: g=g1 & g1 is continuous & g1 is one-to-one by A9,JORDAN7:15; A12:TOP-REAL 2=TopSpaceMetr(Euclid 2) by EUCLID:def 8; A13: Euclid 2= MetrStruct(#REAL 2,Pitag_dist 2#) by EUCLID:def 7; reconsider P'=P,Q'=Q as Subset of TopSpaceMetr(Euclid 2) by A12; A14:P' <> {} & P' is compact & Q' <> {} & Q' is compact by A1,A12,JORDAN5A:1; assume P /\ Q= {}; then P misses Q by XBOOLE_0:def 7; then A15:min_dist_min(P',Q')>0 by A14,Th55; set e=min_dist_min(P',Q')/5; A16:e*5=min_dist_min(P',Q') by XCMPLX_1:88; A17: e>0/5 by A15,REAL_1:73; then consider h being FinSequence of REAL such that A18: h.1=0 & h.len h=1 & 5<=len h & rng h c= the carrier of I[01] & h is increasing &(for i being Nat,R being Subset of I[01], W being Subset of Euclid 2 st 1<=i & i<len h & R=[.h/.i,h/.(i+1).] & W=f1.:(R) holds diameter(W)<e) by A8,A10,UNIFORM1:15; deffunc F(Nat)=f1.(h.$1); ex h1' being FinSequence st len h1'=len h & for i st i in Seg len h holds h1'.i=F(i) from SeqLambda; then consider h1' being FinSequence such that A19:len h1'=len h & for i st i in Seg len h holds h1'.i=f1.(h.i); rng h1' c= the carrier of TOP-REAL 2 proof let y;assume y in rng h1'; then consider x such that A20:x in dom h1' & y=h1'.x by FUNCT_1:def 5; A21:dom h1'=Seg len h1' by FINSEQ_1:def 3; reconsider i=x as Nat by A20; A22: h1'.i=f1.(h.i) by A19,A20,A21; i in dom h by A19,A20,A21,FINSEQ_1:def 3; then A23:h.i in rng h by FUNCT_1:def 5; dom f1= [#](I[01]) by A8,A10,TOPS_2:def 5 .=the carrier of I[01] by PRE_TOPC:12; then A24:h1'.i in rng f by A10,A18,A22,A23,FUNCT_1:def 5; rng f=[#]((TOP-REAL 2)|P) by A8,TOPS_2:def 5 .=P by PRE_TOPC:def 10; hence thesis by A20,A24; end; then reconsider h1=h1' as FinSequence of TOP-REAL 2 by FINSEQ_1:def 4; A25: dom h=Seg len h by FINSEQ_1:def 3; A26:the carrier of TOP-REAL 2=the carrier of Euclid 2 by TOPREAL3:13; A27:for i st 1<=i & i+1<=len h1 holds |.h1/.i-h1/.(i+1).|<e proof let i;assume A28:1<=i & i+1<=len h1; then A29:i<len h1 by NAT_1:38; A30:1<i+1 by A28,NAT_1:38; then A31:i+1 in Seg len h by A19,A28,FINSEQ_1:3; A32: i in Seg len h by A19,A28,A29,FINSEQ_1:3; then A33:h1'.i=f1.(h.i) by A19; then A34:h1/.i=f1.(h.i) by A28,A29,FINSEQ_4:24; A35:h1'.(i+1)=f1.(h.(i+1)) by A19,A31; then A36:h1/.(i+1)=f1.(h.(i+1)) by A28,A30,FINSEQ_4:24; A37:i in dom h by A19,A28,A29,FINSEQ_3:27; then A38:h.i in rng h by FUNCT_1:def 5; A39: dom f1= [#](I[01]) by A8,A10,TOPS_2:def 5 .=the carrier of I[01] by PRE_TOPC:12; then A40:h1'.i in rng f by A10,A18,A33,A38,FUNCT_1:def 5; A41:rng f=[#]((TOP-REAL 2)|P) by A8,TOPS_2:def 5 .=P by PRE_TOPC:def 10; then A42:f1.(h.i) is Element of TOP-REAL 2 by A19,A32, A40; i+1 in dom h by A19,A28,A30,FINSEQ_3:27; then h.(i+1) in rng h by FUNCT_1:def 5; then h1'.(i+1) in rng f by A10,A18,A35,A39,FUNCT_1:def 5; then A43:f1.(h.(i+1)) is Element of TOP-REAL 2 by A19, A31,A41; A44:h.i=h/.i by A19,A28,A29,FINSEQ_4:24; A45:i+1 in dom h by A19,A28,A30,FINSEQ_3:27; then A46:h.(i+1) in rng h by FUNCT_1:def 5; A47:h.(i+1)=h/.(i+1) by A19,A28,A30,FINSEQ_4:24; i<i+1 by NAT_1:38; then A48:0<=h/.i & h/.i <= h/.(i+1) & h/.(i+1)<=1 by A18,A37,A38,A44,A45,A46,A47,BORSUK_1:83,GOBOARD1:def 1,TOPREAL5 :1; reconsider W1=P as Subset of Euclid 2 by A26; reconsider Pa=P as Subset of TOP-REAL 2; A49:Pa is compact by A1,JORDAN5A:1; reconsider Pa as non empty Subset of TOP-REAL 2; reconsider R=[.h/.i,h/.(i+1).] as Subset of I[01] by A18,A38,A44,A46,A47,BORSUK_1:83,RCOMP_1:16; reconsider w1=f1.(h.i),w2=f1.(h.(i+1)) as Point of Euclid 2 by A42,A43,TOPREAL3:13; A50: h.i in R by A44,A48,TOPREAL5:1; h.(i+1) in R by A47,A48,TOPREAL5:1; then A51:w1 in f1.:R & w2 in f1.:R by A39,A50,FUNCT_1:def 12; reconsider W=f1.:R as Subset of Euclid 2 by A26; A52:dom f1=[#](I[01]) & rng f1=P by A8,A10,A41,TOPS_2:def 5; [#](I[01])=[.0,1.] by BORSUK_1:83,PRE_TOPC:12; then P=f1.:([.0,1.]) by A52,RELAT_1:146; then A53:W c= W1 by BORSUK_1:83,RELAT_1:156; p1 in Pa by A1,TOPREAL1:4; then A54: W-bound Pa <= p1`1 & p1`1 <= E-bound Pa & S-bound Pa <= p1`2 & p1`2 <= N-bound Pa by A49,PSCOMP_1:71; set r1=((E-bound Pa) - (W-bound Pa))+((N-bound Pa) - (S-bound Pa))+1; (W-bound Pa)<=(E-bound Pa) by A54,AXIOMS:22; then A55:0<=((E-bound Pa) - (W-bound Pa)) by SQUARE_1:12; (S-bound Pa)<=(N-bound Pa) by A54,AXIOMS:22; then 0<=((N-bound Pa) - (S-bound Pa)) by SQUARE_1:12; then A56:((E-bound Pa) - (W-bound Pa))+((N-bound Pa) - (S-bound Pa)) >=0+0 by A55,REAL_1:55; A57: r1> ((E-bound Pa) - (W-bound Pa))+((N-bound Pa) - (S-bound Pa)) by REAL_1:69; for x,y being Point of Euclid 2 st x in W1 & y in W1 holds dist(x,y)<=r1 proof let x,y be Point of Euclid 2; assume A58: x in W1 & y in W1; then reconsider pw1=x,pw2=y as Point of TOP-REAL 2; A59:dist(x,y)=|.pw1-pw2.| by Th45; A60:|.pw1-pw2.|<=abs(pw1`1-pw2`1)+abs(pw1`2-pw2`2) by Th49; A61:W-bound Pa <= pw1`1 & pw1`1 <= E-bound Pa & S-bound Pa <= pw1`2 & pw1`2 <= N-bound Pa by A49,A58,PSCOMP_1:71; A62:W-bound Pa <= pw2`1 & pw2`1 <= E-bound Pa & S-bound Pa <= pw2`2 & pw2`2 <= N-bound Pa by A49,A58,PSCOMP_1:71; then A63:abs(pw1`1-pw2`1)<=(E-bound Pa) - (W-bound Pa) by A61,Th31; abs(pw1`2-pw2`2)<=(N-bound Pa) - (S-bound Pa) by A61,A62,Th31; then A64:abs(pw1`1-pw2`1)+abs(pw1`2-pw2`2) <=((E-bound Pa) - (W-bound Pa))+((N-bound Pa) - (S-bound Pa)) by A63,REAL_1:55; ((E-bound Pa) - (W-bound Pa))+((N-bound Pa) - (S-bound Pa)) <= ((E-bound Pa) - (W-bound Pa))+((N-bound Pa) - (S-bound Pa)) +1 by REAL_1:69; then abs(pw1`1-pw2`1)+abs(pw1`2-pw2`2)<=r1 by A64,AXIOMS:22; hence thesis by A59,A60,AXIOMS:22; end; then W1 is bounded by A56,A57,TBSP_1:def 9; then A65:W is bounded by A53,TBSP_1:21; A66: diameter(W)<e by A18,A19,A28,A29; dist(w1,w2)<=diameter(W) by A51,A65,TBSP_1:def 10; then dist(w1,w2)<e by A66,AXIOMS:22; hence thesis by A34,A36,Th45; end; A67:len h1>=1 by A18,A19,AXIOMS:22; consider hb being FinSequence of REAL such that A68: hb.1=0 & hb.len hb=1 & 5<=len hb & rng hb c= the carrier of I[01] & hb is increasing &(for i being Nat,R being Subset of I[01], W being Subset of Euclid 2 st 1<=i & i<len hb & R=[.hb/.i,hb/.(i+1).] & W=g1.:(R) holds diameter(W)<e) by A9,A11,A17,UNIFORM1:15; A69: 1<=len hb by A68,AXIOMS:22; then A70:1 in dom hb by FINSEQ_3:27; A71:len hb in dom hb by A69,FINSEQ_3:27; deffunc F(Nat)=g1.(hb.$1); ex h2' being FinSequence st len h2'=len hb & for i st i in Seg len hb holds h2'.i=F(i) from SeqLambda; then consider h2' being FinSequence such that A72:len h2'=len hb & for i st i in Seg len hb holds h2'.i=g1.(hb.i); rng h2' c= the carrier of TOP-REAL 2 proof let y;assume y in rng h2'; then consider x such that A73:x in dom h2' & y=h2'.x by FUNCT_1:def 5; A74:dom h2'=Seg len h2' by FINSEQ_1:def 3; reconsider i=x as Nat by A73; A75: h2'.i=g1.(hb.i) by A72,A73,A74; i in dom hb by A72,A73,A74,FINSEQ_1:def 3; then A76:hb.i in rng hb by FUNCT_1:def 5; dom g1= [#](I[01]) by A9,A11,TOPS_2:def 5 .=the carrier of I[01] by PRE_TOPC:12; then A77:h2'.i in rng g by A11,A68,A75,A76,FUNCT_1:def 5; rng g=[#]((TOP-REAL 2)|Q) by A9,TOPS_2:def 5 .=Q by PRE_TOPC:def 10; hence thesis by A73,A77; end; then reconsider h2=h2' as FinSequence of TOP-REAL 2 by FINSEQ_1:def 4; A78:dom hb=Seg len hb by FINSEQ_1:def 3; A79: for i st i in dom hb holds h2/.i=g1.(hb.i) proof let i;assume A80: i in dom hb; then A81:h2.i=g1.(hb.i) by A72,A78; 1<=i & i<=len hb by A78,A80,FINSEQ_1:3; hence thesis by A72,A81,FINSEQ_4:24; end; A82:len h2>=1 by A68,A72,AXIOMS:22; A83:for i st i in dom h1 holds (h1/.1)`1<=(h1/.i)`1 & (h1/.i)`1<=(h1/.len h1)`1 proof let i;assume A84:i in dom h1; then A85: i in Seg len h by A19,FINSEQ_1:def 3; then A86:i in dom h by FINSEQ_1:def 3; h1.i=f1.(h.i) by A19,A85; then A87:h1/.i=f1.(h.i) by A84,FINSEQ_4:def 4; 1 in dom h by A19,A67,FINSEQ_3:27; then h1.1=f1.(h.1) by A19,A25; then A88:h1/.1=f1.(h.1) by A67,FINSEQ_4:24; len h in dom h by A19,A67,FINSEQ_3:27; then h1.len h1=f1.(h.len h) by A19,A25; then A89:h1/.len h1=f1.(h.len h) by A67,FINSEQ_4:24; A90:h.i in rng h by A86,FUNCT_1:def 5; dom f1= [#](I[01]) by A8,A10,TOPS_2:def 5 .=the carrier of I[01] by PRE_TOPC:12; then A91:h1/.i in rng f by A10,A18,A87,A90,FUNCT_1:def 5; rng f=[#]((TOP-REAL 2)|P) by A8,TOPS_2:def 5 .=P by PRE_TOPC:def 10; hence thesis by A2,A8,A10,A18,A88,A89,A91; end; A92: for i st i in dom (X_axis(h1)) holds (X_axis(h1)).1<=(X_axis(h1)).i & (X_axis(h1)).i<=(X_axis(h1)).(len h1) proof let i;assume i in dom (X_axis(h1)); then i in Seg len (X_axis(h1)) by FINSEQ_1:def 3; then i in Seg len h1 by A67,Th59; then A93:i in dom h1 by FINSEQ_1:def 3; then A94:(X_axis(h1)).i = (h1/.i)`1 by Th61; A95:(X_axis(h1)).1=(h1/.1)`1 by A67,Th59; (X_axis(h1)).len h1=(h1/.len h1)`1 by A67,Th59; hence thesis by A83,A93,A94,A95; end; A96:for i st i in dom h1 holds q1`2<=(h1/.i)`2 & (h1/.i)`2<=q2`2 proof let i;assume A97:i in dom h1; then A98: i in Seg len h1 by FINSEQ_1:def 3; then A99:i in dom h by A19,FINSEQ_1:def 3; h1.i=f1.(h.i) by A19,A98; then A100:h1/.i=f1.(h.i) by A97,FINSEQ_4:def 4; A101:h.i in rng h by A99,FUNCT_1:def 5; dom f1= [#](I[01]) by A8,A10,TOPS_2:def 5 .=the carrier of I[01] by PRE_TOPC:12; then A102:h1/.i in rng f by A10,A18,A100,A101,FUNCT_1:def 5; rng f=[#]((TOP-REAL 2)|P) by A8,TOPS_2:def 5 .=P by PRE_TOPC:def 10; hence thesis by A4,A102; end; for i st i in dom (Y_axis(h1)) holds q1`2<=(Y_axis(h1)).i & (Y_axis(h1)).i<=q2`2 proof let i;assume i in dom (Y_axis(h1)); then i in Seg len (Y_axis(h1)) by FINSEQ_1:def 3; then i in Seg len h1 by A67,Th60; then A103:i in dom h1 by FINSEQ_1:def 3; then (Y_axis(h1)).i = (h1/.i)`2 by Th61; hence thesis by A96,A103; end; then A104:X_axis(h1) lies_between (X_axis(h1)).1, (X_axis(h1)).(len h1) & Y_axis(h1) lies_between q1`2,q2`2 by A92,GOBOARD4:def 1; A105:for i st i in dom h2 holds (h2/.1)`2<=(h2/.i)`2 & (h2/.i)`2<=(h2/.len h2)`2 proof let i;assume i in dom h2; then i in Seg len h2 by FINSEQ_1:def 3; then A106:i in dom hb by A72,FINSEQ_1:def 3; then A107:h2/.i=g1.(hb.i) by A79; 1 in Seg len hb by A72,A82,FINSEQ_1:3; then 1 in dom hb by FINSEQ_1:def 3; then A108:h2/.1=g1.(hb.1) by A79; len hb in Seg len hb by A72,A82,FINSEQ_1:3; then len hb in dom hb by FINSEQ_1:def 3; then A109:h2/.len h2=g1.(hb.len hb) by A72,A79; A110:hb.i in rng hb by A106,FUNCT_1:def 5; dom g1= [#](I[01]) by A9,A11,TOPS_2:def 5 .=the carrier of I[01] by PRE_TOPC:12; then A111:h2/.i in rng g by A11,A68,A107,A110,FUNCT_1:def 5; rng g=[#]((TOP-REAL 2)|Q) by A9,TOPS_2:def 5 .=Q by PRE_TOPC:def 10; hence thesis by A5,A9,A11,A68,A108,A109,A111; end; A112: for i st i in dom (Y_axis(h2)) holds (Y_axis(h2)).1<=(Y_axis(h2)).i & (Y_axis(h2)).i <=(Y_axis(h2)).(len h2) proof let i;assume i in dom (Y_axis(h2)); then i in Seg len (Y_axis(h2)) by FINSEQ_1:def 3; then i in Seg len h2 by A82,Th60; then A113:i in dom h2 by FINSEQ_1:def 3; then A114:(Y_axis(h2)).i = (h2/.i)`2 by Th61; A115:(Y_axis(h2)).1=(h2/.1)`2 by A82,Th60; (Y_axis(h2)).len h2=(h2/.len h2)`2 by A82,Th60; hence thesis by A105,A113,A114,A115; end; A116:for i st i in dom h2 holds p1`1<=(h2/.i)`1 & (h2/.i)`1<=p2`1 proof let i;assume i in dom h2; then i in Seg len h2 by FINSEQ_1:def 3; then A117:i in dom hb by A72,FINSEQ_1:def 3; then A118:h2/.i=g1.(hb.i) by A79; A119:hb.i in rng hb by A117,FUNCT_1:def 5; dom g1= [#](I[01]) by A9,A11,TOPS_2:def 5 .=the carrier of I[01] by PRE_TOPC:12; then A120:h2/.i in rng g by A11,A68,A118,A119,FUNCT_1:def 5; rng g=[#]((TOP-REAL 2)|Q) by A9,TOPS_2:def 5 .=Q by PRE_TOPC:def 10; hence thesis by A3,A120; end; for i st i in dom (X_axis(h2)) holds p1`1<= (X_axis(h2)).i & (X_axis(h2)).i<=p2`1 proof let i;assume i in dom (X_axis(h2)); then i in Seg len (X_axis(h2)) by FINSEQ_1:def 3; then i in Seg len h2 by A82,Th59; then A121:i in dom h2 by FINSEQ_1:def 3; then (X_axis(h2)).i = (h2/.i)`1 by Th61; hence thesis by A116,A121; end; then A122:X_axis(h2) lies_between p1`1, p2`1 & Y_axis(h2) lies_between (Y_axis(h2)).1, (Y_axis(h2)).(len h2) by A112,GOBOARD4:def 1; A123:for i st 1<=i & i+1<=len h2 holds |. (h2/.i)-(h2/.(i+1)) .|<e proof let i;assume A124:1<=i & i+1<=len h2; then A125:i<len h2 by NAT_1:38; A126:1<i+1 by A124,NAT_1:38; then A127:i+1 in Seg len hb by A72,A124,FINSEQ_1:3; i in Seg len hb by A72,A124,A125,FINSEQ_1:3; then A128:h2'.i=g1.(hb.i) by A72; then A129:h2/.i=g1.(hb.i) by A124,A125,FINSEQ_4:24; A130:h2'.(i+1)=g1.(hb.(i+1)) by A72,A127; h2.(i+1)=h2/.(i+1) by A124,A126,FINSEQ_4:24; then A131:h2/.(i+1)=g1.(hb.(i+1)) by A72,A127; i in Seg len hb by A72,A124,A125,FINSEQ_1:3; then A132:i in dom hb by FINSEQ_1:def 3; then A133:hb.i in rng hb by FUNCT_1:def 5; A134: dom g1= [#](I[01]) by A9,A11,TOPS_2:def 5 .=the carrier of I[01] by PRE_TOPC:12; then A135:h2'.i in rng g by A11,A68,A128,A133,FUNCT_1:def 5; A136:rng g=[#]((TOP-REAL 2)|Q) by A9,TOPS_2:def 5 .=Q by PRE_TOPC:def 10; i+1 in Seg len hb by A72,A124,A126,FINSEQ_1:3; then i+1 in dom hb by FINSEQ_1:def 3; then hb.(i+1) in rng hb by FUNCT_1:def 5; then h2'.(i+1) in rng g by A11,A68,A130,A134,FUNCT_1:def 5; then A137:g1.(hb.(i+1)) is Element of TOP-REAL 2 by A72,A127, A136; A138:hb.i=hb/.i by A72,A124,A125,FINSEQ_4:24; i+1 in Seg len hb by A72,A124,A126,FINSEQ_1:3; then A139:i+1 in dom hb by FINSEQ_1:def 3; then A140:hb.(i+1) in rng hb by FUNCT_1:def 5; A141:hb.(i+1)=hb/.(i+1) by A72,A124,A126,FINSEQ_4:24; i<i+1 by NAT_1:38; then A142:0<=hb/.i & hb/.i <= hb/.(i+1) & hb/.(i+1)<=1 by A68,A132,A133,A138,A139,A140,A141,BORSUK_1:83,GOBOARD1:def 1, TOPREAL5:1; reconsider W1=Q as Subset of Euclid 2 by A26; reconsider Qa=Q as Subset of TOP-REAL 2; A143:Qa is compact by A1,JORDAN5A:1; reconsider Qa as non empty Subset of TOP-REAL 2; reconsider R=[.hb/.i,hb/.(i+1).] as Subset of I[01] by A68,A133,A138,A140,A141,BORSUK_1:83, RCOMP_1:16; reconsider w1=g1.(hb.i),w2=g1.(hb.(i+1)) as Point of Euclid 2 by A128,A135,A136,A137,TOPREAL3:13; A144: hb.i in R by A138,A142,TOPREAL5:1; hb.(i+1) in R by A141,A142,TOPREAL5:1; then A145:w1 in g1.:(R) & w2 in g1.:(R) by A134,A144,FUNCT_1:def 12; reconsider W=g1.:(R) as Subset of Euclid 2 by A26; A146:dom g1=[#](I[01]) & rng g1=Q by A9,A11,A136,TOPS_2:def 5; [#](I[01])=[.0,1.] by BORSUK_1:83,PRE_TOPC:12; then Q=g1.:([.0,1.]) by A146,RELAT_1:146; then A147:W c= W1 by BORSUK_1:83,RELAT_1:156; q1 in Qa by A1,TOPREAL1:4; then A148: W-bound Qa <= q1`1 & q1`1 <= E-bound Qa & S-bound Qa <= q1`2 & q1`2 <= N-bound Qa by A143,PSCOMP_1:71; set r1= ((E-bound Qa) - (W-bound Qa))+((N-bound Qa) - (S-bound Qa))+1; (W-bound Qa)<=(E-bound Qa) by A148,AXIOMS:22; then A149:0<=((E-bound Qa) - (W-bound Qa)) by SQUARE_1:12; (S-bound Qa)<=(N-bound Qa) by A148,AXIOMS:22; then 0<=((N-bound Qa) - (S-bound Qa)) by SQUARE_1:12; then A150:((E-bound Qa) - (W-bound Qa))+((N-bound Qa) - (S-bound Qa)) >=0+0 by A149,REAL_1:55; A151: r1> ((E-bound Qa) - (W-bound Qa))+((N-bound Qa) - (S-bound Qa)) by REAL_1:69; for x,y being Point of Euclid 2 st x in W1 & y in W1 holds dist(x,y)<=r1 proof let x,y be Point of Euclid 2; assume A152: x in W1 & y in W1; then reconsider pw1=x,pw2=y as Point of TOP-REAL 2; A153:dist(x,y)=|.pw1-pw2.| by Th45; A154:|.pw1-pw2.|<=abs(pw1`1-pw2`1)+abs(pw1`2-pw2`2) by Th49; A155:W-bound Qa <= pw1`1 & pw1`1 <= E-bound Qa & S-bound Qa <= pw1`2 & pw1`2 <= N-bound Qa by A143,A152,PSCOMP_1:71; A156:W-bound Qa <= pw2`1 & pw2`1 <= E-bound Qa & S-bound Qa <= pw2`2 & pw2`2 <= N-bound Qa by A143,A152,PSCOMP_1:71; then A157:abs(pw1`1-pw2`1)<=(E-bound Qa) - (W-bound Qa) by A155,Th31 ; abs(pw1`2-pw2`2)<=(N-bound Qa) - (S-bound Qa) by A155,A156,Th31; then A158:abs(pw1`1-pw2`1)+abs(pw1`2-pw2`2) <=((E-bound Qa) - (W-bound Qa))+((N-bound Qa) - (S-bound Qa)) by A157,REAL_1:55; ((E-bound Qa) - (W-bound Qa))+((N-bound Qa) - (S-bound Qa)) <= ((E-bound Qa) - (W-bound Qa))+((N-bound Qa) - (S-bound Qa)) +1 by REAL_1:69; then abs(pw1`1-pw2`1)+abs(pw1`2-pw2`2)<=r1 by A158,AXIOMS:22; hence thesis by A153,A154,AXIOMS:22; end; then W1 is bounded by A150,A151,TBSP_1:def 9; then A159:W is bounded by A147,TBSP_1:21; A160: diameter(W)<e by A68,A72,A124,A125; dist(w1,w2)<=diameter(W) by A145,A159,TBSP_1:def 10; then dist(w1,w2)<e by A160,AXIOMS:22; hence thesis by A129,A131,Th45; end; consider f2 being FinSequence of TOP-REAL 2 such that A161: f2 is special & f2.1=h1.1 & f2.len f2=h1.len h1 & len f2>=len h1 & X_axis(f2) lies_between (X_axis(h1)).1, (X_axis(h1)).(len h1) & Y_axis(f2) lies_between q1`2,q2`2 & (for j st j in dom f2 holds ex k st k in dom h1 & |.(f2/.j - h1/.k).|<e) &(for j st 1<=j & j+1<=len f2 holds |.f2/.j - f2/.(j+1).|<e) by A17,A27,A67,A104,Th56; A162:len f2>=1 by A67,A161,AXIOMS:22; consider g2 being FinSequence of TOP-REAL 2 such that A163: g2 is special & g2.1=h2.1 & g2.len g2=h2.len h2 & len g2>=len h2 & Y_axis(g2) lies_between (Y_axis(h2)).1, (Y_axis(h2)).(len h2) & X_axis(g2) lies_between p1`1,p2`1 & (for j st j in dom g2 holds ex k st k in dom h2 & |.((g2/.j) - h2/.k).|<e)& (for j st 1<=j & j+1<=len g2 holds |.(g2/.j) - g2/.(j+1).|<e) by A17,A82,A122,A123,Th57; A164:len g2>=1 by A82,A163,AXIOMS:22; A165:1 in dom h by A19,A67,FINSEQ_3:27; A166:len h in dom h by A19,A67,FINSEQ_3:27; A167: h1.1=f1.(h.1) by A19,A25,A165; A168:h1.1=h1/.1 by A67,FINSEQ_4:24; A169:h1.len h1=h1/.len h1 by A67,FINSEQ_4:24; A170:f2.1=f2/.1 by A162,FINSEQ_4:24; A171:f2.len f2=f2/.len f2 by A162,FINSEQ_4:24; A172:X_axis(f2).1=p1`1 by A8,A10,A18,A161,A162,A167,A170,Th59; (h1/.len h1)=p2 by A8,A10,A18,A19,A25,A166,A169; then A173:X_axis(f2).len f2=p2`1 by A161,A162,A169,A171,Th59; A174:(h2/.1)=q1 by A9,A11,A68,A70,A79; A175:h2.1=h2/.1 by A69,A72,FINSEQ_4:24; A176:h2.len h2=h2/.len h2 by A69,A72,FINSEQ_4:24; g2/.1=g2.1 by A164,FINSEQ_4:24; then A177:Y_axis(g2).1=q1`2 by A163,A164,A174,A175,Th60; A178:g2/.len g2=g2.len g2 by A164,FINSEQ_4:24; (g2.len g2)=q2 by A9,A11,A68,A71,A72,A79,A163,A176; then A179:Y_axis(g2).len g2=q2`2 by A164,A178,Th60; A180:(X_axis(f2)).1=(h1/.1)`1 by A161,A162,A168,A170,Th59 .=(X_axis(h1)).1 by A67,Th59; A181:X_axis(f2).len f2=(h1/.len h1)`1 by A161,A162,A169,A171,Th59 .=X_axis(h1).len h1 by A67,Th59; g2.1=g2/.1 by A164,FINSEQ_4:24; then A182:(Y_axis(g2)).1=(h2/.1)`2 by A163,A164,A175,Th60 .=(Y_axis(h2)).1 by A82,Th60; g2.len g2=g2/.len g2 by A164,FINSEQ_4:24; then A183: Y_axis(g2).len g2=(h2/.len h2)`2 by A163,A164,A176,Th60 .=Y_axis(h2).len h2 by A82,Th60; 5<=len f2 by A18,A19,A161,AXIOMS:22; then A184:2<=len f2 by AXIOMS:22; 5<=len g2 by A68,A72,A163,AXIOMS:22; then A185:2<=len g2 by AXIOMS:22; A186:1<=len h by A18,AXIOMS:22; then 1 in dom h by FINSEQ_3:27; then A187:h1/.1=f1.(h.1) by A19,A25,A168; len h in dom h by A186,FINSEQ_3:27; then A188: f2/.1<>f2/.len f2 by A6,A8,A10,A18,A19,A25,A161,A168,A170,A171, A187; A189:1<=len hb by A68,AXIOMS:22; then 1 in dom hb by FINSEQ_3:27; then A190:h2/.1=g1.(hb.1) by A79; len hb in dom hb by A189,FINSEQ_3:27; then g2.1<>g2.len g2 by A7,A9,A11,A68,A72,A79,A163,A175,A176,A190; then L~f2 meets L~g2 by A161,A163,A170,A171,A172,A173,A177,A179,A180,A181, A182,A183,A184,A185,A188,Th30; then consider s being set such that A191:s in L~f2 & s in L~g2 by XBOOLE_0:3; reconsider ps=s as Point of TOP-REAL 2 by A191; ps in union{ LSeg(f2,i) : 1 <= i & i+1 <= len f2 } by A191,TOPREAL1:def 6; then consider x such that A192: ps in x & x in { LSeg(f2,i) : 1 <= i & i+1 <= len f2 } by TARSKI:def 4; consider i such that A193: x=LSeg(f2,i) &(1 <= i & i+1 <= len f2) by A192; A194:LSeg(f2,i)=LSeg(f2/.i,f2/.(i+1)) by A193,TOPREAL1:def 5; i<len f2 by A193,NAT_1:38; then i in dom f2 by A193,FINSEQ_3:27; then consider k such that A195: k in dom h1 & |.f2/.i - h1/.k.|<e by A161; A196:|.ps-f2/.i.|<=|.f2/.i-f2/.(i+1).| by A192,A193,A194,Th53; |.f2/.i-f2/.(i+1).|<e by A161,A193; then A197:|.ps-f2/.i.|<e by A196,AXIOMS:22; A198:|.ps-h1/.k.|<=|.ps-f2/.i.|+|.f2/.i-h1/.k.| by TOPRNS_1:35; |.ps-f2/.i.|+|.f2/.i-h1/.k.|<e+e by A195,A197,REAL_1:67; then A199:|.ps-h1/.k.|<e+e by A198,AXIOMS:22; A200:k in Seg len h1 by A195,FINSEQ_1:def 3; then A201:k in dom h by A19,FINSEQ_1:def 3; 1<=k & k<=len h1 by A200,FINSEQ_1:3; then h1.k=h1/.k by FINSEQ_4:24; then A202:h1/.k=f1.(h.k) by A19,A200; A203:h.k in rng h by A201,FUNCT_1:def 5; A204: dom f1= [#](I[01]) by A8,A10,TOPS_2:def 5 .=the carrier of I[01] by PRE_TOPC:12; rng f=[#]((TOP-REAL 2)|P) by A8,TOPS_2:def 5 .=P by PRE_TOPC:def 10; then A205: h1/.k in P by A10,A18,A202,A203,A204,FUNCT_1:def 5; reconsider p11=h1/.k as Point of TOP-REAL 2; A206: |.(p11-ps).|<e+e by A199,TOPRNS_1:28; ps in union{ LSeg(g2,j) : 1 <= j & j+1 <= len g2 } by A191,TOPREAL1:def 6; then consider y such that A207: ps in y & y in { LSeg(g2,j) : 1 <= j & j+1 <= len g2 } by TARSKI:def 4; consider j such that A208:y=LSeg(g2,j) &(1 <= j & j+1 <= len g2) by A207; A209: LSeg(g2,j)=LSeg((g2/.j),g2/.(j+1)) & ps in LSeg(g2,j) & y=LSeg(g2,j) &(1 <= j & j+1 <= len g2) by A207,A208,TOPREAL1:def 5; j<len g2 & LSeg(g2,j)=LSeg((g2/.j),g2/.(j+1)) & ps in LSeg(g2,j) & y=LSeg(g2,j) &(1 <= j & j+1 <= len g2) by A207,A208,NAT_1:38,TOPREAL1:def 5; then j in Seg len g2 by FINSEQ_1:3; then j in dom g2 by FINSEQ_1:def 3; then consider k' being Nat such that A210: k' in dom h2 & |.((g2/.j) - h2/.k').|<e by A163; A211:|.ps-(g2/.j).|<=|.(g2/.j)-g2/.(j+1).| by A209,Th53; |.(g2/.j)-g2/.(j+1).|<e by A163,A208; then A212:|.ps-(g2/.j).|<e by A211,AXIOMS:22; A213:|.ps-h2/.k'.|<=|.ps-(g2/.j).|+|.(g2/.j)-h2/.k'.| by TOPRNS_1:35; |.ps-(g2/.j).|+|.(g2/.j)-(h2/.k').|<e+e by A210,A212,REAL_1:67; then A214:|.ps-(h2/.k').|<e+e by A213,AXIOMS:22; k' in Seg len h2 by A210,FINSEQ_1:def 3; then k' in dom hb by A72,FINSEQ_1:def 3; then A215:hb.k' in rng hb & h2/.k'=g1.(hb.k') & k' in dom hb by A79,FUNCT_1:def 5; dom g1= [#](I[01]) by A9,A11,TOPS_2:def 5 .=the carrier of I[01] by PRE_TOPC:12; then A216:Q<>{}(TOP-REAL 2) & h2/.k' in rng g & hb.k' in dom g1 by A11,A68,A215,FUNCT_1:def 5; A217: rng g=[#]((TOP-REAL 2)|Q) by A9,TOPS_2:def 5 .=Q by PRE_TOPC:def 10; reconsider q11=h2/.k' as Point of TOP-REAL 2; reconsider x1=p11,x2=q11 as Point of Euclid 2 by A13,EUCLID:25; min_dist_min(P',Q')<=dist(x1,x2) by A14,A205,A216,A217,WEIERSTR:40; then A218:|.(p11-q11).|<= |.(p11-ps).|+|.(ps-q11).| & min_dist_min(P',Q')<=|.(p11-q11).| by Th45,TOPRNS_1:35; |.(p11-ps).|+|.(ps-q11).| <e+e+(e+e) by A206,A214,REAL_1:67; then |.(p11-q11).|<e+e+(e+e) by A218,AXIOMS:22; then |.(p11-q11).|<e+e+e+e by XCMPLX_1:1; then |.(p11-q11).|<4*e by XCMPLX_1:13; then min_dist_min(P',Q')<4*e by A218,AXIOMS:22; then 4*e-5*e>0 by A16,SQUARE_1:11; then (4-5)*e>0 by XCMPLX_1:40; then (4-5)*e/e>0 by A17,REAL_2:127; then 4-5>0 by A17,XCMPLX_1:90; hence contradiction; end; reserve X,Y for non empty TopSpace; theorem Th63:for f being map of X,Y, P being non empty Subset of Y, f1 being map of X,Y|P st f=f1 & f is continuous holds f1 is continuous proof let f be map of X,Y, P be non empty Subset of Y, f1 be map of X,Y|P; assume A1:f=f1 & f is continuous; A2:rng f1 c= the carrier of (Y|P); for P1 being Subset of (Y|P) st P1 is open holds f1"P1 is open proof let P1 be Subset of (Y|P); assume P1 is open; then P1 in the topology of (Y|P) by PRE_TOPC:def 5; then consider Q being Subset of Y such that A3: Q in the topology of Y & P1 = Q /\ [#](Y|P) by PRE_TOPC:def 9; reconsider Q as Subset of Y; Q is open by A3,PRE_TOPC:def 5; then A4:f"Q is open by A1,TOPS_2:55; A5: [#](Y|P)=P by PRE_TOPC:def 10; A6:f"Q=f1"(rng f1 /\ Q) by A1,RELAT_1:168; Q /\ [#](Y|P) c= Q by XBOOLE_1:17; then A7:f1"P1 c= f"Q by A1,A3,RELAT_1:178; rng f1 c= P by A2,A5,PRE_TOPC:12; then rng f1 /\ Q c= P /\ Q by XBOOLE_1:26; then f1"(rng f1 /\ Q) c= f1"P1 by A3,A5,RELAT_1:178; hence thesis by A4,A6,A7,XBOOLE_0:def 10; end; hence thesis by TOPS_2:55; end; theorem Th64:for f being map of X,Y, P being non empty Subset of Y st X is compact & Y is_T2 & f is continuous one-to-one & P=rng f holds ex f1 being map of X,(Y|P) st f=f1 & f1 is_homeomorphism proof let f be map of X,Y,P be non empty Subset of Y such that A1: X is compact and A2: Y is_T2 and A3: f is continuous one-to-one and A4: P=rng f; A5:f is Function of dom f,rng f by FUNCT_2:3; A6:the carrier of (Y|P)=P by JORDAN1:1; A7:dom f=the carrier of X by FUNCT_2:def 1; then reconsider f2=f as map of X,(Y|P) by A4,A5,A6; A8:dom f2=[#]X by A7,PRE_TOPC:12; A9: rng f2=[#](Y|P) by A4,PRE_TOPC:def 10; A10:Y|P is_T2 by A2,TOPMETR:3; f2 is continuous by A3,Th63; then f2 is_homeomorphism by A1,A3,A8,A9,A10,COMPTS_1:26; hence thesis; end; :: Fashoda Meet Theorem theorem for f,g being map of I[01],TOP-REAL 2,a,b,c,d being real number, O,I being Point of I[01] st O=0 & I=1 & f is continuous one-to-one & g is continuous one-to-one & (f.O)`1=a & (f.I)`1=b & (g.O)`2=c & (g.I)`2=d & (for r being Point of I[01] holds a <=(f.r)`1 & (f.r)`1<=b & a <=(g.r)`1 & (g.r)`1<=b & c <=(f.r)`2 & (f.r)`2<=d & c <=(g.r)`2 & (g.r)`2<=d) holds rng f meets rng g proof let f,g be map of I[01],TOP-REAL 2,a,b,c,d be real number, O,I be Point of I[01]; assume A1: O=0 & I=1 & f is continuous one-to-one & g is continuous one-to-one & (f.O)`1=a & (f.I)`1=b & (g.O)`2=c & (g.I)`2=d & (for r being Point of I[01] holds a<=(f.r)`1 & (f.r)`1<=b & a<=(g.r)`1 & (g.r)`1<=b & c <=(f.r)`2 & (f.r)`2<=d & c <=(g.r)`2 & (g.r)`2<=d); A2:dom f= the carrier of I[01] by FUNCT_2:def 1 .=[#](I[01]) by PRE_TOPC:12; A3:[#](I[01]) =[.0,1.] by BORSUK_1:83,PRE_TOPC:12; then 0 in dom f by A2,TOPREAL5:1; then reconsider P=rng f as non empty Subset of TOP-REAL 2 by FUNCT_1:12; dom g= the carrier of I[01] by FUNCT_2:def 1 .=[#](I[01]) by PRE_TOPC:12; then 0 in dom g by A3,TOPREAL5:1; then reconsider Q=rng g as non empty Subset of TOP-REAL 2 by FUNCT_1:12; A4:I[01] is compact by HEINE:11,TOPMETR:27; A5:TOP-REAL 2 is_T2 by SPPOL_1:26; then consider f1 being map of I[01],((TOP-REAL 2)|P) such that A6: f=f1 & f1 is_homeomorphism by A1,A4,Th64; consider g1 being map of I[01],((TOP-REAL 2)|Q) such that A7:g=g1 & g1 is_homeomorphism by A1,A4,A5,Th64; A8:dom f1= [#](I[01]) by A6,TOPS_2:def 5; reconsider p1=f1.O as Point of TOP-REAL 2 by A1,A6; reconsider p2=f1.I as Point of TOP-REAL 2 by A1,A6; A9:dom g1= [#](I[01]) by A7,TOPS_2:def 5; reconsider q1=g1.O as Point of TOP-REAL 2 by A1,A7; reconsider q2=g1.I as Point of TOP-REAL 2 by A1,A7; A10:P is_an_arc_of p1,p2 by A1,A6,TOPREAL1:def 2; A11:Q is_an_arc_of q1,q2 by A1,A7,TOPREAL1:def 2; A12:for p being Point of TOP-REAL 2 st p in P holds p1`1<=p`1 & p`1<= p2`1 proof let p be Point of TOP-REAL 2; assume p in P; then consider x such that A13:x in dom f1 & p=f1.x by A6,FUNCT_1:def 5; reconsider r=x as Point of I[01] by A8,A13; p=f.r by A6,A13; hence thesis by A1,A6; end; A14:(for p being Point of TOP-REAL 2 st p in Q holds p1`1<=p`1 & p`1<= p2`1) proof let p be Point of TOP-REAL 2; assume p in Q; then consider x such that A15:x in dom g1 & p=g1.x by A7,FUNCT_1:def 5; reconsider r=x as Point of I[01] by A9,A15; p=g.r by A7,A15; hence thesis by A1,A6; end; A16:for p being Point of TOP-REAL 2 st p in P holds q1`2<=p`2 & p`2<= q2`2 proof let p be Point of TOP-REAL 2; assume p in P; then consider x such that A17:x in dom f1 & p=f1.x by A6,FUNCT_1:def 5; reconsider r=x as Point of I[01] by A8,A17; p=f.r by A6,A17; hence thesis by A1,A7; end; for p being Point of TOP-REAL 2 st p in Q holds q1`2<=p`2 & p`2<= q2`2 proof let p be Point of TOP-REAL 2; assume p in Q; then consider x such that A18:x in dom g1 & p=g1.x by A7,FUNCT_1:def 5; reconsider r=x as Point of I[01] by A9,A18; p=g.r by A7,A18; hence thesis by A1,A7; end; hence thesis by A10,A11,A12,A14,A16,Th62; end;

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