:: General {F}ashoda {M}eet {T}heorem for Unit Circle
:: by Yatsuka Nakamura
::
:: Received June 24, 2002
:: Copyright (c) 2002-2026 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;
definitions TARSKI, JORDAN6;
equalities JORDAN6, SQUARE_1, STRUCT_0, RELAT_1, RLTOPSP1, ORDINAL1;
expansions JORDAN6;
theorems TARSKI, XBOOLE_0, XBOOLE_1, RELAT_1, FUNCT_1, FUNCT_2, TOPS_1,
TOPS_2, PRE_TOPC, TOPMETR, JORDAN6, EUCLID, JGRAPH_1, SEQ_4, SQUARE_1,
PSCOMP_1, METRIC_1, JGRAPH_2, RCOMP_1, COMPTS_1, SETWISEO, BORSUK_1,
TOPREAL1, TOPREAL3, TOPREAL5, JGRAPH_3, ABSVALUE, COMPLEX1, JORDAN5A,
JORDAN5B, JORDAN7, HEINE, JGRAPH_4, PCOMPS_1, JORDAN5C, XREAL_0, TSEP_1,
TOPRNS_1, XCMPLX_1, XREAL_1, XXREAL_0, XXREAL_1, XXREAL_2, RLVECT_1;
schemes FUNCT_2, JGRAPH_2;
begin :: Preliminaries
theorem Th1:
for p being Point of TOP-REAL 2 st |.p.|<=1 holds -1<=p`1 & p`1<=
1 & -1<=p`2 & p`2<=1
proof
let p be Point of TOP-REAL 2;
set a=|.p.|;
A1: (|.p.|)^2 =(p`1)^2+(p`2)^2 by JGRAPH_3:1;
then a^2-(p`1)^2>=0 by XREAL_1:63;
then a^2-(p`1)^2+(p`1)^2>=0+(p`1)^2 by XREAL_1:7;
then
A2: -a<=p`1 & p`1<=a by SQUARE_1:47;
a^2-(p`2)^2>=0 by A1,XREAL_1:63;
then a^2-(p`2)^2+(p`2)^2>=0+(p`2)^2 by XREAL_1:7;
then
A3: -a<=p`2 & p`2<=a by SQUARE_1:47;
assume
A4: |.p.|<=1;
then -a>=-1 by XREAL_1:24;
hence thesis by A4,A2,A3,XXREAL_0:2;
end;
theorem Th2:
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
proof
let p be Point of TOP-REAL 2;
assume that
A1: |.p.|<=1 and
A2: p`1<>0 and
A3: p`2<>0;
set a=|.p.|;
A4: (|.p.|)^2 =(p`1)^2+(p`2)^2 by JGRAPH_3:1;
then a^2-(p`1)^2+(p`1)^2>0+(p`1)^2 by A3,SQUARE_1:12,XREAL_1:8;
then
A5: -a<p`1 & p`1<a by SQUARE_1:48;
a^2-(p`2)^2+(p`2)^2>0+(p`2)^2 by A2,A4,SQUARE_1:12,XREAL_1:8;
then
A6: -a<p`2 & p`2<a by SQUARE_1:48;
-a>=-1 by A1,XREAL_1:24;
hence thesis by A1,A5,A6,XXREAL_0:2;
end;
theorem
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 holds Ball
(x,r3) c= Ball(x2,r3)
proof
let a,b,d,e,r3 be Real,PM,PM2 be non empty MetrStruct, x be Element of PM,
x2 be Element of PM2;
assume that
A1: d<=a and
A2: a<=b and
A3: b<=e and
A4: PM=Closed-Interval-MSpace(a,b) and
A5: PM2=Closed-Interval-MSpace(d,e) and
A6: x=x2;
a<=e by A2,A3,XXREAL_0:2;
then
A7: a in [.d,e.] by A1,XXREAL_1:1;
let z be object;
assume z in Ball(x,r3);
then z in {y where y is Element of PM: dist(x,y) < r3 } by METRIC_1:17;
then consider y being Element of PM such that
A8: y=z & dist(x,y)<r3;
the carrier of PM=[.a,b.] by A2,A4,TOPMETR:10;
then
A9: y in [.a,b.];
A10: d<=b by A1,A2,XXREAL_0:2;
then b in [.d,e.] by A3,XXREAL_1:1;
then [.a,b.] c= [.d,e.] by A7,XXREAL_2:def 12;
then reconsider y3=y as Element of PM2 by A3,A5,A10,A9,TOPMETR:10,XXREAL_0:2;
A11: dist(x,y)= (the distance of PM).(x,y) by METRIC_1:def 1;
A12: (the distance of PM).(x,y) = real_dist.(x,y) by A4,METRIC_1:def 13
,TOPMETR:def 1;
real_dist.(x,y)= (the distance of PM2).(x2,y3) by A5,A6,METRIC_1:def 13
,TOPMETR:def 1
.=dist(x2,y3) by METRIC_1:def 1;
then z in {y2 where y2 is Element of PM2: dist(x2,y2)<r3} by A8,A12,A11;
hence thesis by METRIC_1:17;
end;
theorem
for a,b being Real, 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 by TOPMETR:20,23;
theorem Th5:
for X being TopStruct, Y,Z being non empty TopStruct,f being
Function of X,Y, h being Function of Y,Z st h is being_homeomorphism & f is
continuous holds h*f is continuous
proof
let X be TopStruct,Y,Z be non empty TopStruct,f be Function of X,Y, h be
Function of Y,Z;
assume that
A1: h is being_homeomorphism and
A2: f is continuous;
h is continuous by A1,TOPS_2:def 5;
hence thesis by A2,TOPS_2:46;
end;
theorem Th6:
for X,Y,Z being TopStruct, f being Function of X,Y, h being
Function of Y,Z st h is being_homeomorphism & f is one-to-one holds h*f is
one-to-one
proof
let X,Y,Z be TopStruct, f be Function of X,Y, h be Function of Y,Z;
assume that
A1: h is being_homeomorphism and
A2: f is one-to-one;
h is one-to-one by A1,TOPS_2:def 5;
hence thesis by A2;
end;
theorem Th7:
for X being TopStruct,S,V being non empty TopStruct, B being non
empty Subset of S,f being Function of X,S|B, g being Function of S,V, h being
Function of X,V st h=g*f & f is continuous & g is continuous holds h is
continuous
proof
let X be TopStruct,S,V be non empty TopStruct, B be non empty Subset of S, f
be Function of X,S|B, g be Function of S,V, h being Function of X,V;
assume that
A1: h=g*f and
A2: f is continuous and
A3: g is continuous;
now
let P be Subset of V;
A4: (g*f)"P = f"(g"P) by RELAT_1:146;
now
assume P is closed;
then
A5: g"P is closed by A3,PRE_TOPC:def 6;
A6: the carrier of S|B =B by PRE_TOPC:8;
then reconsider F=B /\ g"P as Subset of S|B by XBOOLE_1:17;
A7: rng f /\ (the carrier of S|B)= rng f by XBOOLE_1:28;
[#](S|B)=B by PRE_TOPC:def 5;
then
A8: F is closed by A5,PRE_TOPC:13;
h"P=f"(rng f /\ g"P) by A1,A4,RELAT_1:133
.=f"(rng f /\ ((the carrier of S|B) /\ g"P)) by A7,XBOOLE_1:16
.=f"F by A6,RELAT_1:133;
hence h"P is closed by A2,A8,PRE_TOPC:def 6;
end;
hence P is closed implies h"P is closed;
end;
hence thesis by PRE_TOPC:def 6;
end;
theorem Th8:
for a,b,d,e,s1,s2,t1,t2 being Real,h being Function of
Closed-Interval-TSpace(a,b),Closed-Interval-TSpace(d,e) st h is
being_homeomorphism & h.s1=t1 & h.s2=t2 & h.b=e & d<=e & t1<=t2 & s1 in [.a,b.]
& s2 in [.a,b.] holds s1<=s2
proof
let a,b,d,e,s1,s2,t1,t2 be Real,
h be Function of Closed-Interval-TSpace(a,b)
,Closed-Interval-TSpace(d,e);
assume that
A1: h is being_homeomorphism and
A2: h.s1=t1 and
A3: h.s2=t2 and
A4: h.b=e and
A5: d<=e and
A6: t1<=t2 and
A7: s1 in [.a,b.] and
A8: s2 in [.a,b.];
A9: s1<=b by A7,XXREAL_1:1;
reconsider C=[.d,e.] as non empty Subset of R^1 by A5,TOPMETR:17,XXREAL_1:1;
A10: R^1|C=Closed-Interval-TSpace(d,e) by A5,TOPMETR:19;
A11: a<=s1 by A7,XXREAL_1:1;
then
A12: the carrier of Closed-Interval-TSpace(a,b) =[.a,b.] by A9,TOPMETR:18
,XXREAL_0:2;
then reconsider B1=[.s1,b.] as Subset of Closed-Interval-TSpace(a,b) by A11,
XXREAL_1:34;
A13: dom h=[#](Closed-Interval-TSpace(a,b)) by A1,TOPS_2:def 5
.=[.a,b.] by A11,A9,TOPMETR:18,XXREAL_0:2;
A14: a<=s2 by A8,XXREAL_1:1;
then reconsider B=[.s2,s1.] as Subset of Closed-Interval-TSpace(a,b) by A9
,A12,XXREAL_1:34;
reconsider Bb=[.s2,s1.] as Subset of Closed-Interval-TSpace(a,b) by A14,A9
,A12,XXREAL_1:34;
reconsider f3=h|Bb as Function of Closed-Interval-TSpace(a,b)|B,
Closed-Interval-TSpace(d,e) by PRE_TOPC:9;
assume
A15: s1>s2;
then
A16: Closed-Interval-TSpace(s2,s1) =Closed-Interval-TSpace(a,b)|B by A14,A9,
TOPMETR:23;
then f3 is Function of Closed-Interval-TSpace(s2,s1),R^1 by A10,JORDAN6:3;
then reconsider f=h|B as Function of Closed-Interval-TSpace(s2,s1),R^1;
s2 in B by A15,XXREAL_1:1;
then
A17: f.s2=t2 by A3,FUNCT_1:49;
set t=(t1+t2)/2;
A18: the carrier of Closed-Interval-TSpace(d,e) =[.d,e.] by A5,TOPMETR:18;
h is one-to-one by A1,TOPS_2:def 5;
then t1<>t2 by A2,A3,A7,A8,A13,A15,FUNCT_1:def 4;
then
A19: t1<t2 by A6,XXREAL_0:1;
then t1+t1<t1+t2 by XREAL_1:8;
then
A20: (2*t1)/2<t by XREAL_1:74;
dom f=the carrier of Closed-Interval-TSpace(s2,s1) by FUNCT_2:def 1;
then dom f=[.s2,s1.] by A15,TOPMETR:18;
then s2 in dom f by A15,XXREAL_1:1;
then t2 in rng f3 by A17,FUNCT_1:def 3;
then
A21: t2<=e by A18,XXREAL_1:1;
t1+t2<t2+t2 by A19,XREAL_1:8;
then
A22: (2*t2)/2>t by XREAL_1:74;
then
A23: e>t by A21,XXREAL_0:2;
reconsider B1b=[.s1,b.] as Subset of Closed-Interval-TSpace(a,b) by A11,A12,
XXREAL_1:34;
reconsider f4=h|B1b as Function of Closed-Interval-TSpace(a,b)|B1,
Closed-Interval-TSpace(d,e) by PRE_TOPC:9;
A24: Closed-Interval-TSpace(s1,b) =Closed-Interval-TSpace(a,b)|B1 by A11,A9,
TOPMETR:23;
then f4 is Function of Closed-Interval-TSpace(s1,b),R^1 by A10,JORDAN6:3;
then reconsider f1=h|B1 as Function of Closed-Interval-TSpace(s1,b),R^1;
A25: h is continuous by A1,TOPS_2:def 5;
then f4 is continuous by TOPMETR:7;
then
A26: f1 is continuous by A10,A24,JORDAN6:3;
b in B1 by A9,XXREAL_1:1;
then
A27: f1.b= e by A4,FUNCT_1:49;
s1 in B1 by A9,XXREAL_1:1;
then
A28: f1.s1= t1 by A2,FUNCT_1:49;
s1<b by A2,A4,A9,A19,A21,XXREAL_0:1;
then consider r1 being Real such that
A29: f1.r1 =t and
A30: s1<r1 and
A31: r1 <b by A20,A26,A28,A27,A23,TOPREAL5:6;
A32: r1 in B1 by A30,A31,XXREAL_1:1;
s1 in B by A15,XXREAL_1:1;
then
A33: f.s1=t1 by A2,FUNCT_1:49;
f3 is continuous by A25,TOPMETR:7;
then f is continuous by A10,A16,JORDAN6:3;
then consider r being Real such that
A34: f.r =t and
A35: s2<r and
A36: r <s1 by A15,A17,A33,A20,A22,TOPREAL5:7;
A37: a<r by A14,A35,XXREAL_0:2;
a<r1 by A11,A30,XXREAL_0:2;
then
A38: r1 in [.a,b.] by A31,XXREAL_1:1;
A39: h is one-to-one by A1,TOPS_2:def 5;
r<b by A9,A36,XXREAL_0:2;
then
A40: r in [.a,b.] by A37,XXREAL_1:1;
r in [.s2,s1.] by A35,A36,XXREAL_1:1;
then h.r = t by A34,FUNCT_1:49
.=h.r1 by A29,A32,FUNCT_1:49;
hence contradiction by A13,A39,A36,A40,A30,A38,FUNCT_1:def 4;
end;
theorem Th9:
for a,b,d,e,s1,s2,t1,t2 being Real,h being Function of
Closed-Interval-TSpace(a,b),Closed-Interval-TSpace(d,e) st h is
being_homeomorphism & h.s1=t1 & h.s2=t2 & h.b=d & e>=d & t1>=t2 & s1 in [.a,b.]
& s2 in [.a,b.] holds s1<=s2
proof
let a,b,d,e,s1,s2,t1,t2 be Real,
h be Function of Closed-Interval-TSpace(a,b)
,Closed-Interval-TSpace(d,e);
assume that
A1: h is being_homeomorphism and
A2: h.s1=t1 and
A3: h.s2=t2 and
A4: h.b=d and
A5: e>=d and
A6: t1>=t2 and
A7: s1 in [.a,b.] and
A8: s2 in [.a,b.];
A9: s1<=b by A7,XXREAL_1:1;
reconsider C=[.d,e.] as non empty Subset of R^1 by A5,TOPMETR:17,XXREAL_1:1;
A10: R^1|C=Closed-Interval-TSpace(d,e) by A5,TOPMETR:19;
A11: a<=s1 by A7,XXREAL_1:1;
then
A12: the carrier of Closed-Interval-TSpace(a,b) =[.a,b.] by A9,TOPMETR:18
,XXREAL_0:2;
then reconsider B1=[.s1,b.] as Subset of Closed-Interval-TSpace(a,b) by A11,
XXREAL_1:34;
A13: dom h=[#](Closed-Interval-TSpace(a,b)) by A1,TOPS_2:def 5
.=[.a,b.] by A11,A9,TOPMETR:18,XXREAL_0:2;
A14: a<=s2 by A8,XXREAL_1:1;
then reconsider B=[.s2,s1.] as Subset of Closed-Interval-TSpace(a,b) by A9
,A12,XXREAL_1:34;
reconsider Bb=[.s2,s1.] as Subset of Closed-Interval-TSpace(a,b) by A14,A9
,A12,XXREAL_1:34;
reconsider f3=h|Bb as Function of Closed-Interval-TSpace(a,b)|B,
Closed-Interval-TSpace(d,e) by PRE_TOPC:9;
assume
A15: s1>s2;
then
A16: Closed-Interval-TSpace(s2,s1) =Closed-Interval-TSpace(a,b)|B by A14,A9,
TOPMETR:23;
then f3 is Function of Closed-Interval-TSpace(s2,s1),R^1 by A10,JORDAN6:3;
then reconsider f=h|B as Function of Closed-Interval-TSpace(s2,s1),R^1;
s2 in B by A15,XXREAL_1:1;
then
A17: f.s2=t2 by A3,FUNCT_1:49;
set t=(t1+t2)/2;
A18: the carrier of Closed-Interval-TSpace(d,e) =[.d,e.] by A5,TOPMETR:18;
h is one-to-one by A1,TOPS_2:def 5;
then t1<>t2 by A2,A3,A7,A8,A13,A15,FUNCT_1:def 4;
then
A19: t1>t2 by A6,XXREAL_0:1;
then t1+t1>t1+t2 by XREAL_1:8;
then
A20: (2*t1)/2>t by XREAL_1:74;
dom f=the carrier of Closed-Interval-TSpace(s2,s1) by FUNCT_2:def 1;
then dom f=[.s2,s1.] by A15,TOPMETR:18;
then s2 in dom f by A15,XXREAL_1:1;
then t2 in rng f3 by A17,FUNCT_1:def 3;
then
A21: d<=t2 by A18,XXREAL_1:1;
t1+t2>t2+t2 by A19,XREAL_1:8;
then
A22: (2*t2)/2<t by XREAL_1:74;
then
A23: d<t by A21,XXREAL_0:2;
reconsider B1b=[.s1,b.] as Subset of Closed-Interval-TSpace(a,b) by A11,A12,
XXREAL_1:34;
reconsider f4=h|B1b as Function of Closed-Interval-TSpace(a,b)|B1,
Closed-Interval-TSpace(d,e) by PRE_TOPC:9;
A24: Closed-Interval-TSpace(s1,b) =Closed-Interval-TSpace(a,b)|B1 by A11,A9,
TOPMETR:23;
then f4 is Function of Closed-Interval-TSpace(s1,b),R^1 by A10,JORDAN6:3;
then reconsider f1=h|B1 as Function of Closed-Interval-TSpace(s1,b),R^1;
A25: h is continuous by A1,TOPS_2:def 5;
then f4 is continuous by TOPMETR:7;
then
A26: f1 is continuous by A10,A24,JORDAN6:3;
b in B1 by A9,XXREAL_1:1;
then
A27: f1.b= d by A4,FUNCT_1:49;
s1 in B1 by A9,XXREAL_1:1;
then
A28: f1.s1= t1 by A2,FUNCT_1:49;
s1<b by A2,A4,A9,A19,A21,XXREAL_0:1;
then consider r1 being Real such that
A29: f1.r1 =t and
A30: s1<r1 and
A31: r1 <b by A20,A26,A28,A27,A23,TOPREAL5:7;
A32: r1 in B1 by A30,A31,XXREAL_1:1;
s1 in B by A15,XXREAL_1:1;
then
A33: f.s1=t1 by A2,FUNCT_1:49;
f3 is continuous by A25,TOPMETR:7;
then f is continuous by A10,A16,JORDAN6:3;
then consider r being Real such that
A34: f.r =t and
A35: s2<r and
A36: r <s1 by A15,A17,A33,A20,A22,TOPREAL5:6;
A37: a<r by A14,A35,XXREAL_0:2;
a<r1 by A11,A30,XXREAL_0:2;
then
A38: r1 in [.a,b.] by A31,XXREAL_1:1;
A39: h is one-to-one by A1,TOPS_2:def 5;
r<b by A9,A36,XXREAL_0:2;
then
A40: r in [.a,b.] by A37,XXREAL_1:1;
r in [.s2,s1.] by A35,A36,XXREAL_1:1;
then h.r= t by A34,FUNCT_1:49
.=h.r1 by A29,A32,FUNCT_1:49;
hence contradiction by A13,A39,A36,A40,A30,A38,FUNCT_1:def 4;
end;
theorem
for n being Element of NAT holds -(0.TOP-REAL n)=0.TOP-REAL n
proof
let n be Element of NAT;
0.TOP-REAL n+0.TOP-REAL n=0.TOP-REAL n by RLVECT_1:4;
hence thesis by RLVECT_1:6;
end;
begin :: Fashoda Meet Theorems for Circle in Special Case
theorem Th11:
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
proof
let f,g be Function of I[01],TOP-REAL 2,a,b,c,d be Real, O,I be Point of
I[01];
assume that
A1: O=0 & I=1 & f is continuous one-to-one & g is continuous one-to-one and
A2: a <> b and
A3: c <> d and
A4: (f.O)`1=a and
A5: c <=(f.O)`2 & (f.O)`2 <=d and
A6: (f.I)`1=b & c <=(f.I)`2 & (f.I)`2 <=d & (g.O)`2=c and
A7: a <=(g.O)`1 & (g.O)`1 <=b and
A8: (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) );
c <= d by A5,XXREAL_0:2;
then
A9: c < d by A3,XXREAL_0:1;
a <= b by A7,XXREAL_0:2;
then a < b by A2,XXREAL_0:1;
hence thesis by A1,A4,A5,A6,A7,A8,A9,JGRAPH_2:45;
end;
Lm1: 0 in [.0,1.] by XXREAL_1:1;
Lm2: 1 in [.0,1.] by XXREAL_1:1;
theorem Th12:
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
proof
let f be Function of I[01],TOP-REAL 2;
A1: I[01] is compact by HEINE:4,TOPMETR:20;
A2: dom f=the carrier of I[01] by FUNCT_2:def 1;
then reconsider P=rng f as non empty Subset of TOP-REAL 2 by Lm1,BORSUK_1:40
,FUNCT_1:3;
f.1 in rng f & f.0 in rng f by A2,Lm1,Lm2,BORSUK_1:40,FUNCT_1:3;
then reconsider p1=f.0,p2=f.1 as Point of TOP-REAL 2;
assume f is continuous one-to-one;
then ex f1 being Function of I[01],(TOP-REAL 2)|P st f1=f & f1 is
being_homeomorphism by A1,JGRAPH_1:46;
then P is_an_arc_of p1,p2 by TOPREAL1:def 1;
then P is_an_arc_of p2,p1 by JORDAN5B:14;
then consider f3 being Function of I[01], (TOP-REAL 2)|P such that
A3: f3 is being_homeomorphism and
A4: f3.0 = p2 & f3.1 = p1 by TOPREAL1:def 1;
A5: ex f4 being Function of I[01], (TOP-REAL 2) st f3=f4 & f4 is continuous
& f4 is one-to-one by A3,JORDAN7:15;
rng f3=[#]((TOP-REAL 2)|P) by A3,TOPS_2:def 5
.=P by PRE_TOPC:def 5;
hence thesis by A4,A5;
end;
reserve p,q for Point of TOP-REAL 2;
theorem Th13:
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
proof
let f,g be Function of I[01],TOP-REAL 2, C0,KXP,KXN,KYP,KYN be Subset of
TOP-REAL 2, 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
& 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;
then
ex g2 being Function of I[01],TOP-REAL 2 st g2.0=g.1 & g2 .1=g.0 & rng g2
=rng g & g2 is continuous one-to-one by Th12;
hence thesis by A1,JGRAPH_3:44;
end;
theorem Th14:
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
proof
let f,g be Function of I[01],TOP-REAL 2, C0,KXP,KXN,KYP,KYN be Subset of
TOP-REAL 2, 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
& 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;
A2: dom g=the carrier of I[01] by FUNCT_2:def 1;
reconsider gg=Sq_Circ"*g as Function of I[01],TOP-REAL 2 by FUNCT_2:13
,JGRAPH_3:29;
reconsider ff=Sq_Circ"*f as Function of I[01],TOP-REAL 2 by FUNCT_2:13
,JGRAPH_3:29;
A3: dom gg=the carrier of I[01] by FUNCT_2:def 1;
A4: dom ff=the carrier of I[01] by FUNCT_2:def 1;
A5: (ff.O)`1=-1 & (ff.I)`1=1 & (gg.O)`2=-1 & (gg.I)`2=1
proof
reconsider pz=gg.O as Point of TOP-REAL 2;
reconsider py=ff.I as Point of TOP-REAL 2;
reconsider px=ff.O as Point of TOP-REAL 2;
set q=px;
A6: (|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|)`1 = q`1/sqrt(1+
(q `2/q`1)^2) by EUCLID:52;
reconsider pu=gg.I as Point of TOP-REAL 2;
A7: (|[py`1/sqrt(1+(py`2/py`1)^2),py`2/sqrt(1+(py`2/py`1)^2)]|)`1 = py`1/
sqrt(1+(py`2/py`1)^2) by EUCLID:52;
consider p2 being Point of TOP-REAL 2 such that
A8: f.I=p2 and
A9: |.p2.|=1 and
A10: p2`2<=p2`1 and
A11: p2`2>=-p2`1 by A1;
A12: (ff.I)=(Sq_Circ").(f.I) by A4,FUNCT_1:12;
then
A13: p2=Sq_Circ.py by A8,FUNCT_1:32,JGRAPH_3:22,43;
A14: p2<>0.TOP-REAL 2 by A9,TOPRNS_1:23;
then
A15: Sq_Circ".p2=|[p2`1*sqrt(1+(p2`2/p2`1)^2),p2`2*sqrt(1+(p2`2/p2`1) ^2 )
]| by A10,A11,JGRAPH_3:28;
then
A16: py`1 = p2`1*sqrt(1+(p2`2/p2`1)^2) by A12,A8,EUCLID:52;
(p2`2/p2`1)^2 >=0 by XREAL_1:63;
then
A17: sqrt(1+(p2`2/p2`1)^2)>0 by SQUARE_1:25;
A18: py`2 = p2`2*sqrt(1+(p2`2/p2`1)^2) by A12,A8,A15,EUCLID:52;
A19: now
assume py`1=0 & py`2=0;
then p2`1=0 & p2`2=0 by A16,A18,A17,XCMPLX_1:6;
hence contradiction by A14,EUCLID:53,54;
end;
A20: p2`2<=p2`1 & (-p2`1)*sqrt(1+(p2`2/p2`1)^2) <= p2`2*sqrt(1+(p2`2/p2`1
) ^2 ) or py`2>=py`1 & py`2<=-py`1 by A10,A11,A17,XREAL_1:64;
then p2`2*sqrt(1+(p2`2/p2`1)^2) <= p2`1*sqrt(1+(p2`2/p2`1)^2) & -py `1<=
py `2 or py`2>=py`1 & py`2<=-py`1 by A12,A8,A15,A16,A17,EUCLID:52
,XREAL_1:64;
then
A21: Sq_Circ.py=|[py`1/sqrt(1+(py`2/py`1)^2),py`2/sqrt(1+(py`2/py`1)^2)]|
by A16,A18,A19,JGRAPH_2:3,JGRAPH_3:def 1;
A22: (py`2/py`1)^2 >=0 by XREAL_1:63;
then
A23: sqrt(1+(py`2/py`1)^2)>0 by SQUARE_1:25;
A24: now
assume
A25: py`1=-1;
-p2`2<=--p2`1 by A11,XREAL_1:24;
then -p2`2<0 by A13,A21,A7,A22,A25,SQUARE_1:25,XREAL_1:141;
then --p2`2>-0;
hence contradiction by A10,A13,A21,A23,A25,EUCLID:52;
end;
(|[py`1/sqrt(1+(py`2/py`1)^2),py`2/sqrt(1+(py`2/py`1)^2)]|)`2 = py`2/
sqrt(1+(py`2/py`1)^2) by EUCLID:52;
then
(|.p2.|)^2= (py`1/sqrt(1+(py`2/py`1)^2))^2+(py`2/sqrt(1+(py`2/py`1)^2
) ) ^2 by A13,A21,A7,JGRAPH_3:1
.= (py`1)^2/(sqrt(1+(py`2/py`1)^2))^2+(py`2/sqrt(1+(py`2/py`1)^2))^2
by XCMPLX_1:76
.= (py`1)^2/(sqrt(1+(py`2/py`1)^2))^2 +(py`2)^2/(sqrt(1+(py`2/py`1)^2)
)^2 by XCMPLX_1:76
.= (py`1)^2/(1+(py`2/py`1)^2)+(py`2)^2/(sqrt(1+(py`2/py`1)^2))^2 by A22,
SQUARE_1:def 2
.= (py`1)^2/(1+(py`2/py`1)^2)+(py`2)^2/(1+(py`2/py`1)^2) by A22,
SQUARE_1:def 2
.= ((py`1)^2+(py`2)^2)/(1+(py`2/py`1)^2) by XCMPLX_1:62;
then
((py`1)^2+(py`2)^2)/(1+(py`2/py`1)^2)*(1+(py`2/py`1)^2) =(1)*(1+(py`2
/py`1)^2) by A9;
then ((py`1)^2+(py`2)^2)=(1+(py`2/py`1)^2) by A22,XCMPLX_1:87;
then
A26: (py`1)^2+(py`2)^2-1=(py`2)^2/(py`1)^2 by XCMPLX_1:76;
py`1<>0 by A16,A18,A17,A19,A20,XREAL_1:64;
then ((py`1)^2+(py`2)^2-1)*(py`1)^2=(py`2)^2 by A26,XCMPLX_1:6,87;
then
A27: ((py`1)^2-1)*((py`1)^2+(py`2)^2)=0;
((py`1)^2+(py`2)^2)<>0 by A19,COMPLEX1:1;
then (py`1-1)*(py`1+1)=0 by A27,XCMPLX_1:6;
then
A28: py`1-1=0 or py`1+1=0 by XCMPLX_1:6;
A29: (|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/pz`2)^2)]|)`2 = pz`2
/ sqrt(1+(pz`1/pz`2)^2) by EUCLID:52;
consider p1 being Point of TOP-REAL 2 such that
A30: f.O=p1 and
A31: |.p1.|=1 and
A32: p1`2>=p1`1 and
A33: p1`2<=-p1`1 by A1;
(p1`2/p1`1)^2 >=0 by XREAL_1:63;
then
A34: sqrt(1+(p1`2/p1`1)^2)>0 by SQUARE_1:25;
A35: (ff.O)=(Sq_Circ").(f.O) by A4,FUNCT_1:12;
then
A36: p1=Sq_Circ.px by A30,FUNCT_1:32,JGRAPH_3:22,43;
A37: p1<>0.TOP-REAL 2 by A31,TOPRNS_1:23;
then
Sq_Circ".p1=|[p1`1*sqrt(1+(p1`2/p1`1)^2),p1`2*sqrt(1+(p1`2/p1`1) ^2 )
]| by A32,A33,JGRAPH_3:28;
then
A38: px`1 = p1`1*sqrt(1+(p1`2/p1`1)^2) & px`2 = p1`2*sqrt(1+(p1`2/p1`1)^2)
by A35,A30,EUCLID:52;
A39: now
assume px`1=0 & px`2=0;
then p1`1=0 & p1`2=0 by A38,A34,XCMPLX_1:6;
hence contradiction by A37,EUCLID:53,54;
end;
p1`2<=p1`1 & -p1`1<=p1`2 or p1`2>=p1`1 & p1`2*sqrt(1+(p1`2/p1`1)^2)
<= (-p1`1)*sqrt(1+(p1`2/p1`1)^2) by A32,A33,A34,XREAL_1:64;
then
A40: p1`2<=p1`1 & (-p1`1)*sqrt(1+(p1`2/p1`1)^2) <= p1`2*sqrt(1+(p1`2/ p1`1
)^2) or px`2>=px`1 & px`2<=-px`1 by A38,A34,XREAL_1:64;
then px`2<=px`1 & -px`1<=px`2 or px`2>=px`1 & px`2<=-px`1 by A38,A34,
XREAL_1:64;
then
A41: Sq_Circ.q=|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]| by A39,
JGRAPH_2:3,JGRAPH_3:def 1;
A42: (q`2/q`1)^2 >=0 by XREAL_1:63;
then
A43: sqrt(1+(q`2/q`1)^2)>0 by SQUARE_1:25;
A44: now
assume
A45: px`1=1;
-p1`2>=--p1`1 by A33,XREAL_1:24;
then -p1`2>0 by A36,A41,A6,A43,A45,XREAL_1:139;
then --p1`2<-0;
hence contradiction by A32,A36,A41,A43,A45,EUCLID:52;
end;
(|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|)`2 = q`2/sqrt(1+
(q `2/q`1)^2) by EUCLID:52;
then (|.p1.|)^2= (q`1/sqrt(1+(q`2/q`1)^2))^2+(q`2/sqrt(1+(q`2/q`1)^2)) ^2
by A36,A41,A6,JGRAPH_3:1
.= (q`1)^2/(sqrt(1+(q`2/q`1)^2))^2+(q`2/sqrt(1+(q`2/q`1)^2))^2 by
XCMPLX_1:76
.= (q`1)^2/(sqrt(1+(q`2/q`1)^2))^2+(q`2)^2/(sqrt(1+(q`2/q`1)^2))^2 by
XCMPLX_1:76
.= (q`1)^2/(1+(q`2/q`1)^2)+(q`2)^2/(sqrt(1+(q`2/q`1)^2))^2 by A42,
SQUARE_1:def 2
.= (q`1)^2/(1+(q`2/q`1)^2)+(q`2)^2/(1+(q`2/q`1)^2) by A42,SQUARE_1:def 2
.= ((q`1)^2+(q`2)^2)/(1+(q`2/q`1)^2) by XCMPLX_1:62;
then
((q`1)^2+(q`2)^2)/(1+(q`2/q`1)^2)*(1+(q`2/q`1)^2)=(1)*(1+(q`2/q`1 )^2
) by A31;
then ((q`1)^2+(q`2)^2)=(1+(q`2/q`1)^2) by A42,XCMPLX_1:87;
then
A46: (px`1)^2+(px`2)^2-1=(px`2)^2/(px`1)^2 by XCMPLX_1:76;
px`1<>0 by A38,A34,A39,A40,XREAL_1:64;
then ((px`1)^2+(px`2)^2-1)*(px`1)^2=(px`2)^2 by A46,XCMPLX_1:6,87;
then
A47: ((px`1)^2-1)*((px`1)^2+(px`2)^2)=0;
A48: (|[pu`1/sqrt(1+(pu`1/pu`2)^2),pu`2/sqrt(1+(pu`1/pu`2)^2)]|)`2 = pu`2
/ sqrt(1+(pu`1/pu`2)^2) by EUCLID:52;
consider p4 being Point of TOP-REAL 2 such that
A49: g.I=p4 and
A50: |.p4.|=1 and
A51: p4`2>=p4`1 and
A52: p4`2>=-p4`1 by A1;
(p4`1/p4`2)^2 >=0 by XREAL_1:63;
then
A53: sqrt(1+(p4`1/p4`2)^2)>0 by SQUARE_1:25;
A54: -p4`2<=--p4`1 by A52,XREAL_1:24;
then
A55: p4`1<=p4`2 & (-p4`2)*sqrt(1+(p4`1/p4`2)^2) <= p4`1*sqrt(1+(p4`1/p4`2
) ^2 ) or pu`1>=pu`2 & pu`1<=-pu`2 by A51,A53,XREAL_1:64;
A56: (gg.I)=(Sq_Circ").(g.I) by A3,FUNCT_1:12;
then
A57: p4=Sq_Circ.pu by A49,FUNCT_1:32,JGRAPH_3:22,43;
A58: p4<>0.TOP-REAL 2 by A50,TOPRNS_1:23;
then
A59: Sq_Circ".p4=|[p4`1*sqrt(1+(p4`1/p4`2)^2),p4`2*sqrt(1+(p4`1/p4`2) ^2
)]| by A51,A54,JGRAPH_3:30;
then
A60: pu`2 = p4`2*sqrt(1+(p4`1/p4`2)^2) by A56,A49,EUCLID:52;
A61: pu`1 = p4`1*sqrt(1+(p4`1/p4`2)^2) by A56,A49,A59,EUCLID:52;
A62: now
assume pu`2=0 & pu`1=0;
then p4`2=0 & p4`1=0 by A60,A61,A53,XCMPLX_1:6;
hence contradiction by A58,EUCLID:53,54;
end;
p4`1* sqrt(1+(p4`1/p4`2)^2) <= p4`2*sqrt(1+(p4`1/p4`2)^2) & -pu`2<=
pu `1 or pu`1>=pu`2 & pu`1<=-pu`2 by A56,A49,A59,A60,A53,A55,EUCLID:52
,XREAL_1:64;
then
A63: Sq_Circ.pu=|[pu`1/sqrt(1+(pu`1/pu`2)^2),pu`2/sqrt(1+(pu`1/pu`2)^2)]|
by A60,A61,A62,JGRAPH_2:3,JGRAPH_3:4;
A64: (pu`1/pu`2)^2 >=0 by XREAL_1:63;
then
A65: sqrt(1+(pu`1/pu`2)^2)>0 by SQUARE_1:25;
A66: now
assume
A67: pu`2=-1;
then -p4`1<0 by A52,A57,A63,A48,A64,SQUARE_1:25,XREAL_1:141;
then --p4`1>-0;
hence contradiction by A51,A57,A63,A65,A67,EUCLID:52;
end;
(|[pu`1/sqrt(1+(pu`1/pu`2)^2),pu`2/sqrt(1+(pu`1/pu`2)^2)]|)`1 = pu`1
/ sqrt(1+(pu`1/pu`2)^2) by EUCLID:52;
then (|.p4.|)^2= (pu`2/sqrt(1+(pu`1/pu`2)^2))^2+(pu`1/sqrt(1+(pu`1/pu`2)
^2 ) ) ^2 by A57,A63,A48,JGRAPH_3:1
.= (pu`2)^2/(sqrt(1+(pu`1/pu`2)^2))^2+(pu`1/sqrt(1+(pu`1/pu`2)^2))^2
by XCMPLX_1:76
.= (pu`2)^2/(sqrt(1+(pu`1/pu`2)^2))^2 +(pu`1)^2/(sqrt(1+(pu`1/pu`2)^2)
)^2 by XCMPLX_1:76
.= (pu`2)^2/(1+(pu`1/pu`2)^2)+(pu`1)^2/(sqrt(1+(pu`1/pu`2)^2))^2 by A64,
SQUARE_1:def 2
.= (pu`2)^2/(1+(pu`1/pu`2)^2)+(pu`1)^2/(1+(pu`1/pu`2)^2) by A64,
SQUARE_1:def 2
.= ((pu`2)^2+(pu`1)^2)/(1+(pu`1/pu`2)^2) by XCMPLX_1:62;
then ((pu`2)^2+(pu`1)^2)/(1+(pu`1/pu`2)^2)*(1+(pu`1/pu`2)^2)=(1)*(1+(pu`1
/pu `2)^2) by A50;
then ((pu`2)^2+(pu`1)^2)=(1+(pu`1/pu`2)^2) by A64,XCMPLX_1:87;
then
A68: (pu`2)^2+(pu`1)^2-1=(pu`1)^2/(pu`2)^2 by XCMPLX_1:76;
pu`2<>0 by A60,A61,A53,A62,A55,XREAL_1:64;
then ((pu`2)^2+(pu`1)^2-1)*(pu`2)^2=(pu`1)^2 by A68,XCMPLX_1:6,87;
then
A69: ((pu`2)^2-1)*((pu`2)^2+(pu`1)^2)=0;
((pu`2)^2+(pu`1)^2)<>0 by A62,COMPLEX1:1;
then (pu`2-1)*(pu`2+1)=0 by A69,XCMPLX_1:6;
then
A70: pu`2-1=0 or pu`2+1=0 by XCMPLX_1:6;
consider p3 being Point of TOP-REAL 2 such that
A71: g.O=p3 and
A72: |.p3.|=1 and
A73: p3`2<=p3`1 and
A74: p3`2<=-p3`1 by A1;
A75: p3<>0.TOP-REAL 2 by A72,TOPRNS_1:23;
A76: (gg.O)=(Sq_Circ").(g.O) by A3,FUNCT_1:12;
then
A77: p3=Sq_Circ.pz by A71,FUNCT_1:32,JGRAPH_3:22,43;
A78: -p3`2>=--p3`1 by A74,XREAL_1:24;
then
A79: Sq_Circ".p3=|[p3`1*sqrt(1+(p3`1/p3`2)^2),p3`2*sqrt(1+(p3`1/p3`2) ^2)
]| by A73,A75,JGRAPH_3:30;
then
A80: pz`2 = p3`2*sqrt(1+(p3`1/p3`2)^2) by A76,A71,EUCLID:52;
(p3`1/p3`2)^2 >=0 by XREAL_1:63;
then
A81: sqrt(1+(p3`1/p3`2)^2)>0 by SQUARE_1:25;
A82: pz`1 = p3`1*sqrt(1+(p3`1/p3`2)^2) by A76,A71,A79,EUCLID:52;
A83: now
assume pz`2=0 & pz`1=0;
then p3`2=0 & p3`1=0 by A80,A82,A81,XCMPLX_1:6;
hence contradiction by A75,EUCLID:53,54;
end;
p3`1<=p3`2 & -p3`2<=p3`1 or p3`1>=p3`2 & p3`1*sqrt(1+(p3`1/p3`2)^2)
<= (-p3`2)*sqrt(1+(p3`1/p3`2)^2) by A73,A78,A81,XREAL_1:64;
then
A84: p3`1<=p3`2 & (-p3`2)*sqrt(1+(p3`1/p3`2)^2) <= p3`1*sqrt(1+(p3`1/p3 `2
)^2) or pz`1>=pz`2 & pz`1<=-pz`2 by A80,A82,A81,XREAL_1:64;
then
p3`1*sqrt(1+(p3`1/p3`2)^2) <= p3`2*sqrt(1+(p3`1/p3`2)^2) & -pz `2<=pz
`1 or pz`1>=pz`2 & pz`1<=-pz`2 by A76,A71,A79,A80,A81,EUCLID:52,XREAL_1:64;
then
A85: Sq_Circ.pz=|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/pz`2)^2)]|
by A80,A82,A83,JGRAPH_2:3,JGRAPH_3:4;
A86: (pz`1/pz`2)^2 >=0 by XREAL_1:63;
then
A87: sqrt(1+(pz`1/pz`2)^2)>0 by SQUARE_1:25;
A88: now
assume
A89: pz`2=1;
then -p3`1>0 by A74,A77,A85,A29,A87,XREAL_1:139;
then --p3`1<-0;
hence contradiction by A73,A77,A85,A87,A89,EUCLID:52;
end;
(|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/pz`2)^2)]|)`1 = pz`1
/ sqrt(1+(pz`1/pz`2)^2) by EUCLID:52;
then (|.p3.|)^2= (pz`2/sqrt(1+(pz`1/pz`2)^2))^2+(pz`1/sqrt(1+(pz`1/pz`2)
^2))^2 by A77,A85,A29,JGRAPH_3:1
.= (pz`2)^2/(sqrt(1+(pz`1/pz`2)^2))^2+(pz`1/sqrt(1+(pz`1/pz`2)^2))^2
by XCMPLX_1:76
.= (pz`2)^2/(sqrt(1+(pz`1/pz`2)^2))^2 +(pz`1)^2/(sqrt(1+(pz`1/pz`2)^2)
)^2 by XCMPLX_1:76
.= (pz`2)^2/(1+(pz`1/pz`2)^2)+(pz`1)^2/(sqrt(1+(pz`1/pz`2)^2))^2 by A86,
SQUARE_1:def 2
.= (pz`2)^2/(1+(pz`1/pz`2)^2)+(pz`1)^2/(1+(pz`1/pz`2)^2) by A86,
SQUARE_1:def 2
.= ((pz`2)^2+(pz`1)^2)/(1+(pz`1/pz`2)^2) by XCMPLX_1:62;
then ((pz`2)^2+(pz`1)^2)/(1+(pz`1/pz`2)^2)*(1+(pz`1/pz`2)^2)=(1)*(1+(pz`1
/pz `2 ) ^2 ) by A72;
then ((pz`2)^2+(pz`1)^2)=(1+(pz`1/pz`2)^2) by A86,XCMPLX_1:87;
then
A90: (pz`2)^2+(pz`1)^2-1=(pz`1)^2/(pz`2)^2 by XCMPLX_1:76;
pz`2<>0 by A80,A82,A81,A83,A84,XREAL_1:64;
then ((pz`2)^2+(pz`1)^2-1)*(pz`2)^2=(pz`1)^2 by A90,XCMPLX_1:6,87;
then
A91: ((pz`2)^2-1)*((pz`2)^2+(pz`1)^2)=0;
((pz`2)^2+(pz`1)^2)<>0 by A83,COMPLEX1:1;
then (pz`2-1)*(pz`2+1)=0 by A91,XCMPLX_1:6;
then
A92: pz`2-1=0 or pz`2+1=0 by XCMPLX_1:6;
((px`1)^2+(px`2)^2)<>0 by A39,COMPLEX1:1;
then (px`1-1)*(px`1+1)=0 by A47,XCMPLX_1:6;
then px`1-1=0 or px`1+1=0 by XCMPLX_1:6;
hence thesis by A44,A28,A24,A92,A88,A70,A66;
end;
A93: dom f=the carrier of I[01] by FUNCT_2:def 1;
A94: for r being Point of I[01] holds (-1>=(ff.r)`1 or (ff.r)`1>=1 or -1 >=
(ff.r)`2 or (ff.r)`2>=1) & (-1>=(gg.r)`1 or (gg.r)`1>=1 or -1 >=(gg.r)`2 or (gg
.r)`2>=1)
proof
let r be Point of I[01];
f.r in rng f by A93,FUNCT_1:3;
then f.r in C0 by A1;
then consider p1 being Point of TOP-REAL 2 such that
A95: f.r=p1 and
A96: |.p1.|>=1 by A1;
g.r in rng g by A2,FUNCT_1:3;
then g.r in C0 by A1;
then consider p2 being Point of TOP-REAL 2 such that
A97: g.r=p2 and
A98: |.p2.|>=1 by A1;
A99: (gg.r)=(Sq_Circ").(g.r) by A3,FUNCT_1:12;
A100: now
per cases;
case
p2=0.TOP-REAL 2;
hence contradiction by A98,TOPRNS_1:23;
end;
case
A101: p2<>0.TOP-REAL 2 & (p2`2<=p2`1 & -p2`1<=p2`2 or p2`2>=p2`1 &
p2`2<=-p2`1);
reconsider px=gg.r as Point of TOP-REAL 2;
A102: Sq_Circ".p2=|[p2`1*sqrt(1+(p2`2/p2`1)^2),p2`2*sqrt(1+(p2`2 /p2`1
)^2 )]| by A101,JGRAPH_3:28;
then
A103: px`1 = p2`1*sqrt(1+(p2`2/p2`1)^2) by A99,A97,EUCLID:52;
set q=px;
A104: (px`1)^2 >=0 by XREAL_1:63;
(|.p2.|)^2>=|.p2.| by A98,XREAL_1:151;
then
A105: (|.p2.|)^2>=1 by A98,XXREAL_0:2;
A106: (px`2)^2>=0 by XREAL_1:63;
(p2`2/p2`1)^2 >=0 by XREAL_1:63;
then
A107: sqrt(1+(p2`2/p2`1)^2)>0 by SQUARE_1:25;
A108: px`2 = p2`2*sqrt(1+(p2`2/p2`1)^2) by A99,A97,A102,EUCLID:52;
A109: now
assume px`1=0 & px`2=0;
then p2`1=0 & p2`2=0 by A103,A108,A107,XCMPLX_1:6;
hence contradiction by A101,EUCLID:53,54;
end;
p2`2<=p2`1 & -p2`1<=p2`2 or p2`2>=p2`1 & p2`2*sqrt(1+(p2`2/p2`1)
^2) <= (-p2`1)*sqrt(1+(p2`2/p2`1)^2) by A101,A107,XREAL_1:64;
then
A110: p2`2<=p2`1 & (-p2`1)*sqrt(1+(p2`2/p2`1)^2) <= p2`2*sqrt(1+(p2`2/
p2`1 )^2) or px`2>=px`1 & px`2<=-px`1 by A103,A108,A107,XREAL_1:64;
then
A111: px`1<>0 by A103,A108,A107,A109,XREAL_1:64;
p2`2*sqrt( 1+(p2`2/p2`1)^2) <= p2`1*sqrt(1+(p2`2/p2`1)^2) & -px
`1<= px `2 or px`2>=px`1 & px`2<=-px`1 by A99,A97,A102,A103,A107,A110,EUCLID:52
,XREAL_1:64;
then
A112: Sq_Circ.q=|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]| by A103
,A108,A109,JGRAPH_2:3,JGRAPH_3:def 1;
Sq_Circ".p2=q by A3,A97,FUNCT_1:12;
then
A113: p2=Sq_Circ.px by FUNCT_1:32,JGRAPH_3:22,43;
A114: (q`2/q`1)^2 >=0 by XREAL_1:63;
(|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|)`1 = q`1/
sqrt(1+( q`2/q `1)^2) & (|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|)`2
= q`2/sqrt(1+ ( q`2/q`1)^2) by EUCLID:52;
then (|.p2.|)^2= (q`1/sqrt(1+(q`2/q`1)^2))^2+(q`2/sqrt(1+(q`2/q`1)^2)
) ^2 by A113,A112,JGRAPH_3:1
.= (q`1)^2/(sqrt(1+(q`2/q`1)^2))^2+(q`2/sqrt(1+(q`2/q`1)^2))^2 by
XCMPLX_1:76
.= (q`1)^2/(sqrt(1+(q`2/q`1)^2))^2+(q`2)^2/(sqrt(1+(q`2/q`1)^2))^2
by XCMPLX_1:76
.= (q`1)^2/(1+(q`2/q`1)^2)+(q`2)^2/(sqrt(1+(q`2/q`1)^2))^2 by A114,
SQUARE_1:def 2
.= (q`1)^2/(1+(q`2/q`1)^2)+(q`2)^2/(1+(q`2/q`1)^2) by A114,
SQUARE_1:def 2
.= ((q`1)^2+(q`2)^2)/(1+(q`2/q`1)^2) by XCMPLX_1:62;
then ((q`1)^2+(q`2)^2)/(1+(q`2/q`1)^2)*(1+(q`2/q`1)^2)>=(1)*(1+(q`2/
q `1)^2) by A114,A105,XREAL_1:64;
then ((q`1)^2+(q`2)^2)>=(1+(q`2/q`1)^2) by A114,XCMPLX_1:87;
then (px`1)^2+(px`2)^2>=1+(px`2)^2/(px`1)^2 by XCMPLX_1:76;
then (px`1)^2+(px`2)^2-1>=1+(px`2)^2/(px`1)^2-1 by XREAL_1:9;
then ((px`1)^2+(px`2)^2-1)*(px`1)^2>=(px`2)^2/(px`1)^2*(px`1)^2 by A104
,XREAL_1:64;
then ((px`1)^2+((px`2)^2-1))*(px`1)^2>=(px`2)^2 by A111,XCMPLX_1:6,87;
then (px`1)^2*(px`1)^2+(px`1)^2*((px`2)^2-1)-(px`2)^2>=(px`2)^2-(px
`2 )^2 by XREAL_1:9;
then
A115: ((px`1)^2-1)*((px`1)^2+(px`2)^2)>=0;
((px`1)^2+(px`2)^2)<>0 by A109,COMPLEX1:1;
then (px`1-1)*(px`1+1)>=0 by A104,A115,A106,XREAL_1:132;
hence -1>=(gg.r)`1 or (gg.r)`1>=1 or -1 >=(gg.r)`2 or (gg.r)`2>=1 by
XREAL_1:95;
end;
case
A116: p2<>0.TOP-REAL 2 & not(p2`2<=p2`1 & -p2`1<=p2`2 or p2`2>=p2
`1 & p2`2<=-p2`1);
reconsider pz=gg.r as Point of TOP-REAL 2;
A117: Sq_Circ".p2=|[p2`1*sqrt(1+(p2`1/p2`2)^2),p2`2*sqrt(1+(p2`1 /p2`2
)^2 )]| by A116,JGRAPH_3:28;
then
A118: pz`2 = p2`2*sqrt(1+(p2`1/p2`2)^2) by A99,A97,EUCLID:52;
(p2`1/p2`2)^2 >=0 by XREAL_1:63;
then
A119: sqrt(1+(p2`1/p2`2)^2)>0 by SQUARE_1:25;
A120: now
assume that
A121: pz`2=0 and
pz`1=0;
p2`2=0 by A118,A119,A121,XCMPLX_1:6;
hence contradiction by A116;
end;
A122: pz`1 = p2`1*sqrt(1+(p2`1/p2`2)^2) by A99,A97,A117,EUCLID:52;
p2`1<=p2`2 & -p2`2<=p2`1 or p2`1>=p2`2 & p2`1<=-p2`2 by A116,
JGRAPH_2:13;
then p2`1<=p2`2 & -p2`2<=p2`1 or p2`1>=p2`2 & p2`1*sqrt(1+(p2`1/p2`2)
^2) <= (-p2`2)*sqrt(1+(p2`1/p2`2)^2) by A119,XREAL_1:64;
then
A123: p2`1<=p2`2 & (-p2`2)*sqrt(1+(p2`1/p2`2)^2) <= p2`1*sqrt(1+(p2`1/
p2`2) ^2) or pz`1>=pz`2 & pz`1<=-pz`2 by A118,A122,A119,XREAL_1:64;
then
A124: pz`2<>0 by A118,A122,A119,A120,XREAL_1:64;
p2`1*sqrt(1+(p2`1/p2`2)^2) <= p2`2*sqrt(1+(p2`1/p2`2)^2) & -pz
`2<=pz `1 or pz`1>=pz`2 & pz`1<=-pz`2 by A99,A97,A117,A118,A119,A123,EUCLID:52
,XREAL_1:64;
then
A125: Sq_Circ.pz=|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/pz`2)
^2)]| by A118,A122,A120,JGRAPH_2:3,JGRAPH_3:4;
A126: (pz`1/pz`2)^2 >=0 by XREAL_1:63;
(|.p2.|)^2>=|.p2.| by A98,XREAL_1:151;
then
A127: (|.p2.|)^2>=1 by A98,XXREAL_0:2;
A128: (|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/pz`2)^2)]|)`1 =
pz`1/ sqrt(1+(pz`1/pz`2)^2) by EUCLID:52;
A129: (pz`1)^2>=0 by XREAL_1:63;
A130: (pz`2)^2 >=0 by XREAL_1:63;
p2=Sq_Circ.pz & (|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/
pz`2)^2)]|) `2 = pz`2/ sqrt(1+(pz`1/pz`2)^2) by A99,A97,EUCLID:52,FUNCT_1:32
,JGRAPH_3:22,43;
then ( |.p2.|)^2= (pz`2/sqrt(1+(pz`1/pz`2)^2))^2+(pz`1/sqrt(1+(pz`1/
pz`2)^2))^2 by A125,A128,JGRAPH_3:1
.= (pz`2)^2/(sqrt(1+(pz`1/pz`2)^2))^2+(pz`1/sqrt(1+(pz`1/pz`2)^2))
^2 by XCMPLX_1:76
.= (pz`2)^2/(sqrt(1+(pz`1/pz`2)^2))^2 +(pz`1)^2/(sqrt(1+(pz`1/pz`2
)^2))^2 by XCMPLX_1:76
.= (pz`2)^2/(1+(pz`1/pz`2)^2)+(pz`1)^2/(sqrt(1+(pz`1/pz`2)^2))^2
by A126,SQUARE_1:def 2
.= (pz`2)^2/(1+(pz`1/pz`2)^2)+(pz`1)^2/(1+(pz`1/pz`2)^2) by A126,
SQUARE_1:def 2
.= ((pz`2)^2+(pz`1)^2)/(1+(pz`1/pz`2)^2) by XCMPLX_1:62;
then ((pz`2)^2+(pz`1)^2)/(1+(pz`1/pz`2)^2)*(1+(pz`1/pz`2)^2) >=(1)*(1
+(pz`1/pz`2)^2) by A126,A127,XREAL_1:64;
then ((pz`2)^2+(pz`1)^2)>=(1+(pz`1/pz`2)^2) by A126,XCMPLX_1:87;
then (pz`2)^2+(pz`1)^2>=1+(pz`1)^2/(pz`2)^2 by XCMPLX_1:76;
then (pz`2)^2+(pz`1)^2-1>=1+(pz`1)^2/(pz`2)^2-1 by XREAL_1:9;
then ((pz`2)^2+(pz`1)^2-1)*(pz`2)^2>=(pz`1)^2/(pz`2)^2*(pz`2)^2 by A130
,XREAL_1:64;
then ((pz`2)^2+((pz`1)^2-1))*(pz`2)^2>=(pz`1)^2 by A124,XCMPLX_1:6,87;
then (pz`2)^2*(pz`2)^2+(pz`2)^2*((pz`1)^2-1)-(pz`1)^2 >=(pz`1)^2-(pz
`1)^2 by XREAL_1:9;
then
A131: ((pz`2)^2-1)*((pz`2)^2+(pz`1)^2)>=0;
((pz`2)^2+(pz`1)^2)<>0 by A120,COMPLEX1:1;
then (pz`2-1)*(pz`2+1)>=0 by A130,A131,A129,XREAL_1:132;
hence -1>=(gg.r)`1 or (gg.r)`1>=1 or -1 >=(gg.r)`2 or (gg.r)`2>=1 by
XREAL_1:95;
end;
end;
A132: (ff.r)=(Sq_Circ").(f.r) by A4,FUNCT_1:12;
now
per cases;
case
p1=0.TOP-REAL 2;
hence contradiction by A96,TOPRNS_1:23;
end;
case
A133: p1<>0.TOP-REAL 2 & (p1`2<=p1`1 & -p1`1<=p1`2 or p1`2>=p1`1 &
p1`2<=-p1`1);
reconsider px=ff.r as Point of TOP-REAL 2;
A134: Sq_Circ".p1=|[p1`1*sqrt(1+(p1`2/p1`1)^2),p1`2*sqrt(1+(p1`2 /p1`1
)^2 )]| by A133,JGRAPH_3:28;
then
A135: px`1 = p1`1*sqrt(1+(p1`2/p1`1)^2) by A132,A95,EUCLID:52;
(p1`2/p1`1)^2 >=0 by XREAL_1:63;
then
A136: sqrt(1+(p1`2/p1`1)^2)>0 by SQUARE_1:25;
A137: px`2 = p1`2*sqrt(1+(p1`2/p1`1)^2) by A132,A95,A134,EUCLID:52;
A138: now
assume px`1=0 & px`2=0;
then p1`1=0 & p1`2=0 by A135,A137,A136,XCMPLX_1:6;
hence contradiction by A133,EUCLID:53,54;
end;
p1`2<=p1`1 & -p1`1<=p1`2 or p1`2>=p1`1 & p1`2*sqrt(1+(p1`2/p1`1)
^2) <= (-p1`1)*sqrt(1+(p1`2/p1`1)^2) by A133,A136,XREAL_1:64;
then
A139: p1`2<=p1`1 & (-p1`1)*sqrt(1+(p1`2/p1`1)^2) <= p1`2*sqrt(1+(p1`2/
p1`1 )^2) or px`2>=px`1 & px`2<=-px`1 by A135,A137,A136,XREAL_1:64;
then
A140: px`1<>0 by A135,A137,A136,A138,XREAL_1:64;
(|.p1.|)^2>=|.p1.| by A96,XREAL_1:151;
then
A141: (|.p1.|)^2>=1 by A96,XXREAL_0:2;
A142: (px`1)^2 >=0 by XREAL_1:63;
set q=px;
A143: (|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|)`2 = q`2/
sqrt(1+( q`2/q`1)^2) by EUCLID:52;
A144: (px`2)^2>=0 by XREAL_1:63;
A145: (q`2/q`1)^2 >=0 by XREAL_1:63;
p1`2*sqrt(1+(p1`2/p1`1)^2) <= p1`1*sqrt(1+(p1`2/p1`1)^2) & -px`1
<= px `2 or px`2>=px`1 & px`2<=-px`1 by A132,A95,A134,A135,A136,A139,EUCLID:52
,XREAL_1:64;
then
A146: Sq_Circ.q=|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]| by A135
,A137,A138,JGRAPH_2:3,JGRAPH_3:def 1;
p1=Sq_Circ.px & (|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)
^2)]|)`1 = q `1/sqrt(1+( q`2/q`1)^2) by A132,A95,EUCLID:52,FUNCT_1:32
,JGRAPH_3:22,43;
then (|.p1.|)^2= (q`1/sqrt(1+(q`2/q`1)^2))^2+(q`2/sqrt(1+(q`2/q`1)^2)
) ^2 by A146,A143,JGRAPH_3:1
.= (q`1)^2/(sqrt(1+(q`2/q`1)^2))^2+(q`2/sqrt(1+(q`2/q`1)^2))^2 by
XCMPLX_1:76
.= (q`1)^2/(sqrt(1+(q`2/q`1)^2))^2+(q`2)^2/(sqrt(1+(q`2/q`1)^2))^2
by XCMPLX_1:76
.= (q`1)^2/(1+(q`2/q`1)^2)+(q`2)^2/(sqrt(1+(q`2/q`1)^2))^2 by A145,
SQUARE_1:def 2
.= (q`1)^2/(1+(q`2/q`1)^2)+(q`2)^2/(1+(q`2/q`1)^2) by A145,
SQUARE_1:def 2
.= ((q`1)^2+(q`2)^2)/(1+(q`2/q`1)^2) by XCMPLX_1:62;
then ((q`1)^2+(q`2)^2)/(1+(q`2/q`1)^2)*(1+(q`2/q`1)^2)>=(1)*(1+(q`2/
q `1)^2) by A145,A141,XREAL_1:64;
then ((q`1)^2+(q`2)^2)>=(1+(q`2/q`1)^2) by A145,XCMPLX_1:87;
then (px`1)^2+(px`2)^2>=1+(px`2)^2/(px`1)^2 by XCMPLX_1:76;
then (px`1)^2+(px`2)^2-1>=1+(px`2)^2/(px`1)^2-1 by XREAL_1:9;
then ((px`1)^2+(px`2)^2-1)*(px`1)^2>=(px`2)^2/(px`1)^2*(px`1)^2 by A142
,XREAL_1:64;
then ((px`1)^2+((px`2)^2-1))*(px`1)^2>=(px`2)^2 by A140,XCMPLX_1:6,87;
then (px`1)^2*(px`1)^2+(px`1)^2*((px`2)^2-1)-(px`2)^2>=(px`2)^2-(px
`2 )^2 by XREAL_1:9;
then
A147: ((px`1)^2-1)*((px`1)^2+(px`2)^2)>=0;
((px`1)^2+(px`2)^2)<>0 by A138,COMPLEX1:1;
then (px`1-1)*(px`1+1)>=0 by A142,A147,A144,XREAL_1:132;
hence -1>=(ff.r)`1 or (ff.r)`1>=1 or -1 >=(ff.r)`2 or (ff.r)`2>=1 by
XREAL_1:95;
end;
case
A148: p1<>0.TOP-REAL 2 & not(p1`2<=p1`1 & -p1`1<=p1`2 or p1`2>=p1
`1 & p1`2<=-p1`1);
reconsider pz=ff.r as Point of TOP-REAL 2;
A149: Sq_Circ".p1=|[p1`1*sqrt(1+(p1`1/p1`2)^2),p1`2*sqrt(1+(p1`1 /p1`2
) ^2)]| by A148,JGRAPH_3:28;
then
A150: pz`2 = p1`2*sqrt(1+(p1`1/p1`2)^2) by A132,A95,EUCLID:52;
(p1`1/p1`2)^2 >=0 by XREAL_1:63;
then
A151: sqrt(1+(p1`1/p1`2)^2)>0 by SQUARE_1:25;
A152: now
assume that
A153: pz`2=0 and
pz`1=0;
p1`2=0 by A150,A151,A153,XCMPLX_1:6;
hence contradiction by A148;
end;
A154: pz`1 = p1`1*sqrt(1+(p1`1/p1`2)^2) by A132,A95,A149,EUCLID:52;
p1`1<=p1`2 & -p1`2<=p1`1 or p1`1>=p1`2 & p1`1<=-p1`2 by A148,
JGRAPH_2:13;
then p1`1<=p1`2 & -p1`2<=p1`1 or p1`1>=p1`2 & p1`1*sqrt(1+(p1`1/p1`2)
^2) <= (-p1`2)*sqrt(1+(p1`1/p1`2)^2) by A151,XREAL_1:64;
then
A155: p1`1<=p1`2 & (-p1`2)*sqrt(1+(p1`1/p1`2)^2) <= p1`1*sqrt(1+(p1`1/
p1`2 )^2) or pz`1>=pz`2 & pz`1<=-pz`2 by A150,A154,A151,XREAL_1:64;
then
A156: pz`2<>0 by A150,A154,A151,A152,XREAL_1:64;
p1`1*sqrt(1+(p1`1/p1`2)^2) <= p1`2*sqrt(1+(p1`1/p1`2)^2) & -pz
`2<=pz `1 or pz`1>=pz`2 & pz`1<=-pz`2 by A132,A95,A149,A150,A151,A155,EUCLID:52
,XREAL_1:64;
then
A157: Sq_Circ.pz=|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/pz`2)
^2)]| by A150,A154,A152,JGRAPH_2:3,JGRAPH_3:4;
A158: (pz`1/pz`2)^2 >=0 by XREAL_1:63;
(|.p1.|)^2>=|.p1.| by A96,XREAL_1:151;
then
A159: (|.p1.|)^2>=1 by A96,XXREAL_0:2;
A160: (|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/pz`2)^2)]|)`1 =
pz`1/ sqrt(1+(pz`1/pz`2)^2) by EUCLID:52;
A161: (pz`1)^2>=0 by XREAL_1:63;
A162: (pz`2)^2 >=0 by XREAL_1:63;
p1=Sq_Circ.pz & (|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/
pz`2)^2)]|) `2 = pz`2/ sqrt(1+(pz`1/pz`2)^2) by A132,A95,EUCLID:52,FUNCT_1:32
,JGRAPH_3:22,43;
then ( |.p1.|)^2= (pz`2/sqrt(1+(pz`1/pz`2)^2))^2+(pz`1/sqrt(1+(pz`1/
pz`2)^2))^2 by A157,A160,JGRAPH_3:1
.= (pz`2)^2/(sqrt(1+(pz`1/pz`2)^2))^2+(pz`1/sqrt(1+(pz`1/pz`2)^2))
^2 by XCMPLX_1:76
.= (pz`2)^2/(sqrt(1+(pz`1/pz`2)^2))^2 +(pz`1)^2/(sqrt(1+(pz`1/pz`2
)^2))^2 by XCMPLX_1:76
.= (pz`2)^2/(1+(pz`1/pz`2)^2)+(pz`1)^2/(sqrt(1+(pz`1/pz`2)^2))^2
by A158,SQUARE_1:def 2
.= (pz`2)^2/(1+(pz`1/pz`2)^2)+(pz`1)^2/(1+(pz`1/pz`2)^2) by A158,
SQUARE_1:def 2
.= ((pz`2)^2+(pz`1)^2)/(1+(pz`1/pz`2)^2) by XCMPLX_1:62;
then ((pz`2)^2+(pz`1)^2)/(1+(pz`1/pz`2)^2)*(1+(pz`1/pz`2)^2) >=(1)*(1
+(pz`1/pz`2)^2) by A158,A159,XREAL_1:64;
then ((pz`2)^2+(pz`1)^2)>=(1+(pz`1/pz`2)^2) by A158,XCMPLX_1:87;
then (pz`2)^2+(pz`1)^2>=1+(pz`1)^2/(pz`2)^2 by XCMPLX_1:76;
then (pz`2)^2+(pz`1)^2-1>=1+(pz`1)^2/(pz`2)^2-1 by XREAL_1:9;
then ((pz`2)^2+(pz`1)^2-1)*(pz`2)^2>=(pz`1)^2/(pz`2)^2*(pz`2)^2 by A162
,XREAL_1:64;
then ((pz`2)^2+((pz`1)^2-1))*(pz`2)^2>=(pz`1)^2 by A156,XCMPLX_1:6,87;
then (pz`2)^2*(pz`2)^2+(pz`2)^2*((pz`1)^2-1)-(pz`1)^2 >=(pz`1)^2-(pz
`1)^2 by XREAL_1:9;
then
A163: ((pz`2)^2-1)*((pz`2)^2+(pz`1)^2)>=0;
((pz`2)^2+(pz`1)^2)<>0 by A152,COMPLEX1:1;
then (pz`2-1)*(pz`2+1)>=0 by A162,A163,A161,XREAL_1:132;
hence -1>=(ff.r)`1 or (ff.r)`1>=1 or -1 >=(ff.r)`2 or (ff.r)`2>=1 by
XREAL_1:95;
end;
end;
hence thesis by A100;
end;
-1 <=(ff.O)`2 & (ff.O)`2 <= 1 & -1 <=(ff.I)`2 & (ff.I)`2 <= 1 & -1 <=(
gg.O)`1 & (gg.O)`1 <= 1 & -1 <=(gg.I)`1 & (gg.I)`1 <= 1
proof
reconsider pz=gg.O as Point of TOP-REAL 2;
reconsider py=ff.I as Point of TOP-REAL 2;
reconsider px=ff.O as Point of TOP-REAL 2;
set q=px;
A164: (|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|)`1 = q`1/sqrt(1
+( q`2/q `1)^2) & (|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|)`2 = q`2
/sqrt(1+ ( q`2/q`1)^2) by EUCLID:52;
A165: (q`2/q`1)^2 >=0 by XREAL_1:63;
consider p1 being Point of TOP-REAL 2 such that
A166: f.O=p1 and
A167: |.p1.|=1 and
A168: p1`2>=p1`1 & p1`2<=-p1`1 by A1;
A169: (ff.O)=(Sq_Circ").(f.O) by A4,FUNCT_1:12;
then
A170: p1=Sq_Circ.px by A166,FUNCT_1:32,JGRAPH_3:22,43;
(p1`2/p1`1)^2 >=0 by XREAL_1:63;
then
A171: sqrt(1+(p1`2/p1`1)^2)>0 by SQUARE_1:25;
A172: p1<>0.TOP-REAL 2 by A167,TOPRNS_1:23;
then Sq_Circ".p1=|[p1`1*sqrt(1+(p1`2/p1`1)^2),p1`2*sqrt(1+(p1`2/p1`1) ^2)
]| by A168,JGRAPH_3:28;
then
A173: px`1 = p1`1*sqrt(1+(p1`2/p1`1)^2) & px`2 = p1`2*sqrt(1+(p1`2/p1`1)^2
) by A169,A166,EUCLID:52;
A174: now
assume px`1=0 & px`2=0;
then p1`1=0 & p1`2=0 by A173,A171,XCMPLX_1:6;
hence contradiction by A172,EUCLID:53,54;
end;
p1`2<=p1`1 & -p1`1<=p1`2 or p1`2>=p1`1 & p1`2*sqrt(1+(p1`2/p1`1)^2)
<= (-p1`1)*sqrt(1+(p1`2/p1`1)^2) by A168,A171,XREAL_1:64;
then
A175: p1`2<=p1`1 & (-p1`1)*sqrt(1+(p1`2/p1`1)^2) <= p1`2*sqrt(1+(p1`2 /p1
`1 )^2) or px`2>=px`1 & px`2<=-px`1 by A173,A171,XREAL_1:64;
then px`2<=px`1 & -px`1<=px`2 or px`2>=px`1 & px`2<=-px`1 by A173,A171,
XREAL_1:64;
then Sq_Circ.q=|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]| by A174,
JGRAPH_2:3,JGRAPH_3:def 1;
then (|.p1.|)^2= (q`1/sqrt(1+(q`2/q`1)^2))^2+(q`2/sqrt(1+(q`2/q`1)^2)) ^2
by A170,A164,JGRAPH_3:1
.= (q`1)^2/(sqrt(1+(q`2/q`1)^2))^2+(q`2/sqrt(1+(q`2/q`1)^2))^2 by
XCMPLX_1:76
.= (q`1)^2/(sqrt(1+(q`2/q`1)^2))^2+(q`2)^2/(sqrt(1+(q`2/q`1)^2))^2 by
XCMPLX_1:76
.= (q`1)^2/(1+(q`2/q`1)^2)+(q`2)^2/(sqrt(1+(q`2/q`1)^2))^2 by A165,
SQUARE_1:def 2
.= (q`1)^2/(1+(q`2/q`1)^2)+(q`2)^2/(1+(q`2/q`1)^2) by A165,SQUARE_1:def 2
.= ((q`1)^2+(q`2)^2)/(1+(q`2/q`1)^2) by XCMPLX_1:62;
then ((q`1)^2+(q`2)^2)/(1+(q`2/q`1)^2)*(1+(q`2/q`1)^2)=(1)*(1+(q`2/q`1 )
^2) by A167;
then ((q`1)^2+(q`2)^2)=(1+(q`2/q`1)^2) by A165,XCMPLX_1:87;
then
A176: (px`1)^2+(px`2)^2-1=(px`2)^2/(px`1)^2 by XCMPLX_1:76;
px`1<>0 by A173,A171,A174,A175,XREAL_1:64;
then ((px`1)^2+(px`2)^2-1)*(px`1)^2=(px`2)^2 by A176,XCMPLX_1:6,87;
then
A177: ((px`1)^2-1)*((px`1)^2+(px`2)^2)=0;
((px`1)^2+(px`2)^2)<>0 by A174,COMPLEX1:1;
then (px`1-1)*(px`1+1)=0 by A177,XCMPLX_1:6;
then px`1-1=0 or px`1+1=0 by XCMPLX_1:6;
then px`1=1 or px`1=0-1;
hence -1 <=(ff.O)`2 & (ff.O)`2 <= 1 by A173,A171,A175,XREAL_1:64;
A178: (py`2/py`1)^2 >=0 by XREAL_1:63;
reconsider pu=gg.I as Point of TOP-REAL 2;
A179: (|[py`1/sqrt(1+(py`2/py`1)^2),py`2/sqrt(1+(py`2/py`1)^2)]|)`1 = py`1
/ sqrt(1+ (py`2/py`1)^2) & (|[py`1/sqrt(1+(py`2/py`1)^2),py`2/sqrt(1+(py`2/py`1
)^2)]|)`2 = py`2/ sqrt(1+(py`2/py`1)^2) by EUCLID:52;
A180: (pz`1/pz`2)^2 >=0 by XREAL_1:63;
consider p2 being Point of TOP-REAL 2 such that
A181: f.I=p2 and
A182: |.p2.|=1 and
A183: p2`2<=p2`1 & p2`2>=-p2`1 by A1;
A184: (ff.I)=(Sq_Circ").(f.I) by A4,FUNCT_1:12;
then
A185: p2=Sq_Circ.py by A181,FUNCT_1:32,JGRAPH_3:22,43;
A186: p2<>0.TOP-REAL 2 by A182,TOPRNS_1:23;
then
A187: Sq_Circ".p2=|[p2`1*sqrt(1+(p2`2/p2`1)^2),p2`2*sqrt(1+(p2`2/p2`1) ^2)
]| by A183,JGRAPH_3:28;
then
A188: py`1 = p2`1*sqrt(1+(p2`2/p2`1)^2) by A184,A181,EUCLID:52;
A189: py`2 = p2`2*sqrt(1+(p2`2/p2`1)^2) by A184,A181,A187,EUCLID:52;
(p2`2/p2`1)^2 >=0 by XREAL_1:63;
then
A190: sqrt(1+(p2`2/p2`1)^2)>0 by SQUARE_1:25;
A191: now
assume py`1=0 & py`2=0;
then p2`1=0 & p2`2=0 by A188,A189,A190,XCMPLX_1:6;
hence contradiction by A186,EUCLID:53,54;
end;
A192: p2`2<=p2`1 & (-p2`1)*sqrt(1+(p2`2/p2`1)^2) <= p2`2*sqrt(1+(p2`2/p2`1
) ^2 ) or py`2>=py`1 & py`2<=-py`1 by A183,A190,XREAL_1:64;
then
A193: p2`2*sqrt(1+(p2`2/p2`1)^2) <= p2`1*sqrt(1+(p2`2/p2`1)^2) & -py `1<=
py `2 or py`2>=py`1 & py`2<=-py`1 by A184,A181,A187,A188,A190,EUCLID:52
,XREAL_1:64;
then Sq_Circ.py=|[py`1/sqrt(1+(py`2/py`1)^2),py`2/sqrt(1+(py`2/py`1)^2)]|
by A188,A189,A191,JGRAPH_2:3,JGRAPH_3:def 1;
then (|.p2.|)^2= (py`1/sqrt(1+(py`2/py`1)^2))^2+(py`2/sqrt(1+(py`2/py`1)
^2 ) ) ^2 by A185,A179,JGRAPH_3:1
.= (py`1)^2/(sqrt(1+(py`2/py`1)^2))^2+(py`2/sqrt(1+(py`2/py`1)^2))^2
by XCMPLX_1:76
.= (py`1)^2/(sqrt(1+(py`2/py`1)^2))^2 +(py`2)^2/(sqrt(1+(py`2/py`1)^2)
)^2 by XCMPLX_1:76
.= (py`1)^2/(1+(py`2/py`1)^2)+(py`2)^2/(sqrt(1+(py`2/py`1)^2))^2 by A178,
SQUARE_1:def 2
.= (py`1)^2/(1+(py`2/py`1)^2)+(py`2)^2/(1+(py`2/py`1)^2) by A178,
SQUARE_1:def 2
.= ((py`1)^2+(py`2)^2)/(1+(py`2/py`1)^2) by XCMPLX_1:62;
then ((py`1)^2+(py`2)^2)/(1+(py`2/py`1)^2)*(1+(py`2/py`1)^2) =(1)*(1+(py
`2/py`1)^2) by A182;
then ((py`1)^2+(py`2)^2)=(1+(py`2/py`1)^2) by A178,XCMPLX_1:87;
then
A194: (py`1)^2+(py`2)^2-1=(py`2)^2/(py`1)^2 by XCMPLX_1:76;
py`1<>0 by A188,A189,A190,A191,A192,XREAL_1:64;
then ((py`1)^2+(py`2)^2-1)*(py`1)^2=(py`2)^2 by A194,XCMPLX_1:6,87;
then
A195: ((py`1)^2-1)*((py`1)^2+(py`2)^2)=0;
((py`1)^2+(py`2)^2)<>0 by A191,COMPLEX1:1;
then (py`1-1)*(py`1+1)=0 by A195,XCMPLX_1:6;
then py`1-1=0 or py`1+1=0 by XCMPLX_1:6;
hence -1 <=(ff.I)`2 & (ff.I)`2 <= 1 by A188,A189,A193;
A196: (|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/pz`2)^2)]|)`2 = pz`2
/ sqrt(1+ (pz`1/pz`2)^2) & (|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/pz`2
)^2)]|)`1 = pz`1/ sqrt(1+(pz`1/pz`2)^2) by EUCLID:52;
consider p3 being Point of TOP-REAL 2 such that
A197: g.O=p3 and
A198: |.p3.|=1 and
A199: p3`2<=p3`1 and
A200: p3`2<=-p3`1 by A1;
A201: p3<>0.TOP-REAL 2 by A198,TOPRNS_1:23;
A202: gg.O=(Sq_Circ").(g.O) by A3,FUNCT_1:12;
then
A203: p3=Sq_Circ.pz by A197,FUNCT_1:32,JGRAPH_3:22,43;
A204: -p3`2>=--p3`1 by A200,XREAL_1:24;
then
A205: Sq_Circ".p3=|[p3`1*sqrt(1+(p3`1/p3`2)^2),p3`2*sqrt(1+(p3`1/p3`2) ^2
)]| by A199,A201,JGRAPH_3:30;
then
A206: pz`2 = p3`2*sqrt(1+(p3`1/p3`2)^2) by A202,A197,EUCLID:52;
A207: pz`1 = p3`1*sqrt(1+(p3`1/p3`2)^2) by A202,A197,A205,EUCLID:52;
(p3`1/p3`2)^2 >=0 by XREAL_1:63;
then
A208: sqrt(1+(p3`1/p3`2)^2)>0 by SQUARE_1:25;
A209: now
assume pz`2=0 & pz`1=0;
then p3`2=0 & p3`1=0 by A206,A207,A208,XCMPLX_1:6;
hence contradiction by A201,EUCLID:53,54;
end;
p3`1<=p3`2 & -p3`2<=p3`1 or p3`1>=p3`2 & p3`1*sqrt(1+(p3`1/p3`2)^2)
<= (-p3`2)*sqrt(1+(p3`1/p3`2)^2) by A199,A204,A208,XREAL_1:64;
then
A210: p3`1<=p3`2 & (-p3`2)*sqrt(1+(p3`1/p3`2)^2) <= p3`1*sqrt(1+(p3`1/p3
`2 )^2) or pz`1>=pz`2 & pz`1<=-pz`2 by A206,A207,A208,XREAL_1:64;
then
A211: p3`1*sqrt(1+(p3`1/p3`2)^2) <= p3`2*sqrt(1+(p3`1/p3`2)^2) & -pz `2<=
pz `1 or pz`1>=pz`2 & pz`1<=-pz`2 by A202,A197,A205,A206,A208,EUCLID:52
,XREAL_1:64;
then Sq_Circ.pz=|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/pz`2)^2)]|
by A206,A207,A209,JGRAPH_2:3,JGRAPH_3:4;
then (|.p3.|)^2= (pz`2/sqrt(1+(pz`1/pz`2)^2))^2+(pz`1/sqrt(1+(pz`1/pz`2)
^2))^2 by A203,A196,JGRAPH_3:1
.= (pz`2)^2/(sqrt(1+(pz`1/pz`2)^2))^2+(pz`1/sqrt(1+(pz`1/pz`2)^2))^2
by XCMPLX_1:76
.= (pz`2)^2/(sqrt(1+(pz`1/pz`2)^2))^2 +(pz`1)^2/(sqrt(1+(pz`1/pz`2)^2)
)^2 by XCMPLX_1:76
.= (pz`2)^2/(1+(pz`1/pz`2)^2)+(pz`1)^2/(sqrt(1+(pz`1/pz`2)^2))^2 by A180,
SQUARE_1:def 2
.= (pz`2)^2/(1+(pz`1/pz`2)^2)+(pz`1)^2/(1+(pz`1/pz`2)^2) by A180,
SQUARE_1:def 2
.= ((pz`2)^2+(pz`1)^2)/(1+(pz`1/pz`2)^2) by XCMPLX_1:62;
then ((pz`2)^2+(pz`1)^2)/(1+(pz`1/pz`2)^2)*(1+(pz`1/pz`2)^2)=(1)*(1+(pz`1
/pz `2 ) ^2 ) by A198;
then ((pz`2)^2+(pz`1)^2)=(1+(pz`1/pz`2)^2) by A180,XCMPLX_1:87;
then
A212: (pz`2)^2+(pz`1)^2-1=(pz`1)^2/(pz`2)^2 by XCMPLX_1:76;
pz`2<>0 by A206,A207,A208,A209,A210,XREAL_1:64;
then ((pz`2)^2+(pz`1)^2-1)*(pz`2)^2=(pz`1)^2 by A212,XCMPLX_1:6,87;
then
A213: ((pz`2)^2-1)*((pz`2)^2+(pz`1)^2)=0;
((pz`2)^2+(pz`1)^2)<>0 by A209,COMPLEX1:1;
then (pz`2-1)*(pz`2+1)=0 by A213,XCMPLX_1:6;
then pz`2-1=0 or pz`2+1=0 by XCMPLX_1:6;
hence -1 <=(gg.O)`1 & (gg.O)`1 <= 1 by A206,A207,A211;
A214: (|[pu`1/sqrt(1+(pu`1/pu`2)^2),pu`2/sqrt(1+(pu`1/pu`2)^2)]|)`2 = pu`2
/ sqrt(1+ (pu`1/pu`2)^2) & (|[pu`1/sqrt(1+(pu`1/pu`2)^2),pu`2/sqrt(1+(pu`1/pu`2
)^2)]|)`1 = pu`1/ sqrt(1+(pu`1/pu`2)^2) by EUCLID:52;
A215: (pu`1/pu`2)^2 >=0 by XREAL_1:63;
consider p4 being Point of TOP-REAL 2 such that
A216: g.I=p4 and
A217: |.p4.|=1 and
A218: p4`2>=p4`1 and
A219: p4`2>=-p4`1 by A1;
A220: -p4`2<=--p4`1 by A219,XREAL_1:24;
A221: (gg.I)=(Sq_Circ").(g.I) by A3,FUNCT_1:12;
then
A222: p4=Sq_Circ.pu by A216,FUNCT_1:32,JGRAPH_3:22,43;
A223: p4<>0.TOP-REAL 2 by A217,TOPRNS_1:23;
then
A224: Sq_Circ".p4=|[p4`1*sqrt(1+(p4`1/p4`2)^2),p4`2*sqrt(1+(p4`1/p4`2) ^2)
]| by A218,A220,JGRAPH_3:30;
then
A225: pu`2 = p4`2*sqrt(1+(p4`1/p4`2)^2) by A221,A216,EUCLID:52;
A226: pu`1 = p4`1*sqrt(1+(p4`1/p4`2)^2) by A221,A216,A224,EUCLID:52;
(p4`1/p4`2)^2 >=0 by XREAL_1:63;
then
A227: sqrt(1+(p4`1/p4`2)^2)>0 by SQUARE_1:25;
A228: now
assume pu`2=0 & pu`1=0;
then p4`2=0 & p4`1=0 by A225,A226,A227,XCMPLX_1:6;
hence contradiction by A223,EUCLID:53,54;
end;
A229: p4`1<=p4`2 & (-p4`2)*sqrt(1+(p4`1/p4`2)^2) <= p4`1*sqrt(1+(p4`1/p4`2
) ^2 ) or pu`1>=pu`2 & pu`1<=-pu`2 by A218,A220,A227,XREAL_1:64;
then
A230: p4`1* sqrt(1+(p4`1/p4`2)^2) <= p4`2*sqrt(1+(p4`1/p4`2)^2) & -pu`2<=
pu `1 or pu`1>=pu`2 & pu`1<=-pu`2 by A221,A216,A224,A225,A227,EUCLID:52
,XREAL_1:64;
then Sq_Circ.pu=|[pu`1/sqrt(1+(pu`1/pu`2)^2),pu`2/sqrt(1+(pu`1/pu`2)^2)]|
by A225,A226,A228,JGRAPH_2:3,JGRAPH_3:4;
then (|.p4.|)^2= (pu`2/sqrt(1+(pu`1/pu`2)^2))^2+(pu`1/sqrt(1+(pu`1/pu`2)
^2 ) ) ^2 by A222,A214,JGRAPH_3:1
.= (pu`2)^2/(sqrt(1+(pu`1/pu`2)^2))^2+(pu`1/sqrt(1+(pu`1/pu`2)^2))^2
by XCMPLX_1:76
.= (pu`2)^2/(sqrt(1+(pu`1/pu`2)^2))^2 +(pu`1)^2/(sqrt(1+(pu`1/pu`2)^2)
)^2 by XCMPLX_1:76
.= (pu`2)^2/(1+(pu`1/pu`2)^2)+(pu`1)^2/(sqrt(1+(pu`1/pu`2)^2))^2 by A215,
SQUARE_1:def 2
.= (pu`2)^2/(1+(pu`1/pu`2)^2)+(pu`1)^2/(1+(pu`1/pu`2)^2) by A215,
SQUARE_1:def 2
.= ((pu`2)^2+(pu`1)^2)/(1+(pu`1/pu`2)^2) by XCMPLX_1:62;
then ((pu`2)^2+(pu`1)^2)/(1+(pu`1/pu`2)^2)*(1+(pu`1/pu`2)^2)=(1)*(1+(pu`1
/pu `2)^2) by A217;
then ((pu`2)^2+(pu`1)^2)=(1+(pu`1/pu`2)^2) by A215,XCMPLX_1:87;
then
A231: (pu`2)^2+(pu`1)^2-1=(pu`1)^2/(pu`2)^2 by XCMPLX_1:76;
pu`2<>0 by A225,A226,A227,A228,A229,XREAL_1:64;
then ((pu`2)^2+(pu`1)^2-1)*(pu`2)^2=(pu`1)^2 by A231,XCMPLX_1:6,87;
then
A232: ((pu`2)^2-1)*((pu`2)^2+(pu`1)^2)=0;
((pu`2)^2+(pu`1)^2)<>0 by A228,COMPLEX1:1;
then (pu`2-1)*(pu`2+1)=0 by A232,XCMPLX_1:6;
then pu`2-1=0 or pu`2+1=0 by XCMPLX_1:6;
hence thesis by A225,A226,A230;
end;
then rng ff meets rng gg by A1,A5,A94,Th11,JGRAPH_3:22,42;
then consider y being object such that
A233: y in rng ff and
A234: y in rng gg by XBOOLE_0:3;
consider x1 being object such that
A235: x1 in dom ff and
A236: y=ff.x1 by A233,FUNCT_1:def 3;
consider x2 being object such that
A237: x2 in dom gg and
A238: y=gg.x2 by A234,FUNCT_1:def 3;
A239: dom (Sq_Circ")=the carrier of TOP-REAL 2 & gg.x2=Sq_Circ".(g.x2) by A237,
FUNCT_1:12,FUNCT_2:def 1,JGRAPH_3:29;
x1 in dom f by A235,FUNCT_1:11;
then
A240: f.x1 in rng f by FUNCT_1:def 3;
x2 in dom g by A237,FUNCT_1:11;
then
A241: g.x2 in rng g by FUNCT_1:def 3;
ff.x1=Sq_Circ".(f.x1) by A235,FUNCT_1:12;
then f.x1=g.x2 by A236,A238,A240,A241,A239,FUNCT_1:def 4,JGRAPH_3:22;
hence thesis by A240,A241,XBOOLE_0:3;
end;
theorem Th15:
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
proof
let f,g be Function of I[01],TOP-REAL 2, C0,KXP,KXN,KYP,KYN be Subset of
TOP-REAL 2, 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
& 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;
then
ex g2 being Function of I[01],TOP-REAL 2 st g2.0=g.1 & g2.1=g.0 & rng g2=
rng g & g2 is continuous one-to-one by Th12;
hence thesis by A1,Th14;
end;
theorem Th16:
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
proof
reconsider I=1 as Point of I[01] by BORSUK_1:40,XXREAL_1:1;
reconsider O=0 as Point of I[01] by BORSUK_1:40,XXREAL_1:1;
defpred P[Point of TOP-REAL 2] means |.$1.|=1 & $1`2<=$1`1 & $1`2>=-$1`1;
let f,g be Function of I[01],TOP-REAL 2, C0 be Subset of TOP-REAL 2;
assume
A1: 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;
{q1 where q1 is Point of TOP-REAL 2:P[q1] } is Subset of TOP-REAL 2 from
JGRAPH_2:sch 1;
then reconsider
KXP={q1 where q1 is Point of TOP-REAL 2: |.q1.|=1 & q1`2<=q1`1 &
q1`2>=-q1`1} as Subset of TOP-REAL 2;
A2: (|[0,1]|)`1=0 by EUCLID:52;
defpred P[Point of TOP-REAL 2] means |.$1.|=1 & $1`2>=$1`1 & $1`2<=-$1`1;
{q2 where q2 is Point of TOP-REAL 2: P[q2]} is Subset of TOP-REAL 2 from
JGRAPH_2:sch 1;
then reconsider
KXN={q2 where q2 is Point of TOP-REAL 2: |.q2.|=1 & q2`2>=q2`1 &
q2`2<=-q2`1} as Subset of TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means |.$1.|=1 & $1`2>=$1`1 & $1`2>=-$1`1;
{q3 where q3 is Point of TOP-REAL 2:P[q3]} is Subset of TOP-REAL 2 from
JGRAPH_2:sch 1;
then reconsider
KYP={q3 where q3 is Point of TOP-REAL 2: |.q3.|=1 & q3`2>=q3`1 &
q3`2>=-q3`1} as Subset of TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means |.$1.|=1 & $1`2<=$1`1 & $1`2<=-$1`1;
{q4 where q4 is Point of TOP-REAL 2:P[q4]} is Subset of TOP-REAL 2 from
JGRAPH_2:sch 1;
then reconsider
KYN={q4 where q4 is Point of TOP-REAL 2: |.q4.|=1 & q4`2<=q4`1 &
q4`2<=-q4`1} as Subset of TOP-REAL 2;
A3: (|[0,-1]|)`1=0 by EUCLID:52;
(|[0,-1]|)`2=-1 by EUCLID:52;
then
A4: |.(|[0,-1]|).|=sqrt(0^2+(-1)^2) by A3,JGRAPH_3:1
.=1;
(|[0,-1]|)`2 <=-((|[0,-1]|)`1) by A3,EUCLID:52;
then
A5: g.O in KYN by A1,A3,A4;
A6: (|[-1,0]|)`1=-1 by EUCLID:52;
then
A7: (|[-1,0]|)`2 <=-((|[-1,0]|)`1) by EUCLID:52;
(|[0,1]|)`2=1 by EUCLID:52;
then
A8: |. (|[0,1]|).|=sqrt(0^2+1^2) by A2,JGRAPH_3:1
.=1;
(|[0,1]|)`2 >=-((|[0,1]|)`1) by A2,EUCLID:52;
then
A9: g.I in KYP by A1,A2,A8;
A10: (|[1,0]|)`1=1 & (|[1,0]|)`2=0 by EUCLID:52;
then |.(|[1,0]|).|=sqrt(1^2+0^2) by JGRAPH_3:1
.=1;
then
A11: f.I in KXP by A1,A10;
A12: (|[-1,0]|)`2=0 by EUCLID:52;
then |. (|[-1,0]|).|=sqrt((-1)^2+0^2) by A6,JGRAPH_3:1
.=1;
then f.O in KXN by A1,A6,A12,A7;
hence thesis by A1,A11,A5,A9,Th14;
end;
theorem
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
proof
let p1,p2,p3,p4 be Point of TOP-REAL 2, C0 be Subset of TOP-REAL 2;
assume
A1: 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]|);
then consider h being Function of TOP-REAL 2,TOP-REAL 2 such that
A2: h is being_homeomorphism and
A3: h.:C0 c= C0 and
A4: h.p1=(|[-1,0]|) and
A5: h.p2=(|[0,1]|) and
A6: h.p3=(|[1,0]|) and
A7: h.p4=(|[0,-1]|);
let f,g be Function of I[01],TOP-REAL 2;
assume that
A8: f is continuous one-to-one & g is continuous one-to-one and
A9: f.0=p1 and
A10: f.1=p3 and
A11: g.0=p4 and
A12: g.1=p2 and
A13: rng f c= C0 and
A14: rng g c= C0;
reconsider f2=h*f as Function of I[01],TOP-REAL 2;
0 in dom f2 by Lm1,BORSUK_1:40,FUNCT_2:def 1;
then
A15: f2.0=|[-1,0]| by A4,A9,FUNCT_1:12;
reconsider g2=h*g as Function of I[01],TOP-REAL 2;
0 in dom g2 by Lm1,BORSUK_1:40,FUNCT_2:def 1;
then
A16: g2.0=|[0,-1]| by A7,A11,FUNCT_1:12;
1 in dom g2 by Lm2,BORSUK_1:40,FUNCT_2:def 1;
then
A17: g2.1=|[0,1]| by A5,A12,FUNCT_1:12;
A18: rng f2 c= C0
proof
let y be object;
A19: dom h=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
assume y in rng f2;
then consider x being object such that
A20: x in dom f2 and
A21: y=f2.x by FUNCT_1:def 3;
x in dom f by A20,FUNCT_1:11;
then
A22: f.x in rng f by FUNCT_1:def 3;
y=h.(f.x) by A20,A21,FUNCT_1:12;
then y in h.:C0 by A13,A22,A19,FUNCT_1:def 6;
hence thesis by A3;
end;
A23: rng g2 c= C0
proof
let y be object;
A24: dom h=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
assume y in rng g2;
then consider x being object such that
A25: x in dom g2 and
A26: y=g2.x by FUNCT_1:def 3;
x in dom g by A25,FUNCT_1:11;
then
A27: g.x in rng g by FUNCT_1:def 3;
y=h.(g.x) by A25,A26,FUNCT_1:12;
then y in h.:C0 by A14,A27,A24,FUNCT_1:def 6;
hence thesis by A3;
end;
1 in dom f2 by Lm2,BORSUK_1:40,FUNCT_2:def 1;
then
A28: f2.1=|[1,0]| by A6,A10,FUNCT_1:12;
h is continuous & h is one-to-one by A2,TOPS_2:def 5;
then rng f2 meets rng g2 by A1,A8,A15,A28,A16,A17,A18,A23,Th16;
then consider q5 being object such that
A29: q5 in rng f2 and
A30: q5 in rng g2 by XBOOLE_0:3;
consider x being object such that
A31: x in dom f2 and
A32: q5=f2.x by A29,FUNCT_1:def 3;
x in dom f by A31,FUNCT_1:11;
then
A33: f.x in rng f by FUNCT_1:def 3;
consider u being object such that
A34: u in dom g2 and
A35: q5=g2.u by A30,FUNCT_1:def 3;
A36: q5=h.(g.u) & g.u in dom h by A34,A35,FUNCT_1:11,12;
A37: h is one-to-one by A2,TOPS_2:def 5;
u in dom g by A34,FUNCT_1:11;
then
A38: g.u in rng g by FUNCT_1:def 3;
q5=h.(f.x) & f.x in dom h by A31,A32,FUNCT_1:11,12;
then f.x=g.u by A37,A36,FUNCT_1:def 4;
hence thesis by A33,A38,XBOOLE_0:3;
end;
begin :: Properties of Fan Morphisms
theorem Th18:
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
proof
let cn be Real,q be Point of TOP-REAL 2;
assume that
A1: -1<cn and
A2: cn<1 and
A3: q`2>0;
now
per cases;
case
q`1/|.q.|>=cn;
hence thesis by A2,A3,JGRAPH_4:75;
end;
case
q`1/|.q.|<cn;
hence thesis by A1,A3,JGRAPH_4:76;
end;
end;
hence thesis;
end;
theorem
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
proof
let cn be Real,q be Point of TOP-REAL 2;
assume that
A1: -1<cn & cn<1 and
A2: q`2>=0;
now
per cases by A2;
case
q`2>0;
hence thesis by A1,Th18;
end;
case
q`2=0;
hence thesis by JGRAPH_4:49;
end;
end;
hence thesis;
end;
theorem Th20:
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
proof
let cn be Real,q be Point of TOP-REAL 2;
assume that
A1: -1<cn and
A2: cn<1 and
A3: q`2>=0 & q`1/|.q.|<cn and
A4: |.q.|<>0;
let p be Point of TOP-REAL 2;
assume
A5: p=(cn-FanMorphN).q;
now
per cases;
case
A6: q`2=0;
then |.q.|^2=(q`1)^2+0^2 by JGRAPH_3:1
.=(q`1)^2;
then
A7: |.q.|=q`1 or |.q.|=-(q`1) by SQUARE_1:40;
q=p by A5,A6,JGRAPH_4:49;
hence thesis by A2,A3,A4,A7,XCMPLX_1:60;
end;
case
q`2<>0;
hence thesis by A1,A3,A5,JGRAPH_4:76;
end;
end;
hence thesis;
end;
theorem Th21:
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.|
proof
let cn be Real,q1,q2 be Point of TOP-REAL 2;
assume that
A1: -1<cn & cn<1 and
A2: q1`2>=0 and
A3: q2`2>=0 and
A4: |.q1.|<>0 and
A5: |.q2.|<>0 and
A6: q1`1/|.q1.|<q2`1/|.q2.|;
now
per cases by A2;
case
A7: q1`2>0;
now
per cases by A3;
case
q2`2>0;
hence thesis by A1,A6,A7,JGRAPH_4:79;
end;
case
A8: q2`2=0;
A9: now
(|.q1.|)^2=(q1`1)^2+(q1`2)^2 by JGRAPH_3:1;
then (|.q1.|)^2-(q1`1)^2>=0 by XREAL_1:63;
then (|.q1.|)^2-(q1`1)^2+(q1`1)^2>=0+(q1`1)^2 by XREAL_1:7;
then -|.q1.|<=q1`1 by SQUARE_1:47;
then
A10: (-|.q1.|)/|.q1.|<=q1`1/|.q1.| by XREAL_1:72;
assume |.q2.|=-(q2`1);
then 1=(-(q2`1))/|.q2.| by A5,XCMPLX_1:60;
then q1`1/|.q1.|< -1 by A6,XCMPLX_1:190;
hence contradiction by A4,A10,XCMPLX_1:197;
end;
|.q2.|^2=(q2`1)^2+0^2 by A8,JGRAPH_3:1
.=(q2`1)^2;
then |.q2.|=q2`1 or |.q2.|=-(q2`1) by SQUARE_1:40;
then
A11: q2`1/|.q2.|=1 by A5,A9,XCMPLX_1:60;
thus 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.|
proof
let p1,p2 be Point of TOP-REAL 2;
assume that
A12: p1=(cn-FanMorphN).q1 and
A13: p2=(cn-FanMorphN).q2;
A14: |.p1.|=|.q1.| by A12,JGRAPH_4:66;
A15: (|.p1.|)^2=(p1`1)^2+(p1`2)^2 by JGRAPH_3:1;
A16: p1`2>0 by A1,A7,A12,Th18;
A17: now
assume 1= p1`1/|.p1.|;
then (1)*|.p1.|=p1`1 by A4,A14,XCMPLX_1:87;
hence contradiction by A15,A16,XCMPLX_1:6;
end;
A18: p2=q2 by A8,A13,JGRAPH_4:49;
(|.p1.|)^2-(p1`1)^2>=0 by A15,XREAL_1:63;
then (|.p1.|)^2-(p1`1)^2+(p1`1)^2>=0+(p1`1)^2 by XREAL_1:7;
then p1`1<=|.p1.| by SQUARE_1:47;
then (|.p1.|)/|.p1.|>=p1`1/|.p1.| by XREAL_1:72;
then 1>= p1`1/|.p1.| by A4,A14,XCMPLX_1:60;
hence thesis by A11,A18,A17,XXREAL_0:1;
end;
end;
end;
hence thesis;
end;
case
A19: q1`2=0;
A20: now
(|.q2.|)^2=(q2`1)^2+(q2`2)^2 by JGRAPH_3:1;
then (|.q2.|)^2-(q2`1)^2>=0 by XREAL_1:63;
then (|.q2.|)^2-(q2`1)^2+(q2`1)^2>=0+(q2`1)^2 by XREAL_1:7;
then q2`1<=|.q2.| by SQUARE_1:47;
then
A21: (|.q2.|)/|.q2.|>=q2`1/|.q2.| by XREAL_1:72;
assume |.q1.|=(q1`1);
then q2`1/|.q2.|> 1 by A4,A6,XCMPLX_1:60;
hence contradiction by A5,A21,XCMPLX_1:60;
end;
|.q1.|^2=(q1`1)^2+0^2 by A19,JGRAPH_3:1
.=(q1`1)^2;
then |.q1.|=q1`1 or |.q1.|=-(q1`1) by SQUARE_1:40;
then (-(q1`1))/|.q1.|=1 by A4,A20,XCMPLX_1:60;
then
A22: -(q1`1/|.q1.|)=1 by XCMPLX_1:187;
thus 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.|
proof
let p1,p2 be Point of TOP-REAL 2;
assume that
A23: p1=(cn-FanMorphN).q1 and
A24: p2=(cn-FanMorphN).q2;
A25: |.p2.|=|.q2.| by A24,JGRAPH_4:66;
A26: (|.p2.|)^2=(p2`1)^2+(p2`2)^2 by JGRAPH_3:1;
then (|.p2.|)^2-(p2`1)^2>=0 by XREAL_1:63;
then (|.p2.|)^2-(p2`1)^2+(p2`1)^2>=0+(p2`1)^2 by XREAL_1:7;
then -|.p2.|<=p2`1 by SQUARE_1:47;
then (-|.p2.|)/|.p2.|<=p2`1/|.p2.| by XREAL_1:72;
then
A27: -1 <= p2`1/|.p2.| by A5,A25,XCMPLX_1:197;
A28: now
per cases;
case
q2`2=0;
then p2=q2 by A24,JGRAPH_4:49;
hence p2`1/|.p2.|>-1 by A6,A22;
end;
case
q2`2<>0;
then
A29: p2`2>0 by A1,A3,A24,Th18;
now
assume -1= p2`1/|.p2.|;
then (-1)*|.p2.|=p2`1 by A5,A25,XCMPLX_1:87;
then (|.p2.|)^2=(p2`1)^2;
hence contradiction by A26,A29,XCMPLX_1:6;
end;
hence p2`1/|.p2.|>-1 by A27,XXREAL_0:1;
end;
end;
p1=q1 by A19,A23,JGRAPH_4:49;
hence thesis by A22,A28;
end;
end;
end;
hence thesis;
end;
theorem Th22:
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
proof
let sn be Real,q be Point of TOP-REAL 2;
assume that
A1: -1<sn and
A2: sn<1 and
A3: q`1>0;
now
per cases;
case
q`2/|.q.|>=sn;
hence thesis by A2,A3,JGRAPH_4:106;
end;
case
q`2/|.q.|<sn;
hence thesis by A1,A3,JGRAPH_4:107;
end;
end;
hence thesis;
end;
theorem
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
proof
let sn be Real,q be Point of TOP-REAL 2;
assume that
A1: -1<sn and
A2: sn<1 and
A3: q`1>=0 & q`2/|.q.|<sn and
A4: |.q.|<>0;
let p be Point of TOP-REAL 2;
assume
A5: p=(sn-FanMorphE).q;
now
per cases;
case
A6: q`1=0;
then |.q.|^2=(q`2)^2+0^2 by JGRAPH_3:1
.=(q`2)^2;
then
A7: |.q.|=q`2 or |.q.|=-(q`2) by SQUARE_1:40;
q=p by A5,A6,JGRAPH_4:82;
hence thesis by A2,A3,A4,A7,XCMPLX_1:60;
end;
case
q`1<>0;
hence thesis by A1,A3,A5,JGRAPH_4:107;
end;
end;
hence thesis;
end;
theorem Th24:
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.|
proof
let sn be Real,q1,q2 be Point of TOP-REAL 2;
assume that
A1: -1<sn & sn<1 and
A2: q1`1>=0 and
A3: q2`1>=0 and
A4: |.q1.|<>0 and
A5: |.q2.|<>0 and
A6: q1`2/|.q1.|<q2`2/|.q2.|;
now
per cases by A2;
case
A7: q1`1>0;
now
per cases by A3;
case
q2`1>0;
hence thesis by A1,A6,A7,JGRAPH_4:110;
end;
case
A8: q2`1=0;
A9: now
(|.q1.|)^2=(q1`2)^2+(q1`1)^2 by JGRAPH_3:1;
then (|.q1.|)^2-(q1`2)^2>=0 by XREAL_1:63;
then (|.q1.|)^2-(q1`2)^2+(q1`2)^2>=0+(q1`2)^2 by XREAL_1:7;
then -|.q1.|<=q1`2 by SQUARE_1:47;
then
A10: (-|.q1.|)/|.q1.|<=q1`2/|.q1.| by XREAL_1:72;
assume |.q2.|=-(q2`2);
then 1=(-(q2`2))/|.q2.| by A5,XCMPLX_1:60;
then q1`2/|.q1.|< -1 by A6,XCMPLX_1:190;
hence contradiction by A4,A10,XCMPLX_1:197;
end;
|.q2.|^2=(q2`2)^2+0^2 by A8,JGRAPH_3:1
.=(q2`2)^2;
then |.q2.|=q2`2 or |.q2.|=-(q2`2) by SQUARE_1:40;
then
A11: q2`2/|.q2.|=1 by A5,A9,XCMPLX_1:60;
thus 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.|
proof
let p1,p2 be Point of TOP-REAL 2;
assume that
A12: p1=(sn-FanMorphE).q1 and
A13: p2=(sn-FanMorphE).q2;
A14: |.p1.|=|.q1.| by A12,JGRAPH_4:97;
A15: (|.p1.|)^2=(p1`2)^2+(p1`1)^2 by JGRAPH_3:1;
A16: p1`1>0 by A1,A7,A12,Th22;
A17: now
assume 1= p1`2/|.p1.|;
then (1)*|.p1.|=p1`2 by A4,A14,XCMPLX_1:87;
hence contradiction by A15,A16,XCMPLX_1:6;
end;
A18: p2=q2 by A8,A13,JGRAPH_4:82;
(|.p1.|)^2-(p1`2)^2>=0 by A15,XREAL_1:63;
then (|.p1.|)^2-(p1`2)^2+(p1`2)^2>=0+(p1`2)^2 by XREAL_1:7;
then p1`2<=|.p1.| by SQUARE_1:47;
then (|.p1.|)/|.p1.|>=p1`2/|.p1.| by XREAL_1:72;
then 1>= p1`2/|.p1.| by A4,A14,XCMPLX_1:60;
hence thesis by A11,A18,A17,XXREAL_0:1;
end;
end;
end;
hence thesis;
end;
case
A19: q1`1=0;
A20: now
(|.q2.|)^2=(q2`2)^2+(q2`1)^2 by JGRAPH_3:1;
then (|.q2.|)^2-(q2`2)^2>=0 by XREAL_1:63;
then (|.q2.|)^2-(q2`2)^2+(q2`2)^2>=0+(q2`2)^2 by XREAL_1:7;
then q2`2<=|.q2.| by SQUARE_1:47;
then
A21: (|.q2.|)/|.q2.|>=q2`2/|.q2.| by XREAL_1:72;
assume |.q1.|=(q1`2);
then q2`2/|.q2.|> 1 by A4,A6,XCMPLX_1:60;
hence contradiction by A5,A21,XCMPLX_1:60;
end;
|.q1.|^2=(q1`2)^2+0^2 by A19,JGRAPH_3:1
.=(q1`2)^2;
then |.q1.|=q1`2 or |.q1.|=-(q1`2) by SQUARE_1:40;
then (-(q1`2))/|.q1.|=1 by A4,A20,XCMPLX_1:60;
then
A22: -(q1`2/|.q1.|)=1 by XCMPLX_1:187;
thus 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.|
proof
let p1,p2 be Point of TOP-REAL 2;
assume that
A23: p1=(sn-FanMorphE).q1 and
A24: p2=(sn-FanMorphE).q2;
A25: |.p2.|=|.q2.| by A24,JGRAPH_4:97;
A26: (|.p2.|)^2=(p2`2)^2+(p2`1)^2 by JGRAPH_3:1;
then (|.p2.|)^2-(p2`2)^2>=0 by XREAL_1:63;
then (|.p2.|)^2-(p2`2)^2+(p2`2)^2>=0+(p2`2)^2 by XREAL_1:7;
then -|.p2.|<=p2`2 by SQUARE_1:47;
then (-|.p2.|)/|.p2.|<=p2`2/|.p2.| by XREAL_1:72;
then
A27: -1 <= p2`2/|.p2.| by A5,A25,XCMPLX_1:197;
A28: now
per cases;
case
q2`1=0;
then p2=q2 by A24,JGRAPH_4:82;
hence p2`2/|.p2.|>-1 by A6,A22;
end;
case
q2`1<>0;
then
A29: p2`1>0 by A1,A3,A24,Th22;
now
assume -1= p2`2/|.p2.|;
then (-1)*|.p2.|=p2`2 by A5,A25,XCMPLX_1:87;
then (|.p2.|)^2=(p2`2)^2;
hence contradiction by A26,A29,XCMPLX_1:6;
end;
hence p2`2/|.p2.|>-1 by A27,XXREAL_0:1;
end;
end;
p1=q1 by A19,A23,JGRAPH_4:82;
hence thesis by A22,A28;
end;
end;
end;
hence thesis;
end;
theorem Th25:
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
proof
let cn be Real,q be Point of TOP-REAL 2;
assume that
A1: -1<cn and
A2: cn<1 and
A3: q`2<0;
now
per cases;
case
q`1/|.q.|>=cn;
hence thesis by A2,A3,JGRAPH_4:137;
end;
case
q`1/|.q.|<cn;
hence thesis by A1,A3,JGRAPH_4:138;
end;
end;
hence thesis;
end;
theorem Th26:
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
proof
let cn be Real,q be Point of TOP-REAL 2;
assume that
A1: -1<cn and
A2: cn<1 and
A3: q`2<0 and
A4: q`1/|.q.|>cn;
let p be Point of TOP-REAL 2;
assume
A5: p=(cn-FanMorphS).q;
now
set q1=(|.p.|)*|[cn,-sqrt(1-cn^2)]|;
set p1=(1/|.p.|)*p;
set p2=(cn-FanMorphS).q1;
(|[0,-1]|)`1=0 & (|[0,-1]|)`2=-1 by EUCLID:52;
then
A6: |.p.|*(|[0,-1]|)=|[|.p.|*0,|.p.|*(-1)]| by EUCLID:57
.=|[0,-(|.p.|)]|;
A7: (|[cn,-sqrt(1-cn^2)]|)`1=cn & (|[cn,-sqrt(1-cn^2)]|)`2=-sqrt(1-cn^2)
by EUCLID:52;
then
A8: q1=|[|.p.|*cn,|.p.|*(-sqrt(1-cn^2))]| by EUCLID:57;
then
A9: q1`1=(|.p.|)*cn by EUCLID:52;
assume
A10: p`1=0;
then (|.p.|)^2=(p`2)^2+0^2 by JGRAPH_3:1
.=(p`2)^2;
then
A11: p`2=|.p.| or p`2= - |.p.| by SQUARE_1:40;
then
A12: |.p.| <> 0 by A2,A3,A4,A5,JGRAPH_4:137;
A13: q1`2=-(sqrt(1-cn^2)*(|.p.|)) by A8,EUCLID:52;
1^2>cn^2 by A1,A2,SQUARE_1:50;
then
A14: 1-cn^2>0 by XREAL_1:50;
then sqrt(1-cn^2)>0 by SQUARE_1:25;
then --sqrt(1-cn^2)*(|.p.|)>0 by A12,XREAL_1:129;
then
A15: q1`2<0 by A13;
A16: |.p.|*p1=(|.p.|*(1/|.p.|))*p by RLVECT_1:def 7
.=(1)*p by A12,XCMPLX_1:106
.=p by RLVECT_1:def 8;
A17: p1=|[(1/|.p.|)*p`1,(1/|.p.|)*p`2]| by EUCLID:57;
then p1`2=-(|.p.|*(1/ |.p.|)) by A2,A3,A4,A5,A11,EUCLID:52,JGRAPH_4:137
.=-1 by A12,XCMPLX_1:106;
then
A18: p=|.p.|*(|[0,-1]|) by A10,A16,A17,EUCLID:52;
A19: |.q1.|=|.|.p.|.|*|.(|[cn,-sqrt(1-cn^2)]|).| by TOPRNS_1:7
.=|.|.p.|.|*sqrt((cn)^2+(-sqrt(1-cn^2))^2) by A7,JGRAPH_3:1
.=|.|.p.|.|*sqrt(cn^2+(sqrt(1-cn^2))^2)
.=|.|.p.|.|*sqrt(cn^2+(1-cn^2)) by A14,SQUARE_1:def 2
.=|.p.| by ABSVALUE:def 1;
then
A20: |.p2.|=|.p.| by JGRAPH_4:128;
A21: q1`1/|.q1.|=cn by A12,A9,A19,XCMPLX_1:89;
then
A22: p2`1=0 by A15,JGRAPH_4:142;
then (|.p2.|)^2=(p2`2)^2+0^2 by JGRAPH_3:1
.=(p2`2)^2;
then p2`2=|.p2.| or p2`2= - |.p2.| by SQUARE_1:40;
then
A23: p2=|[0,-(|.p.|)]| by A15,A21,A22,A20,EUCLID:53,JGRAPH_4:142;
(cn-FanMorphS) is one-to-one & dom (cn-FanMorphS)=the carrier of
TOP-REAL 2 by A1,A2,FUNCT_2:def 1,JGRAPH_4:133;
then q1=q by A5,A18,A23,A6,FUNCT_1:def 4;
hence contradiction by A4,A12,A9,A19,XCMPLX_1:89;
end;
hence thesis by A2,A3,A4,A5,JGRAPH_4:137;
end;
theorem Th27:
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.|
proof
let cn be Real,q1,q2 be Point of TOP-REAL 2;
assume that
A1: -1<cn & cn<1 and
A2: q1`2<=0 and
A3: q2`2<=0 and
A4: |.q1.|<>0 and
A5: |.q2.|<>0 and
A6: q1`1/|.q1.|<q2`1/|.q2.|;
now
per cases by A2;
case
A7: q1`2<0;
now
per cases by A3;
case
q2`2<0;
hence thesis by A1,A6,A7,JGRAPH_4:141;
end;
case
A8: q2`2=0;
A9: now
(|.q1.|)^2=(q1`1)^2+(q1`2)^2 by JGRAPH_3:1;
then (|.q1.|)^2-(q1`1)^2>=0 by XREAL_1:63;
then (|.q1.|)^2-(q1`1)^2+(q1`1)^2>=0+(q1`1)^2 by XREAL_1:7;
then -|.q1.|<=q1`1 by SQUARE_1:47;
then
A10: (-|.q1.|)/|.q1.|<=q1`1/|.q1.| by XREAL_1:72;
assume |.q2.|=-(q2`1);
then 1=(-(q2`1))/|.q2.| by A5,XCMPLX_1:60;
then q1`1/|.q1.|< -1 by A6,XCMPLX_1:190;
hence contradiction by A4,A10,XCMPLX_1:197;
end;
|.q2.|^2=(q2`1)^2+0^2 by A8,JGRAPH_3:1
.=(q2`1)^2;
then |.q2.|=q2`1 or |.q2.|=-(q2`1) by SQUARE_1:40;
then
A11: q2`1/|.q2.|=1 by A5,A9,XCMPLX_1:60;
thus 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.|
proof
let p1,p2 be Point of TOP-REAL 2;
assume that
A12: p1=(cn-FanMorphS).q1 and
A13: p2=(cn-FanMorphS).q2;
A14: |.p1.|=|.q1.| by A12,JGRAPH_4:128;
A15: (|.p1.|)^2=(p1`1)^2+(p1`2)^2 by JGRAPH_3:1;
A16: p1`2<0 by A1,A7,A12,Th25;
A17: now
assume 1= p1`1/|.p1.|;
then (1)*|.p1.|=p1`1 by A4,A14,XCMPLX_1:87;
hence contradiction by A15,A16,XCMPLX_1:6;
end;
A18: p2=q2 by A8,A13,JGRAPH_4:113;
(|.p1.|)^2-(p1`1)^2>=0 by A15,XREAL_1:63;
then (|.p1.|)^2-(p1`1)^2+(p1`1)^2>=0+(p1`1)^2 by XREAL_1:7;
then p1`1<=|.p1.| by SQUARE_1:47;
then (|.p1.|)/|.p1.|>=p1`1/|.p1.| by XREAL_1:72;
then 1>= p1`1/|.p1.| by A4,A14,XCMPLX_1:60;
hence thesis by A11,A18,A17,XXREAL_0:1;
end;
end;
end;
hence thesis;
end;
case
A19: q1`2=0;
A20: now
(|.q2.|)^2=(q2`1)^2+(q2`2)^2 by JGRAPH_3:1;
then (|.q2.|)^2-(q2`1)^2>=0 by XREAL_1:63;
then (|.q2.|)^2-(q2`1)^2+(q2`1)^2>=0+(q2`1)^2 by XREAL_1:7;
then q2`1<=|.q2.| by SQUARE_1:47;
then
A21: (|.q2.|)/|.q2.|>=q2`1/|.q2.| by XREAL_1:72;
assume |.q1.|=(q1`1);
then q2`1/|.q2.|> 1 by A4,A6,XCMPLX_1:60;
hence contradiction by A5,A21,XCMPLX_1:60;
end;
|.q1.|^2=(q1`1)^2+0^2 by A19,JGRAPH_3:1
.=(q1`1)^2;
then |.q1.|=q1`1 or |.q1.|=-(q1`1) by SQUARE_1:40;
then (-(q1`1))/|.q1.|=1 by A4,A20,XCMPLX_1:60;
then
A22: -(q1`1/|.q1.|)=1 by XCMPLX_1:187;
thus 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.|
proof
let p1,p2 be Point of TOP-REAL 2;
assume that
A23: p1=(cn-FanMorphS).q1 and
A24: p2=(cn-FanMorphS).q2;
A25: |.p2.|=|.q2.| by A24,JGRAPH_4:128;
A26: (|.p2.|)^2=(p2`1)^2+(p2`2)^2 by JGRAPH_3:1;
then (|.p2.|)^2-(p2`1)^2>=0 by XREAL_1:63;
then (|.p2.|)^2-(p2`1)^2+(p2`1)^2>=0+(p2`1)^2 by XREAL_1:7;
then -|.p2.|<=p2`1 by SQUARE_1:47;
then (-|.p2.|)/|.p2.|<=p2`1/|.p2.| by XREAL_1:72;
then
A27: -1 <= p2`1/|.p2.| by A5,A25,XCMPLX_1:197;
A28: now
per cases;
case
q2`2=0;
then p2=q2 by A24,JGRAPH_4:113;
hence p2`1/|.p2.|>-1 by A6,A22;
end;
case
q2`2<>0;
then
A29: p2`2<0 by A1,A3,A24,Th25;
now
assume -1= p2`1/|.p2.|;
then (-1)*|.p2.|=p2`1 by A5,A25,XCMPLX_1:87;
then (|.p2.|)^2=(p2`1)^2;
hence contradiction by A26,A29,XCMPLX_1:6;
end;
hence p2`1/|.p2.|>-1 by A27,XXREAL_0:1;
end;
end;
p1=q1 by A19,A23,JGRAPH_4:113;
hence thesis by A22,A28;
end;
end;
end;
hence thesis;
end;
begin :: Order of Points on Circle
Lm3: now
let P be compact non empty Subset of TOP-REAL 2;
assume
A1: P={q: |.q.|=1};
A2: [.-1,1.] c= proj1.:P
proof
let y be object;
assume y in [.-1,1.];
then y in {r where r is Real: -1<=r & r<=1 } by RCOMP_1:def 1;
then consider r being Real such that
A3: y=r and
A4: -1<=r & r<=1;
set q=|[r,sqrt(1-r^2)]|;
1^2>=r^2 by A4,SQUARE_1:49;
then
A5: 1-r^2>=0 by XREAL_1:48;
q`1=r & q`2=sqrt(1-r^2) by EUCLID:52;
then |.q.|=sqrt(r^2+(sqrt(1-r^2))^2) by JGRAPH_3:1
.=sqrt(r^2+(1-r^2)) by A5,SQUARE_1:def 2
.=1;
then
A6: dom proj1=the carrier of TOP-REAL 2 & q in P by A1,FUNCT_2:def 1;
proj1.q=q`1 by PSCOMP_1:def 5
.=r by EUCLID:52;
hence thesis by A3,A6,FUNCT_1:def 6;
end;
proj1.:P c= [.-1,1.]
proof
let y be object;
assume y in proj1.:P;
then consider x being object such that
A7: x in dom proj1 and
A8: x in P and
A9: y=proj1.x by FUNCT_1:def 6;
reconsider q=x as Point of TOP-REAL 2 by A7;
ex q2 being Point of TOP-REAL 2 st q2=x & |.q2.|=1 by A1,A8;
then
A10: (q`1)^2+(q`2)^2=1^2 by JGRAPH_3:1;
0<=(q`2)^2 by XREAL_1:63;
then 1-(q`1)^2+(q`1)^2 >=0+(q`1)^2 by A10,XREAL_1:7;
then
A11: -1<=q`1 & q`1<=1 by SQUARE_1:51;
y=q`1 by A9,PSCOMP_1:def 5;
hence thesis by A11,XXREAL_1:1;
end;
hence proj1.:P=[.-1,1.] by A2,XBOOLE_0:def 10;
A12: [.-1,1.] c= proj2.:P
proof
let y be object;
assume y in [.-1,1.];
then y in {r where r is Real: -1<=r & r<=1 } by RCOMP_1:def 1;
then consider r being Real such that
A13: y=r and
A14: -1<=r & r<=1;
set q=|[sqrt(1-r^2),r]|;
1^2>=r^2 by A14,SQUARE_1:49;
then
A15: 1-r^2>=0 by XREAL_1:48;
q`2=r & q`1=sqrt(1-r^2) by EUCLID:52;
then |.q.|=sqrt((sqrt(1-r^2))^2+r^2) by JGRAPH_3:1
.=sqrt((1-r^2)+r^2) by A15,SQUARE_1:def 2
.=1;
then
A16: dom proj2=the carrier of TOP-REAL 2 & q in P by A1,FUNCT_2:def 1;
proj2.q=q`2 by PSCOMP_1:def 6
.=r by EUCLID:52;
hence thesis by A13,A16,FUNCT_1:def 6;
end;
proj2.:P c= [.-1,1.]
proof
let y be object;
assume y in proj2.:P;
then consider x being object such that
A17: x in dom proj2 and
A18: x in P and
A19: y=proj2.x by FUNCT_1:def 6;
reconsider q=x as Point of TOP-REAL 2 by A17;
ex q2 being Point of TOP-REAL 2 st q2=x & |.q2.|=1 by A1,A18;
then
A20: (q`1)^2+(q`2)^2=1^2 by JGRAPH_3:1;
0<=(q`1)^2 by XREAL_1:63;
then 1-(q`2)^2+(q`2)^2 >=0+(q`2)^2 by A20,XREAL_1:7;
then
A21: -1<=q`2 & q`2<=1 by SQUARE_1:51;
y=q`2 by A19,PSCOMP_1:def 6;
hence thesis by A21,XXREAL_1:1;
end;
hence proj2.:P=[.-1,1.] by A12,XBOOLE_0:def 10;
end;
Lm4: for P being compact non empty Subset of TOP-REAL 2 st P={q: |.q.|=1}
holds W-bound(P)=-1
proof
let P be compact non empty Subset of TOP-REAL 2;
assume P={q: |.q.|=1};
then proj1.:P=[.-1,1.] by Lm3;
then (proj1|P).:P=[.-1,1.] by RELAT_1:129;
then the carrier of ((TOP-REAL 2)|P) = P & lower_bound ((proj1|P).:P)=-1 by
JORDAN5A:19,PRE_TOPC:8;
then lower_bound (proj1|P)=-1 by PSCOMP_1:def 1;
hence thesis by PSCOMP_1:def 7;
end;
theorem Th28:
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
proof
let P be compact non empty Subset of TOP-REAL 2;
A1: the carrier of ((TOP-REAL 2)|P) =P by PRE_TOPC:8;
assume
A2: P={q: |.q.|=1};
hence W-bound(P)=-1 by Lm4;
proj1.:P=[.-1,1.] by A2,Lm3;
then (proj1|P).:P=[.-1,1.] by RELAT_1:129;
then upper_bound ((proj1|P).:the carrier of ((TOP-REAL 2)|P))=1
by A1,JORDAN5A:19;
then upper_bound (proj1|P)=1 by PSCOMP_1:def 2;
hence E-bound P=1 by PSCOMP_1:def 9;
proj2.:P=[.-1,1.] by A2,Lm3;
then
A3: (proj2|P).:P=[.-1,1.] by RELAT_1:129;
then lower_bound ((proj2|P).:P)=-1 by JORDAN5A:19;
then lower_bound (proj2|P)=-1 by A1,PSCOMP_1:def 1;
hence S-bound P=-1 by PSCOMP_1:def 10;
upper_bound ((proj2|P).:P)=1 by A3,JORDAN5A:19;
then upper_bound (proj2|P)=1 by A1,PSCOMP_1:def 2;
hence thesis by PSCOMP_1:def 8;
end;
theorem Th29:
for P being compact non empty Subset of TOP-REAL 2 st P={q: |.q
.|=1} holds W-min(P)=|[-1,0]|
proof
let P be compact non empty Subset of TOP-REAL 2;
A1: the carrier of ((TOP-REAL 2)|P) = P by PRE_TOPC:8;
assume
A2: P={q: |.q.|=1};
then
A3: W-bound P=-1 by Lm4;
proj2.:P=[.-1,1.] by A2,Lm3;
then
A4: (proj2|P).:P=[.-1,1.] by RELAT_1:129;
then upper_bound ((proj2|P).:P)=1 by JORDAN5A:19;
then upper_bound (proj2|P)=1 by A1,PSCOMP_1:def 2;
then N-bound P=1 by PSCOMP_1:def 8;
then
A5: NW-corner P=|[-1,1]| by A3,PSCOMP_1:def 12;
lower_bound ((proj2|P).:P)=-1 by A4,JORDAN5A:19;
then lower_bound (proj2|P)=-1 by A1,PSCOMP_1:def 1;
then S-bound P=-1 by PSCOMP_1:def 10;
then
A6: SW-corner P=|[-1,-1]| by A3,PSCOMP_1:def 11;
A7: LSeg(SW-corner P, NW-corner P)/\P c= {|[-1,0]|}
proof
let x be object;
assume
A8: x in LSeg(SW-corner P, NW-corner P)/\P;
then
A9: x in { (1-l)*(SW-corner P) + l*(NW-corner P)
where l is Real: 0 <= l
& l <= 1 } by XBOOLE_0:def 4;
x in P by A8,XBOOLE_0:def 4;
then
A10: ex q2 being Point of TOP-REAL 2 st q2=x & |.q2.|=1 by A2;
consider l being Real such that
A11: x=(1-l)*(SW-corner P)+l*(NW-corner P) and
0<=l and
l<=1 by A9;
reconsider q3=x as Point of TOP-REAL 2 by A11;
x=|[(1-l)*(-1),(1-l)*(-1)]|+(l)*|[-1,1]| by A6,A5,A11,EUCLID:58;
then x=|[(1-l)*(-1),(1-l)*(-1)]|+|[(l)*(-1),(l)*1]| by EUCLID:58;
then
A12: x=|[(1-l)*(-1)+(l)*(-1),(1-l)*(-1)+(l)*1]| by EUCLID:56;
then q3`1=-1 by EUCLID:52;
then
A13: 1=sqrt((-1)^2+(q3`2)^2) by A10,JGRAPH_3:1
.=sqrt(1+(q3`2)^2);
now
assume (q3`2)^2>0;
then 1<1+(q3`2)^2 by XREAL_1:29;
hence contradiction by A13,SQUARE_1:18,27;
end;
then (q3`2)^2=0 by XREAL_1:63;
then
A14: q3`2=0 by XCMPLX_1:6;
q3`2=(1-l)*(-1)+l by A12,EUCLID:52;
hence thesis by A12,A14,TARSKI:def 1;
end;
{|[-1,0]|} c= LSeg(SW-corner P, NW-corner P)/\P
proof
set q=|[-1,0]|;
let x be object;
assume x in {|[-1,0]|};
then
A15: x=|[-1,0]| by TARSKI:def 1;
q`2=0 & q`1=-1 by EUCLID:52;
then |.q.|=sqrt((-1)^2+0^2) by JGRAPH_3:1
.=1;
then
A16: x in P by A2,A15;
q=|[(1/2)*(-1)+(1/2)*(-1),(1/2)*(-1)+(1/2)*1]|;
then q=|[(1/2)*(-1),(1/2)*(-1)]|+|[(1/2)*(-1),(1/2)*1]| by EUCLID:56;
then q=|[(1/2)*(-1),(1/2)*(-1)]|+(1/2)*|[-1,1]| by EUCLID:58;
then q=(1/2)*|[-1,-1]|+(1-(1/2))*|[-1,1]| by EUCLID:58;
then x in LSeg(SW-corner P, NW-corner P) by A6,A5,A15;
hence thesis by A16,XBOOLE_0:def 4;
end;
then LSeg(SW-corner P, NW-corner P)/\P={|[-1,0]|} by A7,XBOOLE_0:def 10;
then
A17: W-most P={|[-1,0]|} by PSCOMP_1:def 15;
(proj2|W-most P).:the carrier of ((TOP-REAL 2)|(W-most P)) =(proj2|(
W-most P)).:(W-most P) by PRE_TOPC:8
.=Im(proj2,|[-1,0]|) by A17,RELAT_1:129
.={proj2.(|[-1,0]|)} by SETWISEO:8
.={(|[-1,0]|)`2} by PSCOMP_1:def 6
.={0} by EUCLID:52;
then
lower_bound ((proj2|W-most P).:the carrier of ((TOP-REAL 2)|(W-most P))
) =0 by SEQ_4:9;
then lower_bound (proj2|W-most P)=0 by PSCOMP_1:def 1;
hence thesis by A3,PSCOMP_1:def 19;
end;
theorem Th30:
for P being compact non empty Subset of TOP-REAL 2 st P={q: |.q
.|=1} holds E-max(P)=|[1,0]|
proof
let P be compact non empty Subset of TOP-REAL 2;
A1: the carrier of ((TOP-REAL 2)|P) =P by PRE_TOPC:8;
assume
A2: P={q: |.q.|=1};
then
A3: E-bound P=1 by Th28;
proj2.:P=[.-1,1.] by A2,Lm3;
then
A4: (proj2|P).:P=[.-1,1.] by RELAT_1:129;
then upper_bound ((proj2|P).:P)=1 by JORDAN5A:19;
then upper_bound (proj2|P)=1 by A1,PSCOMP_1:def 2;
then N-bound P=1 by PSCOMP_1:def 8;
then
A5: NE-corner P=|[1,1]| by A3,PSCOMP_1:def 13;
lower_bound ((proj2|P).:P)=-1 by A4,JORDAN5A:19;
then lower_bound (proj2|P)=-1 by A1,PSCOMP_1:def 1;
then S-bound P=-1 by PSCOMP_1:def 10;
then
A6: SE-corner P=|[1,-1]| by A3,PSCOMP_1:def 14;
A7: LSeg(SE-corner P, NE-corner P)/\P c= {|[1,0]|}
proof
let x be object;
assume
A8: x in LSeg(SE-corner P, NE-corner P)/\P;
then
A9: x in { (1-l)*(SE-corner P) + l*(NE-corner P)
where l is Real: 0 <= l
& l <= 1 } by XBOOLE_0:def 4;
x in P by A8,XBOOLE_0:def 4;
then
A10: ex q2 being Point of TOP-REAL 2 st q2=x & |.q2.|=1 by A2;
consider l being Real such that
A11: x=(1-l)*(SE-corner P)+l*(NE-corner P) and
0<=l and
l<=1 by A9;
reconsider q3=x as Point of TOP-REAL 2 by A11;
x=|[(1-l)*(1),(1-l)*(-1)]|+(l)*|[1,1]| by A6,A5,A11,EUCLID:58;
then x=|[(1-l)*(1),(1-l)*(-1)]|+|[(l)*(1),(l)*1]| by EUCLID:58;
then
A12: x=|[((1-l)+l)*(1),(1-l)*(-1)+(l)*1]| by EUCLID:56;
then
A13: q3`1=1 by EUCLID:52;
now
assume (q3`2)^2>0;
then 1^2<1+(q3`2)^2 by XREAL_1:29;
hence contradiction by A13,A10,JGRAPH_3:1;
end;
then (q3`2)^2=0 by XREAL_1:63;
then
A14: q3`2=0 by XCMPLX_1:6;
q3`2=(1-l)*(-1)+l by A12,EUCLID:52;
hence thesis by A12,A14,TARSKI:def 1;
end;
{|[1,0]|} c= LSeg(SE-corner P, NE-corner P)/\P
proof
set q=|[1,0]|;
let x be object;
assume x in {|[1,0]|};
then
A15: x=|[1,0]| by TARSKI:def 1;
q`2=0 & q`1=1 by EUCLID:52;
then |.q.|=sqrt((1)^2+0^2) by JGRAPH_3:1
.=1;
then
A16: x in P by A2,A15;
q=|[(1/2)*(1)+(1/2)*(1),(1/2)*(-1)+(1/2)*1]|;
then q=|[(1/2)*(1),(1/2)*(-1)]|+|[(1/2)*(1),(1/2)*1]| by EUCLID:56;
then q=|[(1/2)*(1),(1/2)*(-1)]|+(1/2)*|[1,1]| by EUCLID:58;
then q=(1/2)*|[1,-1]|+(1-(1/2))*|[1,1]| by EUCLID:58;
then x in LSeg(SE-corner P, NE-corner P) by A6,A5,A15;
hence thesis by A16,XBOOLE_0:def 4;
end;
then LSeg(SE-corner P, NE-corner P)/\P={|[1,0]|} by A7,XBOOLE_0:def 10;
then
A17: E-most P={|[1,0]|} by PSCOMP_1:def 17;
(proj2|E-most P).:the carrier of ((TOP-REAL 2)|(E-most P)) =(proj2|(
E-most P)).:(E-most P) by PRE_TOPC:8
.=Im(proj2,|[1,0]|) by A17,RELAT_1:129
.={proj2.(|[1,0]|)} by SETWISEO:8
.={(|[1,0]|)`2} by PSCOMP_1:def 6
.={0} by EUCLID:52;
then upper_bound ((proj2|E-most P).:
the carrier of ((TOP-REAL 2)|(E-most P))) =0 by SEQ_4:9;
then upper_bound (proj2|E-most P)=0 by PSCOMP_1:def 2;
hence thesis by A3,PSCOMP_1:def 23;
end;
theorem
for f being Function of TOP-REAL 2,R^1 st (for p being Point of
TOP-REAL 2 holds f.p=proj1.p) holds f is continuous
proof
let f be Function of TOP-REAL 2,R^1;
assume
A1: for p being Point of TOP-REAL 2 holds f.p=proj1.p;
reconsider f as Function of the TopStruct of TOP-REAL 2, R^1;
(TOP-REAL 2)|([#](TOP-REAL 2))=the TopStruct of TOP-REAL 2 by TSEP_1:93;
then f is continuous by A1,JGRAPH_2:29;
hence thesis by PRE_TOPC:32;
end;
theorem Th32:
for f being Function of TOP-REAL 2,R^1 st (for p being Point of
TOP-REAL 2 holds f.p=proj2.p) holds f is continuous
proof
let f be Function of TOP-REAL 2,R^1;
assume
A1: for p being Point of TOP-REAL 2 holds f.p=proj2.p;
reconsider f as Function of the TopStruct of TOP-REAL 2, R^1;
(TOP-REAL 2)|([#](TOP-REAL 2))=the TopStruct of TOP-REAL 2 by TSEP_1:93;
then f is continuous by A1,JGRAPH_2:30;
hence thesis by PRE_TOPC:32;
end;
theorem Th33:
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
proof
let P be compact non empty Subset of TOP-REAL 2;
assume P={q where q is Point of TOP-REAL 2: |.q.|=1};
then P is being_simple_closed_curve by JGRAPH_3:26;
hence thesis by JORDAN6:61;
end;
theorem Th34:
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}
proof
reconsider h2=proj2 as Function of TOP-REAL 2,R^1 by TOPMETR:17;
let P be compact non empty Subset of TOP-REAL 2;
set P4=Lower_Arc(P);
set P1=Upper_Arc(P), P2=Lower_Arc(P), Q=Vertical_Line(0);
set p8= First_Point(Upper_Arc(P),W-min(P),E-max(P), Vertical_Line(0));
set pj= Last_Point(Lower_Arc(P),E-max(P),W-min(P), Vertical_Line(0));
A1: LSeg(|[0,-1]|,|[0,1]|) c= Q
proof
let x be object;
assume x in LSeg(|[0,-1]|,|[0,1]|);
then consider l being Real such that
A2: x=(1-l)*(|[0,-1]|) +l*(|[0,1]|) and
0<=l and
l<=1;
((1-l)*(|[0,-1]|) +l*(|[0,1]|))`1 = ((1-l)*(|[0,-1]|))`1 +(l*(|[0,1]|
))`1 by TOPREAL3:2
.=(1-l)*(|[0,-1]|)`1+(l*(|[0,1]|))`1 by TOPREAL3:4
.=(1-l)*(|[0,-1]|)`1+l*((|[0,1]|))`1 by TOPREAL3:4
.=(1-l)*0+l*((|[0,1]|))`1 by EUCLID:52
.=(1-l)*0+l*0 by EUCLID:52
.=0;
hence thesis by A2;
end;
reconsider R=Upper_Arc(P) as non empty Subset of TOP-REAL 2;
assume
A3: P={q where q is Point of TOP-REAL 2: |.q.|=1};
then
A4: P is being_simple_closed_curve by JGRAPH_3:26;
then
A5: Upper_Arc(P) is_an_arc_of W-min(P),E-max(P) by JORDAN6:def 8;
then consider f being Function of I[01], (TOP-REAL 2)|R such that
A6: f is being_homeomorphism and
A7: f.0 =W-min(P) and
A8: f.1 =E-max(P) by TOPREAL1:def 1;
A9: dom f=the carrier of I[01] & dom h2=the carrier of TOP-REAL 2 by
FUNCT_2:def 1;
A10: ex P2 being non empty Subset of TOP-REAL 2 st P2 is_an_arc_of E-max(P),
W-min(P) & Upper_Arc(P) /\ P2={W-min(P),E-max(P)} & Upper_Arc(P) \/ P2=P &
First_Point(Upper_Arc(P),W-min(P),E-max(P), Vertical_Line((W-bound(P )+E-bound(
P))/2))`2> Last_Point(P2,E-max(P),W-min(P), Vertical_Line((W-bound(P) +E-bound(
P))/2))`2 by A4,JORDAN6:def 8;
then
A11: Upper_Arc(P) c= P by XBOOLE_1:7;
A12: rng f =[#]((TOP-REAL 2)|R) by A6,TOPS_2:def 5
.=R by PRE_TOPC:def 5;
A13: S-bound P=-1 & N-bound P=1 by A3,Th28;
A14: Vertical_Line(0) is closed by JORDAN6:30;
A15: for p being Point of (TOP-REAL 2) holds h2.p=proj2.p;
A16: W-bound P=-1 & E-bound P=1 by A3,Th28;
then
A17: P1 meets Q by A4,A13,A1,JORDAN6:69,XBOOLE_1:64;
A18: P2 meets Q by A4,A16,A13,A1,JORDAN6:70,XBOOLE_1:64;
A19: First_Point(Upper_Arc(P),W-min(P),E-max(P), Vertical_Line((W-bound(P)+
E-bound(P))/2))`2> Last_Point(P4,E-max(P),W-min(P), Vertical_Line((W-bound(P)+
E-bound(P))/2))`2 by A4,JORDAN6:def 9;
Upper_Arc(P) is closed by A5,JORDAN6:11;
then P1 /\ Q is closed by A14,TOPS_1:8;
then
A20: p8 in P1 /\ Q by A5,A17,JORDAN5C:def 1;
then p8 in P1 by XBOOLE_0:def 4;
then consider x8 being object such that
A21: x8 in dom f and
A22: p8=f.x8 by A12,FUNCT_1:def 3;
dom f= [.0,1.] by BORSUK_1:40,FUNCT_2:def 1;
then x8 in {r where r is Real: 0<=r & r<=1 } by A21,RCOMP_1:def 1;
then consider r8 being Real such that
A23: x8=r8 and
A24: 0<=r8 and
A25: r8<=1;
A26: Vertical_Line(0) is closed by JORDAN6:30;
P1 /\ Q c= {|[0,-1]|,|[0,1]|}
proof
let x be object;
assume
A27: x in P1 /\ Q;
then x in P1 by XBOOLE_0:def 4;
then x in P by A10,XBOOLE_0:def 3;
then consider q being Point of TOP-REAL 2 such that
A28: q=x and
A29: |.q.|=1 by A3;
x in Q by A27,XBOOLE_0:def 4;
then
A30: ex p being Point of TOP-REAL 2 st p=x & p`1=0;
then 0^2+(q`2)^2 =1^2 by A28,A29,JGRAPH_3:1;
then q`2=1 or q`2=-1 by SQUARE_1:41;
then x=|[0,-1]| or x=|[0,1]| by A30,A28,EUCLID:53;
hence thesis by TARSKI:def 2;
end;
then p8=|[0,-1]| or p8=|[0,1]| by A20,TARSKI:def 2;
then
A31: p8`2=-1 or p8`2=1 by EUCLID:52;
A32: now
assume r8=0;
then p8=|[-1,0]| by A3,A7,A22,A23,Th29;
hence contradiction by A31,EUCLID:52;
end;
A33: rng (h2*f) c= the carrier of R^1;
A34: the carrier of ((TOP-REAL 2)|R)=R by PRE_TOPC:8;
then rng f c= the carrier of TOP-REAL 2 by XBOOLE_1:1;
then dom (h2*f)=the carrier of I[01] by A9,RELAT_1:27;
then reconsider g0=h2*f as Function of I[01],R^1 by A33,FUNCT_2:2;
A35: f is one-to-one by A6,TOPS_2:def 5;
A36: f is continuous by A6,TOPS_2:def 5;
A37: (ex p being Point of TOP-REAL 2,
t being Real st 0<t & t<1 & f.t=p & p`2
>0) implies for q being Point of TOP-REAL 2 st q in Upper_Arc(P) holds q`2>=0
proof
given p being Point of TOP-REAL 2,t being Real such that
A38: 0<t and
A39: t<1 and
A40: f.t=p and
A41: p`2>0;
now
assume ex q being Point of TOP-REAL 2 st q in Upper_Arc(P) & q`2<0;
then consider q being Point of TOP-REAL 2 such that
A42: q in Upper_Arc(P) and
A43: q`2<0;
rng f =[#]((TOP-REAL 2)|R) by A6,TOPS_2:def 5
.=R by PRE_TOPC:def 5;
then consider x being object such that
A44: x in dom f and
A45: q=f.x by A42,FUNCT_1:def 3;
A46: dom f= [.0,1.] by BORSUK_1:40,FUNCT_2:def 1;
then
A47: x in {r where r is Real: 0<=r & r<=1 } by A44,RCOMP_1:def 1;
t in {v where v is Real: 0<=v & v<=1 } by A38,A39;
then
A48: t in [.0,1.] by RCOMP_1:def 1;
then
A49: (h2*f).t=h2.p by A40,A46,FUNCT_1:13
.=p`2 by PSCOMP_1:def 6;
consider r being Real such that
A50: x=r and
A51: 0<=r and
A52: r<=1 by A47;
A53: (h2*f).r=h2.q by A44,A45,A50,FUNCT_1:13
.=q`2 by PSCOMP_1:def 6;
now
per cases by XXREAL_0:1;
case
A54: r<t;
then reconsider
B=[.r,t.] as non empty Subset of I[01] by A44,A50,A48,BORSUK_1:40
,XXREAL_1:1,XXREAL_2:def 12;
reconsider B0=B as Subset of I[01];
reconsider g=g0|B0 as Function of (I[01]|B0),R^1 by PRE_TOPC:9;
A55: (q`2)*(p`2)<0 by A41,A43,XREAL_1:132;
t in {r4 where r4 is Real: r<=r4 & r4<=t} by A54;
then t in B by RCOMP_1:def 1;
then
A56: p`2=g.t by A49,FUNCT_1:49;
r in {r4 where r4 is Real: r<=r4 & r4<=t} by A54;
then r in B by RCOMP_1:def 1;
then
A57: q`2=g.r by A53,FUNCT_1:49;
g0 is continuous by A36,A15,Th7,Th32;
then
A58: g is continuous by TOPMETR:7;
Closed-Interval-TSpace(r,t)=I[01]|B by A39,A51,A54,TOPMETR:20,23;
then consider r1 being Real such that
A59: g.r1=0 and
A60: r<r1 and
A61: r1<t by A54,A58,A55,A57,A56,TOPREAL5:8;
r1 in {r4 where r4 is Real: r<=r4 & r4<=t} by A60,A61;
then
A62: r1 in B by RCOMP_1:def 1;
r1<1 by A39,A61,XXREAL_0:2;
then r1 in {r2 where r2 is Real: 0<=r2 & r2<=1} by A51,A60;
then
A63: r1 in dom f by A46,RCOMP_1:def 1;
then f.r1 in rng f by FUNCT_1:def 3;
then f.r1 in R by A34;
then f.r1 in P by A11;
then consider q3 being Point of TOP-REAL 2 such that
A64: q3=f.r1 and
A65: |.q3.|=1 by A3;
A66: q3`2=h2.(f.r1) by A64,PSCOMP_1:def 6
.=g0.r1 by A63,FUNCT_1:13
.=0 by A59,A62,FUNCT_1:49;
then
A67: 1^2=(q3`1)^2 +0^2 by A65,JGRAPH_3:1
.=(q3`1)^2;
now
per cases by A67,SQUARE_1:41;
case
A68: q3`1=1;
A69: 1 in dom f by A46,XXREAL_1:1;
q3=|[1,0]| by A66,A68,EUCLID:53
.=E-max(P) by A3,Th30;
hence contradiction by A8,A35,A39,A61,A63,A64,A69,FUNCT_1:def 4;
end;
case
A70: q3`1=-1;
A71: 0 in dom f by A46,XXREAL_1:1;
q3=|[-1,0]| by A66,A70,EUCLID:53
.=W-min(P) by A3,Th29;
hence contradiction by A7,A35,A51,A60,A63,A64,A71,FUNCT_1:def 4;
end;
end;
hence contradiction;
end;
case
A72: t<r;
then reconsider
B=[.t,r.] as non empty Subset of I[01] by A44,A50,A48,BORSUK_1:40
,XXREAL_1:1,XXREAL_2:def 12;
reconsider B0=B as Subset of I[01];
reconsider g=g0|B0 as Function of (I[01]|B0),R^1 by PRE_TOPC:9;
A73: (q`2)*(p`2)<0 by A41,A43,XREAL_1:132;
t in {r4 where r4 is Real: t<=r4 & r4<=r} by A72;
then t in B by RCOMP_1:def 1;
then
A74: p`2=g.t by A49,FUNCT_1:49;
r in {r4 where r4 is Real: t<=r4 & r4<=r} by A72;
then r in B by RCOMP_1:def 1;
then
A75: q`2=g.r by A53,FUNCT_1:49;
g0 is continuous by A36,A15,Th7,Th32;
then
A76: g is continuous by TOPMETR:7;
Closed-Interval-TSpace(t,r)=I[01]|B by A38,A52,A72,TOPMETR:20,23;
then consider r1 being Real such that
A77: g.r1=0 and
A78: t<r1 and
A79: r1<r by A72,A76,A73,A75,A74,TOPREAL5:8;
r1 in {r4 where r4 is Real: t<=r4 & r4<=r} by A78,A79;
then
A80: r1 in B by RCOMP_1:def 1;
r1<1 by A52,A79,XXREAL_0:2;
then r1 in {r2 where r2 is Real: 0<=r2 & r2<=1} by A38,A78;
then
A81: r1 in dom f by A46,RCOMP_1:def 1;
then f.r1 in rng f by FUNCT_1:def 3;
then f.r1 in R by A34;
then f.r1 in P by A11;
then consider q3 being Point of TOP-REAL 2 such that
A82: q3=f.r1 and
A83: |.q3.|=1 by A3;
A84: q3`2=h2.(f.r1) by A82,PSCOMP_1:def 6
.=(h2*f).r1 by A81,FUNCT_1:13
.=0 by A77,A80,FUNCT_1:49;
then
A85: 1^2=(q3`1)^2 +0^2 by A83,JGRAPH_3:1
.=(q3`1)^2;
now
per cases by A85,SQUARE_1:41;
case
A86: q3`1=1;
A87: 1 in dom f by A46,XXREAL_1:1;
q3=|[1,0]| by A84,A86,EUCLID:53
.=E-max(P) by A3,Th30;
hence contradiction by A8,A35,A52,A79,A81,A82,A87,FUNCT_1:def 4;
end;
case
A88: q3`1=-1;
A89: 0 in dom f by A46,XXREAL_1:1;
q3=|[-1,0]| by A84,A88,EUCLID:53
.=W-min(P) by A3,Th29;
hence contradiction by A7,A35,A38,A78,A81,A82,A89,FUNCT_1:def 4;
end;
end;
hence contradiction;
end;
case
t=r;
hence contradiction by A41,A43,A53,A49;
end;
end;
hence contradiction;
end;
hence thesis;
end;
reconsider R=Lower_Arc(P) as non empty Subset of TOP-REAL 2;
A90: Lower_Arc(P) is_an_arc_of E-max(P),W-min(P) by A4,JORDAN6:def 9;
then consider f2 being Function of I[01], (TOP-REAL 2)|R such that
A91: f2 is being_homeomorphism and
A92: f2.0 =E-max(P) and
A93: f2.1 =W-min(P) by TOPREAL1:def 1;
A94: dom f2=the carrier of I[01] & dom h2=the carrier of TOP-REAL 2 by
FUNCT_2:def 1;
A95: rng (h2*f2) c= the carrier of R^1;
A96: the carrier of ((TOP-REAL 2)|R)=R by PRE_TOPC:8;
then rng f2 c= the carrier of TOP-REAL 2 by XBOOLE_1:1;
then dom (h2*f2)=the carrier of I[01] by A94,RELAT_1:27;
then reconsider g1=h2*f2 as Function of I[01],R^1 by A95,FUNCT_2:2;
A97: f2 is one-to-one by A91,TOPS_2:def 5;
A98: Upper_Arc(P) \/ P4=P by A4,JORDAN6:def 9;
then
A99: Lower_Arc(P) c= P by XBOOLE_1:7;
A100: P2 /\ Q c= {|[0,-1]|,|[0,1]|}
proof
let x be object;
assume
A101: x in P2 /\ Q;
then x in P2 by XBOOLE_0:def 4;
then x in P by A98,XBOOLE_0:def 3;
then consider q being Point of TOP-REAL 2 such that
A102: q=x and
A103: |.q.|=1 by A3;
x in Q by A101,XBOOLE_0:def 4;
then
A104: ex p being Point of TOP-REAL 2 st p=x & p`1=0;
then 0^2+(q`2)^2 =1^2 by A102,A103,JGRAPH_3:1;
then q`2=1 or q`2=-1 by SQUARE_1:41;
then x=|[0,-1]| or x=|[0,1]| by A104,A102,EUCLID:53;
hence thesis by TARSKI:def 2;
end;
A105: for p being Point of (TOP-REAL 2) holds h2.p=proj2.p;
A106: f2 is continuous by A91,TOPS_2:def 5;
A107: (ex p being Point of TOP-REAL 2,
t being Real st 0<t & t<1 & f2.t=p & p
`2>0) implies for q being Point of TOP-REAL 2 st q in Lower_Arc(P) holds q`2>=0
proof
given p being Point of TOP-REAL 2,t being Real such that
A108: 0<t and
A109: t<1 and
A110: f2.t=p and
A111: p`2>0;
now
assume ex q being Point of TOP-REAL 2 st q in Lower_Arc(P) & q`2<0;
then consider q being Point of TOP-REAL 2 such that
A112: q in Lower_Arc(P) and
A113: q`2<0;
rng f2 =[#]((TOP-REAL 2)|R) by A91,TOPS_2:def 5
.=R by PRE_TOPC:def 5;
then consider x being object such that
A114: x in dom f2 and
A115: q=f2.x by A112,FUNCT_1:def 3;
A116: dom f2= [.0,1.] by BORSUK_1:40,FUNCT_2:def 1;
then
A117: x in {r where r is Real: 0<=r & r<=1 } by A114,RCOMP_1:def 1;
t in {v where v is Real: 0<=v & v<=1 } by A108,A109;
then
A118: t in [.0,1.] by RCOMP_1:def 1;
then
A119: (h2*f2).t=h2.p by A110,A116,FUNCT_1:13
.=p`2 by PSCOMP_1:def 6;
consider r being Real such that
A120: x=r and
A121: 0<=r and
A122: r<=1 by A117;
A123: (h2*f2).r=h2.q by A114,A115,A120,FUNCT_1:13
.=q`2 by PSCOMP_1:def 6;
now
per cases by XXREAL_0:1;
case
A124: r<t;
then reconsider
B=[.r,t.] as non empty Subset of I[01] by A114,A120,A118,BORSUK_1:40
,XXREAL_1:1,XXREAL_2:def 12;
reconsider B0=B as Subset of I[01];
reconsider g=g1|B0 as Function of (I[01]|B0),R^1 by PRE_TOPC:9;
A125: (q`2)*(p`2)<0 by A111,A113,XREAL_1:132;
t in {r4 where r4 is Real: r<=r4 & r4<=t} by A124;
then t in B by RCOMP_1:def 1;
then
A126: p`2=g.t by A119,FUNCT_1:49;
r in {r4 where r4 is Real: r<=r4 & r4<=t} by A124;
then r in B by RCOMP_1:def 1;
then
A127: q`2=g.r by A123,FUNCT_1:49;
g1 is continuous by A106,A105,Th7,Th32;
then
A128: g is continuous by TOPMETR:7;
Closed-Interval-TSpace(r,t)=I[01]|B by A109,A121,A124,TOPMETR:20,23;
then consider r1 being Real such that
A129: g.r1=0 and
A130: r<r1 and
A131: r1<t by A124,A128,A125,A127,A126,TOPREAL5:8;
r1 in {r4 where r4 is Real: r<=r4 & r4<=t} by A130,A131;
then
A132: r1 in B by RCOMP_1:def 1;
r1<1 by A109,A131,XXREAL_0:2;
then r1 in {r2 where r2 is Real: 0<=r2 & r2<=1} by A121,A130;
then
A133: r1 in dom f2 by A116,RCOMP_1:def 1;
then f2.r1 in rng f2 by FUNCT_1:def 3;
then f2.r1 in R by A96;
then f2.r1 in P by A99;
then consider q3 being Point of TOP-REAL 2 such that
A134: q3=f2.r1 and
A135: |.q3.|=1 by A3;
A136: q3`2=h2.(f2.r1) by A134,PSCOMP_1:def 6
.=(h2*f2).r1 by A133,FUNCT_1:13
.=0 by A129,A132,FUNCT_1:49;
then
A137: 1^2=(q3`1)^2 +0^2 by A135,JGRAPH_3:1
.=(q3`1)^2;
now
per cases by A137,SQUARE_1:41;
case
A138: q3`1=1;
A139: 0 in dom f2 by A116,XXREAL_1:1;
q3=|[1,0]| by A136,A138,EUCLID:53
.=E-max(P) by A3,Th30;
hence contradiction by A92,A97,A121,A130,A133,A134,A139,
FUNCT_1:def 4;
end;
case
A140: q3`1=-1;
A141: 1 in dom f2 by A116,XXREAL_1:1;
q3=|[-1,0]| by A136,A140,EUCLID:53
.=W-min(P) by A3,Th29;
hence contradiction by A93,A97,A109,A131,A133,A134,A141,
FUNCT_1:def 4;
end;
end;
hence contradiction;
end;
case
A142: t<r;
then reconsider
B=[.t,r.] as non empty Subset of I[01] by A114,A120,A118,BORSUK_1:40
,XXREAL_1:1,XXREAL_2:def 12;
reconsider B0=B as Subset of I[01];
reconsider g=g1|B0 as Function of (I[01]|B0),R^1 by PRE_TOPC:9;
A143: (q`2)*(p`2)<0 by A111,A113,XREAL_1:132;
t in {r4 where r4 is Real: t<=r4 & r4<=r} by A142;
then t in B by RCOMP_1:def 1;
then
A144: p`2=g.t by A119,FUNCT_1:49;
r in {r4 where r4 is Real: t<=r4 & r4<=r} by A142;
then r in B by RCOMP_1:def 1;
then
A145: q`2=g.r by A123,FUNCT_1:49;
g1 is continuous by A106,A105,Th7,Th32;
then
A146: g is continuous by TOPMETR:7;
Closed-Interval-TSpace(t,r)=I[01]|B by A108,A122,A142,TOPMETR:20,23;
then consider r1 being Real such that
A147: g.r1=0 and
A148: t<r1 and
A149: r1<r by A142,A146,A143,A145,A144,TOPREAL5:8;
r1 in {r4 where r4 is Real: t<=r4 & r4<=r} by A148,A149;
then
A150: r1 in B by RCOMP_1:def 1;
r1<1 by A122,A149,XXREAL_0:2;
then r1 in {r2 where r2 is Real: 0<=r2 & r2<=1} by A108,A148;
then
A151: r1 in dom f2 by A116,RCOMP_1:def 1;
then f2.r1 in rng f2 by FUNCT_1:def 3;
then f2.r1 in R by A96;
then f2.r1 in P by A99;
then consider q3 being Point of TOP-REAL 2 such that
A152: q3=f2.r1 and
A153: |.q3.|=1 by A3;
A154: q3`2=h2.(f2.r1) by A152,PSCOMP_1:def 6
.=g1.r1 by A151,FUNCT_1:13
.=0 by A147,A150,FUNCT_1:49;
then
A155: 1^2=(q3`1)^2 +0^2 by A153,JGRAPH_3:1
.=(q3`1)^2;
now
per cases by A155,SQUARE_1:41;
case
A156: q3`1=1;
A157: 0 in dom f2 by A116,XXREAL_1:1;
q3=|[1,0]| by A154,A156,EUCLID:53
.=E-max(P) by A3,Th30;
hence contradiction by A92,A97,A108,A148,A151,A152,A157,
FUNCT_1:def 4;
end;
case
A158: q3`1=-1;
A159: 1 in dom f2 by A116,XXREAL_1:1;
q3=|[-1,0]| by A154,A158,EUCLID:53
.=W-min(P) by A3,Th29;
hence contradiction by A93,A97,A122,A149,A151,A152,A159,
FUNCT_1:def 4;
end;
end;
hence contradiction;
end;
case
t=r;
hence contradiction by A111,A113,A123,A119;
end;
end;
hence contradiction;
end;
hence thesis;
end;
W-min(P) in {W-min(P),E-max(P)} by TARSKI:def 2;
then
A160: W-min(P) in Upper_Arc(P) by A10,XBOOLE_0:def 4;
A161: W-bound(P)=-1 & E-bound(P)=1 by A3,Th28;
now
assume r8=1;
then p8=|[1,0]| by A3,A8,A22,A23,Th30;
hence contradiction by A31,EUCLID:52;
end;
then
A162: 1>r8 by A25,XXREAL_0:1;
Lower_Arc(P) is closed by A90,JORDAN6:11;
then P2 /\ Q is closed by A26,TOPS_1:8;
then pj in P2 /\ Q by A90,A18,JORDAN5C:def 2;
then
A163: pj=|[0,-1]| or pj=|[0,1]| by A100,TARSKI:def 2;
E-max(P) in {W-min(P),E-max(P)} by TARSKI:def 2;
then
A164: E-max(P) in Upper_Arc(P) by A10,XBOOLE_0:def 4;
A165: {p where p is Point of TOP-REAL 2:p in P & p`2>=0} c= Upper_Arc(P)
proof
let x be object;
assume x in {p where p is Point of TOP-REAL 2:p in P & p`2>=0};
then consider p being Point of TOP-REAL 2 such that
A166: p=x and
A167: p in P and
A168: p`2>=0;
now
per cases by A168;
case
A169: p`2=0;
ex p8 being Point of TOP-REAL 2 st p8=p & |.p8.|=1 by A3,A167;
then 1=sqrt((p`1)^2+(p`2)^2) by JGRAPH_3:1
.=|.p`1.| by A169,COMPLEX1:72;
then p=|[p`1,p`2]| & (p`1)^2=1^2 by COMPLEX1:75,EUCLID:53;
then p=|[1,0]| or p=|[-1,0]| by A169,SQUARE_1:41;
hence thesis by A3,A164,A160,A166,Th29,Th30;
end;
case
A170: p`2>0;
now
assume not x in Upper_Arc(P);
then
A171: x in Lower_Arc(P) by A98,A166,A167,XBOOLE_0:def 3;
rng f2 =[#]((TOP-REAL 2)|R) by A91,TOPS_2:def 5
.=R by PRE_TOPC:def 5;
then consider x2 being object such that
A172: x2 in dom f2 and
A173: p=f2.x2 by A166,A171,FUNCT_1:def 3;
dom f2= [.0,1.] by BORSUK_1:40,FUNCT_2:def 1;
then x2 in {r where r is Real: 0<=r & r<=1 }
by A172,RCOMP_1:def 1;
then consider t2 being Real such that
A174: x2=t2 and
A175: 0<=t2 and
A176: t2<=1;
A177: (|[0,-1]|)`2=-1 by EUCLID:52;
now
assume t2=1;
then p=|[-1,0]| by A3,A93,A173,A174,Th29;
hence contradiction by A170,EUCLID:52;
end;
then
A178: t2<1 by A176,XXREAL_0:1;
A179: now
assume t2=0;
then p=|[1,0]| by A3,A92,A173,A174,Th30;
hence contradiction by A170,EUCLID:52;
end;
(|[0,-1]|)`1=0 by EUCLID:52;
then |.|[0,-1]|.|=sqrt((0)^2+(-1)^2) by A177,JGRAPH_3:1
.=1;
then
A180: |[0,-1]| in {q where q is Point of TOP-REAL 2: |.q.|=1 };
now
per cases by A3,A98,A180,XBOOLE_0:def 3;
case
|[0,-1]| in Upper_Arc(P);
hence
contradiction by A19,A161,A31,A163,A22,A23,A24,A32,A162,A37,A177,
EUCLID:52;
end;
case
|[0,-1]| in Lower_Arc(P);
hence contradiction by A107,A170,A173,A174,A175,A179,A178,A177;
end;
end;
hence contradiction;
end;
hence thesis;
end;
end;
hence thesis;
end;
Upper_Arc(P) c= {p where p is Point of TOP-REAL 2:p in P & p`2>=0}
proof
let x2 be object;
assume
A181: x2 in Upper_Arc(P);
then reconsider q3=x2 as Point of TOP-REAL 2;
q3`2>=0 by A19,A161,A31,A163,A22,A23,A24,A32,A162,A37,A181,EUCLID:52;
hence thesis by A11,A181;
end;
hence thesis by A165,XBOOLE_0:def 10;
end;
theorem Th35:
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}
proof
reconsider h2=proj2 as Function of TOP-REAL 2,R^1 by TOPMETR:17;
reconsider Q=Vertical_Line(0) as Subset of TOP-REAL 2;
let P be compact non empty Subset of TOP-REAL 2;
set P4=Lower_Arc(P);
reconsider P1=Lower_Arc(P) as Subset of TOP-REAL 2;
reconsider P2=Upper_Arc(P) as Subset of TOP-REAL 2;
set pj= First_Point(Upper_Arc(P),W-min(P),E-max(P), Vertical_Line(0));
set p8= Last_Point(Lower_Arc(P),E-max(P),W-min(P), Vertical_Line(0));
A1: LSeg(|[0,-1]|,|[0,1]|) c= Q
proof
let x be object;
assume x in LSeg(|[0,-1]|,|[0,1]|);
then consider l being Real such that
A2: x=(1-l)*(|[0,-1]|) +l*(|[0,1]|) and
0<=l and
l<=1;
((1-l)*(|[0,-1]|) +l*(|[0,1]|))`1 = ((1-l)*(|[0,-1]|))`1 +(l*(|[0,1]|
))`1 by TOPREAL3:2
.=(1-l)*(|[0,-1]|)`1+(l*(|[0,1]|))`1 by TOPREAL3:4
.=(1-l)*(|[0,-1]|)`1+l*((|[0,1]|))`1 by TOPREAL3:4
.=(1-l)*0+l*((|[0,1]|))`1 by EUCLID:52
.=(1-l)*0+l*0 by EUCLID:52
.=0;
hence thesis by A2;
end;
assume
A3: P={q where q is Point of TOP-REAL 2: |.q.|=1};
then
A4: P is being_simple_closed_curve by JGRAPH_3:26;
then
A5: Upper_Arc(P) \/ P4=P by JORDAN6:def 9;
then
A6: Lower_Arc(P) c= P by XBOOLE_1:7;
A7: P2 /\ Q c= {|[0,-1]|,|[0,1]|}
proof
let x be object;
assume
A8: x in P2 /\ Q;
then x in P2 by XBOOLE_0:def 4;
then x in P by A5,XBOOLE_0:def 3;
then consider q being Point of TOP-REAL 2 such that
A9: q=x and
A10: |.q.|=1 by A3;
x in Q by A8,XBOOLE_0:def 4;
then
A11: ex p being Point of TOP-REAL 2 st p=x & p`1=0;
then 0^2+(q`2)^2 =1^2 by A9,A10,JGRAPH_3:1;
then q`2=1 or q`2=-1 by SQUARE_1:41;
then x=|[0,-1]| or x=|[0,1]| by A11,A9,EUCLID:53;
hence thesis by TARSKI:def 2;
end;
A12: for p being Point of (TOP-REAL 2) holds h2.p=proj2.p;
reconsider R=Lower_Arc(P) as non empty Subset of TOP-REAL 2;
A13: Vertical_Line(0) is closed by JORDAN6:30;
A14: Vertical_Line(0) is closed by JORDAN6:30;
A15: for p being Point of (TOP-REAL 2) holds h2.p=proj2.p;
A16: S-bound P=-1 & N-bound P=1 by A3,Th28;
A17: W-bound P=-1 & E-bound P=1 by A3,Th28;
then
A18: P1 meets Q by A4,A16,A1,JORDAN6:70,XBOOLE_1:64;
A19: P2 meets Q by A4,A17,A16,A1,JORDAN6:69,XBOOLE_1:64;
A20: Upper_Arc(P) /\ P4={W-min(P),E-max(P)} by A4,JORDAN6:def 9;
A21: Lower_Arc(P) is_an_arc_of E-max(P),W-min(P) by A4,JORDAN6:def 9;
then consider f being Function of I[01], (TOP-REAL 2)|R such that
A22: f is being_homeomorphism and
A23: f.0 =E-max(P) and
A24: f.1 =W-min(P) by TOPREAL1:def 1;
A25: dom f=the carrier of I[01] & dom h2=the carrier of TOP-REAL 2 by
FUNCT_2:def 1;
A26: rng f =[#]((TOP-REAL 2)|R) by A22,TOPS_2:def 5
.=R by PRE_TOPC:def 5;
A27: Upper_Arc(P) c= P by A5,XBOOLE_1:7;
A28: rng (h2*f) c= the carrier of R^1;
A29: the carrier of ((TOP-REAL 2)|R)=R by PRE_TOPC:8;
then rng f c= the carrier of TOP-REAL 2 by XBOOLE_1:1;
then dom (h2*f)=the carrier of I[01] by A25,RELAT_1:27;
then reconsider g0=h2*f as Function of I[01],R^1 by A28,FUNCT_2:2;
A30: f is one-to-one by A22,TOPS_2:def 5;
A31: f is continuous by A22,TOPS_2:def 5;
A32: (ex p being Point of TOP-REAL 2,
t being Real st 0<t & t<1 & f.t=p & p`2
<0) implies for q being Point of TOP-REAL 2 st q in Lower_Arc(P) holds q`2<=0
proof
given p being Point of TOP-REAL 2,t being Real such that
A33: 0<t and
A34: t<1 and
A35: f.t=p and
A36: p`2<0;
now
assume ex q being Point of TOP-REAL 2 st q in Lower_Arc(P) & q`2>0;
then consider q being Point of TOP-REAL 2 such that
A37: q in Lower_Arc(P) and
A38: q`2>0;
rng f =[#]((TOP-REAL 2)|R) by A22,TOPS_2:def 5
.=R by PRE_TOPC:def 5;
then consider x being object such that
A39: x in dom f and
A40: q=f.x by A37,FUNCT_1:def 3;
A41: dom f= [.0,1.] by BORSUK_1:40,FUNCT_2:def 1;
then
A42: x in {r where r is Real: 0<=r & r<=1 } by A39,RCOMP_1:def 1;
t in {v where v is Real: 0<=v & v<=1 } by A33,A34;
then
A43: t in [.0,1.] by RCOMP_1:def 1;
then
A44: (h2*f).t=h2.p by A35,A41,FUNCT_1:13
.=p`2 by PSCOMP_1:def 6;
consider r being Real such that
A45: x=r and
A46: 0<=r and
A47: r<=1 by A42;
A48: (h2*f).r=h2.q by A39,A40,A45,FUNCT_1:13
.=q`2 by PSCOMP_1:def 6;
now
per cases by XXREAL_0:1;
case
A49: r<t;
then reconsider
B=[.r,t.] as non empty Subset of I[01] by A39,A45,A43,BORSUK_1:40
,XXREAL_1:1,XXREAL_2:def 12;
reconsider B0=B as Subset of I[01];
reconsider g=g0|B0 as Function of (I[01]|B0),R^1 by PRE_TOPC:9;
A50: (q`2)*(p`2)<0 by A36,A38,XREAL_1:132;
t in {r4 where r4 is Real: r<=r4 & r4<=t} by A49;
then t in B by RCOMP_1:def 1;
then
A51: p`2=g.t by A44,FUNCT_1:49;
r in {r4 where r4 is Real: r<=r4 & r4<=t} by A49;
then r in B by RCOMP_1:def 1;
then
A52: q`2=g.r by A48,FUNCT_1:49;
g0 is continuous by A31,A12,Th7,Th32;
then
A53: g is continuous by TOPMETR:7;
Closed-Interval-TSpace(r,t)=I[01]|B by A34,A46,A49,TOPMETR:20,23;
then consider r1 being Real such that
A54: g.r1=0 and
A55: r<r1 and
A56: r1<t by A49,A53,A50,A52,A51,TOPREAL5:8;
r1 in {r4 where r4 is Real: r<=r4 & r4<=t} by A55,A56;
then
A57: r1 in B by RCOMP_1:def 1;
r1<1 by A34,A56,XXREAL_0:2;
then r1 in {r2 where r2 is Real: 0<=r2 & r2<=1} by A46,A55;
then
A58: r1 in dom f by A41,RCOMP_1:def 1;
then f.r1 in rng f by FUNCT_1:def 3;
then f.r1 in R by A29;
then f.r1 in P by A6;
then consider q3 being Point of TOP-REAL 2 such that
A59: q3=f.r1 and
A60: |.q3.|=1 by A3;
A61: q3`2=h2.(f.r1) by A59,PSCOMP_1:def 6
.=(h2*f).r1 by A58,FUNCT_1:13
.=0 by A54,A57,FUNCT_1:49;
then
A62: 1^2=(q3`1)^2 +0^2 by A60,JGRAPH_3:1
.=(q3`1)^2;
now
per cases by A62,SQUARE_1:41;
case
A63: q3`1=1;
A64: 0 in dom f by A41,XXREAL_1:1;
q3=|[1,0]| by A61,A63,EUCLID:53
.=E-max(P) by A3,Th30;
hence contradiction by A23,A30,A46,A55,A58,A59,A64,FUNCT_1:def 4;
end;
case
A65: q3`1=-1;
A66: 1 in dom f by A41,XXREAL_1:1;
q3=|[-1,0]| by A61,A65,EUCLID:53
.=W-min(P) by A3,Th29;
hence contradiction by A24,A30,A34,A56,A58,A59,A66,FUNCT_1:def 4;
end;
end;
hence contradiction;
end;
case
A67: t<r;
then reconsider
B=[.t,r.] as non empty Subset of I[01] by A39,A45,A43,BORSUK_1:40
,XXREAL_1:1,XXREAL_2:def 12;
reconsider B0=B as Subset of I[01];
reconsider g=g0|B0 as Function of (I[01]|B0),R^1 by PRE_TOPC:9;
A68: (q`2)*(p`2)<0 by A36,A38,XREAL_1:132;
t in {r4 where r4 is Real: t<=r4 & r4<=r} by A67;
then t in B by RCOMP_1:def 1;
then
A69: p`2=g.t by A44,FUNCT_1:49;
r in {r4 where r4 is Real: t<=r4 & r4<=r} by A67;
then r in B by RCOMP_1:def 1;
then
A70: q`2=g.r by A48,FUNCT_1:49;
g0 is continuous by A31,A12,Th7,Th32;
then
A71: g is continuous by TOPMETR:7;
Closed-Interval-TSpace(t,r)=I[01]|B by A33,A47,A67,TOPMETR:20,23;
then consider r1 being Real such that
A72: g.r1=0 and
A73: t<r1 and
A74: r1<r by A67,A71,A68,A70,A69,TOPREAL5:8;
r1 in {r4 where r4 is Real: t<=r4 & r4<=r} by A73,A74;
then
A75: r1 in B by RCOMP_1:def 1;
r1<1 by A47,A74,XXREAL_0:2;
then r1 in {r2 where r2 is Real: 0<=r2 & r2<=1} by A33,A73;
then
A76: r1 in dom f by A41,RCOMP_1:def 1;
then f.r1 in rng f by FUNCT_1:def 3;
then f.r1 in R by A29;
then f.r1 in P by A6;
then consider q3 being Point of TOP-REAL 2 such that
A77: q3=f.r1 and
A78: |.q3.|=1 by A3;
A79: q3`2=h2.(f.r1) by A77,PSCOMP_1:def 6
.=(h2*f).r1 by A76,FUNCT_1:13
.=0 by A72,A75,FUNCT_1:49;
then
A80: 1^2=(q3`1)^2 +0^2 by A78,JGRAPH_3:1
.=(q3`1)^2;
now
per cases by A80,SQUARE_1:41;
case
A81: q3`1=1;
A82: 0 in dom f by A41,XXREAL_1:1;
q3=|[1,0]| by A79,A81,EUCLID:53
.=E-max(P) by A3,Th30;
hence contradiction by A23,A30,A33,A73,A76,A77,A82,FUNCT_1:def 4;
end;
case
A83: q3`1=-1;
A84: 1 in dom f by A41,XXREAL_1:1;
q3=|[-1,0]| by A79,A83,EUCLID:53
.=W-min(P) by A3,Th29;
hence contradiction by A24,A30,A47,A74,A76,A77,A84,FUNCT_1:def 4;
end;
end;
hence contradiction;
end;
case
t=r;
hence contradiction by A36,A38,A48,A44;
end;
end;
hence contradiction;
end;
hence thesis;
end;
reconsider R=Upper_Arc(P) as non empty Subset of TOP-REAL 2;
A85: Upper_Arc(P) is_an_arc_of W-min(P),E-max(P) by A4,JORDAN6:def 8;
then consider f2 being Function of I[01], (TOP-REAL 2)|R such that
A86: f2 is being_homeomorphism and
A87: f2.0 =W-min(P) and
A88: f2.1 =E-max(P) by TOPREAL1:def 1;
A89: dom f2=the carrier of I[01] & dom h2=the carrier of TOP-REAL 2 by
FUNCT_2:def 1;
A90: rng (h2*f2) c= the carrier of R^1;
A91: the carrier of ((TOP-REAL 2)|R)=R by PRE_TOPC:8;
then rng f2 c= the carrier of TOP-REAL 2 by XBOOLE_1:1;
then dom (h2*f2)=the carrier of I[01] by A89,RELAT_1:27;
then reconsider g1=h2*f2 as Function of I[01],R^1 by A90,FUNCT_2:2;
A92: f2 is one-to-one by A86,TOPS_2:def 5;
A93: f2 is continuous by A86,TOPS_2:def 5;
A94: (ex p being Point of TOP-REAL 2,
t being Real st 0<t & t<1 & f2.t=p & p
`2<0) implies for q being Point of TOP-REAL 2 st q in Upper_Arc(P) holds q`2<=0
proof
given p being Point of TOP-REAL 2,t being Real such that
A95: 0<t and
A96: t<1 and
A97: f2.t=p and
A98: p`2<0;
now
assume ex q being Point of TOP-REAL 2 st q in Upper_Arc(P) & q`2>0;
then consider q being Point of TOP-REAL 2 such that
A99: q in Upper_Arc(P) and
A100: q`2>0;
rng f2 =[#]((TOP-REAL 2)|R) by A86,TOPS_2:def 5
.=R by PRE_TOPC:def 5;
then consider x being object such that
A101: x in dom f2 and
A102: q=f2.x by A99,FUNCT_1:def 3;
A103: dom f2= [.0,1.] by BORSUK_1:40,FUNCT_2:def 1;
then
A104: x in {r where r is Real: 0<=r & r<=1 } by A101,RCOMP_1:def 1;
t in {v where v is Real: 0<=v & v<=1 } by A95,A96;
then
A105: t in [.0,1.] by RCOMP_1:def 1;
then
A106: (h2*f2).t=h2.p by A97,A103,FUNCT_1:13
.=p`2 by PSCOMP_1:def 6;
consider r being Real such that
A107: x=r and
A108: 0<=r and
A109: r<=1 by A104;
A110: (h2*f2).r=h2.q by A101,A102,A107,FUNCT_1:13
.=q`2 by PSCOMP_1:def 6;
now
per cases by XXREAL_0:1;
case
A111: r<t;
then reconsider
B=[.r,t.] as non empty Subset of I[01] by A101,A107,A105,BORSUK_1:40
,XXREAL_1:1,XXREAL_2:def 12;
reconsider B0=B as Subset of I[01];
reconsider g=g1|B0 as Function of (I[01]|B0),R^1 by PRE_TOPC:9;
A112: (q`2)*(p`2)<0 by A98,A100,XREAL_1:132;
t in {r4 where r4 is Real: r<=r4 & r4<=t} by A111;
then t in B by RCOMP_1:def 1;
then
A113: p`2=g.t by A106,FUNCT_1:49;
r in {r4 where r4 is Real: r<=r4 & r4<=t} by A111;
then r in B by RCOMP_1:def 1;
then
A114: q`2=g.r by A110,FUNCT_1:49;
g1 is continuous by A93,A15,Th7,Th32;
then
A115: g is continuous by TOPMETR:7;
Closed-Interval-TSpace(r,t)=I[01]|B by A96,A108,A111,TOPMETR:20,23;
then consider r1 being Real such that
A116: g.r1=0 and
A117: r<r1 and
A118: r1<t by A111,A115,A112,A114,A113,TOPREAL5:8;
r1 in {r4 where r4 is Real: r<=r4 & r4<=t} by A117,A118;
then
A119: r1 in B by RCOMP_1:def 1;
r1<1 by A96,A118,XXREAL_0:2;
then r1 in {r2 where r2 is Real: 0<=r2 & r2<=1} by A108,A117;
then
A120: r1 in dom f2 by A103,RCOMP_1:def 1;
then f2.r1 in rng f2 by FUNCT_1:def 3;
then f2.r1 in R by A91;
then f2.r1 in P by A27;
then consider q3 being Point of TOP-REAL 2 such that
A121: q3=f2.r1 and
A122: |.q3.|=1 by A3;
A123: q3`2=h2.(f2.r1) by A121,PSCOMP_1:def 6
.=(h2*f2).r1 by A120,FUNCT_1:13
.=0 by A116,A119,FUNCT_1:49;
then
A124: 1^2=(q3`1)^2 +0^2 by A122,JGRAPH_3:1
.=(q3`1)^2;
now
per cases by A124,SQUARE_1:41;
case
A125: q3`1=1;
A126: 1 in dom f2 by A103,XXREAL_1:1;
q3=|[1,0]| by A123,A125,EUCLID:53
.=E-max(P) by A3,Th30;
hence contradiction by A88,A92,A96,A118,A120,A121,A126,
FUNCT_1:def 4;
end;
case
A127: q3`1=-1;
A128: 0 in dom f2 by A103,XXREAL_1:1;
q3=|[-1,0]| by A123,A127,EUCLID:53
.=W-min(P) by A3,Th29;
hence contradiction by A87,A92,A108,A117,A120,A121,A128,
FUNCT_1:def 4;
end;
end;
hence contradiction;
end;
case
A129: t<r;
then reconsider
B=[.t,r.] as non empty Subset of I[01] by A101,A107,A105,BORSUK_1:40
,XXREAL_1:1,XXREAL_2:def 12;
reconsider B0=B as Subset of I[01];
reconsider g=g1|B0 as Function of (I[01]|B0),R^1 by PRE_TOPC:9;
A130: (q`2)*(p`2)<0 by A98,A100,XREAL_1:132;
t in {r4 where r4 is Real: t<=r4 & r4<=r} by A129;
then t in B by RCOMP_1:def 1;
then
A131: p`2=g.t by A106,FUNCT_1:49;
r in {r4 where r4 is Real: t<=r4 & r4<=r} by A129;
then r in B by RCOMP_1:def 1;
then
A132: q`2=g.r by A110,FUNCT_1:49;
g1 is continuous by A93,A15,Th7,Th32;
then
A133: g is continuous by TOPMETR:7;
Closed-Interval-TSpace(t,r)=I[01]|B by A95,A109,A129,TOPMETR:20,23;
then consider r1 being Real such that
A134: g.r1=0 and
A135: t<r1 and
A136: r1<r by A129,A133,A130,A132,A131,TOPREAL5:8;
r1 in {r4 where r4 is Real: t<=r4 & r4<=r} by A135,A136;
then
A137: r1 in B by RCOMP_1:def 1;
r1<1 by A109,A136,XXREAL_0:2;
then r1 in {r2 where r2 is Real: 0<=r2 & r2<=1} by A95,A135;
then
A138: r1 in dom f2 by A103,RCOMP_1:def 1;
then f2.r1 in rng f2 by FUNCT_1:def 3;
then f2.r1 in R by A91;
then f2.r1 in P by A27;
then consider q3 being Point of TOP-REAL 2 such that
A139: q3=f2.r1 and
A140: |.q3.|=1 by A3;
A141: q3`2=h2.(f2.r1) by A139,PSCOMP_1:def 6
.=(h2*f2).r1 by A138,FUNCT_1:13
.=0 by A134,A137,FUNCT_1:49;
then
A142: 1^2=(q3`1)^2 +0^2 by A140,JGRAPH_3:1
.=(q3`1)^2;
now
per cases by A142,SQUARE_1:41;
case
A143: q3`1=1;
A144: 1 in dom f2 by A103,XXREAL_1:1;
q3=|[1,0]| by A141,A143,EUCLID:53
.=E-max(P) by A3,Th30;
hence contradiction by A88,A92,A109,A136,A138,A139,A144,
FUNCT_1:def 4;
end;
case
A145: q3`1=-1;
A146: 0 in dom f2 by A103,XXREAL_1:1;
q3=|[-1,0]| by A141,A145,EUCLID:53
.=W-min(P) by A3,Th29;
hence contradiction by A87,A92,A95,A135,A138,A139,A146,
FUNCT_1:def 4;
end;
end;
hence contradiction;
end;
case
t=r;
hence contradiction by A98,A100,A110,A106;
end;
end;
hence contradiction;
end;
hence thesis;
end;
A147: W-bound(P)=-1 & E-bound(P)=1 by A3,Th28;
Lower_Arc(P) is closed by A21,JORDAN6:11;
then P1 /\ Q is closed by A13,TOPS_1:8;
then
A148: p8 in P1 /\ Q by A21,A18,JORDAN5C:def 2;
then p8 in P1 by XBOOLE_0:def 4;
then consider x8 being object such that
A149: x8 in dom f and
A150: p8=f.x8 by A26,FUNCT_1:def 3;
dom f= [.0,1.] by BORSUK_1:40,FUNCT_2:def 1;
then x8 in {r where r is Real: 0<=r & r<=1 } by A149,RCOMP_1:def 1;
then consider r8 being Real such that
A151: x8=r8 and
A152: 0<=r8 and
A153: r8<=1;
P1 /\ Q c= {|[0,-1]|,|[0,1]|}
proof
let x be object;
assume
A154: x in P1 /\ Q;
then x in P1 by XBOOLE_0:def 4;
then x in P by A5,XBOOLE_0:def 3;
then consider q being Point of TOP-REAL 2 such that
A155: q=x and
A156: |.q.|=1 by A3;
x in Q by A154,XBOOLE_0:def 4;
then
A157: ex p being Point of TOP-REAL 2 st p=x & p`1=0;
then 0^2+(q`2)^2 =1^2 by A155,A156,JGRAPH_3:1;
then q`2=1 or q`2=-1 by SQUARE_1:41;
then x=|[0,-1]| or x=|[0,1]| by A157,A155,EUCLID:53;
hence thesis by TARSKI:def 2;
end;
then p8=|[0,-1]| or p8=|[0,1]| by A148,TARSKI:def 2;
then
A158: p8`2=-1 or p8`2=1 by EUCLID:52;
A159: now
assume r8=0;
then p8=|[1,0]| by A3,A23,A150,A151,Th30;
hence contradiction by A158,EUCLID:52;
end;
Upper_Arc(P) is closed by A85,JORDAN6:11;
then P2 /\ Q is closed by A14,TOPS_1:8;
then pj in P2 /\ Q by A85,A19,JORDAN5C:def 1;
then
A160: pj=|[0,-1]| or pj=|[0,1]| by A7,TARSKI:def 2;
W-min(P) in {W-min(P),E-max(P)} by TARSKI:def 2;
then
A161: W-min(P) in Lower_Arc(P) by A20,XBOOLE_0:def 4;
A162: First_Point(Upper_Arc(P),W-min(P),E-max(P), Vertical_Line((W-bound(P)+
E-bound(P))/2))`2> Last_Point(P4,E-max(P),W-min(P), Vertical_Line((W-bound(P)+
E-bound(P))/2))`2 by A4,JORDAN6:def 9;
now
assume r8=1;
then p8=|[-1,0]| by A3,A24,A150,A151,Th29;
hence contradiction by A158,EUCLID:52;
end;
then
A163: 1>r8 by A153,XXREAL_0:1;
E-max(P) in {W-min(P),E-max(P)} by TARSKI:def 2;
then
A164: E-max(P) in Lower_Arc(P) by A20,XBOOLE_0:def 4;
A165: {p where p is Point of TOP-REAL 2:p in P & p`2<=0} c= Lower_Arc(P)
proof
let x be object;
assume x in {p where p is Point of TOP-REAL 2:p in P & p`2<=0};
then consider p being Point of TOP-REAL 2 such that
A166: p=x and
A167: p in P and
A168: p`2<=0;
now
per cases by A168;
case
A169: p`2=0;
ex p8 being Point of TOP-REAL 2 st p8=p & |.p8.|=1 by A3,A167;
then 1=sqrt((p`1)^2+(p`2)^2) by JGRAPH_3:1
.=|.p`1.| by A169,COMPLEX1:72;
then p=|[p`1,p`2]| & (p`1)^2=1^2 by COMPLEX1:75,EUCLID:53;
then p=|[1,0]| or p=|[-1,0]| by A169,SQUARE_1:41;
hence thesis by A3,A164,A161,A166,Th29,Th30;
end;
case
A170: p`2<0;
now
assume not x in Lower_Arc(P);
then
A171: x in Upper_Arc(P) by A5,A166,A167,XBOOLE_0:def 3;
rng f2 =[#]((TOP-REAL 2)|R) by A86,TOPS_2:def 5
.=R by PRE_TOPC:def 5;
then consider x2 being object such that
A172: x2 in dom f2 and
A173: p=f2.x2 by A166,A171,FUNCT_1:def 3;
dom f2= [.0,1.] by BORSUK_1:40,FUNCT_2:def 1;
then x2 in {r where r is Real: 0<=r & r<=1 }
by A172,RCOMP_1:def 1;
then consider t2 being Real such that
A174: x2=t2 and
A175: 0<=t2 and
A176: t2<=1;
A177: (|[0,1]|)`2=1 by EUCLID:52;
now
assume t2=1;
then p=|[1,0]| by A3,A88,A173,A174,Th30;
hence contradiction by A170,EUCLID:52;
end;
then
A178: t2<1 by A176,XXREAL_0:1;
A179: now
assume t2=0;
then p=|[-1,0]| by A3,A87,A173,A174,Th29;
hence contradiction by A170,EUCLID:52;
end;
(|[0,1]|)`1=0 by EUCLID:52;
then |.|[0,1]|.|=sqrt((0)^2+(1)^2) by A177,JGRAPH_3:1
.=1;
then
A180: |[0,1]| in {q where q is Point of TOP-REAL 2: |.q.|=1};
now
per cases by A3,A5,A180,XBOOLE_0:def 3;
case
|[0,1]| in Lower_Arc(P);
hence
contradiction by A162,A147,A158,A160,A150,A151,A152,A159,A163,A32,A177,
EUCLID:52;
end;
case
|[0,1]| in Upper_Arc(P);
hence contradiction by A94,A170,A173,A174,A175,A179,A178,A177;
end;
end;
hence contradiction;
end;
hence thesis;
end;
end;
hence thesis;
end;
Lower_Arc(P) c= {p where p is Point of TOP-REAL 2:p in P & p`2<=0}
proof
let x2 be object;
assume
A181: x2 in Lower_Arc(P);
then reconsider q3=x2 as Point of TOP-REAL 2;
q3`2<=0 by A162,A147,A158,A160,A150,A151,A152,A159,A163,A32,A181,EUCLID:52;
hence thesis by A6,A181;
end;
hence thesis by A165,XBOOLE_0:def 10;
end;
theorem Th36:
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
proof
let a,b,d,e be Real;
assume that
A1: a<=b and
A2: e>0;
set S=Closed-Interval-TSpace(a,b);
defpred P[object,object] means
(for r being Real st $1=r holds $2=e*r+d);
set X=the carrier of Closed-Interval-TSpace(a,b);
A3: X=[.a,b.] by A1,TOPMETR:18;
then reconsider B=the carrier of S as Subset of R^1 by TOPMETR:17;
A4: R^1|B= S by A1,A3,TOPMETR:19;
set T=Closed-Interval-TSpace(e*a+d,e*b+d);
set Y=the carrier of Closed-Interval-TSpace(e*a+d,e*b+d);
A5: e*a<=e*b by A1,A2,XREAL_1:64;
then
A6: Y=[.e*a+d,e*b+d.] by TOPMETR:18,XREAL_1:7;
then reconsider C=the carrier of T as Subset of R^1 by TOPMETR:17;
defpred P1[object,object] means
for r being Real st r=$1 holds $2=e*r+d;
T=TopSpaceMetr(Closed-Interval-MSpace(e*a+d,e*b+d)) by TOPMETR:def 7;
then
A7: T is T_2 by PCOMPS_1:34;
A8: for x being object st x in X ex y being object st y in Y & P[x,y]
proof
let x be object;
assume
A9: x in X;
then reconsider r1=x as Real;
set y1=e*r1+d;
r1<=b by A3,A9,XXREAL_1:1;
then e*r1<=e*b by A2,XREAL_1:64;
then
A10: y1 <=e*b+d by XREAL_1:7;
a<=r1 by A3,A9,XXREAL_1:1;
then e*a<=e*r1 by A2,XREAL_1:64;
then e*a+d<=y1 by XREAL_1:7;
then ( for r being Real st x=r holds y1=e*r+d)& y1 in Y by A6,A10,
XXREAL_1:1;
hence thesis;
end;
ex f being Function of X,Y st
for x being object st x in X holds P[x,f.x]
from FUNCT_2:sch 1(A8);
then consider f1 being Function of X,Y such that
A11: for x being object st x in X holds P[x,f1.x];
reconsider f2=f1 as Function of Closed-Interval-TSpace(a,b),
Closed-Interval-TSpace(e*a+d,e*b+d);
A12: for r being Real st r in [.a,b.] holds f2.r=e*r+d by A3,A11;
A13: dom f2=the carrier of S by FUNCT_2:def 1;
[#]T c= rng f2
proof
let y be object;
assume
A14: y in [#]T;
then reconsider ry=y as Real;
ry<=e*b+d by A6,A14,XXREAL_1:1;
then e*b+d-d>=ry-d by XREAL_1:9;
then b*e/e>=(ry-d)/e by A2,XREAL_1:72;
then
A15: b>=(ry-d)/e by A2,XCMPLX_1:89;
e*a+d <= ry by A6,A14,XXREAL_1:1;
then e*a+d-d<=ry-d by XREAL_1:9;
then a*e/e<=(ry-d)/e by A2,XREAL_1:72;
then a<=(ry-d)/e by A2,XCMPLX_1:89;
then
A16: (ry-d)/e in [.a,b.] by A15,XXREAL_1:1;
then f2.((ry-d)/e)=e*((ry-d)/e)+d by A3,A11
.=ry-d+d by A2,XCMPLX_1:87
.=ry;
hence thesis by A3,A13,A16,FUNCT_1:3;
end;
then
A17: rng f2 = [#]T by XBOOLE_0:def 10;
then reconsider f3=f1 as Function of S,R^1 by A6,A13,FUNCT_2:2,TOPMETR:17;
for x1,x2 being object st x1 in dom f2 & x2 in dom f2 & f2.x1=f2.x2
holds x1=x2
proof
let x1,x2 be object;
assume that
A18: x1 in dom f2 and
A19: x2 in dom f2 and
A20: f2.x1=f2.x2;
reconsider r2=x2 as Real by A19;
reconsider r1=x1 as Real by A18;
f2.x1=e*r1+d by A11,A18;
then e*r1+d-d=e*r2+d-d by A11,A19,A20
.=e*r2;
then r1*e/e=r2 by A2,XCMPLX_1:89;
hence thesis by A2,XCMPLX_1:89;
end;
then
A21: dom f2=[#]S & f2 is one-to-one by FUNCT_1:def 4,FUNCT_2:def 1;
A22: for x being object st x in the carrier of R^1
ex y being object st y in the carrier of R^1 & P1[x,y]
proof
let x be object;
assume x in the carrier of R^1;
then reconsider rx=x as Real;
reconsider ry=e*rx+d as Element of REAL by XREAL_0:def 1;
for r being Real st r=x holds ry=e*r+d;
hence thesis by TOPMETR:17;
end;
ex f4 being Function of the carrier of R^1,the carrier of R^1 st for x
being object st x in the carrier of R^1 holds P1[x,f4.x]
from FUNCT_2:sch 1(A22);
then consider
f4 being Function of the carrier of R^1,the carrier of R^1 such
that
A23: for x being object st x in the carrier of R^1 holds P1[x,f4.x];
reconsider f5=f4 as Function of R^1,R^1;
A24: for x being Real holds f5.x = e*x + d
by A23,TOPMETR:17,XREAL_0:def 1;
A25: (dom f5) /\ B =REAL /\ B by FUNCT_2:def 1,TOPMETR:17
.=B by TOPMETR:17,XBOOLE_1:28;
A26: for x being object st x in dom f3 holds f3.x=f5.x
proof
let x be object;
assume
A27: x in dom f3;
then reconsider rx=x as Element of REAL by A3,A13;
f4.x=e*rx+d by A23,TOPMETR:17;
hence thesis by A11,A27;
end;
dom f3=B by FUNCT_2:def 1;
then f3=f5|B by A25,A26,FUNCT_1:46;
then
A28: f3 is continuous by A24,A4,TOPMETR:7,21;
A29: S is compact by A1,HEINE:4;
R^1|C=T by A5,A6,TOPMETR:19,XREAL_1:7;
then f2 is being_homeomorphism by A21,A17,A28,A29,A7,COMPTS_1:17,TOPMETR:6;
hence thesis by A12;
end;
theorem Th37:
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
proof
let a,b,d,e be Real;
assume that
A1: a<=b and
A2: e<0;
set S=Closed-Interval-TSpace(a,b);
defpred P[object,object] means
(for r being Real st $1=r holds $2=e*r+d);
set X=the carrier of Closed-Interval-TSpace(a,b);
A3: X=[.a,b.] by A1,TOPMETR:18;
then reconsider B=the carrier of S as Subset of R^1 by TOPMETR:17;
A4: R^1|B= S by A1,A3,TOPMETR:19;
set T=Closed-Interval-TSpace(e*b+d,e*a+d);
set Y=the carrier of Closed-Interval-TSpace(e*b+d,e*a+d);
A5: e*a>=e*b by A1,A2,XREAL_1:65;
then
A6: Y=[.e*b+d,e*a+d.] by TOPMETR:18,XREAL_1:7;
then reconsider C=the carrier of T as Subset of R^1 by TOPMETR:17;
defpred P1[object,object] means
for r being Real st r=$1 holds $2=e*r+d;
T=TopSpaceMetr(Closed-Interval-MSpace(e*b+d,e*a+d)) by TOPMETR:def 7;
then
A7: T is T_2 by PCOMPS_1:34;
A8: for x being object st x in X ex y being object st y in Y & P[x,y]
proof
let x be object;
assume
A9: x in X;
then reconsider r1=x as Real;
set y1=e*r1+d;
r1<=b by A3,A9,XXREAL_1:1;
then e*r1>=e*b by A2,XREAL_1:65;
then
A10: y1 >=e*b+d by XREAL_1:7;
a<=r1 by A3,A9,XXREAL_1:1;
then e*a>=e*r1 by A2,XREAL_1:65;
then e*a+d>=y1 by XREAL_1:7;
then ( for r being Real st x=r holds y1=e*r+d)& y1 in Y by A6,A10,
XXREAL_1:1;
hence thesis;
end;
ex f being Function of X,Y st
for x being object st x in X holds P[x,f.x]
from FUNCT_2:sch 1(A8);
then consider f1 being Function of X,Y such that
A11: for x being object st x in X holds P[x,f1.x];
reconsider f2=f1 as Function of Closed-Interval-TSpace(a,b),
Closed-Interval-TSpace(e*b+d,e*a+d);
A12: for r being Real st r in [.a,b.] holds f2.r=e*r+d by A3,A11;
A13: dom f2=the carrier of S by FUNCT_2:def 1;
[#]T c= rng f2
proof
let y be object;
assume
A14: y in [#]T;
then reconsider ry=y as Real;
ry<=e*a+d by A6,A14,XXREAL_1:1;
then e*a+d-d>=ry-d by XREAL_1:9;
then a*e/e<=(ry-d)/e by A2,XREAL_1:73;
then
A15: a<=(ry-d)/e by A2,XCMPLX_1:89;
e*b+d <= ry by A6,A14,XXREAL_1:1;
then e*b+d-d<=ry-d by XREAL_1:9;
then b*e/e>=(ry-d)/e by A2,XREAL_1:73;
then b>=(ry-d)/e by A2,XCMPLX_1:89;
then
A16: (ry-d)/e in [.a,b.] by A15,XXREAL_1:1;
then f2.((ry-d)/e)=e*((ry-d)/e)+d by A3,A11
.=ry-d+d by A2,XCMPLX_1:87
.=ry;
hence thesis by A3,A13,A16,FUNCT_1:3;
end;
then
A17: rng f2 = [#]T by XBOOLE_0:def 10;
then reconsider f3=f1 as Function of S,R^1 by A6,A13,FUNCT_2:2,TOPMETR:17;
for x1,x2 being object st x1 in dom f2 & x2 in dom f2 & f2.x1=f2.x2
holds x1=x2
proof
let x1,x2 be object;
assume that
A18: x1 in dom f2 and
A19: x2 in dom f2 and
A20: f2.x1=f2.x2;
reconsider r2=x2 as Real by A19;
reconsider r1=x1 as Real by A18;
f2.x1=e*r1+d by A11,A18;
then e*r1+d-d=e*r2+d-d by A11,A19,A20
.=e*r2;
then r1*e/e=r2 by A2,XCMPLX_1:89;
hence thesis by A2,XCMPLX_1:89;
end;
then
A21: dom f2=[#]S & f2 is one-to-one by FUNCT_1:def 4,FUNCT_2:def 1;
A22: for x being object st x in the carrier of R^1
ex y being object st y in the carrier of R^1 & P1[x,y]
proof
let x be object;
assume x in the carrier of R^1;
then reconsider rx=x as Real;
reconsider ry=e*rx+d as Element of REAL by XREAL_0:def 1;
for r being Real st r=x holds ry=e*r+d;
hence thesis by TOPMETR:17;
end;
ex f4 being Function of the carrier of R^1,the carrier of R^1 st for x
being object st x in the carrier of R^1 holds P1[x,f4.x]
from FUNCT_2:sch 1(A22);
then consider
f4 being Function of the carrier of R^1,the carrier of R^1 such
that
A23: for x being object st x in the carrier of R^1 holds P1[x,f4.x];
reconsider f5=f4 as Function of R^1,R^1;
A24: for x being Real holds f5.x = e*x + d
by XREAL_0:def 1,TOPMETR:17,A23;
A25: (dom f5) /\ B =REAL /\ B by FUNCT_2:def 1,TOPMETR:17
.=B by TOPMETR:17,XBOOLE_1:28;
A26: for x being object st x in dom f3 holds f3.x=f5.x
proof
let x be object;
assume
A27: x in dom f3;
then reconsider rx=x as Element of REAL by A3,A13;
f4.x=e*rx+d by A23,TOPMETR:17;
hence thesis by A11,A27;
end;
dom f3=B by FUNCT_2:def 1;
then f3=f5|B by A25,A26,FUNCT_1:46;
then
A28: f3 is continuous by A24,A4,TOPMETR:7,21;
A29: S is compact by A1,HEINE:4;
R^1|C=T by A5,A6,TOPMETR:19,XREAL_1:7;
then f2 is being_homeomorphism by A21,A17,A28,A29,A7,COMPTS_1:17,TOPMETR:6;
hence thesis by A12;
end;
theorem Th38:
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
proof
consider f being Function of I[01], Closed-Interval-TSpace((-2)*1+1,(-2)*0+1
) such that
A1: f is being_homeomorphism and
A2: for r being Real st r in [.0,1.] holds f.r=(-2)*r+1
by Th37,TOPMETR:20;
1 in [.0,1.] by XXREAL_1:1;
then
A3: f.1=-1 by A2;
f.0=(-2)*0+1 by A2,Lm1;
hence thesis by A1,A2,A3;
end;
theorem Th39:
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
proof
consider f being Function of I[01], Closed-Interval-TSpace(2*0+-1,2*1+-1)
such that
A1: f is being_homeomorphism and
A2: for r being Real st r in [.0,1.] holds f.r=2*r+-1
by Th36,TOPMETR:20;
A3: for r being Real st r in [.0,1.] holds f.r=2*r-1
proof
let r be Real;
assume r in [.0,1.];
hence f.r=2*r+-1 by A2
.=2*r-1;
end;
1 in [.0,1.] by XXREAL_1:1;
then
A4: f.1=2*1-1 by A3
.=1;
f.0=2*0-1 by A3,Lm1
.=-1;
hence thesis by A1,A3,A4;
end;
Lm5: now
reconsider B=[.-1,1.] as non empty Subset of R^1 by TOPMETR:17,XXREAL_1:1;
reconsider g=proj1 as Function of TOP-REAL 2,R^1 by TOPMETR:17;
let P be compact non empty Subset of TOP-REAL 2;
set K0=Lower_Arc(P);
reconsider g2=g|K0 as Function of (TOP-REAL 2)|K0,R^1 by PRE_TOPC:9;
A1: for p being Point of (TOP-REAL 2)|K0 holds g2.p=proj1.p
proof
let p be Point of (TOP-REAL 2)|K0;
p in the carrier of (TOP-REAL 2)|K0;
then p in K0 by PRE_TOPC:8;
hence thesis by FUNCT_1:49;
end;
assume
A2: P={p where p is Point of TOP-REAL 2: |.p.|=1};
then
A3: K0 c= P by Th33;
A4: dom g2=the carrier of ((TOP-REAL 2)|K0) by FUNCT_2:def 1;
then
A5: dom g2=K0 by PRE_TOPC:8;
rng g2 c= the carrier of Closed-Interval-TSpace(-1,1)
proof
let x be object;
assume x in rng g2;
then consider z being object such that
A6: z in dom g2 and
A7: x=g2.z by FUNCT_1:def 3;
z in P by A5,A3,A6;
then consider p being Point of TOP-REAL 2 such that
A8: p=z and
A9: |.p.|=1 by A2;
1^2=(p`1)^2+(p`2)^2 by A9,JGRAPH_3:1;
then 1-(p`1)^2>=0 by XREAL_1:63;
then -(1-(p`1)^2)<=0;
then (p`1)^2-1<=0;
then
A10: -1<=p`1 & p`1<=1 by SQUARE_1:43;
x=proj1.p by A1,A6,A7,A8
.=p`1 by PSCOMP_1:def 5;
then x in [.-1,1.] by A10,XXREAL_1:1;
hence thesis by TOPMETR:18;
end;
then reconsider
g3=g2 as Function of (TOP-REAL 2)|K0, Closed-Interval-TSpace(-1,1
) by A4,FUNCT_2:2;
dom g3=[#]((TOP-REAL 2)|K0) by FUNCT_2:def 1;
then
A11: dom g3=K0 by PRE_TOPC:def 5;
A12: for x,y being object st x in dom g3 & y in dom g3 & g3.x=g3.y holds x=y
proof
let x,y be object;
assume that
A13: x in dom g3 and
A14: y in dom g3 and
A15: g3.x=g3.y;
reconsider p2=y as Point of TOP-REAL 2 by A11,A14;
A16: g3.y=proj1.p2 by A1,A14
.=p2`1 by PSCOMP_1:def 5;
reconsider p1=x as Point of TOP-REAL 2 by A11,A13;
A17: g3.x=proj1.p1 by A1,A13
.=p1`1 by PSCOMP_1:def 5;
p2 in P by A3,A11,A14;
then ex p22 being Point of TOP-REAL 2 st p2=p22 & |.p22.|=1 by A2;
then
A18: 1^2= (p2`1)^2+(p2`2)^2 by JGRAPH_3:1;
p2 in {p3 where p3 is Point of TOP-REAL 2:p3 in P & p3`2<=0} by A2,A11,A14
,Th35;
then
A19: ex p44 being Point of TOP-REAL 2 st p2=p44 & p44 in P & p44`2<=0;
p1 in {p3 where p3 is Point of TOP-REAL 2:p3 in P & p3`2<=0} by A2,A11,A13
,Th35;
then
A20: ex p33 being Point of TOP-REAL 2 st p1=p33 & p33 in P & p33`2<=0;
p1 in P by A3,A11,A13;
then ex p11 being Point of TOP-REAL 2 st p1=p11 & |.p11.|=1 by A2;
then 1^2= (p1`1)^2+(p1`2)^2 by JGRAPH_3:1;
then (-(p1`2))^2 =(p2`2)^2 by A15,A17,A16,A18;
then -(p1`2)=sqrt((-(p2`2))^2) by A20,SQUARE_1:22;
then -(p1`2)=-(p2`2) by A19,SQUARE_1:22;
then p1=|[p2`1,p2`2]| by A15,A17,A16,EUCLID:53
.=p2 by EUCLID:53;
hence thesis;
end;
A21: [#](Closed-Interval-TSpace(-1,1)) c= rng g3
proof
let x be object;
assume x in [#](Closed-Interval-TSpace(-1,1));
then
A22: x in [.-1,1.] by TOPMETR:18;
then reconsider r=x as Real;
-1<=r & r<=1 by A22,XXREAL_1:1;
then 1^2>=r^2 by SQUARE_1:49;
then
A23: 1-r^2>=0 by XREAL_1:48;
set q=|[r,-sqrt(1-r^2)]|;
A24: q`2=-sqrt(1-r^2) by EUCLID:52;
|.q.|=sqrt((q`1)^2+(q`2)^2) by JGRAPH_3:1
.=sqrt(r^2+(q`2)^2) by EUCLID:52
.=sqrt(r^2+(-sqrt(1-r^2))^2) by EUCLID:52
.=sqrt(r^2+(sqrt(1-r^2))^2);
then |.q.|=sqrt(r^2+(1-r^2)) by A23,SQUARE_1:def 2
.=1;
then
A25: q in P by A2;
0<=sqrt(1-r^2) by A23,SQUARE_1:def 2;
then q in {p where p is Point of TOP-REAL 2:p in P & p`2<=0} by A25,A24;
then
A26: q in dom g3 by A2,A11,Th35;
then g3.q=proj1.q by A1
.=q`1 by PSCOMP_1:def 5
.=r by EUCLID:52;
hence thesis by A26,FUNCT_1:def 3;
end;
A27: Closed-Interval-TSpace(-1,1)=R^1|B by TOPMETR:19;
g2 is continuous by A1,JGRAPH_2:29;
hence proj1|K0 is continuous Function of (TOP-REAL 2)|K0,
Closed-Interval-TSpace(-1,1) & proj1|K0 is one-to-one & rng (proj1|K0)=[#](
Closed-Interval-TSpace(-1,1)) by A21,A27,A12,FUNCT_1:def 4,JGRAPH_1:45
,XBOOLE_0:def 10;
end;
Lm6: now
reconsider B=[.-1,1.] as non empty Subset of R^1 by TOPMETR:17,XXREAL_1:1;
reconsider g=proj1 as Function of TOP-REAL 2,R^1 by TOPMETR:17;
let P be compact non empty Subset of TOP-REAL 2;
set K0=Upper_Arc(P);
reconsider g2=g|K0 as Function of (TOP-REAL 2)|K0,R^1 by PRE_TOPC:9;
A1: for p being Point of (TOP-REAL 2)|K0 holds g2.p=proj1.p
proof
let p be Point of (TOP-REAL 2)|K0;
p in the carrier of (TOP-REAL 2)|K0;
then p in K0 by PRE_TOPC:8;
hence thesis by FUNCT_1:49;
end;
assume
A2: P={p where p is Point of TOP-REAL 2: |.p.|=1};
then
A3: K0 c= P by Th33;
A4: dom g2=the carrier of ((TOP-REAL 2)|K0) by FUNCT_2:def 1;
then
A5: dom g2=K0 by PRE_TOPC:8;
rng g2 c= the carrier of Closed-Interval-TSpace(-1,1)
proof
let x be object;
assume x in rng g2;
then consider z being object such that
A6: z in dom g2 and
A7: x=g2.z by FUNCT_1:def 3;
z in P by A5,A3,A6;
then consider p being Point of TOP-REAL 2 such that
A8: p=z and
A9: |.p.|=1 by A2;
1^2=(p`1)^2+(p`2)^2 by A9,JGRAPH_3:1;
then 1-(p`1)^2>=0 by XREAL_1:63;
then -(1-(p`1)^2)<=0;
then (p`1)^2-1<=0;
then
A10: -1<=p`1 & p`1<=1 by SQUARE_1:43;
x=proj1.p by A1,A6,A7,A8
.=p`1 by PSCOMP_1:def 5;
then x in [.-1,1.] by A10,XXREAL_1:1;
hence thesis by TOPMETR:18;
end;
then reconsider
g3=g2 as Function of (TOP-REAL 2)|K0, Closed-Interval-TSpace(-1,1
) by A4,FUNCT_2:2;
dom g3=[#]((TOP-REAL 2)|K0) by FUNCT_2:def 1;
then
A11: dom g3=K0 by PRE_TOPC:def 5;
A12: for x,y being object st x in dom g3 & y in dom g3 & g3.x=g3.y holds x=y
proof
let x,y be object;
assume that
A13: x in dom g3 and
A14: y in dom g3 and
A15: g3.x=g3.y;
reconsider p2=y as Point of TOP-REAL 2 by A11,A14;
A16: g3.y=proj1.p2 by A1,A14
.=p2`1 by PSCOMP_1:def 5;
reconsider p1=x as Point of TOP-REAL 2 by A11,A13;
A17: g3.x=proj1.p1 by A1,A13
.=p1`1 by PSCOMP_1:def 5;
p2 in P by A3,A11,A14;
then ex p22 being Point of TOP-REAL 2 st p2=p22 & |.p22.|=1 by A2;
then
A18: 1^2= (p2`1)^2+(p2`2)^2 by JGRAPH_3:1;
p2 in {p3 where p3 is Point of TOP-REAL 2:p3 in P & p3`2>=0} by A2,A11,A14
,Th34;
then
A19: ex p44 being Point of TOP-REAL 2 st p2=p44 & p44 in P & p44`2>=0;
p1 in P by A3,A11,A13;
then ex p11 being Point of TOP-REAL 2 st p1=p11 & |.p11.|=1 by A2;
then
A20: 1^2= (p1`1)^2+(p1`2)^2 by JGRAPH_3:1;
p1 in {p3 where p3 is Point of TOP-REAL 2:p3 in P & p3`2>=0} by A2,A11,A13
,Th34;
then ex p33 being Point of TOP-REAL 2 st p1=p33 & p33 in P & p33`2>=0;
then p1`2=sqrt(((p2`2))^2) by A15,A17,A16,A18,A20,SQUARE_1:22;
then (p1`2)=(p2`2) by A19,SQUARE_1:22;
then p1=|[p2`1,p2`2]| by A15,A17,A16,EUCLID:53
.=p2 by EUCLID:53;
hence thesis;
end;
A21: [#](Closed-Interval-TSpace(-1,1)) c= rng g3
proof
let x be object;
assume x in [#](Closed-Interval-TSpace(-1,1));
then
A22: x in [.-1,1.] by TOPMETR:18;
then reconsider r=x as Real;
-1<=r & r<=1 by A22,XXREAL_1:1;
then 1^2>=r^2 by SQUARE_1:49;
then
A23: 1-r^2>=0 by XREAL_1:48;
set q=|[r,sqrt(1-r^2)]|;
A24: q`2=sqrt(1-r^2) by EUCLID:52;
|.q.|=sqrt((q`1)^2+(q`2)^2) by JGRAPH_3:1
.=sqrt(r^2+(q`2)^2) by EUCLID:52
.=sqrt(r^2+(sqrt(1-r^2))^2) by EUCLID:52;
then |.q.|=sqrt(r^2+(1-r^2)) by A23,SQUARE_1:def 2
.=1;
then
A25: q in P by A2;
0<=sqrt(1-r^2) by A23,SQUARE_1:def 2;
then q in {p where p is Point of TOP-REAL 2:p in P & p`2>=0} by A25,A24;
then
A26: q in dom g3 by A2,A11,Th34;
then g3.q=proj1.q by A1
.=q`1 by PSCOMP_1:def 5
.=r by EUCLID:52;
hence thesis by A26,FUNCT_1:def 3;
end;
A27: Closed-Interval-TSpace(-1,1)=R^1|B by TOPMETR:19;
g2 is continuous by A1,JGRAPH_2:29;
hence proj1|K0 is continuous Function of (TOP-REAL 2)|K0,
Closed-Interval-TSpace(-1,1) & proj1|K0 is one-to-one & rng (proj1|K0)=[#](
Closed-Interval-TSpace(-1,1)) by A21,A27,A12,FUNCT_1:def 4,JGRAPH_1:45
,XBOOLE_0:def 10;
end;
theorem Th40:
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)
proof
reconsider g=proj1 as Function of TOP-REAL 2,R^1 by TOPMETR:17;
let P be compact non empty Subset of TOP-REAL 2;
set P4=Lower_Arc(P);
set K0=Lower_Arc(P);
reconsider g2=g|K0 as Function of (TOP-REAL 2)|K0,R^1 by PRE_TOPC:9;
A1: for p being Point of (TOP-REAL 2)|K0 holds g2.p=proj1.p
proof
let p be Point of (TOP-REAL 2)|K0;
p in the carrier of (TOP-REAL 2)|K0;
then p in K0 by PRE_TOPC:8;
hence thesis by FUNCT_1:49;
end;
assume
A2: P={p where p is Point of TOP-REAL 2: |.p.|=1};
then reconsider g3=g2 as continuous Function of (TOP-REAL 2)|K0,
Closed-Interval-TSpace(-1,1) by Lm5;
A3: rng g3=[#](Closed-Interval-TSpace(-1,1)) by A2,Lm5;
A4: P is being_simple_closed_curve by A2,JGRAPH_3:26;
then
A5: Upper_Arc(P) /\ P4={W-min(P),E-max(P)} by JORDAN6:def 9;
E-max(P) in {W-min(P),E-max(P)} by TARSKI:def 2;
then
A6: E-max(P) in Lower_Arc(P) by A5,XBOOLE_0:def 4;
Closed-Interval-TSpace(-1,1) =TopSpaceMetr(Closed-Interval-MSpace(-1,1)
) by TOPMETR:def 7;
then
A7: Closed-Interval-TSpace(-1,1) is T_2 by PCOMPS_1:34;
A8: g3 is one-to-one by A2,Lm5;
A9: dom g3=[#]((TOP-REAL 2)|K0) by FUNCT_2:def 1;
then
A10: dom g3=K0 by PRE_TOPC:def 5;
A11: g3 is onto by A3,FUNCT_2:def 3;
A12: for q be Point of TOP-REAL 2 st q in Lower_Arc(P) holds (g3/").(q`1)=q
proof
reconsider g4=g3 as Function;
let q be Point of TOP-REAL 2;
A13: q in dom g4 implies q = (g4").(g4.q) & q = (g4"*g4).q by A8,FUNCT_1:34;
assume
A14: q in Lower_Arc(P);
then g3.q=proj1.q by A1,A10
.=q`1 by PSCOMP_1:def 5;
hence thesis by A11,A9,A8,A14,A13,PRE_TOPC:def 5,TOPS_2:def 4;
end;
W-min(P) in {W-min(P),E-max(P)} by TARSKI:def 2;
then
A15: W-min(P) in Lower_Arc(P) by A5,XBOOLE_0:def 4;
A16: E-max(P)=|[1,0]| by A2,Th30;
A17: W-min(P)=|[-1,0]| by A2,Th29;
Lower_Arc(P) is_an_arc_of E-max(P),W-min(P) by A4,JORDAN6:def 9;
then K0 is non empty compact by JORDAN5A:1;
then
A18: g3/" is being_homeomorphism by A3,A8,A7,COMPTS_1:17,TOPS_2:56;
A19: g3/".1=g3/".((|[1,0]|)`1) by EUCLID:52
.=E-max(P) by A6,A12,A16;
g3/".(-1)=g3/".((|[-1,0]|)`1) by EUCLID:52
.=W-min(P) by A15,A12,A17;
hence thesis by A18,A12,A19;
end;
theorem Th41:
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)
proof
reconsider g=proj1 as Function of TOP-REAL 2,R^1 by TOPMETR:17;
let P be compact non empty Subset of TOP-REAL 2;
set P4=Lower_Arc(P);
set K0=Upper_Arc(P);
reconsider g2=g|K0 as Function of (TOP-REAL 2)|K0,R^1 by PRE_TOPC:9;
A1: for p being Point of (TOP-REAL 2)|K0 holds g2.p=proj1.p
proof
let p be Point of (TOP-REAL 2)|K0;
p in the carrier of (TOP-REAL 2)|K0;
then p in K0 by PRE_TOPC:8;
hence thesis by FUNCT_1:49;
end;
assume
A2: P={p where p is Point of TOP-REAL 2: |.p.|=1};
then reconsider g3=g2 as continuous Function of (TOP-REAL 2)|K0,
Closed-Interval-TSpace(-1,1) by Lm6;
A3: rng g3=[#](Closed-Interval-TSpace(-1,1)) by A2,Lm6;
A4: P is being_simple_closed_curve by A2,JGRAPH_3:26;
then
A5: Upper_Arc(P) /\ P4={W-min(P),E-max(P)} by JORDAN6:def 9;
E-max(P) in {W-min(P),E-max(P)} by TARSKI:def 2;
then
A6: E-max(P) in Upper_Arc(P) by A5,XBOOLE_0:def 4;
Closed-Interval-TSpace(-1,1) =TopSpaceMetr(Closed-Interval-MSpace(-1,1)
) by TOPMETR:def 7;
then
A7: Closed-Interval-TSpace(-1,1) is T_2 by PCOMPS_1:34;
A8: g3 is one-to-one by A2,Lm6;
A9: dom g3=[#]((TOP-REAL 2)|K0) by FUNCT_2:def 1;
then
A10: dom g3=K0 by PRE_TOPC:def 5;
A11: g3 is onto by A3,FUNCT_2:def 3;
A12: for q be Point of TOP-REAL 2 st q in Upper_Arc(P) holds (g3/").(q`1)=q
proof
reconsider g4=g3 as Function;
let q be Point of TOP-REAL 2;
A13: q in dom g4 implies q = (g4").(g4.q) & q = (g4"*g4).q by A8,FUNCT_1:34;
assume
A14: q in Upper_Arc(P);
then g3.q=proj1.q by A1,A10
.=q`1 by PSCOMP_1:def 5;
hence thesis by A11,A9,A8,A14,A13,PRE_TOPC:def 5,TOPS_2:def 4;
end;
W-min(P) in {W-min(P),E-max(P)} by TARSKI:def 2;
then
A15: W-min(P) in Upper_Arc(P) by A5,XBOOLE_0:def 4;
A16: E-max(P)=|[1,0]| by A2,Th30;
A17: W-min(P)=|[-1,0]| by A2,Th29;
Upper_Arc(P) is_an_arc_of W-min(P),E-max(P) by A4,JORDAN6:def 8;
then K0 is non empty compact by JORDAN5A:1;
then
A18: g3/" is being_homeomorphism by A3,A8,A7,COMPTS_1:17,TOPS_2:56;
A19: g3/".(1)=g3/".((|[1,0]|)`1) by EUCLID:52
.=E-max(P) by A6,A12,A16;
g3/".(-1)=g3/".((|[-1,0]|)`1) by EUCLID:52
.=W-min(P) by A15,A12,A17;
hence thesis by A18,A12,A19;
end;
theorem Th42:
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
r1<r2 iff q1`1>q2`1)& f.0 = E-max(P) & f.1 = W-min(P)
proof
let P be compact non empty Subset of TOP-REAL 2;
reconsider T= (TOP-REAL 2)|Lower_Arc(P) as non empty TopSpace;
consider g being Function of I[01],Closed-Interval-TSpace(-1,1) such that
A1: g is being_homeomorphism and
A2: for r being Real st r in [.0,1.] holds g.r=(-2)*r+1 and
A3: g.0=1 and
A4: g.1=-1 by Th38;
assume
A5: P={p where p is Point of TOP-REAL 2: |.p.|=1};
then consider
f1 being Function of Closed-Interval-TSpace(-1,1),(TOP-REAL 2)|
Lower_Arc(P) such that
A6: f1 is being_homeomorphism and
A7: for q being Point of TOP-REAL 2 st q in Lower_Arc(P) holds f1.(q`1) =q and
A8: f1.(-1)=W-min(P) and
A9: f1.1=E-max(P) by Th40;
reconsider h=f1*g as Function of I[01],(TOP-REAL 2)|Lower_Arc(P);
A10: dom h=[.0,1.] by BORSUK_1:40,FUNCT_2:def 1;
then 0 in dom h by XXREAL_1:1;
then
A11: h.0=E-max(P) by A9,A3,FUNCT_1:12;
A12: for q1,q2 being Point of TOP-REAL 2, r1,r2 being Real st h.r1=q1 & h.r2
=q2 & r1 in [.0,1.] & r2 in [.0,1.] holds r1<r2 iff q1`1>q2`1
proof
let q1,q2 be Point of TOP-REAL 2,r1,r2 be Real;
assume that
A13: h.r1=q1 and
A14: h.r2=q2 and
A15: r1 in [.0,1.] and
A16: r2 in [.0,1.];
A17: now
set s1=(-2)*r2+1,s2=(-2)*r1+1;
set p1=|[s1,-sqrt(1-s1^2)]|,p2=|[s2,-sqrt(1-s2^2)]|;
A18: (|[s1,-sqrt(1-s1^2)]|)`2=-sqrt(1-s1^2) by EUCLID:52;
r2<=1 by A16,XXREAL_1:1;
then (-2)*r2>=(-2)*1 by XREAL_1:65;
then (-2)*r2+1>=(-2)*1+1 by XREAL_1:7;
then
A19: -1<=s1;
r2>=0 by A16,XXREAL_1:1;
then (-2)*r2+1<=(-2)*0+1 by XREAL_1:7;
then s1^2<=1^2 by A19,SQUARE_1:49;
then
A20: 1-s1^2>=0 by XREAL_1:48;
then
A21: sqrt(1-s1^2)>=0 by SQUARE_1:def 2;
|.p1.|=sqrt((p1`1)^2+(p1`2)^2) by JGRAPH_3:1
.=sqrt((s1)^2+(sqrt(1-s1^2))^2) by A18,EUCLID:52
.=sqrt((s1)^2+(1-s1^2)) by A20,SQUARE_1:def 2
.=1;
then p1 in P by A5;
then p1 in {p3 where p3 is Point of TOP-REAL 2: p3 in P & p3`2<=0} by A18
,A21;
then
A22: (|[s1,-sqrt(1-s1^2)]|)`1=s1 & |[s1,-sqrt(1-s1^2)]| in Lower_Arc(P)
by A5,Th35,EUCLID:52;
g.r2=(-2)*r2+1 & dom h=[.0,1.] by A2,A16,BORSUK_1:40,FUNCT_2:def 1;
then h.r2=f1.s1 by A16,FUNCT_1:12
.=p1 by A7,A22;
then
A23: q2`1=s1 by A14,EUCLID:52;
A24: (|[s2,-sqrt(1-s2^2)]|)`1=s2 by EUCLID:52;
r1<=1 by A15,XXREAL_1:1;
then (-2)*r1>=(-2)*1 by XREAL_1:65;
then (-2)*r1+1>=(-2)*1+1 by XREAL_1:7;
then
A25: -1<=s2;
r1>=0 by A15,XXREAL_1:1;
then (-2)*r1+1<=(-2)*0+1 by XREAL_1:7;
then s2^2<=1^2 by A25,SQUARE_1:49;
then
A26: 1-s2^2>=0 by XREAL_1:48;
then
A27: sqrt(1-s2^2)>=0 by SQUARE_1:def 2;
assume r2<r1;
then
A28: (-2)*r2 > (-2)*r1 by XREAL_1:69;
A29: (|[s2,-sqrt(1-s2^2)]|)`2=-sqrt(1-s2^2) by EUCLID:52;
|.p2.|=sqrt((p2`1)^2+(p2`2)^2) by JGRAPH_3:1
.=sqrt((s2)^2+(sqrt(1-s2^2))^2) by A29,EUCLID:52
.=sqrt((s2)^2+(1-s2^2)) by A26,SQUARE_1:def 2
.=1;
then p2 in P by A5;
then p2 in {p3 where p3 is Point of TOP-REAL 2: p3 in P & p3`2<=0} by A29
,A27;
then
A30: |[s2,-sqrt(1-s2^2)]| in Lower_Arc(P) by A5,Th35;
g.r1=(-2)*r1+1 & dom h=[.0,1.] by A2,A15,BORSUK_1:40,FUNCT_2:def 1;
then h.r1=f1.s2 by A15,FUNCT_1:12
.=p2 by A7,A24,A30;
hence q2`1>q1`1 by A13,A28,A23,A24,XREAL_1:8;
end;
A31: now
assume
A32: q1`1>q2`1;
now
assume
A33: r1>=r2;
now
per cases by A33,XXREAL_0:1;
case
r1>r2;
hence contradiction by A17,A32;
end;
case
r1=r2;
hence contradiction by A13,A14,A32;
end;
end;
hence contradiction;
end;
hence r1<r2;
end;
now
assume r1<r2;
then (-2)*r1 > (-2)*r2 by XREAL_1:69;
then
A34: (-2)*r1 +1 > (-2)*r2 +1 by XREAL_1:8;
set s1=(-2)*r1+1,s2=(-2)*r2+1;
set p1=|[s1,-sqrt(1-s1^2)]|,p2=|[s2,-sqrt(1-s2^2)]|;
A35: (|[s1,-sqrt(1-s1^2)]|)`2=-sqrt(1-s1^2) by EUCLID:52;
r1<=1 by A15,XXREAL_1:1;
then (-2)*r1>=(-2)*1 by XREAL_1:65;
then (-2)*r1+1>=(-2)*1+1 by XREAL_1:7;
then
A36: -1<=s1;
r1>=0 by A15,XXREAL_1:1;
then (-2)*r1+1<=(-2)*0+1 by XREAL_1:7;
then s1^2<=1^2 by A36,SQUARE_1:49;
then
A37: 1-s1^2>=0 by XREAL_1:48;
then
A38: sqrt(1-s1^2)>=0 by SQUARE_1:def 2;
|.p1.|=sqrt((p1`1)^2+(p1`2)^2) by JGRAPH_3:1
.=sqrt((s1)^2+(sqrt(1-s1^2))^2) by A35,EUCLID:52
.=sqrt((s1)^2+(1-s1^2)) by A37,SQUARE_1:def 2
.=1;
then p1 in P by A5;
then p1 in {p3 where p3 is Point of TOP-REAL 2: p3 in P & p3`2<=0} by A35
,A38;
then
A39: (|[s1,-sqrt(1-s1^2)]|)`1=s1 & |[s1,-sqrt(1-s1^2)]| in Lower_Arc(P)
by A5,Th35,EUCLID:52;
g.r1=(-2)*r1+1 & dom h=[.0,1.] by A2,A15,BORSUK_1:40,FUNCT_2:def 1;
then h.r1=f1.s1 by A15,FUNCT_1:12
.=p1 by A7,A39;
then
A40: q1`1=s1 by A13,EUCLID:52;
A41: (|[s2,-sqrt(1-s2^2)]|)`2=-sqrt(1-s2^2) by EUCLID:52;
r2<=1 by A16,XXREAL_1:1;
then (-2)*r2>=(-2)*1 by XREAL_1:65;
then (-2)*r2+1>=(-2)*1+1 by XREAL_1:7;
then
A42: -1<=s2;
r2>=0 by A16,XXREAL_1:1;
then (-2)*r2+1<=(-2)*0+1 by XREAL_1:7;
then s2^2<=1^2 by A42,SQUARE_1:49;
then
A43: 1-s2^2>=0 by XREAL_1:48;
then
A44: sqrt(1-s2^2)>=0 by SQUARE_1:def 2;
|.p2.|=sqrt((p2`1)^2+(p2`2)^2) by JGRAPH_3:1
.=sqrt((s2)^2+(sqrt(1-s2^2))^2) by A41,EUCLID:52
.=sqrt((s2)^2+(1-s2^2)) by A43,SQUARE_1:def 2
.=1;
then p2 in P by A5;
then p2 in {p3 where p3 is Point of TOP-REAL 2: p3 in P & p3`2<=0} by A41
,A44;
then
A45: (|[s2,-sqrt(1-s2^2)]|)`1=s2 & |[s2,-sqrt(1-s2^2)]| in Lower_Arc(P)
by A5,Th35,EUCLID:52;
g.r2=(-2)*r2+1 & dom h=[.0,1.] by A2,A16,BORSUK_1:40,FUNCT_2:def 1;
then h.r2=f1.s2 by A16,FUNCT_1:12
.=p2 by A7,A45;
hence q1`1>q2`1 by A14,A34,A40,EUCLID:52;
end;
hence thesis by A31;
end;
1 in dom h by A10,XXREAL_1:1;
then
A46: h.1=W-min(P) by A8,A4,FUNCT_1:12;
reconsider f2=f1 as Function of Closed-Interval-TSpace(-1,1),T;
f2*g is being_homeomorphism by A6,A1,TOPS_2:57;
hence thesis by A12,A11,A46;
end;
theorem Th43:
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
r1<r2 iff q1`1<q2`1)& f.0 = W-min(P) & f.1 = E-max(P)
proof
let P be compact non empty Subset of TOP-REAL 2;
reconsider T= (TOP-REAL 2)|Upper_Arc(P) as non empty TopSpace;
consider g being Function of I[01],Closed-Interval-TSpace(-1,1) such that
A1: g is being_homeomorphism and
A2: for r being Real st r in [.0,1.] holds g.r=2*r-1 and
A3: g.0=-1 and
A4: g.1=1 by Th39;
assume
A5: P={p where p is Point of TOP-REAL 2: |.p.|=1};
then consider
f1 being Function of Closed-Interval-TSpace(-1,1),(TOP-REAL 2)|
Upper_Arc(P) such that
A6: f1 is being_homeomorphism and
A7: for q being Point of TOP-REAL 2 st q in Upper_Arc(P) holds f1.(q`1) =q and
A8: f1.(-1)=W-min(P) and
A9: f1.1=E-max(P) by Th41;
reconsider h=f1*g as Function of I[01],(TOP-REAL 2)|Upper_Arc(P);
A10: dom h=[.0,1.] by BORSUK_1:40,FUNCT_2:def 1;
then 0 in dom h by XXREAL_1:1;
then
A11: h.0=W-min(P) by A8,A3,FUNCT_1:12;
A12: for q1,q2 being Point of TOP-REAL 2, r1,r2 being Real st h.r1=q1 & h.r2
=q2 & r1 in [.0,1.] & r2 in [.0,1.] holds r1<r2 iff q1`1<q2`1
proof
let q1,q2 be Point of TOP-REAL 2,r1,r2 be Real;
assume that
A13: h.r1=q1 and
A14: h.r2=q2 and
A15: r1 in [.0,1.] and
A16: r2 in [.0,1.];
A17: now
r2<=1 by A16,XXREAL_1:1;
then 2*r2<=2*1 by XREAL_1:64;
then
A18: 2*r2-1<=2*1-1 by XREAL_1:9;
r1>=0 by A15,XXREAL_1:1;
then
A19: 2*r1-1>=2*0-1 by XREAL_1:9;
set s1=2*r1-1,s2=2*r2-1;
set p1=|[s1,sqrt(1-s1^2)]|,p2=|[s2,sqrt(1-s2^2)]|;
A20: (|[s1,sqrt(1-s1^2)]|)`1=s1 by EUCLID:52;
r2>=0 by A16,XXREAL_1:1;
then
A21: 2*r2-1>=2*0-1 by XREAL_1:9;
2*0-1=-1;
then s2^2<=1^2 by A18,A21,SQUARE_1:49;
then
A22: 1-s2^2>=0 by XREAL_1:48;
then
A23: sqrt(1-s2^2)>=0 by SQUARE_1:def 2;
r1<=1 by A15,XXREAL_1:1;
then 2*r1<=2*1 by XREAL_1:64;
then
A24: 2*r1-1<=2*1-1 by XREAL_1:9;
assume r1>r2;
then
A25: 2*r1 > 2*r2 by XREAL_1:68;
2*0-1=-1;
then s1^2<=1^2 by A24,A19,SQUARE_1:49;
then
A26: 1-s1^2>=0 by XREAL_1:48;
then
A27: sqrt(1-s1^2)>=0 by SQUARE_1:def 2;
A28: (|[s1,sqrt(1-s1^2)]|)`2=sqrt(1-s1^2) by EUCLID:52;
then |.p1.|=sqrt((s1)^2+(sqrt(1-s1^2))^2) by A20,JGRAPH_3:1
.=sqrt((s1)^2+(1-s1^2)) by A26,SQUARE_1:def 2
.=1;
then p1 in P by A5;
then p1 in {p3 where p3 is Point of TOP-REAL 2: p3 in P & p3`2>=0} by A28
,A27;
then
A29: |[s1,sqrt(1-s1^2)]| in Upper_Arc(P) by A5,Th34;
g.r1=2*r1-1 & dom h=[.0,1.] by A2,A15,BORSUK_1:40,FUNCT_2:def 1;
then h.r1=f1.s1 by A15,FUNCT_1:12
.=p1 by A7,A20,A29;
then
A30: q1`1=s1 by A13,EUCLID:52;
A31: (|[s2,sqrt(1-s2^2)]|)`1=s2 by EUCLID:52;
A32: (|[s2,sqrt(1-s2^2)]|)`2=sqrt(1-s2^2) by EUCLID:52;
then |.p2.|=sqrt((s2)^2+(sqrt(1-s2^2))^2) by A31,JGRAPH_3:1
.=sqrt((s2)^2+(1-s2^2)) by A22,SQUARE_1:def 2
.=1;
then p2 in P by A5;
then p2 in {p3 where p3 is Point of TOP-REAL 2: p3 in P & p3`2>=0} by A32
,A23;
then
A33: |[s2,sqrt(1-s2^2)]| in Upper_Arc(P) by A5,Th34;
g.r2=2*r2-1 & dom h=[.0,1.] by A2,A16,BORSUK_1:40,FUNCT_2:def 1;
then h.r2=f1.s2 by A16,FUNCT_1:12
.=p2 by A7,A31,A33;
hence q1`1>q2`1 by A14,A25,A30,A31,XREAL_1:14;
end;
A34: now
assume
A35: q1`1<q2`1;
now
assume
A36: r1>=r2;
now
per cases by A36,XXREAL_0:1;
case
r1>r2;
hence contradiction by A17,A35;
end;
case
r1=r2;
hence contradiction by A13,A14,A35;
end;
end;
hence contradiction;
end;
hence r1<r2;
end;
now
assume r2>r1;
then
A37: 2*r2 > 2*r1 by XREAL_1:68;
set s1=2*r2-1,s2=2*r1-1;
set p1=|[s1,sqrt(1-s1^2)]|,p2=|[s2,sqrt(1-s2^2)]|;
A38: (|[s1,sqrt(1-s1^2)]|)`1=s1 by EUCLID:52;
r2>=0 by A16,XXREAL_1:1;
then 2*r2-1>=2*0-1 by XREAL_1:9;
then
A39: -1<=s1;
r2<=1 by A16,XXREAL_1:1;
then 2*r2<=2*1 by XREAL_1:64;
then 2*r2-1<=2*1-1 by XREAL_1:9;
then s1^2<=1^2 by A39,SQUARE_1:49;
then
A40: 1-s1^2>=0 by XREAL_1:48;
then
A41: sqrt(1-s1^2)>=0 by SQUARE_1:def 2;
A42: (|[s1,sqrt(1-s1^2)]|)`2=sqrt(1-s1^2) by EUCLID:52;
then |.p1.|=sqrt((s1)^2+(sqrt(1-s1^2))^2) by A38,JGRAPH_3:1
.=sqrt((s1)^2+(1-s1^2)) by A40,SQUARE_1:def 2
.=1;
then p1 in P by A5;
then p1 in {p3 where p3 is Point of TOP-REAL 2: p3 in P & p3`2>=0} by A42
,A41;
then
A43: |[s1,sqrt(1-s1^2)]| in Upper_Arc(P) by A5,Th34;
g.r2=2*r2-1 & dom h=[.0,1.] by A2,A16,BORSUK_1:40,FUNCT_2:def 1;
then h.r2=f1.s1 by A16,FUNCT_1:12
.=p1 by A7,A38,A43;
then
A44: q2`1=s1 by A14,EUCLID:52;
A45: (|[s2,sqrt(1-s2^2)]|)`1=s2 by EUCLID:52;
r1>=0 by A15,XXREAL_1:1;
then 2*r1-1>=2*0-1 by XREAL_1:9;
then
A46: -1<=s2;
r1<=1 by A15,XXREAL_1:1;
then 2*r1<=2*1 by XREAL_1:64;
then 2*r1-1<=2*1-1 by XREAL_1:9;
then s2^2<=1^2 by A46,SQUARE_1:49;
then
A47: 1-s2^2>=0 by XREAL_1:48;
then
A48: sqrt(1-s2^2)>=0 by SQUARE_1:def 2;
A49: (|[s2,sqrt(1-s2^2)]|)`2=sqrt(1-s2^2) by EUCLID:52;
then |.p2.|=sqrt((s2)^2+(sqrt(1-s2^2))^2) by A45,JGRAPH_3:1
.=sqrt((s2)^2+(1-s2^2)) by A47,SQUARE_1:def 2
.=1;
then p2 in P by A5;
then p2 in {p3 where p3 is Point of TOP-REAL 2: p3 in P & p3`2>=0} by A49
,A48;
then
A50: |[s2,sqrt(1-s2^2)]| in Upper_Arc(P) by A5,Th34;
g.r1=2*r1-1 & dom h=[.0,1.] by A2,A15,BORSUK_1:40,FUNCT_2:def 1;
then h.r1=f1.s2 by A15,FUNCT_1:12
.=p2 by A7,A45,A50;
hence q2`1>q1`1 by A13,A37,A44,A45,XREAL_1:14;
end;
hence thesis by A34;
end;
1 in dom h by A10,XXREAL_1:1;
then
A51: h.1=E-max(P) by A9,A4,FUNCT_1:12;
reconsider f2=f1 as Function of Closed-Interval-TSpace(-1,1),T;
f2*g is being_homeomorphism by A6,A1,TOPS_2:57;
hence thesis by A12,A11,A51;
end;
theorem Th44:
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)
proof
let p1,p2 be Point of TOP-REAL 2, P be compact non empty Subset of TOP-REAL
2;
assume that
A1: P={p where p is Point of TOP-REAL 2: |.p.|=1} and
A2: p2 in Upper_Arc(P) and
A3: LE p1,p2,P;
set P4b=Lower_Arc(P);
A4: p1 in Upper_Arc(P) & p2 in Lower_Arc(P)& not p2=W-min(P) or p1 in
Upper_Arc(P) & p2 in Upper_Arc(P) & LE p1,p2,Upper_Arc(P),W-min(P),E-max(P) or
p1 in Lower_Arc(P) & p2 in Lower_Arc(P)& not p2=W-min(P) & LE p1,p2,Lower_Arc(P
),E-max(P),W-min(P) by A3;
A5: P is being_simple_closed_curve by A1,JGRAPH_3:26;
then
A6: Lower_Arc(P) is_an_arc_of E-max(P),W-min(P) by JORDAN6:def 9;
A7: Upper_Arc(P) /\ P4b={W-min(P),E-max(P)} by A5,JORDAN6:def 9;
then E-max(P) in Upper_Arc(P) /\ P4b by TARSKI:def 2;
then
A8: E-max(P) in Upper_Arc(P) by XBOOLE_0:def 4;
now
assume
A9: not p1 in Upper_Arc(P);
then p2 in Upper_Arc(P) /\ P4b by A2,A4,XBOOLE_0:def 4;
then
A10: p2=E-max(P) by A7,A4,A9,TARSKI:def 2;
then LE p2,p1,Lower_Arc(P),E-max(P),W-min(P) by A6,A4,A9,JORDAN5C:10;
hence contradiction by A6,A8,A4,A9,A10,JORDAN5C:12;
end;
hence thesis;
end;
theorem Th45:
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 & p1`2<0 & p2`2<0 holds p1`1>p2`1 & p1`2<p2`2
proof
let p1,p2 be Point of TOP-REAL 2, P be compact non empty Subset of TOP-REAL
2;
assume that
A1: P={p where p is Point of TOP-REAL 2: |.p.|=1} and
A2: LE p1,p2,P and
A3: p1<>p2 and
A4: p1`1<0 and
A5: p1`2<0 and
A6: p2`2<0;
consider f being Function of I[01],(TOP-REAL 2)|Lower_Arc(P) such that
A7: f is being_homeomorphism and
A8: 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 and
A9: f.0=E-max(P) & f.1=W-min(P) by A1,Th42;
A10: rng f=[#]((TOP-REAL 2)|Lower_Arc(P)) by A7,TOPS_2:def 5
.=Lower_Arc(P) by PRE_TOPC:def 5;
A11: Upper_Arc(P)={p where p is Point of TOP-REAL 2:p in P & p`2>=0} by A1,Th34
;
A12: now
assume p1 in Upper_Arc(P);
then ex p being Point of TOP-REAL 2 st p1=p & p in P & p`2>= 0 by A11;
hence contradiction by A5;
end;
then
A13: LE p1,p2,Lower_Arc(P),E-max(P),W-min(P) by A2;
p2 in Lower_Arc(P) by A2,A12;
then consider x2 being object such that
A14: x2 in dom f and
A15: p2=f.x2 by A10,FUNCT_1:def 3;
A16: dom f=[#](I[01]) by A7,TOPS_2:def 5
.=[.0,1.] by BORSUK_1:40;
reconsider r22=x2 as Real by A14;
A17: 0<=r22 & r22<=1 by A14,A16,XXREAL_1:1;
p1 in Lower_Arc(P) by A2,A12;
then consider x1 being object such that
A18: x1 in dom f and
A19: p1=f.x1 by A10,FUNCT_1:def 3;
reconsider r11=x1 as Real by A18;
r11<=1 by A18,A16,XXREAL_1:1;
then
A20: r11<=r22 by A13,A7,A9,A19,A15,A17,JORDAN5C:def 3;
A21: P is being_simple_closed_curve by A1,JGRAPH_3:26;
then p1 in P by A2,JORDAN7:5;
then ex p3 being Point of TOP-REAL 2 st p3=p1 & |.p3.|=1 by A1;
then 1^2=(p1`1)^2+(p1`2)^2 by JGRAPH_3:1;
then 1^2-(p1`1)^2=(-(p1`2))^2;
then -(p1`2)=sqrt(1^2-(-(p1`1))^2) by A5,SQUARE_1:22;
then
A22: (p1`2)=-sqrt(1^2-(-(p1`1))^2);
p2 in P by A2,A21,JORDAN7:5;
then ex p4 being Point of TOP-REAL 2 st p4=p2 & |.p4.|=1 by A1;
then
A23: 1^2=(p2`1)^2+(p2`2)^2 by JGRAPH_3:1;
then 1^2-(p2`1)^2=(-(p2`2))^2;
then -(p2`2)=sqrt(1^2-(-(p2`1))^2) by A6,SQUARE_1:22;
then
A24: (p2`2)=-sqrt(1^2-(-(p2`1))^2);
A25: r11<r22 iff p1`1>p2`1 by A8,A18,A19,A14,A15,A16;
then -(p1`1)< -(p2`1) by A3,A19,A15,A20,XREAL_1:24,XXREAL_0:1;
then (-(p1`1))^2 < (-(p2`1))^2 by A4,SQUARE_1:16;
then 1^2- (-(p1`1))^2 > 1^2-(-(p2`1))^2 by XREAL_1:15;
then sqrt(1^2- (-(p1`1))^2) > sqrt(1^2-(-(p2`1))^2) by A23,SQUARE_1:27
,XREAL_1:63;
hence thesis by A19,A15,A25,A20,A22,A24,XREAL_1:24,XXREAL_0:1;
end;
theorem Th46:
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 & p2`1<0 & p1`2>=0 & p2`2>=0 holds p1`1<p2`1 & p1`2<p2`2
proof
let p1,p2 be Point of TOP-REAL 2, P be compact non empty Subset of TOP-REAL
2;
assume that
A1: P={p where p is Point of TOP-REAL 2: |.p.|=1} and
A2: LE p1,p2,P and
A3: p1<>p2 and
A4: p2`1<0 and
A5: p1`2>=0 and
A6: p2`2>=0;
set P4=Lower_Arc(P);
A7: P is being_simple_closed_curve by A1,JGRAPH_3:26;
then
A8: Upper_Arc(P) /\ P4={W-min(P),E-max(P)} by JORDAN6:def 9;
A9: p1 in P by A2,A7,JORDAN7:5;
A10: now
assume p2=W-min(P);
then LE p2,p1,P by A7,A9,JORDAN7:3;
hence contradiction by A1,A2,A3,JGRAPH_3:26,JORDAN6:57;
end;
A11: p2 in P by A2,A7,JORDAN7:5;
then ex p4 being Point of TOP-REAL 2 st p4=p2 & |.p4.|=1 by A1;
then 1^2=(p2`1)^2+(p2`2)^2 by JGRAPH_3:1;
then
A12: (p2`2)=sqrt(1^2-(-(p2`1))^2) by A6,SQUARE_1:22;
A13: Upper_Arc(P)={p where p is Point of TOP-REAL 2:p in P & p`2>=0} by A1,Th34
;
A14: now
assume
A15: p2 in Lower_Arc(P);
p2 in Upper_Arc(P) by A6,A11,A13;
then p2 in {W-min(P),E-max(P)} by A8,A15,XBOOLE_0:def 4;
then
A16: p2=W-min(P) or p2=E-max(P) by TARSKI:def 2;
E-max(P)=|[1,0]| by A1,Th30;
hence contradiction by A4,A10,A16,EUCLID:52;
end;
then
A17: LE p1,p2,Upper_Arc(P),W-min(P),E-max(P) by A2;
A18: ex p3 being Point of TOP-REAL 2 st p3=p1 & |.p3.|=1 by A1,A9;
then 1^2=(p1`1)^2+(p1`2)^2 by JGRAPH_3:1;
then
A19: (p1`2)=sqrt(1^2-(-(p1`1))^2) by A5,SQUARE_1:22;
1^2=(p1`1)^2+(p1`2)^2 by A18,JGRAPH_3:1;
then
A20: 1^2-(-(p1`1))^2>=0 by XREAL_1:63;
consider f being Function of I[01],(TOP-REAL 2)|Upper_Arc(P) such that
A21: f is being_homeomorphism and
A22: 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 and
A23: f.0=W-min(P) & f.1=E-max(P) by A1,Th43;
A24: rng f=[#]((TOP-REAL 2)|Upper_Arc(P)) by A21,TOPS_2:def 5
.=Upper_Arc(P) by PRE_TOPC:def 5;
p2 in Upper_Arc(P) by A2,A14;
then consider x2 being object such that
A25: x2 in dom f and
A26: p2=f.x2 by A24,FUNCT_1:def 3;
A27: dom f=[#](I[01]) by A21,TOPS_2:def 5
.=[.0,1.] by BORSUK_1:40;
reconsider r22=x2 as Real by A25;
A28: 0<=r22 & r22<=1 by A25,A27,XXREAL_1:1;
p1 in Upper_Arc(P) by A2,A14;
then consider x1 being object such that
A29: x1 in dom f and
A30: p1=f.x1 by A24,FUNCT_1:def 3;
reconsider r11=x1 as Real by A29;
r11<=1 by A29,A27,XXREAL_1:1;
then
A31: r11<=r22 by A17,A21,A23,A30,A26,A28,JORDAN5C:def 3;
A32: r11<r22 iff p1`1<p2`1 by A22,A29,A30,A25,A26,A27;
then -(p1`1)> -(p2`1) by A3,A30,A26,A31,XREAL_1:24,XXREAL_0:1;
then (-(p1`1))^2 > (-(p2`1))^2 by A4,SQUARE_1:16;
then 1^2- (-(p1`1))^2 < 1^2-(-(p2`1))^2 by XREAL_1:15;
hence thesis by A30,A26,A32,A31,A19,A12,A20,SQUARE_1:27,XXREAL_0:1;
end;
theorem Th47:
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 & p2`2>=0 holds p1`1<p2`1
proof
let p1,p2 be Point of TOP-REAL 2, P be compact non empty Subset of TOP-REAL
2;
assume that
A1: P={p where p is Point of TOP-REAL 2: |.p.|=1} and
A2: LE p1,p2,P and
A3: p1<>p2 and
A4: p2`2>=0;
A5: P is being_simple_closed_curve by A1,JGRAPH_3:26;
then
A6: p1 in P by A2,JORDAN7:5;
set P4=Lower_Arc(P);
A7: Upper_Arc(P)={p where p is Point of TOP-REAL 2:p in P & p`2>=0} by A1,Th34;
A8: Upper_Arc(P) /\ P4={W-min(P),E-max(P)} by A5,JORDAN6:def 9;
A9: p2 in P by A2,A5,JORDAN7:5;
A10: now
A11: now
assume p2=W-min(P);
then LE p2,p1,P by A5,A6,JORDAN7:3;
hence contradiction by A1,A2,A3,JGRAPH_3:26,JORDAN6:57;
end;
assume
A12: p2 in Lower_Arc(P);
p2 in Upper_Arc(P) by A4,A9,A7;
then p2 in {W-min(P),E-max(P)} by A8,A12,XBOOLE_0:def 4;
then p2=W-min(P) or p2=E-max(P) by TARSKI:def 2;
then
A13: p2= |[1,0]| by A1,A11,Th30;
then
A14: p2`1=1 by EUCLID:52;
A15: ex p8 being Point of TOP-REAL 2 st p8=p1 & |.p8.|=1 by A1,A6;
A16: now
assume
A17: p1`1=1;
1^2=(p1`1)^2+(p1`2)^2 by A15,JGRAPH_3:1;
then p1`2=0 by A17,XCMPLX_1:6;
hence contradiction by A3,A13,A17,EUCLID:53;
end;
p1`1<=1 by A15,Th1;
hence thesis by A14,A16,XXREAL_0:1;
end;
now
assume p2=W-min(P);
then LE p2,p1,P by A5,A6,JORDAN7:3;
hence contradiction by A1,A2,A3,JGRAPH_3:26,JORDAN6:57;
end;
then
A18: p1 in Upper_Arc(P) & p2 in Upper_Arc(P)& not p2=W-min(P) & LE p1,p2,
Upper_Arc(P),W-min(P),E-max(P) or p1`1<p2`1 by A2,A10;
consider f being Function of I[01],(TOP-REAL 2)|Upper_Arc(P) such that
A19: f is being_homeomorphism and
A20: 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 and
A21: f.0=W-min(P) & f.1=E-max(P) by A1,Th43;
A22: rng f=[#]((TOP-REAL 2)|Upper_Arc(P)) by A19,TOPS_2:def 5
.=Upper_Arc(P) by PRE_TOPC:def 5;
now
per cases;
case
A23: not p1`1 < p2`1;
then consider x1 being object such that
A24: x1 in dom f and
A25: p1=f.x1 by A18,A22,FUNCT_1:def 3;
consider x2 being object such that
A26: x2 in dom f and
A27: p2=f.x2 by A18,A22,A23,FUNCT_1:def 3;
A28: dom f=[#](I[01]) by A19,TOPS_2:def 5
.=[.0,1.] by BORSUK_1:40;
reconsider r22=x2 as Real by A26;
A29: 0<=r22 & r22<=1 by A26,A28,XXREAL_1:1;
reconsider r11=x1 as Real by A24;
A30: r11<r22 iff p1`1<p2`1 by A20,A24,A25,A26,A27,A28;
r11<=1 by A24,A28,XXREAL_1:1;
then r11<=r22 or p1`1<p2`1 by A18,A19,A21,A25,A27,A29,JORDAN5C:def 3;
hence thesis by A3,A25,A27,A30,XXREAL_0:1;
end;
case
p1`1<p2`1;
hence thesis;
end;
end;
hence thesis;
end;
theorem Th48:
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 & p1<>W-min(P) holds p1`1>p2`1
proof
let p1,p2 be Point of TOP-REAL 2, P be compact non empty Subset of TOP-REAL
2;
assume that
A1: P={p where p is Point of TOP-REAL 2: |.p.|=1} and
A2: LE p1,p2,P and
A3: p1<>p2 and
A4: p1`2<=0 and
A5: p1<>W-min(P);
A6: P is being_simple_closed_curve by A1,JGRAPH_3:26;
then
A7: p2 in P by A2,JORDAN7:5;
set P4=Lower_Arc(P);
A8: Lower_Arc(P)={p where p is Point of TOP-REAL 2:p in P & p`2<=0} by A1,Th35;
A9: Upper_Arc(P) /\ P4={W-min(P),E-max(P)} by A6,JORDAN6:def 9;
A10: p1 in P by A2,A6,JORDAN7:5;
now
assume
A11: p1 in Upper_Arc(P);
p1 in Lower_Arc(P) by A4,A10,A8;
then p1 in {W-min(P),E-max(P)} by A9,A11,XBOOLE_0:def 4;
then p1=W-min(P) or p1=E-max(P) by TARSKI:def 2;
then
A12: p1= |[1,0]| by A1,A5,Th30;
then
A13: p1`1=1 by EUCLID:52;
A14: ex p9 being Point of TOP-REAL 2 st p9=p2 & |.p9.|=1 by A1,A7;
A15: now
assume
A16: p2`1=1;
1^2 =(p2`1)^2+(p2`2)^2 by A14,JGRAPH_3:1;
then p2`2=0 by A16,XCMPLX_1:6;
hence contradiction by A3,A12,A16,EUCLID:53;
end;
p2`1<=1 by A14,Th1;
hence thesis by A13,A15,XXREAL_0:1;
end;
then
A17: p1 in Lower_Arc(P) & p2 in Lower_Arc(P)& not p2=W-min(P) & LE p1,p2,
Lower_Arc(P),E-max(P),W-min(P) or p1`1>p2`1 by A2;
consider f being Function of I[01],(TOP-REAL 2)|Lower_Arc(P) such that
A18: f is being_homeomorphism and
A19: 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 and
A20: f.0=E-max(P) & f.1=W-min(P) by A1,Th42;
A21: rng f=[#]((TOP-REAL 2)|Lower_Arc(P)) by A18,TOPS_2:def 5
.=Lower_Arc(P) by PRE_TOPC:def 5;
now
per cases;
case
A22: not p1`1 > p2`1;
then consider x1 being object such that
A23: x1 in dom f and
A24: p1=f.x1 by A17,A21,FUNCT_1:def 3;
consider x2 being object such that
A25: x2 in dom f and
A26: p2=f.x2 by A17,A21,A22,FUNCT_1:def 3;
A27: dom f=[#](I[01]) by A18,TOPS_2:def 5
.=[.0,1.] by BORSUK_1:40;
reconsider r22=x2 as Real by A25;
A28: 0<=r22 & r22<=1 by A25,A27,XXREAL_1:1;
reconsider r11=x1 as Real by A23;
A29: r11<r22 iff p1`1>p2`1 by A19,A23,A24,A25,A26,A27;
r11<=1 by A23,A27,XXREAL_1:1;
then r11<=r22 or p1`1>p2`1 by A17,A18,A20,A24,A26,A28,JORDAN5C:def 3;
hence thesis by A3,A24,A26,A29,XXREAL_0:1;
end;
case
p1`1>p2`1;
hence thesis;
end;
end;
hence thesis;
end;
theorem Th49:
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
proof
let p1,p2 be Point of TOP-REAL 2, P be compact non empty Subset of TOP-REAL
2;
assume that
A1: P={p where p is Point of TOP-REAL 2: |.p.|=1} and
A2: p2`2>=0 or p2`1>=0 and
A3: LE p1,p2,P;
A4: Upper_Arc(P)={p where p is Point of TOP-REAL 2:p in P & p`2>=0} by A1,Th34;
A5: P is being_simple_closed_curve by A1,JGRAPH_3:26;
then
A6: p2 in P by A3,JORDAN7:5;
A7: Lower_Arc(P) is_an_arc_of E-max(P),W-min(P) by A5,JORDAN6:def 9;
per cases by A2;
suppose
p2`2>=0;
then p2 in Upper_Arc(P) by A6,A4;
then p1 in Upper_Arc(P) by A1,A3,Th44;
then ex p8 being Point of TOP-REAL 2 st p8=p1 & p8 in P & p8 `2>=0 by A4;
hence thesis;
end;
suppose
A8: p2`2<0 & p2`1>=0;
then not ex p8 being Point of TOP-REAL 2 st p8=p2 & p8 in P & p8`2>=0;
then
A9: not p2 in Upper_Arc(P) by A4;
now
per cases by A3,A9;
case
p1 in Upper_Arc(P) & p2 in Lower_Arc(P)& not p2=W-min(P);
then ex p8 being Point of TOP-REAL 2 st p8=p1 & p8 in P & p8 `2>=0 by
A4;
hence thesis;
end;
case
A10: p1 in Lower_Arc(P) & p2 in Lower_Arc(P)& not p2=W-min(P)& LE
p1,p2,Lower_Arc(P),E-max(P),W-min(P);
now
assume
A11: p1=W-min(P);
then LE p2,p1,Lower_Arc(P),E-max(P),W-min(P) by A7,A10,JORDAN5C:10;
hence contradiction by A7,A10,A11,JORDAN5C:12;
end;
hence thesis by A1,A3,A8,Th48;
end;
end;
hence thesis;
end;
end;
theorem Th50:
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
proof
let p1,p2 be Point of TOP-REAL 2, P be compact non empty Subset of TOP-REAL
2;
assume that
A1: P={p where p is Point of TOP-REAL 2: |.p.|=1} and
A2: LE p1,p2,P and
A3: p1<>p2 and
A4: p1`1>=0 and
A5: p2`1>=0;
A6: Upper_Arc(P)={p where p is Point of TOP-REAL 2:p in P & p`2>=0} by A1,Th34;
A7: P is being_simple_closed_curve by A1,JGRAPH_3:26;
then
A8: p2 in P by A2,JORDAN7:5;
then
A9: ex p3 being Point of TOP-REAL 2 st p3=p2 & |.p3.|=1 by A1;
W-min(P)=|[-1,0]| by A1,Th29;
then
A10: (W-min(P))`2=0 by EUCLID:52;
A11: Lower_Arc(P)={p where p is Point of TOP-REAL 2:p in P & p`2<=0} by A1,Th35
;
A12: p1 in P by A2,A7,JORDAN7:5;
then
A13: ex p4 being Point of TOP-REAL 2 st p4=p1 & |.p4.|=1 by A1;
now
per cases;
case
A14: p1`2>=0 & p2`2>=0;
then p1`1<p2`1 by A1,A2,A3,Th47;
then (p1`1)^2 < ((p2`1))^2 by A4,SQUARE_1:16;
then
A15: 1^2- ((p1`1))^2 > 1^2-((p2`1))^2 by XREAL_1:15;
1^2=(p1`1)^2+(p1`2)^2 by A13,JGRAPH_3:1;
then
A16: p1`2=sqrt(1^2-((p1`1))^2) by A14,SQUARE_1:22;
A17: 1^2=(p2`1)^2+(p2`2)^2 by A9,JGRAPH_3:1;
then (p2`2)=sqrt(1^2-((p2`1))^2) by A14,SQUARE_1:22;
hence thesis by A15,A16,A17,SQUARE_1:27,XREAL_1:63;
end;
case
p1`2>=0 & p2`2<0;
hence thesis;
end;
case
A18: p1`2<0 & p2`2>=0;
then p1 in Lower_Arc(P) & p2 in Upper_Arc(P) by A12,A8,A6,A11;
then LE p2,p1,P by A10,A18;
hence contradiction by A1,A2,A3,JGRAPH_3:26,JORDAN6:57;
end;
case
A19: p1`2<0 & p2`2<0;
ex p3 being Point of TOP-REAL 2 st p3=p1 & |.p3.|=1 by A1,A12;
then
A20: 1^2=(p1`1)^2+(p1`2)^2 by JGRAPH_3:1;
then 1^2-((p1`1))^2=(-(p1`2))^2;
then
A21: -(p1`2)=sqrt(1^2-((p1`1))^2) by A19,SQUARE_1:22;
not ex p being Point of TOP-REAL 2 st p=p1 & p in P & p`2>=0 by A19;
then
A22: not p1 in Upper_Arc(P) by A6;
then
A23: LE p1,p2,Lower_Arc(P),E-max(P),W-min(P) by A2;
ex p4 being Point of TOP-REAL 2 st p4=p2 & |.p4.|=1 by A1,A8;
then 1^2=(p2`1)^2+(p2`2)^2 by JGRAPH_3:1;
then 1^2-((p2`1))^2=(-(p2`2))^2;
then
A24: -(p2`2)=sqrt(1^2-((p2`1))^2) by A19,SQUARE_1:22;
consider f being Function of I[01],(TOP-REAL 2)|Lower_Arc(P) such that
A25: f is being_homeomorphism and
A26: 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
and
A27: f.0=E-max(P) & f.1=W-min(P) by A1,Th42;
A28: rng f=[#]((TOP-REAL 2)|Lower_Arc(P)) by A25,TOPS_2:def 5
.=Lower_Arc(P) by PRE_TOPC:def 5;
p2 in Lower_Arc(P) by A2,A22;
then consider x2 being object such that
A29: x2 in dom f and
A30: p2=f.x2 by A28,FUNCT_1:def 3;
A31: dom f=[#](I[01]) by A25,TOPS_2:def 5
.=[.0,1.] by BORSUK_1:40;
reconsider r22=x2 as Real by A29;
A32: 0<=r22 & r22<=1 by A29,A31,XXREAL_1:1;
p1 in Lower_Arc(P) by A2,A22;
then consider x1 being object such that
A33: x1 in dom f and
A34: p1=f.x1 by A28,FUNCT_1:def 3;
reconsider r11=x1 as Real by A33;
A35: r11<r22 iff p1`1>p2`1 by A26,A33,A34,A29,A30,A31;
r11<=1 by A33,A31,XXREAL_1:1;
then r11<=r22 by A23,A25,A27,A34,A30,A32,JORDAN5C:def 3;
then (p1`1) ^2 > ((p2`1))^2 by A3,A5,A34,A30,A35,SQUARE_1:16,XXREAL_0:1;
then 1^2- ((p1`1))^2 < 1^2-((p2`1))^2 by XREAL_1:15;
then sqrt(1^2- ((p1`1))^2) < sqrt(1^2-((p2`1))^2) by A20,SQUARE_1:27
,XREAL_1:63;
hence thesis by A21,A24,XREAL_1:24;
end;
end;
hence thesis;
end;
theorem Th51:
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
proof
let p1,p2 be Point of TOP-REAL 2, P be compact non empty Subset of TOP-REAL
2;
assume that
A1: P={p where p is Point of TOP-REAL 2: |.p.|=1} and
A2: p1 in P and
A3: p2 in P and
A4: p1`1<0 and
A5: p2`1<0 and
A6: p1`2<0 and
A7: p2`2<0 and
A8: p1`1>=p2`1 or p1`2<=p2`2;
A9: ex p3 being Point of TOP-REAL 2 st p3=p2 & |.p3.|=1 by A1,A3;
set P4=Lower_Arc(P);
A10: P is being_simple_closed_curve by A1,JGRAPH_3:26;
then
A11: Upper_Arc(P) \/ P4=P by JORDAN6:def 9;
A12: Upper_Arc(P)={p where p is Point of TOP-REAL 2:p in P & p`2>=0} by A1,Th34
;
A13: now
assume not p1 in Lower_Arc(P);
then p1 in Upper_Arc(P) by A2,A11,XBOOLE_0:def 3;
then ex p being Point of TOP-REAL 2 st p1=p & p in P & p`2>= 0 by A12;
hence contradiction by A6;
end;
A14: now
assume not p2 in Lower_Arc(P);
then p2 in Upper_Arc(P) by A3,A11,XBOOLE_0:def 3;
then ex p being Point of TOP-REAL 2 st p2=p & p in P & p`2>= 0 by A12;
hence contradiction by A7;
end;
A15: ex p3 being Point of TOP-REAL 2 st p3=p1 & |.p3.|=1 by A1,A2;
A16: now
assume p1`2<=p2`2;
then -p1`2>=-p2`2 by XREAL_1:24;
then (-(p1`2))^2 >= (-(p2`2))^2 by A7,SQUARE_1:15;
then
A17: 1^2- (-(p1`2))^2 <= 1^2-(-(p2`2))^2 by XREAL_1:13;
A18: 1^2=(p1`1)^2+(p1`2)^2 by A15,JGRAPH_3:1;
then 1^2-(-(p1`2))^2>=0 by XREAL_1:63;
then
A19: sqrt(1^2- (-(p1`2))^2) <= sqrt(1^2-(-(p2`2))^2) by A17,SQUARE_1:26;
1^2=(p2`1)^2+(p2`2)^2 by A9,JGRAPH_3:1;
then 1^2-(-(p2`2))^2=(-(p2`1))^2;
then
A20: -(p2`1)=sqrt(1^2-(-(p2`2))^2) by A5,SQUARE_1:22;
1^2-(-(p1`2))^2=(-(p1`1))^2 by A18;
then -(p1`1)=sqrt(1^2-(-(p1`2))^2) by A4,SQUARE_1:22;
hence p1`1>=p2`1 by A20,A19,XREAL_1:24;
end;
A21: Upper_Arc(P) /\ P4={W-min(P),E-max(P)} by A10,JORDAN6:def 9;
A22: Lower_Arc(P) is_an_arc_of E-max(P),W-min(P) by A10,JORDAN6:def 9;
A23: W-min(P)=|[-1,0]| by A1,Th29;
for g being Function of I[01], (TOP-REAL 2)|P4,
s1, s2 being Real st g
is being_homeomorphism & g.0 = E-max(P) & g.1 = W-min(P) & g.s1 = p1 & 0 <= s1
& s1 <= 1 & g.s2 = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2
proof
W-min(P) in {W-min(P),E-max(P)} by TARSKI:def 2;
then
A24: W-min(P) in Lower_Arc(P) by A21,XBOOLE_0:def 4;
set K0=Lower_Arc(P);
reconsider g0=proj1 as Function of TOP-REAL 2,R^1 by TOPMETR:17;
reconsider g2=g0|K0 as Function of (TOP-REAL 2)|K0,R^1 by PRE_TOPC:9;
Closed-Interval-TSpace(-1,1) =TopSpaceMetr(Closed-Interval-MSpace(-1,
1)) by TOPMETR:def 7;
then
A25: Closed-Interval-TSpace(-1,1) is T_2 by PCOMPS_1:34;
reconsider g3=g2 as continuous Function of (TOP-REAL 2)|K0,
Closed-Interval-TSpace(-1,1) by A1,Lm5;
let g be Function of I[01], (TOP-REAL 2)|P4, s1, s2 be Real;
assume that
A26: g is being_homeomorphism and
g.0 = E-max(P) and
A27: g.1 =W-min(P) and
A28: g.s1 = p1 and
A29: 0 <= s1 & s1 <= 1 and
A30: g.s2 = p2 and
A31: 0 <= s2 & s2 <= 1;
A32: s2 in [.0,1.] by A31,XXREAL_1:1;
reconsider h=g3*g as Function of Closed-Interval-TSpace(0,1),
Closed-Interval-TSpace(-1,1) by TOPMETR:20;
A33: dom g3=[#]((TOP-REAL 2)|K0) & rng g3=[#](Closed-Interval-TSpace(-1,1)
) by A1,Lm5,FUNCT_2:def 1;
g3 is one-to-one & K0 is non empty compact by A1,A22,Lm5,JORDAN5A:1;
then g3 is being_homeomorphism by A33,A25,COMPTS_1:17;
then
A34: h is being_homeomorphism by A26,TOPMETR:20,TOPS_2:57;
A35: dom g=[#](I[01]) by A26,TOPS_2:def 5
.=[.0,1.] by BORSUK_1:40;
then
A36: 1 in dom g by XXREAL_1:1;
A37: -1=(|[-1,0]|)`1 by EUCLID:52
.=proj1.(|[-1,0]|) by PSCOMP_1:def 5
.=g3.(g.1) by A23,A27,A24,FUNCT_1:49
.= h.1 by A36,FUNCT_1:13;
A38: s1 in [.0,1.] by A29,XXREAL_1:1;
A39: p2`1=proj1.p2 by PSCOMP_1:def 5
.=g3.(g.s2) by A14,A30,FUNCT_1:49
.= h.s2 by A35,A32,FUNCT_1:13;
p1`1=g0.p1 by PSCOMP_1:def 5
.=g3.(g.s1) by A13,A28,FUNCT_1:49
.= h.s1 by A35,A38,FUNCT_1:13;
hence thesis by A8,A16,A34,A38,A32,A37,A39,Th9;
end;
then
A40: LE p1,p2,Lower_Arc(P),E-max(P),W-min(P) by A13,A14,JORDAN5C:def 3;
now
assume
A41: p2=W-min(P);
W-min(P)=|[-1,0]| by A1,Th29;
hence contradiction by A7,A41,EUCLID:52;
end;
hence thesis by A13,A14,A40;
end;
theorem
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
proof
let p1,p2 be Point of TOP-REAL 2, P be compact non empty Subset of TOP-REAL
2;
assume that
A1: P={p where p is Point of TOP-REAL 2: |.p.|=1} and
A2: p1 in P and
A3: p2 in P and
A4: p1`1>0 and
A5: p2`1>0 and
A6: p1`2<0 and
A7: p2`2<0 and
A8: p1`1>=p2`1 or p1`2>=p2`2;
A9: ex p3 being Point of TOP-REAL 2 st p3=p2 & |.p3.|=1 by A1,A3;
set P4=Lower_Arc(P);
A10: P is being_simple_closed_curve by A1,JGRAPH_3:26;
then
A11: Upper_Arc(P) \/ P4=P by JORDAN6:def 9;
A12: Upper_Arc(P)={p where p is Point of TOP-REAL 2:p in P & p`2>=0} by A1,Th34
;
A13: now
assume not p1 in Lower_Arc(P);
then p1 in Upper_Arc(P) by A2,A11,XBOOLE_0:def 3;
then ex p being Point of TOP-REAL 2 st p1=p & p in P & p`2>= 0 by A12;
hence contradiction by A6;
end;
A14: now
assume not p2 in Lower_Arc(P);
then p2 in Upper_Arc(P) by A3,A11,XBOOLE_0:def 3;
then ex p being Point of TOP-REAL 2 st p2=p & p in P & p`2>= 0 by A12;
hence contradiction by A7;
end;
A15: ex p3 being Point of TOP-REAL 2 st p3=p1 & |.p3.|=1 by A1,A2;
A16: now
assume p1`2>=p2`2;
then -p1`2<=-p2`2 by XREAL_1:24;
then (-(p1`2))^2 <= (-(p2`2))^2 by A6,SQUARE_1:15;
then
A17: 1^2- (-(p1`2))^2 >= 1^2-(-(p2`2))^2 by XREAL_1:13;
1^2=(p2`1)^2+(p2`2)^2 by A9,JGRAPH_3:1;
then
A18: (p2`1)=sqrt(1^2-(-(p2`2))^2) & 1^2-(-(p2`2))^2>=0 by A5,SQUARE_1:22 ;
1^2=(p1`1)^2+(p1`2)^2 by A15,JGRAPH_3:1;
then (p1`1)=sqrt(1^2-(-(p1`2))^2) by A4,SQUARE_1:22;
hence p1`1>=p2`1 by A17,A18,SQUARE_1:26;
end;
A19: Upper_Arc(P) /\ P4={W-min(P),E-max(P)} by A10,JORDAN6:def 9;
A20: Lower_Arc(P) is_an_arc_of E-max(P),W-min(P) by A10,JORDAN6:def 9;
A21: W-min(P)=|[-1,0]| by A1,Th29;
for g being Function of I[01], (TOP-REAL 2)|P4,
s1, s2 being Real st g
is being_homeomorphism & g.0 = E-max(P) & g.1 = W-min(P) & g.s1 = p1 & 0 <= s1
& s1 <= 1 & g.s2 = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2
proof
W-min(P) in {W-min(P),E-max(P)} by TARSKI:def 2;
then
A22: W-min(P) in Lower_Arc(P) by A19,XBOOLE_0:def 4;
set K0=Lower_Arc(P);
reconsider g0=proj1 as Function of TOP-REAL 2,R^1 by TOPMETR:17;
reconsider g2=g0|K0 as Function of (TOP-REAL 2)|K0,R^1 by PRE_TOPC:9;
Closed-Interval-TSpace(-1,1) =TopSpaceMetr(Closed-Interval-MSpace(-1,
1)) by TOPMETR:def 7;
then
A23: Closed-Interval-TSpace(-1,1) is T_2 by PCOMPS_1:34;
reconsider g3=g2 as continuous Function of (TOP-REAL 2)|K0,
Closed-Interval-TSpace(-1,1) by A1,Lm5;
let g be Function of I[01], (TOP-REAL 2)|P4, s1, s2 be Real;
assume that
A24: g is being_homeomorphism and
g.0 = E-max(P) and
A25: g.1 =W-min(P) and
A26: g.s1 = p1 and
A27: 0 <= s1 & s1 <= 1 and
A28: g.s2 = p2 and
A29: 0 <= s2 & s2 <= 1;
A30: s2 in [.0,1.] by A29,XXREAL_1:1;
reconsider h=g3*g as Function of Closed-Interval-TSpace(0,1),
Closed-Interval-TSpace(-1,1) by TOPMETR:20;
A31: dom g3=[#]((TOP-REAL 2)|K0) & rng g3=[#](Closed-Interval-TSpace(-1,1)
) by A1,Lm5,FUNCT_2:def 1;
g3 is one-to-one & K0 is non empty compact by A1,A20,Lm5,JORDAN5A:1;
then g3 is being_homeomorphism by A31,A23,COMPTS_1:17;
then
A32: h is being_homeomorphism by A24,TOPMETR:20,TOPS_2:57;
A33: dom g=[#](I[01]) by A24,TOPS_2:def 5
.=[.0,1.] by BORSUK_1:40;
then
A34: 1 in dom g by XXREAL_1:1;
A35: -1=(|[-1,0]|)`1 by EUCLID:52
.=proj1.(|[-1,0]|) by PSCOMP_1:def 5
.=g3.(g.1) by A21,A25,A22,FUNCT_1:49
.= h.1 by A34,FUNCT_1:13;
A36: s1 in [.0,1.] by A27,XXREAL_1:1;
A37: p2`1=proj1.p2 by PSCOMP_1:def 5
.=g3.p2 by A14,FUNCT_1:49
.= h.s2 by A28,A33,A30,FUNCT_1:13;
p1`1=g0.p1 by PSCOMP_1:def 5
.=g3.(g.s1) by A13,A26,FUNCT_1:49
.= h.s1 by A33,A36,FUNCT_1:13;
hence thesis by A8,A16,A32,A36,A30,A35,A37,Th9;
end;
then
A38: LE p1,p2,Lower_Arc(P),E-max(P),W-min(P) by A13,A14,JORDAN5C:def 3;
now
assume
A39: p2=W-min(P);
W-min(P)=|[-1,0]| by A1,Th29;
hence contradiction by A5,A39,EUCLID:52;
end;
hence thesis by A13,A14,A38;
end;
theorem Th53:
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
proof
let p1,p2 be Point of TOP-REAL 2, P be compact non empty Subset of TOP-REAL
2;
assume that
A1: P={p where p is Point of TOP-REAL 2: |.p.|=1} and
A2: p1 in P and
A3: p2 in P and
A4: p1`1<0 and
A5: p2`1<0 and
A6: p1`2>=0 and
A7: p2`2>=0 and
A8: p1`1<=p2`1 or p1`2<=p2`2;
A9: ex p3 being Point of TOP-REAL 2 st p3=p2 & |.p3.|=1 by A1,A3;
set P4b=Upper_Arc(P);
set P4=Lower_Arc(P);
A10: P is being_simple_closed_curve by A1,JGRAPH_3:26;
then
A11: Upper_Arc(P) /\ P4={W-min(P),E-max(P)} by JORDAN6:def 9;
A12: Upper_Arc(P)={p where p is Point of TOP-REAL 2:p in P & p`2>=0} by A1,Th34
;
then
A13: p1 in Upper_Arc(P) by A2,A6;
A14: p2 in Upper_Arc(P) by A3,A7,A12;
A15: ex p3 being Point of TOP-REAL 2 st p3=p1 & |.p3.|=1 by A1,A2;
A16: now
assume p1`2<=p2`2;
then (p1`2) ^2 <= ((p2`2))^2 by A6,SQUARE_1:15;
then
A17: 1^2- ((p1`2))^2 >= 1^2-((p2`2))^2 by XREAL_1:13;
A18: 1^2=(p2`1)^2+(p2`2)^2 by A9,JGRAPH_3:1;
then 1^2-((p2`2))^2>=0 by XREAL_1:63;
then
A19: sqrt(1^2- ((p1`2))^2) >= sqrt(1^2-((p2`2))^2) by A17,SQUARE_1:26;
1^2=(p1`1)^2+(p1`2)^2 by A15,JGRAPH_3:1;
then 1^2-(p1`2)^2=(-(p1`1))^2;
then
A20: -(p1`1)=sqrt(1^2-((p1`2))^2) by A4,SQUARE_1:22;
1^2-(p2`2)^2=(-(p2`1))^2 by A18;
then -(p2`1)=sqrt(1^2-((p2`2))^2) by A5,SQUARE_1:22;
hence p1`1<=p2`1 by A20,A19,XREAL_1:24;
end;
A21: E-max(P)=|[1,0]| by A1,Th30;
A22: Upper_Arc(P) is_an_arc_of W-min(P),E-max(P) by A10,JORDAN6:def 8;
for g being Function of I[01], (TOP-REAL 2)|P4b,
s1, s2 being Real st g
is being_homeomorphism & g.0 = W-min(P) & g.1 = E-max(P) & g.s1 = p1 & 0 <= s1
& s1 <= 1 & g.s2 = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2
proof
E-max(P) in {W-min(P),E-max(P)} by TARSKI:def 2;
then
A23: E-max(P) in Upper_Arc(P) by A11,XBOOLE_0:def 4;
set K0=Upper_Arc(P);
reconsider g0=proj1 as Function of TOP-REAL 2,R^1 by TOPMETR:17;
reconsider g2=g0|K0 as Function of (TOP-REAL 2)|K0,R^1 by PRE_TOPC:9;
Closed-Interval-TSpace(-1,1) =TopSpaceMetr(Closed-Interval-MSpace(-1,
1)) by TOPMETR:def 7;
then
A24: Closed-Interval-TSpace(-1,1) is T_2 by PCOMPS_1:34;
reconsider g3=g2 as continuous Function of (TOP-REAL 2)|K0,
Closed-Interval-TSpace(-1,1) by A1,Lm6;
let g be Function of I[01], (TOP-REAL 2)|P4b, s1, s2 be Real;
assume that
A25: g is being_homeomorphism and
g.0 = W-min(P) and
A26: g.1 = E-max(P) and
A27: g.s1 = p1 and
A28: 0 <= s1 & s1 <= 1 and
A29: g.s2 = p2 and
A30: 0 <= s2 & s2 <= 1;
A31: s2 in [.0,1.] by A30,XXREAL_1:1;
reconsider h=g3*g as Function of Closed-Interval-TSpace(0,1),
Closed-Interval-TSpace(-1,1) by TOPMETR:20;
A32: dom g3=[#]((TOP-REAL 2)|K0) & rng g3=[#](Closed-Interval-TSpace(-1,1)
) by A1,Lm6,FUNCT_2:def 1;
g3 is one-to-one & K0 is non empty compact by A1,A22,Lm6,JORDAN5A:1;
then g3 is being_homeomorphism by A32,A24,COMPTS_1:17;
then
A33: h is being_homeomorphism by A25,TOPMETR:20,TOPS_2:57;
A34: dom g=[#](I[01]) by A25,TOPS_2:def 5
.=[.0,1.] by BORSUK_1:40;
then
A35: 1 in dom g by XXREAL_1:1;
A36: 1=(|[1,0]|)`1 by EUCLID:52
.=g0.(|[1,0]|) by PSCOMP_1:def 5
.=g3.(|[1,0]|) by A21,A23,FUNCT_1:49
.= h.1 by A21,A26,A35,FUNCT_1:13;
A37: s1 in [.0,1.] by A28,XXREAL_1:1;
A38: p2`1=g0.p2 by PSCOMP_1:def 5
.=g3.p2 by A14,FUNCT_1:49
.= h.s2 by A29,A34,A31,FUNCT_1:13;
p1`1=g0.p1 by PSCOMP_1:def 5
.=g3.(g.s1) by A13,A27,FUNCT_1:49
.= h.s1 by A34,A37,FUNCT_1:13;
hence thesis by A8,A16,A33,A37,A31,A36,A38,Th8;
end;
then
A39: LE p1,p2,Upper_Arc(P),W-min(P),E-max(P) by A13,A14,JORDAN5C:def 3;
p1 in Upper_Arc(P) by A2,A6,A12;
hence thesis by A14,A39;
end;
theorem Th54:
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
proof
let p1,p2 be Point of TOP-REAL 2, P be compact non empty Subset of TOP-REAL
2;
assume that
A1: P={p where p is Point of TOP-REAL 2: |.p.|=1} and
A2: p1 in P and
A3: p2 in P and
A4: p1`2>=0 and
A5: p2`2>=0 and
A6: p1`1<=p2`1;
A7: Upper_Arc(P)={p where p is Point of TOP-REAL 2:p in P & p`2>=0} by A1,Th34;
then
A8: p1 in Upper_Arc(P) by A2,A4;
A9: p2 in Upper_Arc(P) by A3,A5,A7;
set P4b=Upper_Arc(P);
set P4=Lower_Arc(P);
A10: P is being_simple_closed_curve by A1,JGRAPH_3:26;
then
A11: Upper_Arc(P) /\ P4={W-min(P),E-max(P)} by JORDAN6:def 9;
A12: E-max(P)=|[1,0]| by A1,Th30;
A13: Upper_Arc(P) is_an_arc_of W-min(P),E-max(P) by A10,JORDAN6:def 8;
for g being Function of I[01], (TOP-REAL 2)|P4b,
s1, s2 being Real st g
is being_homeomorphism & g.0 = W-min(P) & g.1 = E-max(P) & g.s1 = p1 & 0 <= s1
& s1 <= 1 & g.s2 = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2
proof
E-max(P) in {W-min(P),E-max(P)} by TARSKI:def 2;
then
A14: E-max(P) in Upper_Arc(P) by A11,XBOOLE_0:def 4;
set K0=Upper_Arc(P);
reconsider g0=proj1 as Function of TOP-REAL 2,R^1 by TOPMETR:17;
reconsider g2=g0|K0 as Function of (TOP-REAL 2)|K0,R^1 by PRE_TOPC:9;
Closed-Interval-TSpace(-1,1) =TopSpaceMetr(Closed-Interval-MSpace(-1,
1)) by TOPMETR:def 7;
then
A15: Closed-Interval-TSpace(-1,1) is T_2 by PCOMPS_1:34;
reconsider g3=g2 as continuous Function of (TOP-REAL 2)|K0,
Closed-Interval-TSpace(-1,1) by A1,Lm6;
let g be Function of I[01], (TOP-REAL 2)|P4b, s1, s2 be Real;
assume that
A16: g is being_homeomorphism and
g.0 = W-min(P) and
A17: g.1 = E-max(P) and
A18: g.s1 = p1 and
A19: 0 <= s1 & s1 <= 1 and
A20: g.s2 = p2 and
A21: 0 <= s2 & s2 <= 1;
A22: s2 in [.0,1.] by A21,XXREAL_1:1;
reconsider h=g3*g as Function of Closed-Interval-TSpace(0,1),
Closed-Interval-TSpace(-1,1) by TOPMETR:20;
A23: dom g3=[#]((TOP-REAL 2)|K0) & rng g3=[#](Closed-Interval-TSpace(-1,1)
) by A1,Lm6,FUNCT_2:def 1;
g3 is one-to-one & K0 is non empty compact by A1,A13,Lm6,JORDAN5A:1;
then g3 is being_homeomorphism by A23,A15,COMPTS_1:17;
then
A24: h is being_homeomorphism by A16,TOPMETR:20,TOPS_2:57;
A25: dom g=[#](I[01]) by A16,TOPS_2:def 5
.=[.0,1.] by BORSUK_1:40;
then
A26: 1 in dom g by XXREAL_1:1;
A27: 1=(|[1,0]|)`1 by EUCLID:52
.=g0.(|[1,0]|) by PSCOMP_1:def 5
.=g3.(|[1,0]|) by A12,A14,FUNCT_1:49
.= h.1 by A12,A17,A26,FUNCT_1:13;
A28: s1 in [.0,1.] by A19,XXREAL_1:1;
A29: p2`1=g0.p2 by PSCOMP_1:def 5
.=g3.p2 by A9,FUNCT_1:49
.= h.s2 by A20,A25,A22,FUNCT_1:13;
p1`1=g0.p1 by PSCOMP_1:def 5
.=g3.p1 by A8,FUNCT_1:49
.= h.s1 by A18,A25,A28,FUNCT_1:13;
hence thesis by A6,A24,A28,A22,A27,A29,Th8;
end;
then
A30: LE p1,p2,Upper_Arc(P),W-min(P),E-max(P) by A8,A9,JORDAN5C:def 3;
p1 in Upper_Arc(P) by A2,A4,A7;
hence thesis by A9,A30;
end;
theorem Th55:
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
proof
let p1,p2 be Point of TOP-REAL 2, P be compact non empty Subset of TOP-REAL
2;
assume that
A1: P={p where p is Point of TOP-REAL 2: |.p.|=1} and
A2: p1 in P and
A3: p2 in P and
A4: p1`1>=0 and
A5: p2`1>=0 and
A6: p1`2>=p2`2;
A7: ex p3 being Point of TOP-REAL 2 st p3=p1 & |.p3.|=1 by A1,A2;
A8: W-min(P)=|[-1,0]| by A1,Th29;
A9: ex p3 being Point of TOP-REAL 2 st p3=p2 & |.p3.|=1 by A1,A3;
A10: Upper_Arc(P)={p where p is Point of TOP-REAL 2:p in P & p`2>=0} by A1,Th34
;
set P4b=Lower_Arc(P);
A11: P is being_simple_closed_curve by A1,JGRAPH_3:26;
then
A12: Upper_Arc(P) /\ P4b={W-min(P),E-max(P)} by JORDAN6:def 9;
A13: Upper_Arc(P) \/ P4b=P by A11,JORDAN6:def 9;
A14: Lower_Arc(P) is_an_arc_of E-max(P),W-min(P) by A11,JORDAN6:def 9;
now
per cases;
case
A15: p1 in Upper_Arc(P) & p2 in Upper_Arc(P);
1^2=(p1`1)^2+(p1`2)^2 by A7,JGRAPH_3:1;
then
A16: (p1`1)=sqrt(1^2-((p1`2))^2) & 1^2-((p1`2))^2>=0 by A4,SQUARE_1:22 ;
1^2=(p2`1)^2+(p2`2)^2 by A9,JGRAPH_3:1;
then
A17: (p2`1)=sqrt(1^2-((p2`2))^2) by A5,SQUARE_1:22;
A18: ex p22 being Point of TOP-REAL 2 st p2=p22 & p22 in P & p22`2>=0 by A10
,A15;
then (p1`2) ^2 >= ((p2`2))^2 by A6,SQUARE_1:15;
then 1^2- ((p1`2))^2 <= 1^2-((p2`2))^2 by XREAL_1:13;
hence thesis by A1,A2,A6,A18,A17,A16,Th54,SQUARE_1:26;
end;
case
A19: p1 in Upper_Arc(P) & not p2 in Upper_Arc(P);
A20: now
assume
A21: p2=W-min(P);
W-min(P)=|[-1,0]| by A1,Th29;
then p2`2=0 by A21,EUCLID:52;
hence contradiction by A3,A10,A19;
end;
p2 in Lower_Arc(P) by A3,A13,A19,XBOOLE_0:def 3;
hence thesis by A19,A20;
end;
case
A22: not p1 in Upper_Arc(P) & p2 in Upper_Arc(P);
then
ex p9 being Point of TOP-REAL 2 st p2=p9 & p9 in P & p9 `2>=0 by A10;
hence contradiction by A2,A6,A10,A22;
end;
case
A23: not p1 in Upper_Arc(P) & not p2 in Upper_Arc(P);
A24: -p1`2<=-p2`2 by A6,XREAL_1:24;
p1`2<0 by A2,A10,A23;
then (-(p1`2))^2 <= (-(p2`2))^2 by A24,SQUARE_1:15;
then
A25: 1^2- (-(p1`2))^2 >= 1^2-(-(p2`2))^2 by XREAL_1:13;
1^2=(p2`1)^2+(p2`2)^2 by A9,JGRAPH_3:1;
then
A26: (p2`1)=sqrt(1^2-(-(p2`2))^2) & 1^2-(-(p2`2))^2>=0 by A5,SQUARE_1:22 ;
A27: p2 in Lower_Arc(P) by A3,A13,A23,XBOOLE_0:def 3;
A28: now
assume
A29: p2=W-min(P);
W-min(P)=|[-1,0]| by A1,Th29;
then p2`2=0 by A29,EUCLID:52;
hence contradiction by A3,A10,A23;
end;
A30: p1 in Lower_Arc(P) by A2,A13,A23,XBOOLE_0:def 3;
1^2=(p1`1)^2+(p1`2)^2 by A7,JGRAPH_3:1;
then p1`1=sqrt(1^2-(-(p1`2))^2) by A4,SQUARE_1:22;
then
A31: p1`1>=p2`1 by A25,A26,SQUARE_1:26;
for g being Function of I[01], (TOP-REAL 2)|P4b,
s1, s2 being Real
st g is being_homeomorphism & g.0 = E-max(P) & g.1 = W-min(P) & g.s1 = p1 & 0
<= s1 & s1 <= 1 & g.s2 = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2
proof
W-min(P) in {W-min(P),E-max(P)} by TARSKI:def 2;
then
A32: W-min(P) in Lower_Arc(P) by A12,XBOOLE_0:def 4;
set K0=Lower_Arc(P);
reconsider g0=proj1 as Function of TOP-REAL 2,R^1 by TOPMETR:17;
reconsider g2=g0|K0 as Function of (TOP-REAL 2)|K0,R^1 by PRE_TOPC:9;
Closed-Interval-TSpace(-1,1) =TopSpaceMetr(Closed-Interval-MSpace
(-1,1)) by TOPMETR:def 7;
then
A33: Closed-Interval-TSpace(-1,1) is T_2 by PCOMPS_1:34;
reconsider g3=g2 as continuous Function of (TOP-REAL 2)|K0,
Closed-Interval-TSpace(-1,1) by A1,Lm5;
let g be Function of I[01], (TOP-REAL 2)|P4b, s1, s2 be Real;
assume that
A34: g is being_homeomorphism and
g.0 = E-max(P) and
A35: g.1 = W-min(P) and
A36: g.s1 = p1 and
A37: 0 <= s1 & s1 <= 1 and
A38: g.s2 = p2 and
A39: 0 <= s2 & s2 <= 1;
A40: s2 in [.0,1.] by A39,XXREAL_1:1;
reconsider h=g3*g as Function of Closed-Interval-TSpace(0,1),
Closed-Interval-TSpace(-1,1) by TOPMETR:20;
A41: dom g3=[#]((TOP-REAL 2)|K0) & rng g3=[#](Closed-Interval-TSpace(-
1,1)) by A1,Lm5,FUNCT_2:def 1;
g3 is one-to-one & K0 is non empty compact by A1,A14,Lm5,JORDAN5A:1;
then g3 is being_homeomorphism by A41,A33,COMPTS_1:17;
then
A42: h is being_homeomorphism by A34,TOPMETR:20,TOPS_2:57;
A43: dom g=[#](I[01]) by A34,TOPS_2:def 5
.=[.0,1.] by BORSUK_1:40;
then
A44: 1 in dom g by XXREAL_1:1;
A45: -1=(|[-1,0]|)`1 by EUCLID:52
.=proj1.(|[-1,0]|) by PSCOMP_1:def 5
.=g3.(|[-1,0]|) by A8,A32,FUNCT_1:49
.= h.1 by A8,A35,A44,FUNCT_1:13;
A46: s1 in [.0,1.] by A37,XXREAL_1:1;
A47: p2`1=g0.p2 by PSCOMP_1:def 5
.=g3.p2 by A27,FUNCT_1:49
.= h.s2 by A38,A43,A40,FUNCT_1:13;
p1`1=g0.p1 by PSCOMP_1:def 5
.=g3.p1 by A30,FUNCT_1:49
.= h.s1 by A36,A43,A46,FUNCT_1:13;
hence thesis by A31,A42,A46,A40,A45,A47,Th9;
end;
then
A48: LE p1,p2,Lower_Arc(P),E-max(P),W-min(P) by A30,A27,JORDAN5C:def 3;
p1 in Lower_Arc(P) by A2,A13,A23,XBOOLE_0:def 3;
hence thesis by A27,A28,A48;
end;
end;
hence thesis;
end;
theorem Th56:
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
proof
let p1,p2 be Point of TOP-REAL 2, P be compact non empty Subset of TOP-REAL
2;
assume that
A1: P={p where p is Point of TOP-REAL 2: |.p.|=1} and
A2: p1 in P and
A3: p2 in P and
A4: p1`2<=0 and
A5: p2`2<=0 and
A6: p2<>W-min(P) and
A7: p1`1>=p2`1;
A8: Lower_Arc(P)={p where p is Point of TOP-REAL 2:p in P & p`2<=0} by A1,Th35;
then
A9: p1 in Lower_Arc(P) by A2,A4;
set P4=Lower_Arc(P);
A10: P is being_simple_closed_curve by A1,JGRAPH_3:26;
then
A11: Upper_Arc(P) /\ P4={W-min(P),E-max(P)} by JORDAN6:def 9;
A12: W-min(P)=|[-1,0]| by A1,Th29;
A13: p2 in Lower_Arc(P) by A3,A5,A8;
A14: Lower_Arc(P) is_an_arc_of E-max(P),W-min(P) by A10,JORDAN6:def 9;
for g being Function of I[01], (TOP-REAL 2)|P4,
s1, s2 being Real st g
is being_homeomorphism & g.0 = E-max(P) & g.1 = W-min(P) & g.s1 = p1 & 0 <= s1
& s1 <= 1 & g.s2 = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2
proof
W-min(P) in {W-min(P),E-max(P)} by TARSKI:def 2;
then
A15: W-min(P) in Lower_Arc(P) by A11,XBOOLE_0:def 4;
set K0=Lower_Arc(P);
reconsider g0=proj1 as Function of TOP-REAL 2,R^1 by TOPMETR:17;
reconsider g2=g0|K0 as Function of (TOP-REAL 2)|K0,R^1 by PRE_TOPC:9;
Closed-Interval-TSpace(-1,1) =TopSpaceMetr(Closed-Interval-MSpace(-1,
1)) by TOPMETR:def 7;
then
A16: Closed-Interval-TSpace(-1,1) is T_2 by PCOMPS_1:34;
reconsider g3=g2 as continuous Function of (TOP-REAL 2)|K0,
Closed-Interval-TSpace(-1,1) by A1,Lm5;
let g be Function of I[01], (TOP-REAL 2)|P4, s1, s2 be Real;
assume that
A17: g is being_homeomorphism and
g.0 = E-max(P) and
A18: g.1 = W-min(P) and
A19: g.s1 = p1 and
A20: 0 <= s1 & s1 <= 1 and
A21: g.s2 = p2 and
A22: 0 <= s2 & s2 <= 1;
A23: s2 in [.0,1.] by A22,XXREAL_1:1;
reconsider h=g3*g as Function of Closed-Interval-TSpace(0,1),
Closed-Interval-TSpace(-1,1) by TOPMETR:20;
A24: dom g3=[#]((TOP-REAL 2)|K0) & rng g3=[#](Closed-Interval-TSpace(-1,1)
) by A1,Lm5,FUNCT_2:def 1;
g3 is one-to-one & K0 is non empty compact by A1,A14,Lm5,JORDAN5A:1;
then g3 is being_homeomorphism by A24,A16,COMPTS_1:17;
then
A25: h is being_homeomorphism by A17,TOPMETR:20,TOPS_2:57;
A26: dom g=[#](I[01]) by A17,TOPS_2:def 5
.=[.0,1.] by BORSUK_1:40;
then
A27: 1 in dom g by XXREAL_1:1;
A28: -1=(|[-1,0]|)`1 by EUCLID:52
.=proj1.(|[-1,0]|) by PSCOMP_1:def 5
.=g3.(|[-1,0]|) by A12,A15,FUNCT_1:49
.= h.1 by A12,A18,A27,FUNCT_1:13;
A29: s1 in [.0,1.] by A20,XXREAL_1:1;
A30: p2`1=g0.p2 by PSCOMP_1:def 5
.=g3.p2 by A13,FUNCT_1:49
.= h.s2 by A21,A26,A23,FUNCT_1:13;
p1`1=g0.p1 by PSCOMP_1:def 5
.=g3.p1 by A9,FUNCT_1:49
.= h.s1 by A19,A26,A29,FUNCT_1:13;
hence thesis by A7,A25,A29,A23,A28,A30,Th9;
end;
then
A31: LE p1,p2,Lower_Arc(P),E-max(P),W-min(P) by A9,A13,JORDAN5C:def 3;
p1 in Lower_Arc(P) & p2 in Lower_Arc(P) by A2,A3,A4,A5,A8;
hence thesis by A6,A31;
end;
theorem Th57:
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
proof
let cn be Real,q be Point of TOP-REAL 2;
assume that
A1: -1<cn and
A2: cn<1 and
A3: q`2<=0;
let p be Point of TOP-REAL 2;
assume
A4: p=(cn-FanMorphS).q;
per cases by A3;
suppose
A5: q`2<0;
now
per cases;
case
q`1/|.q.|<cn;
hence thesis by A1,A4,A5,JGRAPH_4:138;
end;
case
q`1/|.q.|>=cn;
hence thesis by A2,A4,A5,JGRAPH_4:137;
end;
end;
hence thesis;
end;
suppose
q`2=0;
hence thesis by A4,JGRAPH_4:113;
end;
end;
theorem Th58:
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
proof
let cn be Real,p1,p2,q1,q2 be Point of TOP-REAL 2, P be compact non empty
Subset of TOP-REAL 2;
assume that
A1: -1<cn & cn<1 and
A2: P={p where p is Point of TOP-REAL 2: |.p.|=1} and
A3: LE p1,p2,P and
A4: q1=(cn-FanMorphS).p1 and
A5: q2=(cn-FanMorphS).p2;
A6: P is being_simple_closed_curve by A2,JGRAPH_3:26;
W-min(P)=|[-1,0]| by A2,Th29;
then
A7: (W-min(P))`2=0 by EUCLID:52;
then
A8: (cn-FanMorphS).(W-min(P))=W-min(P) by JGRAPH_4:113;
p2 in the carrier of TOP-REAL 2;
then
A9: p2 in dom ((cn-FanMorphS)) by FUNCT_2:def 1;
W-min(P) in the carrier of TOP-REAL 2;
then
A10: W-min(P) in dom ((cn-FanMorphS)) by FUNCT_2:def 1;
A11: Lower_Arc(P) c= P by A2,Th33;
A12: (cn-FanMorphS) is one-to-one by A1,JGRAPH_4:133;
A13: Upper_Arc(P) c= P by A2,Th33;
A14: now
per cases by A3;
case
p1 in Upper_Arc(P);
hence p1 in P by A13;
end;
case
p1 in Lower_Arc(P);
hence p1 in P by A11;
end;
end;
A15: now
assume
A16: q2=W-min(P);
then p2=W-min(P) by A5,A8,A10,A9,A12,FUNCT_1:def 4;
then LE p2,p1,P by A6,A14,JORDAN7:3;
then
A17: q1=q2 by A2,A3,A4,A5,JGRAPH_3:26,JORDAN6:57;
W-min(P) in Lower_Arc(P) by A6,JORDAN7:1;
then LE q1,q2,P by A2,A11,A16,A17,JGRAPH_3:26,JORDAN6:56;
hence q1 in Upper_Arc(P) & q2 in Lower_Arc(P)& not q2=W-min(P) or q1 in
Upper_Arc(P) & q2 in Upper_Arc(P) & LE q1,q2,Upper_Arc(P),W-min(P),E-max(P) or
q1 in Lower_Arc(P) & q2 in Lower_Arc(P)& not q2=W-min(P) & LE q1,q2,Lower_Arc(P
),E-max(P),W-min(P);
end;
A18: Upper_Arc(P)={p where p is Point of TOP-REAL 2:p in P & p`2>=0} by A2,Th34
;
A19: Lower_Arc(P)={p where p is Point of TOP-REAL 2:p in P & p`2<=0} by A2,Th35
;
per cases by A3;
suppose
A20: p1 in Upper_Arc(P) & p2 in Lower_Arc(P)& not p2=W-min(P);
A21: |.q2.|=|.p2.| by A5,JGRAPH_4:128;
A22: ex p9 being Point of TOP-REAL 2 st p9=p2 & p9 in P & p9 `2<=0 by A19,A20;
then ex p10 being Point of TOP-REAL 2 st p10=p2 & |.p10.|=1 by A2;
then
A23: q2 in P by A2,A21;
A24: ex p8 being Point of TOP-REAL 2 st p8=p1 & p8 in P & p8 `2>=0 by A18,A20;
q2`2<=0 by A1,A5,A22,Th57;
hence q1 in Upper_Arc(P) & q2 in Lower_Arc(P)& not q2=W-min(P) or q1 in
Upper_Arc(P) & q2 in Upper_Arc(P) & LE q1,q2,Upper_Arc(P),W-min(P),E-max(P) or
q1 in Lower_Arc(P) & q2 in Lower_Arc(P)& not q2=W-min(P) & LE q1,q2,Lower_Arc(P
),E-max(P),W-min(P) by A4,A19,A15,A20,A24,A23,JGRAPH_4:113;
end;
suppose
A25: p1 in Upper_Arc(P) & p2 in Upper_Arc(P) & LE p1,p2,Upper_Arc(P),
W-min(P),E-max(P);
then ex p8 being Point of TOP-REAL 2 st p8=p1 & p8 in P & p8 `2>=0 by A18;
then
A26: p1=(cn-FanMorphS).p1 by JGRAPH_4:113;
ex p9 being Point of TOP-REAL 2 st p9=p2 & p9 in P & p9 `2>=0 by A18,A25;
hence q1 in Upper_Arc(P) & q2 in Lower_Arc(P)& not q2=W-min(P) or q1 in
Upper_Arc(P) & q2 in Upper_Arc(P) & LE q1,q2,Upper_Arc(P),W-min(P),E-max(P) or
q1 in Lower_Arc(P) & q2 in Lower_Arc(P)& not q2=W-min(P) & LE q1,q2,Lower_Arc(P
),E-max(P),W-min(P) by A4,A5,A25,A26,JGRAPH_4:113;
end;
suppose
A27: p1 in Lower_Arc(P) & p2 in Lower_Arc(P)& not p2=W-min(P) & LE p1,
p2,Lower_Arc(P),E-max(P),W-min(P) & not p1 in Upper_Arc(P);
then
A28: ex p8 being Point of TOP-REAL 2 st p8=p1 & p8 in P & p8 `2<=0 by A19;
then
A29: ex p10 being Point of TOP-REAL 2 st p10=p1 & |.p10.|=1 by A2;
A30: ex p9 being Point of TOP-REAL 2 st p9=p2 & p9 in P & p9 `2<=0 by A19,A27;
then
A31: ex p11 being Point of TOP-REAL 2 st p11=p2 & |.p11.|=1 by A2;
A32: q2`2<=0 by A1,A5,A30,Th57;
A33: |.q2.|=|.p2.| by A5,JGRAPH_4:128;
then
A34: q2 in P by A2,A31;
A35: q1`2<=0 by A1,A4,A28,Th57;
A36: |.q1.|=|.p1.| by A4,JGRAPH_4:128;
then
A37: q1 in P by A2,A29;
now
per cases;
case
A38: p1=W-min(P);
then p1=(cn-FanMorphS).p1 by A7,JGRAPH_4:113;
then LE q1,q2,P by A4,A6,A34,A38,JORDAN7:3;
hence
q1 in Upper_Arc(P) & q2 in Lower_Arc(P)& not q2=W-min(P) or q1 in
Upper_Arc(P) & q2 in Upper_Arc(P) & LE q1,q2,Upper_Arc(P),W-min(P),E-max(P) or
q1 in Lower_Arc(P) & q2 in Lower_Arc(P)& not q2=W-min(P) & LE q1,q2,Lower_Arc(P
),E-max(P),W-min(P);
end;
case
A39: p1<>W-min(P);
now
per cases by A2,A3,A28,A39,Th48;
case
A40: p1`1=p2`1;
A41: p2=|[p2`1,p2`2]| by EUCLID:53;
A42: now
assume
A43: p1`2=-p2`2;
then p2`2=0 by A28,A30,XREAL_1:58;
hence p1=p2 by A40,A41,A43,EUCLID:53;
end;
(p1`1)^2+(p1`2)^2=1^2 by A29,JGRAPH_3:1
.=(p1`1)^2+(p2`2)^2 by A31,A40,JGRAPH_3:1;
then
A44: p1`2=p2`2 or p1`2=-p2`2 by SQUARE_1:40;
p1=|[p1`1,p1`2]| by EUCLID:53;
then LE q1,q2,P by A2,A4,A5,A34,A40,A44,A41,A42,JGRAPH_3:26
,JORDAN6:56;
hence q1 in Upper_Arc(P) & q2 in Lower_Arc(P)& not q2=W-min(P) or
q1 in Upper_Arc(P) & q2 in Upper_Arc(P) & LE q1,q2,Upper_Arc(P),W-min(P),E-max(
P) or q1 in Lower_Arc(P) & q2 in Lower_Arc(P)& not q2=W-min(P) & LE q1,q2,
Lower_Arc(P),E-max(P),W-min(P);
end;
case
p1`1>p2`1;
then p1`1/|.p1.|>p2`1/|.p2.| by A29,A31;
then
A45: q1`1/|.q1.|>=q2`1/|.q2.| by A1,A4,A5,A28,A30,A29,A31,Th27;
q2<> W-min(P) by A5,A8,A10,A9,A12,A27,FUNCT_1:def 4;
then LE q1,q2,P by A2,A36,A33,A35,A32,A29,A31,A37,A34,A45,Th56;
hence q1 in Upper_Arc(P) & q2 in Lower_Arc(P)& not q2=W-min(P) or
q1 in Upper_Arc(P) & q2 in Upper_Arc(P) & LE q1,q2,Upper_Arc(P),W-min(P),E-max(
P) or q1 in Lower_Arc(P) & q2 in Lower_Arc(P)& not q2=W-min(P) & LE q1,q2,
Lower_Arc(P),E-max(P),W-min(P);
end;
end;
hence
q1 in Upper_Arc(P) & q2 in Lower_Arc(P)& not q2=W-min(P) or q1 in
Upper_Arc(P) & q2 in Upper_Arc(P) & LE q1,q2,Upper_Arc(P),W-min(P),E-max(P) or
q1 in Lower_Arc(P) & q2 in Lower_Arc(P)& not q2=W-min(P) & LE q1,q2,Lower_Arc(P
),E-max(P),W-min(P);
end;
end;
hence q1 in Upper_Arc(P) & q2 in Lower_Arc(P)& not q2=W-min(P) or q1 in
Upper_Arc(P) & q2 in Upper_Arc(P) & LE q1,q2,Upper_Arc(P),W-min(P),E-max(P) or
q1 in Lower_Arc(P) & q2 in Lower_Arc(P)& not q2=W-min(P) & LE q1,q2,Lower_Arc(P
),E-max(P),W-min(P);
end;
end;
theorem Th59:
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 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
proof
let p1,p2,p3,p4 be Point of TOP-REAL 2, P be compact non empty Subset of
TOP-REAL 2;
assume that
A1: P={p where p is Point of TOP-REAL 2: |.p.|=1} and
A2: LE p1,p2,P and
A3: LE p2,p3,P and
A4: LE p3,p4,P and
A5: p1`1<0 and
A6: p1`2>=0 and
A7: p2`1<0 and
A8: p2`2>=0 and
A9: p3`1<0 and
A10: p3`2>=0 and
A11: p4`1<0 and
A12: p4`2>=0;
consider r being Real such that
A13: p4`1<r and
A14: r<0 by A11,XREAL_1:5;
reconsider r1=r as Real;
set s=sqrt(1-r1^2);
A15: P is being_simple_closed_curve by A1,JGRAPH_3:26;
then p4 in P by A4,JORDAN7:5;
then
A16: ex p being Point of TOP-REAL 2 st p=p4 & |.p.|=1 by A1;
then -1<=p4`1 by Th1;
then -1<=r1 by A13,XXREAL_0:2;
then r1^2<=1^2 by A14,SQUARE_1:49;
then
A17: 1-r1^2>=0 by XREAL_1:48;
then
A18: s^2=1-r1^2 by SQUARE_1:def 2;
then
A19: 1-s^2+s^2>0+s^2 by A14,SQUARE_1:12,XREAL_1:8;
then
A20: -1<s by SQUARE_1:52;
A21: s<1 by A19,SQUARE_1:52;
then consider f1 being Function of TOP-REAL 2,TOP-REAL 2 such that
A22: f1=s-FanMorphW and
A23: f1 is being_homeomorphism by A20,JGRAPH_4:41;
set q11=f1.p1, q22=f1.p2, q33=f1.p3, q44=f1.p4;
A24: s>=0 by A17,SQUARE_1:def 2;
p3 in P by A3,A15,JORDAN7:5;
then
A25: ex p33 being Point of TOP-REAL 2 st p33=p3 & |.p33.|=1 by A1;
then p3`2/|.p3.|<p4`2/|.p4.| or p3=p4 by A1,A4,A11,A12,A16,Th46;
then
A26: q33`2/|.q33.|<q44`2/|.q44.| or p3=p4 by A9,A11,A20,A21,A22,JGRAPH_4:46;
(p4`1)^2>r1^2 by A13,A14,SQUARE_1:44;
then
A27: 1-(p4`1)^2<1-r1^2 by XREAL_1:15;
A28: p3`1<p4`1 or p3=p4 by A1,A4,A9,A10,A12,Th46;
then -(p3`1)>= -(p4`1) by XREAL_1:24;
then (-(p3`1))^2>=(-(p4`1))^2 by A11,SQUARE_1:15;
then 1-((p3`1))^2<=1-((p4`1))^2 by XREAL_1:10;
then
A29: 1-(p3`1)^2< s^2 by A27,A18,XXREAL_0:2;
p2`1<p3`1 or p2=p3 by A1,A3,A7,A8,A10,Th46;
then
A30: p2`1<=p4`1 by A28,XXREAL_0:2;
then -(p2`1)>= -(p4`1) by XREAL_1:24;
then (-(p2`1))^2>=(-(p4`1))^2 by A11,SQUARE_1:15;
then 1-((p2`1))^2<=1-((p4`1))^2 by XREAL_1:10;
then
A31: 1-(p2`1)^2< s^2 by A27,A18,XXREAL_0:2;
p1`1<p2`1 or p1=p2 by A1,A2,A5,A6,A8,Th46;
then p1`1<=p4`1 by A30,XXREAL_0:2;
then -(p1`1)>= -(p4`1) by XREAL_1:24;
then (-(p1`1))^2>=(-(p4`1))^2 by A11,SQUARE_1:15;
then 1-((p1`1))^2<=1-((p4`1))^2 by XREAL_1:10;
then
A32: 1-(p1`1)^2< s^2 by A27,A18,XXREAL_0:2;
1^2=(p3`1)^2+(p3`2)^2 by A25,JGRAPH_3:1;
then
A33: p3`2/|.p3.|<s by A25,A24,A29,SQUARE_1:48;
then
A34: q33`1<0 by A9,A20,A22,JGRAPH_4:43;
p2 in P by A2,A15,JORDAN7:5;
then
A35: ex p22 being Point of TOP-REAL 2 st p22=p2 & |.p22.|=1 by A1;
then
A36: |.q22.|=1 by A22,JGRAPH_4:33;
then
A37: q22 in P by A1;
p2`2/|.p2.|<p3`2/|.p3.| or p2=p3 by A1,A3,A9,A10,A35,A25,Th46;
then
A38: q22`2/|.q22.|<q33`2/|.q33.| or p2=p3 by A7,A9,A20,A21,A22,JGRAPH_4:46;
A39: |.q33.|=1 by A25,A22,JGRAPH_4:33;
then
A40: q33 in P by A1;
1^2=(p2`1)^2+(p2`2)^2 by A35,JGRAPH_3:1;
then
A41: p2`2/|.p2.|<s by A35,A24,A31,SQUARE_1:48;
then
A42: q22`2<0 by A7,A20,A22,JGRAPH_4:43;
A43: q22`1<0 by A7,A20,A22,A41,JGRAPH_4:43;
1^2=(p4`1)^2+(p4`2)^2 by A16,JGRAPH_3:1;
then p4`2/|.p4.|<s by A27,A16,A18,A24,SQUARE_1:48;
then
A44: q44`1<0 & q44`2<0 by A11,A20,A22,JGRAPH_4:43;
p1 in P by A2,A15,JORDAN7:5;
then
A45: ex p11 being Point of TOP-REAL 2 st p11=p1 & |.p11.|=1 by A1;
then p1`2/|.p1.|<p2`2/|.p2.| or p1=p2 by A1,A2,A7,A8,A35,Th46;
then
A46: q11`2/|.q11.|<q22`2/|.q22.| or p1=p2 by A5,A7,A20,A21,A22,JGRAPH_4:46;
1^2=(p1`1)^2+(p1`2)^2 by A45,JGRAPH_3:1;
then
A47: p1`2/|.p1.|<s by A45,A24,A32,SQUARE_1:48;
then
A48: q11`1<0 by A5,A20,A22,JGRAPH_4:43;
A49: |.q11.|=1 by A45,A22,JGRAPH_4:33;
then q11 in P by A1;
then
A50: LE q11,q22,P by A1,A49,A36,A37,A48,A43,A42,A46,Th51;
A51: q22`1<0 & q22`2<0 by A7,A20,A22,A41,JGRAPH_4:43;
A52: q11`1<0 & q11`2<0 by A5,A20,A22,A47,JGRAPH_4:43;
A53: for q being Point of TOP-REAL 2 holds |.(f1.q).|=|.q.| by A22,JGRAPH_4:33;
q33`1<0 & q33`2<0 by A9,A20,A22,A33,JGRAPH_4:43;
then
A54: LE q22,q33,P by A1,A36,A37,A39,A40,A43,A38,Th51;
A55: q33`2<0 by A9,A20,A22,A33,JGRAPH_4:43;
A56: |.q44.|=1 by A16,A22,JGRAPH_4:33;
then q44 in P by A1;
then LE q33,q44,P by A1,A39,A40,A56,A34,A44,A26,Th51;
hence thesis by A23,A53,A52,A51,A34,A55,A44,A50,A54;
end;
theorem Th60:
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
proof
let p1,p2,p3,p4 be Point of TOP-REAL 2, P be compact non empty Subset of
TOP-REAL 2;
assume that
A1: P={p where p is Point of TOP-REAL 2: |.p.|=1} and
A2: LE p1,p2,P and
A3: LE p2,p3,P and
A4: LE p3,p4,P and
A5: p1`2>=0 and
A6: p2`2>=0 and
A7: p3`2>=0 and
A8: p4`2>0;
A9: P is being_simple_closed_curve by A1,JGRAPH_3:26;
then p4 in P by A4,JORDAN7:5;
then
A10: ex p being Point of TOP-REAL 2 st p=p4 & |.p.|=1 by A1;
A11: now
assume p4`1=1;
then 1^2=1+(p4`2)^2 by A10,JGRAPH_3:1;
hence contradiction by A8,XCMPLX_1:6;
end;
p4`1<=1 by A10,Th1;
then p4`1<1 by A11,XXREAL_0:1;
then consider r being Real such that
A12: p4`1<r and
A13: r<1 by XREAL_1:5;
reconsider r1=r as Real;
-1<=p4`1 by A10,Th1;
then
A14: -1<r1 by A12,XXREAL_0:2;
then consider f1 being Function of TOP-REAL 2,TOP-REAL 2 such that
A15: f1=r1-FanMorphN and
A16: f1 is being_homeomorphism by A13,JGRAPH_4:74;
set q11=f1.p1, q22=f1.p2, q33=f1.p3, q44=f1.p4;
A17: for q being Point of TOP-REAL 2 holds |.(f1.q).|=|.q.| by A15,JGRAPH_4:66;
A18: p3`1<p4`1 or p3=p4 by A1,A4,A8,Th47;
then
A19: p3`1<r1 by A12,XXREAL_0:2;
p3 in P by A3,A9,JORDAN7:5;
then
A20: ex p33 being Point of TOP-REAL 2 st p33=p3 & |.p33.|=1 by A1;
then p3`1/|.p3.|<p4`1/|.p4.| or p3=p4 by A1,A4,A8,A10,Th47;
then
A21: q33`1/|.q33.|<q44`1/|.q44.| or p3=p4 by A7,A8,A10,A20,A13,A14,A15,Th21;
A22: p3`1/|.p3.|<r1 by A20,A12,A18,XXREAL_0:2;
then
A23: q33`2>=0 by A7,A20,A13,A14,A15,Th20;
A24: p1`1<p2`1 or p1=p2 by A1,A2,A6,Th47;
p4`1/|.p4.|<r1 by A10,A12;
then
A25: q44`1<0 & q44`2>0 by A8,A14,A15,JGRAPH_4:76;
p2 in P by A2,A9,JORDAN7:5;
then
A26: ex p22 being Point of TOP-REAL 2 st p22=p2 & |.p22.|=1 by A1;
then
A27: |.q22.|=1 by A15,JGRAPH_4:66;
then
A28: q22 in P by A1;
A29: p2`1<p3`1 or p2=p3 by A1,A3,A7,Th47;
then
A30: p2`1/|.p2.|<r1 by A26,A19,XXREAL_0:2;
then
A31: q22`2>=0 by A6,A26,A13,A14,A15,Th20;
p1 in P by A2,A9,JORDAN7:5;
then
A32: ex p11 being Point of TOP-REAL 2 st p11=p1 & |.p11.|=1 by A1;
then p1`1/|.p1.|<p2`1/|.p2.| or p1=p2 by A1,A2,A6,A26,Th47;
then
A33: q11`1/|.q11.|<q22`1/|.q22.| or p1=p2 by A5,A6,A32,A26,A13,A14,A15,Th21;
p2`1<r1 by A29,A19,XXREAL_0:2;
then
A34: p1`1/|.p1.|<r1 by A32,A24,XXREAL_0:2;
then
A35: q11`2>=0 by A5,A32,A13,A14,A15,Th20;
A36: q22`1<0 by A6,A26,A13,A14,A15,A30,Th20;
A37: |.q11.|=1 by A32,A15,JGRAPH_4:66;
then q11 in P by A1;
then
A38: LE q11,q22,P by A1,A37,A27,A28,A31,A36,A35,A33,Th53;
A39: |.q33.|=1 by A20,A15,JGRAPH_4:66;
then
A40: q33 in P by A1;
A41: q33`1<0 by A7,A20,A13,A14,A15,A22,Th20;
A42: q22`1<0 & q22`2>=0 by A6,A26,A13,A14,A15,A30,Th20;
A43: q11`1<0 & q11`2>=0 or q11`1<0 & q11`2=0 by A5,A32,A13,A14,A15,A34,Th20;
A44: |.q44.|=1 by A10,A15,JGRAPH_4:66;
then q44 in P by A1;
then
A45: LE q33,q44,P by A1,A39,A40,A44,A25,A23,A21,Th53;
p2`1/|.p2.|<p3`1/|.p3.| or p2=p3 by A1,A3,A7,A26,A20,Th47;
then q22`1/|.q22.|<q33`1/|.q33.| or p2=p3 by A6,A7,A26,A20,A13,A14,A15,Th21;
then LE q22,q33,P by A1,A27,A28,A39,A40,A31,A23,A41,Th53;
hence thesis by A16,A17,A25,A43,A42,A38,A23,A41,A45;
end;
theorem Th61:
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
proof
let p1,p2,p3,p4 be Point of TOP-REAL 2, P be compact non empty Subset of
TOP-REAL 2;
assume that
A1: P={p where p is Point of TOP-REAL 2: |.p.|=1} and
A2: LE p1,p2,P & LE p2,p3,P & LE p3,p4,P & p1`2>=0 & p2`2>=0 & p3`2>=0 &
p4`2>0;
consider f being Function of TOP-REAL 2,TOP-REAL 2, q1,q2,q3,q4 being Point
of TOP-REAL 2 such that
A3: f is being_homeomorphism and
A4: for q being Point of TOP-REAL 2 holds |.(f.q).|=|.q.| and
A5: q1=f.p1 & q2=f.p2 and
A6: q3=f.p3 & q4=f.p4 and
A7: 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 by A1,A2,Th60;
consider f2 being Function of TOP-REAL 2,TOP-REAL 2, q1b,q2b,q3b,q4b being
Point of TOP-REAL 2 such that
A8: f2 is being_homeomorphism and
A9: for q being Point of TOP-REAL 2 holds |.(f2.q).|=|.q.| and
A10: q1b=f2.q1 & q2b=f2.q2 and
A11: q3b=f2.q3 & q4b=f2.q4 and
A12: q1b`1<0 & q1b`2<0 & q2b`1<0 & q2b`2<0 & q3b`1<0 & q3b`2<0 & q4b`1<
0 & q4b`2<0 & LE q1b,q2b,P & LE q2b,q3b,P & LE q3b,q4b,P by A1,A7,Th59;
reconsider f3=f2*f as Function of TOP-REAL 2,TOP-REAL 2;
A13: f3 is being_homeomorphism by A3,A8,TOPS_2:57;
A14: dom f=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
then
A15: f3.p3=q3b & f3.p4=q4b by A6,A11,FUNCT_1:13;
A16: for q being Point of TOP-REAL 2 holds |.(f3.q).|=|.q.|
proof
let q be Point of TOP-REAL 2;
dom f=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
then f3.q=f2.(f.q) by FUNCT_1:13;
hence |.f3.q.|=|.(f.q).| by A9
.=|.q.| by A4;
end;
f3.p1=q1b & f3.p2=q2b by A5,A10,A14,FUNCT_1:13;
hence thesis by A12,A13,A16,A15;
end;
theorem Th62:
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
proof
let p1,p2,p3,p4 be Point of TOP-REAL 2, P be compact non empty Subset of
TOP-REAL 2;
assume that
A1: P={p where p is Point of TOP-REAL 2: |.p.|=1} and
A2: LE p1,p2,P and
A3: LE p2,p3,P and
A4: LE p3,p4,P and
A5: p1`2>=0 or p1`1>=0 and
A6: p2`2>=0 or p2`1>=0 and
A7: p3`2>=0 or p3`1>=0 and
A8: p4`2>0 or p4`1>0;
A9: P is being_simple_closed_curve by A1,JGRAPH_3:26;
then
A10: p4 in P by A4,JORDAN7:5;
then
A11: ex p44 being Point of TOP-REAL 2 st p44=p4 & |.p44.|=1 by A1;
then
A12: -1<=p4`2 by Th1;
now
assume
A13: p4`2=-1;
1^2=(p4`1)^2+(p4`2)^2 by A11,JGRAPH_3:1
.=(p4`1)^2+1 by A13;
hence contradiction by A8,A13,XCMPLX_1:6;
end;
then p4`2> -1 by A12,XXREAL_0:1;
then consider r being Real such that
A14: -1<r and
A15: r<p4`2 by XREAL_1:5;
reconsider r1=r as Real;
p4`2<=1 by A11,Th1;
then
A16: r1<1 by A15,XXREAL_0:2;
then consider f1 being Function of TOP-REAL 2,TOP-REAL 2 such that
A17: f1=r1-FanMorphE and
A18: f1 is being_homeomorphism by A14,JGRAPH_4:105;
set q11=f1.p1, q22=f1.p2, q33=f1.p3, q44=f1.p4;
A19: |.q44.|=1 by A11,A17,JGRAPH_4:97;
then
A20: q44 in P by A1;
A21: p1 in P by A2,A9,JORDAN7:5;
then
A22: ex p11 being Point of TOP-REAL 2 st p11=p1 & |.p11.|=1 by A1;
then
A23: |.q11.|=1 by A17,JGRAPH_4:97;
then
A24: q11 in P by A1;
A25: p3 in P by A3,A9,JORDAN7:5;
then
A26: ex p33 being Point of TOP-REAL 2 st p33=p3 & |.p33.|=1 by A1;
then
A27: |.q33.|=1 by A17,JGRAPH_4:97;
then
A28: q33 in P by A1;
A29: p2 in P by A2,A9,JORDAN7:5;
then
A30: ex p22 being Point of TOP-REAL 2 st p22=p2 & |.p22.|=1 by A1;
then
A31: |.q22.|=1 by A17,JGRAPH_4:97;
then
A32: q22 in P by A1;
now
per cases;
case
A33: p4`2<=0;
A34: Upper_Arc(P) ={p7 where p7 is Point of TOP-REAL 2:p7 in P & p7`2>=0
} by A1,Th34;
A35: p4`2/|.p4.|>r1 by A11,A15;
then
A36: q44`1>0 by A8,A15,A17,A33,JGRAPH_4:106;
A37: now
set q8=|[sqrt(1-r1^2),r1]|;
assume
A38: q44`2=0;
1^2=(q44`1)^2+(q44`2)^2 by A19,JGRAPH_3:1
.=(q44`1)^2 by A38;
then q44`1=-1 or q44`1=1 by SQUARE_1:41;
then
A39: q44=|[1,0]| by A8,A15,A17,A33,A35,A38,EUCLID:53,JGRAPH_4:106;
set r8=f1.q8;
1^2>r1^2 by A14,A16,SQUARE_1:50;
then
A40: 1-r1^2>0 by XREAL_1:50;
A41: q8`1=sqrt(1-r1^2) by EUCLID:52;
then
A42: q8`1>0 by A40,SQUARE_1:25;
q8`2=r1 by EUCLID:52;
then |.q8.|=sqrt((sqrt(1-r1^2))^2+r1^2)by A41,JGRAPH_3:1;
then
A43: |.q8.|=sqrt((1-r1^2)+r1^2) by A40,SQUARE_1:def 2
.=1;
then
A44: q8`2/|.q8.|=r1 by EUCLID:52;
then
A45: r8`2=0 by A17,A42,JGRAPH_4:111;
|.r8.|=1 by A17,A43,JGRAPH_4:97;
then 1^2=(r8`1)^2+(r8`2)^2 by JGRAPH_3:1
.=(r8`1)^2 by A45;
then r8`1=-1 or r8`1=1 by SQUARE_1:41;
then
A46: f1.(|[sqrt(1-r1^2),r1]|)=|[1,0]| by A17,A44,A42,A45,EUCLID:53
,JGRAPH_4:111;
f1 is one-to-one & dom f1=the carrier of TOP-REAL 2 by A14,A16,A17,
FUNCT_2:def 1,JGRAPH_4:102;
then p4=|[sqrt(1-r1^2),r1]| by A39,A46,FUNCT_1:def 4;
hence contradiction by A15,EUCLID:52;
end;
A47: q44`2>=0 by A8,A15,A17,A33,A35,JGRAPH_4:106;
A48: Lower_Arc(P) ={p7 where p7 is Point of TOP-REAL 2:p7 in P & p7`2<=0
} by A1,Th35;
A49: now
per cases;
case
A50: p3`1<=0;
then
A51: q33=p3 by A17,JGRAPH_4:82;
A52: now
per cases by A50;
case
A53: p3`1=0;
A54: now
assume q33`2=-1;
then -1>=p4`2 by A1,A4,A7,A8,A33,A51,Th50;
hence contradiction by A14,A15,XXREAL_0:2;
end;
1^2=0^2+(q33`2)^2 by A26,A51,A53,JGRAPH_3:1
.=(q33`2)^2;
hence q33`2>=0 by A54,SQUARE_1:41;
end;
case
p3`1<0;
hence q33`2>=0 by A7,A17,JGRAPH_4:82;
end;
end;
now
per cases;
case
A55: p2<> W-min(P);
A56: now
A57: p3 in Upper_Arc(P) by A25,A34,A51,A52;
assume
A58: p2`2<0;
then p2 in Lower_Arc(P) by A29,A48;
then LE p3,p2,P by A55,A57;
hence contradiction by A1,A3,A51,A52,A58,JGRAPH_3:26,JORDAN6:57
;
end;
A59: p2`1<=p3`1 by A1,A3,A51,A52,Th47;
then
A60: q22=p2 by A17,A50,JGRAPH_4:82;
now
per cases;
case
A61: p1<> W-min(P);
A62: now
A63: p2 in Upper_Arc(P) by A29,A34,A56;
assume
A64: p1`2<0;
then p1 in Lower_Arc(P) by A21,A48;
then LE p2,p1,P by A61,A63;
hence contradiction by A1,A2,A56,A64,JGRAPH_3:26,JORDAN6:57
;
end;
p1`1<=p2`1 by A1,A2,A56,Th47;
hence
q11`2>=0 & q22`2>=0 & q33`2>=0 & q44`2>0 & LE q11,q22,P
& LE q22,q33,P & LE q33,q44,P by A1,A2,A3,A17,A28,A20,A36,A47,A37,A51,A52,A56
,A59,A60,A62,Th54,JGRAPH_4:82;
end;
case
A65: p1=W-min(P);
A66: W-min(P)=|[-1,0]| by A1,Th29;
then p1`1=-1 by A65,EUCLID:52;
then p1=q11 by A17,JGRAPH_4:82;
hence
q11`2>=0 & q22`2>=0 & q33`2>=0 & q44`2>0 & LE q11,q22,P
& LE q22,q33,P & LE q33,q44,P by A1,A2,A3,A25,A17,A20,A36,A47,A37,A51,A52,A56
,A59,A65,A66,Th54,EUCLID:52,JGRAPH_4:82;
end;
end;
hence q11`2>=0 & q22`2>=0 & q33`2>=0 & q44`2>0 & LE q11,q22,P &
LE q22,q33,P & LE q33,q44,P;
end;
case
A67: p2=W-min(P);
W-min(P)=|[-1,0]| by A1,Th29;
then
A68: p2`1=-1 by A67,EUCLID:52;
then p2=q22 & p1`1<=p2`1 by A1,A2,A6,A17,Th47,JGRAPH_4:82;
hence q11`2>=0 & q22`2>=0 & q33`2>=0 & q44`2>0 & LE q11,q22,P &
LE q22,q33,P & LE q33,q44,P by A1,A2,A3,A5,A6,A14,A15,A17,A28,A20,A33,A36,A47
,A37,A51,A52,A68,Th54,JGRAPH_4:82;
end;
end;
hence q11`2>=0 & q22`2>=0 & q33`2>=0 & q44`2>0 & LE q11,q22,P & LE
q22,q33,P & LE q33,q44,P;
end;
case
A69: p3`1>0;
A70: now
per cases;
case
A71: p3<>p4;
A72: now
A73: LE p2,p4,P by A1,A3,A4,JGRAPH_3:26,JORDAN6:58;
assume that
A74: p2`1=0 and
A75: p2`2=-1;
p2`2<=p4`2 by A11,A75,Th1;
then LE p4,p2,P by A1,A8,A29,A10,A33,A74,Th55;
hence contradiction by A1,A8,A74,A75,A73,JGRAPH_3:26,JORDAN6:57
;
end;
p3`2>p4`2 by A1,A4,A8,A33,A69,A71,Th50;
then
A76: p3`2/|.p3.|>=r1 by A26,A15,XXREAL_0:2;
then
A77: q33`1>0 by A16,A17,A69,JGRAPH_4:106;
A78: q33`2>=0 by A16,A17,A69,A76,JGRAPH_4:106;
A79: now
assume p2`1=0;
then 1^2=0^2+(p2`2)^2 by A30,JGRAPH_3:1;
hence p2`2=1 or p2`2=-1 by SQUARE_1:40;
end;
A80: now
per cases by A6,A79,A72;
case
A81: p2`1<=0 & p2`2>=0;
then q22=p2 by A17,JGRAPH_4:82;
hence q22`2>=0 & LE q22,q33,P by A1,A29,A28,A77,A78,A81,Th54;
end;
case
A82: p2`1>0;
then
A83: q22`1>0 by A14,A16,A17,Th22;
now
per cases;
case
p2=p3;
hence
q22`2>=0 & LE q22,q33,P by A9,A16,A17,A28,A69,A76,
JGRAPH_4:106,JORDAN6:56;
end;
case
p2<>p3;
then p2`2/|.p2.|>p3`2/|.p3.| by A1,A3,A30,A26,A69,A82
,Th50;
then q22`2/|.q22.|>q33`2/|.q33.| by A30,A26,A14,A16,A17
,A69,A82,Th24;
hence q22`2>=0 & LE q22,q33,P by A1,A16,A17,A31,A32,A27
,A28,A69,A76,A77,A83,Th55,JGRAPH_4:106;
end;
end;
hence q22`2>=0 & LE q22,q33,P;
end;
end;
p3`2/|.p3.|>p4`2/|.p4.| by A1,A4,A8,A11,A26,A33,A69,A71,Th50;
then q33`2/|.q33.|>q44`2/|.q44.| by A8,A11,A26,A14,A15,A17,A33
,A69,Th24;
then (q33`2) ^2 > ((q44`2))^2 by A27,A19,A47,SQUARE_1:16;
then
A84: 1^2- ((q33`2))^2 < 1^2-((q44`2))^2 by XREAL_1:15;
1^2=(q44`1)^2+(q44`2)^2 by A19,JGRAPH_3:1;
then
A85: (q44`1)=sqrt(1^2-((q44`2))^2) by A36,SQUARE_1:22;
A86: 1^2=(q33`1)^2+(q33`2)^2 by A27,JGRAPH_3:1;
then (q33`1)=sqrt(1^2-((q33`2))^2) by A77,SQUARE_1:22;
then q33`1< q44`1 by A86,A85,A84,SQUARE_1:27,XREAL_1:63;
hence
q22`2>=0 & LE q22,q33,P & q33`2>=0 & LE q33,q44,P by A1,A28,A20
,A47,A78,A80,Th54;
end;
case
A87: p3=p4;
A88: now
A89: LE p2,p4,P by A1,A3,A4,JGRAPH_3:26,JORDAN6:58;
assume
A90: p2`1=0 & p2`2=-1;
then LE p4,p2,P by A1,A8,A29,A10,A12,A33,Th55;
hence contradiction by A1,A8,A90,A89,JGRAPH_3:26,JORDAN6:57;
end;
A91: now
assume p2`1=0;
then 1^2=0^2+(p2`2)^2 by A30,JGRAPH_3:1;
hence p2`2=1 or p2`2=-1 by SQUARE_1:40;
end;
A92: p3`2/|.p3.|>=r1 by A26,A15,A87;
then
A93: q33`1>0 by A16,A17,A69,JGRAPH_4:106;
A94: q33`2>=0 by A16,A17,A69,A92,JGRAPH_4:106;
now
per cases by A6,A91,A88;
case
A95: p2`1<=0 & p2`2>=0;
then q22=p2 by A17,JGRAPH_4:82;
hence q22`2>=0 & LE q22,q33,P by A1,A29,A28,A93,A94,A95,Th54;
end;
case
A96: p2`1>0;
then
A97: q22`1>0 by A14,A16,A17,Th22;
now
per cases;
case
p2=p3;
hence
q22`2>=0 & LE q22,q33,P by A9,A16,A17,A28,A69,A92,
JGRAPH_4:106,JORDAN6:56;
end;
case
p2<>p3;
then p2`2/|.p2.|>p3`2/|.p3.| by A1,A3,A30,A26,A69,A96
,Th50;
then q22`2/|.q22.|>q33`2/|.q33.| by A30,A26,A14,A16,A17
,A69,A96,Th24;
hence q22`2>=0 & LE q22,q33,P by A1,A16,A17,A31,A32,A27
,A28,A69,A92,A93,A97,Th55,JGRAPH_4:106;
end;
end;
hence q22`2>=0 & LE q22,q33,P;
end;
end;
hence
q22`2>=0 & LE q22,q33,P & q33`2>=0 & LE q33,q44,P by A1,A28,A36
,A47,A87,Th54;
end;
end;
A98: now
LE p1,p3,P by A1,A2,A3,JGRAPH_3:26,JORDAN6:58;
then
A99: LE p1,p4,P by A1,A4,JGRAPH_3:26,JORDAN6:58;
assume
A100: p1`1=0 & p1`2=-1;
then LE p4,p1,P by A1,A8,A21,A10,A12,A33,Th55;
hence contradiction by A1,A8,A100,A99,JGRAPH_3:26,JORDAN6:57;
end;
A101: now
assume p2`1=0;
then 1^2=0^2+(p2`2)^2 by A30,JGRAPH_3:1;
hence p2`2=1 or p2`2=-1 by SQUARE_1:40;
end;
A102: now
A103: LE p2,p4,P by A1,A3,A4,JGRAPH_3:26,JORDAN6:58;
assume that
A104: p2`1=0 and
A105: p2`2=-1;
p2`2<=p4`2 by A11,A105,Th1;
then LE p4,p2,P by A1,A8,A29,A10,A33,A104,Th55;
hence contradiction by A1,A8,A104,A105,A103,JGRAPH_3:26,JORDAN6:57;
end;
A106: now
assume p1`1=0;
then 1^2=0^2+(p1`2)^2 by A22,JGRAPH_3:1;
hence p1`2=1 or p1`2=-1 by SQUARE_1:40;
end;
now
per cases by A5,A106,A98;
case
A107: p1`1<=0 & p1`2>=0;
then
A108: p1=q11 by A17,JGRAPH_4:82;
A109: q11`2>=0 by A17,A107,JGRAPH_4:82;
now
per cases by A6,A101,A102;
case
p2`1<=0 & p2`2>=0;
hence q11`2>=0 & LE q11,q22,P by A2,A17,A107,A108,JGRAPH_4:82
;
end;
case
p2`1>0;
then q11`1<q22`1 by A14,A16,A17,A107,A108,Th22;
hence q11`2>=0 & LE q11,q22,P by A1,A24,A32,A70,A109,Th54;
end;
end;
hence q11`2>=0 & LE q11,q22,P;
end;
case
A110: p1`1>0;
then
A111: q11`1>0 by A14,A16,A17,Th22;
now
per cases by A6,A101,A102;
case
A112: p2`1<=0 & p2`2>=0;
now
A113: p2 in Upper_Arc(P) by A29,A34,A112;
assume
A114: p1`2<0;
W-min(P)=|[-1,0]| by A1,Th29;
then
A115: p1<>W-min(P) by A114,EUCLID:52;
p1 in Lower_Arc(P) by A21,A48,A114;
then LE p2,p1,P by A113,A115;
hence contradiction by A1,A2,A110,A112,JGRAPH_3:26
,JORDAN6:57;
end;
then LE p2,p1,P by A1,A21,A29,A110,A112,Th54;
then q11=q22 by A1,A2,JGRAPH_3:26,JORDAN6:57;
hence q11`2>=0 & LE q11,q22,P by A9,A17,A24,A112,JGRAPH_4:82
,JORDAN6:56;
end;
case
A116: p2`1>0;
then
A117: q22`1>0 by A14,A16,A17,Th22;
now
per cases;
case
p1=p2;
hence q11`2>=0 & LE q11,q22,P by A1,A24,A70,JGRAPH_3:26
,JORDAN6:56;
end;
case
p1<>p2;
then p1`2/|.p1.|>p2`2/|.p2.| by A1,A2,A22,A30,A110,A116
,Th50;
then q11`2/|.q11.|>q22`2/|.q22.| by A22,A30,A14,A16,A17
,A110,A116,Th24;
hence
q11`2>=0 & LE q11,q22,P by A1,A23,A24,A31,A32,A70,A111
,A117,Th55;
end;
end;
hence q11`2>=0 & LE q11,q22,P;
end;
end;
hence q11`2>=0 & LE q11,q22,P;
end;
end;
hence q11`2>=0 & q22`2>=0 & q33`2>=0 & q44`2>0 & LE q11,q22,P & LE
q22,q33,P & LE q33,q44,P by A8,A15,A17,A33,A35,A37,A70,JGRAPH_4:106;
end;
end;
for q being Point of TOP-REAL 2 holds |.(f1.q).|=|.q.| by A17,JGRAPH_4:97
;
hence thesis by A18,A49;
end;
case
A118: p4`2>0;
A119: Lower_Arc(P)={p8 where p8 is Point of TOP-REAL 2:p8 in P & p8`2<=0
} by A1,Th35;
A120: now
assume p4 in Lower_Arc(P);
then ex p9 being Point of TOP-REAL 2 st p9=p4 & p9 in P & p9 `2<=0 by
A119;
hence contradiction by A118;
end;
A121: Upper_Arc(P)={p7 where p7 is Point of TOP-REAL 2:p7 in P & p7`2>=0
} by A1,Th34;
p3 in Upper_Arc(P) & p4 in Lower_Arc(P)& not p4=W-min(P) or p3 in
Upper_Arc(P) & p4 in Upper_Arc(P) & LE p3,p4,Upper_Arc(P),W-min(P),E-max(P) or
p3 in Lower_Arc(P) & p4 in Lower_Arc(P)& not p4=W-min(P) & LE p3,p4,Lower_Arc(P
),E-max(P),W-min(P) by A4;
then
A122: ex p33 being Point of TOP-REAL 2 st p33=p3 & p33 in P & p33`2>=0 by A121
,A120;
set f4=id (TOP-REAL 2);
A123: f4.p3=p3 & f4.p4=p4;
A124: for q being Point of TOP-REAL 2 holds |.(f4.q).|=|.q.|;
A125: LE p2,p4,P by A1,A3,A4,JGRAPH_3:26,JORDAN6:58;
then p2 in Upper_Arc(P) & p4 in Lower_Arc(P)& not p4=W-min(P) or p2 in
Upper_Arc(P) & p4 in Upper_Arc(P) & LE p2,p4,Upper_Arc(P),W-min(P),E-max(P) or
p2 in Lower_Arc(P) & p4 in Lower_Arc(P)& not p4=W-min(P) & LE p2,p4,Lower_Arc(P
),E-max(P),W-min(P);
then
A126: ex p22 being Point of TOP-REAL 2 st p22=p2 & p22 in P & p22`2>=0 by A121
,A120;
LE p1,p4,P by A1,A2,A125,JGRAPH_3:26,JORDAN6:58;
then p1 in Upper_Arc(P) & p4 in Lower_Arc(P)& not p4=W-min(P) or p1 in
Upper_Arc(P) & p4 in Upper_Arc(P) & LE p1,p4,Upper_Arc(P),W-min(P),E-max(P) or
p1 in Lower_Arc(P) & p4 in Lower_Arc(P)& not p4=W-min(P) & LE p1,p4,Lower_Arc(P
),E-max(P),W-min(P);
then
A127: ex p11 being Point of TOP-REAL 2 st p11=p1 & p11 in P & p11`2>=0 by A121
,A120;
f4.p1=p1 & f4.p2=p2;
hence thesis by A2,A3,A4,A118,A122,A126,A127,A123,A124;
end;
end;
hence thesis;
end;
theorem Th63:
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
proof
let p1,p2,p3,p4 be Point of TOP-REAL 2, P be compact non empty Subset of
TOP-REAL 2;
assume that
A1: P={p where p is Point of TOP-REAL 2: |.p.|=1} and
A2: 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);
consider f being Function of TOP-REAL 2,TOP-REAL 2, q1,q2,q3,q4 being Point
of TOP-REAL 2 such that
A3: f is being_homeomorphism and
A4: for q being Point of TOP-REAL 2 holds |.(f.q).|=|.q.| and
A5: q1=f.p1 & q2=f.p2 and
A6: q3=f.p3 & q4=f.p4 and
A7: q1`2>=0 & q2`2>=0 & q3`2>=0 & q4`2>0 & LE q1,q2,P & LE q2,q3,P & LE
q3,q4,P by A1,A2,Th62;
consider f2 being Function of TOP-REAL 2,TOP-REAL 2, q1b,q2b,q3b,q4b being
Point of TOP-REAL 2 such that
A8: f2 is being_homeomorphism and
A9: for q being Point of TOP-REAL 2 holds |.(f2.q).|=|.q.| and
A10: q1b=f2.q1 & q2b=f2.q2 and
A11: q3b=f2.q3 & q4b=f2.q4 and
A12: q1b`1<0 & q1b`2<0 & q2b`1<0 & q2b`2<0 & q3b`1<0 & q3b`2<0 & q4b`1<
0 & q4b`2<0 & LE q1b,q2b,P & LE q2b,q3b,P & LE q3b,q4b,P by A1,A7,Th61;
reconsider f3=f2*f as Function of TOP-REAL 2,TOP-REAL 2;
A13: f3 is being_homeomorphism by A3,A8,TOPS_2:57;
A14: dom f=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
then
A15: f3.p3=q3b & f3.p4=q4b by A6,A11,FUNCT_1:13;
A16: for q being Point of TOP-REAL 2 holds |.(f3.q).|=|.q.|
proof
let q be Point of TOP-REAL 2;
dom f=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
then f3.q=f2.(f.q) by FUNCT_1:13;
hence |.f3.q.|=|.(f.q).| by A9
.=|.q.| by A4;
end;
f3.p1=q1b & f3.p2=q2b by A5,A10,A14,FUNCT_1:13;
hence thesis by A12,A13,A16,A15;
end;
theorem
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
proof
let p1,p2,p3,p4 be Point of TOP-REAL 2, P be compact non empty Subset of
TOP-REAL 2;
assume that
A1: P={p where p is Point of TOP-REAL 2: |.p.|=1} and
A2: p4=W-min(P) and
A3: LE p1,p2,P and
A4: LE p2,p3,P and
A5: LE p3,p4,P;
A6: Upper_Arc(P) ={p7 where p7 is Point of TOP-REAL 2:p7 in P & p7`2>=0} by A1
,Th34;
A7: W-min(P)=|[-1,0]| by A1,Th29;
then
A8: (W-min(P))`2=0 by EUCLID:52;
A9: P is being_simple_closed_curve by A1,JGRAPH_3:26;
then p4 in P by A5,JORDAN7:5;
then
A10: p4 in Upper_Arc(P) by A2,A6,A8;
A11: Upper_Arc(P) is_an_arc_of W-min(P),E-max(P) by A9,JORDAN6:def 8;
A12: p3 in Upper_Arc(P) by A1,A5,A10,Th44;
then LE p4,p3,Upper_Arc(P),W-min(P),E-max(P) by A2,A11,JORDAN5C:10;
then LE p4,p3,P by A10,A12;
then
A13: p3=p4 by A1,A5,JGRAPH_3:26,JORDAN6:57;
A14: LE p2,p4,P by A1,A4,A5,JGRAPH_3:26,JORDAN6:58;
A15: p2 in Upper_Arc(P) by A1,A4,A12,Th44;
then LE p4,p2,Upper_Arc(P),W-min(P),E-max(P) by A2,A11,JORDAN5C:10;
then LE p4,p2,P by A10,A15;
then
A16: p2=p4 by A1,A14,JGRAPH_3:26,JORDAN6:57;
A17: (W-min(P))`1=-1 by A7,EUCLID:52;
A18: p1 in Upper_Arc(P) by A1,A3,A15,Th44;
then LE p4,p1,Upper_Arc(P),W-min(P),E-max(P) by A2,A11,JORDAN5C:10;
then LE p4,p1,P by A10,A18;
then p1=p4 by A1,A3,A16,JGRAPH_3:26,JORDAN6:57;
hence thesis by A1,A2,A3,A17,A8,A13,A16,Th59;
end;
theorem Th65:
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
proof
let p1,p2,p3,p4 be Point of TOP-REAL 2, P be compact non empty Subset of
TOP-REAL 2;
assume that
A1: P={p where p is Point of TOP-REAL 2: |.p.|=1} and
A2: LE p1,p2,P and
A3: LE p2,p3,P and
A4: LE p3,p4,P;
A5: Lower_Arc(P) ={p7 where p7 is Point of TOP-REAL 2:p7 in P & p7`2<=0} by A1
,Th35;
A6: W-min(P)=|[-1,0]| by A1,Th29;
then
A7: (W-min(P))`2=0 by EUCLID:52;
A8: P is being_simple_closed_curve by A1,JGRAPH_3:26;
then
A9: p1 in P by A2,JORDAN7:5;
A10: Upper_Arc(P) is_an_arc_of W-min(P),E-max(P) by A8,JORDAN6:def 8;
A11: Upper_Arc(P) ={p7 where p7 is Point of TOP-REAL 2:p7 in P & p7`2>=0} by A1
,Th34;
A12: p4 in P by A4,A8,JORDAN7:5;
then
A13: ex p44 being Point of TOP-REAL 2 st p44=p4 & |.p44.|=1 by A1;
then
A14: p4`1<=1 by Th1;
A15: -1<=p4`1 by A13,Th1;
now
per cases;
case
A16: p4`1=-1;
1^2=(p4`1)^2+(p4`2)^2 by A13,JGRAPH_3:1
.=(p4`2)^2+1 by A16;
then
A17: p4`2=0 by XCMPLX_1:6;
then
A18: p4 in Upper_Arc(P) by A12,A11;
A19: p4=W-min(P) by A6,A16,A17,EUCLID:53;
A20: now
per cases;
case
A21: p1 in Upper_Arc(P);
then LE p4,p1,Upper_Arc(P),W-min(P),E-max(P) by A10,A19,JORDAN5C:10;
hence LE p4,p1,P by A18,A21;
end;
case
not p1 in Upper_Arc(P);
then
A22: p1`2<0 by A9,A11;
then p1 in Lower_Arc(P) by A9,A5;
hence LE p4,p1,P by A7,A18,A22;
end;
end;
then
A23: LE p4,p2,P by A1,A2,JGRAPH_3:26,JORDAN6:58;
then LE p4,p3,P by A1,A3,JGRAPH_3:26,JORDAN6:58;
then
A24: p3=p4 by A1,A4,JGRAPH_3:26,JORDAN6:57;
LE p2,p4,P by A1,A3,A4,JGRAPH_3:26,JORDAN6:58;
then
A25: p2=p4 by A1,A23,JGRAPH_3:26,JORDAN6:57;
LE p1,p3,P by A1,A2,A3,JGRAPH_3:26,JORDAN6:58;
then LE p1,p4,P by A1,A4,JGRAPH_3:26,JORDAN6:58;
then p4=p1 by A1,A20,JGRAPH_3:26,JORDAN6:57;
hence thesis by A1,A2,A16,A17,A25,A24,Th59;
end;
case
A26: p4`1<>-1;
then p4`1> -1 by A15,XXREAL_0:1;
then consider r being Real such that
A27: -1<r and
A28: r<p4`1 by XREAL_1:5;
reconsider r1=r as Real;
A29: r1<1 by A14,A28,XXREAL_0:2;
then consider f1 being Function of TOP-REAL 2,TOP-REAL 2 such that
A30: f1=r1-FanMorphS and
A31: f1 is being_homeomorphism by A27,JGRAPH_4:136;
set q11=f1.p1, q22=f1.p2, q33=f1.p3, q44=f1.p4;
now
per cases;
case
A32: p4`1>0 or p4`2>=0;
A33: now
assume that
A34: p4`2=0 and
A35: p4`1<=0;
1^2 =(p4`1)^2+(p4`2)^2 by A13,JGRAPH_3:1
.=(p4`1)^2 by A34;
hence contradiction by A26,A35,SQUARE_1:40;
end;
A36: p3`1>=0 or p3`2>=0 by A1,A4,A32,Th49;
then
A37: p2`1>=0 or p2`2>=0 by A1,A3,Th49;
then p1`1>=0 or p1`2>=0 by A1,A2,Th49;
hence thesis by A1,A2,A3,A4,A32,A36,A37,A33,Th63;
end;
case
A38: p4`1<=0 & p4`2<0;
p4`1/|.p4.|>r1 by A13,A28;
then
A39: q44`1>0 by A27,A28,A30,A38,Th26;
A40: LE q33,q44,P by A1,A4,A27,A29,A30,Th58;
W-min(P)=|[-1,0]| by A1,Th29;
then
A41: (W-min(P))`2=0 by EUCLID:52;
A42: now
per cases;
case
q33`2>=0;
hence q33`2>=0 or q33`1>=0;
end;
case
q33`2<0;
thus q33`2>=0 or q33`1>=0 by A1,A39,A40,A41,Th48;
end;
end;
A43: LE q22,q33,P by A1,A3,A27,A29,A30,Th58;
A44: now
per cases;
case
q22`2>=0;
hence q22`2>=0 or q22`1>=0;
end;
case
q22`2<0;
thus q22`2>=0 or q22`1>=0 by A1,A8,A39,A40,A43,A41,Th48,
JORDAN6:58;
end;
end;
A45: LE q11,q22,P by A1,A2,A27,A29,A30,Th58;
A46: LE q22,q44,P by A1,A40,A43,JGRAPH_3:26,JORDAN6:58;
now
per cases;
case
q11`2>=0;
hence q11`2>=0 or q11`1>=0;
end;
case
q11`2<0;
thus q11`2>=0 or q11`1>=0 by A1,A8,A39,A46,A45,A41,Th48,
JORDAN6:58;
end;
end;
then consider
f2 being Function of TOP-REAL 2,TOP-REAL 2, q81,q82,q83,q84
being Point of TOP-REAL 2 such that
A47: f2 is being_homeomorphism and
A48: for q being Point of TOP-REAL 2 holds |.(f2.q).|=|.q.| and
A49: q81=f2.q11 & q82=f2.q22 and
A50: q83=f2.q33 & q84=f2.q44 and
A51: q81`1<0 & q81`2<0 & q82`1<0 & q82`2<0 & q83`1<0 & q83`2<0 &
q84`1< 0 & q84`2<0 & LE q81,q82,P & LE q82,q83,P & LE q83,q84,P by A1,A39,A40
,A43,A45,A42,A44,Th63;
reconsider f3=f2*f1 as Function of TOP-REAL 2,TOP-REAL 2;
A52: dom f1=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
then
A53: f3.p1=q81 & f3.p2=q82 by A49,FUNCT_1:13;
A54: for q being Point of TOP-REAL 2 holds |.(f3.q).|=|.q.|
proof
let q be Point of TOP-REAL 2;
dom f1=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
then f3.q=f2.(f1.q) by FUNCT_1:13;
hence |.f3.q.|=|.(f1.q).| by A48
.=|.q.| by A30,JGRAPH_4:128;
end;
A55: f3.p3=q83 & f3.p4=q84 by A50,A52,FUNCT_1:13;
f3 is being_homeomorphism by A31,A47,TOPS_2:57;
hence thesis by A51,A54,A53,A55;
end;
end;
hence thesis;
end;
end;
hence thesis;
end;
begin :: General Fashoda Meet Theorems
theorem Th66:
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
proof
let p1,p2,p3,p4 be Point of TOP-REAL 2, P be compact non empty Subset of
TOP-REAL 2;
assume that
A1: P={p where p is Point of TOP-REAL 2: |.p.|=1} and
A2: LE p1,p2,P and
A3: LE p2,p3,P and
A4: LE p3,p4,P and
A5: p1<>p2 and
A6: p2<>p3 and
A7: p3<>p4 and
A8: p1`1<0 and
A9: p2`1<0 and
A10: p3`1<0 and
A11: p4`1<0 and
A12: p1`2<0 and
A13: p2`2<0 and
A14: p3`2<0;
set q2=((p1`2)-FanMorphW).p2;
set q3=((p1`2)-FanMorphW).p3;
A15: p1`2<p2`2 by A1,A2,A5,A8,A12,Th45;
set q1=((p1`2)-FanMorphW).p1;
A16: P is being_simple_closed_curve by A1,JGRAPH_3:26;
then p1 in P by A2,JORDAN7:5;
then
A17: ex p11 being Point of TOP-REAL 2 st p11=p1 & |.p11.|=1 by A1;
then
A18: p1`2/|.p1.|=p1`2;
then
A19: q1`2=0 by A8,JGRAPH_4:47;
A20: |.q1.|=1 by A17,JGRAPH_4:33;
then
A21: q1`2/|.q1.|=q1`2;
p2 in P by A2,A16,JORDAN7:5;
then
A22: ex p22 being Point of TOP-REAL 2 st p22=p2 & |.p22.|=1 by A1;
then
A23: p2`2/|.p2.|=p2`2;
then
A24: q2`1<0 by A9,A12,A15,JGRAPH_4:42;
A25: |.q2.|=1 by A22,JGRAPH_4:33;
then
A26: q2`2/|.q2.|=q2`2;
then
A27: q1`2<q2`2 by A8,A9,A12,A15,A18,A23,A21,JGRAPH_4:44;
then
A28: q2`1<1 by A25,A19,Th2;
p3 in P by A3,A16,JORDAN7:5;
then
A29: ex p33 being Point of TOP-REAL 2 st p33=p3 & |.p33.|=1 by A1;
then
A30: |.q3.|=1 by JGRAPH_4:33;
then
A31: q3`2/|.q3.|=q3`2;
set r3=((q2`1)-FanMorphN).q3;
set r2=((q2`1)-FanMorphN).q2;
A32: q3`1/|.q3.|=q3`1 by A30;
A33: p2`2<p3`2 by A1,A3,A6,A9,A13,Th45;
then
A34: p3`2/|.p3.|>p1`2 by A15,A29,XXREAL_0:2;
then
A35: q3`1<0 by A10,A12,JGRAPH_4:42;
A36: 1^2=(q3`1)^2+(q3`2)^2 by A30,JGRAPH_3:1;
A37: 1^2=(q2`1)^2+(q2`2)^2 by A25,JGRAPH_3:1;
p3`2/|.p3.|>p2`2 by A1,A3,A6,A9,A13,A29,Th45;
then
A38: q2`2<q3`2 by A9,A10,A12,A15,A23,A26,A31,A34,JGRAPH_4:44;
then (q3`2)^2>(q2`2)^2 by A19,A27,SQUARE_1:16;
then (-(q2`1))^2>(q3`1)^2 by A37,A36,XREAL_1:8;
then
A39: --(q2`1)<(q3`1) by A24,SQUARE_1:48;
A40: 0<q3`2 by A8,A10,A12,A18,A21,A31,A19,A34,JGRAPH_4:44;
then
A41: r3`2>0 by A39,A28,A32,JGRAPH_4:75;
A42: |.r3.|=1 by A30,JGRAPH_4:66;
then
A43: r3`1/|.r3.|=r3`1;
A44: -1<p1`2 by A8,A12,A17,Th2;
then consider f1 being Function of TOP-REAL 2,TOP-REAL 2 such that
A45: f1=((p1`2)-FanMorphW) and
A46: f1 is being_homeomorphism by A12,JGRAPH_4:41;
A47: -1<q2`1 by A12,A44,A25,A19,A27,Th2;
then consider f2 being Function of TOP-REAL 2,TOP-REAL 2 such that
A48: f2=((q2`1)-FanMorphN) and
A49: f2 is being_homeomorphism by A28,JGRAPH_4:74;
A50: q2`1/|.q2.|=q2`1 by A25;
then
A51: r2`1=0 by A19,A27,JGRAPH_4:80;
A52: |.r2.|=1 by A25,JGRAPH_4:66;
then
A53: r2`1/|.r2.|=r2`1;
then
A54: r2`1<r3`1 by A19,A27,A38,A39,A47,A28,A50,A32,A43,JGRAPH_4:79;
then
A55: -1<r3`2 by A12,A44,A42,A51,Th2;
q1`2<q2`2 by A8,A9,A12,A15,A18,A23,A21,A26,JGRAPH_4:44;
then
A56: r2`1<r3`1 by A19,A40,A39,A47,A28,A50,A32,A53,A43,JGRAPH_4:79;
set q4=((p1`2)-FanMorphW).p4;
p4 in P by A4,A16,JORDAN7:5;
then
A57: ex p44 being Point of TOP-REAL 2 st p44=p4 & |.p44.|=1 by A1;
then
A58: |.q4.|=1 by JGRAPH_4:33;
then
A59: q4`2/|.q4.|=q4`2;
p3`2<p4`2 by A1,A4,A7,A10,A14,Th45;
then p4`2/|.p4.|>p2`2 by A33,A57,XXREAL_0:2;
then
A60: p4`2/|.p4.|>p1`2 by A15,XXREAL_0:2;
p4`2/|.p4.|>p3`2 by A1,A4,A7,A10,A14,A57,Th45;
then q3`2<q4`2 by A10,A11,A12,A29,A31,A59,A34,A60,JGRAPH_4:44;
then
A61: (q4`2)^2>(q3`2)^2 by A19,A27,A38,SQUARE_1:16;
1^2=(q4`1)^2+(q4`2)^2 by A58,JGRAPH_3:1;
then (-(q3`1))^2>(q4`1)^2 by A36,A61,XREAL_1:8;
then --(q3`1)<(q4`1) by A35,SQUARE_1:48;
then
A62: q4`1/|.q4.|>q3`1 by A58;
set r4=((q2`1)-FanMorphN).q4;
A63: 1^2=(r3`1)^2+(r3`2)^2 by A42,JGRAPH_3:1;
A64: |.r4.|=1 by A58,JGRAPH_4:66;
then
A65: r4`1/|.r4.|=r4`1;
set r1=((q2`1)-FanMorphN).q1;
(|.q1.|)^2 =(q1`1)^2+(q1`2)^2 by JGRAPH_3:1;
then
A66: q1`1=-1 or q1`1=1 by A20,A19,SQUARE_1:40;
then
A67: r1`1=-1 by A8,A18,A19,JGRAPH_4:47,49;
A68: 1^2=(r4`1)^2+(r4`2)^2 by A64,JGRAPH_3:1;
0<q4`2 by A8,A11,A12,A18,A21,A59,A19,A60,JGRAPH_4:44;
then
A69: r3`1<r4`1 by A40,A47,A28,A32,A43,A65,A62,JGRAPH_4:79;
then (r4`1)^2>(r3`1)^2 by A51,A56,SQUARE_1:16;
then (r3`2)^2-(r4`2)^2+(r4`2)^2>0+(r4`2)^2 by A63,A68,XREAL_1:8;
then
A70: r3`2>r4`2 by A41,SQUARE_1:48;
set s4=((r3`2)-FanMorphE).r4;
set s1=((r3`2)-FanMorphE).r1;
r1`2=0 by A19,JGRAPH_4:49;
then
A71: s1`2=0 by A67,JGRAPH_4:82;
set t4=((s4`1)-FanMorphS).s4;
set s3=((r3`2)-FanMorphE).r3;
set s2=((r3`2)-FanMorphE).r2;
A72: (|.s3.|)^2 =(s3`1)^2+(s3`2)^2 by JGRAPH_3:1;
A73: r3`2/|.r3.|=r3`2 by A42;
then
A74: s3`2=0 by A51,A56,JGRAPH_4:111;
(|.r2.|)^2 =(r2`1)^2+(r2`2)^2 by JGRAPH_3:1;
then
A75: r2`2=-1 or r2`2=1 by A52,A51,SQUARE_1:40;
then r2=|[0,1]| by A19,A27,A50,A51,EUCLID:53,JGRAPH_4:80;
then
A76: s2=|[0,1]| by A51,JGRAPH_4:82;
s2`2=1 by A19,A27,A50,A51,A75,JGRAPH_4:80,82;
then
A77: ((s4`1)-FanMorphS).s2=|[0,1]| by A76,JGRAPH_4:113;
A78: r3`2<1 by A42,A51,A54,Th2;
then consider f3 being Function of TOP-REAL 2,TOP-REAL 2 such that
A79: f3=((r3`2)-FanMorphE) and
A80: f3 is being_homeomorphism by A55,JGRAPH_4:105;
A81: dom (f2*f1)=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
A82: r4`2/|.r4.|=r4`2 by A64;
then
A83: s3`2/|.s3.|>s4`2/|.s4.| by A51,A56,A69,A70,A55,A78,A73,JGRAPH_4:110;
A84: |.s4.|=1 by A64,JGRAPH_4:97;
then
A85: s4`1/|.s4.|=s4`1;
then
A86: t4`1=0 by A84,A74,A83,JGRAPH_4:142;
s4`2<0 by A51,A56,A69,A70,A55,A82,JGRAPH_4:107;
then
A87: s4`1<1 by A84,Th2;
-1<s4`1 by A51,A56,A69,A70,A55,A82,JGRAPH_4:107;
then consider f4 being Function of TOP-REAL 2,TOP-REAL 2 such that
A88: f4=((s4`1)-FanMorphS) and
A89: f4 is being_homeomorphism by A87,JGRAPH_4:136;
reconsider g=f4*(f3*(f2*f1)) as Function of TOP-REAL 2,TOP-REAL 2;
A90: dom (f3*(f2*f1))=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
f2*f1 is being_homeomorphism by A46,A49,TOPS_2:57;
then f3*(f2*f1) is being_homeomorphism by A80,TOPS_2:57;
then
A91: g is being_homeomorphism by A89,TOPS_2:57;
A92: dom g=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
then
A93: g.p2=(f4.((f3*(f2*f1)).p2)) by FUNCT_1:12
.=f4.((f3.((f2*f1).p2))) by A90,FUNCT_1:12
.=|[0,1]| by A45,A48,A79,A88,A77,A81,FUNCT_1:12;
|.s3.|=1 by A42,JGRAPH_4:97;
then s3`1=-1 or s3`1=1 by A74,A72,SQUARE_1:40;
then s3=|[1,0]| by A51,A56,A73,A74,EUCLID:53,JGRAPH_4:111;
then
A94: ((s4`1)-FanMorphS).s3=|[1,0]| by A74,JGRAPH_4:113;
q1=|[-1,0]| by A8,A18,A19,A66,EUCLID:53,JGRAPH_4:47;
then r1=|[-1,0]| by A19,JGRAPH_4:49;
then s1=|[-1,0]| by A67,JGRAPH_4:82;
then
A95: ((s4`1)-FanMorphS).s1=|[-1,0]| by A71,JGRAPH_4:113;
A96: (|.t4.|)^2 =(t4`1)^2+(t4`2)^2 by JGRAPH_3:1;
|.t4.|=1 by A84,JGRAPH_4:128;
then t4`2=-1 or t4`2=1 by A86,A96,SQUARE_1:40;
then
A97: t4=|[0,-1]| by A84,A74,A83,A85,A86,EUCLID:53,JGRAPH_4:142;
A98: for q being Point of TOP-REAL 2 holds |.(g.q).|=|.q.|
proof
let q be Point of TOP-REAL 2;
A99: |.((f2*f1).q).|=|.(f2.(f1.q)).| by A81,FUNCT_1:12
.=|.(f1.q).| by A48,JGRAPH_4:66
.=|.q.| by A45,JGRAPH_4:33;
A100: |.((f3*(f2*f1)).q).|=|.(f3.((f2*f1).q)).| by A90,FUNCT_1:12
.=|.q.| by A79,A99,JGRAPH_4:97;
thus |.(g.q).|=|.(f4.((f3*(f2*f1)).q)).| by A92,FUNCT_1:12
.=|.q.| by A88,A100,JGRAPH_4:128;
end;
A101: g.p3= (f4.((f3*(f2*f1)).p3)) by A92,FUNCT_1:12
.=f4.((f3.((f2*f1).p3))) by A90,FUNCT_1:12
.=|[1,0]| by A45,A48,A79,A88,A94,A81,FUNCT_1:12;
A102: g.p4= (f4.((f3*(f2*f1)).p4)) by A92,FUNCT_1:12
.=f4.((f3.((f2*f1).p4))) by A90,FUNCT_1:12
.=|[0,-1]| by A45,A48,A79,A88,A97,A81,FUNCT_1:12;
g.p1=(f4.((f3*(f2*f1)).p1)) by A92,FUNCT_1:12
.=f4.((f3.((f2*f1).p1))) by A90,FUNCT_1:12
.=|[-1,0]| by A45,A48,A79,A88,A95,A81,FUNCT_1:12;
hence thesis by A91,A98,A93,A101,A102;
end;
theorem Th67:
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
proof
let p1,p2,p3,p4 be Point of TOP-REAL 2, P be compact non empty Subset of
TOP-REAL 2;
assume that
A1: P={p where p is Point of TOP-REAL 2: |.p.|=1} and
A2: LE p1,p2,P & LE p2,p3,P & LE p3,p4,P and
A3: p1<>p2 & p2<>p3 and
A4: p3<>p4;
consider f being Function of TOP-REAL 2,TOP-REAL 2, q1,q2,q3,q4 being Point
of TOP-REAL 2 such that
A5: f is being_homeomorphism and
A6: for q being Point of TOP-REAL 2 holds |.(f.q).|=|.q.| and
A7: q1=f.p1 & q2=f.p2 and
A8: q3=f.p3 and
A9: q4=f.p4 and
A10: q1`1<0 & q1`2<0 & q2`1<0 & q2`2<0 & q3`1<0 & q3`2<0 & q4`1<0 and
q4`2<0 and
A11: LE q1,q2,P & LE q2,q3,P & LE q3,q4,P by A1,A2,Th65;
A12: dom f=the carrier of TOP-REAL 2 & f is one-to-one by A5,FUNCT_2:def 1
,TOPS_2:def 5;
then
A13: q3<>q4 by A4,A8,A9,FUNCT_1:def 4;
q1<>q2 & q2<>q3 by A3,A7,A8,A12,FUNCT_1:def 4;
then consider f2 being Function of TOP-REAL 2,TOP-REAL 2 such that
A14: f2 is being_homeomorphism and
A15: for q being Point of TOP-REAL 2 holds |.(f2.q).|=|.q.| and
A16: |[-1,0]|=f2.q1 & |[0,1]|=f2.q2 and
A17: |[1,0]|=f2.q3 & |[0,-1]|=f2.q4 by A1,A10,A11,A13,Th66;
reconsider f3=f2*f as Function of TOP-REAL 2,TOP-REAL 2;
A18: f3 is being_homeomorphism by A5,A14,TOPS_2:57;
A19: dom f3=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
then
A20: f3.p1=|[-1,0]| & f3.p2=|[0,1]| by A7,A16,FUNCT_1:12;
A21: for q being Point of TOP-REAL 2 holds |.(f3.q).|=|.q.|
proof
let q be Point of TOP-REAL 2;
|.(f3.q).|=|.f2.(f.q).| by A19,FUNCT_1:12
.=|.(f.q).| by A15
.=|.q.| by A6;
hence thesis;
end;
f3.p3=|[1,0]| & f3.p4=|[0,-1]| by A8,A9,A17,A19,FUNCT_1:12;
hence thesis by A18,A21,A20;
end;
Lm7: (|[-1,0]|)`1 =-1 by EUCLID:52;
Lm8: (|[-1,0]|)`2=0 by EUCLID:52;
Lm9: (|[1,0]|)`1 =1 & (|[1,0]|)`2=0 by EUCLID:52;
Lm10: (|[0,-1]|)`1 =0 by EUCLID:52;
Lm11: (|[0,-1]|)`2=-1 by EUCLID:52;
Lm12: (|[0,1]|)`1 =0 by EUCLID:52;
Lm13: (|[0,1]|)`2=1 by EUCLID:52;
Lm14: now
thus |.(|[-1,0]|).|=sqrt((-1)^2+0^2) by Lm7,Lm8,JGRAPH_3:1
.=1;
thus |.(|[1,0]|).|=sqrt(1^2+0^2) by Lm9,JGRAPH_3:1
.=1;
thus |.(|[0,-1]|).|=sqrt(0^2+(-1)^2) by Lm10,Lm11,JGRAPH_3:1
.=1;
thus |.(|[0,1]|).|=sqrt(0^2+1^2) by Lm12,Lm13,JGRAPH_3:1
.=1;
end;
Lm15: 0 in [.0,1.] by XXREAL_1:1;
Lm16: 1 in [.0,1.] by XXREAL_1:1;
theorem
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
proof
let p1,p2,p3,p4 be Point of TOP-REAL 2, P be compact non empty Subset of
TOP-REAL 2,C0 be Subset of TOP-REAL 2;
assume
A1: P={p where p is Point of TOP-REAL 2: |.p.|=1} & LE p1,p2,P & LE p2,
p3, P & LE p3,p4,P;
let f,g be Function of I[01],TOP-REAL 2;
assume
A2: 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;
A3: dom g=the carrier of I[01] by FUNCT_2:def 1;
A4: dom f=the carrier of I[01] by FUNCT_2:def 1;
per cases;
suppose
A5: not (p1<>p2 & p2<>p3 & p3<>p4);
now
per cases by A5;
case
A6: p1=p2;
p1 in rng f & p2 in rng g by A2,A4,A3,Lm15,BORSUK_1:40,FUNCT_1:def 3;
hence rng f meets rng g by A6,XBOOLE_0:3;
end;
case
A7: p2=p3;
p3 in rng f & p2 in rng g by A2,A4,A3,Lm15,Lm16,BORSUK_1:40
,FUNCT_1:def 3;
hence rng f meets rng g by A7,XBOOLE_0:3;
end;
case
A8: p3=p4;
p3 in rng f & p4 in rng g by A2,A4,A3,Lm16,BORSUK_1:40,FUNCT_1:def 3;
hence rng f meets rng g by A8,XBOOLE_0:3;
end;
end;
hence thesis;
end;
suppose
p1<>p2 & p2<>p3 & p3<>p4;
then consider h being Function of TOP-REAL 2,TOP-REAL 2 such that
A9: h is being_homeomorphism and
A10: for q being Point of TOP-REAL 2 holds |.(h.q).|=|.q.| and
A11: |[-1,0]|=h.p1 and
A12: |[0,1]|=h.p2 and
A13: |[1,0]|=h.p3 and
A14: |[0,-1]|=h.p4 by A1,Th67;
A15: h is one-to-one by A9,TOPS_2:def 5;
reconsider h1=h as Function;
reconsider O=0,I=1 as Point of I[01] by BORSUK_1:40,XXREAL_1:1;
defpred P[Point of TOP-REAL 2] means |.$1.|=1 & $1`2<=$1`1 & $1`2>=-$1`1;
{q1 where q1 is Point of TOP-REAL 2:P[q1]} is Subset of TOP-REAL 2
from JGRAPH_2:sch 1;
then reconsider
KXP={q1 where q1 is Point of TOP-REAL 2: |.q1.|=1 & q1`2<=q1`1
& q1`2>=-q1`1} as Subset of TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means |.$1.|=1 & $1`2>=$1`1 & $1`2<=-$1`1;
A16: dom h=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
{q2 where q2 is Point of TOP-REAL 2:P[q2]} is Subset of TOP-REAL 2
from JGRAPH_2:sch 1;
then reconsider
KXN={q2 where q2 is Point of TOP-REAL 2: |.q2.|=1 & q2`2>=q2`1
& q2`2<=-q2`1} as Subset of TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means |.$1.|=1 & $1`2>=$1`1 & $1`2>=-$1`1;
{q3 where q3 is Point of TOP-REAL 2:P[q3]} is Subset of TOP-REAL 2
from JGRAPH_2:sch 1;
then reconsider
KYP={q3 where q3 is Point of TOP-REAL 2: |.q3.|=1 & q3`2>=q3`1
& q3`2>=-q3`1} as Subset of TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means |.$1.|=1 & $1`2<=$1`1 & $1`2<=-$1`1;
{q4 where q4 is Point of TOP-REAL 2:P[q4]} is Subset of TOP-REAL 2
from JGRAPH_2:sch 1;
then reconsider
KYN={q4 where q4 is Point of TOP-REAL 2: |.q4.|=1 & q4`2<=q4`1
& q4`2<=-q4`1} as Subset of TOP-REAL 2;
A17: -(|[0,1]|)`1= 0 by Lm12;
reconsider g2=h*g as Function of I[01],TOP-REAL 2;
A18: -(|[0,-1]|)`1= 0 by Lm10;
A19: dom g2=the carrier of I[01] by FUNCT_2:def 1;
then g2.0= |[0,1]| by A2,A12,Lm15,BORSUK_1:40,FUNCT_1:12;
then
A20: g2.O in KYP by A17,Lm13,Lm14;
A21: rng g2 c= C0
proof
let y be object;
assume y in rng g2;
then consider x being object such that
A22: x in dom g2 and
A23: y=g2.x by FUNCT_1:def 3;
A24: g.x in rng g by A3,A22,FUNCT_1:def 3;
then reconsider qg=g.x as Point of TOP-REAL 2;
g.x in C0 by A2,A24;
then
A25: ex q5 being Point of TOP-REAL 2 st q5=g.x & |.q5.|<=1 by A2;
A26: |.(h.qg).|=|.qg.| by A10;
g2.x=h.(g.x) by A22,FUNCT_1:12;
hence thesis by A2,A23,A25,A26;
end;
reconsider f2=h*f as Function of I[01],TOP-REAL 2;
A27: -(|[-1,0]|)`1=1 by Lm7;
A28: dom f2=the carrier of I[01] by FUNCT_2:def 1;
then f2.1= |[1,0]| by A2,A13,Lm16,BORSUK_1:40,FUNCT_1:12;
then
A29: f2.I in KXP by Lm9,Lm14;
A30: rng f2 c= C0
proof
let y be object;
assume y in rng f2;
then consider x being object such that
A31: x in dom f2 and
A32: y=f2.x by FUNCT_1:def 3;
A33: f.x in rng f by A4,A31,FUNCT_1:def 3;
then reconsider qf=f.x as Point of TOP-REAL 2;
f.x in C0 by A2,A33;
then
A34: ex q5 being Point of TOP-REAL 2 st q5=f.x & |.q5.|<=1 by A2;
A35: |.(h.qf).|=|.qf.| by A10;
f2.x=h.(f.x) by A31,FUNCT_1:12;
hence thesis by A2,A32,A34,A35;
end;
g2.1= |[0,-1]| by A2,A14,A19,Lm16,BORSUK_1:40,FUNCT_1:12;
then
A36: g2.I in KYN by A18,Lm11,Lm14;
f2.0= |[-1,0]| by A2,A11,A28,Lm15,BORSUK_1:40,FUNCT_1:12;
then
A37: f2.O in KXN by A27,Lm8,Lm14;
f2 is continuous one-to-one & g2 is continuous one-to-one by A2,A9,Th5,Th6;
then rng f2 meets rng g2 by A2,A30,A21,A37,A29,A36,A20,Th13;
then consider x2 being object such that
A38: x2 in rng f2 and
A39: x2 in rng g2 by XBOOLE_0:3;
consider z3 being object such that
A40: z3 in dom g2 and
A41: x2=g2.z3 by A39,FUNCT_1:def 3;
A42: g.z3 in rng g by A3,A40,FUNCT_1:def 3;
h1".x2=h1".(h.(g.z3)) by A40,A41,FUNCT_1:12
.=g.z3 by A15,A16,A42,FUNCT_1:34;
then
A43: h1".x2 in rng g by A3,A40,FUNCT_1:def 3;
consider z2 being object such that
A44: z2 in dom f2 and
A45: x2=f2.z2 by A38,FUNCT_1:def 3;
A46: f.z2 in rng f by A4,A44,FUNCT_1:def 3;
h1".x2=h1".(h.(f.z2)) by A44,A45,FUNCT_1:12
.=f.z2 by A15,A16,A46,FUNCT_1:34;
then h1".x2 in rng f by A4,A44,FUNCT_1:def 3;
hence thesis by A43,XBOOLE_0:3;
end;
end;
theorem
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
proof
let p1,p2,p3,p4 be Point of TOP-REAL 2, P be compact non empty Subset of
TOP-REAL 2,C0 be Subset of TOP-REAL 2;
assume
A1: P={p where p is Point of TOP-REAL 2: |.p.|=1} & LE p1,p2,P & LE p2,
p3, P & LE p3,p4,P;
let f,g be Function of I[01],TOP-REAL 2;
assume
A2: 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;
A3: dom g=the carrier of I[01] by FUNCT_2:def 1;
A4: dom f=the carrier of I[01] by FUNCT_2:def 1;
per cases;
suppose
A5: not (p1<>p2 & p2<>p3 & p3<>p4);
now
per cases by A5;
case
A6: p1=p2;
p1 in rng f & p2 in rng g by A2,A4,A3,Lm15,Lm16,BORSUK_1:40
,FUNCT_1:def 3;
hence rng f meets rng g by A6,XBOOLE_0:3;
end;
case
A7: p2=p3;
p3 in rng f & p2 in rng g by A2,A4,A3,Lm16,BORSUK_1:40,FUNCT_1:def 3;
hence rng f meets rng g by A7,XBOOLE_0:3;
end;
case
A8: p3=p4;
p3 in rng f & p4 in rng g by A2,A4,A3,Lm15,Lm16,BORSUK_1:40
,FUNCT_1:def 3;
hence rng f meets rng g by A8,XBOOLE_0:3;
end;
end;
hence thesis;
end;
suppose
p1<>p2 & p2<>p3 & p3<>p4;
then consider h being Function of TOP-REAL 2,TOP-REAL 2 such that
A9: h is being_homeomorphism and
A10: for q being Point of TOP-REAL 2 holds |.(h.q).|=|.q.| and
A11: |[-1,0]|=h.p1 and
A12: |[0,1]|=h.p2 and
A13: |[1,0]|=h.p3 and
A14: |[0,-1]|=h.p4 by A1,Th67;
A15: h is one-to-one by A9,TOPS_2:def 5;
reconsider h1=h as Function;
reconsider O=0,I=1 as Point of I[01] by BORSUK_1:40,XXREAL_1:1;
defpred P[Point of TOP-REAL 2] means |.$1.|=1 & $1`2<=$1`1 & $1`2>=-$1`1;
{q1 where q1 is Point of TOP-REAL 2:P[q1]} is Subset of TOP-REAL 2
from JGRAPH_2:sch 1;
then reconsider
KXP={q1 where q1 is Point of TOP-REAL 2: |.q1.|=1 & q1`2<=q1`1
& q1`2>=-q1`1} as Subset of TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means |.$1.|=1 & $1`2>=$1`1 & $1`2<=-$1`1;
A16: dom h=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
{q2 where q2 is Point of TOP-REAL 2:P[q2]} is Subset of TOP-REAL 2
from JGRAPH_2:sch 1;
then reconsider
KXN={q2 where q2 is Point of TOP-REAL 2: |.q2.|=1 & q2`2>=q2`1
& q2`2<=-q2`1} as Subset of TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means |.$1.|=1 & $1`2>=$1`1 & $1`2>=-$1`1;
{q3 where q3 is Point of TOP-REAL 2:P[q3]} is Subset of TOP-REAL 2
from JGRAPH_2:sch 1;
then reconsider
KYP={q3 where q3 is Point of TOP-REAL 2: |.q3.|=1 & q3`2>=q3`1
& q3`2>=-q3`1} as Subset of TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means |.$1.|=1 & $1`2<=$1`1 & $1`2<=-$1`1;
{q4 where q4 is Point of TOP-REAL 2:P[q4]} is Subset of TOP-REAL 2
from JGRAPH_2:sch 1;
then reconsider
KYN={q4 where q4 is Point of TOP-REAL 2: |.q4.|=1 & q4`2<=q4`1
& q4`2<=-q4`1} as Subset of TOP-REAL 2;
A17: -(|[0,1]|)`1= 0 by Lm12;
reconsider g2=h*g as Function of I[01],TOP-REAL 2;
A18: -(|[0,-1]|)`1= 0 by Lm10;
A19: dom g2=the carrier of I[01] by FUNCT_2:def 1;
then g2.0= |[0,-1]| by A2,A14,Lm15,BORSUK_1:40,FUNCT_1:12;
then
A20: g2.O in KYN by A18,Lm11,Lm14;
A21: rng g2 c= C0
proof
let y be object;
assume y in rng g2;
then consider x being object such that
A22: x in dom g2 and
A23: y=g2.x by FUNCT_1:def 3;
A24: g.x in rng g by A3,A22,FUNCT_1:def 3;
then reconsider qg=g.x as Point of TOP-REAL 2;
g.x in C0 by A2,A24;
then
A25: ex q5 being Point of TOP-REAL 2 st q5=g.x & |.q5.|<=1 by A2;
A26: |.(h.qg).|=|.qg.| by A10;
g2.x=h.(g.x) by A22,FUNCT_1:12;
hence thesis by A2,A23,A25,A26;
end;
reconsider f2=h*f as Function of I[01],TOP-REAL 2;
A27: -(|[-1,0]|)`1=1 by Lm7;
A28: dom f2=the carrier of I[01] by FUNCT_2:def 1;
then f2.1= |[1,0]| by A2,A13,Lm16,BORSUK_1:40,FUNCT_1:12;
then
A29: f2.I in KXP by Lm9,Lm14;
A30: rng f2 c= C0
proof
let y be object;
assume y in rng f2;
then consider x being object such that
A31: x in dom f2 and
A32: y=f2.x by FUNCT_1:def 3;
A33: f.x in rng f by A4,A31,FUNCT_1:def 3;
then reconsider qf=f.x as Point of TOP-REAL 2;
f.x in C0 by A2,A33;
then
A34: ex q5 being Point of TOP-REAL 2 st q5=f.x & |.q5.|<=1 by A2;
A35: |.(h.qf).|=|.qf.| by A10;
f2.x=h.(f.x) by A31,FUNCT_1:12;
hence thesis by A2,A32,A34,A35;
end;
g2.1= |[0,1]| by A2,A12,A19,Lm16,BORSUK_1:40,FUNCT_1:12;
then
A36: g2.I in KYP by A17,Lm13,Lm14;
f2.0= |[-1,0]| by A2,A11,A28,Lm15,BORSUK_1:40,FUNCT_1:12;
then
A37: f2.O in KXN by A27,Lm8,Lm14;
f2 is continuous one-to-one & g2 is continuous one-to-one by A2,A9,Th5,Th6;
then rng f2 meets rng g2 by A2,A30,A21,A37,A29,A20,A36,JGRAPH_3:44;
then consider x2 being object such that
A38: x2 in rng f2 and
A39: x2 in rng g2 by XBOOLE_0:3;
consider z3 being object such that
A40: z3 in dom g2 and
A41: x2=g2.z3 by A39,FUNCT_1:def 3;
A42: g.z3 in rng g by A3,A40,FUNCT_1:def 3;
h1".x2=h1".(h.(g.z3)) by A40,A41,FUNCT_1:12
.=g.z3 by A15,A16,A42,FUNCT_1:34;
then
A43: h1".x2 in rng g by A3,A40,FUNCT_1:def 3;
consider z2 being object such that
A44: z2 in dom f2 and
A45: x2=f2.z2 by A38,FUNCT_1:def 3;
A46: f.z2 in rng f by A4,A44,FUNCT_1:def 3;
h1".x2=h1".(h.(f.z2)) by A44,A45,FUNCT_1:12
.=f.z2 by A15,A16,A46,FUNCT_1:34;
then h1".x2 in rng f by A4,A44,FUNCT_1:def 3;
hence thesis by A43,XBOOLE_0:3;
end;
end;
theorem
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
proof
let p1,p2,p3,p4 be Point of TOP-REAL 2, P be compact non empty Subset of
TOP-REAL 2,C0 be Subset of TOP-REAL 2;
assume
A1: P={p where p is Point of TOP-REAL 2: |.p.|=1} & LE p1,p2,P & LE p2,
p3, P & LE p3,p4,P;
let f,g be Function of I[01],TOP-REAL 2;
assume
A2: 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;
A3: dom g=the carrier of I[01] by FUNCT_2:def 1;
A4: dom f=the carrier of I[01] by FUNCT_2:def 1;
per cases;
suppose
A5: not (p1<>p2 & p2<>p3 & p3<>p4);
now
per cases by A5;
case
A6: p1=p2;
p1 in rng f & p2 in rng g by A2,A4,A3,Lm15,Lm16,BORSUK_1:40
,FUNCT_1:def 3;
hence rng f meets rng g by A6,XBOOLE_0:3;
end;
case
A7: p2=p3;
p3 in rng f & p2 in rng g by A2,A4,A3,Lm16,BORSUK_1:40,FUNCT_1:def 3;
hence rng f meets rng g by A7,XBOOLE_0:3;
end;
case
A8: p3=p4;
p3 in rng f & p4 in rng g by A2,A4,A3,Lm15,Lm16,BORSUK_1:40
,FUNCT_1:def 3;
hence rng f meets rng g by A8,XBOOLE_0:3;
end;
end;
hence thesis;
end;
suppose
p1<>p2 & p2<>p3 & p3<>p4;
then consider h being Function of TOP-REAL 2,TOP-REAL 2 such that
A9: h is being_homeomorphism and
A10: for q being Point of TOP-REAL 2 holds |.(h.q).|=|.q.| and
A11: |[-1,0]|=h.p1 and
A12: |[0,1]|=h.p2 and
A13: |[1,0]|=h.p3 and
A14: |[0,-1]|=h.p4 by A1,Th67;
A15: h is one-to-one by A9,TOPS_2:def 5;
reconsider h1=h as Function;
reconsider O=0,I=1 as Point of I[01] by BORSUK_1:40,XXREAL_1:1;
defpred P[Point of TOP-REAL 2] means |.$1.|=1 & $1`2<=$1`1 & $1`2>=-$1`1;
{q1 where q1 is Point of TOP-REAL 2:P[q1]} is Subset of TOP-REAL 2
from JGRAPH_2:sch 1;
then reconsider
KXP={q1 where q1 is Point of TOP-REAL 2: |.q1.|=1 & q1`2<=q1`1
& q1`2>=-q1`1} as Subset of TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means |.$1.|=1 & $1`2>=$1`1 & $1`2<=-$1`1;
A16: dom h=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
{q2 where q2 is Point of TOP-REAL 2:P[q2]} is Subset of TOP-REAL 2
from JGRAPH_2:sch 1;
then reconsider
KXN={q2 where q2 is Point of TOP-REAL 2: |.q2.|=1 & q2`2>=q2`1
& q2`2<=-q2`1} as Subset of TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means |.$1.|=1 & $1`2>=$1`1 & $1`2>=-$1`1;
{q3 where q3 is Point of TOP-REAL 2:P[q3]} is Subset of TOP-REAL 2
from JGRAPH_2:sch 1;
then reconsider
KYP={q3 where q3 is Point of TOP-REAL 2: |.q3.|=1 & q3`2>=q3`1
& q3`2>=-q3`1} as Subset of TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means |.$1.|=1 & $1`2<=$1`1 & $1`2<=-$1`1;
{q4 where q4 is Point of TOP-REAL 2:P[q4]} is Subset of TOP-REAL 2
from JGRAPH_2:sch 1;
then reconsider
KYN={q4 where q4 is Point of TOP-REAL 2: |.q4.|=1 & q4`2<=q4`1
& q4`2<=-q4`1} as Subset of TOP-REAL 2;
A17: -(|[0,1]|)`1= 0 by Lm12;
reconsider g2=h*g as Function of I[01],TOP-REAL 2;
A18: -(|[0,-1]|)`1= 0 by Lm10;
A19: dom g2=the carrier of I[01] by FUNCT_2:def 1;
then g2.0= |[0,-1]| by A2,A14,Lm15,BORSUK_1:40,FUNCT_1:12;
then
A20: g2.O in KYN by A18,Lm11,Lm14;
A21: rng g2 c= C0
proof
let y be object;
assume y in rng g2;
then consider x being object such that
A22: x in dom g2 and
A23: y=g2.x by FUNCT_1:def 3;
A24: g.x in rng g by A3,A22,FUNCT_1:def 3;
then reconsider qg=g.x as Point of TOP-REAL 2;
g.x in C0 by A2,A24;
then
A25: ex q5 being Point of TOP-REAL 2 st q5=g.x & |.q5.|>=1 by A2;
A26: |.(h.qg).|=|.qg.| by A10;
g2.x=h.(g.x) by A22,FUNCT_1:12;
hence thesis by A2,A23,A25,A26;
end;
reconsider f2=h*f as Function of I[01],TOP-REAL 2;
A27: -(|[-1,0]|)`1=1 by Lm7;
A28: dom f2=the carrier of I[01] by FUNCT_2:def 1;
then f2.1= |[1,0]| by A2,A13,Lm16,BORSUK_1:40,FUNCT_1:12;
then
A29: f2.I in KXP by Lm9,Lm14;
A30: rng f2 c= C0
proof
let y be object;
assume y in rng f2;
then consider x being object such that
A31: x in dom f2 and
A32: y=f2.x by FUNCT_1:def 3;
A33: f.x in rng f by A4,A31,FUNCT_1:def 3;
then reconsider qf=f.x as Point of TOP-REAL 2;
f.x in C0 by A2,A33;
then
A34: ex q5 being Point of TOP-REAL 2 st q5=f.x & |.q5.|>=1 by A2;
A35: |.(h.qf).|=|.qf.| by A10;
f2.x=h.(f.x) by A31,FUNCT_1:12;
hence thesis by A2,A32,A34,A35;
end;
g2.1= |[0,1]| by A2,A12,A19,Lm16,BORSUK_1:40,FUNCT_1:12;
then
A36: g2.I in KYP by A17,Lm13,Lm14;
f2.0= |[-1,0]| by A2,A11,A28,Lm15,BORSUK_1:40,FUNCT_1:12;
then
A37: f2.O in KXN by A27,Lm8,Lm14;
f2 is continuous one-to-one & g2 is continuous one-to-one by A2,A9,Th5,Th6;
then rng f2 meets rng g2 by A2,A30,A21,A37,A29,A20,A36,Th14;
then consider x2 being object such that
A38: x2 in rng f2 and
A39: x2 in rng g2 by XBOOLE_0:3;
consider z3 being object such that
A40: z3 in dom g2 and
A41: x2=g2.z3 by A39,FUNCT_1:def 3;
A42: g.z3 in rng g by A3,A40,FUNCT_1:def 3;
h1".x2=h1".(h.(g.z3)) by A40,A41,FUNCT_1:12
.=g.z3 by A15,A16,A42,FUNCT_1:34;
then
A43: h1".x2 in rng g by A3,A40,FUNCT_1:def 3;
consider z2 being object such that
A44: z2 in dom f2 and
A45: x2=f2.z2 by A38,FUNCT_1:def 3;
A46: f.z2 in rng f by A4,A44,FUNCT_1:def 3;
h1".x2=h1".(h.(f.z2)) by A44,A45,FUNCT_1:12
.=f.z2 by A15,A16,A46,FUNCT_1:34;
then h1".x2 in rng f by A4,A44,FUNCT_1:def 3;
hence thesis by A43,XBOOLE_0:3;
end;
end;
theorem
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
proof
let p1,p2,p3,p4 be Point of TOP-REAL 2, P be compact non empty Subset of
TOP-REAL 2,C0 be Subset of TOP-REAL 2;
assume
A1: P={p where p is Point of TOP-REAL 2: |.p.|=1} & LE p1,p2,P & LE p2,
p3, P & LE p3,p4,P;
let f,g be Function of I[01],TOP-REAL 2;
assume
A2: 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;
A3: dom g=the carrier of I[01] by FUNCT_2:def 1;
A4: dom f=the carrier of I[01] by FUNCT_2:def 1;
per cases;
suppose
A5: not (p1<>p2 & p2<>p3 & p3<>p4);
now
per cases by A5;
case
A6: p1=p2;
p1 in rng f & p2 in rng g by A2,A4,A3,Lm15,BORSUK_1:40,FUNCT_1:def 3;
hence rng f meets rng g by A6,XBOOLE_0:3;
end;
case
A7: p2=p3;
p3 in rng f & p2 in rng g by A2,A4,A3,Lm15,Lm16,BORSUK_1:40
,FUNCT_1:def 3;
hence rng f meets rng g by A7,XBOOLE_0:3;
end;
case
A8: p3=p4;
p3 in rng f & p4 in rng g by A2,A4,A3,Lm16,BORSUK_1:40,FUNCT_1:def 3;
hence rng f meets rng g by A8,XBOOLE_0:3;
end;
end;
hence thesis;
end;
suppose
p1<>p2 & p2<>p3 & p3<>p4;
then consider h being Function of TOP-REAL 2,TOP-REAL 2 such that
A9: h is being_homeomorphism and
A10: for q being Point of TOP-REAL 2 holds |.(h.q).|=|.q.| and
A11: |[-1,0]|=h.p1 and
A12: |[0,1]|=h.p2 and
A13: |[1,0]|=h.p3 and
A14: |[0,-1]|=h.p4 by A1,Th67;
reconsider f2=h*f,g2=h*g as Function of I[01],TOP-REAL 2;
A15: -(|[0,-1]|)`1= 0 by Lm10;
A16: rng g2 c= C0
proof
let y be object;
assume y in rng g2;
then consider x being object such that
A17: x in dom g2 and
A18: y=g2.x by FUNCT_1:def 3;
A19: g.x in rng g by A3,A17,FUNCT_1:def 3;
then reconsider qg=g.x as Point of TOP-REAL 2;
g.x in C0 by A2,A19;
then
A20: ex q5 being Point of TOP-REAL 2 st q5=g.x & |.q5.|>=1 by A2;
A21: |.(h.qg).|=|.qg.| by A10;
g2.x=h.(g.x) by A17,FUNCT_1:12;
hence thesis by A2,A18,A20,A21;
end;
A22: rng f2 c= C0
proof
let y be object;
assume y in rng f2;
then consider x being object such that
A23: x in dom f2 and
A24: y=f2.x by FUNCT_1:def 3;
A25: f.x in rng f by A4,A23,FUNCT_1:def 3;
then reconsider qf=f.x as Point of TOP-REAL 2;
f.x in C0 by A2,A25;
then
A26: ex q5 being Point of TOP-REAL 2 st q5=f.x & |.q5.|>=1 by A2;
A27: |.(h.qf).|=|.qf.| by A10;
f2.x=h.(f.x) by A23,FUNCT_1:12;
hence thesis by A2,A24,A26,A27;
end;
reconsider h1=h as Function;
reconsider O=0,I=1 as Point of I[01] by BORSUK_1:40,XXREAL_1:1;
defpred P[Point of TOP-REAL 2] means |.$1.|=1 & $1`2<=$1`1 & $1`2>=-$1`1;
{q1 where q1 is Point of TOP-REAL 2:P[q1]} is Subset of TOP-REAL 2
from JGRAPH_2:sch 1;
then reconsider
KXP={q1 where q1 is Point of TOP-REAL 2: |.q1.|=1 & q1`2<=q1`1
& q1`2>=-q1`1} as Subset of TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means |.$1.|=1 & $1`2>=$1`1 & $1`2<=-$1`1;
A28: dom h=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
{q2 where q2 is Point of TOP-REAL 2:P[q2]} is Subset of TOP-REAL 2
from JGRAPH_2:sch 1;
then reconsider
KXN={q2 where q2 is Point of TOP-REAL 2: |.q2.|=1 & q2`2>=q2`1
& q2`2<=-q2`1} as Subset of TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means |.$1.|=1 & $1`2>=$1`1 & $1`2>=-$1`1;
{q3 where q3 is Point of TOP-REAL 2:P[q3]} is Subset of TOP-REAL 2
from JGRAPH_2:sch 1;
then reconsider
KYP={q3 where q3 is Point of TOP-REAL 2: |.q3.|=1 & q3`2>=q3`1
& q3`2>=-q3`1} as Subset of TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means |.$1.|=1 & $1`2<=$1`1 & $1`2<=-$1`1;
{q4 where q4 is Point of TOP-REAL 2:P[q4]} is Subset of TOP-REAL 2
from JGRAPH_2:sch 1;
then reconsider
KYN={q4 where q4 is Point of TOP-REAL 2: |.q4.|=1 & q4`2<=q4`1
& q4`2<=-q4`1} as Subset of TOP-REAL 2;
A29: -(|[-1,0]|)`1=1 by Lm7;
A30: -(|[0,1]|)`1= 0 by Lm12;
A31: dom g2=the carrier of I[01] by FUNCT_2:def 1;
then g2.0 = |[0,1]| by A2,A12,Lm15,BORSUK_1:40,FUNCT_1:12;
then
A32: g2.O in KYP by A30,Lm13,Lm14;
g2.1 = |[0,-1]| by A2,A14,A31,Lm16,BORSUK_1:40,FUNCT_1:12;
then
A33: g2.I in KYN by A15,Lm11,Lm14;
A34: dom f2=the carrier of I[01] by FUNCT_2:def 1;
then f2.1 = |[1,0]| by A2,A13,Lm16,BORSUK_1:40,FUNCT_1:12;
then
A35: f2.I in KXP by Lm9,Lm14;
f2.0 = |[-1,0]| by A2,A11,A34,Lm15,BORSUK_1:40,FUNCT_1:12;
then
A36: f2.O in KXN by A29,Lm8,Lm14;
A37: h is one-to-one by A9,TOPS_2:def 5;
f2 is continuous one-to-one & g2 is continuous one-to-one by A2,A9,Th5,Th6;
then rng f2 meets rng g2 by A2,A22,A16,A36,A35,A33,A32,Th15;
then consider x2 being object such that
A38: x2 in rng f2 and
A39: x2 in rng g2 by XBOOLE_0:3;
consider z3 being object such that
A40: z3 in dom g2 and
A41: x2=g2.z3 by A39,FUNCT_1:def 3;
A42: g.z3 in rng g by A3,A40,FUNCT_1:def 3;
h1".x2=h1".(h.(g.z3)) by A40,A41,FUNCT_1:12
.=g.z3 by A37,A28,A42,FUNCT_1:34;
then
A43: h1".x2 in rng g by A3,A40,FUNCT_1:def 3;
consider z2 being object such that
A44: z2 in dom f2 and
A45: x2=f2.z2 by A38,FUNCT_1:def 3;
A46: f.z2 in rng f by A4,A44,FUNCT_1:def 3;
h1".x2=h1".(h.(f.z2)) by A44,A45,FUNCT_1:12
.=f.z2 by A37,A28,A46,FUNCT_1:34;
then h1".x2 in rng f by A4,A44,FUNCT_1:def 3;
hence thesis by A43,XBOOLE_0:3;
end;
end;