Volume 14, 2002

University of Bialystok

Copyright (c) 2002 Association of Mizar Users

### The abstract of the Mizar article:

### General Fashoda Meet Theorem for Unit Circle

**by****Yatsuka Nakamura**- Received June 24, 2002
- MML identifier: JGRAPH_5

- [ Mizar article, MML identifier index ]

environ vocabulary FUNCT_1, BOOLE, ABSVALUE, EUCLID, PRE_TOPC, SQUARE_1, RELAT_1, SUBSET_1, ARYTM_3, METRIC_1, RCOMP_1, FUNCT_5, TOPMETR, COMPTS_1, JGRAPH_4, ORDINAL2, TOPS_2, ARYTM_1, COMPLEX1, MCART_1, PCOMPS_1, JGRAPH_3, BORSUK_1, TOPREAL1, TOPREAL2, JORDAN3, PSCOMP_1, REALSET1, JORDAN5C, JORDAN6, ARYTM, SEQ_1; notation ORDINAL1, NUMBERS, XCMPLX_0, XREAL_0, REAL_1, XBOOLE_0, ABSVALUE, EUCLID, TARSKI, RELAT_1, TOPS_2, FUNCT_1, FUNCT_2, NAT_1, STRUCT_0, TOPMETR, PCOMPS_1, COMPTS_1, METRIC_1, SQUARE_1, RCOMP_1, PSCOMP_1, BINOP_1, PRE_TOPC, JGRAPH_1, JGRAPH_3, TOPREAL1, JORDAN5C, JORDAN6, TOPREAL2, JGRAPH_4, GRCAT_1; constructors REAL_1, ABSVALUE, TOPREAL1, TOPS_2, RCOMP_1, PSCOMP_1, TOPREAL2, WELLFND1, JGRAPH_3, JORDAN5C, JORDAN6, JGRAPH_4, GRCAT_1, BORSUK_3, TOPRNS_1; clusters XREAL_0, STRUCT_0, RELSET_1, FUNCT_1, EUCLID, PRE_TOPC, TOPMETR, SQUARE_1, PSCOMP_1, BORSUK_1, METRIC_1, BORSUK_2, BORSUK_3, MEMBERED; requirements REAL, NUMERALS, BOOLE, SUBSET, ARITHM; begin :: Preliminaries reserve x,a for real number; theorem :: JGRAPH_5:1 a>=0 & (x-a)*(x+a)>=0 implies -a>=x or x>=a; theorem :: JGRAPH_5:2 a<=0 & x<a implies x^2>a^2; theorem :: JGRAPH_5:3 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_5:4 for p being Point of TOP-REAL 2 st |.p.|<=1 & p`1<>0 & p`2<>0 holds -1<p`1 & p`1<1 & -1<p`2 & p`2<1; theorem :: JGRAPH_5:5 for a,b,d,e,r3 being Real,PM,PM2 being non empty MetrStruct, x being Element of PM, x2 being Element of PM2 st d<=a & a<=b & b<=e & PM=Closed-Interval-MSpace(a,b) & PM2=Closed-Interval-MSpace(d,e) & x=x2 & x in the carrier of PM & x2 in the carrier of PM2 holds Ball(x,r3) c= Ball(x2,r3); theorem :: JGRAPH_5:6 for a,b,d,e being real number, B being Subset of Closed-Interval-TSpace(d,e) st d<=a & a<=b & b<=e & B=[.a,b.] holds Closed-Interval-TSpace(a,b)=Closed-Interval-TSpace(d,e)|B; theorem :: JGRAPH_5:7 for a,b being real number, B being Subset of I[01] st 0<=a & a<=b & b<=1 & B=[.a,b.] holds Closed-Interval-TSpace(a,b)=I[01]|B; theorem :: JGRAPH_5:8 for X being TopStruct, Y,Z being non empty TopStruct,f being map of X,Y, h being map of Y,Z st h is_homeomorphism & f is continuous holds h*f is continuous; theorem :: JGRAPH_5:9 for X,Y,Z being TopStruct, f being map of X,Y, h being map of Y,Z st h is_homeomorphism & f is one-to-one holds h*f is one-to-one; theorem :: JGRAPH_5:10 for X being TopStruct,S,V being non empty TopStruct, B being non empty Subset of S,f being map of X,S|B, g being map of S,V, h being map of X,V st h=g*f & f is continuous & g is continuous holds h is continuous; theorem :: JGRAPH_5:11 for a,b,d,e,s1,s2,t1,t2 being Real,h being map of Closed-Interval-TSpace(a,b),Closed-Interval-TSpace(d,e) st h is_homeomorphism & h.s1=t1 & h.s2=t2 & h.a=d & h.b=e & d<=e & t1<=t2 & s1 in [.a,b.] & s2 in [.a,b.] holds s1<=s2; theorem :: JGRAPH_5:12 for a,b,d,e,s1,s2,t1,t2 being Real,h being map of Closed-Interval-TSpace(a,b),Closed-Interval-TSpace(d,e) st h is_homeomorphism & h.s1=t1 & h.s2=t2 & h.a=e & h.b=d & e>=d & t1>=t2 & s1 in [.a,b.] & s2 in [.a,b.] holds s1<=s2; theorem :: JGRAPH_5:13 for n being Nat holds -(0.REAL n)=0.REAL n; begin :: Fashoda Meet Theorems for Circle in Special Case theorem :: JGRAPH_5:14 for f,g being map 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_5:15 for f being map of I[01],TOP-REAL 2 st f is continuous one-to-one ex f2 being map 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_5:16 for f,g being map 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_5:17 for f,g being map 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_5:18 for f,g being map 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_5:19 for f,g being map 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_5:20 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 map of TOP-REAL 2,TOP-REAL 2 st h is_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 map 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_5:21 for cn being Real,q being Point of TOP-REAL 2 st -1<cn & cn<1 & q`2>0 holds (for p being Point of TOP-REAL 2 st p=(cn-FanMorphN).q holds p`2>0); theorem :: JGRAPH_5:22 for cn being Real,q being Point of TOP-REAL 2 st -1<cn & cn<1 & q`2>=0 holds (for p being Point of TOP-REAL 2 st p=(cn-FanMorphN).q holds p`2>=0); theorem :: JGRAPH_5:23 for cn being Real,q being Point of TOP-REAL 2 st -1<cn & cn<1 & q`2>=0 & q`1/|.q.|<cn & |.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_5:24 for cn being Real,q1,q2 being Point of TOP-REAL 2 st -1<cn & cn<1 & q1`2>=0 & q2`2>=0 & |.q1.|<>0 & |.q2.|<>0 & q1`1/|.q1.|<q2`1/|.q2.| holds (for p1,p2 being Point of TOP-REAL 2 st p1=(cn-FanMorphN).q1 & p2=(cn-FanMorphN).q2 holds p1`1/|.p1.|<p2`1/|.p2.|); theorem :: JGRAPH_5:25 for sn being Real,q being Point of TOP-REAL 2 st -1<sn & sn<1 & q`1>0 holds (for p being Point of TOP-REAL 2 st p=(sn-FanMorphE).q holds p`1>0); theorem :: JGRAPH_5:26 for sn being Real,q being Point of TOP-REAL 2 st -1<sn & sn<1 & q`1>=0 & q`2/|.q.|<sn & |.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_5:27 for sn being Real,q1,q2 being Point of TOP-REAL 2 st -1<sn & sn<1 & q1`1>=0 & q2`1>=0 & |.q1.|<>0 & |.q2.|<>0 & q1`2/|.q1.|<q2`2/|.q2.| holds (for p1,p2 being Point of TOP-REAL 2 st p1=(sn-FanMorphE).q1 & p2=(sn-FanMorphE).q2 holds p1`2/|.p1.|<p2`2/|.p2.|); theorem :: JGRAPH_5:28 for cn being Real,q being Point of TOP-REAL 2 st -1<cn & cn<1 & q`2<0 holds (for p being Point of TOP-REAL 2 st p=(cn-FanMorphS).q holds p`2<0); theorem :: JGRAPH_5:29 for cn being Real,q being Point of TOP-REAL 2 st -1<cn & cn<1 & q`2<0 & q`1/|.q.|>cn holds (for p being Point of TOP-REAL 2 st p=(cn-FanMorphS).q holds p`2<0 & p`1>0); theorem :: JGRAPH_5:30 for cn being Real,q1,q2 being Point of TOP-REAL 2 st -1<cn & cn<1 & q1`2<=0 & q2`2<=0 & |.q1.|<>0 & |.q2.|<>0 & q1`1/|.q1.|<q2`1/|.q2.| holds (for p1,p2 being Point of TOP-REAL 2 st p1=(cn-FanMorphS).q1 & p2=(cn-FanMorphS).q2 holds p1`1/|.p1.|<p2`1/|.p2.|); begin theorem :: JGRAPH_5:31 for P being compact non empty Subset of TOP-REAL 2 st P={q: |.q.|=1} holds W-bound(P)=-1 & E-bound(P)=1 & S-bound(P)=-1 & N-bound(P)=1; theorem :: JGRAPH_5:32 for P being compact non empty Subset of TOP-REAL 2 st P={q: |.q.|=1} holds W-min(P)=|[-1,0]|; theorem :: JGRAPH_5:33 for P being compact non empty Subset of TOP-REAL 2 st P={q: |.q.|=1} holds E-max(P)=|[1,0]|; theorem :: JGRAPH_5:34 for f being map of TOP-REAL 2,R^1 st (for p being Point of TOP-REAL 2 holds f.p=proj1.p) holds f is continuous; theorem :: JGRAPH_5:35 for f being map of TOP-REAL 2,R^1 st (for p being Point of TOP-REAL 2 holds f.p=proj2.p) holds f is continuous; theorem :: JGRAPH_5:36 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 Upper_Arc(P) c= P & Lower_Arc(P) c= P; theorem :: JGRAPH_5:37 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 Upper_Arc(P)={p where p is Point of TOP-REAL 2:p in P & p`2>=0}; theorem :: JGRAPH_5:38 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_5:39 for a,b,d,e being Real st a<=b & e>0 ex f being map of Closed-Interval-TSpace(a,b),Closed-Interval-TSpace(e*a+d,e*b+d) st f is_homeomorphism & for r being Real st r in [.a,b.] holds f.r=e*r+d; theorem :: JGRAPH_5:40 for a,b,d,e being Real st a<=b & e<0 ex f being map of Closed-Interval-TSpace(a,b),Closed-Interval-TSpace(e*b+d,e*a+d) st f is_homeomorphism & for r being Real st r in [.a,b.] holds f.r=e*r+d; theorem :: JGRAPH_5:41 ex f being map of I[01],Closed-Interval-TSpace(-1,1) st f is_homeomorphism & (for r being Real st r in [.0,1.] holds f.r=(-2)*r+1) & f.0=1 & f.1=-1; theorem :: JGRAPH_5:42 ex f being map of I[01],Closed-Interval-TSpace(-1,1) st f is_homeomorphism & (for r being Real st r in [.0,1.] holds f.r=2*r-1) & f.0=-1 & f.1=1; theorem :: JGRAPH_5:43 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 map of Closed-Interval-TSpace(-1,1),(TOP-REAL 2)|Lower_Arc(P) st f is_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_5:44 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 map of Closed-Interval-TSpace(-1,1),(TOP-REAL 2)|Upper_Arc(P) st f is_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_5:45 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 map of I[01],(TOP-REAL 2)|Lower_Arc(P) st f is_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 r1<r2 iff q1`1>q2`1)& f.0 = E-max(P) & f.1 = W-min(P); theorem :: JGRAPH_5:46 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 map of I[01],(TOP-REAL 2)|Upper_Arc(P) st f is_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 r1<r2 iff q1`1<q2`1)& f.0 = W-min(P) & f.1 = E-max(P); theorem :: JGRAPH_5:47 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 in Upper_Arc(P) & LE p1,p2,P holds p1 in Upper_Arc(P); theorem :: JGRAPH_5:48 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 & p1`2<0 & p2`2<0 holds p1`1>p2`1 & p1`2<p2`2; 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} & LE p1,p2,P & p1<>p2 & p1`1<0 & p2`1<0 & p1`2>=0 & p2`2>=0 holds p1`1<p2`1 & p1`2<p2`2; 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`2>=0 & p2`2>=0 holds p1`1<p2`1; 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} & LE p1,p2,P & p1<>p2 & p1`2<=0 & p2`2<=0 & p1<>W-min(P) holds p1`1>p2`1; 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} & (p2`2>=0 or p2`1>=0) & LE p1,p2,P holds p1`2>=0 or p1`1>=0; 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} & LE p1,p2,P & p1<>p2 & p1`1>=0 & p2`1>=0 holds p1`2>p2`2; 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`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_5:55 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<0 & p2`2<0 & (p1`1>=p2`1 or 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`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_5:57 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_5:58 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_5:59 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_5:60 for cn being Real,q being Point of TOP-REAL 2 st -1<cn & cn<1 & q`2<=0 holds (for p being Point of TOP-REAL 2 st p=(cn-FanMorphS).q holds p`2<=0); theorem :: JGRAPH_5:61 for cn being Real,p1,p2,q1,q2 being Point of TOP-REAL 2, P being compact non empty Subset of TOP-REAL 2 st -1<cn & cn<1 & P={p where p is Point of TOP-REAL 2: |.p.|=1} & LE p1,p2,P & q1=(cn-FanMorphS).p1 & q2=(cn-FanMorphS).p2 holds LE q1,q2,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`1<0 & p1`2>=0 & p2`1<0 & p2`2>=0 & p3`1<0 & p3`2>=0 & p4`1<0 & p4`2>=0 ex f being map of TOP-REAL 2,TOP-REAL 2, q1,q2,q3,q4 being Point of TOP-REAL 2 st f is_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_5:63 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 & p2`2>=0 & p3`2>=0 & p4`2>0 ex f being map of TOP-REAL 2,TOP-REAL 2, q1,q2,q3,q4 being Point of TOP-REAL 2 st f is_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} & 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 map of TOP-REAL 2,TOP-REAL 2, q1,q2,q3,q4 being Point of TOP-REAL 2 st f is_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 & (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 map of TOP-REAL 2,TOP-REAL 2, q1,q2,q3,q4 being Point of TOP-REAL 2 st f is_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_5:66 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`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 map of TOP-REAL 2,TOP-REAL 2, q1,q2,q3,q4 being Point of TOP-REAL 2 st f is_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_5:67 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} & p4=W-min(P) & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P ex f being map of TOP-REAL 2,TOP-REAL 2, q1,q2,q3,q4 being Point of TOP-REAL 2 st f is_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_5:68 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 map of TOP-REAL 2,TOP-REAL 2, q1,q2,q3,q4 being Point of TOP-REAL 2 st f is_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_5:69 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 & p4`2<0 ex f being map of TOP-REAL 2,TOP-REAL 2 st f is_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_5:70 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 map of TOP-REAL 2,TOP-REAL 2 st f is_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_5:71 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 map 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_5:72 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 map 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_5:73 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 map 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_5:74 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 map 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);

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