:: General {F}ashoda {M}eet {T}heorem for Unit Circle :: by Yatsuka Nakamura :: :: Received June 24, 2002 :: Copyright (c) 2002-2021 Association of Mizar Users :: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland). :: This code can be distributed under the GNU General Public Licence :: version 3.0 or later, or the Creative Commons Attribution-ShareAlike :: License version 3.0 or later, subject to the binding interpretation :: detailed in file COPYING.interpretation. :: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these :: licenses, or see http://www.gnu.org/licenses/gpl.html and :: http://creativecommons.org/licenses/by-sa/3.0/. environ vocabularies NUMBERS, PRE_TOPC, EUCLID, COMPLEX1, XXREAL_0, ARYTM_1, MCART_1, SQUARE_1, ARYTM_3, CARD_1, REAL_1, XBOOLE_0, METRIC_1, SUBSET_1, TOPMETR, TARSKI, XXREAL_1, STRUCT_0, FUNCT_1, BORSUK_1, RELAT_1, TOPS_2, ORDINAL2, RCOMP_1, SUPINF_2, TOPREAL1, JGRAPH_3, JGRAPH_4, PSCOMP_1, SEQ_4, RLTOPSP1, JORDAN6, TOPREAL2, JORDAN5C, PCOMPS_1, VALUED_1, JORDAN3, FUNCT_2; notations ORDINAL1, NUMBERS, XREAL_0, XCMPLX_0, COMPLEX1, REAL_1, XBOOLE_0, SUBSET_1, TARSKI, RELAT_1, TOPS_2, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, SEQ_4, STRUCT_0, RLVECT_1, RLTOPSP1, EUCLID, TOPMETR, PCOMPS_1, COMPTS_1, METRIC_1, SQUARE_1, RCOMP_1, PSCOMP_1, BINOP_1, PRE_TOPC, JGRAPH_3, TOPREAL1, JORDAN5C, JORDAN6, TOPREAL2, JGRAPH_4, XXREAL_0; constructors REAL_1, SQUARE_1, COMPLEX1, RCOMP_1, TOPS_2, COMPTS_1, TOPREAL1, JORDAN5C, JORDAN6, JGRAPH_3, JGRAPH_4, PSCOMP_1, SEQ_4, BINOP_2, PCOMPS_1, BINOP_1; registrations XBOOLE_0, FUNCT_1, RELSET_1, FUNCT_2, XXREAL_0, XREAL_0, MEMBERED, STRUCT_0, PRE_TOPC, METRIC_1, BORSUK_1, EUCLID, TOPMETR, TOPREAL1, BORSUK_3, COMPTS_1, XXREAL_2, SQUARE_1, ORDINAL1; requirements REAL, NUMERALS, BOOLE, SUBSET, ARITHM; begin :: Preliminaries theorem :: JGRAPH_5:1 for p being Point of TOP-REAL 2 st |.p.|<=1 holds -1<=p`1 & p`1<= 1 & -1<=p`2 & p`2<=1; theorem :: JGRAPH_5:2 for p being Point of TOP-REAL 2 st |.p.|<=1 & p`1<>0 & p`2<>0 holds -1=d & t1>=t2 & s1 in [.a,b.] & s2 in [.a,b.] holds s1<=s2; theorem :: JGRAPH_5:10 for n being Element of NAT holds -(0.TOP-REAL n)=0.TOP-REAL n; begin :: Fashoda Meet Theorems for Circle in Special Case theorem :: JGRAPH_5:11 for f,g being Function of I[01],TOP-REAL 2,a,b,c,d being Real, O ,I being Point of I[01] st O=0 & I=1 & f is continuous one-to-one & g is continuous one-to-one & a <> b & c <> d & (f.O)`1=a & c <=(f.O)`2 & (f.O)`2 <=d & (f.I)`1=b & c <=(f.I)`2 & (f.I)`2 <=d & (g.O)`2=c & a <=(g.O)`1 & (g.O)`1 <=b & (g.I)`2=d & a <=(g.I)`1 & (g.I)`1 <=b & (for r being Point of I[01] holds (a >=(f.r)`1 or (f.r)`1>=b or c >=(f.r)`2 or (f.r)`2>=d) & (a >=(g.r)`1 or (g.r)`1 >=b or c >=(g.r)`2 or (g.r)`2>=d)) holds rng f meets rng g; theorem :: JGRAPH_5:12 for f being Function of I[01],TOP-REAL 2 st f is continuous one-to-one ex f2 being Function of I[01],TOP-REAL 2 st f2.0=f.1 & f2.1=f.0 & rng f2=rng f & f2 is continuous & f2 is one-to-one; reserve p,q for Point of TOP-REAL 2; theorem :: JGRAPH_5:13 for f,g being Function of I[01],TOP-REAL 2, C0,KXP,KXN,KYP,KYN being Subset of TOP-REAL 2, O,I being Point of I[01] st O=0 & I=1 & f is continuous one-to-one & g is continuous one-to-one & C0={p: |.p.|<=1}& KXP={q1 where q1 is Point of TOP-REAL 2: |.q1.|=1 & q1`2<=q1`1 & q1`2>=-q1`1} & KXN={q2 where q2 is Point of TOP-REAL 2: |.q2.|=1 & q2`2>=q2`1 & q2`2<=-q2`1} & KYP={q3 where q3 is Point of TOP-REAL 2: |.q3.|=1 & q3`2>=q3`1 & q3`2>=-q3`1} & KYN={q4 where q4 is Point of TOP-REAL 2: |.q4.|=1 & q4`2<=q4`1 & q4`2<=-q4`1} & f.O in KXN & f.I in KXP & g.O in KYP & g.I in KYN & rng f c= C0 & rng g c= C0 holds rng f meets rng g; theorem :: JGRAPH_5:14 for f,g being Function of I[01],TOP-REAL 2, C0,KXP,KXN,KYP,KYN being Subset of TOP-REAL 2, O,I being Point of I[01] st O=0 & I=1 & f is continuous one-to-one & g is continuous one-to-one & C0={p: |.p.|>=1}& KXP={q1 where q1 is Point of TOP-REAL 2: |.q1.|=1 & q1`2<=q1`1 & q1`2>=-q1`1} & KXN={q2 where q2 is Point of TOP-REAL 2: |.q2.|=1 & q2`2>=q2`1 & q2`2<=-q2`1} & KYP={q3 where q3 is Point of TOP-REAL 2: |.q3.|=1 & q3`2>=q3`1 & q3`2>=-q3`1} & KYN={q4 where q4 is Point of TOP-REAL 2: |.q4.|=1 & q4`2<=q4`1 & q4`2<=-q4`1} & f.O in KXN & f.I in KXP & g.O in KYN & g.I in KYP & rng f c= C0 & rng g c= C0 holds rng f meets rng g; theorem :: JGRAPH_5:15 for f,g being Function of I[01],TOP-REAL 2, C0,KXP,KXN,KYP,KYN being Subset of TOP-REAL 2, O,I being Point of I[01] st O=0 & I=1 & f is continuous one-to-one & g is continuous one-to-one & C0={p: |.p.|>=1}& KXP={q1 where q1 is Point of TOP-REAL 2: |.q1.|=1 & q1`2<=q1`1 & q1`2>=-q1`1} & KXN={q2 where q2 is Point of TOP-REAL 2: |.q2.|=1 & q2`2>=q2`1 & q2`2<=-q2`1} & KYP={q3 where q3 is Point of TOP-REAL 2: |.q3.|=1 & q3`2>=q3`1 & q3`2>=-q3`1} & KYN={q4 where q4 is Point of TOP-REAL 2: |.q4.|=1 & q4`2<=q4`1 & q4`2<=-q4`1} & f.O in KXN & f.I in KXP & g.O in KYP & g.I in KYN & rng f c= C0 & rng g c= C0 holds rng f meets rng g; theorem :: JGRAPH_5:16 for f,g being Function of I[01],TOP-REAL 2, C0 being Subset of TOP-REAL 2 st C0={q: |.q.|>=1} & f is continuous one-to-one & g is continuous one-to-one & f.0=|[-1,0]| & f.1=|[1,0]| & g.1=|[0,1]| & g.0=|[0,-1]| & rng f c= C0 & rng g c= C0 holds rng f meets rng g; theorem :: JGRAPH_5:17 for p1,p2,p3,p4 being Point of TOP-REAL 2, C0 being Subset of TOP-REAL 2 st C0={p: |.p.|>=1} & |.p1.|=1 & |.p2.|=1 & |.p3.|=1 & |.p4.|=1 & (ex h being Function of TOP-REAL 2,TOP-REAL 2 st h is being_homeomorphism & h.:C0 c= C0 & h .p1=|[-1,0]| & h.p2=|[0,1]| & h.p3=|[1,0]| & h.p4=|[0,-1]|) holds for f,g being Function of I[01],TOP-REAL 2 st f is continuous one-to-one & g is continuous one-to-one & f.0=p1 & f.1=p3 & g.0=p4 & g.1=p2 & rng f c= C0 & rng g c= C0 holds rng f meets rng g; begin :: Properties of Fan Morphisms theorem :: JGRAPH_5:18 for cn being Real,q being Point of TOP-REAL 2 st -10 holds for p being Point of TOP-REAL 2 st p=(cn-FanMorphN).q holds p`2>0 ; theorem :: JGRAPH_5:19 for cn being Real,q being Point of TOP-REAL 2 st -1=0 holds for p being Point of TOP-REAL 2 st p=(cn-FanMorphN).q holds p`2>=0; theorem :: JGRAPH_5:20 for cn being Real,q being Point of TOP-REAL 2 st -1=0 & q`1/|.q.|0 holds for p being Point of TOP-REAL 2 st p=(cn -FanMorphN).q holds p`2>=0 & p`1<0; theorem :: JGRAPH_5:21 for cn being Real,q1,q2 being Point of TOP-REAL 2 st -1=0 & q2`2>=0 & |.q1.|<>0 & |.q2.|<>0 & q1`1/|.q1.|0 holds for p being Point of TOP-REAL 2 st p=(sn-FanMorphE).q holds p`1>0 ; theorem :: JGRAPH_5:23 for sn being Real,q being Point of TOP-REAL 2 st -1=0 & q`2/|.q.|0 holds for p being Point of TOP-REAL 2 st p=(sn -FanMorphE).q holds p`1>=0 & p`2<0; theorem :: JGRAPH_5:24 for sn being Real,q1,q2 being Point of TOP-REAL 2 st -1=0 & q2`1>=0 & |.q1.|<>0 & |.q2.|<>0 & q1`2/|.q1.|cn holds for p being Point of TOP-REAL 2 st p=(cn-FanMorphS). q holds p`2<0 & p`1>0; theorem :: JGRAPH_5:27 for cn being Real,q1,q2 being Point of TOP-REAL 2 st -10 & |.q2.|<>0 & q1`1/|.q1.|=0}; theorem :: JGRAPH_5:35 for P being compact non empty Subset of TOP-REAL 2 st P={q where q is Point of TOP-REAL 2: |.q.|=1} holds Lower_Arc(P)={p where p is Point of TOP-REAL 2:p in P & p`2<=0}; theorem :: JGRAPH_5:36 for a,b,d,e being Real st a<=b & e>0 ex f being Function of Closed-Interval-TSpace(a,b),Closed-Interval-TSpace(e*a+d,e*b+d) st f is being_homeomorphism & for r being Real st r in [.a,b.] holds f.r=e*r+d; theorem :: JGRAPH_5:37 for a,b,d,e being Real st a<=b & e<0 ex f being Function of Closed-Interval-TSpace(a,b),Closed-Interval-TSpace(e*b+d,e*a+d) st f is being_homeomorphism & for r being Real st r in [.a,b.] holds f.r=e*r+d; theorem :: JGRAPH_5:38 ex f being Function of I[01],Closed-Interval-TSpace(-1,1) st f is being_homeomorphism & (for r being Real st r in [.0,1.] holds f.r=(-2)*r+1) & f.0=1 & f.1=-1; theorem :: JGRAPH_5:39 ex f being Function of I[01],Closed-Interval-TSpace(-1,1) st f is being_homeomorphism & (for r being Real st r in [.0,1.] holds f.r=2*r-1) & f .0=-1 & f.1=1; theorem :: JGRAPH_5:40 for P being compact non empty Subset of TOP-REAL 2 st P={p where p is Point of TOP-REAL 2: |.p.|=1} ex f being Function of Closed-Interval-TSpace(-1,1),(TOP-REAL 2)|Lower_Arc(P) st f is being_homeomorphism & (for q being Point of TOP-REAL 2 st q in Lower_Arc(P) holds f.(q`1)=q) & f.(-1)=W-min(P) & f.1=E-max(P); theorem :: JGRAPH_5:41 for P being compact non empty Subset of TOP-REAL 2 st P={p where p is Point of TOP-REAL 2: |.p.|=1} ex f being Function of Closed-Interval-TSpace(-1,1),(TOP-REAL 2)|Upper_Arc(P) st f is being_homeomorphism & (for q being Point of TOP-REAL 2 st q in Upper_Arc(P) holds f.(q`1)=q) & f.(-1)=W-min(P) & f.1=E-max(P); theorem :: JGRAPH_5:42 for P being compact non empty Subset of TOP-REAL 2 st P={p where p is Point of TOP-REAL 2: |.p.|=1} ex f being Function of I[01],(TOP-REAL 2)| Lower_Arc(P) st f is being_homeomorphism & (for q1,q2 being Point of TOP-REAL 2 , r1,r2 being Real st f.r1=q1 & f.r2=q2 & r1 in [.0,1.] & r2 in [.0,1.] holds r1q2`1)& f.0 = E-max(P) & f.1 = W-min(P); theorem :: JGRAPH_5:43 for P being compact non empty Subset of TOP-REAL 2 st P={p where p is Point of TOP-REAL 2: |.p.|=1} ex f being Function of I[01],(TOP-REAL 2)| Upper_Arc(P) st f is being_homeomorphism & (for q1,q2 being Point of TOP-REAL 2 , r1,r2 being Real st f.r1=q1 & f.r2=q2 & r1 in [.0,1.] & r2 in [.0,1.] holds r1p2 & p1`1<0 & p1`2<0 & p2`2<0 holds p1`1>p2`1 & p1`2p2 & p2`1<0 & p1`2>=0 & p2`2>=0 holds p1`1p2 & p2`2>=0 holds p1`1p2 & p1`2<=0 & p1<>W-min(P) holds p1`1>p2`1; theorem :: JGRAPH_5:49 for p1,p2 being Point of TOP-REAL 2, P being compact non empty Subset of TOP-REAL 2 st P={p where p is Point of TOP-REAL 2: |.p.|=1} & (p2`2>= 0 or p2`1>=0) & LE p1,p2,P holds p1`2>=0 or p1`1>=0; theorem :: JGRAPH_5:50 for p1,p2 being Point of TOP-REAL 2, P being compact non empty Subset of TOP-REAL 2 st P={p where p is Point of TOP-REAL 2: |.p.|=1} & LE p1, p2,P & p1<>p2 & p1`1>=0 & p2`1>=0 holds p1`2>p2`2; theorem :: JGRAPH_5:51 for p1,p2 being Point of TOP-REAL 2, P being compact non empty Subset of TOP-REAL 2 st P={p where p is Point of TOP-REAL 2: |.p.|=1} & p1 in P & p2 in P & p1`1<0 & p2`1<0 & p1`2<0 & p2`2<0 & (p1`1>=p2`1 or p1`2<=p2`2) holds LE p1,p2,P; theorem :: JGRAPH_5:52 for p1,p2 being Point of TOP-REAL 2, P being compact non empty Subset of TOP-REAL 2 st P={p where p is Point of TOP-REAL 2: |.p.|=1} & p1 in P & p2 in P & p1`1>0 & p2`1>0 & p1`2<0 & p2`2<0 & (p1`1>=p2`1 or p1`2>=p2`2) holds LE p1,p2,P; theorem :: JGRAPH_5:53 for p1,p2 being Point of TOP-REAL 2, P being compact non empty Subset of TOP-REAL 2 st P={p where p is Point of TOP-REAL 2: |.p.|=1} & p1 in P & p2 in P & p1`1<0 & p2`1<0 & p1`2>=0 & p2`2>=0 & (p1`1<=p2`1 or p1`2<=p2`2) holds LE p1,p2,P; theorem :: JGRAPH_5:54 for p1,p2 being Point of TOP-REAL 2, P being compact non empty Subset of TOP-REAL 2 st P={p where p is Point of TOP-REAL 2: |.p.|=1} & p1 in P & p2 in P & p1`2>=0 & p2`2>=0 & p1`1<=p2`1 holds LE p1,p2,P; theorem :: JGRAPH_5:55 for p1,p2 being Point of TOP-REAL 2, P being compact non empty Subset of TOP-REAL 2 st P={p where p is Point of TOP-REAL 2: |.p.|=1} & p1 in P & p2 in P & p1`1>=0 & p2`1>=0 & p1`2>=p2`2 holds LE p1,p2,P; theorem :: JGRAPH_5:56 for p1,p2 being Point of TOP-REAL 2, P being compact non empty Subset of TOP-REAL 2 st P={p where p is Point of TOP-REAL 2: |.p.|=1} & p1 in P & p2 in P & p1`2<=0 & p2`2<=0 & p2<>W-min(P) & p1`1>=p2`1 holds LE p1,p2,P; theorem :: JGRAPH_5:57 for cn being Real,q being Point of TOP-REAL 2 st -1=0 & p2`1<0 & p2`2>=0 & p3 `1<0 & p3`2>=0 & p4`1<0 & p4`2>=0 ex f being Function of TOP-REAL 2,TOP-REAL 2, q1,q2,q3,q4 being Point of TOP-REAL 2 st f is being_homeomorphism & (for q being Point of TOP-REAL 2 holds |.(f.q).|=|.q.|)& q1=f.p1 & q2=f.p2 & q3=f.p3 & q4=f.p4 & q1`1<0 & q1`2<0 & q2`1<0 & q2`2<0 & q3`1<0 & q3`2<0 & q4`1<0 & q4`2<0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P; theorem :: JGRAPH_5:60 for p1,p2,p3,p4 being Point of TOP-REAL 2, P being compact non empty Subset of TOP-REAL 2 st P={p where p is Point of TOP-REAL 2: |.p.|=1} & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P & p1`2>=0 & p2`2>=0 & p3`2>=0 & p4`2>0 ex f being Function of TOP-REAL 2,TOP-REAL 2, q1,q2,q3,q4 being Point of TOP-REAL 2 st f is being_homeomorphism & (for q being Point of TOP-REAL 2 holds |.(f.q) .|=|.q.|)& q1=f.p1 & q2=f.p2 & q3=f.p3 & q4=f.p4 & q1`1<0 & q1`2>=0 & q2`1<0 & q2`2>=0 & q3`1<0 & q3`2>=0 & q4`1<0 & q4`2>=0 & LE q1,q2,P & LE q2,q3,P & LE q3 ,q4,P; theorem :: JGRAPH_5:61 for p1,p2,p3,p4 being Point of TOP-REAL 2, P being compact non empty Subset of TOP-REAL 2 st P={p where p is Point of TOP-REAL 2: |.p.|=1} & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P & p1`2>=0 & p2`2>=0 & p3`2>=0 & p4`2>0 ex f being Function of TOP-REAL 2,TOP-REAL 2, q1,q2,q3,q4 being Point of TOP-REAL 2 st f is being_homeomorphism & (for q being Point of TOP-REAL 2 holds |.(f.q) .|=|.q.|)& q1=f.p1 & q2=f.p2 & q3=f.p3 & q4=f.p4 & q1`1<0 & q1`2<0 & q2`1<0 & q2`2<0 & q3`1<0 & q3`2<0 & q4`1<0 & q4`2<0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4 ,P; theorem :: JGRAPH_5:62 for p1,p2,p3,p4 being Point of TOP-REAL 2, P being compact non empty Subset of TOP-REAL 2 st P={p where p is Point of TOP-REAL 2: |.p.|=1} & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P & (p1`2>=0 or p1`1>=0)& (p2`2>=0 or p2`1>= 0) & (p3`2>=0 or p3`1>=0)& (p4`2>0 or p4`1>0) ex f being Function of TOP-REAL 2 ,TOP-REAL 2, q1,q2,q3,q4 being Point of TOP-REAL 2 st f is being_homeomorphism & (for q being Point of TOP-REAL 2 holds |.(f.q).|=|.q.|)& q1=f.p1 & q2=f.p2 & q3=f.p3 & q4=f.p4 & q1`2>=0 & q2`2>=0 & q3`2>=0 & q4`2>0 & LE q1,q2,P & LE q2, q3,P & LE q3,q4,P; theorem :: JGRAPH_5:63 for p1,p2,p3,p4 being Point of TOP-REAL 2, P being compact non empty Subset of TOP-REAL 2 st P={p where p is Point of TOP-REAL 2: |.p.|=1} & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P & (p1`2>=0 or p1`1>=0)& (p2`2>=0 or p2`1>= 0) & (p3`2>=0 or p3`1>=0)& (p4`2>0 or p4`1>0) ex f being Function of TOP-REAL 2 ,TOP-REAL 2, q1,q2,q3,q4 being Point of TOP-REAL 2 st f is being_homeomorphism & (for q being Point of TOP-REAL 2 holds |.(f.q).|=|.q.|)& q1=f.p1 & q2=f.p2 & q3=f.p3 & q4=f.p4 & q1`1<0 & q1`2<0 & q2`1<0 & q2`2<0 & q3`1<0 & q3`2<0 & q4`1< 0 & q4`2<0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P; theorem :: JGRAPH_5:64 for p1,p2,p3,p4 being Point of TOP-REAL 2, P being compact non empty Subset of TOP-REAL 2 st P={p where p is Point of TOP-REAL 2: |.p.|=1} & p4= W-min(P) & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P ex f being Function of TOP-REAL 2,TOP-REAL 2, q1,q2,q3,q4 being Point of TOP-REAL 2 st f is being_homeomorphism & (for q being Point of TOP-REAL 2 holds |.(f.q).|=|.q.|)& q1=f.p1 & q2=f.p2 & q3=f.p3 & q4=f.p4 & q1`1<0 & q1`2<0 & q2`1<0 & q2`2<0 & q3`1<0 & q3`2<0 & q4`1< 0 & q4`2<0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P; theorem :: JGRAPH_5:65 for p1,p2,p3,p4 being Point of TOP-REAL 2, P being compact non empty Subset of TOP-REAL 2 st P={p where p is Point of TOP-REAL 2: |.p.|=1} & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P ex f being Function of TOP-REAL 2,TOP-REAL 2, q1,q2,q3,q4 being Point of TOP-REAL 2 st f is being_homeomorphism & (for q being Point of TOP-REAL 2 holds |.(f.q).|=|.q.|)& q1=f.p1 & q2=f.p2 & q3=f.p3 & q4=f.p4 & q1`1<0 & q1`2<0 & q2`1<0 & q2`2<0 & q3`1<0 & q3`2<0 & q4`1<0 & q4`2<0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P; begin :: General Fashoda Meet Theorems theorem :: JGRAPH_5:66 for p1,p2,p3,p4 being Point of TOP-REAL 2, P being compact non empty Subset of TOP-REAL 2 st P={p where p is Point of TOP-REAL 2: |.p.|=1} & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P & p1<>p2 & p2<>p3 & p3<>p4 & p1`1<0 & p2`1 <0 & p3`1<0 & p4`1<0 & p1`2<0 & p2`2<0 & p3`2<0 ex f being Function of TOP-REAL 2,TOP-REAL 2 st f is being_homeomorphism & (for q being Point of TOP-REAL 2 holds |.(f.q).|=|.q.|)& |[-1,0]|=f.p1 & |[0,1]|=f.p2 & |[1,0]|=f.p3 & |[0,-1]|= f.p4; theorem :: JGRAPH_5:67 for p1,p2,p3,p4 being Point of TOP-REAL 2, P being compact non empty Subset of TOP-REAL 2 st P={p where p is Point of TOP-REAL 2: |.p.|=1} & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P & p1<>p2 & p2<>p3 & p3<>p4 ex f being Function of TOP-REAL 2,TOP-REAL 2 st f is being_homeomorphism & (for q being Point of TOP-REAL 2 holds |.(f.q).|=|.q.|)& |[-1,0]|=f.p1 & |[0,1]|=f.p2 & |[1, 0]|=f.p3 & |[0,-1]|=f.p4; theorem :: JGRAPH_5:68 for p1,p2,p3,p4 being Point of TOP-REAL 2, P being compact non empty Subset of TOP-REAL 2,C0 being Subset of TOP-REAL 2 st P={p where p is Point of TOP-REAL 2: |.p.|=1} & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P holds for f,g being Function of I[01],TOP-REAL 2 st f is continuous one-to-one & g is continuous one-to-one & C0={p: |.p.|<=1}& f.0=p1 & f.1=p3 & g.0=p2 & g.1=p4 & rng f c= C0 & rng g c= C0 holds rng f meets rng g; theorem :: JGRAPH_5:69 for p1,p2,p3,p4 being Point of TOP-REAL 2, P being compact non empty Subset of TOP-REAL 2,C0 being Subset of TOP-REAL 2 st P={p where p is Point of TOP-REAL 2: |.p.|=1} & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P holds for f,g being Function of I[01],TOP-REAL 2 st f is continuous one-to-one & g is continuous one-to-one & C0={p: |.p.|<=1}& f.0=p1 & f.1=p3 & g.0=p4 & g.1=p2 & rng f c= C0 & rng g c= C0 holds rng f meets rng g; theorem :: JGRAPH_5:70 for p1,p2,p3,p4 being Point of TOP-REAL 2, P being compact non empty Subset of TOP-REAL 2,C0 being Subset of TOP-REAL 2 st P={p where p is Point of TOP-REAL 2: |.p.|=1} & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P holds for f,g being Function of I[01],TOP-REAL 2 st f is continuous one-to-one & g is continuous one-to-one & C0={p: |.p.|>=1}& f.0=p1 & f.1=p3 & g.0=p4 & g.1=p2 & rng f c= C0 & rng g c= C0 holds rng f meets rng g; theorem :: JGRAPH_5:71 for p1,p2,p3,p4 being Point of TOP-REAL 2, P being compact non empty Subset of TOP-REAL 2,C0 being Subset of TOP-REAL 2 st P={p where p is Point of TOP-REAL 2: |.p.|=1} & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P holds for f,g being Function of I[01],TOP-REAL 2 st f is continuous one-to-one & g is continuous one-to-one & C0={p: |.p.|>=1}& f.0=p1 & f.1=p3 & g.0=p2 & g.1=p4 & rng f c= C0 & rng g c= C0 holds rng f meets rng g;