:: General {F}ashoda Meet Theorem for Unit Circle and Square
:: by Yatsuka Nakamura
::
:: Received February 25, 2003
:: Copyright (c) 2003-2026 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, XXREAL_0, CARD_1, RELAT_1, SQUARE_1, ARYTM_3, ARYTM_1,
PRE_TOPC, EUCLID, RLTOPSP1, MCART_1, REAL_1, SUBSET_1, BORSUK_1,
XXREAL_1, RCOMP_1, TOPMETR, STRUCT_0, XBOOLE_0, FUNCT_1, ORDINAL2,
TARSKI, TOPS_1, METRIC_1, SUPINF_2, COMPLEX1, JGRAPH_3, PSCOMP_1,
PCOMPS_1, JORDAN3, TOPS_2, JORDAN17, JGRAPH_2, TOPREAL2, TOPREAL1, SEQ_4,
FINSEQ_1, PARTFUN1, NAT_1, JORDAN6, JORDAN5C, FUNCT_4, RELAT_2, JGRAPH_6;
notations XBOOLE_0, ORDINAL1, XCMPLX_0, COMPLEX1, TARSKI, RELAT_1, TOPS_2,
FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, SUBSET_1, FINSEQ_1, FINSEQ_4,
SEQ_4, STRUCT_0, TOPMETR, COMPTS_1, METRIC_1, SQUARE_1, RCOMP_1,
PRE_TOPC, PCOMPS_1, RLVECT_1, RLTOPSP1, EUCLID, FUNCT_4, JGRAPH_3,
JORDAN5C, JORDAN6, TOPREAL2, CONNSP_1, TOPS_1, JORDAN17, JGRAPH_2,
NUMBERS, XREAL_0, TOPREAL1, PSCOMP_1, REAL_1, NAT_1, SPPOL_2, XXREAL_0;
constructors FUNCT_4, REAL_1, SQUARE_1, COMPLEX1, RCOMP_1, FINSEQ_4, TOPS_1,
CONNSP_1, TOPS_2, COMPTS_1, TOPMETR, TOPREAL1, SPPOL_2, PSCOMP_1,
JORDAN5C, JORDAN6, JGRAPH_2, JGRAPH_3, JORDAN17, PCOMPS_1, SEQ_4,
FUNCSDOM, CONVEX1, BINOP_2;
registrations XBOOLE_0, RELSET_1, FUNCT_2, NUMBERS, XXREAL_0, XREAL_0,
SQUARE_1, MEMBERED, STRUCT_0, PRE_TOPC, TOPS_1, BORSUK_1, EUCLID,
TOPMETR, TOPREAL1, TOPREAL2, SPPOL_2, PSCOMP_1, JGRAPH_2, JGRAPH_3,
XXREAL_2, COMPTS_1, RLTOPSP1, MEASURE6, FUNCT_1, VALUED_0, ORDINAL1;
requirements REAL, BOOLE, SUBSET, NUMERALS, ARITHM;
definitions TARSKI, XBOOLE_0, TOPREAL1;
equalities TOPREAL1, SQUARE_1, STRUCT_0, RLTOPSP1, RLVECT_1;
expansions TARSKI, XBOOLE_0;
theorems TARSKI, XBOOLE_0, XBOOLE_1, RELAT_1, FUNCT_1, FUNCT_2, TOPS_1,
TOPS_2, PRE_TOPC, TOPMETR, JORDAN6, EUCLID, JGRAPH_1, SQUARE_1, PSCOMP_1,
METRIC_1, JGRAPH_2, RCOMP_1, COMPTS_1, BORSUK_1, TOPREAL1, TOPREAL3,
TOPREAL5, JGRAPH_3, ABSVALUE, JORDAN7, HEINE, JGRAPH_4, JORDAN5C,
JGRAPH_5, GOBOARD6, TOPREAL2, TREAL_1, FINSEQ_1, FINSEQ_3, FINSEQ_4,
ZFMISC_1, NAT_1, ENUMSET1, UNIFORM1, BORSUK_4, TSEP_1, FUNCT_4, JORDAN17,
GOBOARD7, TOPRNS_1, COMPLEX1, XCMPLX_0, XCMPLX_1, SPPOL_2, SEQ_4,
XREAL_1, XXREAL_0, XXREAL_1, XXREAL_2, RLTOPSP1, RLVECT_1, RELSET_1,
RLVECT_4;
schemes FUNCT_2, JGRAPH_2, FRAENKEL, DOMAIN_1, CLASSES1;
begin
Lm1: for a,b being Real st b<=0 & a<=b
holds a*sqrt(1+b^2)<= b*sqrt(1+a^2)
proof
let a,b be Real;
assume that
A1: b<=0 and
A2: a<=b;
A3: a^2>=0 by XREAL_1:63;
then
A4: (-b)*sqrt(1+a^2)=sqrt((-b)^2*(1+a^2)) by A1,SQUARE_1:54;
A5: (-a)*sqrt(1+b^2)=sqrt((-a)^2*(1+b^2)) by A1,A2,SQUARE_1:54;
a<b or a=b by A2,XXREAL_0:1;
then b^2<a^2 or a=b by A1,SQUARE_1:44;
then
A6: b^2*1+b^2*a^2<=a^2*1+a^2*b^2 by XREAL_1:7;
(b)^2>=0 by XREAL_1:63;
then -(a*sqrt(1+b^2))>= -(b*sqrt(1+a^2)) by A3,A4,A5,A6,SQUARE_1:26;
hence thesis by XREAL_1:24;
end;
Lm2: for a,b being Real st a<=0 & a<=b
holds a*sqrt(1+b^2)<= b*sqrt(1+a^2)
proof
let a,b be Real;
assume that
A1: a<=0 and
A2: a<=b;
now per cases;
case b<=0;
hence thesis by A2,Lm1;
end;
case
A3: b>0;
(b)^2 >=0 by XREAL_1:63;
then sqrt(1+(b)^2)>0 by SQUARE_1:25;
then
A4: a*sqrt(1+b^2)<=0 by A1;
(a)^2 >=0 by XREAL_1:63;
then sqrt(1+(a)^2)>0 by SQUARE_1:25;
hence thesis by A3,A4;
end;
end;
hence thesis;
end;
Lm3: for a,b being Real st a>=0 & a>=b
holds a*sqrt(1+b^2)>= b*sqrt(1+a^2)
proof
let a,b be Real;
assume that
A1: a>=0 and
A2: a>=b;
-a <= -b by A2,XREAL_1:24;
then (-a)*sqrt(1+(-b)^2)<= (-b)*sqrt(1+(-a)^2) by A1,Lm2;
then -(a*sqrt(1+(b)^2))<= -(b*sqrt(1+(a)^2));
hence thesis by XREAL_1:24;
end;
theorem Th1:
for a,c,d being Real,p being Point of TOP-REAL 2
st c <=d & p in LSeg(|[a,c]|,|[a,d]|) holds p`1=a & c <=p`2 & p`2<=d
proof
let a,c,d be Real,p be Point of TOP-REAL 2;
assume that
A1: c <=d and
A2: p in LSeg(|[a,c]|,|[a,d]|);
thus p`1=a by A2,TOPREAL3:11;
A3: (|[a,c]|)`2=c by EUCLID:52;
(|[a,d]|)`2=d by EUCLID:52;
hence thesis by A1,A2,A3,TOPREAL1:4;
end;
theorem Th2:
for a,c,d being Real,p being Point of TOP-REAL 2
st c <d & p`1=a & c <=p`2 & p`2<=d holds p in LSeg(|[a,c]|,|[a,d]|)
proof
let a,c,d be Real,p be Point of TOP-REAL 2;
assume that
A1: c <d and
A2: p`1=a and
A3: c <=p`2 and
A4: p`2<=d;
A5: d-c>0 by A1,XREAL_1:50;
reconsider w=(p`2-c)/(d-c) as Real;
A6: (1-w)*(|[a,c]|)+w*(|[a,d]|) =|[(1-w)*a,(1-w)*c]|+w*(|[a,d]|) by EUCLID:58
.=|[(1-w)*a,(1-w)*c]|+(|[w*a,w*d]|) by EUCLID:58
.=|[(1-w)*a+w*a,(1-w)*c+w*d]| by EUCLID:56
.= |[a,c+w*(d-c)]|
.= |[a,c+(p`2-c)]| by A5,XCMPLX_1:87
.= p by A2,EUCLID:53;
A7: p`2-c>=0 by A3,XREAL_1:48;
p`2-c <=d-c by A4,XREAL_1:9;
then w<=(d-c)/(d-c) by A5,XREAL_1:72;
then w<=1 by A5,XCMPLX_1:60;
hence thesis by A5,A6,A7;
end;
theorem Th3:
for a,b,d being Real,p being Point of TOP-REAL 2
st a <=b & p in LSeg(|[a,d]|,|[b,d]|) holds p`2=d & a <=p`1 & p`1<=b
proof
let a,b,d be Real,p be Point of TOP-REAL 2;
assume that
A1: a <=b and
A2: p in LSeg(|[a,d]|,|[b,d]|);
thus p`2=d by A2,TOPREAL3:12;
A3: (|[a,d]|)`1=a by EUCLID:52;
(|[b,d]|)`1=b by EUCLID:52;
hence thesis by A1,A2,A3,TOPREAL1:3;
end;
theorem Th4: :: BORSUK_4:48
for a,b being Real,B being Subset of I[01] st
B=[.a,b.] holds B is closed
proof
let a,b be Real,B be Subset of I[01];
assume
A1: B=[.a,b.];
reconsider B2=B as Subset of R^1 by BORSUK_1:40,TOPMETR:17,XBOOLE_1:1;
A2: B2 is closed by A1,TREAL_1:1;
reconsider I1=[.0,1.] as Subset of R^1 by TOPMETR:17;
A3: [#](R^1|I1)=the carrier of I[01] by BORSUK_1:40,PRE_TOPC:def 5;
A4: I[01]=R^1|I1 by TOPMETR:19,20;
B=B2 /\ [#](R^1|I1) by A3,XBOOLE_1:28;
hence thesis by A2,A4,PRE_TOPC:13;
end;
theorem Th5:
for X being TopStruct,Y,Z being non empty TopStruct,
f being Function of X,Y, g being Function of X,Z holds dom f=dom g
& dom f=the carrier of X & dom f=[#]X
proof
let X be TopStruct,Y,Z be non empty TopStruct,
f be Function of X,Y, g be Function of X,Z;
dom f=the carrier of X by FUNCT_2:def 1;
hence thesis by FUNCT_2:def 1;
end;
theorem Th6:
for X being non empty TopSpace,B being non empty Subset of X
ex f being Function of X|B,X st (for p being Point of X|B holds f.p=p) &
f is continuous
proof
let X be non empty TopSpace,B be non empty Subset of X;
defpred P[set,set] means (for p being Point of X|B holds $2=$1);
A1: [#](X|B)= B by PRE_TOPC:def 5;
A2: for x being Element of X|B ex y being Element of X st P[x,y]
proof
let x be Element of X|B;
x in B by A1;
then reconsider px=x as Point of X;
set y0=px;
P[x,y0];
hence thesis;
end;
ex g being Function of the carrier of X|B,the carrier of X st
for x being Element of X|B holds P[x,g.x] from FUNCT_2:sch 3(A2);
then consider g being Function of the carrier of X|B,the carrier of X such
that
A3: for x being Element of X|B holds P[x,g.x];
A4: for p being Point of X|B holds g.p=p by A3;
A5: for r0 being Point of X|B,V being Subset of X
st g.r0 in V & V is open holds
ex W being Subset of X|B st r0 in W & W is open & g.:W c= V
proof
let r0 be Point of X|B,V be Subset of X;
assume that
A6: g.r0 in V and
A7: V is open;
reconsider W2=V /\ [#](X|B) as Subset of X|B;
g.r0=r0 by A3;
then
A8: r0 in W2 by A6,XBOOLE_0:def 4;
A9: W2 is open by A7,TOPS_2:24;
g.:W2 c= V
proof
let y be object;
assume y in g.:W2;
then consider x being object such that
A10: x in dom g and
A11: x in W2 and
A12: y=g.x by FUNCT_1:def 6;
reconsider px=x as Point of X|B by A10;
g.px=px by A3;
hence thesis by A11,A12,XBOOLE_0:def 4;
end;
hence thesis by A8,A9;
end;
reconsider g1=g as Function of X|B,X;
g1 is continuous by A5,JGRAPH_2:10;
hence thesis by A4;
end;
theorem Th7:
for X being non empty TopSpace,
f1 being Function of X,R^1,a being Real st f1 is continuous
holds ex g being Function of X,R^1
st (for p being Point of X,r1 being Real st
f1.p=r1 holds g.p=r1-a) & g is continuous
proof
let X be non empty TopSpace, f1 be Function of X,R^1,a be Real;
assume f1 is continuous;
then consider g1 being Function of X,R^1 such that
A1: for p being Point of X,r1 being Real st f1.p=r1 holds g1.p=r1+-a and
A2: g1 is continuous by JGRAPH_2:24;
for p being Point of X,r1 being Real st f1.p=r1 holds g1.p=r1-a
proof
let p be Point of X,r1 be Real;
assume f1.p=r1;
then g1.p=r1+-a by A1;
hence thesis;
end;
hence thesis by A2;
end;
theorem Th8:
for X being non empty TopSpace,
f1 being Function of X,R^1,a being Real st f1 is continuous
holds ex g being Function of X,R^1
st (for p being Point of X,r1 being Real st
f1.p=r1 holds g.p=a-r1) & g is continuous
proof
let X be non empty TopSpace, f1 be Function of X,R^1,a be Real;
assume f1 is continuous;
then consider g1 being Function of X,R^1 such that
A1: for p being Point of X,r1 being Real st f1.p=r1 holds g1.p=r1-a and
A2: g1 is continuous by Th7;
consider g2 being Function of X,R^1 such that
A3: for p being Point of X,r1 being Real st g1.p=r1 holds g2.p= -r1 and
A4: g2 is continuous by A2,JGRAPH_4:8;
for p being Point of X,r1 being Real st f1.p=r1 holds g2.p=a-r1
proof
let p be Point of X,r1 be Real;
assume f1.p=r1;
then g1.p=r1-a by A1;
then g2.p=-(r1-a) by A3
.=a-r1;
hence thesis;
end;
hence thesis by A4;
end;
theorem Th9:
for X being non empty TopSpace, n being Nat,
p being Point of TOP-REAL n,
f being Function of X,R^1 st f is continuous ex g being Function
of X,TOP-REAL n st (for r being Point of X holds g.r=(f.r)*p) &
g is continuous
proof
let X be non empty TopSpace, n be Nat, p be Point of TOP-REAL n,
f be Function of X,R^1;
assume
A1: f is continuous;
defpred P[set,set] means $2=(f.$1)*p;
A2: for x being Element of X ex y being Element of TOP-REAL n st P[x,y];
ex g being Function of the carrier of X,the carrier of TOP-REAL n st
for x being Element of X holds P[x,g.x] from FUNCT_2:sch 3(A2);
then consider g being Function of the carrier of X,the carrier of TOP-REAL n
such that
A3: for x being Element of X holds P[x,g.x];
reconsider g as Function of X,TOP-REAL n;
for r0 being Point of X,V being Subset of TOP-REAL n
st g.r0 in V & V is open holds
ex W being Subset of X st r0 in W & W is open & g.:W c= V
proof
let r0 be Point of X,V be Subset of TOP-REAL n;
assume that
A4: g.r0 in V and
A5: V is open;
A6: g.r0 in Int V by A4,A5,TOPS_1:23;
reconsider u=g.r0 as Point of Euclid n by TOPREAL3:8;
consider s being Real such that
A7: s>0 and
A8: Ball(u,s) c= V by A6,GOBOARD6:5;
now per cases;
case
A9: p <> 0.TOP-REAL n;
then
A10: |.p.| <> 0 by TOPRNS_1:24;
set r2=s/|.p.|;
reconsider G1=].f.r0-r2,f.r0+r2.[ as Subset of R^1 by TOPMETR:17;
A11: f.r0<f.r0+r2 by A7,A10,XREAL_1:29,139;
then f.r0-r2<f.r0 by XREAL_1:19;
then
A12: f.r0 in G1 by A11,XXREAL_1:4;
G1 is open by JORDAN6:35;
then consider W2 being Subset of X such that
A13: r0 in W2 and
A14: W2 is open and
A15: f.:W2 c= G1 by A1,A12,JGRAPH_2:10;
g.:W2 c= V
proof
let y be object;
assume y in g.:W2;
then consider r being object such that
A16: r in dom g and
A17: r in W2 and
A18: y=g.r by FUNCT_1:def 6;
reconsider r as Point of X by A16;
dom f=the carrier of X by FUNCT_2:def 1;
then f.r in f.:W2 by A17,FUNCT_1:def 6;
then
A19: |.f.r - f.r0.| <r2 by A15,RCOMP_1:1;
reconsider t=f.r,t0=f.r0 as Real;
A20: |.t0 - t.|=|.t-t0.| by UNIFORM1:11;
reconsider v=g.r as Point of Euclid n by TOPREAL3:8;
g.r0=(f.r0)*p by A3;
then
A21: |.g.r0 -g.r.| = |.(f.r0)*p -(f.r)*p.| by A3
.= |.((f.r0)-(f.r))*p.| by RLVECT_1:35
.=|.t0-t.|*|.p.| by TOPRNS_1:7;
|.f.r - f.r0.|*|.p.| < r2*|.p.| by A10,A19,XREAL_1:68;
then |.g.r0 -g.r .|<s by A9,A20,A21,TOPRNS_1:24,XCMPLX_1:87;
then dist(u,v)<s by JGRAPH_1:28;
then g.r in Ball(u,s) by METRIC_1:11;
hence thesis by A8,A18;
end;
hence thesis by A13,A14;
end;
case
A22: p =0.TOP-REAL n;
A23: for r being Point of X holds g.r=0.TOP-REAL n
proof
let r be Point of X;
thus g.r=(f.r)*p by A3
.=0.TOP-REAL n by A22,RLVECT_1:10;
end;
then
A24: 0.TOP-REAL n in V by A4;
set W2=[#]X;
g.:W2 c= V
proof
let y be object;
assume y in g.:W2;
then ex x being object st ( x in dom g)&( x in W2)&( y=g.x) by
FUNCT_1:def 6;
hence thesis by A23,A24;
end;
hence thesis;
end;
end;
hence thesis;
end;
then g is continuous by JGRAPH_2:10;
hence thesis by A3;
end;
theorem Th10:
Sq_Circ.(|[-1,0]|)= |[-1,0]|
proof
set p= |[-1,0]|;
A1: p`1=-1 by EUCLID:52;
A2: p`2=0 by EUCLID:52;
A3: p<>0.TOP-REAL 2 by A1,EUCLID:52,54;
p`2<=-p`1 by A1,EUCLID:52;
then Sq_Circ.p= |[p`1/sqrt(1+(p`2/p`1)^2),p`2/sqrt(1+(p`2/p`1)^2)]|
by A1,A2,A3,JGRAPH_3:def 1;
hence thesis by A2,EUCLID:52;
end;
theorem
for P being compact non empty Subset of TOP-REAL 2
st P={p where p is Point of TOP-REAL 2: |.p.|=1}
holds Sq_Circ.(|[-1,0]|)=W-min(P) by Th10,JGRAPH_5:29;
theorem Th12:
for X being non empty TopSpace, n being Nat,
g1,g2 being Function of X,TOP-REAL n st g1 is continuous & g2 is continuous
ex g being Function of X,TOP-REAL n st
(for r being Point of X holds g.r=g1.r + g2.r) & g is continuous
proof
let X being non empty TopSpace,n be Nat, g1,g2 be Function
of X,TOP-REAL n;
assume that
A1: g1 is continuous and
A2: g2 is continuous;
defpred P[set,set] means (for r1,r2 being Element of TOP-REAL n
st g1.$1=r1 & g2.$1=r2 holds $2=r1+r2);
A3: for x being Element of X ex y being Element of TOP-REAL n st P[x,y]
proof
let x be Element of X;
set rr1=g1.x;
set rr2=g2.x;
set r3=rr1+rr2;
for s1,s2 being Point of TOP-REAL n st g1.x=s1 & g2.x=s2 holds r3=s1+s2;
hence thesis;
end;
ex f being Function of the carrier of X,the carrier of TOP-REAL n
st for x being Element of X holds P[x,f.x] from FUNCT_2:sch 3(A3);
then consider f being Function of the carrier of X,the carrier of TOP-REAL n
such that
A4: for x being Element of X holds for r1,r2 being Element of TOP-REAL n
st g1.x=r1 & g2.x=r2 holds f.x=r1+r2;
reconsider g0=f as Function of X,TOP-REAL n;
A5: for r being Point of X holds g0.r=g1.r + g2.r by A4;
for p being Point of X,V being Subset of TOP-REAL n
st g0.p in V & V is open holds
ex W being Subset of X st p in W & W is open & g0.:W c= V
proof
let p be Point of X,V be Subset of TOP-REAL n;
assume that
A6: g0.p in V and
A7: V is open;
A8: g0.p in Int V by A6,A7,TOPS_1:23;
reconsider r=g0.p as Point of Euclid n by TOPREAL3:8;
consider r0 being Real such that
A9: r0>0 and
A10: Ball(r,r0) c= V by A8,GOBOARD6:5;
reconsider r01=g1.p as Point of Euclid n by TOPREAL3:8;
reconsider G1=Ball(r01,r0/2) as Subset of TOP-REAL n by TOPREAL3:8;
reconsider r02=g2.p as Point of Euclid n by TOPREAL3:8;
reconsider G2=Ball(r02,r0/2) as Subset of TOP-REAL n by TOPREAL3:8;
A11: g1.p in G1 by A9,GOBOARD6:1,XREAL_1:215;
A12: the TopStruct of TOP-REAL n = TopSpaceMetr Euclid n by EUCLID:def 8;
then reconsider GG1 = G1, GG2 = G2 as Subset of TopSpaceMetr Euclid n;
GG1 is open by TOPMETR:14;
then G1 is open by A12,PRE_TOPC:30;
then consider W1 being Subset of X such that
A13: p in W1 and
A14: W1 is open and
A15: g1.:W1 c= G1 by A1,A11,JGRAPH_2:10;
A16: g2.p in G2 by A9,GOBOARD6:1,XREAL_1:215;
GG2 is open by TOPMETR:14;
then G2 is open by A12,PRE_TOPC:30;
then consider W2 being Subset of X such that
A17: p in W2 and
A18: W2 is open and
A19: g2.:W2 c= G2 by A2,A16,JGRAPH_2:10;
set W=W1 /\ W2;
A20: p in W by A13,A17,XBOOLE_0:def 4;
g0.:W c= Ball(r,r0)
proof
let x be object;
assume x in g0.:W;
then consider z being object such that
A21: z in dom g0 and
A22: z in W and
A23: g0.z=x by FUNCT_1:def 6;
A24: z in W1 by A22,XBOOLE_0:def 4;
reconsider pz=z as Point of X by A21;
dom g1=the carrier of X by FUNCT_2:def 1;
then
A25: g1.pz in g1.:W1 by A24,FUNCT_1:def 6;
reconsider aa1=g1.pz as Point of TOP-REAL n;
reconsider bb1=aa1 as Point of Euclid n by TOPREAL3:8;
dist(r01,bb1)<r0/2 by A15,A25,METRIC_1:11;
then
A26: |.g1.p - g1.pz .|<r0/2 by JGRAPH_1:28;
A27: z in W2 by A22,XBOOLE_0:def 4;
dom g2=the carrier of X by FUNCT_2:def 1;
then
A28: g2.pz in g2.:W2 by A27,FUNCT_1:def 6;
reconsider aa2=g2.pz as Point of TOP-REAL n;
reconsider bb2=aa2 as Point of Euclid n by TOPREAL3:8;
dist(r02,bb2)<r0/2 by A19,A28,METRIC_1:11;
then
A29: |.g2.p - g2.pz .|<r0/2 by JGRAPH_1:28;
A30: aa1+aa2=x by A4,A23;
reconsider bb0=aa1+aa2 as Point of Euclid n by TOPREAL3:8;
A31: g0.pz= g1.pz+g2.pz by A4;
(g1.p +g2.p)-(g1.pz+g2.pz)=g1.p+g2.p-g1.pz-g2.pz by RLVECT_1:27
.= g1.p+g2.p+-g1.pz-g2.pz
.= g1.p+g2.p+-g1.pz+-g2.pz
.= g1.p+-g1.pz+g2.p+-g2.pz by RLVECT_1:def 3
.= g1.p+-g1.pz+(g2.p+-g2.pz) by RLVECT_1:def 3
.= g1.p-g1.pz+(g2.p+-g2.pz)
.= g1.p-g1.pz+(g2.p-g2.pz);
then
A32: |.(g1.p +g2.p)-(g1.pz+g2.pz).|
<= |.g1.p-g1.pz.| + |.g2.p-g2.pz.| by TOPRNS_1:29;
|.g1.p-g1.pz.| + |.g2.p-g2.pz.| < r0/2 +r0/2 by A26,A29,XREAL_1:8;
then |.(g1.p +g2.p)-(g1.pz+g2.pz).|<r0 by A32,XXREAL_0:2;
then |.g0.p - g0.pz .|<r0 by A4,A31;
then dist(r,bb0)<r0 by A23,A30,JGRAPH_1:28;
hence thesis by A30,METRIC_1:11;
end;
hence thesis by A10,A14,A18,A20,XBOOLE_1:1;
end;
then g0 is continuous by JGRAPH_2:10;
hence thesis by A5;
end;
theorem Th13:
for X being non empty TopSpace, n being Nat,
p1,p2 being Point of TOP-REAL n, f1,f2 being Function of X,R^1 st
f1 is continuous & f2 is continuous ex g being Function of X,TOP-REAL n st
(for r being Point of X holds g.r=(f1.r)*p1+(f2.r)*p2) & g is continuous
proof
let X be non empty TopSpace, n be Nat,
p1,p2 be Point of TOP-REAL n, f1,f2 be Function of X,R^1;
assume that
A1: f1 is continuous and
A2: f2 is continuous;
consider g1 being Function of X,TOP-REAL n such that
A3: for r being Point of X holds g1.r=(f1.r)*p1 and
A4: g1 is continuous by A1,Th9;
consider g2 being Function of X,TOP-REAL n such that
A5: for r being Point of X holds g2.r=(f2.r)*p2 and
A6: g2 is continuous by A2,Th9;
consider g being Function of X,TOP-REAL n such that
A7: for r being Point of X holds g.r=g1.r + g2.r and
A8: g is continuous by A4,A6,Th12;
for r being Point of X holds g.r=(f1.r)*p1+(f2.r)*p2
proof
let r be Point of X;
g.r=g1.r+g2.r by A7;
then g.r=g1.r+(f2.r)*p2 by A5;
hence thesis by A3;
end;
hence thesis by A8;
end;
begin :: General Fashoda Theorem for Unit Circle
Lm4: (|[-1,0]|)`1 =-1 by EUCLID:52;
Lm5: (|[-1,0]|)`2=0 by EUCLID:52;
Lm6: (|[1,0]|)`1 =1 by EUCLID:52;
Lm7: (|[1,0]|)`2=0 by EUCLID:52;
Lm8: (|[0,-1]|)`1 =0 by EUCLID:52;
Lm9: (|[0,-1]|)`2=-1 by EUCLID:52;
Lm10: (|[0,1]|)`1 =0 by EUCLID:52;
Lm11: (|[0,1]|)`2=1 by EUCLID:52;
Lm12: now
thus |.(|[-1,0]|).|=sqrt((-1)^2+0^2) by Lm4,Lm5,JGRAPH_3:1
.=1;
thus |.(|[1,0]|).|=sqrt(1^2+0^2) by Lm6,Lm7,JGRAPH_3:1
.=1;
thus |.(|[0,-1]|).|=sqrt(0^2+(-1)^2) by Lm8,Lm9,JGRAPH_3:1
.=1;
thus |.(|[0,1]|).|=sqrt(0^2+1^2) by Lm10,Lm11,JGRAPH_3:1
.=1;
end;
Lm13: 0 in [.0,1.] by XXREAL_1:1;
Lm14: 1 in [.0,1.] by XXREAL_1:1;
reserve p,p1,p2,p3,q,q1,q2 for Point of TOP-REAL 2,
i for Nat,
lambda for Real;
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 KXP & f.I in KXN &
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 & f is one-to-one &
g is continuous & g is 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 KXP & f.I in KXN &
g.O in KYP & g.I in KYN & rng f c= C0 & rng g c= C0;
then 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) by
JGRAPH_5:12;
hence thesis by A1,JGRAPH_5:13;
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 & f is one-to-one &
g is continuous & g is 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 KXP & f.I in KXN &
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 & f is one-to-one &
g is continuous & g is 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 KXP & f.I in KXN &
g.O in KYN & g.I in KYP & 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)&( g2 is one-to-one) by
JGRAPH_5:12;
hence thesis by A1,Th14;
end;
theorem Th16:
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={p8 where p8 is Point of TOP-REAL 2: |.p8.|<=1}&
f.0=p3 & f.1=p1 & 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 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;
let f,g be Function of I[01],TOP-REAL 2;
assume that
A5: f is continuous one-to-one and
A6: g is continuous one-to-one and
A7: C0={p8 where p8 is Point of TOP-REAL 2: |.p8.|<=1} and
A8: f.0=p3 and
A9: f.1=p1 and
A10: g.0=p2 and
A11: g.1=p4 and
A12: rng f c= C0 and
A13: rng g c= C0;
A14: dom f=the carrier of I[01] by FUNCT_2:def 1;
A15: dom g=the carrier of I[01] by FUNCT_2:def 1;
per cases;
suppose
A16: not (p1<>p2 & p2<>p3 & p3<>p4);
now per cases by A16;
case
A17: p1=p2;
A18: p1 in rng f by A9,A14,Lm14,BORSUK_1:40,FUNCT_1:def 3;
p2 in rng g by A10,A15,Lm13,BORSUK_1:40,FUNCT_1:def 3;
hence rng f meets rng g by A17,A18,XBOOLE_0:3;
end;
case
A19: p2=p3;
A20: p3 in rng f by A8,A14,Lm13,BORSUK_1:40,FUNCT_1:def 3;
p2 in rng g by A10,A15,Lm13,BORSUK_1:40,FUNCT_1:def 3;
hence rng f meets rng g by A19,A20,XBOOLE_0:3;
end;
case
A21: p3=p4;
A22: p3 in rng f by A8,A14,Lm13,BORSUK_1:40,FUNCT_1:def 3;
p4 in rng g by A11,A15,Lm14,BORSUK_1:40,FUNCT_1:def 3;
hence rng f meets rng g by A21,A22,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
A23: h is being_homeomorphism and
A24: for q being Point of TOP-REAL 2 holds |.(h.q).|=|.q.| and
A25: |[-1,0]|=h.p1 and
A26: |[0,1]|=h.p2 and
A27: |[1,0]|=h.p3 and
A28: |[0,-1]|=h.p4 by A1,A2,A3,A4,JGRAPH_5:67;
A29: h is one-to-one by A23,TOPS_2:def 5;
reconsider f2=h*f as Function of I[01],TOP-REAL 2;
reconsider g2=h*g as Function of I[01],TOP-REAL 2;
A30: dom f2=the carrier of I[01] by FUNCT_2:def 1;
A31: dom g2=the carrier of I[01] by FUNCT_2:def 1;
A32: f2.1= |[-1,0]| by A9,A25,A30,Lm14,BORSUK_1:40,FUNCT_1:12;
A33: g2.1= |[0,-1]| by A11,A28,A31,Lm14,BORSUK_1:40,FUNCT_1:12;
A34: f2.0= |[1,0]| by A8,A27,A30,Lm13,BORSUK_1:40,FUNCT_1:12;
A35: g2.0= |[0,1]| by A10,A26,A31,Lm13,BORSUK_1:40,FUNCT_1:12;
A36: f2 is continuous one-to-one by A5,A23,JGRAPH_5:5,6;
A37: g2 is continuous one-to-one by A6,A23,JGRAPH_5:5,6;
A38: rng f2 c= C0
proof
let y be object;
assume y in rng f2;
then consider x being object such that
A39: x in dom f2 and
A40: y=f2.x by FUNCT_1:def 3;
A41: f2.x=h.(f.x) by A39,FUNCT_1:12;
A42: f.x in rng f by A14,A39,FUNCT_1:def 3;
then
A43: f.x in C0 by A12;
reconsider qf=f.x as Point of TOP-REAL 2 by A42;
A44: ex q5 being Point of TOP-REAL 2 st ( q5=f.x)&( |.q5.|<=1) by A7,A43;
|.(h.qf).|=|.qf.| by A24;
hence thesis by A7,A40,A41,A44;
end;
A45: rng g2 c= C0
proof
let y be object;
assume y in rng g2;
then consider x being object such that
A46: x in dom g2 and
A47: y=g2.x by FUNCT_1:def 3;
A48: g2.x=h.(g.x) by A46,FUNCT_1:12;
A49: g.x in rng g by A15,A46,FUNCT_1:def 3;
then
A50: g.x in C0 by A13;
reconsider qg=g.x as Point of TOP-REAL 2 by A49;
A51: ex q5 being Point of TOP-REAL 2 st ( q5=g.x)&( |.q5.|<=1) by A7,A50;
|.(h.qg).|=|.qg.| by A24;
hence thesis by A7,A47,A48,A51;
end;
defpred Q[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:Q[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 Q[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:Q[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 Q[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:Q[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 Q[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:Q[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;
reconsider O=0,I=1 as Point of I[01] by BORSUK_1:40,XXREAL_1:1;
-(|[-1,0]|)`1=1 by Lm4;
then
A52: f2.I in KXN by A32,Lm5,Lm12;
A53: f2.O in KXP by A34,Lm6,Lm7,Lm12;
-(|[0,-1]|)`1= 0 by Lm8;
then
A54: g2.I in KYN by A33,Lm9,Lm12;
-(|[0,1]|)`1= 0 by Lm10;
then g2.O in KYP by A35,Lm11,Lm12;
then rng f2 meets rng g2 by A7,A36,A37,A38,A45,A52,A53,A54,Th14;
then consider x2 being object such that
A55: x2 in rng f2 and
A56: x2 in rng g2 by XBOOLE_0:3;
consider z2 being object such that
A57: z2 in dom f2 and
A58: x2=f2.z2 by A55,FUNCT_1:def 3;
consider z3 being object such that
A59: z3 in dom g2 and
A60: x2=g2.z3 by A56,FUNCT_1:def 3;
A61: dom h=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
A62: g.z3 in rng g by A15,A59,FUNCT_1:def 3;
A63: f.z2 in rng f by A14,A57,FUNCT_1:def 3;
reconsider h1=h as Function;
A64: h1".x2=h1".(h.(f.z2)) by A57,A58,FUNCT_1:12
.=f.z2 by A29,A61,A63,FUNCT_1:34;
A65: h1".x2=h1".(h.(g.z3)) by A59,A60,FUNCT_1:12
.=g.z3 by A29,A61,A62,FUNCT_1:34;
A66: h1".x2 in rng f by A14,A57,A64,FUNCT_1:def 3;
h1".x2 in rng g by A15,A59,A65,FUNCT_1:def 3;
hence thesis by A66,XBOOLE_0:3;
end;
end;
theorem Th17:
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={p8 where p8 is Point of TOP-REAL 2: |.p8.|<=1}&
f.0=p3 & f.1=p1 & 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 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;
let f,g be Function of I[01],TOP-REAL 2;
assume that
A5: f is continuous one-to-one and
A6: g is continuous one-to-one and
A7: C0={p8 where p8 is Point of TOP-REAL 2: |.p8.|<=1} and
A8: f.0=p3 and
A9: f.1=p1 and
A10: g.0=p4 and
A11: g.1=p2 and
A12: rng f c= C0 and
A13: rng g c= C0;
A14: dom f=the carrier of I[01] by FUNCT_2:def 1;
A15: dom g=the carrier of I[01] by FUNCT_2:def 1;
per cases;
suppose
A16: not (p1<>p2 & p2<>p3 & p3<>p4);
now per cases by A16;
case
A17: p1=p2;
A18: p1 in rng f by A9,A14,Lm14,BORSUK_1:40,FUNCT_1:def 3;
p2 in rng g by A11,A15,Lm14,BORSUK_1:40,FUNCT_1:def 3;
hence rng f meets rng g by A17,A18,XBOOLE_0:3;
end;
case
A19: p2=p3;
A20: p3 in rng f by A8,A14,Lm13,BORSUK_1:40,FUNCT_1:def 3;
p2 in rng g by A11,A15,Lm14,BORSUK_1:40,FUNCT_1:def 3;
hence rng f meets rng g by A19,A20,XBOOLE_0:3;
end;
case
A21: p3=p4;
A22: p3 in rng f by A8,A14,Lm13,BORSUK_1:40,FUNCT_1:def 3;
p4 in rng g by A10,A15,Lm13,BORSUK_1:40,FUNCT_1:def 3;
hence rng f meets rng g by A21,A22,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
A23: h is being_homeomorphism and
A24: for q being Point of TOP-REAL 2 holds |.(h.q).|=|.q.| and
A25: |[-1,0]|=h.p1 and
A26: |[0,1]|=h.p2 and
A27: |[1,0]|=h.p3 and
A28: |[0,-1]|=h.p4 by A1,A2,A3,A4,JGRAPH_5:67;
A29: h is one-to-one by A23,TOPS_2:def 5;
reconsider f2=h*f as Function of I[01],TOP-REAL 2;
reconsider g2=h*g as Function of I[01],TOP-REAL 2;
A30: dom f2=the carrier of I[01] by FUNCT_2:def 1;
A31: dom g2=the carrier of I[01] by FUNCT_2:def 1;
A32: f2.1= |[-1,0]| by A9,A25,A30,Lm14,BORSUK_1:40,FUNCT_1:12;
A33: g2.1= |[0,1]| by A11,A26,A31,Lm14,BORSUK_1:40,FUNCT_1:12;
A34: f2.0= |[1,0]| by A8,A27,A30,Lm13,BORSUK_1:40,FUNCT_1:12;
A35: g2.0= |[0,-1]| by A10,A28,A31,Lm13,BORSUK_1:40,FUNCT_1:12;
A36: f2 is continuous one-to-one by A5,A23,JGRAPH_5:5,6;
A37: g2 is continuous one-to-one by A6,A23,JGRAPH_5:5,6;
A38: rng f2 c= C0
proof
let y be object;
assume y in rng f2;
then consider x being object such that
A39: x in dom f2 and
A40: y=f2.x by FUNCT_1:def 3;
A41: f2.x=h.(f.x) by A39,FUNCT_1:12;
A42: f.x in rng f by A14,A39,FUNCT_1:def 3;
then
A43: f.x in C0 by A12;
reconsider qf=f.x as Point of TOP-REAL 2 by A42;
A44: ex q5 being Point of TOP-REAL 2 st ( q5=f.x)&( |.q5.|<=1) by A7,A43;
|.(h.qf).|=|.qf.| by A24;
hence thesis by A7,A40,A41,A44;
end;
A45: rng g2 c= C0
proof
let y be object;
assume y in rng g2;
then consider x being object such that
A46: x in dom g2 and
A47: y=g2.x by FUNCT_1:def 3;
A48: g2.x=h.(g.x) by A46,FUNCT_1:12;
A49: g.x in rng g by A15,A46,FUNCT_1:def 3;
then
A50: g.x in C0 by A13;
reconsider qg=g.x as Point of TOP-REAL 2 by A49;
A51: ex q5 being Point of TOP-REAL 2 st ( q5=g.x)&( |.q5.|<=1) by A7,A50;
|.(h.qg).|=|.qg.| by A24;
hence thesis by A7,A47,A48,A51;
end;
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;
{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;
reconsider O=0,I=1 as Point of I[01] by BORSUK_1:40,XXREAL_1:1;
-(|[-1,0]|)`1=1 by Lm4;
then
A52: f2.I in KXN by A32,Lm5,Lm12;
A53: f2.O in KXP by A34,Lm6,Lm7,Lm12;
-(|[0,-1]|)`1= 0 by Lm8;
then
A54: g2.I in KYP by A33,Lm10,Lm11,Lm12;
-(|[0,1]|)`1= 0 by Lm10;
then g2.O in KYN by A35,Lm8,Lm9,Lm12;
then rng f2 meets rng g2 by A7,A36,A37,A38,A45,A52,A53,A54,Th15;
then consider x2 being object such that
A55: x2 in rng f2 and
A56: x2 in rng g2 by XBOOLE_0:3;
consider z2 being object such that
A57: z2 in dom f2 and
A58: x2=f2.z2 by A55,FUNCT_1:def 3;
consider z3 being object such that
A59: z3 in dom g2 and
A60: x2=g2.z3 by A56,FUNCT_1:def 3;
A61: dom h=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
A62: g.z3 in rng g by A15,A59,FUNCT_1:def 3;
A63: f.z2 in rng f by A14,A57,FUNCT_1:def 3;
reconsider h1=h as Function;
A64: h1".x2=h1".(h.(f.z2)) by A57,A58,FUNCT_1:12
.=f.z2 by A29,A61,A63,FUNCT_1:34;
A65: h1".x2=h1".(h.(g.z3)) by A59,A60,FUNCT_1:12
.=g.z3 by A29,A61,A62,FUNCT_1:34;
A66: h1".x2 in rng f by A14,A57,A64,FUNCT_1:def 3;
h1".x2 in rng g by A15,A59,A65,FUNCT_1:def 3;
hence thesis by A66,XBOOLE_0:3;
end;
end;
theorem Th18:
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} &
p1,p2,p3,p4 are_in_this_order_on 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={p8 where p8 is Point of TOP-REAL 2: |.p8.|<=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 that
A1: P={p where p is Point of TOP-REAL 2: |.p.|=1} and
A2: p1,p2,p3,p4 are_in_this_order_on P;
per cases by A2,JORDAN17:def 1;
suppose LE p1,p2,P & LE p2,p3,P & LE p3,p4,P;
hence thesis by A1,JGRAPH_5:68;
end;
suppose LE p2,p3,P & LE p3,p4,P & LE p4,p1,P;
hence thesis by A1,JGRAPH_5:69;
end;
suppose LE p3,p4,P & LE p4,p1,P & LE p1,p2,P;
hence thesis by A1,Th17;
end;
suppose LE p4,p1,P & LE p1,p2,P & LE p2,p3,P;
hence thesis by A1,Th16;
end;
end;
begin :: General Rectangles and Circles
notation
let a,b,c,d be Real;
synonym rectangle(a,b,c,d) for [.a,b,c,d.];
end;
Lm15: for a,b,c,d being Real st a <= b & c <= d holds
rectangle(a,b,c,d) =
{p: p`1=a & c <=p`2 & p`2<=d or p`2=d & a<=p`1 & p`1<=b or
p`1=b & c <=p`2 & p`2<=d or p`2=c & a<=p`1 & p`1<=b}
proof
let a,b,c,d being Real;
set X = {p: p`1=a & c <=p`2 & p`2<=d or p`2=d & a<=p`1 & p`1<=b or
p`1=b & c <=p`2 & p`2<=d or p`2=c & a<=p`1 & p`1<=b};
assume that
A1: a <= b and
A2: c <= d;
A3: rectangle(a,b,c,d) = {p2: p2`1 = a & p2`2 <= d & p2`2 >= c or
p2`1 <= b & p2`1 >= a & p2`2 = d or p2`1 <= b & p2`1 >= a & p2`2 = c or
p2`1 = b & p2`2 <= d & p2`2 >= c} by A1,A2,SPPOL_2:54;
hereby
let x be object;
assume x in rectangle(a,b,c,d);
then ex p2 st x = p2 & (p2`1 = a & p2`2 <= d & p2`2 >= c or
p2`1 <= b & p2`1 >= a & p2`2 = d or p2`1 <= b & p2`1 >= a & p2`2 = c or
p2`1 = b & p2`2 <= d & p2`2 >= c) by A3;
hence x in X;
end;
let x be object;
assume x in X;
then ex p st x = p & (p`1=a & c <=p`2 & p`2<=d or p`2=d & a<=p`1 & p`1<=b or
p`1=b & c <=p`2 & p`2<=d or p`2=c & a<=p`1 & p`1<=b);
hence thesis by A3;
end;
theorem Th19:
for a,b,c,d being Real, p being Point of TOP-REAL 2
st a<=b & c <=d & p in rectangle(a,b,c,d) holds
a<=p`1 & p`1<=b & c <=p`2 & p`2<=d
proof
let a,b,c,d be Real, p be Point of TOP-REAL 2;
assume that
A1: a<=b and
A2: c <=d and
A3: p in rectangle(a,b,c,d);
p in {p2: p2`1 = a & p2`2 <= d & p2`2 >= c or
p2`1 <= b & p2`1 >= a & p2`2 = d or p2`1 <= b & p2`1 >= a & p2`2 = c or
p2`1 = b & p2`2 <= d & p2`2 >= c} by A1,A2,A3,SPPOL_2:54;
then
A4: ex p2 st ( p2=p)&( p2`1 = a & p2`2 <= d & p2`2 >= c or p2`1
<= b & p2`1 >= a & p2`2 = d or p2`1 <= b & p2`1 >= a & p2`2 = c or p2`1 = b &
p2`2 <= d & p2`2 >= c);
per cases by A4;
suppose p`1=a & c <=p`2 & p`2<=d;
hence thesis by A1;
end;
suppose p`2=d & a<=p`1 & p`1<=b;
hence thesis by A2;
end;
suppose p`1=b & c <=p`2 & p`2<=d;
hence thesis by A1;
end;
suppose p`2=c & a<=p`1 & p`1<=b;
hence thesis by A2;
end;
end;
definition
let a,b,c,d be Real;
func inside_of_rectangle(a,b,c,d) -> Subset of TOP-REAL 2 equals
{p: a <p`1 & p`1< b & c <p`2 & p`2< d};
coherence
proof
defpred P[Point of TOP-REAL 2] means
a <$1`1 & $1`1< b & c <$1`2 & $1`2< d;
{p: P[p]} c= the carrier of TOP-REAL 2 from FRAENKEL:sch 10;
hence thesis;
end;
end;
definition
let a,b,c,d be Real;
func closed_inside_of_rectangle(a,b,c,d) -> Subset of TOP-REAL 2 equals
{p: a <=p`1 & p`1<= b & c <=p`2 & p`2<= d};
coherence
proof
defpred P[Point of TOP-REAL 2] means
a <=$1`1 & $1`1<= b & c <=$1`2 & $1`2<= d;
{p: P[p]} c= the carrier of TOP-REAL 2 from FRAENKEL:sch 10;
hence thesis;
end;
end;
definition
let a,b,c,d be Real;
func outside_of_rectangle(a,b,c,d) -> Subset of TOP-REAL 2 equals
{p: not(a <=p`1 & p`1<= b & c <=p`2 & p`2<= d)};
coherence
proof
defpred P[Point of TOP-REAL 2] means
not(a <=$1`1 & $1`1<= b & c <=$1`2 & $1`2<= d);
{p: P[p] } c= the carrier of TOP-REAL 2 from FRAENKEL:sch 10;
hence thesis;
end;
end;
definition
let a,b,c,d be Real;
func closed_outside_of_rectangle(a,b,c,d) -> Subset of TOP-REAL 2 equals
{p: not(a <p`1 & p`1< b & c <p`2 & p`2< d)};
coherence
proof
defpred P[Point of TOP-REAL 2] means
not(a <$1`1 & $1`1< b & c <$1`2 & $1`2< d);
{p: P[p] } c= the carrier of TOP-REAL 2 from FRAENKEL:sch 10;
hence thesis;
end;
end;
theorem Th20:
for a,b,r being Real,Kb,Cb being Subset of TOP-REAL 2 st
r>=0 & Kb={q: |.q.|=1}&
Cb={p2 where p2 is Point of TOP-REAL 2: |.(p2- |[a,b]|).|=r} holds
AffineMap(r,a,r,b).:Kb=Cb
proof
let a,b,r be Real,Kb,Cb be Subset of TOP-REAL 2;
assume
A1: r>=0 & Kb={q: |.q.|=1}&
Cb={p2 where p2 is Point of TOP-REAL 2: |.(p2- |[a,b]|).|=r};
reconsider rr=r as Real;
A2: AffineMap(r,a,r,b).:Kb c= Cb
proof
let y be object;
assume y in AffineMap(r,a,r,b).:Kb;
then consider x being object such that
x in dom (AffineMap(r,a,r,b)) and
A3: x in Kb and
A4: y=(AffineMap(r,a,r,b)).x by FUNCT_1:def 6;
consider p being Point of TOP-REAL 2 such that
A5: x=p and
A6: |.p.|=1 by A1,A3;
A7: (AffineMap(r,a,r,b)).p=|[r*(p`1)+a,r*(p`2)+b]| by JGRAPH_2:def 2;
then reconsider q=y as Point of TOP-REAL 2 by A4,A5;
A8: q- |[a,b]|= |[r*(p`1)+a-a,r*(p`2)+b-b]| by A4,A5,A7,EUCLID:62
.= r*(|[(p`1),(p`2)]|) by EUCLID:58
.= r*p by EUCLID:53;
|.r*p.|=|.rr.|*(|.p.|) by TOPRNS_1:7
.=r by A1,A6,ABSVALUE:def 1;
hence thesis by A1,A8;
end;
Cb c= AffineMap(r,a,r,b).:Kb
proof
let y be object;
assume y in Cb;
then consider p2 being Point of TOP-REAL 2 such that
A9: y=p2 and
A10: |.(p2- |[a,b]|).|=r by A1;
now per cases by A1;
case
A11: r>0;
set p1=(1/r)*(p2- |[a,b]|);
|.p1.|=|.1/rr.|*|.(p2- |[a,b]|).| by TOPRNS_1:7
.=(1/r)*r by A10,ABSVALUE:def 1
.= 1 by A11,XCMPLX_1:87;
then
A12: p1 in Kb by A1;
A13: p1=|[(1/r)*(p2- |[a,b]|)`1,(1/r)*(p2- |[a,b]|)`2]| by EUCLID:57;
then
A14: p1`1=(1/r)*(p2- |[a,b]|)`1 by EUCLID:52;
A15: p1`2=(1/r)*(p2- |[a,b]|)`2 by A13,EUCLID:52;
A16: r*p1`1=r*(1/r)*(p2- |[a,b]|)`1 by A14
.=1*(p2- |[a,b]|)`1 by A11,XCMPLX_1:87
.=p2`1 -(|[a,b]|)`1 by TOPREAL3:3
.=p2`1 - a by EUCLID:52;
A17: r*p1`2=r*(1/r)*(p2- |[a,b]|)`2 by A15
.=1*(p2- |[a,b]|)`2 by A11,XCMPLX_1:87
.=p2`2 -(|[a,b]|)`2 by TOPREAL3:3
.=p2`2 - b by EUCLID:52;
A18: (AffineMap(r,a,r,b)).p1= |[r*(p1`1)+a,r*(p1`2)+b]| by JGRAPH_2:def 2
.=p2 by A16,A17,EUCLID:53;
dom (AffineMap(r,a,r,b))=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
hence thesis by A9,A12,A18,FUNCT_1:def 6;
end;
case
A19: r=0;
set p1= |[1,0]|;
A20: p1`1=1 by EUCLID:52;
p1`2=0 by EUCLID:52;
then |.p1.|=sqrt(1^2+0^2) by A20,JGRAPH_3:1
.=1;
then
A21: p1 in Kb by A1;
A22: (AffineMap(r,a,r,b)).p1= |[(0)*(p1`1)+a,(0)*(p1`2)+b]|
by A19,JGRAPH_2:def 2
.=p2 by A10,A19,TOPRNS_1:28;
dom (AffineMap(r,a,r,b))=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
hence thesis by A9,A21,A22,FUNCT_1:def 6;
end;
end;
hence thesis;
end;
hence thesis by A2;
end;
theorem Th21:
for P,Q being Subset of TOP-REAL 2 st
(ex f being Function of (TOP-REAL 2)|P,(TOP-REAL 2)|Q st
f is being_homeomorphism) & P is being_simple_closed_curve holds
Q is being_simple_closed_curve
proof
let P,Q be Subset of TOP-REAL 2;
assume that
A1: ex f being Function of (TOP-REAL 2)|P,(TOP-REAL 2)|Q st f is
being_homeomorphism and
A2: P is being_simple_closed_curve;
consider f being Function of (TOP-REAL 2)|P,(TOP-REAL 2)|Q such that
A3: f is being_homeomorphism by A1;
consider g being Function of (TOP-REAL 2)|R^2-unit_square,
(TOP-REAL 2)|P such that
A4: g is being_homeomorphism by A2,TOPREAL2:def 1;
A5: (|[1,0]|)`1=1 by EUCLID:52;
(|[1,0]|)`2=0 by EUCLID:52;
then
A6: |[1,0]| in R^2-unit_square by A5,TOPREAL1:14;
A7: dom g=[#]((TOP-REAL 2)|R^2-unit_square) by A4,TOPS_2:def 5;
A8: rng g=[#]((TOP-REAL 2)|P) by A4,TOPS_2:def 5;
dom g= R^2-unit_square by A7,PRE_TOPC:def 5;
then
A9: g.(|[1,0]|) in rng g by A6,FUNCT_1:3;
then
A10: g.(|[1,0]|) in P by A8,PRE_TOPC:def 5;
reconsider P1=P as non empty Subset of TOP-REAL 2 by A9;
dom f=[#]((TOP-REAL 2)|P) by A3,TOPS_2:def 5;
then dom f= P by PRE_TOPC:def 5;
then f.(g.(|[1,0]|)) in rng f by A10,FUNCT_1:3;
then reconsider Q1=Q as non empty Subset of TOP-REAL 2;
reconsider f1=f as Function of (TOP-REAL 2)|P1,(TOP-REAL 2)|Q1;
reconsider g1=g as Function of (TOP-REAL 2)|R^2-unit_square,(TOP-REAL 2)|P1;
reconsider h=f1*g1 as Function of (TOP-REAL 2)|R^2-unit_square,
(TOP-REAL 2)|Q1;
h is being_homeomorphism by A3,A4,TOPS_2:57;
hence thesis by TOPREAL2:def 1;
end;
theorem Th22:
for P being Subset of TOP-REAL 2 st
P is being_simple_closed_curve holds P is compact;
theorem Th23:
for a,b,r be Real, Cb being Subset of TOP-REAL 2 st r>0 &
Cb={p where p is Point of TOP-REAL 2: |.(p- |[a,b]|).|=r}
holds Cb is being_simple_closed_curve
proof
let a,b,r be Real, Cb be Subset of TOP-REAL 2;
assume that
A1: r>0 and
A2: Cb={p where p is Point of TOP-REAL 2: |.(p- |[a,b]|).|=r};
A3: (|[r,0]|)`1=r by EUCLID:52;
A4: (|[r,0]|)`2=0 by EUCLID:52;
|.(|[r+a,b]| - |[a,b]|).|=|.(|[r+a,0+b]| - |[a,b]|).|
.=|.(|[r,0]|+|[a,b]|- |[a,b]|).| by EUCLID:56
.= |.(|[r,0]|+(|[a,b]|- |[a,b]|)).| by RLVECT_1:def 3
.=|.|[r,0]|+(0.TOP-REAL 2).| by RLVECT_1:5
.=|.|[r,0]|.| by RLVECT_1:4
.=sqrt(r^2+0^2) by A3,A4,JGRAPH_3:1
.=r by A1,SQUARE_1:22;
then |[r+a,b]| in Cb by A2;
then reconsider Cbb=Cb as non empty Subset of TOP-REAL 2;
set v= |[1,0]|;
A5: v`1=1 by EUCLID:52;
v`2=0 by EUCLID:52;
then |.v.|=sqrt(1^2+0^2) by A5,JGRAPH_3:1
.=1;
then
A6: |[1,0]| in {q: |.q.|=1};
defpred P[Point of TOP-REAL 2] means |.$1.|=1;
{q where q is Element of TOP-REAL 2:P[q]}
is Subset of TOP-REAL 2 from DOMAIN_1:sch 7;
then reconsider Kb= {q: |.q.|=1} as non empty Subset of TOP-REAL 2 by A6;
A7: the carrier of (TOP-REAL 2)|Kb=Kb by PRE_TOPC:8;
set SC= AffineMap(r,a,r,b);
A8: SC is one-to-one by A1,JGRAPH_2:44;
A9: dom SC = the carrier of TOP-REAL 2 by FUNCT_2:def 1;
A10: dom (SC|Kb)=(dom SC)/\ Kb by RELAT_1:61
.=the carrier of ((TOP-REAL 2)|Kb) by A7,A9,XBOOLE_1:28;
A11: rng (SC|Kb) c= (SC|Kb).:(the carrier of ((TOP-REAL 2)|Kb))
proof
let u be object;
assume u in rng (SC|Kb);
then ex z being object st ( z in dom ((SC|Kb)))&( u=(SC|Kb).z) by
FUNCT_1:def 3;
hence thesis by A10,FUNCT_1:def 6;
end;
(SC|Kb).: (the carrier of ((TOP-REAL 2)|Kb)) = SC.:Kb by A7,RELAT_1:129
.= Cb by A1,A2,Th20
.=the carrier of (TOP-REAL 2)|Cbb by PRE_TOPC:8;
then reconsider f0=SC|Kb
as Function of (TOP-REAL 2)|Kb, (TOP-REAL 2)|Cbb by A10,A11,FUNCT_2:2;
rng (SC|Kb) c= the carrier of (TOP-REAL 2);
then reconsider f00=f0 as Function
of (TOP-REAL 2)|Kb,TOP-REAL 2 by A10,FUNCT_2:2;
A12: rng f0 = (SC|Kb).: (the carrier of ((TOP-REAL 2)|Kb)) by RELSET_1:22
.= SC.:Kb by A7,RELAT_1:129
.= Cb by A1,A2,Th20;
A13: f0 is one-to-one by A8,FUNCT_1:52;
Kb is compact by Th22,JGRAPH_3:26;
then ex f1 being Function of (TOP-REAL 2)|Kb,(TOP-REAL 2)|Cbb st
f00=f1 & f1 is being_homeomorphism by A12,A13,JGRAPH_1:46,TOPMETR:7;
hence thesis by Th21,JGRAPH_3:26;
end;
definition
let a,b,r be Real;
func circle(a,b,r) -> Subset of TOP-REAL 2 equals
{p where p is Point of TOP-REAL 2: |.p- |[a,b]| .|=r};
coherence
proof
defpred P[Point of TOP-REAL 2] means |.$1- |[a,b]| .|=r;
{p where p is Point of TOP-REAL 2:P[p]}
c= the carrier of TOP-REAL 2 from FRAENKEL:sch 10;
hence thesis;
end;
end;
registration
let a,b,r be Real;
cluster circle(a,b,r) -> compact;
coherence
proof
set Cb = circle(a,b,r);
per cases;
suppose
A1: r < 0;
Cb = {}
proof
hereby
let x be object;
assume x in Cb;
then ex p being Point of TOP-REAL 2 st x = p & |.p- |[a,b]| .| = r;
hence x in {} by A1;
end;
thus thesis;
end;
hence thesis;
end;
suppose r > 0;
hence thesis by Th22,Th23;
end;
suppose
A2: r = 0;
Cb = {|[a,b]|}
proof
hereby
let x be object;
assume x in Cb;
then consider p being Point of TOP-REAL 2 such that
A3: x = p and
A4: |.p- |[a,b]| .| = r;
p = |[a,b]| by A2,A4,TOPRNS_1:28;
hence x in {|[a,b]|} by A3,TARSKI:def 1;
end;
let x be object;
assume x in {|[a,b]|};
then
A5: x = |[a,b]| by TARSKI:def 1;
|. |[a,b]| - |[a,b]| .| = 0 by TOPRNS_1:28;
hence thesis by A2,A5;
end;
hence thesis;
end;
end;
end;
registration
let a,b be Real;
let r be non negative Real;
cluster circle(a,b,r) -> non empty;
coherence
proof
set Cb = circle(a,b,r);
A1: (|[r,0]|)`1=r by EUCLID:52;
A2: (|[r,0]|)`2=0 by EUCLID:52;
|.(|[r+a,b]| - |[a,b]|).|=|.(|[r+a,0+b]| - |[a,b]|).|
.=|.(|[r,0]|+|[a,b]|- |[a,b]|).| by EUCLID:56
.= |.(|[r,0]|+(|[a,b]|- |[a,b]|)).| by RLVECT_1:def 3
.=|.|[r,0]|+(0.TOP-REAL 2).| by RLVECT_1:5
.=|.|[r,0]|.| by RLVECT_1:4
.=sqrt(r^2+0^2) by A1,A2,JGRAPH_3:1
.= r by SQUARE_1:22;
then |[r+a,b]| in Cb;
hence thesis;
end;
end;
definition
let a,b,r be Real;
func inside_of_circle(a,b,r) -> Subset of TOP-REAL 2 equals
{p where p is Point of TOP-REAL 2: |.p- |[a,b]| .|<r};
coherence
proof
defpred P[Point of TOP-REAL 2] means |.$1- |[a,b]| .|<r;
{p where p is Point of TOP-REAL 2: P[p]}
c= the carrier of TOP-REAL 2 from FRAENKEL:sch 10;
hence thesis;
end;
end;
definition
let a,b,r be Real;
func closed_inside_of_circle(a,b,r) -> Subset of TOP-REAL 2 equals
{p where p is Point of TOP-REAL 2: |.p- |[a,b]| .|<=r};
coherence
proof
defpred P[Point of TOP-REAL 2] means |.$1- |[a,b]| .|<=r;
{p where p is Point of TOP-REAL 2: P[p]}
c= the carrier of TOP-REAL 2 from FRAENKEL:sch 10;
hence thesis;
end;
end;
definition
let a,b,r be Real;
func outside_of_circle(a,b,r) -> Subset of TOP-REAL 2 equals
{p where p is Point of TOP-REAL 2: |.p- |[a,b]| .|>r};
coherence
proof
defpred P[Point of TOP-REAL 2] means |.$1- |[a,b]| .|>r;
{p where p is Point of TOP-REAL 2: P[p]}
c= the carrier of TOP-REAL 2 from FRAENKEL:sch 10;
hence thesis;
end;
end;
definition
let a,b,r be Real;
func closed_outside_of_circle(a,b,r) -> Subset of TOP-REAL 2 equals
{p where p is Point of TOP-REAL 2: |.p- |[a,b]| .|>=r};
coherence
proof
defpred P[Point of TOP-REAL 2] means |.$1- |[a,b]| .|>=r;
{p where p is Point of TOP-REAL 2: P[p]}
c= the carrier of TOP-REAL 2 from FRAENKEL:sch 10;
hence thesis;
end;
end;
theorem Th24:
for r being Real holds inside_of_circle(0,0,r)={p : |.p.|<r} &
(r>0 implies circle(0,0,r)={p2 : |.p2.|=r}) &
outside_of_circle(0,0,r)={p3 : |.p3.|>r} &
closed_inside_of_circle(0,0,r)={q : |.q.|<=r} &
closed_outside_of_circle(0,0,r)={q2 : |.q2.|>=r}
proof
let r be Real;
defpred P[Point of TOP-REAL 2] means |.$1- |[0,0]| .|<r;
defpred Q[Point of TOP-REAL 2] means |.$1.|<r;
deffunc F(set)=$1;
A1: for p holds P[p] iff Q[p] by EUCLID:54,RLVECT_1:13;
inside_of_circle(0,0,r) = {F(p) where p is Point of TOP-REAL 2: P[p]}
.= {F(p2) where p2 is Point of TOP-REAL 2: Q[p2]} from FRAENKEL:sch 3(A1
);
hence inside_of_circle(0,0,r)={p : |.p.|<r};
defpred P[Point of TOP-REAL 2] means |.$1- |[0,0]| .|=r;
defpred Q[Point of TOP-REAL 2] means |.$1.|=r;
A2: for p holds P[p] iff Q[p] by EUCLID:54,RLVECT_1:13;
hereby
assume r>0;
circle(0,0,r)= {F(p): P[p]}
.= {F(p2) where p2 is Point of TOP-REAL 2: Q[p2]} from FRAENKEL:sch 3(A2);
hence circle(0,0,r)={p2 : |.p2.|=r};
end;
defpred P[Point of TOP-REAL 2] means |.$1- |[0,0]| .|>r;
defpred Q[Point of TOP-REAL 2] means |.$1.|>r;
A3: for p holds P[p] iff Q[p] by EUCLID:54,RLVECT_1:13;
outside_of_circle(0,0,r)= {F(p): P[p]}
.= {F(p2) where p2 is Point of TOP-REAL 2: Q[p2]} from FRAENKEL:sch 3(A3
);
hence outside_of_circle(0,0,r)={p3 : |.p3.|>r};
defpred P[Point of TOP-REAL 2] means |.$1- |[0,0]| .|<=r;
defpred Q[Point of TOP-REAL 2] means |.$1.|<=r;
A4: for p holds P[p] iff Q[p] by EUCLID:54,RLVECT_1:13;
closed_inside_of_circle(0,0,r)= {F(p): P[p]}
.= {F(p2) where p2 is Point of TOP-REAL 2: Q[p2]} from FRAENKEL:sch 3(A4
);
hence closed_inside_of_circle(0,0,r)={p : |.p.|<=r};
defpred P[Point of TOP-REAL 2] means |.$1- |[0,0]| .|>=r;
defpred Q[Point of TOP-REAL 2] means |.$1.|>=r;
A5: for p holds P[p] iff Q[p] by EUCLID:54,RLVECT_1:13;
closed_outside_of_circle(0,0,r)= {F(p): P[p]}
.= {F(p2) where p2 is Point of TOP-REAL 2: Q[p2]} from FRAENKEL:sch 3(A5
);
hence thesis;
end;
theorem Th25:
for Kb,Cb being Subset of TOP-REAL 2 st
Kb={p: -1 <p`1 & p`1< 1 & -1 <p`2 & p`2< 1}&
Cb={p2 where p2 is Point of TOP-REAL 2: |.p2.|<1} holds Sq_Circ.:Kb=Cb
proof
let Kb,Cb be Subset of TOP-REAL 2;
assume
A1: Kb={p: -1 <p`1 & p`1< 1 & -1 <p`2 & p`2< 1}&
Cb={p2 where p2 is Point of TOP-REAL 2: |.p2.|<1};
thus Sq_Circ.:Kb c= Cb
proof
let y be object;
assume y in Sq_Circ.:Kb;
then consider x being object such that
x in dom Sq_Circ and
A2: x in Kb and
A3: y=Sq_Circ.x by FUNCT_1:def 6;
consider q being Point of TOP-REAL 2 such that
A4: q=x and
A5: -1 <q`1 and
A6: q`1< 1 and
A7: -1 <q`2 and
A8: q`2< 1 by A1,A2;
now per cases;
case
A9: q=0.TOP-REAL 2;
then
A10: Sq_Circ.q=q by JGRAPH_3:def 1;
|.q.|=0 by A9,TOPRNS_1:23;
hence ex p2 being Point of TOP-REAL 2 st p2=y & |.p2.|<1 by A3,A4,A10;
end;
case
A11: q<>0.TOP-REAL 2 & (q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=-q`1);
then
A12: Sq_Circ.q=|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2 )]|
by JGRAPH_3:def 1;
A13: (|[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;
A14: (|[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;
A15: 1+(q`2/q`1)^2>0 by XREAL_1:34,63;
A16: now
assume
A17: q`1=0;
then q`2=0 by A11;
hence contradiction by A11,A17,EUCLID:53,54;
end;
then
A18: (q`1)^2>0 by SQUARE_1:12;
(q`1)^2<1^2 by A5,A6,SQUARE_1:50;
then
A19: sqrt((q`1)^2)<1 by A18,SQUARE_1:18,27;
|.(|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|).|^2
=((q`1)/sqrt(1+(q`2/(q`1))^2))^2+(q`2/sqrt(1+(q`2/(q`1))^2))^2
by A13,A14,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 A15,SQUARE_1:def 2
.=(q`1)^2/(1+(q`2/q`1)^2)+(q`2)^2/(1+(q`2/q`1)^2)
by A15,SQUARE_1:def 2
.=((q`1)^2+(q`2)^2)/(1+(q`2/q`1)^2) by XCMPLX_1:62
.=((q`1)^2+(q`2)^2)/(1+(q`2)^2/(q`1)^2) by XCMPLX_1:76
.=((q`1)^2+(q`2)^2)/((q`1)^2/(q`1)^2+(q`2)^2/(q`1)^2)
by A18,XCMPLX_1:60
.=((q`1)^2+(q`2)^2)/(((q`1)^2+(q`2)^2)/(q`1)^2)
by XCMPLX_1:62
.=(q`1)^2*(((q`1)^2+(q`2)^2)/((q`1)^2+(q`2)^2))
by XCMPLX_1:81
.=(q`1)^2*1 by A16,COMPLEX1:1,XCMPLX_1:60
.=(q`1)^2;
then |.(|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|).|<1
by A19,SQUARE_1:22;
hence ex p2 being Point of TOP-REAL 2 st p2=y & |.p2.|<1 by A3,A4,A12;
end;
case
A20: q<>0.TOP-REAL 2 & not(q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=-q`1);
then
A21: Sq_Circ.q=|[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2 )]|
by JGRAPH_3:def 1;
A22: (|[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2)]|)`1 = q`1/sqrt(1+(q
`1/q`2)^2) by EUCLID:52;
A23: (|[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2)]|)`2 = q`2/sqrt(1+(q
`1/q`2)^2) by EUCLID:52;
A24: 1+(q`1/q`2)^2>0 by XREAL_1:34,63;
A25: q`2 <> 0 by A20;
then
A26: (q`2)^2>0 by SQUARE_1:12;
(q`2)^2<1^2 by A7,A8,SQUARE_1:50;
then
A27: sqrt((q`2)^2)<1 by A26,SQUARE_1:18,27;
|.(|[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2)]|).|^2
=((q`1)/sqrt(1+(q`1/(q`2))^2))^2+(q`2/sqrt(1+(q`1/(q`2))^2))^2
by A22,A23,JGRAPH_3:1
.=(q`1)^2/(sqrt(1+(q`1/(q`2))^2))^2+(q`2/sqrt(1+(q`1/(q`2))^2))^2
by XCMPLX_1:76
.=(q`1)^2/(sqrt(1+(q`1/(q`2))^2))^2
+(q`2)^2/(sqrt(1+(q`1/(q`2))^2))^2 by XCMPLX_1:76
.=(q`1)^2/(1+(q`1/q`2)^2)+(q`2)^2/(sqrt(1+(q`1/q`2)^2))^2
by A24,SQUARE_1:def 2
.=(q`1)^2/(1+(q`1/q`2)^2)+(q`2)^2/(1+(q`1/q`2)^2)
by A24,SQUARE_1:def 2
.=((q`1)^2+(q`2)^2)/(1+(q`1/q`2)^2) by XCMPLX_1:62
.=((q`1)^2+(q`2)^2)/(1+(q`1)^2/(q`2)^2) by XCMPLX_1:76
.=((q`1)^2+(q`2)^2)/((q`1)^2/(q`2)^2+(q`2)^2/(q`2)^2)
by A26,XCMPLX_1:60
.=((q`1)^2+(q`2)^2)/(((q`1)^2+(q`2)^2)/(q`2)^2)
by XCMPLX_1:62
.=(q`2)^2*(((q`1)^2+(q`2)^2)/((q`1)^2+(q`2)^2))
by XCMPLX_1:81
.=(q`2)^2*1 by A25,COMPLEX1:1,XCMPLX_1:60
.=(q`2)^2;
then |.(|[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2)]|).|<1
by A27,SQUARE_1:22;
hence ex p2 being Point of TOP-REAL 2 st p2=y & |.p2.|<1 by A3,A4,A21;
end;
end;
hence thesis by A1;
end;
let y be object;
assume y in Cb;
then consider p2 being Point of TOP-REAL 2 such that
A28: p2=y and
A29: |.p2.|<1 by A1;
set q=p2;
now per cases;
case
A30: q=0.TOP-REAL 2;
then
A31: q`1=0 by EUCLID:52,54;
q`2=0 by A30,EUCLID:52,54;
then
A32: y in Kb by A1,A28,A31;
A33: Sq_Circ".y=y by A28,A30,JGRAPH_3:28;
A34: dom Sq_Circ=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
y=Sq_Circ.y by A28,A33,FUNCT_1:35,JGRAPH_3:43;
hence ex x being set st x in dom Sq_Circ & x in Kb & y=Sq_Circ.x
by A32,A34;
end;
case
A35: q<>0.TOP-REAL 2 & (q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=-q`1);
set px=|[q`1*sqrt(1+(q`2/q`1)^2),q`2*sqrt(1+(q`2/q`1)^2)]|;
A36: px`1 = q`1*sqrt(1+(q`2/q`1)^2) by EUCLID:52;
A37: px`2 = q`2*sqrt(1+(q`2/q`1)^2) by EUCLID:52;
1+(q`2/q`1)^2>0 by XREAL_1:34,63;
then
A38: sqrt(1+(q`2/q`1)^2)>0 by SQUARE_1:25;
A39: 1+(px`2/px`1)^2>0 by XREAL_1:34,63;
A40: px`2/px`1=q`2/q`1 by A36,A37,A38,XCMPLX_1:91;
A41: q`1=q`1*sqrt(1+(q`2/q`1)^2)/(sqrt(1+(q`2/q`1)^2))by A38,XCMPLX_1:89
.=px`1/(sqrt(1+(q`2/q`1)^2)) by EUCLID:52;
A42: q`2=q`2*sqrt(1+(q`2/q`1)^2)/(sqrt(1+(q`2/q`1)^2))by A38,XCMPLX_1:89
.=px`2/(sqrt(1+(q`2/q`1)^2)) by EUCLID:52;
A43: |.q.|^2=q`1^2+q`2^2 by JGRAPH_3:1;
A44: |.q.|^2<1^2 by A29,SQUARE_1:16;
A45: now
assume that
A46: px`1=0 and
A47: px`2=0;
A48: q`1*sqrt(1+(q`2/q`1)^2)=0 by A46,EUCLID:52;
A49: q`2*sqrt(1+(q`2/q`1)^2)=0 by A47,EUCLID:52;
A50: q`1=0 by A38,A48,XCMPLX_1:6;
q`2=0 by A38,A49,XCMPLX_1:6;
hence contradiction by A35,A50,EUCLID:53,54;
end;
q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 &
q`2*sqrt(1+(q`2/q`1)^2) <= (-q`1)*sqrt(1+(q`2/q`1)^2)
by A35,A38,XREAL_1:64;
then
A51: q`2<=q`1 & (-q`1)*sqrt(1+(q`2/q`1)^2) <= q`2*sqrt(1+(q`2/q`1) ^2)
or px`2>=px`1 & px`2<=-px`1 by A36,A37,A38,XREAL_1:64;
then px`2<=px`1 & -px`1<=px`2 or px`2>=px`1 & px`2<=-px`1
by A36,A37,A38,XREAL_1:64;
then
A52: Sq_Circ
.px=|[px`1/sqrt(1+(px`2/px`1)^2),px`2/sqrt(1+(px`2/px`1 )^2) ]|
by A45,JGRAPH_2:3,JGRAPH_3:def 1;
px`2<=px`1 & --px`1>=-px`2 or px`2>=px`1 & px`2<=-px`1
by A36,A37,A38,A51,XREAL_1:24,64;
then
A53: px`2<=px`1 & px`1>=-px`2 or px`2>=px`1 & -px`2>=--px`1 by XREAL_1:24;
A54: px`1/sqrt(1+(px`2/px`1)^2)=q`1 by A36,A38,A40,XCMPLX_1:89;
A55: px`2/sqrt(1+(px`2/px`1)^2)=q`2 by A37,A38,A40,XCMPLX_1:89;
A56: dom Sq_Circ=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
not px`1=0 by A36,A37,A38,A45,A51,XREAL_1:64;
then
A57: (px`1)^2>0 by SQUARE_1:12;
A58: (px`2)^2>=0 by XREAL_1:63;
(px`1)^2/(sqrt(1+(px`2/px`1)^2))^2+(px`2/sqrt(1+(px`2/px`1)^2)) ^2 < 1
by A40,A41,A42,A43,A44,XCMPLX_1:76;
then (px`1)^2/(sqrt(1+(px`2/px`1)^2))^2+(px`2)^2/(sqrt(1+(px`2/px`1)^2))
^2<1 by XCMPLX_1:76;
then (px`1)^2/(1+(px`2/px`1)^2)+(px`2)^2/(sqrt(1+(px`2/px`1)^2))^2<1
by A39,SQUARE_1:def 2;
then (px`1)^2/(1+(px`2/px`1)^2)+(px`2)^2/(1+(px`2/px`1)^2)<1
by A39,SQUARE_1:def 2;
then ((px`1)^2/(1+(px`2/px`1)^2)
+(px`2)^2/(1+(px`2/px`1)^2))*(1+(px`2/px`1)^2)<1 *(1+(px`2/px`1)^2)
by A39,XREAL_1:68;
then (px`1)^2/(1+(px`2/px`1)^2)*(1+(px`2/px`1)^2)
+(px`2)^2/(1+(px`2/px`1)^2)*(1+(px`2/px`1)^2)<1 *(1+(px`2/px`1)^2);
then (px`1)^2+(px`2)^2/(1+(px`2/px`1)^2)*(1+(px`2/px`1)^2)<1 *(1+(px`2/
px `1)^2) by A39,XCMPLX_1:87;
then
A59: (px`1)^2+(px`2)^2<1 *(1+(px`2/px`1)^2) by A39,XCMPLX_1:87;
1 *(1+(px`2/px`1)^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 A59,XREAL_1:9;
then ((px`1)^2+(px`2)^2-1)*(px`1)^2<((px`2)^2/(px`1)^2)*(px`1)^2
by A57,XREAL_1:68;
then
A60: ((px`1)^2+((px`2)^2-1))*(px`1)^2<(px`2)^2 by A57,XCMPLX_1:87;
(px`1)^2*(px`1)^2+((px`1)^2*(px`2)^2-(px`1)^2*1)-(px`2)^2
= ((px`1)^2-1)*((px`1)^2+(px`2)^2);
then (px`1)^2-1<0 or ((px`1)^2+(px`2)^2)<0 by A60,XREAL_1:49;
then
A61: (px`1)^2-1+1<0+1 by A58,XREAL_1:6;
then
A62: px`1<1^2 by SQUARE_1:48;
A63: px`1>-1^2 by A61,SQUARE_1:48;
px`2<1 & 1>-px`2 or px`2>=px`1 & -px`2>=px`1 by A53,A62,XXREAL_0:2;
then px`2<1 & -1< --px`2 or px`2>-1 & -px`2> -1
by A63,XREAL_1:24,XXREAL_0:2;
then px`2<1 & -1<px`2 or px`2>-1 & --px`2< --1 by XREAL_1:24;
then px in Kb by A1,A62,A63;
hence ex x being set st x in dom Sq_Circ & x in Kb & y=Sq_Circ.x
by A28,A52,A54,A55,A56,EUCLID:53;
end;
case
A64: q<>0.TOP-REAL 2 & not(q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=-q`1);
set px=|[q`1*sqrt(1+(q`1/q`2)^2),q`2*sqrt(1+(q`1/q`2)^2)]|;
A65: q`1<=q`2 & -q`2<=q`1 or q`1>=q`2 & q`1<=-q`2 by A64,JGRAPH_2:13;
A66: px`2 = q`2*sqrt(1+(q`1/q`2)^2) by EUCLID:52;
A67: px`1 = q`1*sqrt(1+(q`1/q`2)^2) by EUCLID:52;
1+(q`1/q`2)^2>0 by XREAL_1:34,63;
then
A68: sqrt(1+(q`1/q`2)^2)>0 by SQUARE_1:25;
A69: 1+(px`1/px`2)^2>0 by XREAL_1:34,63;
A70: px`1/px`2=q`1/q`2 by A66,A67,A68,XCMPLX_1:91;
A71: q`2=q`2*sqrt(1+(q`1/q`2)^2)/(sqrt(1+(q`1/q`2)^2))by A68,XCMPLX_1:89
.=px`2/(sqrt(1+(q`1/q`2)^2)) by EUCLID:52;
A72: q`1=q`1*sqrt(1+(q`1/q`2)^2)/(sqrt(1+(q`1/q`2)^2))by A68,XCMPLX_1:89
.=px`1/(sqrt(1+(q`1/q`2)^2)) by EUCLID:52;
A73: |.q.|^2=q`2^2+q`1^2 by JGRAPH_3:1;
A74: |.q.|^2<1^2 by A29,SQUARE_1:16;
A75: now
assume that
A76: px`2=0 and
A77: px`1=0;
A78: q`2*sqrt(1+(q`1/q`2)^2)=0 by A76,EUCLID:52;
q`1*sqrt(1+(q`1/q`2)^2)=0 by A77,EUCLID:52;
then q`1=0 by A68,XCMPLX_1:6;
hence contradiction by A64,A78;
end;
q`1<=q`2 & -q`2<=q`1 or q`1>=q`2 &
q`1*sqrt(1+(q`1/q`2)^2) <= (-q`2)*sqrt(1+(q`1/q`2)^2)
by A65,A68,XREAL_1:64;
then
A79: q`1<=q`2 & (-q`2)*sqrt(1+(q`1/q`2)^2) <= q`1*sqrt(1+(q`1/q`2) ^2)
or px`1>=px`2 & px`1<=-px`2 by A66,A67,A68,XREAL_1:64;
then px`1<=px`2 & -px`2<=px`1 or px`1>=px`2 & px`1<=-px`2
by A66,A67,A68,XREAL_1:64;
then
A80: Sq_Circ
.px=|[px`1/sqrt(1+(px`1/px`2)^2),px`2/sqrt(1+(px`1/px`2 )^2) ]|
by A75,JGRAPH_2:3,JGRAPH_3:4;
px`1<=px`2 & --px`2>=-px`1 or px`1>=px`2 & px`1<=-px`2
by A66,A67,A68,A79,XREAL_1:24,64;
then
A81: px`1<=px`2 & px`2>=-px`1 or px`1>=px`2 & -px`1>=--px`2 by XREAL_1:24;
A82: px`2/sqrt(1+(px`1/px`2)^2)=q`2 by A66,A68,A70,XCMPLX_1:89;
A83: px`1/sqrt(1+(px`1/px`2)^2)=q`1 by A67,A68,A70,XCMPLX_1:89;
A84: dom Sq_Circ=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
not px`2=0 by A66,A67,A68,A75,A79,XREAL_1:64;
then
A85: (px`2)^2>0 by SQUARE_1:12;
A86: (px`1)^2>=0 by XREAL_1:63;
(px`2)^2/(sqrt(1+(px`1/px`2)^2))^2+(px`1/sqrt(1+(px`1/px`2)^2)) ^2 < 1
by A70,A71,A72,A73,A74,XCMPLX_1:76;
then (px`2)^2/(sqrt(1+(px`1/px`2)^2))^2+(px`1)^2/(sqrt(1+(px`1/px`2)^2))
^2<1 by XCMPLX_1:76;
then (px`2)^2/(1+(px`1/px`2)^2)+(px`1)^2/(sqrt(1+(px`1/px`2)^2))^2<1
by A69,SQUARE_1:def 2;
then (px`2)^2/(1+(px`1/px`2)^2)+(px`1)^2/(1+(px`1/px`2)^2)<1
by A69,SQUARE_1:def 2;
then ((px`2)^2/(1+(px`1/px`2)^2)
+(px`1)^2/(1+(px`1/px`2)^2))*(1+(px`1/px`2)^2)<1 *(1+(px`1/px`2)^2)
by A69,XREAL_1:68;
then (px`2)^2/(1+(px`1/px`2)^2)*(1+(px`1/px`2)^2)
+(px`1)^2/(1+(px`1/px`2)^2)*(1+(px`1/px`2)^2)<1 *(1+(px`1/px`2)^2);
then (px`2)^2+(px`1)^2/(1+(px`1/px`2)^2)*(1+(px`1/px`2)^2)<1 *(1+(px`1/
px `2)^2) by A69,XCMPLX_1:87;
then
A87: (px`2)^2+(px`1)^2<1 *(1+(px`1/px`2)^2) by A69,XCMPLX_1:87;
1 *(1+(px`1/px`2)^2) =1+(px`1)^2/(px`2)^2 by XCMPLX_1:76;
then (px`2)^2+(px`1)^2-1<1+(px`1)^2/(px`2)^2-1 by A87,XREAL_1:9;
then ((px`2)^2+(px`1)^2-1)*(px`2)^2<((px`1)^2/(px`2)^2)*(px`2)^2
by A85,XREAL_1:68;
then
A88: ((px`2)^2+((px`1)^2-1))*(px`2)^2<(px`1)^2 by A85,XCMPLX_1:87;
(px`2)^2*(px`2)^2+((px`2)^2*(px`1)^2-(px`2)^2*1)-(px`1)^2
= ((px`2)^2-1)*((px`2)^2+(px`1)^2);
then (px`2)^2-1<0 or ((px`2)^2+(px`1)^2)<0 by A88,XREAL_1:49;
then
A89: (px`2)^2-1+1<0+1 by A86,XREAL_1:6;
then
A90: px`2<1^2 by SQUARE_1:48;
A91: px`2>-1^2 by A89,SQUARE_1:48;
px`1<1 & 1>-px`1 or px`1>=px`2 & -px`1>=px`2 by A81,A90,XXREAL_0:2;
then px`1<1 & -1< --px`1 or px`1>-1 & -px`1> -1
by A91,XREAL_1:24,XXREAL_0:2;
then px`1<1 & -1<px`1 or px`1>-1 & --px`1< --1 by XREAL_1:24;
then px in Kb by A1,A90,A91;
hence ex x being set st x in dom Sq_Circ & x in Kb & y=Sq_Circ.x
by A28,A80,A82,A83,A84,EUCLID:53;
end;
end;
hence thesis by FUNCT_1:def 6;
end;
theorem Th26:
for Kb,Cb being Subset of TOP-REAL 2 st
Kb={p: not(-1 <=p`1 & p`1<= 1 & -1 <=p`2 & p`2<= 1)}&
Cb={p2 where p2 is Point of TOP-REAL 2: |.p2.|>1} holds Sq_Circ.:Kb=Cb
proof
let Kb,Cb be Subset of TOP-REAL 2;
assume
A1: Kb={p: not(-1 <=p`1 & p`1<= 1 & -1 <=p`2 & p`2<= 1)}&
Cb={p2 where p2 is Point of TOP-REAL 2: |.p2.|>1};
thus Sq_Circ.:Kb c= Cb
proof
let y be object;
assume y in Sq_Circ.:Kb;
then consider x being object such that
x in dom Sq_Circ and
A2: x in Kb and
A3: y=Sq_Circ.x by FUNCT_1:def 6;
consider q being Point of TOP-REAL 2 such that
A4: q=x and
A5: not(-1 <=q`1 & q`1<= 1 & -1 <=q`2 & q`2<= 1) by A1,A2;
now per cases;
case q=0.TOP-REAL 2;
hence contradiction by A5,EUCLID:52,54;
end;
case
A6: q<>0.TOP-REAL 2 & (q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=-q`1);
then
A7: Sq_Circ.q=|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2 )]|
by JGRAPH_3:def 1;
A8: not(-1 <=q`2 & q`2<= 1) implies -1>q`1 or q`1>1
proof
assume
A9: not(-1 <=q`2 & q`2<= 1);
now per cases by A9;
case
A10: -1>q`2;
then -q`1< -1 or q`2>=q`1 & q`2<= -q`1 by A6,XXREAL_0:2;
hence thesis by A10,XREAL_1:24,XXREAL_0:2;
end;
case q`2>1;
then 1<q`1 or 1< -q`1 by A6,XXREAL_0:2;
then 1<q`1 or --q`1< -1 by XREAL_1:24;
hence thesis;
end;
end;
hence thesis;
end;
A11: (|[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;
A12: (|[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;
A13: 1+(q`2/q`1)^2>0 by XREAL_1:34,63;
A14: now
assume
A15: q`1=0;
then q`2=0 by A6;
hence contradiction by A6,A15,EUCLID:53,54;
end;
then
A16: (q`1)^2>0 by SQUARE_1:12;
(q`1)^2>1^2 by A5,A8,SQUARE_1:47;
then
A17: sqrt((q`1)^2)>1 by SQUARE_1:18,27;
|.(|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|).|^2
=((q`1)/sqrt(1+(q`2/(q`1))^2))^2+(q`2/sqrt(1+(q`2/(q`1))^2))^2
by A11,A12,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 A13,SQUARE_1:def 2
.=(q`1)^2/(1+(q`2/q`1)^2)+(q`2)^2/(1+(q`2/q`1)^2)
by A13,SQUARE_1:def 2
.=((q`1)^2+(q`2)^2)/(1+(q`2/q`1)^2) by XCMPLX_1:62
.=((q`1)^2+(q`2)^2)/(1+(q`2)^2/(q`1)^2) by XCMPLX_1:76
.=((q`1)^2+(q`2)^2)/((q`1)^2/(q`1)^2+(q`2)^2/(q`1)^2)
by A16,XCMPLX_1:60
.=((q`1)^2+(q`2)^2)/(((q`1)^2+(q`2)^2)/(q`1)^2) by XCMPLX_1:62
.=(q`1)^2*(((q`1)^2+(q`2)^2)/((q`1)^2+(q`2)^2)) by XCMPLX_1:81
.=(q`1)^2*1 by A14,COMPLEX1:1,XCMPLX_1:60
.=(q`1)^2;
then |.(|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|).|>1
by A17,SQUARE_1:22;
hence ex p2 being Point of TOP-REAL 2 st p2=y & |.p2.|>1 by A3,A4,A7;
end;
case
A18: q<>0.TOP-REAL 2 & not(q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=-q`1);
then
A19: Sq_Circ.q=|[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2 )]|
by JGRAPH_3:def 1;
A20: (|[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2)]|)`1 = q`1/sqrt(1+(q
`1/q`2)^2) by EUCLID:52;
A21: (|[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2)]|)`2 = q`2/sqrt(1+(q
`1/q`2)^2) by EUCLID:52;
A22: 1+(q`1/q`2)^2>0 by XREAL_1:34,63;
A23: q`2 <> 0 by A18;
then
A24: (q`2)^2>0 by SQUARE_1:12;
not(-1 <=q`1 & q`1<= 1) implies -1>q`2 or q`2>1
proof
assume
A25: not(-1 <=q`1 & q`1<= 1);
now per cases by A25;
case
A26: -1>q`1;
then q`2< -1 or q`1<q`2 & -q`2< --q`1 by A18,XREAL_1:24,XXREAL_0:2;
then -q`2< -1 or -1>q`2 by A26,XXREAL_0:2;
hence thesis by XREAL_1:24;
end;
case
A27: q`1>1;
--q`1< -q`2 & q`2<q`1 or q`2>q`1 & q`2> -q`1 by A18,XREAL_1:24;
then 1< -q`2 or q`2>q`1 & q`2> -q`1 by A27,XXREAL_0:2;
then -1> --q`2 or 1<q`2 by A27,XREAL_1:24,XXREAL_0:2;
hence thesis;
end;
end;
hence thesis;
end;
then (q`2)^2>1^2 by A5,SQUARE_1:47;
then
A28: sqrt((q`2)^2)>1 by SQUARE_1:18,27;
|.(|[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2)]|).|^2
=((q`1)/sqrt(1+(q`1/(q`2))^2))^2+(q`2/sqrt(1+(q`1/(q`2))^2))^2
by A20,A21,JGRAPH_3:1
.=(q`1)^2/(sqrt(1+(q`1/(q`2))^2))^2+(q`2/sqrt(1+(q`1/(q`2))^2))^2
by XCMPLX_1:76
.=(q`1)^2/(sqrt(1+(q`1/(q`2))^2))^2
+(q`2)^2/(sqrt(1+(q`1/(q`2))^2))^2 by XCMPLX_1:76
.=(q`1)^2/(1+(q`1/q`2)^2)+(q`2)^2/(sqrt(1+(q`1/q`2)^2))^2
by A22,SQUARE_1:def 2
.=(q`1)^2/(1+(q`1/q`2)^2)+(q`2)^2/(1+(q`1/q`2)^2)
by A22,SQUARE_1:def 2
.=((q`1)^2+(q`2)^2)/(1+(q`1/q`2)^2) by XCMPLX_1:62
.=((q`1)^2+(q`2)^2)/(1+(q`1)^2/(q`2)^2) by XCMPLX_1:76
.=((q`1)^2+(q`2)^2)/((q`1)^2/(q`2)^2+(q`2)^2/(q`2)^2)
by A24,XCMPLX_1:60
.=((q`1)^2+(q`2)^2)/(((q`1)^2+(q`2)^2)/(q`2)^2)
by XCMPLX_1:62
.=(q`2)^2*(((q`1)^2+(q`2)^2)/((q`1)^2+(q`2)^2))
by XCMPLX_1:81
.=(q`2)^2*1 by A23,COMPLEX1:1,XCMPLX_1:60
.=(q`2)^2;
then |.(|[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2)]|).|>1
by A28,SQUARE_1:22;
hence ex p2 being Point of TOP-REAL 2 st p2=y & |.p2.|>1 by A3,A4,A19;
end;
end;
hence thesis by A1;
end;
let y be object;
assume y in Cb;
then consider p2 being Point of TOP-REAL 2 such that
A29: p2=y and
A30: |.p2.|>1 by A1;
set q=p2;
now per cases;
case q=0.TOP-REAL 2;
hence contradiction by A30,TOPRNS_1:23;
end;
case
A31: q<>0.TOP-REAL 2 & (q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=-q`1);
set px=|[q`1*sqrt(1+(q`2/q`1)^2),q`2*sqrt(1+(q`2/q`1)^2)]|;
A32: px`1 = q`1*sqrt(1+(q`2/q`1)^2) by EUCLID:52;
A33: px`2 = q`2*sqrt(1+(q`2/q`1)^2) by EUCLID:52;
1+(q`2/q`1)^2>0 by XREAL_1:34,63;
then
A34: sqrt(1+(q`2/q`1)^2)>0 by SQUARE_1:25;
A35: 1+(px`2/px`1)^2>0 by XREAL_1:34,63;
A36: px`2/px`1=q`2/q`1 by A32,A33,A34,XCMPLX_1:91;
A37: q`1=q`1*sqrt(1+(q`2/q`1)^2)/(sqrt(1+(q`2/q`1)^2))by A34,XCMPLX_1:89
.=px`1/(sqrt(1+(q`2/q`1)^2)) by EUCLID:52;
A38: q`2=q`2*sqrt(1+(q`2/q`1)^2)/(sqrt(1+(q`2/q`1)^2))by A34,XCMPLX_1:89
.=px`2/(sqrt(1+(q`2/q`1)^2)) by EUCLID:52;
A39: |.q.|^2=q`1^2+q`2^2 by JGRAPH_3:1;
A40: |.q.|^2>1^2 by A30,SQUARE_1:16;
A41: now
assume that
A42: px`1=0 and
A43: px`2=0;
A44: q`1*sqrt(1+(q`2/q`1)^2)=0 by A42,EUCLID:52;
A45: q`2*sqrt(1+(q`2/q`1)^2)=0 by A43,EUCLID:52;
A46: q`1=0 by A34,A44,XCMPLX_1:6;
q`2=0 by A34,A45,XCMPLX_1:6;
hence contradiction by A31,A46,EUCLID:53,54;
end;
q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 &
q`2*sqrt(1+(q`2/q`1)^2) <= (-q`1)*sqrt(1+(q`2/q`1)^2)
by A31,A34,XREAL_1:64;
then
A47: q`2<=q`1 & (-q`1)*sqrt(1+(q`2/q`1)^2) <= q`2*sqrt(1+(q`2/q`1) ^2)
or px`2>=px`1 & px`2<=-px`1 by A32,A33,A34,XREAL_1:64;
then px`2<=px`1 & -px`1<=px`2 or px`2>=px`1 & px`2<=-px`1
by A32,A33,A34,XREAL_1:64;
then
A48: Sq_Circ
.px=|[px`1/sqrt(1+(px`2/px`1)^2),px`2/sqrt(1+(px`2/px`1 )^2) ]|
by A41,JGRAPH_2:3,JGRAPH_3:def 1;
A49: px`1/sqrt(1+(px`2/px`1)^2)=q`1 by A32,A34,A36,XCMPLX_1:89;
A50: px`2/sqrt(1+(px`2/px`1)^2)=q`2 by A33,A34,A36,XCMPLX_1:89;
A51: dom Sq_Circ=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
not px`1=0 by A32,A33,A34,A41,A47,XREAL_1:64;
then
A52: (px`1)^2>0 by SQUARE_1:12;
A53: (px`2)^2>=0 by XREAL_1:63;
(px`1)^2/(sqrt(1+(px`2/px`1)^2))^2+(px`2/sqrt(1+(px`2/px`1)^2)) ^2 > 1
by A36,A37,A38,A39,A40,XCMPLX_1:76;
then (px`1)^2/(sqrt(1+(px`2/px`1)^2))^2+(px`2)^2/(sqrt(1+(px`2/px`1)^2))
^2>1 by XCMPLX_1:76;
then (px`1)^2/(1+(px`2/px`1)^2)+(px`2)^2/(sqrt(1+(px`2/px`1)^2))^2>1
by A35,SQUARE_1:def 2;
then (px`1)^2/(1+(px`2/px`1)^2)+(px`2)^2/(1+(px`2/px`1)^2)>1
by A35,SQUARE_1:def 2;
then ((px`1)^2/(1+(px`2/px`1)^2)
+(px`2)^2/(1+(px`2/px`1)^2))*(1+(px`2/px`1)^2)>1 *(1+(px`2/px`1)^2)
by A35,XREAL_1:68;
then (px`1)^2/(1+(px`2/px`1)^2)*(1+(px`2/px`1)^2)
+(px`2)^2/(1+(px`2/px`1)^2)*(1+(px`2/px`1)^2)>1 *(1+(px`2/px`1)^2);
then (px`1)^2+(px`2)^2/(1+(px`2/px`1)^2)*(1+(px`2/px`1)^2)>1 *(1+(px`2/
px `1)^2) by A35,XCMPLX_1:87;
then
A54: (px`1)^2+(px`2)^2>1 *(1+(px`2/px`1)^2) by A35,XCMPLX_1:87;
1 *(1+(px`2/px`1)^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 A54,XREAL_1:9;
then ((px`1)^2+(px`2)^2-1)*(px`1)^2>((px`2)^2/(px`1)^2)*(px`1)^2
by A52,XREAL_1:68;
then
A55: ((px`1)^2+((px`2)^2-1))*(px`1)^2>(px`2)^2 by A52,XCMPLX_1:87;
(px`1)^2*(px`1)^2+((px`1)^2*(px`2)^2-(px`1)^2*1)-(px`2)^2
= ((px`1)^2-1)*((px`1)^2+(px`2)^2);
then (px`1)^2-1>0 or (px`1)^2+(px`2)^2<0 by A55,XREAL_1:50;
then (px`1)^2-1+1>0+1 by A52,A53,XREAL_1:6;
then px`1>1^2 or px`1< -1 by SQUARE_1:49;
then px in Kb by A1;
hence ex x being set st x in dom Sq_Circ & x in Kb & y=Sq_Circ.x
by A29,A48,A49,A50,A51,EUCLID:53;
end;
case
A56: q<>0.TOP-REAL 2 & not(q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=-q`1);
set px=|[q`1*sqrt(1+(q`1/q`2)^2),q`2*sqrt(1+(q`1/q`2)^2)]|;
A57: q`1<=q`2 & -q`2<=q`1 or q`1>=q`2 & q`1<=-q`2 by A56,JGRAPH_2:13;
A58: px`2 = q`2*sqrt(1+(q`1/q`2)^2) by EUCLID:52;
A59: px`1 = q`1*sqrt(1+(q`1/q`2)^2) by EUCLID:52;
1+(q`1/q`2)^2>0 by XREAL_1:34,63;
then
A60: sqrt(1+(q`1/q`2)^2)>0 by SQUARE_1:25;
A61: 1+(px`1/px`2)^2>0 by XREAL_1:34,63;
A62: px`1/px`2=q`1/q`2 by A58,A59,A60,XCMPLX_1:91;
A63: q`2=q`2*sqrt(1+(q`1/q`2)^2)/(sqrt(1+(q`1/q`2)^2))by A60,XCMPLX_1:89
.=px`2/(sqrt(1+(q`1/q`2)^2)) by EUCLID:52;
A64: q`1=q`1*sqrt(1+(q`1/q`2)^2)/(sqrt(1+(q`1/q`2)^2))by A60,XCMPLX_1:89
.=px`1/(sqrt(1+(q`1/q`2)^2)) by EUCLID:52;
A65: |.q.|^2=q`2^2+q`1^2 by JGRAPH_3:1;
A66: |.q.|^2>1^2 by A30,SQUARE_1:16;
A67: now
assume that
A68: px`2=0 and
A69: px`1=0;
A70: q`2*sqrt(1+(q`1/q`2)^2)=0 by A68,EUCLID:52;
q`1*sqrt(1+(q`1/q`2)^2)=0 by A69,EUCLID:52;
then q`1=0 by A60,XCMPLX_1:6;
hence contradiction by A56,A70;
end;
q`1<=q`2 & -q`2<=q`1 or q`1>=q`2 &
q`1*sqrt(1+(q`1/q`2)^2) <= (-q`2)*sqrt(1+(q`1/q`2)^2)
by A57,A60,XREAL_1:64;
then
A71: q`1<=q`2 & (-q`2)*sqrt(1+(q`1/q`2)^2) <= q`1*sqrt(1+(q`1/q`2) ^2)
or px`1>=px`2 & px`1<=-px`2 by A58,A59,A60,XREAL_1:64;
then px`1<=px`2 & -px`2<=px`1 or px`1>=px`2 & px`1<=-px`2
by A58,A59,A60,XREAL_1:64;
then
A72: Sq_Circ
.px=|[px`1/sqrt(1+(px`1/px`2)^2),px`2/sqrt(1+(px`1/px`2 )^2) ]|
by A67,JGRAPH_2:3,JGRAPH_3:4;
A73: px`2/sqrt(1+(px`1/px`2)^2)=q`2 by A58,A60,A62,XCMPLX_1:89;
A74: px`1/sqrt(1+(px`1/px`2)^2)=q`1 by A59,A60,A62,XCMPLX_1:89;
A75: dom Sq_Circ=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
not px`2=0 by A58,A59,A60,A67,A71,XREAL_1:64;
then
A76: (px`2)^2>0 by SQUARE_1:12;
A77: (px`1)^2>=0 by XREAL_1:63;
(px`2)^2/(sqrt(1+(px`1/px`2)^2))^2+(px`1/sqrt(1+(px`1/px`2)^2)) ^2 > 1
by A62,A63,A64,A65,A66,XCMPLX_1:76;
then (px`2)^2/(sqrt(1+(px`1/px`2)^2))^2+(px`1)^2/(sqrt(1+(px`1/px`2)^2))
^2>1 by XCMPLX_1:76;
then (px`2)^2/(1+(px`1/px`2)^2)+(px`1)^2/(sqrt(1+(px`1/px`2)^2))^2>1
by A61,SQUARE_1:def 2;
then (px`2)^2/(1+(px`1/px`2)^2)+(px`1)^2/(1+(px`1/px`2)^2)>1
by A61,SQUARE_1:def 2;
then ((px`2)^2/(1+(px`1/px`2)^2)
+(px`1)^2/(1+(px`1/px`2)^2))*(1+(px`1/px`2)^2)>1 *(1+(px`1/px`2)^2)
by A61,XREAL_1:68;
then (px`2)^2/(1+(px`1/px`2)^2)*(1+(px`1/px`2)^2)
+(px`1)^2/(1+(px`1/px`2)^2)*(1+(px`1/px`2)^2)>1 *(1+(px`1/px`2)^2);
then (px`2)^2+(px`1)^2/(1+(px`1/px`2)^2)*(1+(px`1/px`2)^2)>1 *(1+(px`1/
px `2)^2) by A61,XCMPLX_1:87;
then
A78: (px`2)^2+(px`1)^2>1 *(1+(px`1/px`2)^2) by A61,XCMPLX_1:87;
1 *(1+(px`1/px`2)^2) =1+(px`1)^2/(px`2)^2 by XCMPLX_1:76;
then (px`2)^2+(px`1)^2-1>1+(px`1)^2/(px`2)^2-1 by A78,XREAL_1:9;
then ((px`2)^2+(px`1)^2-1)*(px`2)^2>((px`1)^2/(px`2)^2)*(px`2)^2
by A76,XREAL_1:68;
then
A79: ((px`2)^2+((px`1)^2-1))*(px`2)^2>(px`1)^2 by A76,XCMPLX_1:87;
(px`2)^2*(px`2)^2+((px`2)^2*(px`1)^2-(px`2)^2*1)-(px`1)^2
= ((px`2)^2-1)*((px`2)^2+(px`1)^2);
then (px`2)^2-1>0 or (px`1)^2+(px`2)^2<0 by A79,XREAL_1:50;
then (px`2)^2-1+1>0+1 by A76,A77,XREAL_1:6;
then px`2>1^2 or px`2< -1 by SQUARE_1:49;
then px in Kb by A1;
hence ex x being set st x in dom Sq_Circ & x in Kb & y=Sq_Circ.x
by A29,A72,A73,A74,A75,EUCLID:53;
end;
end;
hence thesis by FUNCT_1:def 6;
end;
theorem Th27:
for Kb,Cb being Subset of TOP-REAL 2 st
Kb={p: -1 <=p`1 & p`1<= 1 & -1 <=p`2 & p`2<= 1}&
Cb={p2 where p2 is Point of TOP-REAL 2: |.p2.|<=1} holds Sq_Circ.:Kb=Cb
proof
let Kb,Cb be Subset of TOP-REAL 2;
assume
A1: Kb={p: -1 <=p`1 & p`1<= 1 & -1 <=p`2 & p`2<= 1}&
Cb={p2 where p2 is Point of TOP-REAL 2: |.p2.|<=1};
thus Sq_Circ.:Kb c= Cb
proof
let y be object;
assume y in Sq_Circ.:Kb;
then consider x being object such that
x in dom Sq_Circ and
A2: x in Kb and
A3: y=Sq_Circ.x by FUNCT_1:def 6;
consider q being Point of TOP-REAL 2 such that
A4: q=x and
A5: -1 <=q`1 and
A6: q`1<= 1 and
A7: -1 <=q`2 and
A8: q`2<= 1 by A1,A2;
now per cases;
case
A9: q=0.TOP-REAL 2;
then
A10: Sq_Circ.q=q by JGRAPH_3:def 1;
|.q.|=0 by A9,TOPRNS_1:23;
hence ex p2 being Point of TOP-REAL 2 st p2=y & |.p2.|<=1 by A3,A4,A10;
end;
case
A11: q<>0.TOP-REAL 2 & (q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=-q`1);
then
A12: Sq_Circ.q=|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2 )]|
by JGRAPH_3:def 1;
A13: (|[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;
A14: (|[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;
A15: 1+(q`2/q`1)^2>0 by XREAL_1:34,63;
A16: now
assume
A17: q`1=0;
then q`2=0 by A11;
hence contradiction by A11,A17,EUCLID:53,54;
end;
then
A18: (q`1)^2>0 by SQUARE_1:12;
(q`1)^2<=1^2 by A5,A6,SQUARE_1:49;
then
A19: sqrt((q`1)^2)<=1 by A18,SQUARE_1:18,26;
|.(|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|).|^2
=((q`1)/sqrt(1+(q`2/(q`1))^2))^2+(q`2/sqrt(1+(q`2/(q`1))^2))^2
by A13,A14,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 A15,SQUARE_1:def 2
.=(q`1)^2/(1+(q`2/q`1)^2)+(q`2)^2/(1+(q`2/q`1)^2)
by A15,SQUARE_1:def 2
.=((q`1)^2+(q`2)^2)/(1+(q`2/q`1)^2) by XCMPLX_1:62
.=((q`1)^2+(q`2)^2)/(1+(q`2)^2/(q`1)^2) by XCMPLX_1:76
.=((q`1)^2+(q`2)^2)/((q`1)^2/(q`1)^2+(q`2)^2/(q`1)^2)
by A18,XCMPLX_1:60
.=((q`1)^2+(q`2)^2)/(((q`1)^2+(q`2)^2)/(q`1)^2)
by XCMPLX_1:62
.=(q`1)^2*(((q`1)^2+(q`2)^2)/((q`1)^2+(q`2)^2))
by XCMPLX_1:81
.=(q`1)^2*1 by A16,COMPLEX1:1,XCMPLX_1:60
.=(q`1)^2;
then |.(|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|).|<=1
by A19,SQUARE_1:22;
hence ex p2 being Point of TOP-REAL 2 st p2=y & |.p2.|<=1 by A3,A4,A12;
end;
case
A20: q<>0.TOP-REAL 2 & not(q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=-q`1);
then
A21: Sq_Circ.q=|[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2 )]|
by JGRAPH_3:def 1;
A22: (|[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2)]|)`1 = q`1/sqrt(1+(q
`1/q`2)^2) by EUCLID:52;
A23: (|[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2)]|)`2 = q`2/sqrt(1+(q
`1/q`2)^2) by EUCLID:52;
A24: 1+(q`1/q`2)^2>0 by XREAL_1:34,63;
A25: q`2 <>0 by A20;
then
A26: (q`2)^2>0 by SQUARE_1:12;
(q`2)^2<=1^2 by A7,A8,SQUARE_1:49;
then
A27: sqrt((q`2)^2)<=1 by A26,SQUARE_1:18,26;
|.(|[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2)]|).|^2
=((q`1)/sqrt(1+(q`1/(q`2))^2))^2+(q`2/sqrt(1+(q`1/(q`2))^2))^2
by A22,A23,JGRAPH_3:1
.=(q`1)^2/(sqrt(1+(q`1/(q`2))^2))^2+(q`2/sqrt(1+(q`1/(q`2))^2))^2
by XCMPLX_1:76
.=(q`1)^2/(sqrt(1+(q`1/(q`2))^2))^2
+(q`2)^2/(sqrt(1+(q`1/(q`2))^2))^2 by XCMPLX_1:76
.=(q`1)^2/(1+(q`1/q`2)^2)+(q`2)^2/(sqrt(1+(q`1/q`2)^2))^2
by A24,SQUARE_1:def 2
.=(q`1)^2/(1+(q`1/q`2)^2)+(q`2)^2/(1+(q`1/q`2)^2)
by A24,SQUARE_1:def 2
.=((q`1)^2+(q`2)^2)/(1+(q`1/q`2)^2) by XCMPLX_1:62
.=((q`1)^2+(q`2)^2)/(1+(q`1)^2/(q`2)^2) by XCMPLX_1:76
.=((q`1)^2+(q`2)^2)/((q`1)^2/(q`2)^2+(q`2)^2/(q`2)^2)
by A26,XCMPLX_1:60
.=((q`1)^2+(q`2)^2)/(((q`1)^2+(q`2)^2)/(q`2)^2)
by XCMPLX_1:62
.=(q`2)^2*(((q`1)^2+(q`2)^2)/((q`1)^2+(q`2)^2)) by XCMPLX_1:81
.=(q`2)^2*1 by A25,COMPLEX1:1,XCMPLX_1:60
.=(q`2)^2;
then |.(|[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2)]|).|<=1
by A27,SQUARE_1:22;
hence ex p2 being Point of TOP-REAL 2 st p2=y & |.p2.|<=1 by A3,A4,A21;
end;
end;
hence thesis by A1;
end;
let y be object;
assume y in Cb;
then consider p2 being Point of TOP-REAL 2 such that
A28: p2=y and
A29: |.p2.|<=1 by A1;
set q=p2;
now per cases;
case
A30: q=0.TOP-REAL 2;
then
A31: q`1=0 by EUCLID:52,54;
q`2=0 by A30,EUCLID:52,54;
then
A32: y in Kb by A1,A28,A31;
A33: Sq_Circ".y=y by A28,A30,JGRAPH_3:28;
A34: dom Sq_Circ=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
y=Sq_Circ.y by A28,A33,FUNCT_1:35,JGRAPH_3:43;
hence ex x being set st x in dom Sq_Circ & x in Kb & y=Sq_Circ.x
by A32,A34;
end;
case
A35: q<>0.TOP-REAL 2 & (q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=-q`1);
set px=|[q`1*sqrt(1+(q`2/q`1)^2),q`2*sqrt(1+(q`2/q`1)^2)]|;
A36: px`1 = q`1*sqrt(1+(q`2/q`1)^2) by EUCLID:52;
A37: px`2 = q`2*sqrt(1+(q`2/q`1)^2) by EUCLID:52;
1+(q`2/q`1)^2>0 by XREAL_1:34,63;
then
A38: sqrt(1+(q`2/q`1)^2)>0 by SQUARE_1:25;
A39: 1+(px`2/px`1)^2>0 by XREAL_1:34,63;
A40: px`2/px`1=q`2/q`1 by A36,A37,A38,XCMPLX_1:91;
A41: q`1=q`1*sqrt(1+(q`2/q`1)^2)/(sqrt(1+(q`2/q`1)^2))by A38,XCMPLX_1:89
.=px`1/(sqrt(1+(q`2/q`1)^2)) by EUCLID:52;
A42: q`2=q`2*sqrt(1+(q`2/q`1)^2)/(sqrt(1+(q`2/q`1)^2))by A38,XCMPLX_1:89
.=px`2/(sqrt(1+(q`2/q`1)^2)) by EUCLID:52;
A43: |.q.|^2=q`1^2+q`2^2 by JGRAPH_3:1;
A44: |.q.|^2<=1^2 by A29,SQUARE_1:15;
A45: now
assume that
A46: px`1=0 and
A47: px`2=0;
A48: q`1*sqrt(1+(q`2/q`1)^2)=0 by A46,EUCLID:52;
A49: q`2*sqrt(1+(q`2/q`1)^2)=0 by A47,EUCLID:52;
A50: q`1=0 by A38,A48,XCMPLX_1:6;
q`2=0 by A38,A49,XCMPLX_1:6;
hence contradiction by A35,A50,EUCLID:53,54;
end;
q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 &
q`2*sqrt(1+(q`2/q`1)^2) <= (-q`1)*sqrt(1+(q`2/q`1)^2)
by A35,A38,XREAL_1:64;
then
A51: q`2<=q`1 & (-q`1)*sqrt(1+(q`2/q`1)^2) <= q`2*sqrt(1+(q`2/q`1) ^2)
or px`2>=px`1 & px`2<=-px`1 by A36,A37,A38,XREAL_1:64;
then px`2<=px`1 & -px`1<=px`2 or px`2>=px`1 & px`2<=-px`1
by A36,A37,A38,XREAL_1:64;
then
A52: Sq_Circ
.px=|[px`1/sqrt(1+(px`2/px`1)^2),px`2/sqrt(1+(px`2/px`1 )^2) ]|
by A45,JGRAPH_2:3,JGRAPH_3:def 1;
px`2<=px`1 & --px`1>=-px`2 or px`2>=px`1 & px`2<=-px`1
by A36,A37,A38,A51,XREAL_1:24,64;
then
A53: px`2<=px`1 & px`1>=-px`2 or px`2>=px`1 & -px`2>=--px`1 by XREAL_1:24;
A54: px`1/sqrt(1+(px`2/px`1)^2)=q`1 by A36,A38,A40,XCMPLX_1:89;
A55: px`2/sqrt(1+(px`2/px`1)^2)=q`2 by A37,A38,A40,XCMPLX_1:89;
A56: dom Sq_Circ=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
not px`1=0 by A36,A37,A38,A45,A51,XREAL_1:64;
then
A57: (px`1)^2>0 by SQUARE_1:12;
A58: (px`2)^2>=0 by XREAL_1:63;
(px`1)^2/(sqrt(1+(px`2/px`1)^2))^2+(px`2/sqrt(1+(px`2/px`1)^2)) ^2 <=1
by A40,A41,A42,A43,A44,XCMPLX_1:76;
then (px`1)^2/(sqrt(1+(px`2/px`1)^2))^2+(px`2)^2/(sqrt(1+(px`2/px`1)^2))
^2<=1 by XCMPLX_1:76;
then (px`1)^2/(1+(px`2/px`1)^2)+(px`2)^2/(sqrt(1+(px`2/px`1)^2))^2<=1
by A39,SQUARE_1:def 2;
then (px`1)^2/(1+(px`2/px`1)^2)+(px`2)^2/(1+(px`2/px`1)^2)<=1
by A39,SQUARE_1:def 2;
then ((px`1)^2/(1+(px`2/px`1)^2)
+(px`2)^2/(1+(px`2/px`1)^2))*(1+(px`2/px`1)^2)<=1 *(1+(px`2/px`1)^2)
by A39,XREAL_1:64;
then (px`1)^2/(1+(px`2/px`1)^2)*(1+(px`2/px`1)^2)
+(px`2)^2/(1+(px`2/px`1)^2)*(1+(px`2/px`1)^2)<=1 *(1+(px`2/px`1)^2);
then (px`1)^2+(px`2)^2/(1+(px`2/px`1)^2)*(1+(px`2/px`1)^2)
<=1 *(1+(px`2/px`1)^2) by A39,XCMPLX_1:87;
then
A59: (px`1)^2+(px`2)^2<=1 *(1+(px`2/px`1)^2) by A39,XCMPLX_1:87;
1 *(1+(px`2/px`1)^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 A59,XREAL_1:9;
then ((px`1)^2+(px`2)^2-1)*(px`1)^2<=((px`2)^2/(px`1)^2)*(px`1)^2
by A57,XREAL_1:64;
then ((px`1)^2+((px`2)^2-1))*(px`1)^2<=(px`2)^2 by A57,XCMPLX_1:87;
then
A60: (px`1)^2*(px`1)^2+(px`1)^2*((px`2)^2-1)-(px`2)^2<=0 by XREAL_1:47;
(px`1)^2*(px`1)^2+((px`1)^2*(px`2)^2-(px`1)^2*1)-(px`2)^2
= ((px`1)^2-1)*((px`1)^2+(px`2)^2);
then (px`1)^2-1<=0 & (px`1)^2-1>=0 or
(px`1)^2-1<=0 & (px`1)^2+(px`2)^2>=0 or
(px`1)^2+(px`2)^2<=0 & (px`1)^2-1>=0 or
(px`1)^2+(px`2)^2<=0 & (px`1)^2+(px`2)^2>=0 by A60,XREAL_1:129,130;
then
A61: (px`1)^2-1+1<=0+1 by A58,XREAL_1:7;
then
A62: px`1<=1^2 by SQUARE_1:47;
A63: px`1>= -1^2 by A61,SQUARE_1:47;
then px`2<=1 & 1>= -px`2 or px`2>= -1 & -px`2>=px`1 by A53,A62,XXREAL_0:2
;
then px`2<=1 & -1<= --px`2 or px`2>= -1 & -px`2>= -1
by A63,XREAL_1:24,XXREAL_0:2;
then px`2<=1 & -1<=px`2 or px`2>= -1 & --px`2<= --1 by XREAL_1:24;
then px in Kb by A1,A62,A63;
hence ex x being set st x in dom Sq_Circ & x in Kb & y=Sq_Circ.x
by A28,A52,A54,A55,A56,EUCLID:53;
end;
case
A64: q<>0.TOP-REAL 2 & not(q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=-q`1);
set px=|[q`1*sqrt(1+(q`1/q`2)^2),q`2*sqrt(1+(q`1/q`2)^2)]|;
A65: q`1<=q`2 & -q`2<=q`1 or q`1>=q`2 & q`1<=-q`2 by A64,JGRAPH_2:13;
A66: px`2 = q`2*sqrt(1+(q`1/q`2)^2) by EUCLID:52;
A67: px`1 = q`1*sqrt(1+(q`1/q`2)^2) by EUCLID:52;
1+(q`1/q`2)^2>0 by XREAL_1:34,63;
then
A68: sqrt(1+(q`1/q`2)^2)>0 by SQUARE_1:25;
A69: 1+(px`1/px`2)^2>0 by XREAL_1:34,63;
A70: px`1/px`2=q`1/q`2 by A66,A67,A68,XCMPLX_1:91;
A71: q`2=q`2*sqrt(1+(q`1/q`2)^2)/(sqrt(1+(q`1/q`2)^2))by A68,XCMPLX_1:89
.=px`2/(sqrt(1+(q`1/q`2)^2)) by EUCLID:52;
A72: q`1=q`1*sqrt(1+(q`1/q`2)^2)/(sqrt(1+(q`1/q`2)^2))by A68,XCMPLX_1:89
.=px`1/(sqrt(1+(q`1/q`2)^2)) by EUCLID:52;
A73: |.q.|^2=q`2^2+q`1^2 by JGRAPH_3:1;
A74: |.q.|^2<=1^2 by A29,SQUARE_1:15;
A75: now
assume that
A76: px`2=0 and
A77: px`1=0;
A78: q`2*sqrt(1+(q`1/q`2)^2)=0 by A76,EUCLID:52;
q`1*sqrt(1+(q`1/q`2)^2)=0 by A77,EUCLID:52;
then q`1=0 by A68,XCMPLX_1:6;
hence contradiction by A64,A78;
end;
q`1<=q`2 & -q`2<=q`1 or q`1>=q`2 &
q`1*sqrt(1+(q`1/q`2)^2) <= (-q`2)*sqrt(1+(q`1/q`2)^2)
by A65,A68,XREAL_1:64;
then
A79: q`1<=q`2 & (-q`2)*sqrt(1+(q`1/q`2)^2) <= q`1*sqrt(1+(q`1/q`2) ^2)
or px`1>=px`2 & px`1<=-px`2 by A66,A67,A68,XREAL_1:64;
then px`1<=px`2 & -px`2<=px`1 or px`1>=px`2 & px`1<=-px`2
by A66,A67,A68,XREAL_1:64;
then
A80: Sq_Circ
.px=|[px`1/sqrt(1+(px`1/px`2)^2),px`2/sqrt(1+(px`1/px`2 )^2) ]|
by A75,JGRAPH_2:3,JGRAPH_3:4;
px`1<=px`2 & --px`2>=-px`1 or px`1>=px`2 & px`1<=-px`2
by A66,A67,A68,A79,XREAL_1:24,64;
then
A81: px`1<=px`2 & px`2>=-px`1 or px`1>=px`2 & -px`1>=--px`2 by XREAL_1:24;
A82: px`2/sqrt(1+(px`1/px`2)^2)=q`2 by A66,A68,A70,XCMPLX_1:89;
A83: px`1/sqrt(1+(px`1/px`2)^2)=q`1 by A67,A68,A70,XCMPLX_1:89;
A84: dom Sq_Circ=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
not px`2=0 by A66,A67,A68,A75,A79,XREAL_1:64;
then
A85: (px`2)^2>0 by SQUARE_1:12;
A86: (px`1)^2>=0 by XREAL_1:63;
(px`2)^2/(sqrt(1+(px`1/px`2)^2))^2+(px`1/sqrt(1+(px`1/px`2)^2)) ^2 <= 1
by A70,A71,A72,A73,A74,XCMPLX_1:76;
then
(px`2)^2/(sqrt(1+(px`1/px`2)^2))^2+(px`1)^2/(sqrt(1+(px`1/px`2)^2))
^2<=1 by XCMPLX_1:76;
then (px`2)^2/(1+(px`1/px`2)^2)+(px`1)^2/(sqrt(1+(px`1/px`2)^2))^2<=1
by A69,SQUARE_1:def 2;
then (px`2)^2/(1+(px`1/px`2)^2)+(px`1)^2/(1+(px`1/px`2)^2)<=1
by A69,SQUARE_1:def 2;
then ((px`2)^2/(1+(px`1/px`2)^2)
+(px`1)^2/(1+(px`1/px`2)^2))*(1+(px`1/px`2)^2)<=1 *(1+(px`1/px`2)^2)
by A69,XREAL_1:64;
then (px`2)^2/(1+(px`1/px`2)^2)*(1+(px`1/px`2)^2)
+(px`1)^2/(1+(px`1/px`2)^2)*(1+(px`1/px`2)^2)<=1 *(1+(px`1/px`2)^2);
then (px`2)^2+(px`1)^2/(1+(px`1/px`2)^2)*(1+(px`1/px`2)^2)
<=1 *(1+(px`1/px`2)^2) by A69,XCMPLX_1:87;
then
A87: (px`2)^2+(px`1)^2<=1 *(1+(px`1/px`2)^2) by A69,XCMPLX_1:87;
1 *(1+(px`1/px`2)^2) =1+(px`1)^2/(px`2)^2 by XCMPLX_1:76;
then (px`2)^2+(px`1)^2-1<=1+(px`1)^2/(px`2)^2-1 by A87,XREAL_1:9;
then ((px`2)^2+(px`1)^2-1)*(px`2)^2<=((px`1)^2/(px`2)^2)*(px`2)^2
by A85,XREAL_1:64;
then ((px`2)^2+((px`1)^2-1))*(px`2)^2<=(px`1)^2 by A85,XCMPLX_1:87;
then
A88: (px`2)^2*(px`2)^2+(px`2)^2*((px`1)^2-1)-(px`1)^2<=0 by XREAL_1:47;
(px`2)^2*(px`2)^2+((px`2)^2*(px`1)^2-(px`2)^2*1)-(px`1)^2
= ((px`2)^2-1)*((px`2)^2+(px`1)^2);
then (px`2)^2-1<=0 or ((px`2)^2+(px`1)^2)<=0 by A88,XREAL_1:129;
then
A89: (px`2)^2-1+1<=0+1 by A86,XREAL_1:7;
then
A90: px`2<=1^2 by SQUARE_1:47;
A91: px`2>= -1^2 by A89,SQUARE_1:47;
then px`1<=1 & 1>= -px`1 or px`1>= -1 & -px`1>=px`2 by A81,A90,XXREAL_0:2
;
then px`1<=1 & -1<= --px`1 or px`1>= -1 & -px`1>= -1
by A91,XREAL_1:24,XXREAL_0:2;
then px`1<=1 & -1<=px`1 or px`1>= -1 & --px`1<= --1 by XREAL_1:24;
then px in Kb by A1,A90,A91;
hence ex x being set st x in dom Sq_Circ & x in Kb & y=Sq_Circ.x
by A28,A80,A82,A83,A84,EUCLID:53;
end;
end;
hence thesis by FUNCT_1:def 6;
end;
theorem Th28:
for Kb,Cb being Subset of TOP-REAL 2 st
Kb={p: not(-1 <p`1 & p`1< 1 & -1 <p`2 & p`2< 1)}&
Cb={p2 where p2 is Point of TOP-REAL 2: |.p2.|>=1} holds Sq_Circ.:Kb=Cb
proof
let Kb,Cb be Subset of TOP-REAL 2;
assume
A1: Kb={p: not(-1 <p`1 & p`1< 1 & -1 <p`2 & p`2< 1)}&
Cb={p2 where p2 is Point of TOP-REAL 2: |.p2.|>=1};
thus Sq_Circ.:Kb c= Cb
proof
let y be object;
assume y in Sq_Circ.:Kb;
then consider x being object such that
x in dom Sq_Circ and
A2: x in Kb and
A3: y=Sq_Circ.x by FUNCT_1:def 6;
consider q being Point of TOP-REAL 2 such that
A4: q=x and
A5: not(-1 <q`1 & q`1< 1 & -1 <q`2 & q`2< 1) by A1,A2;
now per cases;
case
A6: q=0.TOP-REAL 2;
then
A7: q`1=0 by EUCLID:52,54;
q`2=0 by A6,EUCLID:52,54;
hence contradiction by A5,A7;
end;
case
A8: q<>0.TOP-REAL 2 & (q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=-q`1);
then
A9: Sq_Circ.q=|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2 )]|
by JGRAPH_3:def 1;
A10: not(-1 <q`2 & q`2< 1) implies -1>=q`1 or q`1>=1
proof
assume
A11: not(-1 <q`2 & q`2< 1);
now per cases by A11;
case
A12: -1>=q`2;
then -q`1<= -1 or q`2>=q`1 & q`2<= -q`1 by A8,XXREAL_0:2;
hence thesis by A12,XREAL_1:24,XXREAL_0:2;
end;
case q`2>=1;
then 1<=q`1 or 1<= -q`1 by A8,XXREAL_0:2;
then 1<=q`1 or --q`1<= -1 by XREAL_1:24;
hence thesis;
end;
end;
hence thesis;
end;
A13: (|[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;
A14: (|[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;
A15: 1+(q`2/q`1)^2>0 by XREAL_1:34,63;
A16: now
assume
A17: q`1=0;
then q`2=0 by A8;
hence contradiction by A8,A17,EUCLID:53,54;
end;
then
A18: (q`1)^2>0 by SQUARE_1:12;
(q`1)^2>=1^2 by A5,A10,SQUARE_1:48;
then
A19: sqrt((q`1)^2)>=1 by SQUARE_1:18,26;
|.(|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|).|^2
=((q`1)/sqrt(1+(q`2/(q`1))^2))^2+(q`2/sqrt(1+(q`2/(q`1))^2))^2
by A13,A14,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 A15,SQUARE_1:def 2
.=(q`1)^2/(1+(q`2/q`1)^2)+(q`2)^2/(1+(q`2/q`1)^2)
by A15,SQUARE_1:def 2
.=((q`1)^2+(q`2)^2)/(1+(q`2/q`1)^2) by XCMPLX_1:62
.=((q`1)^2+(q`2)^2)/(1+(q`2)^2/(q`1)^2) by XCMPLX_1:76
.=((q`1)^2+(q`2)^2)/((q`1)^2/(q`1)^2+(q`2)^2/(q`1)^2)
by A18,XCMPLX_1:60
.=((q`1)^2+(q`2)^2)/(((q`1)^2+(q`2)^2)/(q`1)^2)
by XCMPLX_1:62
.=(q`1)^2*(((q`1)^2+(q`2)^2)/((q`1)^2+(q`2)^2))
by XCMPLX_1:81
.=(q`1)^2*1 by A16,COMPLEX1:1,XCMPLX_1:60
.=(q`1)^2;
then |.(|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|).|>=1
by A19,SQUARE_1:22;
hence ex p2 being Point of TOP-REAL 2 st p2=y & |.p2.|>=1 by A3,A4,A9;
end;
case
A20: q<>0.TOP-REAL 2 & not(q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=-q`1);
then
A21: Sq_Circ.q=|[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2 )]|
by JGRAPH_3:def 1;
A22: (|[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2)]|)`1 = q`1/sqrt(1+(q
`1/q`2)^2) by EUCLID:52;
A23: (|[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2)]|)`2 = q`2/sqrt(1+(q
`1/q`2)^2) by EUCLID:52;
A24: 1+(q`1/q`2)^2>0 by XREAL_1:34,63;
A25: q`2 <> 0 by A20;
then
A26: (q`2)^2>0 by SQUARE_1:12;
not(-1 <q`1 & q`1< 1) implies -1>=q`2 or q`2>=1
proof
assume
A27: not(-1 <q`1 & q`1< 1);
now per cases by A27;
case
A28: -1>=q`1;
then q`2<= -1 or q`1<q`2 & -q`2<= --q`1 by A20,XREAL_1:24,XXREAL_0:2;
then -q`2<= -1 or -1>=q`2 by A28,XXREAL_0:2;
hence thesis by XREAL_1:24;
end;
case
A29: q`1>=1;
--q`1<= -q`2 & q`2<=q`1 or q`2>=q`1 & q`2>= -q`1
by A20,XREAL_1:24;
then 1<= -q`2 or q`2>=q`1 & q`2>= -q`1 by A29,XXREAL_0:2;
then -1>= --q`2 or 1<=q`2 by A29,XREAL_1:24,XXREAL_0:2;
hence thesis;
end;
end;
hence thesis;
end;
then (q`2)^2>=1^2 by A5,SQUARE_1:48;
then
A30: sqrt((q`2)^2)>=1 by SQUARE_1:18,26;
|.(|[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2)]|).|^2
=((q`1)/sqrt(1+(q`1/(q`2))^2))^2+(q`2/sqrt(1+(q`1/(q`2))^2))^2
by A22,A23,JGRAPH_3:1
.=(q`1)^2/(sqrt(1+(q`1/(q`2))^2))^2+(q`2/sqrt(1+(q`1/(q`2))^2))^2
by XCMPLX_1:76
.=(q`1)^2/(sqrt(1+(q`1/(q`2))^2))^2
+(q`2)^2/(sqrt(1+(q`1/(q`2))^2))^2 by XCMPLX_1:76
.=(q`1)^2/(1+(q`1/q`2)^2)+(q`2)^2/(sqrt(1+(q`1/q`2)^2))^2
by A24,SQUARE_1:def 2
.=(q`1)^2/(1+(q`1/q`2)^2)+(q`2)^2/(1+(q`1/q`2)^2)
by A24,SQUARE_1:def 2
.=((q`1)^2+(q`2)^2)/(1+(q`1/q`2)^2) by XCMPLX_1:62
.=((q`1)^2+(q`2)^2)/(1+(q`1)^2/(q`2)^2) by XCMPLX_1:76
.=((q`1)^2+(q`2)^2)/((q`1)^2/(q`2)^2+(q`2)^2/(q`2)^2)
by A26,XCMPLX_1:60
.=((q`1)^2+(q`2)^2)/(((q`1)^2+(q`2)^2)/(q`2)^2) by XCMPLX_1:62
.=(q`2)^2*(((q`1)^2+(q`2)^2)/((q`1)^2+(q`2)^2)) by XCMPLX_1:81
.=(q`2)^2*1 by A25,COMPLEX1:1,XCMPLX_1:60
.=(q`2)^2;
then |.(|[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2)]|).|>=1
by A30,SQUARE_1:22;
hence ex p2 being Point of TOP-REAL 2 st p2=y & |.p2.|>=1 by A3,A4,A21;
end;
end;
hence thesis by A1;
end;
let y be object;
assume y in Cb;
then consider p2 being Point of TOP-REAL 2 such that
A31: p2=y and
A32: |.p2.|>=1 by A1;
set q=p2;
now per cases;
case q=0.TOP-REAL 2;
hence contradiction by A32,TOPRNS_1:23;
end;
case
A33: q<>0.TOP-REAL 2 & (q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=-q`1);
set px=|[q`1*sqrt(1+(q`2/q`1)^2),q`2*sqrt(1+(q`2/q`1)^2)]|;
A34: px`1 = q`1*sqrt(1+(q`2/q`1)^2) by EUCLID:52;
A35: px`2 = q`2*sqrt(1+(q`2/q`1)^2) by EUCLID:52;
1+(q`2/q`1)^2>0 by XREAL_1:34,63;
then
A36: sqrt(1+(q`2/q`1)^2)>0 by SQUARE_1:25;
A37: 1+(px`2/px`1)^2>0 by XREAL_1:34,63;
A38: px`2/px`1=q`2/q`1 by A34,A35,A36,XCMPLX_1:91;
A39: q`1=q`1*sqrt(1+(q`2/q`1)^2)/(sqrt(1+(q`2/q`1)^2))by A36,XCMPLX_1:89
.=px`1/(sqrt(1+(q`2/q`1)^2)) by EUCLID:52;
A40: q`2=q`2*sqrt(1+(q`2/q`1)^2)/(sqrt(1+(q`2/q`1)^2))by A36,XCMPLX_1:89
.=px`2/(sqrt(1+(q`2/q`1)^2)) by EUCLID:52;
A41: |.q.|^2=q`1^2+q`2^2 by JGRAPH_3:1;
A42: |.q.|^2>=1^2 by A32,SQUARE_1:15;
A43: now
assume that
A44: px`1=0 and
A45: px`2=0;
A46: q`1*sqrt(1+(q`2/q`1)^2)=0 by A44,EUCLID:52;
A47: q`2*sqrt(1+(q`2/q`1)^2)=0 by A45,EUCLID:52;
A48: q`1=0 by A36,A46,XCMPLX_1:6;
q`2=0 by A36,A47,XCMPLX_1:6;
hence contradiction by A33,A48,EUCLID:53,54;
end;
q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 &
q`2*sqrt(1+(q`2/q`1)^2) <= (-q`1)*sqrt(1+(q`2/q`1)^2)
by A33,A36,XREAL_1:64;
then
A49: q`2<=q`1 & (-q`1)*sqrt(1+(q`2/q`1)^2) <= q`2*sqrt(1+(q`2/q`1) ^2)
or px`2>=px`1 & px`2<=-px`1 by A34,A35,A36,XREAL_1:64;
then px`2<=px`1 & -px`1<=px`2 or px`2>=px`1 & px`2<=-px`1
by A34,A35,A36,XREAL_1:64;
then
A50: Sq_Circ
.px=|[px`1/sqrt(1+(px`2/px`1)^2),px`2/sqrt(1+(px`2/px`1 )^2) ]|
by A43,JGRAPH_2:3,JGRAPH_3:def 1;
A51: px`1/sqrt(1+(px`2/px`1)^2)=q`1 by A34,A36,A38,XCMPLX_1:89;
A52: px`2/sqrt(1+(px`2/px`1)^2)=q`2 by A35,A36,A38,XCMPLX_1:89;
A53: dom Sq_Circ=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
not px`1=0 by A34,A35,A36,A43,A49,XREAL_1:64;
then
A54: (px`1)^2>0 by SQUARE_1:12;
then
A55: (px`1)^2+(px`2)^2>0 by XREAL_1:34,63;
(px`1)^2/(sqrt(1+(px`2/px`1)^2))^2+(px`2/sqrt(1+(px`2/px`1)^2)) ^2 >= 1
by A38,A39,A40,A41,A42,XCMPLX_1:76;
then
(px`1)^2/(sqrt(1+(px`2/px`1)^2))^2+(px`2)^2/(sqrt(1+(px`2/px`1)^2))
^2>=1 by XCMPLX_1:76;
then (px`1)^2/(1+(px`2/px`1)^2)+(px`2)^2/(sqrt(1+(px`2/px`1)^2))^2>=1
by A37,SQUARE_1:def 2;
then (px`1)^2/(1+(px`2/px`1)^2)+(px`2)^2/(1+(px`2/px`1)^2)>=1
by A37,SQUARE_1:def 2;
then ((px`1)^2/(1+(px`2/px`1)^2)
+(px`2)^2/(1+(px`2/px`1)^2))*(1+(px`2/px`1)^2)>=1 *(1+(px`2/px`1)^2)
by A37,XREAL_1:64;
then (px`1)^2/(1+(px`2/px`1)^2)*(1+(px`2/px`1)^2)
+(px`2)^2/(1+(px`2/px`1)^2)*(1+(px`2/px`1)^2)>=1 *(1+(px`2/px`1)^2);
then (px`1)^2+(px`2)^2/(1+(px`2/px`1)^2)*(1+(px`2/px`1)^2)
>=1 *(1+(px`2/px`1)^2) by A37,XCMPLX_1:87;
then
A56: (px`1)^2+(px`2)^2>=1 *(1+(px`2/px`1)^2) by A37,XCMPLX_1:87;
1 *(1+(px`2/px`1)^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 A56,XREAL_1:9;
then ((px`1)^2+(px`2)^2-1)*(px`1)^2>=((px`2)^2/(px`1)^2)*(px`1)^2
by A54,XREAL_1:64;
then ((px`1)^2+((px`2)^2-1))*(px`1)^2>=(px`2)^2 by A54,XCMPLX_1:87;
then
A57: (px`1)^2*(px`1)^2+(px`1)^2*((px`2)^2-1)-(px`2)^2>=0 by XREAL_1:48;
(px`1)^2*(px`1)^2+((px`1)^2*(px`2)^2-(px`1)^2*1)-(px`2)^2
= ((px`1)^2-1)*((px`1)^2+(px`2)^2);
then (px`1)^2-1>=0 by A55,A57,XREAL_1:132;
then (px`1)^2-1+1>=0+1 by XREAL_1:7;
then px`1>=1^2 or px`1<= -1 by SQUARE_1:50;
then px in Kb by A1;
hence ex x being set st x in dom Sq_Circ & x in Kb & y=Sq_Circ.x
by A31,A50,A51,A52,A53,EUCLID:53;
end;
case
A58: q<>0.TOP-REAL 2 & not(q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=-q`1);
set px=|[q`1*sqrt(1+(q`1/q`2)^2),q`2*sqrt(1+(q`1/q`2)^2)]|;
A59: q`1<=q`2 & -q`2<=q`1 or q`1>=q`2 & q`1<=-q`2 by A58,JGRAPH_2:13;
A60: px`2 = q`2*sqrt(1+(q`1/q`2)^2) by EUCLID:52;
A61: px`1 = q`1*sqrt(1+(q`1/q`2)^2) by EUCLID:52;
1+(q`1/q`2)^2>0 by XREAL_1:34,63;
then
A62: sqrt(1+(q`1/q`2)^2)>0 by SQUARE_1:25;
A63: 1+(px`1/px`2)^2>0 by XREAL_1:34,63;
A64: px`1/px`2=q`1/q`2 by A60,A61,A62,XCMPLX_1:91;
A65: q`2=q`2*sqrt(1+(q`1/q`2)^2)/(sqrt(1+(q`1/q`2)^2))by A62,XCMPLX_1:89
.=px`2/(sqrt(1+(q`1/q`2)^2)) by EUCLID:52;
A66: q`1=q`1*sqrt(1+(q`1/q`2)^2)/(sqrt(1+(q`1/q`2)^2))by A62,XCMPLX_1:89
.=px`1/(sqrt(1+(q`1/q`2)^2)) by EUCLID:52;
A67: |.q.|^2=q`2^2+q`1^2 by JGRAPH_3:1;
A68: |.q.|^2>=1^2 by A32,SQUARE_1:15;
A69: now
assume that
A70: px`2=0 and
A71: px`1=0;
A72: q`2*sqrt(1+(q`1/q`2)^2)=0 by A70,EUCLID:52;
q`1*sqrt(1+(q`1/q`2)^2)=0 by A71,EUCLID:52;
then q`1=0 by A62,XCMPLX_1:6;
hence contradiction by A58,A72;
end;
q`1<=q`2 & -q`2<=q`1 or q`1>=q`2 &
q`1*sqrt(1+(q`1/q`2)^2) <= (-q`2)*sqrt(1+(q`1/q`2)^2)
by A59,A62,XREAL_1:64;
then
A73: q`1<=q`2 & (-q`2)*sqrt(1+(q`1/q`2)^2) <= q`1*sqrt(1+(q`1/q`2) ^2)
or px`1>=px`2 & px`1<=-px`2 by A60,A61,A62,XREAL_1:64;
then px`1<=px`2 & -px`2<=px`1 or px`1>=px`2 & px`1<=-px`2
by A60,A61,A62,XREAL_1:64;
then
A74: Sq_Circ
.px=|[px`1/sqrt(1+(px`1/px`2)^2),px`2/sqrt(1+(px`1/px`2 )^2) ]|
by A69,JGRAPH_2:3,JGRAPH_3:4;
A75: px`2/sqrt(1+(px`1/px`2)^2)=q`2 by A60,A62,A64,XCMPLX_1:89;
A76: px`1/sqrt(1+(px`1/px`2)^2)=q`1 by A61,A62,A64,XCMPLX_1:89;
A77: dom Sq_Circ=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
not px`2=0 by A60,A61,A62,A69,A73,XREAL_1:64;
then
A78: (px`2)^2>0 by SQUARE_1:12;
A79: (px`1)^2>=0 by XREAL_1:63;
(px`2)^2/(sqrt(1+(px`1/px`2)^2))^2+(px`1/sqrt(1+(px`1/px`2)^2)) ^2 >= 1
by A64,A65,A66,A67,A68,XCMPLX_1:76;
then
(px`2)^2/(sqrt(1+(px`1/px`2)^2))^2+(px`1)^2/(sqrt(1+(px`1/px`2)^2))
^2>=1 by XCMPLX_1:76;
then (px`2)^2/(1+(px`1/px`2)^2)+(px`1)^2/(sqrt(1+(px`1/px`2)^2))^2>=1
by A63,SQUARE_1:def 2;
then (px`2)^2/(1+(px`1/px`2)^2)+(px`1)^2/(1+(px`1/px`2)^2)>=1
by A63,SQUARE_1:def 2;
then ((px`2)^2/(1+(px`1/px`2)^2)
+(px`1)^2/(1+(px`1/px`2)^2))*(1+(px`1/px`2)^2)>=1 *(1+(px`1/px`2)^2)
by A63,XREAL_1:64;
then (px`2)^2/(1+(px`1/px`2)^2)*(1+(px`1/px`2)^2)
+(px`1)^2/(1+(px`1/px`2)^2)*(1+(px`1/px`2)^2)>=1 *(1+(px`1/px`2)^2);
then (px`2)^2+(px`1)^2/(1+(px`1/px`2)^2)*(1+(px`1/px`2)^2)
>=1 *(1+(px`1/px`2)^2) by A63,XCMPLX_1:87;
then
A80: (px`2)^2+(px`1)^2>=1 *(1+(px`1/px`2)^2) by A63,XCMPLX_1:87;
1 *(1+(px`1/px`2)^2) =1+(px`1)^2/(px`2)^2 by XCMPLX_1:76;
then (px`2)^2+(px`1)^2-1>=1+(px`1)^2/(px`2)^2-1 by A80,XREAL_1:9;
then ((px`2)^2+(px`1)^2-1)*(px`2)^2>=((px`1)^2/(px`2)^2)*(px`2)^2
by A78,XREAL_1:64;
then ((px`2)^2+((px`1)^2-1))*(px`2)^2>=(px`1)^2 by A78,XCMPLX_1:87;
then
A81: (px`2)^2*(px`2)^2+(px`2)^2*((px`1)^2-1)-(px`1)^2>=0 by XREAL_1:48;
(px`2)^2*(px`2)^2+((px`2)^2*(px`1)^2-(px`2)^2*1)-(px`1)^2
= ((px`2)^2-1)*((px`2)^2+(px`1)^2);
then (px`2)^2-1>=0 & (px`1)^2+(px`2)^2>=0 or
(px`2)^2-1<=0 & (px`1)^2+(px`2)^2<=0 by A81,XREAL_1:132;
then (px`2)^2-1+1>=0+1 by A78,A79,XREAL_1:7;
then px`2>=1^2 or px`2<= -1 by SQUARE_1:50;
then px in Kb by A1;
hence ex x being set st x in dom Sq_Circ & x in Kb & y=Sq_Circ.x
by A31,A74,A75,A76,A77,EUCLID:53;
end;
end;
hence thesis by FUNCT_1:def 6;
end;
theorem
for P0,P1,P01,P11,K0,K1,K01,K11 being Subset of TOP-REAL 2,
P being non empty compact Subset of TOP-REAL 2,
f being Function of TOP-REAL 2,TOP-REAL 2 st P= circle(0,0,1) &
P0= inside_of_circle(0,0,1) & P1= outside_of_circle(0,0,1) &
P01= closed_inside_of_circle(0,0,1) & P11= closed_outside_of_circle(0,0,1) &
K0=inside_of_rectangle(-1,1,-1,1) & K1=outside_of_rectangle(-1,1,-1,1) &
K01=closed_inside_of_rectangle(-1,1,-1,1) &
K11=closed_outside_of_rectangle(-1,1,-1,1) & f=Sq_Circ
holds f.:rectangle(-1,1,-1,1)=P & f".:P=rectangle(-1,1,-1,1) &
f.:K0=P0 & f".:P0=K0 & f.:K1=P1 & f".:P1=K1 &
f.:K01=P01 & f.:K11=P11 & f".:P01=K01 & f".:P11=K11
proof
let P0,P1,P01,P11,K0,K1,K01,K11 be Subset of TOP-REAL 2,
P being non empty compact Subset of TOP-REAL 2,
f be Function of TOP-REAL 2,TOP-REAL 2;
assume that
A1: P= circle(0,0,1) and
A2: P0= inside_of_circle(0,0,1) and
A3: P1= outside_of_circle(0,0,1) and
A4: P01= closed_inside_of_circle(0,0,1) and
A5: P11= closed_outside_of_circle(0,0,1) and
A6: K0=inside_of_rectangle(-1,1,-1,1) and
A7: K1=outside_of_rectangle(-1,1,-1,1) and
A8: K01=closed_inside_of_rectangle(-1,1,-1,1) and
A9: K11=closed_outside_of_rectangle(-1,1,-1,1) and
A10: f=Sq_Circ;
set K=rectangle(-1,1,-1,1);
A11: P0={p: |.p.| <1} by A2,Th24;
A12: P01={p: |.p.| <=1} by A4,Th24;
A13: P1={p: |.p.| >1} by A3,Th24;
A14: P11={p: |.p.| >=1} by A5,Th24;
defpred P[Point of TOP-REAL 2] means
$1`1 = -1 & $1`2 <= 1 & $1`2 >= -1 or $1`1 <= 1 & $1`1 >= -1 & $1`2 = 1 or
$1`1 <= 1 & $1`1 >= -1 & $1`2 = -1 or $1`1 = 1 & $1`2 <= 1 & $1`2 >= -1;
defpred Q[Point of TOP-REAL 2] means
-1=$1`1 & -1<=$1`2 & $1`2<=1 or $1`1=1 & -1<=$1`2 & $1`2<=1
or -1=$1`2 & -1<=$1`1 & $1`1<=1 or 1=$1`2 & -1<=$1`1 & $1`1<=1;
deffunc F(set)=$1;
A15: for p being Element of TOP-REAL 2 holds P[p] iff Q[p];
A16: K= {F(p): P[p]} by SPPOL_2:54
.= {F(q):Q[q]} from FRAENKEL:sch 3(A15);
defpred Q[Point of TOP-REAL 2] means |.$1.|=1;
defpred P[Point of TOP-REAL 2] means |.$1- |[0,0]| .|=1;
A17: for p holds P[p] iff Q[p] by EUCLID:54,RLVECT_1:13;
P= {F(p): P[p]} by A1
.= {F(p2) where p2 is Point of TOP-REAL 2: Q[p2]} from FRAENKEL:sch 3(A17
);
then
A18: f.:K=P by A10,A16,JGRAPH_3:23;
A19: dom f=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
A20: f.:K0 = P0 by A6,A10,A11,Th25;
f.:K1 = P1 by A7,A10,A13,Th26;
hence f.:K=P & f".:P=K & f.:K0=P0 & f".:P0=K0 & f.:K1=P1 & f".:P1=K1
by A10,A18,A19,A20,FUNCT_1:107;
A21: f.:K01 = P01 by A8,A10,A12,Th27;
f.:K11 = P11 by A9,A10,A14,Th28;
hence thesis by A10,A19,A21,FUNCT_1:107;
end;
begin :: Order of Points on Rectangle
theorem Th30:
for a,b,c,d being Real st a <= b & c <= d holds
LSeg(|[ a,c ]|, |[ a,d ]|) = { p1 : p1`1 = a & p1`2 <= d & p1`2 >= c}
& LSeg(|[ a,d ]|, |[ b,d ]|) = { p2 : p2`1 <= b & p2`1 >= a & p2`2 = d}
& LSeg(|[ a,c ]|, |[ b,c ]|) = { q1 : q1`1 <= b & q1`1 >= a & q1`2 = c}
& LSeg(|[ b,c ]|, |[ b,d ]|) = { q2 : q2`1 = b & q2`2 <= d & q2`2 >= c}
proof
let a,b,c,d be Real;
assume that
A1: a <= b and
A2: c <= d;
set L1 = { p : p`1 = a & p`2 <= d & p`2 >= c},
L2 = { p : p`1 <= b & p`1 >= a & p`2 = d},
L3 = { p : p`1 <= b & p`1 >= a & p`2 = c},
L4 = { p : p`1 = b & p`2 <= d & p`2 >= c};
set p0 = |[ a,c ]|, p01 = |[ a,d ]|, p10 = |[ b,c ]|, p1 = |[ b,d ]|;
A3: p01`1 = a by EUCLID:52;
A4: p01`2 = d by EUCLID:52;
A5: p10`1 = b by EUCLID:52;
A6: p10`2 = c by EUCLID:52;
A7: L1 c= LSeg(p0,p01)
proof
let a2 be object;
assume a2 in L1;
then consider p such that
A8: a2 = p and
A9: p`1 = a and
A10: p`2 <= d and
A11: p`2 >= c;
now per cases;
case
A12: d <>c;
reconsider lambda = (p`2-c)/(d-c) as Real;
d>=c by A10,A11,XXREAL_0:2;
then d>c by A12,XXREAL_0:1;
then
A13: d-c>0 by XREAL_1:50;
A14: p`2-c>=0 by A11,XREAL_1:48;
d-c>=p`2-c by A10,XREAL_1:9;
then (d-c)/(d-c)>=(p`2-c)/(d-c) by A13,XREAL_1:72;
then
A15: 1>=lambda by A13,XCMPLX_1:60;
A16: (1-lambda)*c+lambda*d
=((d-c)/(d-c)- (p`2-c)/(d-c))*c+(p`2-c)/(d-c)*d by A13,XCMPLX_1:60
.=(((d-c)- (p`2-c))/(d-c))*c+(p`2-c)/(d-c)*d by XCMPLX_1:120
.=c*((d- p`2)/(d-c))+d*(p`2-c)/(d-c) by XCMPLX_1:74
.=c*(d- p`2)/(d-c)+d*(p`2-c)/(d-c) by XCMPLX_1:74
.=((c*d- c*p`2)+(d*p`2-d*c))/(d-c) by XCMPLX_1:62
.=(d- c)*p`2/(d-c)
.=p`2*((d- c)/(d-c)) by XCMPLX_1:74
.=p`2*1 by A13,XCMPLX_1:60
.=p`2;
(1-lambda)*p0 + lambda*p01
=|[(1-lambda)*a,(1-lambda)*c]| + lambda*(|[a,d]|) by EUCLID:58
.=|[(1-lambda)*a,(1-lambda)*c]| +(|[ lambda*a, lambda*d]|)
by EUCLID:58
.=|[(1-lambda)*a+lambda*a,(1-lambda)*c+lambda*d]| by EUCLID:56
.= p by A9,A16,EUCLID:53;
hence thesis by A8,A13,A14,A15;
end;
case d =c;
then
A17: p`2=c by A10,A11,XXREAL_0:1;
reconsider lambda = 0 as Real;
(1-lambda)*p0 + lambda*p01 =
|[(1-lambda)*a,(1-lambda)*c]| + lambda*(|[a,d]|) by EUCLID:58
.=|[(1-lambda)*a,(1-lambda)*c]| +(|[ lambda*a, lambda*d]|)
by EUCLID:58
.=|[(1-lambda)*a+lambda*a,(1-lambda)*c+lambda*d]|
by EUCLID:56
.= p by A9,A17,EUCLID:53;
hence thesis by A8;
end;
end;
hence thesis;
end;
LSeg(p0,p01) c= L1
proof
let a2 be object;
assume a2 in LSeg(p0,p01);
then consider lambda such that
A18: a2 = (1-lambda)*p0 + lambda*p01 and
A19: 0 <= lambda and
A20: lambda <= 1;
set q = (1-lambda)*p0 + lambda*p01;
A21: q`1= ((1-lambda)*p0)`1 + (lambda*p01)`1 by TOPREAL3:2
.= (1-lambda)*(p0)`1 + (lambda*p01)`1 by TOPREAL3:4
.= (1-lambda)*(p0)`1 + lambda*(p01)`1 by TOPREAL3:4
.=(1-lambda)*a +lambda*a by A3,EUCLID:52
.=a;
A22: q`2= ((1-lambda)*p0)`2 + (lambda*p01)`2 by TOPREAL3:2
.= (1-lambda)*(p0)`2 + (lambda*p01)`2 by TOPREAL3:4
.= (1-lambda)*(p0)`2 + lambda*(p01)`2 by TOPREAL3:4
.= (1-lambda)*c + lambda*d by A4,EUCLID:52;
then
A23: q`2 <= d by A2,A20,XREAL_1:172;
q`2 >= c by A2,A19,A20,A22,XREAL_1:173;
hence thesis by A18,A21,A23;
end;
hence L1 = LSeg(p0,p01) by A7;
A24: L2 c= LSeg(p01,p1)
proof
let a2 be object;
assume a2 in L2;
then consider p such that
A25: a2 = p and
A26: p`1 <= b and
A27: p`1 >= a and
A28: p`2=d;
now per cases;
case
A29: b <>a;
reconsider lambda = (p`1-a)/(b-a) as Real;
b>=a by A26,A27,XXREAL_0:2;
then b>a by A29,XXREAL_0:1;
then
A30: b-a>0 by XREAL_1:50;
A31: p`1-a>=0 by A27,XREAL_1:48;
b-a>=p`1-a by A26,XREAL_1:9;
then (b-a)/(b-a)>=(p`1-a)/(b-a) by A30,XREAL_1:72;
then
A32: 1>=lambda by A30,XCMPLX_1:60;
A33: (1-lambda)*a+lambda*b
=((b-a)/(b-a)- (p`1-a)/(b-a))*a+(p`1-a)/(b-a)*b by A30,XCMPLX_1:60
.=(((b-a)- (p`1-a))/(b-a))*a+(p`1-a)/(b-a)*b by XCMPLX_1:120
.=a*((b- p`1)/(b-a))+b*(p`1-a)/(b-a) by XCMPLX_1:74
.=a*(b- p`1)/(b-a)+b*(p`1-a)/(b-a) by XCMPLX_1:74
.=((a*b- a*p`1)+(b*p`1-b*a))/(b-a) by XCMPLX_1:62
.=(b- a)*p`1/(b-a)
.=p`1*((b- a)/(b-a)) by XCMPLX_1:74
.=p`1*1 by A30,XCMPLX_1:60
.=p`1;
(1-lambda)*p01 + lambda*p1
=|[(1-lambda)*a,(1-lambda)*d]| + lambda*(|[b,d]|) by EUCLID:58
.=|[(1-lambda)*a,(1-lambda)*d]| +(|[ lambda*b, lambda*d]|)
by EUCLID:58
.=|[(1-lambda)*a+lambda*b,(1-lambda)*d+lambda*d]|
by EUCLID:56
.= p by A28,A33,EUCLID:53;
hence thesis by A25,A30,A31,A32;
end;
case b =a;
then
A34: p`1=a by A26,A27,XXREAL_0:1;
reconsider lambda = 0 as Real;
(1-lambda)*p01 + lambda*p1
=|[(1-lambda)*a,(1-lambda)*d]| + lambda*(|[b,d]|) by EUCLID:58
.=|[(1-lambda)*a,(1-lambda)*d]| +(|[ lambda*b, lambda*d]|)
by EUCLID:58
.=|[(1-lambda)*a+lambda*b,(1-lambda)*d+lambda*d]|
by EUCLID:56
.= p by A28,A34,EUCLID:53;
hence thesis by A25;
end;
end;
hence thesis;
end;
LSeg(p01,p1) c= L2
proof
let a2 be object;
assume a2 in LSeg(p01,p1);
then consider lambda such that
A35: a2 = (1-lambda)*p01 + lambda*p1 and
A36: 0 <= lambda and
A37: lambda <= 1;
set q = (1-lambda)*p01 + lambda*p1;
A38: q`2= ((1-lambda)*p01)`2 + (lambda*p1)`2 by TOPREAL3:2
.= (1-lambda)*(p01)`2 + (lambda*p1)`2 by TOPREAL3:4
.= (1-lambda)*(p01)`2 + lambda*(p1)`2 by TOPREAL3:4
.=(1-lambda)*d +lambda*d by A4,EUCLID:52
.=d;
A39: q`1= ((1-lambda)*p01)`1 + (lambda*p1)`1 by TOPREAL3:2
.= (1-lambda)*(p01)`1 + (lambda*p1)`1 by TOPREAL3:4
.= (1-lambda)*(p01)`1 + lambda*(p1)`1 by TOPREAL3:4
.= (1-lambda)*a + lambda*b by A3,EUCLID:52;
then
A40: q`1 <= b by A1,A37,XREAL_1:172;
q`1 >= a by A1,A36,A37,A39,XREAL_1:173;
hence thesis by A35,A38,A40;
end;
hence L2 = LSeg(p01,p1) by A24;
A41: L3 c= LSeg(p0,p10)
proof
let a2 be object;
assume a2 in L3;
then consider p such that
A42: a2 = p and
A43: p`1 <= b and
A44: p`1 >= a and
A45: p`2=c;
now per cases;
case
A46: b <>a;
reconsider lambda = (p`1-a)/(b-a) as Real;
b>=a by A43,A44,XXREAL_0:2;
then b>a by A46,XXREAL_0:1;
then
A47: b-a>0 by XREAL_1:50;
A48: p`1-a>=0 by A44,XREAL_1:48;
b-a>=p`1-a by A43,XREAL_1:9;
then (b-a)/(b-a)>=(p`1-a)/(b-a) by A47,XREAL_1:72;
then
A49: 1>=lambda by A47,XCMPLX_1:60;
A50: (1-lambda)*a+lambda*b
=((b-a)/(b-a)- (p`1-a)/(b-a))*a+(p`1-a)/(b-a)*b by A47,XCMPLX_1:60
.=(((b-a)- (p`1-a))/(b-a))*a+(p`1-a)/(b-a)*b by XCMPLX_1:120
.=a*((b- p`1)/(b-a))+b*(p`1-a)/(b-a) by XCMPLX_1:74
.=a*(b- p`1)/(b-a)+b*(p`1-a)/(b-a) by XCMPLX_1:74
.=((a*b- a*p`1)+(b*p`1-b*a))/(b-a) by XCMPLX_1:62
.=(b- a)*p`1/(b-a)
.=p`1*((b- a)/(b-a)) by XCMPLX_1:74
.=p`1*1 by A47,XCMPLX_1:60
.=p`1;
(1-lambda)*p0 + lambda*p10
=|[(1-lambda)*a,(1-lambda)*c]| + lambda*(|[b,c]|) by EUCLID:58
.=|[(1-lambda)*a,(1-lambda)*c]| +(|[ lambda*b, lambda*c]|)
by EUCLID:58
.=|[(1-lambda)*a+lambda*b,(1-lambda)*c+lambda*c]| by EUCLID:56
.= p by A45,A50,EUCLID:53;
hence thesis by A42,A47,A48,A49;
end;
case b =a;
then
A51: p`1=a by A43,A44,XXREAL_0:1;
reconsider lambda = 0 as Real;
(1-lambda)*p0 + lambda*p10 =
|[(1-lambda)*a,(1-lambda)*c]| + lambda*(|[b,c]|) by EUCLID:58
.=|[(1-lambda)*a,(1-lambda)*c]| +(|[ lambda*b, lambda*c]|)
by EUCLID:58
.=|[(1-lambda)*a+lambda*b,(1-lambda)*c+lambda*c]|
by EUCLID:56
.= p by A45,A51,EUCLID:53;
hence thesis by A42;
end;
end;
hence thesis;
end;
LSeg(p0,p10) c= L3
proof
let a2 be object;
assume a2 in LSeg(p0,p10);
then consider lambda such that
A52: a2 = (1-lambda)*p0 + lambda*p10 and
A53: 0 <= lambda and
A54: lambda <= 1;
set q = (1-lambda)*p0 + lambda*p10;
A55: q`2= ((1-lambda)*p0)`2 + (lambda*p10)`2 by TOPREAL3:2
.= (1-lambda)*(p0)`2 + (lambda*p10)`2 by TOPREAL3:4
.= (1-lambda)*(p0)`2 + lambda*(p10)`2 by TOPREAL3:4
.=(1-lambda)*c +lambda*c by A6,EUCLID:52
.=c;
A56: q`1= ((1-lambda)*p0)`1 + (lambda*p10)`1 by TOPREAL3:2
.= (1-lambda)*(p0)`1 + (lambda*p10)`1 by TOPREAL3:4
.= (1-lambda)*(p0)`1 + lambda*(p10)`1 by TOPREAL3:4
.= (1-lambda)*a + lambda*b by A5,EUCLID:52;
then
A57: q`1 <= b by A1,A54,XREAL_1:172;
q`1 >= a by A1,A53,A54,A56,XREAL_1:173;
hence thesis by A52,A55,A57;
end;
hence L3 = LSeg(p0,p10) by A41;
A58: L4 c= LSeg(p10,p1)
proof
let a2 be object;
assume a2 in L4;
then consider p such that
A59: a2 = p and
A60: p`1 = b and
A61: p`2 <= d and
A62: p`2 >= c;
now per cases;
case
A63: d <>c;
reconsider lambda = (p`2-c)/(d-c) as Real;
d>=c by A61,A62,XXREAL_0:2;
then d>c by A63,XXREAL_0:1;
then
A64: d-c>0 by XREAL_1:50;
A65: p`2-c>=0 by A62,XREAL_1:48;
d-c>=p`2-c by A61,XREAL_1:9;
then (d-c)/(d-c)>=(p`2-c)/(d-c) by A64,XREAL_1:72;
then
A66: 1>=lambda by A64,XCMPLX_1:60;
A67: (1-lambda)*c+lambda*d
=((d-c)/(d-c)- (p`2-c)/(d-c))*c+(p`2-c)/(d-c)*d by A64,XCMPLX_1:60
.=(((d-c)- (p`2-c))/(d-c))*c+(p`2-c)/(d-c)*d by XCMPLX_1:120
.=c*((d- p`2)/(d-c))+d*(p`2-c)/(d-c) by XCMPLX_1:74
.=c*(d- p`2)/(d-c)+d*(p`2-c)/(d-c) by XCMPLX_1:74
.=((c*d- c*p`2)+(d*p`2-d*c))/(d-c) by XCMPLX_1:62
.=(d- c)*p`2/(d-c)
.=p`2*((d- c)/(d-c)) by XCMPLX_1:74
.=p`2*1 by A64,XCMPLX_1:60
.=p`2;
(1-lambda)*p10 + lambda*p1
=|[(1-lambda)*b,(1-lambda)*c]| + lambda*(|[b,d]|) by EUCLID:58
.=|[(1-lambda)*b,(1-lambda)*c]| +(|[ lambda*b, lambda*d]|)
by EUCLID:58
.=|[(1-lambda)*b+lambda*b,(1-lambda)*c+lambda*d]|
by EUCLID:56
.= p by A60,A67,EUCLID:53;
hence thesis by A59,A64,A65,A66;
end;
case d =c;
then
A68: p`2=c by A61,A62,XXREAL_0:1;
reconsider lambda = 0 as Real;
(1-lambda)*p10 + lambda*p1
=|[(1-lambda)*b,(1-lambda)*c]| + lambda*(|[b,d]|) by EUCLID:58
.=|[(1-lambda)*b,(1-lambda)*c]| +(|[ lambda*b, lambda*d]|)
by EUCLID:58
.=|[(1-lambda)*b+lambda*b,(1-lambda)*c+lambda*d]|
by EUCLID:56
.= p by A60,A68,EUCLID:53;
hence thesis by A59;
end;
end;
hence thesis;
end;
LSeg(p10,p1) c= L4
proof
let a2 be object;
assume a2 in LSeg(p10,p1);
then consider lambda such that
A69: a2 = (1-lambda)*p10 + lambda*p1 and
A70: 0 <= lambda and
A71: lambda <= 1;
set q = (1-lambda)*p10 + lambda*p1;
A72: q`1= ((1-lambda)*p10)`1 + (lambda*p1)`1 by TOPREAL3:2
.= (1-lambda)*(p10)`1 + (lambda*p1)`1 by TOPREAL3:4
.= (1-lambda)*(p10)`1 + lambda*(p1)`1 by TOPREAL3:4
.=(1-lambda)*b +lambda*b by A5,EUCLID:52
.=b;
A73: q`2= ((1-lambda)*p10)`2 + (lambda*p1)`2 by TOPREAL3:2
.= (1-lambda)*(p10)`2 + (lambda*p1)`2 by TOPREAL3:4
.= (1-lambda)*(p10)`2 + lambda*(p1)`2 by TOPREAL3:4
.= (1-lambda)*c + lambda*d by A6,EUCLID:52;
then
A74: q`2 <= d by A2,A71,XREAL_1:172;
q`2 >= c by A2,A70,A71,A73,XREAL_1:173;
hence thesis by A69,A72,A74;
end;
hence L4 = LSeg(p10,p1) by A58;
end;
theorem Th31:
for a,b,c,d being Real st a <= b & c <= d holds
LSeg(|[a,c]|,|[a,d]|) /\ LSeg(|[a,c]|,|[b,c]|) = {|[a,c]|}
proof
let a,b,c,d be Real;
assume that
A1: a <= b and
A2: c <= d;
for ax being object
holds ax in LSeg(|[a,c]|,|[a,d]|) /\ LSeg(|[a,c]|,|[b,c]|) iff ax = |[a,c]|
proof
let ax be object;
thus ax in LSeg(|[a,c]|,|[a,d]|) /\ LSeg(|[a,c]|,|[b,c]|)
implies ax = |[a,c]|
proof
assume
A3: ax in LSeg(|[a,c]|,|[a,d]|) /\ LSeg(|[a,c]|,|[b,c]|);
then
A4: ax in LSeg(|[a,c]|,|[a,d]|) by XBOOLE_0:def 4;
ax in LSeg(|[a,c]|,|[b,c]|) by A3,XBOOLE_0:def 4;
then ax in { p2 : p2`1 <= b & p2`1 >= a & p2`2 = c } by A1,Th30;
then
A5: ex p2 st p2 = ax & p2`1 <= b & p2`1 >= a & p2`2 = c;
ax in { p2 : p2`1 = a & p2`2 <= d & p2`2 >= c } by A2,A4,Th30;
then ex p st p = ax & p`1 = a & p`2 <= d & p`2 >= c;
hence thesis by A5,EUCLID:53;
end;
assume
A6: ax = |[a,c]|;
then
A7: ax in LSeg(|[a,c]|,|[a,d]|) by RLTOPSP1:68;
ax in LSeg(|[a,c]|,|[b,c]|) by A6,RLTOPSP1:68;
hence thesis by A7,XBOOLE_0:def 4;
end;
hence thesis by TARSKI:def 1;
end;
theorem Th32:
for a,b,c,d being Real st a <= b & c <= d holds
LSeg(|[a,c]|,|[b,c]|) /\ LSeg(|[b,c]|,|[b,d]|) = {|[b,c]|}
proof
let a,b,c,d be Real;
assume that
A1: a <= b and
A2: c <= d;
for ax being object
holds ax in LSeg(|[a,c]|,|[b,c]|) /\ LSeg(|[b,c]|,|[b,d]|) iff ax = |[b,c]|
proof
let ax be object;
thus ax in LSeg(|[a,c]|,|[b,c]|) /\ LSeg(|[b,c]|,|[b,d]|)
implies ax = |[b,c]|
proof
assume
A3: ax in LSeg(|[a,c]|,|[b,c]|) /\ LSeg(|[b,c]|,|[b,d]|);
then
A4: ax in LSeg(|[a,c]|,|[b,c]|) by XBOOLE_0:def 4;
A5: ax in LSeg(|[b,c]|,|[b,d]|) by A3,XBOOLE_0:def 4;
ax in { q1 : q1`1 <= b & q1`1 >= a & q1`2 = c} by A1,A4,Th30;
then
A6: ex p2 st p2 = ax & p2`1 <= b & p2`1 >= a & p2`2 = c;
ax in { q2 : q2`1 = b & q2`2 <= d & q2`2 >= c} by A2,A5,Th30;
then ex p st p = ax & p`1 = b & p`2 <= d & p`2 >= c;
hence thesis by A6,EUCLID:53;
end;
assume
A7: ax = |[b,c]|;
then
A8: ax in LSeg(|[a,c]|,|[b,c]|) by RLTOPSP1:68;
ax in LSeg(|[b,c]|,|[b,d]|) by A7,RLTOPSP1:68;
hence thesis by A8,XBOOLE_0:def 4;
end;
hence thesis by TARSKI:def 1;
end;
theorem Th33:
for a,b,c,d being Real st a <= b & c <= d holds
LSeg(|[a,d]|,|[b,d]|) /\ LSeg(|[b,c]|,|[b,d]|) = {|[b,d]|}
proof
let a,b,c,d be Real;
assume that
A1: a <= b and
A2: c <= d;
for ax being object
holds ax in LSeg(|[a,d]|,|[b,d]|) /\ LSeg(|[b,c]|,|[b,d]|) iff ax = |[b,d]|
proof
let ax be object;
thus ax in LSeg(|[a,d]|,|[b,d]|) /\ LSeg(|[b,c]|,|[b,d]|)
implies ax = |[b,d]|
proof
assume
A3: ax in LSeg(|[a,d]|,|[b,d]|) /\ LSeg(|[b,c]|,|[b,d]|);
then
A4: ax in LSeg(|[b,c]|,|[b,d]|) by XBOOLE_0:def 4;
ax in LSeg(|[a,d]|,|[b,d]|) by A3,XBOOLE_0:def 4;
then ax in { p : p`1 <= b & p`1 >= a & p`2 = d} by A1,Th30;
then
A5: ex p st p = ax & p`1 <= b & p`1 >= a & p`2 = d;
ax in { p : p`1 = b & p`2 <= d & p`2 >= c} by A2,A4,Th30;
then ex p2 st p2 = ax & p2`1 = b & p2`2 <= d & p2`2 >= c;
hence thesis by A5,EUCLID:53;
end;
assume
A6: ax = |[b,d]|;
then
A7: ax in LSeg(|[a,d]|,|[b,d]|) by RLTOPSP1:68;
ax in LSeg(|[b,c]|,|[b,d]|) by A6,RLTOPSP1:68;
hence thesis by A7,XBOOLE_0:def 4;
end;
hence thesis by TARSKI:def 1;
end;
theorem Th34:
for a,b,c,d being Real st a <= b & c <= d holds
LSeg(|[a,c]|,|[a,d]|) /\ LSeg(|[a,d]|,|[b,d]|) = {|[a,d]|}
proof
let a,b,c,d be Real;
assume that
A1: a <= b and
A2: c <= d;
for ax being object
holds ax in LSeg(|[a,c]|,|[a,d]|) /\ LSeg(|[a,d]|,|[b,d]|) iff ax = |[a,d]|
proof
let ax be object;
thus ax in LSeg(|[a,c]|,|[a,d]|) /\ LSeg(|[a,d]|,|[b,d]|)
implies ax = |[a,d]|
proof
assume
A3: ax in LSeg(|[a,c]|,|[a,d]|) /\ LSeg(|[a,d]|,|[b,d]|);
then
A4: ax in LSeg(|[a,c]|,|[a,d]|) by XBOOLE_0:def 4;
ax in LSeg(|[a,d]|,|[b,d]|) by A3,XBOOLE_0:def 4;
then ax in { p2 : p2`1 <= b & p2`1 >= a & p2`2 = d } by A1,Th30;
then
A5: ex p2 st p2 = ax & p2`1 <= b & p2`1 >= a & p2`2 = d;
ax in { p2 : p2`1 = a & p2`2 <= d & p2`2 >= c } by A2,A4,Th30;
then ex p st p = ax & p`1 = a & p`2 <= d & p`2 >= c;
hence thesis by A5,EUCLID:53;
end;
assume
A6: ax = |[a,d]|;
then
A7: ax in LSeg(|[a,c]|,|[a,d]|) by RLTOPSP1:68;
ax in LSeg(|[a,d]|,|[b,d]|) by A6,RLTOPSP1:68;
hence thesis by A7,XBOOLE_0:def 4;
end;
hence thesis by TARSKI:def 1;
end;
theorem Th35:
{q: -1=q`1 & -1<=q`2 & q`2<=1 or q`1=1 & -1<=q`2 & q`2<=1
or -1=q`2 & -1<=q`1 & q`1<=1 or 1=q`2 & -1<=q`1 & q`1<=1}
= {p: p`1=-1 & -1 <=p`2 & p`2<=1 or p`2=1 & -1<=p`1 & p`1<=1 or
p`1=1 & -1 <=p`2 & p`2<=1 or p`2=-1 & -1<=p`1 & p`1<=1}
proof
thus {q: -1=q`1 & -1<=q`2 & q`2<=1 or q`1=1 & -1<=q`2 & q`2<=1
or -1=q`2 & -1<=q`1 & q`1<=1 or 1=q`2 & -1<=q`1 & q`1<=1}
c= {p: p`1=-1 & -1 <=p`2 & p`2<=1 or p`2=1 & -1<=p`1 & p`1<=1 or
p`1=1 & -1 <=p`2 & p`2<=1 or p`2=-1 & -1<=p`1 & p`1<=1}
proof
let x be object;
assume x in {q: -1=q`1 & -1<=q`2 & q`2<=1 or q`1=1 & -1<=q`2 & q`2<=1
or -1=q`2 & -1<=q`1 & q`1<=1 or 1=q`2 & -1<=q`1 & q`1<=1};
then ex q st ( x=q)&( -1=q`1 & -1<=q`2 & q`2<=1 or q`1=1 & -1<=q
`2 & q`2<=1 or -1=q`2 & -1<=q`1 & q`1<=1 or 1=q`2 & -1<=q`1 & q`1<=1);
hence thesis;
end;
thus {p: p`1=-1 & -1 <=p`2 & p`2<=1 or p`2=1 & -1<=p`1 & p`1<=1 or
p`1=1 & -1 <=p`2 & p`2<=1 or p`2=-1 & -1<=p`1 & p`1<=1}
c= {q: -1=q`1 & -1<=q`2 & q`2<=1 or q`1=1 & -1<=q`2 & q`2<=1
or -1=q`2 & -1<=q`1 & q`1<=1 or 1=q`2 & -1<=q`1 & q`1<=1}
proof
let x be object;
assume x in {p: p`1=-1 & -1 <=p`2 & p`2<=1 or p`2=1 & -1<=p`1 & p`1<=1 or
p`1=1 & -1 <=p`2 & p`2<=1 or p`2=-1 & -1<=p`1 & p`1<=1};
then ex p st ( p=x)&( p`1=-1 & -1 <=p`2 & p`2<=1 or p`2=1 & -1<=
p`1 & p`1<=1 or p`1=1 & -1 <=p`2 & p`2<=1 or p`2=-1 & -1<=p`1 & p`1<=1);
hence thesis;
end;
end;
theorem Th36:
for a,b,c,d being Real st a<=b & c <=d
holds W-bound rectangle(a,b,c,d) = a
proof
let a,b,c,d be Real;
assume that
A1: a<=b and
A2: c <=d;
set X = rectangle(a,b,c,d);
reconsider Z = (proj1|X).:the carrier of ((TOP-REAL 2)|X)
as Subset of REAL;
A3: X = the carrier of ((TOP-REAL 2)|X) by PRE_TOPC:8;
A4: for p be Real st p in Z holds p >= a
proof
let p be Real;
assume p in Z;
then consider p0 being object such that
A5: p0 in the carrier of (TOP-REAL 2)|X and
p0 in the carrier of (TOP-REAL 2)|X and
A6: p = (proj1|X).p0 by FUNCT_2:64;
reconsider p0 as Point of TOP-REAL 2 by A3,A5;
X= {q : q`1 = a & q`2 <= d & q`2 >= c or q`1 <= b & q`1 >= a & q`2 = d or
q`1 <= b & q`1 >= a & q`2 = c or q`1 = b & q`2 <= d & q`2 >= c}
by A1,A2,SPPOL_2:54;
then ex q being Point of TOP-REAL 2 st p0 = q &
(q`1 = a & q`2 <= d & q`2 >= c or q`1 <= b & q`1 >= a & q`2 = d or
q`1 <= b & q`1 >= a & q`2 = c or q`1 = b & q`2 <= d & q`2 >= c) by A3,A5;
hence thesis by A1,A3,A5,A6,PSCOMP_1:22;
end;
A7: for q being Real st
for p being Real st p in Z holds p >= q holds a >= q
proof
let q be Real such that
A8: for p being Real st p in Z holds p >= q;
|[a,c]| in LSeg(|[ a,c ]|, |[ b,c ]|) by RLTOPSP1:68;
then
A9: |[a,c]| in LSeg(|[ a,c ]|, |[ b,c ]|) \/ LSeg(|[ b,c ]|, |[ b,d ]| )
by XBOOLE_0:def 3;
X= (LSeg(|[ a,c ]|, |[ a,d ]|) \/ LSeg(|[ a,d ]|, |[ b,d ]|))
\/ (LSeg(|[ a,c ]|, |[ b,c ]|) \/ LSeg(|[ b,c ]|, |[ b,d ]|))
by SPPOL_2:def 3;
then
A10: |[a,c]| in X by A9,XBOOLE_0:def 3;
then (proj1|X). |[a,c]| = |[a,c]|`1 by PSCOMP_1:22
.= a by EUCLID:52;
hence thesis by A3,A8,A10,FUNCT_2:35;
end;
thus W-bound X = lower_bound (proj1|X) by PSCOMP_1:def 7
.= lower_bound Z by PSCOMP_1:def 1
.= a by A4,A7,SEQ_4:44;
end;
theorem Th37:
for a,b,c,d being Real st a<=b & c <=d
holds N-bound rectangle(a,b,c,d) = d
proof
let a,b,c,d be Real;
assume that
A1: a<=b and
A2: c <=d;
set X = rectangle(a,b,c,d);
reconsider Z = (proj2|X).:the carrier of ((TOP-REAL 2)|X)
as Subset of REAL;
A3: X = the carrier of ((TOP-REAL 2)|X) by PRE_TOPC:8;
A4: for p be Real st p in Z holds p <= d
proof
let p be Real;
assume p in Z;
then consider p0 being object such that
A5: p0 in the carrier of (TOP-REAL 2)|X and
p0 in the carrier of (TOP-REAL 2)|X and
A6: p = (proj2|X).p0 by FUNCT_2:64;
reconsider p0 as Point of TOP-REAL 2 by A3,A5;
X= {q : q`1 = a & q`2 <= d & q`2 >= c or q`1 <= b & q`1 >= a & q`2 = d or
q`1 <= b & q`1 >= a & q`2 = c or q`1 = b & q`2 <= d & q`2 >= c}
by A1,A2,SPPOL_2:54;
then ex q being Point of TOP-REAL 2 st p0 = q &
(q`1 = a & q`2 <= d & q`2 >= c or q`1 <= b & q`1 >= a & q`2 = d or
q`1 <= b & q`1 >= a & q`2 = c or q`1 = b & q`2 <= d & q`2 >= c) by A3,A5;
hence thesis by A2,A3,A5,A6,PSCOMP_1:23;
end;
A7: for q being Real st for p being Real
st p in Z holds p <= q holds d <= q
proof
let q be Real such that
A8: for p be Real st p in Z holds p <= q;
|[b,d]| in LSeg(|[ b,c ]|, |[ b,d ]|) by RLTOPSP1:68;
then
A9: |[b,d]| in LSeg(|[ a,c ]|, |[ b,c ]|) \/ LSeg(|[ b,c ]|, |[ b,d ]| )
by XBOOLE_0:def 3;
X= (LSeg(|[ a,c ]|, |[ a,d ]|) \/ LSeg(|[ a,d ]|, |[ b,d ]|))
\/ (LSeg(|[ a,c ]|, |[ b,c ]|) \/ LSeg(|[ b,c ]|, |[ b,d ]|))
by SPPOL_2:def 3;
then
A10: |[b,d]| in X by A9,XBOOLE_0:def 3;
then (proj2|X). |[b,d]| = |[b,d]|`2 by PSCOMP_1:23
.= d by EUCLID:52;
hence thesis by A3,A8,A10,FUNCT_2:35;
end;
thus N-bound X = upper_bound (proj2|X) by PSCOMP_1:def 8
.= upper_bound Z by PSCOMP_1:def 2
.= d by A4,A7,SEQ_4:46;
end;
theorem Th38:
for a,b,c,d being Real st a<=b & c <=d
holds E-bound rectangle(a,b,c,d) = b
proof
let a,b,c,d be Real;
assume that
A1: a<=b and
A2: c <=d;
set X = rectangle(a,b,c,d);
reconsider Z = (proj1|X).:the carrier of ((TOP-REAL 2)|X)
as Subset of REAL;
A3: X = the carrier of ((TOP-REAL 2)|X) by PRE_TOPC:8;
A4: for p be Real st p in Z holds p <= b
proof
let p be Real;
assume p in Z;
then consider p0 being object such that
A5: p0 in the carrier of (TOP-REAL 2)|X and
p0 in the carrier of (TOP-REAL 2)|X and
A6: p = (proj1|X).p0 by FUNCT_2:64;
reconsider p0 as Point of TOP-REAL 2 by A3,A5;
X= {q : q`1 = a & q`2 <= d & q`2 >= c or q`1 <= b & q`1 >= a & q`2 = d or
q`1 <= b & q`1 >= a & q`2 = c or q`1 = b & q`2 <= d & q`2 >= c}
by A1,A2,SPPOL_2:54;
then ex q being Point of TOP-REAL 2 st p0 = q &
(q`1 = a & q`2 <= d & q`2 >= c or q`1 <= b & q`1 >= a & q`2 = d or
q`1 <= b & q`1 >= a & q`2 = c or q`1 = b & q`2 <= d & q`2 >= c) by A3,A5;
hence thesis by A1,A3,A5,A6,PSCOMP_1:22;
end;
A7: for q being Real st for p being Real
st p in Z holds p <= q holds b <= q
proof
let q be Real such that
A8: for p be Real st p in Z holds p <= q;
|[b,d]| in LSeg(|[ b,c ]|, |[ b,d ]|) by RLTOPSP1:68;
then
A9: |[b,d]| in LSeg(|[ a,c ]|, |[ b,c ]|) \/ LSeg(|[ b,c ]|, |[ b,d ]| )
by XBOOLE_0:def 3;
X= (LSeg(|[ a,c ]|, |[ a,d ]|) \/ LSeg(|[ a,d ]|, |[ b,d ]|))
\/ (LSeg(|[ a,c ]|, |[ b,c ]|) \/ LSeg(|[ b,c ]|, |[ b,d ]|))
by SPPOL_2:def 3;
then
A10: |[b,d]| in X by A9,XBOOLE_0:def 3;
then (proj1|X). |[b,d]| = |[b,d]|`1 by PSCOMP_1:22
.= b by EUCLID:52;
hence thesis by A3,A8,A10,FUNCT_2:35;
end;
thus E-bound X = upper_bound (proj1|X) by PSCOMP_1:def 9
.= upper_bound Z by PSCOMP_1:def 2
.= b by A4,A7,SEQ_4:46;
end;
theorem Th39:
for a,b,c,d being Real st a<=b & c <=d
holds S-bound rectangle(a,b,c,d) = c
proof
let a,b,c,d be Real;
assume that
A1: a<=b and
A2: c <=d;
set X = rectangle(a,b,c,d);
reconsider Z = (proj2|X).:the carrier of ((TOP-REAL 2)|X)
as Subset of REAL;
A3: X = the carrier of ((TOP-REAL 2)|X) by PRE_TOPC:8;
A4: for p be Real st p in Z holds p >= c
proof
let p be Real;
assume p in Z;
then consider p0 being object such that
A5: p0 in the carrier of (TOP-REAL 2)|X and
p0 in the carrier of (TOP-REAL 2)|X and
A6: p = (proj2|X).p0 by FUNCT_2:64;
reconsider p0 as Point of TOP-REAL 2 by A3,A5;
X= {q : q`1 = a & q`2 <= d & q`2 >= c or q`1 <= b & q`1 >= a & q`2 = d or
q`1 <= b & q`1 >= a & q`2 = c or q`1 = b & q`2 <= d & q`2 >= c}
by A1,A2,SPPOL_2:54;
then ex q being Point of TOP-REAL 2 st p0 = q &
(q`1 = a & q`2 <= d & q`2 >= c or q`1 <= b & q`1 >= a & q`2 = d or
q`1 <= b & q`1 >= a & q`2 = c or q`1 = b & q`2 <= d & q`2 >= c) by A3,A5;
hence thesis by A2,A3,A5,A6,PSCOMP_1:23;
end;
A7: for q being Real st
for p being Real st p in Z holds p >= q holds c >= q
proof
let q be Real such that
A8: for p being Real st p in Z holds p >= q;
|[b,c]| in LSeg(|[ b,c ]|, |[ b,d ]|) by RLTOPSP1:68;
then
A9: |[b,c]| in LSeg(|[ a,c ]|, |[ b,c ]|) \/ LSeg(|[ b,c ]|, |[ b,d ]| )
by XBOOLE_0:def 3;
X= (LSeg(|[ a,c ]|, |[ a,d ]|) \/ LSeg(|[ a,d ]|, |[ b,d ]|))
\/ (LSeg(|[ a,c ]|, |[ b,c ]|) \/ LSeg(|[ b,c ]|, |[ b,d ]|))
by SPPOL_2:def 3;
then
A10: |[b,c]| in X by A9,XBOOLE_0:def 3;
then (proj2|X). |[b,c]| = |[b,c]|`2 by PSCOMP_1:23
.= c by EUCLID:52;
hence thesis by A3,A8,A10,FUNCT_2:35;
end;
thus S-bound X = lower_bound (proj2|X) by PSCOMP_1:def 10
.= lower_bound Z by PSCOMP_1:def 1
.= c by A4,A7,SEQ_4:44;
end;
theorem Th40:
for a,b,c,d being Real st a<=b & c <=d holds
NW-corner rectangle(a,b,c,d) = |[a,d]|
proof
let a,b,c,d be Real;
assume that
A1: a<=b and
A2: c <=d;
set K=rectangle(a,b,c,d);
A3: NW-corner K= |[W-bound K, N-bound K]| by PSCOMP_1:def 12;
W-bound K=a by A1,A2,Th36;
hence thesis by A1,A2,A3,Th37;
end;
theorem Th41:
for a,b,c,d being Real st a<=b & c <=d holds
NE-corner rectangle(a,b,c,d) = |[b,d]|
proof
let a,b,c,d be Real;
set K=rectangle(a,b,c,d);
assume that
A1: a<=b and
A2: c <=d;
A3: NE-corner K= |[E-bound K, N-bound K]| by PSCOMP_1:def 13;
E-bound K=b by A1,A2,Th38;
hence thesis by A1,A2,A3,Th37;
end;
theorem
for a,b,c,d being Real st a<=b & c <=d
holds SW-corner rectangle(a,b,c,d) = |[a,c]|
proof
let a,b,c,d be Real;
set K=rectangle(a,b,c,d);
assume that
A1: a<=b and
A2: c <=d;
A3: SW-corner K= |[W-bound K, S-bound K]| by PSCOMP_1:def 11;
W-bound K=a by A1,A2,Th36;
hence thesis by A1,A2,A3,Th39;
end;
theorem
for a,b,c,d being Real st a<=b & c <=d
holds SE-corner rectangle(a,b,c,d) = |[b,c]|
proof
let a,b,c,d be Real;
set K=rectangle(a,b,c,d);
assume that
A1: a<=b and
A2: c <=d;
A3: SE-corner K= |[E-bound K, S-bound K]| by PSCOMP_1:def 14;
E-bound K=b by A1,A2,Th38;
hence thesis by A1,A2,A3,Th39;
end;
theorem Th44:
for a,b,c,d being Real st a<=b & c <=d
holds W-most rectangle(a,b,c,d) = LSeg(|[a,c]|,|[a,d]|)
proof
let a,b,c,d be Real;
set K = rectangle(a,b,c,d);
assume that
A1: a<=b and
A2: c <=d;
K= LSeg(|[ a,c ]|, |[ a,d ]|) \/ LSeg(|[ a,d ]|, |[ b,d ]|)
\/ (LSeg(|[ a,c ]|, |[ b,c ]|) \/ LSeg(|[ b,c ]|, |[ b,d ]|))
by SPPOL_2:def 3;
then
A3: LSeg(|[ a,c ]|, |[ a,d ]|) \/ LSeg(|[ a,d ]|, |[ b,d ]|) c= K
by XBOOLE_1:7;
A4: LSeg(|[a,c]|,|[a,d]|) c=
LSeg(|[ a,c ]|, |[ a,d ]|) \/ LSeg(|[ a,d ]|, |[ b,d ]|) by XBOOLE_1:7;
A5: SW-corner K= |[W-bound K, S-bound K]| by PSCOMP_1:def 11;
A6: NW-corner K = |[a,d]| by A1,A2,Th40;
A7: W-bound K=a by A1,A2,Th36;
A8: S-bound K= c by A1,A2,Th39;
thus W-most K = LSeg(SW-corner K, NW-corner K)/\K by PSCOMP_1:def 15
.= LSeg(|[a,c]|,|[a,d]|) by A3,A4,A5,A6,A7,A8,XBOOLE_1:1,28;
end;
theorem Th45:
for a,b,c,d being Real st a<=b & c <=d
holds E-most rectangle(a,b,c,d) = LSeg(|[b,c]|,|[b,d]|)
proof
let a,b,c,d be Real;
set K = rectangle(a,b,c,d);
assume that
A1: a<=b and
A2: c <=d;
K= LSeg(|[ a,c ]|, |[ a,d ]|) \/ LSeg(|[ a,d ]|, |[ b,d ]|)
\/ (LSeg(|[ a,c ]|, |[ b,c ]|) \/ LSeg(|[ b,c ]|, |[ b,d ]|))
by SPPOL_2:def 3;
then
A3: LSeg(|[ a,c ]|, |[ b,c ]|) \/ LSeg(|[ b,c ]|, |[ b,d ]|) c= K
by XBOOLE_1:7;
A4: LSeg(|[b,c]|,|[b,d]|) c=
LSeg(|[ a,c ]|, |[ b,c ]|) \/ LSeg(|[ b,c ]|, |[ b,d ]|) by XBOOLE_1:7;
A5: SE-corner K= |[E-bound K, S-bound K]| by PSCOMP_1:def 14;
A6: NE-corner K = |[b,d]| by A1,A2,Th41;
A7: E-bound K=b by A1,A2,Th38;
A8: S-bound K= c by A1,A2,Th39;
thus E-most K = LSeg(SE-corner K, NE-corner K)/\K by PSCOMP_1:def 17
.= LSeg(|[b,c]|,|[b,d]|) by A3,A4,A5,A6,A7,A8,XBOOLE_1:1,28;
end;
theorem Th46:
for a,b,c,d being Real st a<=b & c <=d
holds W-min rectangle(a,b,c,d)=|[a,c]| & E-max rectangle(a,b,c,d)=|[b,d]|
proof
let a,b,c,d be Real;
set K = rectangle(a,b,c,d);
assume that
A1: a<=b and
A2: c <=d;
A3: lower_bound (proj2|LSeg(|[a,c]|,|[a,d]|)) = c
proof
set X = LSeg(|[a,c]|,|[a,d]|);
reconsider Z = (proj2|X).:the carrier of ((TOP-REAL 2)|X)
as Subset of REAL;
A4: X = the carrier of ((TOP-REAL 2)|X) by PRE_TOPC:8;
A5: for p be Real st p in Z holds p >= c
proof
let p be Real;
assume p in Z;
then consider p0 being object such that
A6: p0 in the carrier of (TOP-REAL 2)|X and
p0 in the carrier of (TOP-REAL 2)|X and
A7: p = (proj2|X).p0 by FUNCT_2:64;
reconsider p0 as Point of TOP-REAL 2 by A4,A6;
A8: |[a,c]|`2 = c by EUCLID:52;
|[a,d]|`2 = d by EUCLID:52;
then p0`2 >= c by A2,A4,A6,A8,TOPREAL1:4;
hence thesis by A4,A6,A7,PSCOMP_1:23;
end;
A9: for q being Real st
for p being Real st p in Z holds p >= q holds c >= q
proof
let q be Real such that
A10: for p being Real st p in Z holds p >= q;
A11: |[a,c]| in X by RLTOPSP1:68;
(proj2|X). |[a,c]| = |[a,c]|`2 by PSCOMP_1:23,RLTOPSP1:68
.= c by EUCLID:52;
hence thesis by A4,A10,A11,FUNCT_2:35;
end;
thus lower_bound (proj2|X) = lower_bound Z by PSCOMP_1:def 1
.= c by A5,A9,SEQ_4:44;
end;
A12: W-most K = LSeg(|[a,c]|,|[a,d]|) by A1,A2,Th44;
A13: W-bound K = a by A1,A2,Th36;
A14: upper_bound (proj2|LSeg(|[b,c]|,|[b,d]|)) = d
proof
set X = LSeg(|[b,c]|,|[b,d]|);
reconsider Z = (proj2|X).:the carrier of ((TOP-REAL 2)|X)
as Subset of REAL;
A15: X = the carrier of ((TOP-REAL 2)|X) by PRE_TOPC:8;
A16: for p be Real st p in Z holds p <= d
proof
let p be Real;
assume p in Z;
then consider p0 being object such that
A17: p0 in the carrier of (TOP-REAL 2)|X and
p0 in the carrier of (TOP-REAL 2)|X and
A18: p = (proj2|X).p0 by FUNCT_2:64;
reconsider p0 as Point of TOP-REAL 2 by A15,A17;
A19: |[b,c]|`2 = c by EUCLID:52;
|[b,d]|`2 = d by EUCLID:52;
then p0`2 <= d by A2,A15,A17,A19,TOPREAL1:4;
hence thesis by A15,A17,A18,PSCOMP_1:23;
end;
A20: for q being Real st
for p being Real st p in Z holds p <= q holds d <= q
proof
let q be Real such that
A21: for p being Real st p in Z holds p <= q;
A22: |[b,d]| in X by RLTOPSP1:68;
(proj2|X). |[b,d]| = |[b,d]|`2 by PSCOMP_1:23,RLTOPSP1:68
.= d by EUCLID:52;
hence thesis by A15,A21,A22,FUNCT_2:35;
end;
thus upper_bound (proj2|X) = upper_bound Z by PSCOMP_1:def 2
.= d by A16,A20,SEQ_4:46;
end;
A23: E-most K = LSeg(|[b,c]|,|[b,d]|) by A1,A2,Th45;
E-bound K = b by A1,A2,Th38;
hence thesis by A3,A12,A13,A14,A23,PSCOMP_1:def 19,def 23;
end;
theorem Th47:
for a,b,c,d being Real st a<b & c <d
holds LSeg(|[a,c]|,|[a,d]|) \/ LSeg(|[a,d]|,|[b,d]|)
is_an_arc_of W-min rectangle(a,b,c,d), E-max rectangle(a,b,c,d) &
LSeg(|[a,c]|,|[b,c]|) \/ LSeg(|[b,c]|,|[b,d]|)
is_an_arc_of E-max rectangle(a,b,c,d), W-min rectangle(a,b,c,d)
proof
let a,b,c,d be Real;
set K = rectangle(a,b,c,d);
assume that
A1: a<b and
A2: c <d;
A3: W-min(K)= |[a,c]| by A1,A2,Th46;
A4: E-max(K)= |[b,d]| by A1,A2,Th46;
(|[a,c]|)`2=c by EUCLID:52;
then
A5: |[a,c]| <> |[a,d]| by A2,EUCLID:52;
set p1= |[a,c]|,p2= |[a,d]|,q1=|[b,d]|;
A6: LSeg(p1,p2) /\ LSeg(p2,q1) ={p2} by A1,A2,Th34;
(|[a,c]|)`1=a by EUCLID:52;
then
A7: |[a,c]| <> |[b,c]| by A1,EUCLID:52;
set q2=|[b,c]|;
LSeg(q1,q2) /\ LSeg(q2,p1) ={q2} by A1,A2,Th32;
hence thesis by A3,A4,A5,A6,A7,TOPREAL1:12;
end;
theorem Th48:
for a,b,c,d being Real, f1,f2 being FinSequence of TOP-REAL 2,
p0,p1,p01,p10 being Point of TOP-REAL 2 st a < b & c < d &
p0=|[a,c]| & p1=|[b,d]| & p01=|[a,d]| & p10=|[b,c]| & f1=<*p0,p01,p1*>
& f2=<*p0,p10,p1*> holds f1 is being_S-Seq &
L~f1 = LSeg(p0,p01) \/ LSeg(p01,p1) &
f2 is being_S-Seq & L~f2 = LSeg(p0,p10) \/ LSeg(p10,p1) &
rectangle(a,b,c,d) = L~f1 \/ L~f2 & L~f1 /\ L~f2 = {p0,p1} & f1/.1 = p0 &
f1/.len f1=p1 & f2/.1 = p0 & f2/.len f2 = p1
proof
let a,b,c,d be Real, f1,f2 be FinSequence of TOP-REAL 2,
p0,p1,p01,p10 be Point of TOP-REAL 2;
assume that
A1: a < b and
A2: c < d and
A3: p0=|[a,c]| and
A4: p1=|[b,d]| and
A5: p01=|[a,d]| and
A6: p10=|[b,c]| and
A7: f1=<*p0,p01,p1*> and
A8: f2=<*p0,p10,p1*>;
set P = rectangle(a,b,c,d);
set L1 = { p : p`1 = a & p`2 <= d & p`2 >= c};
set L2 = { p : p`1 <= b & p`1 >= a & p`2 = d};
set L3 = { p : p`1 <= b & p`1 >= a & p`2 = c};
set L4 = { p : p`1 = b & p`2 <= d & p`2 >= c};
A9: p1`1 = b by A4,EUCLID:52;
A10: p1`2 = d by A4,EUCLID:52;
A11: p10`1 = b by A6,EUCLID:52;
A12: p10`2 = c by A6,EUCLID:52;
A13: p0`1 = a by A3,EUCLID:52;
A14: p0`2 = c by A3,EUCLID:52;
A15: len f1 = 1 + 2 by A7,FINSEQ_1:45;
A16: f1/.1 = p0 by A7,FINSEQ_4:18;
A17: f1/.2 = p01 by A7,FINSEQ_4:18;
A18: f1/.3 = p1 by A7,FINSEQ_4:18;
thus f1 is being_S-Seq
proof
A19: p0 <> p01 by A2,A5,A14,EUCLID:52;
p01 <> p1 by A1,A5,A9,EUCLID:52;
hence f1 is one-to-one by A1,A7,A9,A13,A19,FINSEQ_3:95;
thus len f1 >= 2 by A15;
thus f1 is unfolded
proof
let i be Nat;
assume that
A20: 1 <= i and
A21: i + 2 <= len f1;
i <= 1 by A15,A21,XREAL_1:6;
then
A22: i = 1 by A20,XXREAL_0:1;
reconsider n2=1+1 as Nat;
n2 in Seg len f1 by A15,FINSEQ_1:1;
then
A23: LSeg(f1,1) = LSeg(p0,p01) by A15,A16,A17,TOPREAL1:def 3;
A24: LSeg(f1,n2) = LSeg(p01,p1) by A15,A17,A18,TOPREAL1:def 3;
for x being object holds x in LSeg(p0,p01) /\ LSeg(p01,p1) iff x = p01
proof
let x be object;
thus x in LSeg(p0,p01) /\ LSeg(p01,p1) implies x = p01
proof
assume
A25: x in LSeg(p0,p01) /\ LSeg(p01,p1);
then
A26: x in LSeg(p0,p01) by XBOOLE_0:def 4;
A27: x in LSeg(p01,p1) by A25,XBOOLE_0:def 4;
A28: x in {p : p`1 = a & p`2 <= d & p`2 >= c} by A2,A3,A5,A26,Th30;
A29: x in {p2 : p2`1 <= b & p2`1 >= a & p2`2 = d} by A1,A4,A5,A27,Th30;
A30: ex p st p = x & p`1 = a & p`2 <= d & p`2 >= c by A28;
ex p2 st p2=x & p2`1<=b & p2`1>=a & p2`2=d by A29;
hence thesis by A5,A30,EUCLID:53;
end;
assume
A31: x = p01;
then
A32: x in LSeg(p0,p01) by RLTOPSP1:68;
x in LSeg(p01,p1) by A31,RLTOPSP1:68;
hence thesis by A32,XBOOLE_0:def 4;
end;
hence thesis by A17,A22,A23,A24,TARSKI:def 1;
end;
thus f1 is s.n.c.
proof
let i,j be Nat such that
A33: i+1 < j;
now per cases;
suppose 1 <= i;
then
A34: 1+1 <= i+1 by XREAL_1:6;
now per cases;
case 1 <= j & j+1 <= len f1;
then j <= 2 by A15,XREAL_1:6;
hence contradiction by A33,A34,XXREAL_0:2;
end;
case not (1 <= j & j+1 <= len f1);
then LSeg(f1,j) = {} by TOPREAL1:def 3;
hence LSeg(f1,i) /\ LSeg(f1,j) = {};
end;
end;
hence LSeg(f1,i) /\ LSeg(f1,j) = {};
end;
suppose not (1 <= i & i+1 <= len f1);
then LSeg(f1,i) = {} by TOPREAL1:def 3;
hence LSeg(f1,i) /\ LSeg(f1,j) = {};
end;
end;
hence LSeg(f1,i) /\ LSeg(f1,j) = {};
end;
let i be Nat;
assume that
A35: 1 <= i and
A36: i + 1 <= len f1;
A37: i <= 1 + 1 by A15,A36,XREAL_1:6;
now per cases by A35,A37,NAT_1:9;
suppose
A38: i = 1;
then (f1/.i)`1 = p0`1 by A7,FINSEQ_4:18
.= a by A3,EUCLID:52
.= (f1/.(i+1))`1 by A5,A17,A38,EUCLID:52;
hence thesis;
end;
suppose
A39: i = 2;
then (f1/.i)`2 = p01`2 by A7,FINSEQ_4:18
.= d by A5,EUCLID:52
.= (f1/.(i+1))`2 by A4,A18,A39,EUCLID:52;
hence thesis;
end;
end;
hence thesis;
end;
A40: 1+1 in Seg len f1 by A15,FINSEQ_1:1;
A41: 1+1 <= len f1 by A15;
LSeg(p0,p01) = LSeg(f1,1) by A15,A16,A17,A40,TOPREAL1:def 3;
then
A42: LSeg(p0,p01) in {LSeg(f1,i): 1 <= i & i+1 <= len f1} by A41;
LSeg(p01,p1) = LSeg(f1,2) by A15,A17,A18,TOPREAL1:def 3;
then LSeg(p01,p1) in {LSeg(f1,i): 1 <= i & i+1 <= len f1} by A15;
then
A43: {LSeg(p0,p01),LSeg(p01,p1)} c= {LSeg(f1,i):1<=i & i+1<=len f1}
by A42,ZFMISC_1:32;
{LSeg(f1,i): 1 <= i & i+1 <= len f1} c= {LSeg(p0,p01),LSeg(p01,p1)}
proof
let a be object;
assume a in {LSeg(f1,i): 1 <= i & i+1 <= len f1};
then consider i such that
A44: a = LSeg(f1,i) and
A45: 1<=i and
A46: i+1<=len f1;
i+1 <= 2 + 1 by A7,A46,FINSEQ_1:45;
then i <= 1 + 1 by XREAL_1:6;
then i = 1 or i = 2 by A45,NAT_1:9;
then a = LSeg(p0,p01) or a = LSeg(p01,p1) by A16,A17,A18,A44,A46,
TOPREAL1:def 3;
hence thesis by TARSKI:def 2;
end;
then L~f1 = union {LSeg(p0,p01),LSeg(p01,p1)} by A43,XBOOLE_0:def 10;
hence
A47: L~f1 = (LSeg(p0,p01) \/ LSeg(p01,p1)) by ZFMISC_1:75;
then
A48: L~f1=L1 \/ LSeg(p01,p1) by A2,A3,A5,Th30
.=L1 \/ L2 by A1,A4,A5,Th30;
A49: len f2 = 1 + 2 by A8,FINSEQ_1:45;
A50: f2/.1 = p0 by A8,FINSEQ_4:18;
A51: f2/.2 = p10 by A8,FINSEQ_4:18;
A52: f2/.3 = p1 by A8,FINSEQ_4:18;
thus f2 is being_S-Seq
proof
thus f2 is one-to-one by A1,A2,A8,A9,A10,A11,A12,A13,FINSEQ_3:95;
thus len f2 >= 2 by A49;
thus f2 is unfolded
proof
let i be Nat;
assume that
A53: 1 <= i and
A54: i + 2 <= len f2;
i <= 1 by A49,A54,XREAL_1:6;
then
A55: i = 1 by A53,XXREAL_0:1;
1+1 in Seg len f2 by A49,FINSEQ_1:1;
then
A56: LSeg(f2,1) = LSeg(p0,p10) by A49,A50,A51,TOPREAL1:def 3;
A57: LSeg(f2,1+1) = LSeg(p10,p1) by A49,A51,A52,TOPREAL1:def 3;
for x being object holds x in LSeg(p0,p10) /\ LSeg(p10,p1) iff x = p10
proof
let x be object;
thus x in LSeg(p0,p10) /\ LSeg(p10,p1) implies x = p10
proof
assume
A58: x in LSeg(p0,p10) /\ LSeg(p10,p1);
then
A59: x in LSeg(p0,p10) by XBOOLE_0:def 4;
A60: x in LSeg(p10,p1) by A58,XBOOLE_0:def 4;
A61: x in { p : p`1 <= b & p`1 >= a & p`2 = c} by A1,A3,A6,A59,Th30;
A62: x in { p2 : p2`1 = b & p2`2 <= d & p2`2 >= c} by A2,A4,A6,A60,Th30;
A63: ex p st p = x & p`1 <= b & p`1 >= a & p`2 = c by A61;
ex p2 st p2=x & p2`1=b & p2`2<=d & p2`2>=c by A62;
hence thesis by A6,A63,EUCLID:53;
end;
assume
A64: x = p10;
then
A65: x in LSeg(p0,p10) by RLTOPSP1:68;
x in LSeg(p10,p1) by A64,RLTOPSP1:68;
hence thesis by A65,XBOOLE_0:def 4;
end;
hence thesis by A51,A55,A56,A57,TARSKI:def 1;
end;
thus f2 is s.n.c.
proof
let i,j be Nat such that
A66: i+1 < j;
now per cases;
suppose 1 <= i;
then
A67: 1+1 <= i+1 by XREAL_1:6;
now per cases;
case 1 <= j & j+1 <= len f2;
then j <= 2 by A49,XREAL_1:6;
hence contradiction by A66,A67,XXREAL_0:2;
end;
case not (1 <= j & j+1 <= len f2);
then LSeg(f2,j) = {} by TOPREAL1:def 3;
hence LSeg(f2,i) /\ LSeg(f2,j) = {};
end;
end;
hence LSeg(f2,i) /\ LSeg(f2,j) = {};
end;
suppose not (1 <= i & i+1 <= len f2);
then LSeg(f2,i) = {} by TOPREAL1:def 3;
hence LSeg(f2,i) /\ LSeg(f2,j) = {};
end;
end;
hence LSeg(f2,i) /\ LSeg(f2,j) = {};
end;
let i be Nat;
assume that
A68: 1 <= i and
A69: i + 1 <= len f2;
A70: i <= 1 + 1 by A49,A69,XREAL_1:6;
per cases by A68,A70,NAT_1:9;
suppose
A71: i = 1;
then (f2/.i)`2 = p0`2 by A8,FINSEQ_4:18
.= c by A3,EUCLID:52
.= (f2/.(i+1))`2 by A6,A51,A71,EUCLID:52;
hence thesis;
end;
suppose
A72: i = 2;
then (f2/.i)`1 = p10`1 by A8,FINSEQ_4:18
.= b by A6,EUCLID:52
.= (f2/.(i+1))`1 by A4,A52,A72,EUCLID:52;
hence thesis;
end;
end;
A73: 1+1 in Seg len f2 by A49,FINSEQ_1:1;
A74: 1+1 <= len f2 by A49;
LSeg(p0,p10) = LSeg(f2,1) by A49,A50,A51,A73,TOPREAL1:def 3;
then
A75: LSeg(p0,p10) in {LSeg(f2,i): 1 <= i & i+1 <= len f2} by A74;
LSeg(p10,p1) = LSeg(f2,2) by A49,A51,A52,TOPREAL1:def 3;
then LSeg(p10,p1) in {LSeg(f2,i): 1 <= i & i+1 <= len f2} by A49;
then
A76: {LSeg(p0,p10),LSeg(p10,p1)} c= {LSeg(f2,i):1<=i & i+1<=len f2}
by A75,ZFMISC_1:32;
{LSeg(f2,i): 1 <= i & i+1 <= len f2} c= {LSeg(p0,p10),LSeg(p10,p1)}
proof
let ax be object;
assume ax in {LSeg(f2,i): 1 <= i & i+1 <= len f2};
then consider i such that
A77: ax = LSeg(f2,i) and
A78: 1<=i and
A79: i+1<=len f2;
i+1 <= 2 + 1 by A8,A79,FINSEQ_1:45;
then i <= 1 + 1 by XREAL_1:6;
then i = 1 or i = 2 by A78,NAT_1:9;
then ax = LSeg(p0,p10) or ax = LSeg(p10,p1) by A50,A51,A52,A77,A79,
TOPREAL1:def 3;
hence thesis by TARSKI:def 2;
end;
then
A80: L~f2 = union {LSeg(p0,p10),LSeg(p10,p1)} by A76,XBOOLE_0:def 10;
hence L~f2 = LSeg(p0,p10) \/ LSeg(p10,p1) by ZFMISC_1:75;
L~f2 = (LSeg(p0,p10) \/ LSeg(p10,p1)) by A80,ZFMISC_1:75;
then
A81: L~f2=L3 \/ LSeg(p10,p1) by A1,A3,A6,Th30
.=L3 \/ L4 by A2,A4,A6,Th30;
P = LSeg(p0,p01) \/ LSeg(p01,p1) \/ (LSeg(p0,p10) \/ LSeg(p10,p1))
by A3,A4,A5,A6,SPPOL_2:def 3;
hence P = L~f1 \/ L~f2 by A47,A80,ZFMISC_1:75;
now
assume L2 meets L3;
then consider x being object such that
A82: x in L2 and
A83: x in L3 by XBOOLE_0:3;
A84: ex p st p = x & p`1 <= b & p`1 >= a & p`2 = d by A82;
ex p2 st p2 = x & p2`1 <= b & p2`1 >= a & p2`2 = c by A83;
hence contradiction by A2,A84;
end;
then
A85: L2 /\ L3 = {};
A86: LSeg(|[ a,c ]|, |[ a,d ]|) = { p3 : p3`1 = a & p3`2 <= d & p3`2 >= c}
by A2,Th30;
A87: LSeg(|[ a,d ]|, |[ b,d ]|) = { p2 : p2`1 <= b & p2`1 >= a & p2`2 = d}
by A1,Th30;
A88: LSeg(|[ a,c ]|, |[ b,c ]|) = { q1 : q1`1 <= b & q1`1 >= a & q1`2 = c}
by A1,Th30;
A89: LSeg(|[ b,c ]|, |[ b,d ]|) = { q2 : q2`1 = b & q2`2 <= d & q2`2 >= c}
by A2,Th30;
A90: LSeg(|[a,c]|,|[a,d]|) /\ LSeg(|[a,c]|,|[b,c]|) = {|[a,c]|} by A1,A2,Th31;
A91: LSeg(|[a,d]|,|[b,d]|) /\ LSeg(|[b,c]|,|[b,d]|) = {|[b,d]|} by A1,A2,Th33;
now
assume L1 meets L4;
then consider x being object such that
A92: x in L1 and
A93: x in L4 by XBOOLE_0:3;
A94: ex p st p = x & p`1 = a & p`2 <= d & p`2 >= c by A92;
ex p2 st p2 = x & p2`1 = b & p2`2 <= d & p2`2 >= c by A93;
hence contradiction by A1,A94;
end;
then
A95: L1 /\ L4 = {};
thus L~f1 /\ L~f2
= (L1 \/ L2) /\ L3 \/ (L1 \/ L2) /\ L4 by A48,A81,XBOOLE_1:23
.= L1 /\ L3 \/ L2 /\ L3 \/ (L1 \/ L2) /\ L4 by XBOOLE_1:23
.= L1 /\ L3 \/ (L1 /\ L4 \/ L2 /\ L4) by A85,XBOOLE_1:23
.= {p0, p1} by A3,A4,A86,A87,A88,A89,A90,A91,A95,ENUMSET1:1;
thus thesis by A7,A8,A15,A49,FINSEQ_4:18;
end;
theorem Th49:
for P1,P2 being Subset of TOP-REAL 2, a,b,c,d being Real,
f1,f2 being FinSequence of TOP-REAL 2,p1,p2 being Point of TOP-REAL 2 st
a < b & c < d & p1=|[a,c]| & p2=|[b,d]| & f1=<*|[a,c]|,|[a,d]|,|[b,d]|*>
& f2=<*|[a,c]|,|[b,c]|,|[b,d]|*> & P1=L~f1 & P2=L~f2
holds P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2
& P1 is non empty & P2 is non empty &
rectangle(a,b,c,d) = P1 \/ P2 & P1 /\ P2 = {p1,p2}
proof
let P1,P2 be Subset of TOP-REAL 2, a,b,c,d be Real,
f1,f2 be FinSequence of TOP-REAL 2,p1,p2 be Point of TOP-REAL 2;
assume that
A1: a < b and
A2: c < d and
A3: p1=|[a,c]| and
A4: p2=|[b,d]| and
A5: f1=<*|[a,c]|,|[a,d]|,|[b,d]|*> and
A6: f2=<*|[a,c]|,|[b,c]|,|[b,d]|*> and
A7: P1=L~f1 and
A8: P2=L~f2;
(|[a,c]|)`2=c by EUCLID:52;
then
A9: |[a,c]|<>|[a,d]| or |[a,d]|<>|[b,d]| by A2,EUCLID:52;
A10: P1=LSeg(|[a,c]|,|[a,d]|) \/ LSeg(|[a,d]|,|[b,d]|) by A1,A2,A5,A6,A7,Th48;
A11: LSeg(|[a,c]|,|[a,d]|) /\ LSeg(|[a,d]|,|[b,d]|)={|[a,d]|} by A1,A2,Th34;
(|[b,c]|)`2=c by EUCLID:52;
then
A12: |[a,c]|<>|[b,c]| or |[b,c]|<>|[b,d]| by A2,EUCLID:52;
A13: P2=LSeg(|[a,c]|,|[b,c]|) \/ LSeg(|[b,c]|,|[b,d]|) by A1,A2,A5,A6,A8,Th48;
LSeg(|[a,c]|,|[b,c]|) /\ LSeg(|[b,c]|,|[b,d]|)={|[b,c]|} by A1,A2,Th32;
hence thesis by A1,A2,A3,A4,A5,A6,A7,A8,A9,A10,A11,A12,A13,Th48,TOPREAL1:12;
end;
theorem Th50:
for a,b,c,d being Real st a < b & c < d
holds rectangle(a,b,c,d) is being_simple_closed_curve
proof
let a,b,c,d be Real;
assume that
A1: a < b and
A2: c < d;
set P=rectangle(a,b,c,d);
set p1=|[a,c]|,p2=|[b,d]|;
reconsider f1=<*|[a,c]|,|[a,d]|,|[b,d]|*> as FinSequence of TOP-REAL 2;
reconsider f2=<*|[a,c]|,|[b,c]|,|[b,d]|*> as FinSequence of TOP-REAL 2;
set P1=L~f1,P2=L~f2;
A3: a < b & c < d &
P={p: p`1=a & c <=p`2 & p`2<=d or p`2=d & a<=p`1 & p`1<=b or
p`1=b & c <=p`2 & p`2<=d or p`2=c & a<=p`1 & p`1<=b} &
p1=|[a,c]| & p2=|[b,d]| & f1=<*|[a,c]|,|[a,d]|,|[b,d]|*>
& f2=<*|[a,c]|,|[b,c]|,|[b,d]|*> & P1=L~f1 & P2=L~f2
implies P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2
& P1 is non empty & P2 is non empty &
P = P1 \/ P2 & P1 /\ P2 = {p1,p2} by Th49;
(|[a,c]|)`1=a by EUCLID:52;
then
A4: p1<>p2 by A1,EUCLID:52;
p1 in P1 /\ P2 by A1,A2,A3,Lm15,TARSKI:def 2;
then p1 in P1 by XBOOLE_0:def 4;
then
A5: p1 in P by A1,A2,A3,Lm15,XBOOLE_0:def 3;
p2 in P1 /\ P2 by A1,A2,A3,Lm15,TARSKI:def 2;
then p2 in P1 by XBOOLE_0:def 4;
then p2 in P by A1,A2,A3,Lm15,XBOOLE_0:def 3;
hence thesis by A1,A2,A3,A4,A5,Lm15,TOPREAL2:6;
end;
theorem Th51:
for a,b,c,d being Real st a<b & c <d holds
Upper_Arc rectangle(a,b,c,d) = LSeg(|[a,c]|,|[a,d]|) \/ LSeg(|[a,d]|,|[b,d]|)
proof
let a,b,c,d be Real;
set K = rectangle(a,b,c,d);
assume that
A1: a<b and
A2: c <d;
A3: K is being_simple_closed_curve by A1,A2,Th50;
set P=K;
A4: W-min(K)= |[a,c]| by A1,A2,Th46;
A5: E-max(K)= |[b,d]| by A1,A2,Th46;
reconsider U= LSeg(|[a,c]|,|[a,d]|) \/ LSeg(|[a,d]|,|[b,d]|)
as non empty Subset of TOP-REAL 2;
A6: U is_an_arc_of W-min(P),E-max(P) by A1,A2,Th47;
reconsider P3= LSeg(|[a,c]|,|[b,c]|) \/ LSeg(|[b,c]|,|[b,d]|)
as non empty Subset of TOP-REAL 2;
A7: P3 is_an_arc_of E-max(P),W-min(P) by A1,A2,Th47;
reconsider f1=<* |[a,c]|,|[a,d]|,|[b,d]| *>,
f2=<* |[a,c]|,|[b,c]|,|[b,d]| *> as FinSequence of TOP-REAL 2;
set p0=|[a,c]|,p01=|[a,d]|,p10=|[b,c]|,p1=|[b,d]|;
A8: a < b & c < d & p0=|[a,c]| & p1=|[b,d]| & p01=|[a,d]| & p10=|[b,c]|
& f1=<*p0,p01,p1*> & f2=<*p0,p10,p1*> implies f1 is being_S-Seq &
L~f1 = LSeg(p0,p01) \/ LSeg(p01,p1) & f2 is being_S-Seq &
L~f2 = LSeg(p0,p10) \/ LSeg(p10,p1) & K = L~f1 \/ L~f2 &
L~f1 /\ L~f2 = {p0,p1} & f1/.1 = p0 &
f1/.len f1=p1 & f2/.1 = p0 & f2/.len f2 = p1 by Th48;
A9: Vertical_Line((W-bound(P)+E-bound(P))/2)
= Vertical_Line((a+E-bound(P))/2) by A1,A2,Th36
.= Vertical_Line((a+b)/2) by A1,A2,Th38;
set Q=Vertical_Line((W-bound(P)+E-bound(P))/2);
reconsider a2=a,b2=b,c2=c,d2=d as Real;
A10: U /\ Vertical_Line((W-bound(P)+E-bound(P))/2) ={|[(a+b)/2,d]|}
proof
thus
U /\ Vertical_Line((W-bound(P)+E-bound(P))/2) c= {|[(a+b)/2,d]|}
proof
let x be object;
assume
A11: x in U /\ Vertical_Line((W-bound(P)+E-bound(P))/2);
then
A12: x in U by XBOOLE_0:def 4;
x in Vertical_Line((W-bound(P)+E-bound(P))/2) by A11,XBOOLE_0:def 4;
then x in {p where p is Point of TOP-REAL 2: p`1=(a+b)/2}
by A9,JORDAN6:def 6;
then consider p such that
A13: x=p and
A14: p`1=(a+b)/2;
now
assume p in LSeg(|[a,c]|,|[a,d]|);
then p`1=a by TOPREAL3:11;
hence contradiction by A1,A14;
end;
then p in LSeg(|[a2,d2]|,|[b2,d2]|) by A12,A13,XBOOLE_0:def 3;
then p`2=d by TOPREAL3:12;
then x=|[(a+b)/2,d]| by A13,A14,EUCLID:53;
hence thesis by TARSKI:def 1;
end;
let x be object;
assume x in {|[(a+b)/2,d]|};
then
A15: x= |[(a+b)/2,d]| by TARSKI:def 1;
(|[(a+b)/2,d]|)`1= (a+b)/2 by EUCLID:52;
then x in {p where p is Point of TOP-REAL 2: p`1=(a+b)/2} by A15;
then
A16: x in Vertical_Line((W-bound(P)+E-bound(P))/2) by A9,JORDAN6:def 6;
A17: (|[b,d]|)`1=b by EUCLID:52;
A18: (|[b,d]|)`2=d by EUCLID:52;
A19: (|[a,d]|)`1=a by EUCLID:52;
(|[a,d]|)`2=d by EUCLID:52;
then x in LSeg(|[b,d]|,|[a,d]|) by A1,A15,A17,A18,A19,TOPREAL3:13;
then x in U by XBOOLE_0:def 3;
hence thesis by A16,XBOOLE_0:def 4;
end;
then |[(a+b)/2,d]| in U /\ Q by TARSKI:def 1;
then U meets Q;
then First_Point(U,W-min(P),E-max(P),
Vertical_Line((W-bound(P)+E-bound(P))/2)) in {|[(a+b)/2,d]|}
by A6,A10,JORDAN5C:def 1;
then First_Point(U,W-min(P),E-max(P),
Vertical_Line((W-bound(P)+E-bound(P))/2)) = |[(a+b)/2,d]| by TARSKI:def 1;
then
A20: First_Point(U,W-min(P),E-max(P),
Vertical_Line((W-bound(P)+E-bound(P))/2))`2=d by EUCLID:52;
A21: P3 /\ Vertical_Line((W-bound(P)+E-bound(P))/2) ={|[(a+b)/2,c]|}
proof
thus
P3 /\ Vertical_Line((W-bound(P)+E-bound(P))/2) c= {|[(a+b)/2,c]|}
proof
let x be object;
assume
A22: x in P3 /\ Vertical_Line((W-bound(P)+E-bound(P))/2);
then
A23: x in P3 by XBOOLE_0:def 4;
x in Vertical_Line((W-bound(P)+E-bound(P))/2) by A22,XBOOLE_0:def 4;
then x in {p where p is Point of TOP-REAL 2: p`1=(a+b)/2}
by A9,JORDAN6:def 6;
then consider p such that
A24: x=p and
A25: p`1=(a+b)/2;
now
assume p in LSeg(|[b,c]|,|[b,d]|);
then p`1= b by TOPREAL3:11;
hence contradiction by A1,A25;
end;
then p in LSeg(|[a2,c2]|,|[b2,c2]|) by A23,A24,XBOOLE_0:def 3;
then p`2= c by TOPREAL3:12;
then x=|[(a+b)/2,c]| by A24,A25,EUCLID:53;
hence thesis by TARSKI:def 1;
end;
let x be object;
assume x in {|[(a+b)/2,c]|};
then
A26: x= |[(a+b)/2,c]| by TARSKI:def 1;
(|[(a+b)/2,c]|)`1= (a+b)/2 by EUCLID:52;
then x in {p where p is Point of TOP-REAL 2: p`1=(a+b)/2} by A26;
then
A27: x in Vertical_Line((W-bound(P)+E-bound(P))/2) by A9,JORDAN6:def 6;
A28: (|[b,c]|)`1=b by EUCLID:52;
A29: (|[b,c]|)`2=c by EUCLID:52;
A30: (|[a,c]|)`1=a by EUCLID:52;
(|[a,c]|)`2=c by EUCLID:52;
then |[(b+a)/2,c]| in LSeg(|[a,c]|,|[b,c]|) by A1,A28,A29,A30,TOPREAL3:13;
then x in P3 by A26,XBOOLE_0:def 3;
hence thesis by A27,XBOOLE_0:def 4;
end;
then |[(a+b)/2,c]| in P3 /\ Q by TARSKI:def 1;
then P3 meets Q;
then Last_Point(P3,E-max(P),W-min(P),
Vertical_Line((W-bound(P)+E-bound(P))/2)) in {|[(a+b)/2,c]|}
by A7,A21,JORDAN5C:def 2;
then Last_Point(P3,E-max(P),W-min(P),
Vertical_Line((W-bound(P)+E-bound(P))/2)) = |[(a+b)/2,c]| by TARSKI:def 1;
then Last_Point(P3,E-max(P),W-min(P),
Vertical_Line((W-bound(P)+E-bound(P))/2))`2 = c by EUCLID:52;
hence thesis by A1,A2,A3,A4,A5,A6,A7,A8,A20,JORDAN6:def 8;
end;
theorem Th52:
for a,b,c,d being Real st a<b & c <d holds
Lower_Arc rectangle(a,b,c,d)= LSeg(|[a,c]|,|[b,c]|) \/ LSeg(|[b,c]|,|[b,d]|)
proof
let a,b,c,d be Real;
set K = rectangle(a,b,c,d);
assume that
A1: a<b and
A2: c <d;
A3: K is being_simple_closed_curve by A1,A2,Th50;
set P=K;
A4: W-min(K)= |[a,c]| by A1,A2,Th46;
A5: E-max(K)= |[b,d]| by A1,A2,Th46;
reconsider U= LSeg(|[a,c]|,|[a,d]|) \/ LSeg(|[a,d]|,|[b,d]|)
as non empty Subset of TOP-REAL 2;
A6: U is_an_arc_of W-min(P),E-max(P) by A1,A2,Th47;
reconsider P3= LSeg(|[a,c]|,|[b,c]|) \/ LSeg(|[b,c]|,|[b,d]|)
as non empty Subset of TOP-REAL 2;
A7: P3 is_an_arc_of E-max(P),W-min(P) by A1,A2,Th47;
reconsider f1=<* |[a,c]|,|[a,d]|,|[b,d]| *>,
f2=<* |[a,c]|,|[b,c]|,|[b,d]| *> as FinSequence of TOP-REAL 2;
set p0=|[a,c]|,p01=|[a,d]|,p10=|[b,c]|,p1=|[b,d]|;
A8: a < b & c < d & p0=|[a,c]| & p1=|[b,d]| & p01=|[a,d]| & p10=|[b,c]|
& f1=<*p0,p01,p1*> & f2=<*p0,p10,p1*> implies f1 is being_S-Seq &
L~f1 = LSeg(p0,p01) \/ LSeg(p01,p1) & f2 is being_S-Seq &
L~f2 = LSeg(p0,p10) \/ LSeg(p10,p1) & K = L~f1 \/ L~f2 &
L~f1 /\ L~f2 = {p0,p1} & f1/.1 = p0 &
f1/.len f1=p1 & f2/.1 = p0 & f2/.len f2 = p1 by Th48;
A9: Vertical_Line((W-bound(P)+E-bound(P))/2)
= Vertical_Line((a+E-bound(P))/2) by A1,A2,Th36
.= Vertical_Line((a+b)/2) by A1,A2,Th38;
set Q=Vertical_Line((W-bound(P)+E-bound(P))/2);
reconsider a2=a,b2=b,c2=c,d2=d as Real;
A10: U /\ Vertical_Line((W-bound(P)+E-bound(P))/2) ={|[(a+b)/2,d]|}
proof
thus
U /\ Vertical_Line((W-bound(P)+E-bound(P))/2) c= {|[(a+b)/2,d]|}
proof
let x be object;
assume
A11: x in U /\ Vertical_Line((W-bound(P)+E-bound(P))/2);
then
A12: x in U by XBOOLE_0:def 4;
x in Vertical_Line((W-bound(P)+E-bound(P))/2) by A11,XBOOLE_0:def 4;
then x in {p where p is Point of TOP-REAL 2: p`1=(a+b)/2}
by A9,JORDAN6:def 6;
then consider p such that
A13: x=p and
A14: p`1=(a+b)/2;
now
assume p in LSeg(|[a,c]|,|[a,d]|);
then p`1=a by TOPREAL3:11;
hence contradiction by A1,A14;
end;
then p in LSeg(|[a2,d2]|,|[b2,d2]|) by A12,A13,XBOOLE_0:def 3;
then p`2=d by TOPREAL3:12;
then x=|[(a+b)/2,d]| by A13,A14,EUCLID:53;
hence thesis by TARSKI:def 1;
end;
let x be object;
assume x in {|[(a+b)/2,d]|};
then
A15: x= |[(a+b)/2,d]| by TARSKI:def 1;
(|[(a+b)/2,d]|)`1= (a+b)/2 by EUCLID:52;
then x in {p where p is Point of TOP-REAL 2: p`1=(a+b)/2} by A15;
then
A16: x in Vertical_Line((W-bound(P)+E-bound(P))/2) by A9,JORDAN6:def 6;
A17: (|[b,d]|)`1=b by EUCLID:52;
A18: (|[b,d]|)`2=d by EUCLID:52;
A19: (|[a,d]|)`1=a by EUCLID:52;
(|[a,d]|)`2=d by EUCLID:52;
then x in LSeg(|[b,d]|,|[a,d]|) by A1,A15,A17,A18,A19,TOPREAL3:13;
then x in U by XBOOLE_0:def 3;
hence thesis by A16,XBOOLE_0:def 4;
end;
then |[(a+b)/2,d]| in U /\ Q by TARSKI:def 1;
then U meets Q;
then First_Point(U,W-min(P),E-max(P),
Vertical_Line((W-bound(P)+E-bound(P))/2)) in {|[(a+b)/2,d]|}
by A6,A10,JORDAN5C:def 1;
then First_Point(U,W-min(P),E-max(P),
Vertical_Line((W-bound(P)+E-bound(P))/2)) = |[(a+b)/2,d]| by TARSKI:def 1;
then
A20: First_Point(U,W-min(P),E-max(P),
Vertical_Line((W-bound(P)+E-bound(P))/2))`2=d by EUCLID:52;
A21: P3 /\ Vertical_Line((W-bound(P)+E-bound(P))/2) ={|[(a+b)/2,c]|}
proof
thus
P3 /\ Vertical_Line((W-bound(P)+E-bound(P))/2) c= {|[(a+b)/2,c]|}
proof
let x be object;
assume
A22: x in P3 /\ Vertical_Line((W-bound(P)+E-bound(P))/2);
then
A23: x in P3 by XBOOLE_0:def 4;
x in Vertical_Line((W-bound(P)+E-bound(P))/2) by A22,XBOOLE_0:def 4;
then x in {p where p is Point of TOP-REAL 2: p`1=(a+b)/2}
by A9,JORDAN6:def 6;
then consider p such that
A24: x=p and
A25: p`1=(a+b)/2;
now
assume p in LSeg(|[b,c]|,|[b,d]|);
then p`1= b by TOPREAL3:11;
hence contradiction by A1,A25;
end;
then p in LSeg(|[a2,c2]|,|[b2,c2]|) by A23,A24,XBOOLE_0:def 3;
then p`2= c by TOPREAL3:12;
then x=|[(a+b)/2,c]| by A24,A25,EUCLID:53;
hence thesis by TARSKI:def 1;
end;
let x be object;
assume x in {|[(a+b)/2,c]|};
then
A26: x= |[(a+b)/2,c]| by TARSKI:def 1;
(|[(a+b)/2,c]|)`1= (a+b)/2 by EUCLID:52;
then x in {p where p is Point of TOP-REAL 2: p`1=(a+b)/2} by A26;
then
A27: x in Vertical_Line((W-bound(P)+E-bound(P))/2) by A9,JORDAN6:def 6;
A28: (|[b,c]|)`1=b by EUCLID:52;
A29: (|[b,c]|)`2=c by EUCLID:52;
A30: (|[a,c]|)`1=a by EUCLID:52;
(|[a,c]|)`2=c by EUCLID:52;
then |[(a+b)/2,c]| in LSeg(|[a,c]|,|[b,c]|) by A1,A28,A29,A30,TOPREAL3:13;
then x in P3 by A26,XBOOLE_0:def 3;
hence thesis by A27,XBOOLE_0:def 4;
end;
then |[(a+b)/2,c]| in P3 /\ Q by TARSKI:def 1;
then P3 meets Q;
then Last_Point(P3,E-max(P),W-min(P),
Vertical_Line((W-bound(P)+E-bound(P))/2)) in {|[(a+b)/2,c]|}
by A7,A21,JORDAN5C:def 2;
then Last_Point(P3,E-max(P),W-min(P),
Vertical_Line((W-bound(P)+E-bound(P))/2)) = |[(a+b)/2,c]| by TARSKI:def 1;
then
A31: Last_Point(P3,E-max(P),W-min(P),
Vertical_Line((W-bound(P)+E-bound(P))/2))`2 = c by EUCLID:52;
A32: P3 is_an_arc_of E-max(P),W-min(P) by A1,A2,Th47;
A33: Upper_Arc(P) /\ P3={W-min(P),E-max(P)} by A1,A2,A4,A5,A8,Th51;
A34: Upper_Arc(P) \/ P3=P by A1,A2,A8,Th51;
First_Point(Upper_Arc(P),W-min(P),E-max(P), Vertical_Line((W-bound(P)+
E-bound(P))/2))`2> Last_Point(P3,E-max(P),W-min(P), Vertical_Line((W-bound(P)+
E-bound(P))/2))`2 by A1,A2,A20,A31,Th51;
hence thesis by A3,A32,A33,A34,JORDAN6:def 9;
end;
theorem Th53:
for a,b,c,d being Real st a<b & c <d
ex f being Function of I[01],(TOP-REAL 2)|Upper_Arc rectangle(a,b,c,d)
st f is being_homeomorphism
& f.0=W-min rectangle(a,b,c,d) & f.1=E-max rectangle(a,b,c,d) &
rng f=Upper_Arc rectangle(a,b,c,d) &
(for r being Real st r in [.0,1/2 .] holds
f.r=(1-2*r)*|[a,c]|+(2*r)*|[a,d]|)&
(for r being Real st r in [.1/2,1 .] holds
f.r=(1-(2*r-1))*|[a,d]|+(2*r-1)*|[b,d]|)&
(for p being Point of TOP-REAL 2 st p in LSeg(|[a,c]|,|[a,d]|)
holds 0<=((p`2)-c)/(d-c)/2 & ((p`2)-c)/(d-c)/2<=1
& f.(((p`2)-c)/(d-c)/2)=p)&
for p being Point of TOP-REAL 2 st p in LSeg(|[a,d]|,|[b,d]|)
holds 0<=((p`1)-a)/(b-a)/2 + 1/2 & ((p`1)-a)/(b-a)/2 + 1/2<=1
& f.(((p`1)-a)/(b-a)/2 + 1/2)=p
proof
let a,b,c,d be Real;
set K = rectangle(a,b,c,d);
assume that
A1: a<b and
A2: c <d;
defpred P[object,object] means for r being Real st $1=r holds
(r in [.0,1/2 .] implies $2=(1-2*r)*|[a,c]|+(2*r)*|[a,d]|) &
(r in [.1/2,1 .] implies $2=(1-(2*r-1))*|[a,d]|+(2*r-1)*|[b,d]|);
A3: [.0,1.]=[.0,1/2 .] \/ [.1/2,1 .] by XXREAL_1:165;
A4: for x being object st x in [.0,1.] ex y being object st P[x,y]
proof
let x be object;
assume
A5: x in [.0,1.];
now per cases by A3,A5,XBOOLE_0:def 3;
case
A6: x in [.0,1/2.];
then reconsider r=x as Real;
A7: r<=1/2 by A6,XXREAL_1:1;
set y0= (1-2*r)*|[a,c]|+(2*r)*|[a,d]|;
r in [.1/2,1.] implies y0=(1-(2*r-1))*|[a,d]|+(2*r-1)*|[b,d]|
proof
assume r in [.1/2,1.];
then 1/2 <=r by XXREAL_1:1;
then
A8: r=1/2 by A7,XXREAL_0:1;
then
A9: y0= (0)*|[a,c]|+|[a,d]| by RLVECT_1:def 8
.= (0.TOP-REAL 2) + |[a,d]| by RLVECT_1:10
.= |[a,d]| by RLVECT_1:4;
(1-(2*r-1))*|[a,d]|+(2*r-1)*|[b,d]|
= (1)*|[a,d]|+0.TOP-REAL 2 by A8,RLVECT_1:10
.= |[a,d]|+0.TOP-REAL 2 by RLVECT_1:def 8
.= |[a,d]| by RLVECT_1:4;
hence thesis by A9;
end;
then for r2 being Real st x=r2 holds
(r2 in [.0,1/2.] implies y0=(1-2*r2)*|[a,c]|+(2*r2)*|[a,d]|) &
(r2 in [.1/2,1.] implies y0=(1-(2*r2-1))*|[a,d]|+(2*r2-1)*|[b,d]|);
hence thesis;
end;
case
A10: x in [.1/2,1.];
then reconsider r=x as Real;
A11: 1/2<=r by A10,XXREAL_1:1;
set y0= (1-(2*r-1))*|[a,d]|+(2*r-1)*|[b,d]|;
r in [.0,1/2.] implies y0=(1-2*r)*|[a,c]|+(2*r)*|[a,d]|
proof
assume r in [.0,1/2.];
then r<=1/2 by XXREAL_1:1;
then
A12: r=1/2 by A11,XXREAL_0:1;
then
A13: y0= |[a,d]|+(0)*|[b,d]| by RLVECT_1:def 8
.= |[a,d]|+(0.TOP-REAL 2) by RLVECT_1:10
.= |[a,d]| by RLVECT_1:4;
(1-2*r)*|[a,c]|+(2*r)*|[a,d]|
= 0.TOP-REAL 2+(1)*|[a,d]| by A12,RLVECT_1:10
.= 0.TOP-REAL 2+|[a,d]| by RLVECT_1:def 8
.= |[a,d]| by RLVECT_1:4;
hence thesis by A13;
end;
then for r2 being Real st x=r2 holds
(r2 in [.0,1/2.] implies y0=(1-2*r2)*|[a,c]|+(2*r2)*|[a,d]|) &
(r2 in [.1/2,1.] implies y0=(1-(2*r2-1))*|[a,d]|+(2*r2-1)*|[b,d]|);
hence thesis;
end;
end;
hence thesis;
end;
ex f2 being Function st dom f2 = [.0,1.] &
for x being object st x in [.0,1.] holds P[x,f2.x] from CLASSES1:sch 1(A4);
then consider f2 being Function such that
A14: dom f2 = [.0,1.] and
A15: for x being object st x in [.0,1.] holds P[x,f2.x];
rng f2 c= the carrier of (TOP-REAL 2)|(Upper_Arc(K))
proof
let y be object;
assume y in rng f2;
then consider x being object such that
A16: x in dom f2 and
A17: y=f2.x by FUNCT_1:def 3;
now per cases by A3,A14,A16,XBOOLE_0:def 3;
case
A18: x in [.0,1/2.];
then reconsider r=x as Real;
A19: 0<=r by A18,XXREAL_1:1;
r<=1/2 by A18,XXREAL_1:1;
then
A20: r*2<=1/2*2 by XREAL_1:64;
f2.x= (1-2*r)*|[a,c]|+(2*r)*|[a,d]| by A14,A15,A16,A18;
then
A21: y
in { (1-lambda)*|[a,c]| + lambda*|[a,d]| where lambda is Real:
0 <= lambda & lambda <= 1 } by A17,A19,A20;
Upper_Arc(K)= LSeg(|[a,c]|,|[a,d]|) \/ LSeg(|[a,d]|,|[b,d]|)
by A1,A2,Th51;
then y in Upper_Arc(K) by A21,XBOOLE_0:def 3;
hence thesis by PRE_TOPC:8;
end;
case
A22: x in [.1/2,1.];
then reconsider r=x as Real;
A23: 1/2<=r by A22,XXREAL_1:1;
A24: r<=1 by A22,XXREAL_1:1;
r*2>=1/2*2 by A23,XREAL_1:64;
then
A25: 2*r-1>=0 by XREAL_1:48;
2*1>=2*r by A24,XREAL_1:64;
then
A26: 1+1-1>=2*r-1 by XREAL_1:9;
f2.x= (1-(2*r-1))*|[a,d]|+(2*r-1)*|[b,d]| by A14,A15,A16,A22;
then
A27: y
in { (1-lambda)*|[a,d]| + lambda*|[b,d]| where lambda is Real:
0 <= lambda & lambda <= 1 } by A17,A25,A26;
Upper_Arc(K)= LSeg(|[a,c]|,|[a,d]|) \/ LSeg(|[a,d]|,|[b,d]|)
by A1,A2,Th51;
then y in Upper_Arc(K) by A27,XBOOLE_0:def 3;
hence thesis by PRE_TOPC:8;
end;
end;
hence thesis;
end;
then reconsider f3=f2 as Function of I[01],(TOP-REAL 2)|(Upper_Arc(K))
by A14,BORSUK_1:40,FUNCT_2:2;
A28: 0 in [.0,1.] by XXREAL_1:1;
0 in [.0,1/2.] by XXREAL_1:1;
then
A29: f3.0= (1-2*0)*|[a,c]|+(2*0)*|[a,d]| by A15,A28
.= (1)*|[a,c]|+0.TOP-REAL 2 by RLVECT_1:10
.= |[a,c]|+0.TOP-REAL 2 by RLVECT_1:def 8
.= |[a,c]| by RLVECT_1:4
.= W-min(K) by A1,A2,Th46;
A30: 1 in [.0,1.] by XXREAL_1:1;
1 in [.1/2,1.] by XXREAL_1:1;
then
A31: f3.1= (1-(2*1-1))*|[a,d]|+(2*1-1)*|[b,d]| by A15,A30
.= (0)*|[a,d]|+|[b,d]| by RLVECT_1:def 8
.= (0.TOP-REAL 2) + |[b,d]| by RLVECT_1:10
.= |[b,d]| by RLVECT_1:4
.= E-max(K) by A1,A2,Th46;
A32: for r being Real st r in [.0,1/2.] holds
f3.r=(1-2*r)*|[a,c]|+(2*r)*|[a,d]|
proof
let r be Real;
assume
A33: r in [.0,1/2.];
then
A34: 0<=r by XXREAL_1:1;
r<=1/2 by A33,XXREAL_1:1;
then r<=1 by XXREAL_0:2;
then r in [.0,1.] by A34,XXREAL_1:1;
hence thesis by A15,A33;
end;
A35: for r being Real st r in [.1/2,1.] holds
f3.r=(1-(2*r-1))*|[a,d]|+(2*r-1)*|[b,d]|
proof
let r be Real;
assume
A36: r in [.1/2,1.];
then
A37: 1/2<=r by XXREAL_1:1;
r<=1 by A36,XXREAL_1:1;
then r in [.0,1.] by A37,XXREAL_1:1;
hence thesis by A15,A36;
end;
A38: for p being Point of TOP-REAL 2 st p in LSeg(|[a,c]|,|[a,d]|)
holds 0<=((p`2)-c)/(d-c)/2 & ((p`2)-c)/(d-c)/2<=1
& f3.(((p`2)-c)/(d-c)/2)=p
proof
let p be Point of TOP-REAL 2;
assume
A39: p in LSeg(|[a,c]|,|[a,d]|);
A40: (|[a,c]|)`2= c by EUCLID:52;
A41: (|[a,d]|)`2= d by EUCLID:52;
then
A42: c <=p`2 by A2,A39,A40,TOPREAL1:4;
A43: p`2<=d by A2,A39,A40,A41,TOPREAL1:4;
A44: d-c>0 by A2,XREAL_1:50;
A45: (p`2) -c >=0 by A42,XREAL_1:48;
A46: d-c>0 by A2,XREAL_1:50;
(p`2) -c <=d-c by A43,XREAL_1:9;
then ((p`2) -c)/(d-c) <=(d-c)/(d-c) by A46,XREAL_1:72;
then ((p`2) -c)/(d-c) <=1 by A46,XCMPLX_1:60;
then
A47: ((p`2) -c)/(d-c)/2 <=1/2 by XREAL_1:72;
set r=((p`2)-c)/(d-c)/2;
r in [.0,1/2.] by A44,A45,A47,XXREAL_1:1;
then f3.(((p`2)-c)/(d-c)/2)=(1-2*r)*|[a,c]|+(2*r)*|[a,d]| by A32
.=|[(1-2*r)*a,(1-2*r)*c]|+(2*r)*|[a,d]| by EUCLID:58
.=|[(1-2*r)*a,(1-2*r)*c]|+|[(2*r)*a,(2*r)*d]| by EUCLID:58
.=|[1*a-(2*r)*a+(2*r)*a,(1-2*r)*c+(2*r)*d]| by EUCLID:56
.=|[a,1*c+(((p`2)-c)/(d-c))*(d-c)]|
.=|[a,1*c+((p`2)-c)]| by A46,XCMPLX_1:87
.=|[p`1,p`2]| by A39,TOPREAL3:11
.= p by EUCLID:53;
hence thesis by A44,A45,A47,XXREAL_0:2;
end;
A48: for p being Point of TOP-REAL 2 st p in LSeg(|[a,d]|,|[b,d]|)
holds 0<=((p`1)-a)/(b-a)/2+1/2 & ((p`1)-a)/(b-a)/2+1/2<=1
& f3.(((p`1)-a)/(b-a)/2+1/2)=p
proof
let p be Point of TOP-REAL 2;
assume
A49: p in LSeg(|[a,d]|,|[b,d]|);
A50: (|[a,d]|)`1= a by EUCLID:52;
A51: (|[b,d]|)`1= b by EUCLID:52;
then
A52: a <=p`1 by A1,A49,A50,TOPREAL1:3;
A53: p`1<=b by A1,A49,A50,A51,TOPREAL1:3;
A54: b-a>0 by A1,XREAL_1:50;
A55: (p`1) -a >=0 by A52,XREAL_1:48;
then
A56: ((p`1) -a)/(b-a)/2+1/2 >=0+1/2 by A54,XREAL_1:7;
A57: b-a>0 by A1,XREAL_1:50;
(p`1) -a <=b-a by A53,XREAL_1:9;
then ((p`1) -a)/(b-a) <=(b-a)/(b-a) by A57,XREAL_1:72;
then ((p`1) -a)/(b-a) <=1 by A57,XCMPLX_1:60;
then ((p`1) -a)/(b-a)/2 <=1/2 by XREAL_1:72;
then
A58: ((p`1) -a)/(b-a)/2+1/2 <=1/2+1/2 by XREAL_1:7;
set r=((p`1)-a)/(b-a)/2+1/2;
r in [.1/2,1.] by A56,A58,XXREAL_1:1;
then f3.(((p`1)-a)/(b-a)/2+1/2)=(1-(2*r-1))*|[a,d]|+(2*r-1)*|[b,d]| by A35
.=|[(1-(2*r-1))*a,(1-(2*r-1))*d]|+((2*r-1))*|[b,d]| by EUCLID:58
.=|[(1-(2*r-1))*a,(1-(2*r-1))*d]|+|[((2*r-1))*b,((2*r-1))*d]|
by EUCLID:58
.=|[(1-(2*r-1))*a+((2*r-1))*b,1*d-(2*r-1)*d+((2*r-1))*d]| by EUCLID:56
.=|[1*a+(((p`1)-a)/(b-a))*(b-a),d]|
.=|[1*a+((p`1)-a),d]| by A57,XCMPLX_1:87
.=|[p`1,p`2]| by A49,TOPREAL3:12
.= p by EUCLID:53;
hence thesis by A54,A55,A58;
end;
reconsider B00=[.0,1.] as Subset of R^1 by TOPMETR:17;
reconsider B01=B00 as non empty Subset of R^1 by XXREAL_1:1;
I[01]=(R^1)|B01 by TOPMETR:19,20;
then consider h1 being Function of I[01],R^1 such that
A59: for p being Point of I[01] holds h1.p=p and
A60: h1 is continuous by Th6;
consider h2 being Function of I[01],R^1 such that
A61: for p being Point of I[01],r1 being Real st h1.p=r1 holds h2.p
=2*r1 and
A62: h2 is continuous by A60,JGRAPH_2:23;
consider h5 being Function of I[01],R^1 such that
A63: for p being Point of I[01],r1 being Real st h2.p=r1 holds h5.p
=1-r1 and
A64: h5 is continuous by A62,Th8;
consider h3 being Function of I[01],R^1 such that
A65: for p being Point of I[01],r1 being Real st h2.p=r1 holds h3.p
=r1-1 and
A66: h3 is continuous by A62,Th7;
consider h4 being Function of I[01],R^1 such that
A67: for p being Point of I[01],r1 being Real st h3.p=r1 holds h4.p
=1-r1 and
A68: h4 is continuous by A66,Th8;
consider g1 being Function of I[01],TOP-REAL 2 such that
A69: for r being Point of I[01] holds g1.r=(h5.r)*|[a,c]|+(h2.r)*|[a,d]| and
A70: g1 is continuous by A62,A64,Th13;
A71: for r being Point of I[01],s being Real st r=s holds
g1.r=(1-2*s)*|[a,c]|+(2*s)*|[a,d]|
proof
let r be Point of I[01],s be Real;
assume
A72: r=s;
g1.r=(h5.r)*|[a,c]|+(h2.r)*|[a,d]| by A69
.=(1-2*(h1.r))*|[a,c]|+(h2.r)*|[a,d]| by A61,A63
.=(1-2*(h1.r))*|[a,c]|+(2*(h1.r))*|[a,d]| by A61
.=(1-2*s)*|[a,c]|+(2*(h1.r))*|[a,d]| by A59,A72
.=(1-2*s)*|[a,c]|+(2*s)*|[a,d]| by A59,A72;
hence thesis;
end;
consider g2 being Function of I[01],TOP-REAL 2 such that
A73: for r being Point of I[01] holds g2.r=(h4.r)*|[a,d]|+(h3.r)*|[b,d]| and
A74: g2 is continuous by A66,A68,Th13;
A75: for r being Point of I[01],s being Real st r=s holds
g2.r=(1-(2*s-1))*|[a,d]|+(2*s-1)*|[b,d]|
proof
let r be Point of I[01],s be Real;
assume
A76: r=s;
g2.r=(h4.r)*|[a,d]|+(h3.r)*|[b,d]| by A73
.=(1-((h2.r)-1))*|[a,d]|+(h3.r)*|[b,d]| by A65,A67
.=(1-((h2.r)-1))*|[a,d]|+((h2.r)-1)*|[b,d]| by A65
.=(1-(2*(h1.r)-1))*|[a,d]|+((h2.r)-1)*|[b,d]| by A61
.=(1-(2*(h1.r)-1))*|[a,d]|+(2*(h1.r)-1)*|[b,d]| by A61
.=(1-(2*s-1))*|[a,d]|+(2*(h1.r)-1)*|[b,d]| by A59,A76
.=(1-(2*s-1))*|[a,d]|+(2*s-1)*|[b,d]| by A59,A76;
hence thesis;
end;
reconsider B11=[.0,1/2.] as non empty Subset of I[01]
by A3,BORSUK_1:40,XBOOLE_1:7,XXREAL_1:1;
A77: dom (g1|B11)=dom g1 /\ B11 by RELAT_1:61
.= (the carrier of I[01]) /\ B11 by FUNCT_2:def 1
.=B11 by XBOOLE_1:28
.=the carrier of (I[01]|B11) by PRE_TOPC:8;
rng (g1|B11) c= the carrier of TOP-REAL 2;
then reconsider g11=g1|B11 as Function
of I[01]|B11,TOP-REAL 2 by A77,FUNCT_2:2;
A78: TOP-REAL 2 is SubSpace of TOP-REAL 2 by TSEP_1:2;
then
A79: g11 is continuous by A70,BORSUK_4:44;
reconsider B22=[.1/2,1.] as non empty Subset of I[01] by A3,BORSUK_1:40
,XBOOLE_1:7,XXREAL_1:1;
A80: dom (g2|B22)=dom g2 /\ B22 by RELAT_1:61
.= (the carrier of I[01]) /\ B22 by FUNCT_2:def 1
.=B22 by XBOOLE_1:28
.=the carrier of (I[01]|B22) by PRE_TOPC:8;
rng (g2|B22) c= the carrier of TOP-REAL 2;
then reconsider g22=g2|B22 as Function
of I[01]|B22,TOP-REAL 2 by A80,FUNCT_2:2;
A81: g22 is continuous by A74,A78,BORSUK_4:44;
A82: B11=[#](I[01]|B11) by PRE_TOPC:def 5;
A83: B22=[#](I[01]|B22) by PRE_TOPC:def 5;
A84: B11 is closed by Th4;
A85: B22 is closed by Th4;
A86: [#](I[01]|B11) \/ [#](I[01]|B22)=[#]I[01]
by A82,A83,BORSUK_1:40,XXREAL_1:165;
for p being object st p in ([#](I[01]|B11)) /\ ([#](I[01]|B22))
holds g11.p = g22.p
proof
let p be object;
assume
A87: p in ([#](I[01]|B11)) /\ ([#](I[01]|B22));
then
A88: p in [#](I[01]|B11) by XBOOLE_0:def 4;
A89: p in [#](I[01]|B22) by A87;
A90: p in B11 by A88,PRE_TOPC:def 5;
A91: p in B22 by A89,PRE_TOPC:def 5;
reconsider rp=p as Real by A90;
A92: rp<=1/2 by A90,XXREAL_1:1;
rp>=1/2 by A91,XXREAL_1:1;
then rp=1/2 by A92,XXREAL_0:1;
then
A93: 2*rp=1;
thus g11.p=g1.p by A90,FUNCT_1:49
.= (1-1)*|[a,c]|+(1)*|[a,d]| by A71,A90,A93
.=0.TOP-REAL 2 +1*|[a,d]| by RLVECT_1:10
.=(1-0)*|[a,d]| +(1-1)*|[b,d]| by RLVECT_1:10
.=g2.p by A75,A90,A93
.=g22.p by A91,FUNCT_1:49;
end;
then consider h being Function of I[01],TOP-REAL 2 such that
A94: h=g11+*g22 and
A95: h is continuous by A79,A81,A82,A83,A84,A85,A86,JGRAPH_2:1;
A96: dom f3=dom h by Th5;
A97: dom f3=the carrier of I[01] by Th5;
for x being object st x in dom f2 holds f3.x=h.x
proof
let x be object;
assume
A98: x in dom f2;
then reconsider rx=x as Real by A97;
A99: 0<=rx by A96,A98,BORSUK_1:40,XXREAL_1:1;
A100: rx<=1 by A96,A98,BORSUK_1:40,XXREAL_1:1;
now per cases;
case
A101: rx<1/2;
then
A102: rx in [.0,1/2.] by A99,XXREAL_1:1;
not rx in dom g22 by A83,A101,XXREAL_1:1;
then h.rx=g11.rx by A94,FUNCT_4:11
.=g1.rx by A102,FUNCT_1:49
.=(1-(2*rx))*|[a,c]|+(2*rx)*|[a,d]| by A71,A96,A98
.=f3.rx by A32,A102;
hence thesis;
end;
case rx >=1/2;
then
A103: rx in [.1/2,1.] by A100,XXREAL_1:1;
then rx in [#](I[01]|B22) by PRE_TOPC:def 5;
then h.rx=g22.rx by A80,A94,FUNCT_4:13
.=g2.rx by A103,FUNCT_1:49
.=(1-(2*rx-1))*|[a,d]|+(2*rx-1)*|[b,d]| by A75,A96,A98
.=f3.rx by A35,A103;
hence thesis;
end;
end;
hence thesis;
end;
then
A104: f2=h by A96,FUNCT_1:2;
for x1,x2 being object st x1 in dom f3 & x2 in dom f3 & f3.x1=f3.x2
holds x1=x2
proof
let x1,x2 be object;
assume that
A105: x1 in dom f3 and
A106: x2 in dom f3 and
A107: f3.x1=f3.x2;
reconsider r1=x1 as Real by A105;
reconsider r2=x2 as Real by A106;
A108: LSeg(|[a,c]|,|[a,d]|) /\ LSeg(|[a,d]|,|[b,d]|) = {|[a,d]|} by A1,A2,Th34;
now per cases by A3,A14,A105,A106,XBOOLE_0:def 3;
case
A109: x1 in [.0,1/2.] & x2 in [.0,1/2.];
then f3.r1=(1-2*r1)*|[a,c]|+(2*r1)*|[a,d]| by A32;
then (1-2*r2)*|[a,c]|+(2*r2)*|[a,d]|= (1-2*r1)*|[a,c]|+(2*r1)*|[a,d]|
by A32,A107,A109;
then (1-2*r2)*|[a,c]|+(2*r2)*|[a,d]| -(2*r1)*|[a,d]|
= (1-2*r1)*|[a,c]| by RLVECT_4:1;
then (1-2*r2)*|[a,c]|+((2*r2)*|[a,d]| -(2*r1)*|[a,d]|)
= (1-2*r1)*|[a,c]| by RLVECT_1:def 3;
then (1-2*r2)*|[a,c]|+(2*r2-2*r1)*(|[a,d]|)
= (1-2*r1)*|[a,c]| by RLVECT_1:35;
then (2*r2-2*r1)*(|[a,d]|)+((1-2*r2)*|[a,c]|-(1-2*r1)*|[a,c]|)
= (1-2*r1)*|[a,c]|-(1-2*r1)*|[a,c]| by RLVECT_1:def 3;
then (2*r2-2*r1)*(|[a,d]|)+((1-2*r2)*|[a,c]|-(1-2*r1)*|[a,c]|)
= 0.TOP-REAL 2 by RLVECT_1:5;
then (2*r2-2*r1)*(|[a,d]|)+((1-2*r2)-(1-2*r1))*|[a,c]|
= 0.TOP-REAL 2 by RLVECT_1:35;
then (2*r2-2*r1)*(|[a,d]|)+(-(2*r2-2*r1))*|[a,c]| = 0.TOP-REAL 2;
then (2*r2-2*r1)*(|[a,d]|)+-((2*r2-2*r1)*|[a,c]|)
= 0.TOP-REAL 2 by RLVECT_1:79;
then (2*r2-2*r1)*(|[a,d]|)-((2*r2-2*r1)*|[a,c]|)
= 0.TOP-REAL 2;
then (2*r2-2*r1)*((|[a,d]|)-(|[a,c]|)) = 0.TOP-REAL 2 by RLVECT_1:34;
then (2*r2-2*r1)=0 or (|[a,d]|)-(|[a,c]|)=0.TOP-REAL 2 by RLVECT_1:11;
then (2*r2-2*r1)=0 or |[a,d]|=|[a,c]| by RLVECT_1:21;
then (2*r2-2*r1)=0 or d =|[a,c]|`2 by EUCLID:52;
hence thesis by A2,EUCLID:52;
end;
case
A110: x1 in [.0,1/2.] & x2 in [.1/2,1.];
then
A111: f3.r1=(1-2*r1)*|[a,c]|+(2*r1)*|[a,d]| by A32;
A112: 0<=r1 by A110,XXREAL_1:1;
r1<=1/2 by A110,XXREAL_1:1;
then r1*2<=1/2*2 by XREAL_1:64;
then
A113: f3.r1 in LSeg(|[a,c]|,|[a,d]|) by A111,A112;
A114: f3.r2=(1-(2*r2-1))*|[a,d]|+(2*r2-1)*|[b,d]| by A35,A110;
A115: 1/2<=r2 by A110,XXREAL_1:1;
A116: r2<=1 by A110,XXREAL_1:1;
r2*2>=1/2*2 by A115,XREAL_1:64;
then
A117: 2*r2-1>=0 by XREAL_1:48;
2*1>=2*r2 by A116,XREAL_1:64;
then 1+1-1>=2*r2-1 by XREAL_1:9;
then f3.r2 in { (1-lambda)*|[a,d]| + lambda*|[b,d]|
where lambda is Real :
0 <= lambda & lambda <= 1 } by A114,A117;
then f3.r1 in LSeg(|[a,c]|,|[a,d]|) /\ LSeg(|[a,d]|,|[b,d]|)
by A107,A113,XBOOLE_0:def 4;
then
A118: f3.r1= |[a,d]| by A108,TARSKI:def 1;
then (1-2*r1)*|[a,c]|+(2*r1)*|[a,d]|+-(|[a,d]|)=0.TOP-REAL 2
by A111,RLVECT_1:5;
then (1-2*r1)*|[a,c]|+(2*r1)*|[a,d]|+(-1)*|[a,d]|=0.TOP-REAL 2
by RLVECT_1:16;
then (1-2*r1)*|[a,c]|+((2*r1)*|[a,d]|+(-1)*|[a,d]|)=0.TOP-REAL 2
by RLVECT_1:def 3;
then (1-2*r1)*|[a,c]|+((2*r1)+(-1))*|[a,d]|=0.TOP-REAL 2 by
RLVECT_1:def 6;
then (1-2*r1)*|[a,c]|+(-(1-(2*r1)))*|[a,d]|=0.TOP-REAL 2;
then (1-2*r1)*|[a,c]|+-((1-(2*r1))*|[a,d]|)=0.TOP-REAL 2 by RLVECT_1:79
;
then (1-2*r1)*|[a,c]|-((1-(2*r1))*|[a,d]|)=0.TOP-REAL 2;
then (1-2*r1)*(|[a,c]|-(|[a,d]|))=0.TOP-REAL 2 by RLVECT_1:34;
then 1-2*r1=0 or (|[a,c]|-(|[a,d]|))=0.TOP-REAL 2 by RLVECT_1:11;
then 1-2*r1=0 or |[a,c]|=|[a,d]| by RLVECT_1:21;
then
A119: 1-2*r1=0 or c =|[a,d]|`2 by EUCLID:52;
(1-(2*r2-1))*|[a,d]|+(2*r2-1)*|[b,d]|+-(|[a,d]|)=0.TOP-REAL 2
by A107,A114,A118,RLVECT_1:5;
then
(1-(2*r2-1))*|[a,d]|+((2*r2-1))*|[b,d]|+(-1)*|[a,d]|=0.TOP-REAL 2
by RLVECT_1:16;
then (2*r2-1) *|[b,d]|+((1-(2*r2-1))*|[a,d]|+(-1)*|[a,d]|)
=0.TOP-REAL 2
by RLVECT_1:def 3;
then (2*r2-1) *|[b,d]|+((1-(2*r2-1))+(-1))*|[a,d]|=0.TOP-REAL 2
by RLVECT_1:def 6;
then (2*r2-1) *|[b,d]|+(-(2*r2-1))*|[a,d]|=0.TOP-REAL 2;
then (2*r2-1) *|[b,d]|+-((2*r2-1)*|[a,d]|)=0.TOP-REAL 2 by RLVECT_1:79;
then (2*r2-1) *|[b,d]|-((2*r2-1)*|[a,d]|)=0.TOP-REAL 2;
then (2*r2-1) *(|[b,d]|-(|[a,d]|))=0.TOP-REAL 2 by RLVECT_1:34;
then (2*r2-1)=0 or (|[b,d]|-(|[a,d]|))=0.TOP-REAL 2 by RLVECT_1:11;
then (2*r2-1)=0 or |[b,d]|=|[a,d]| by RLVECT_1:21;
then (2*r2-1)=0 or b =|[a,d]|`1 by EUCLID:52;
hence thesis by A1,A2,A119,EUCLID:52;
end;
case
A120: x1 in [.1/2,1.] & x2 in [.0,1/2.];
then
A121: f3.r2=(1-2*r2)*|[a,c]|+(2*r2)*|[a,d]| by A32;
A122: 0<=r2 by A120,XXREAL_1:1;
r2<=1/2 by A120,XXREAL_1:1;
then r2*2<=1/2*2 by XREAL_1:64;
then
A123: f3.r2 in LSeg(|[a,c]|,|[a,d]|) by A121,A122;
A124: f3.r1=(1-(2*r1-1))*|[a,d]|+(2*r1-1)*|[b,d]| by A35,A120;
A125: 1/2<=r1 by A120,XXREAL_1:1;
A126: r1<=1 by A120,XXREAL_1:1;
r1*2>=1/2*2 by A125,XREAL_1:64;
then
A127: 2*r1-1>=0 by XREAL_1:48;
2*1>=2*r1 by A126,XREAL_1:64;
then 1+1-1>=2*r1-1 by XREAL_1:9;
then f3.r1 in { (1-lambda)*|[a,d]| + lambda*|[b,d]|
where lambda is Real:
0 <= lambda & lambda <= 1 } by A124,A127;
then f3.r2 in LSeg(|[a,c]|,|[a,d]|) /\ LSeg(|[a,d]|,|[b,d]|)
by A107,A123,XBOOLE_0:def 4;
then
A128: f3.r2= |[a,d]| by A108,TARSKI:def 1;
then (1-2*r2)*|[a,c]|+(2*r2)*|[a,d]|+-(|[a,d]|)=0.TOP-REAL 2
by A121,RLVECT_1:5;
then (1-2*r2)*|[a,c]|+(2*r2)*|[a,d]|+(-1)*|[a,d]|=0.TOP-REAL 2
by RLVECT_1:16;
then (1-2*r2)*|[a,c]|+((2*r2)*|[a,d]|+(-1)*|[a,d]|)=0.TOP-REAL 2
by RLVECT_1:def 3;
then (1-2*r2)*|[a,c]|+((2*r2)+(-1))*|[a,d]|=0.TOP-REAL 2 by
RLVECT_1:def 6;
then (1-2*r2)*|[a,c]|+(-(1-(2*r2)))*|[a,d]|=0.TOP-REAL 2;
then (1-2*r2)*|[a,c]|+-((1-(2*r2))*|[a,d]|)=0.TOP-REAL 2 by RLVECT_1:79
;
then (1-2*r2)*|[a,c]|-((1-(2*r2))*|[a,d]|)=0.TOP-REAL 2;
then (1-2*r2)*(|[a,c]|-(|[a,d]|))=0.TOP-REAL 2 by RLVECT_1:34;
then 1-2*r2=0 or (|[a,c]|-(|[a,d]|))=0.TOP-REAL 2 by RLVECT_1:11;
then 1-2*r2=0 or |[a,c]|=|[a,d]| by RLVECT_1:21;
then
A129: 1-2*r2=0 or c =|[a,d]|`2 by EUCLID:52;
(1-(2*r1-1))*|[a,d]|+(2*r1-1)*|[b,d]|+-(|[a,d]|)=0.TOP-REAL 2
by A107,A124,A128,RLVECT_1:5;
then
(1-(2*r1-1))*|[a,d]|+((2*r1-1))*|[b,d]|+(-1)*|[a,d]|=0.TOP-REAL 2
by RLVECT_1:16;
then (2*r1-1) *|[b,d]|+((1-(2*r1-1))*|[a,d]|+(-1)*|[a,d]|)
=0.TOP-REAL 2
by RLVECT_1:def 3;
then (2*r1-1) *|[b,d]|+(-1+(1-(2*r1-1)))*|[a,d]|=0.TOP-REAL 2 by
RLVECT_1:def 6;
then (2*r1-1) *|[b,d]|+(-(2*r1-1))*|[a,d]|=0.TOP-REAL 2;
then (2*r1-1) *|[b,d]|+-((2*r1-1)*|[a,d]|)=0.TOP-REAL 2 by RLVECT_1:79;
then (2*r1-1) *|[b,d]|-((2*r1-1)*|[a,d]|)=0.TOP-REAL 2;
then (2*r1-1) *(|[b,d]|-(|[a,d]|))=0.TOP-REAL 2 by RLVECT_1:34;
then (2*r1-1)=0 or (|[b,d]|-(|[a,d]|))=0.TOP-REAL 2 by RLVECT_1:11;
then (2*r1-1)=0 or |[b,d]|=|[a,d]| by RLVECT_1:21;
then (2*r1-1)=0 or b =|[a,d]|`1 by EUCLID:52;
hence thesis by A1,A2,A129,EUCLID:52;
end;
case
A130: x1 in [.1/2,1.] & x2 in [.1/2,1.];
then f3.r1=(1-(2*r1-1))*|[a,d]|+((2*r1-1))*|[b,d]| by A35;
then (1-(2*r2-1))*|[a,d]|+((2*r2-1))*|[b,d]|
= (1-(2*r1-1))*|[a,d]|+((2*r1-1))*|[b,d]| by A35,A107,A130;
then (1-(2*r2-1))*|[a,d]|+((2*r2-1))*|[b,d]| -((2*r1-1))*|[b,d]|
= (1-(2*r1-1))*|[a,d]| by RLVECT_4:1;
then (1-(2*r2-1))*|[a,d]|+(((2*r2-1))*|[b,d]| -((2*r1-1))*|[b,d]|)
= (1-(2*r1-1))*|[a,d]| by RLVECT_1:def 3;
then (1-(2*r2-1))*|[a,d]|+((2*r2-1)-(2*r1-1))*(|[b,d]|)
= (1-(2*r1-1))*|[a,d]| by RLVECT_1:35;
then ((2*r2-1)-(2*r1-1))*(|[b,d]|)
+((1-(2*r2-1))*|[a,d]|-(1-(2*r1-1))*|[a,d]|)
= (1-(2*r1-1))*|[a,d]|-(1-(2*r1-1))*|[a,d]| by RLVECT_1:def 3;
then ((2*r2-1)-(2*r1-1))*(|[b,d]|)
+((1-(2*r2-1))*|[a,d]|-(1-(2*r1-1))*|[a,d]|) = 0.TOP-REAL 2
by RLVECT_1:5;
then
((2*r2-1)-(2*r1-1))*(|[b,d]|)+((1-(2*r2-1))-(1-(2*r1-1)))*|[a,d]|
= 0.TOP-REAL 2 by RLVECT_1:35;
then ((2*r2-1)-(2*r1-1))*(|[b,d]|)+(-((2*r2-1)-(2*r1-1)))*|[a,d]|
= 0.TOP-REAL 2;
then ((2*r2-1)-(2*r1-1))*(|[b,d]|)+-(((2*r2-1)-(2*r1-1))*|[a,d]|)
= 0.TOP-REAL 2 by RLVECT_1:79;
then ((2*r2-1)-(2*r1-1))*(|[b,d]|)-(((2*r2-1)-(2*r1-1))*|[a,d]|)
= 0.TOP-REAL 2;
then ((2*r2-1)-(2*r1-1))*((|[b,d]|)-(|[a,d]|))
= 0.TOP-REAL 2 by RLVECT_1:34;
then ((2*r2-1)-(2*r1-1))=0 or (|[b,d]|)-(|[a,d]|)=0.TOP-REAL 2
by RLVECT_1:11;
then ((2*r2-1)-(2*r1-1))=0 or |[b,d]|=|[a,d]| by RLVECT_1:21;
then ((2*r2-1)-(2*r1-1))=0 or b =|[a,d]|`1 by EUCLID:52;
hence thesis by A1,EUCLID:52;
end;
end;
hence thesis;
end;
then
A131: f3 is one-to-one by FUNCT_1:def 4;
[#]((TOP-REAL 2)|(Upper_Arc(K))) c= rng f3
proof
let y be object;
assume y in [#]((TOP-REAL 2)|(Upper_Arc(K)));
then
A132: y in Upper_Arc(K) by PRE_TOPC:def 5;
then reconsider q=y as Point of TOP-REAL 2;
A133: Upper_Arc(K)=LSeg(|[a,c]|,|[a,d]|) \/ LSeg(|[a,d]|,|[b,d]|) by A1,A2,Th51
;
now per cases by A132,A133,XBOOLE_0:def 3;
case
A134: q in LSeg(|[a,c]|,|[a,d]|);
then
A135: 0<=((q`2)-c)/(d-c)/2 by A38;
A136: ((q`2)-c)/(d-c)/2<=1 by A38,A134;
A137: f3.(((q`2)-c)/(d-c)/2)=q by A38,A134;
((q`2)-c)/(d-c)/2 in [.0,1.] by A135,A136,XXREAL_1:1;
hence thesis by A14,A137,FUNCT_1:def 3;
end;
case
A138: q in LSeg(|[a,d]|,|[b,d]|);
then
A139: 0<=((q`1)-a)/(b-a)/2+1/2 by A48;
A140: ((q`1)-a)/(b-a)/2+1/2<=1 by A48,A138;
A141: f3.(((q`1)-a)/(b-a)/2+1/2)=q by A48,A138;
((q`1)-a)/(b-a)/2+1/2 in [.0,1.] by A139,A140,XXREAL_1:1;
hence thesis by A14,A141,FUNCT_1:def 3;
end;
end;
hence thesis;
end;
then
A142: rng f3=[#]((TOP-REAL 2)|(Upper_Arc(K)));
I[01] is compact by HEINE:4,TOPMETR:20;
then
A143: f3 is being_homeomorphism
by A95,A104,A131,A142,COMPTS_1:17,JGRAPH_1:45;
rng f3=Upper_Arc(K) by A142,PRE_TOPC:def 5;
hence thesis by A29,A31,A32,A35,A38,A48,A143;
end;
theorem Th54:
for a,b,c,d being Real st a<b & c <d
ex f being Function of I[01],(TOP-REAL 2)|(Lower_Arc rectangle(a,b,c,d))
st f is being_homeomorphism
& f.0=E-max rectangle(a,b,c,d) & f.1=W-min rectangle(a,b,c,d) &
rng f=Lower_Arc rectangle(a,b,c,d) &
(for r being Real st r in [.0,1/2.] holds f.r=(1-2*r)*|[b,d]|+(2*r)*|[b,c]|)&
(for r being Real st r in [.1/2,1.] holds
f.r=(1-(2*r-1))*|[b,c]|+(2*r-1)*|[a,c]|)&
(for p being Point of TOP-REAL 2 st p in LSeg(|[b,d]|,|[b,c]|)
holds 0<=((p`2)-d)/(c-d)/2 & ((p`2)-d)/(c-d)/2<=1
& f.(((p`2)-d)/(c-d)/2)=p)&
for p being Point of TOP-REAL 2 st p in LSeg(|[b,c]|,|[a,c]|)
holds 0<=((p`1)-b)/(a-b)/2+1/2 & ((p`1)-b)/(a-b)/2+1/2<=1
& f.(((p`1)-b)/(a-b)/2+1/2)=p
proof
let a,b,c,d be Real;
set K = rectangle(a,b,c,d);
assume that
A1: a<b and
A2: c <d;
defpred P[object,object] means for r being Real st $1=r holds
(r in [.0,1/2.] implies $2=(1-2*r)*|[b,d]|+(2*r)*|[b,c]|) &
(r in [.1/2,1.] implies $2=(1-(2*r-1))*|[b,c]|+(2*r-1)*|[a,c]|);
A3: [.0,1.]=[.0,1/2.] \/ [.1/2,1.] by XXREAL_1:165;
A4: for x being object st x in [.0,1.] ex y being object st P[x,y]
proof
let x be object;
assume
A5: x in [.0,1.];
now per cases by A3,A5,XBOOLE_0:def 3;
case
A6: x in [.0,1/2.];
then reconsider r=x as Real;
A7: r<=1/2 by A6,XXREAL_1:1;
set y0= (1-2*r)*|[b,d]|+(2*r)*|[b,c]|;
r in [.1/2,1.] implies y0=(1-(2*r-1))*|[b,c]|+(2*r-1)*|[a,c]|
proof
assume r in [.1/2,1.];
then 1/2 <=r by XXREAL_1:1;
then
A8: r=1/2 by A7,XXREAL_0:1;
then
A9: y0= (0)*|[b,d]|+|[b,c]| by RLVECT_1:def 8
.= (0.TOP-REAL 2) + |[b,c]| by RLVECT_1:10
.= |[b,c]| by RLVECT_1:4;
(1-(2*r-1))*|[b,c]|+(2*r-1)*|[a,c]|
= (1)*|[b,c]|+0.TOP-REAL 2 by A8,RLVECT_1:10
.= |[b,c]|+0.TOP-REAL 2 by RLVECT_1:def 8
.= |[b,c]| by RLVECT_1:4;
hence thesis by A9;
end;
then for r2 being Real st x=r2 holds
(r2 in [.0,1/2.] implies y0=(1-2*r2)*|[b,d]|+(2*r2)*|[b,c]|) &
(r2 in [.1/2,1.] implies y0=(1-(2*r2-1))*|[b,c]|+(2*r2-1)*|[a,c]|);
hence thesis;
end;
case
A10: x in [.1/2,1.];
then reconsider r=x as Real;
A11: 1/2<=r by A10,XXREAL_1:1;
set y0= (1-(2*r-1))*|[b,c]|+(2*r-1)*|[a,c]|;
r in [.0,1/2.] implies y0=(1-2*r)*|[b,d]|+(2*r)*|[b,c]|
proof
assume r in [.0,1/2.];
then r<=1/2 by XXREAL_1:1;
then
A12: r=1/2 by A11,XXREAL_0:1;
then
A13: y0= |[b,c]|+(0)*|[a,c]| by RLVECT_1:def 8
.= |[b,c]|+(0.TOP-REAL 2) by RLVECT_1:10
.= |[b,c]| by RLVECT_1:4;
(1-2*r)*|[b,d]|+(2*r)*|[b,c]|
= 0.TOP-REAL 2+(1)*|[b,c]| by A12,RLVECT_1:10
.= 0.TOP-REAL 2+|[b,c]| by RLVECT_1:def 8
.= |[b,c]| by RLVECT_1:4;
hence thesis by A13;
end;
then for r2 being Real st x=r2 holds
(r2 in [.0,1/2.] implies y0=(1-2*r2)*|[b,d]|+(2*r2)*|[b,c]|) &
(r2 in [.1/2,1.] implies y0=(1-(2*r2-1))*|[b,c]|+(2*r2-1)*|[a,c]|);
hence thesis;
end;
end;
hence thesis;
end;
ex f2 being Function st dom f2 = [.0,1.] &
for x being object st x in [.0,1.] holds P[x,f2.x] from CLASSES1:sch 1(A4);
then consider f2 being Function such that
A14: dom f2 = [.0,1.] and
A15: for x being object st x in [.0,1.] holds P[x,f2.x];
rng f2 c= the carrier of (TOP-REAL 2)|(Lower_Arc(K))
proof
let y be object;
assume y in rng f2;
then consider x being object such that
A16: x in dom f2 and
A17: y=f2.x by FUNCT_1:def 3;
now per cases by A3,A14,A16,XBOOLE_0:def 3;
case
A18: x in [.0,1/2.];
then reconsider r=x as Real;
A19: 0<=r by A18,XXREAL_1:1;
r<=1/2 by A18,XXREAL_1:1;
then
A20: r*2<=1/2*2 by XREAL_1:64;
f2.x= (1-2*r)*|[b,d]|+(2*r)*|[b,c]| by A14,A15,A16,A18;
then
A21: y in LSeg(|[b,d]|,|[b,c]|) by A17,A19,A20;
Lower_Arc(K)= LSeg(|[a,c]|,|[b,c]|) \/ LSeg(|[b,c]|,|[b,d]|)
by A1,A2,Th52;
then y in Lower_Arc(K) by A21,XBOOLE_0:def 3;
hence thesis by PRE_TOPC:8;
end;
case
A22: x in [.1/2,1.];
then reconsider r=x as Real;
A23: 1/2<=r by A22,XXREAL_1:1;
A24: r<=1 by A22,XXREAL_1:1;
r*2>=1/2*2 by A23,XREAL_1:64;
then
A25: 2*r-1>=0 by XREAL_1:48;
2*1>=2*r by A24,XREAL_1:64;
then
A26: 1+1-1>=2*r-1 by XREAL_1:9;
f2.x= (1-(2*r-1))*|[b,c]|+(2*r-1)*|[a,c]| by A14,A15,A16,A22;
then
A27: y in LSeg(|[b,c]|,|[a,c]|) by A17,A25,A26;
Lower_Arc(K)= LSeg(|[a,c]|,|[b,c]|) \/ LSeg(|[b,c]|,|[b,d]|)
by A1,A2,Th52;
then y in Lower_Arc(K) by A27,XBOOLE_0:def 3;
hence thesis by PRE_TOPC:8;
end;
end;
hence thesis;
end;
then reconsider f3=f2 as Function of I[01],(TOP-REAL 2)|(Lower_Arc(K))
by A14,BORSUK_1:40,FUNCT_2:2;
A28: 0 in [.0,1.] by XXREAL_1:1;
0 in [.0,1/2.] by XXREAL_1:1;
then
A29: f3.0= (1-2*0)*|[b,d]|+(2*0)*|[b,c]| by A15,A28
.= (1)*|[b,d]|+0.TOP-REAL 2 by RLVECT_1:10
.= |[b,d]|+0.TOP-REAL 2 by RLVECT_1:def 8
.= |[b,d]| by RLVECT_1:4
.= E-max(K) by A1,A2,Th46;
A30: 1 in [.0,1.] by XXREAL_1:1;
1 in [.1/2,1.] by XXREAL_1:1;
then
A31: f3.1= (1-(2*1-1))*|[b,c]|+(2*1-1)*|[a,c]| by A15,A30
.= (0)*|[b,c]|+|[a,c]| by RLVECT_1:def 8
.= (0.TOP-REAL 2) + |[a,c]| by RLVECT_1:10
.= |[a,c]| by RLVECT_1:4
.= W-min(K) by A1,A2,Th46;
A32: for r being Real st r in [.0,1/2.] holds
f3.r=(1-2*r)*|[b,d]|+(2*r)*|[b,c]|
proof
let r be Real;
assume
A33: r in [.0,1/2.];
then
A34: 0<=r by XXREAL_1:1;
r<=1/2 by A33,XXREAL_1:1;
then r<=1 by XXREAL_0:2;
then r in [.0,1.] by A34,XXREAL_1:1;
hence thesis by A15,A33;
end;
A35: for r being Real st r in [.1/2,1.] holds
f3.r=(1-(2*r-1))*|[b,c]|+(2*r-1)*|[a,c]|
proof
let r be Real;
assume
A36: r in [.1/2,1.];
then
A37: 1/2<=r by XXREAL_1:1;
r<=1 by A36,XXREAL_1:1;
then r in [.0,1.] by A37,XXREAL_1:1;
hence thesis by A15,A36;
end;
A38: for p being Point of TOP-REAL 2 st p in LSeg(|[b,d]|,|[b,c]|)
holds 0<=((p`2)-d)/(c-d)/2 & ((p`2)-d)/(c-d)/2<=1
& f3.(((p`2)-d)/(c-d)/2)=p
proof
let p be Point of TOP-REAL 2;
assume
A39: p in LSeg(|[b,d]|,|[b,c]|);
A40: (|[b,d]|)`2= d by EUCLID:52;
A41: (|[b,c]|)`2= c by EUCLID:52;
then
A42: c <=p`2 by A2,A39,A40,TOPREAL1:4;
A43: p`2<=d by A2,A39,A40,A41,TOPREAL1:4;
d-c>0 by A2,XREAL_1:50;
then
A44: -(d-c)< -0 by XREAL_1:24;
d-(p`2) >=0 by A43,XREAL_1:48;
then
A45: -(d-(p`2)) <= -0;
(p`2) -d >=c-d by A42,XREAL_1:9;
then ((p`2) -d)/(c-d) <=(c-d)/(c-d) by A44,XREAL_1:73;
then ((p`2) -d)/(c-d) <=1 by A44,XCMPLX_1:60;
then
A46: ((p`2) -d)/(c-d)/2 <=1/2 by XREAL_1:72;
set r=((p`2)-d)/(c-d)/2;
r in [.0,1/2.] by A44,A45,A46,XXREAL_1:1;
then f3.(((p`2)-d)/(c-d)/2)=(1-2*r)*|[b,d]|+(2*r)*|[b,c]| by A32
.=|[(1-2*r)*b,(1-2*r)*d]|+(2*r)*|[b,c]| by EUCLID:58
.=|[(1-2*r)*b,(1-2*r)*d]|+|[(2*r)*b,(2*r)*c]| by EUCLID:58
.=|[1*b-(2*r)*b+(2*r)*b,(1-2*r)*d+(2*r)*c]| by EUCLID:56
.=|[b,1*d+(((p`2)-d)/(c-d))*(c-d)]|
.=|[b,1*d+((p`2)-d)]| by A44,XCMPLX_1:87
.=|[p`1,p`2]| by A39,TOPREAL3:11
.= p by EUCLID:53;
hence thesis by A44,A45,A46,XXREAL_0:2;
end;
A47: for p being Point of TOP-REAL 2 st p in LSeg(|[b,c]|,|[a,c]|)
holds 0<=((p`1)-b)/(a-b)/2+1/2 & ((p`1)-b)/(a-b)/2+1/2<=1
& f3.(((p`1)-b)/(a-b)/2+1/2)=p
proof
let p be Point of TOP-REAL 2;
assume
A48: p in LSeg(|[b,c]|,|[a,c]|);
A49: (|[b,c]|)`1= b by EUCLID:52;
A50: (|[a,c]|)`1= a by EUCLID:52;
then
A51: a <=p`1 by A1,A48,A49,TOPREAL1:3;
A52: p`1<=b by A1,A48,A49,A50,TOPREAL1:3;
b-a>0 by A1,XREAL_1:50;
then
A53: -(b-a)< -0 by XREAL_1:24;
b-(p`1) >=0 by A52,XREAL_1:48;
then
A54: -(b-(p`1)) <= -0;
then
A55: ((p`1) -b)/(a-b)/2+1/2 >=0+1/2 by A53,XREAL_1:7;
(p`1) -b >=a-b by A51,XREAL_1:9;
then ((p`1) -b)/(a-b) <=(a-b)/(a-b) by A53,XREAL_1:73;
then ((p`1) -b)/(a-b) <=1 by A53,XCMPLX_1:60;
then ((p`1) -b)/(a-b)/2 <=1/2 by XREAL_1:72;
then
A56: ((p`1) -b)/(a-b)/2+1/2 <=1/2+1/2 by XREAL_1:7;
set r=((p`1)-b)/(a-b)/2+1/2;
r in [.1/2,1.] by A55,A56,XXREAL_1:1;
then f3.(((p`1)-b)/(a-b)/2+1/2)=(1-(2*r-1))*|[b,c]|+(2*r-1)*|[a,c]| by A35
.=|[(1-(2*r-1))*b,(1-(2*r-1))*c]|+((2*r-1))*|[a,c]| by EUCLID:58
.=|[(1-(2*r-1))*b,(1-(2*r-1))*c]|+|[((2*r-1))*a,((2*r-1))*c]|
by EUCLID:58
.=|[(1-(2*r-1))*b+((2*r-1))*a,1*c-(2*r-1)*c+((2*r-1))*c]| by EUCLID:56
.=|[1*b+(((p`1)-b)/(a-b))*(a-b),c]|
.=|[1*b+((p`1)-b),c]| by A53,XCMPLX_1:87
.=|[p`1,p`2]| by A48,TOPREAL3:12
.= p by EUCLID:53;
hence thesis by A53,A54,A56;
end;
reconsider B00=[.0,1.] as Subset of R^1 by TOPMETR:17;
reconsider B01=B00 as non empty Subset of R^1 by XXREAL_1:1;
I[01]=(R^1)|B01 by TOPMETR:19,20;
then consider h1 being Function of I[01],R^1 such that
A57: for p being Point of I[01] holds h1.p=p and
A58: h1 is continuous by Th6;
consider h2 being Function of I[01],R^1 such that
A59: for p being Point of I[01],r1 being Real st h1.p=r1 holds h2.p
=2*r1 and
A60: h2 is continuous by A58,JGRAPH_2:23;
consider h5 being Function of I[01],R^1 such that
A61: for p being Point of I[01],r1 being Real st h2.p=r1 holds h5.p
=1-r1 and
A62: h5 is continuous by A60,Th8;
consider h3 being Function of I[01],R^1 such that
A63: for p being Point of I[01],r1 being Real st h2.p=r1 holds h3.p
=r1-1 and
A64: h3 is continuous by A60,Th7;
consider h4 being Function of I[01],R^1 such that
A65: for p being Point of I[01],r1 being Real st h3.p=r1 holds h4.p
=1-r1 and
A66: h4 is continuous by A64,Th8;
consider g1 being Function of I[01],TOP-REAL 2 such that
A67: for r being Point of I[01] holds g1.r=(h5.r)*|[b,d]|+(h2.r)*|[b,c]| and
A68: g1 is continuous by A60,A62,Th13;
A69: for r being Point of I[01],s being Real st r=s holds
g1.r=(1-2*s)*|[b,d]|+(2*s)*|[b,c]|
proof
let r be Point of I[01],s be Real;
assume
A70: r=s;
g1.r=(h5.r)*|[b,d]|+(h2.r)*|[b,c]| by A67
.=(1-2*(h1.r))*|[b,d]|+(h2.r)*|[b,c]| by A59,A61
.=(1-2*(h1.r))*|[b,d]|+(2*(h1.r))*|[b,c]| by A59
.=(1-2*s)*|[b,d]|+(2*(h1.r))*|[b,c]| by A57,A70
.=(1-2*s)*|[b,d]|+(2*s)*|[b,c]| by A57,A70;
hence thesis;
end;
consider g2 being Function of I[01],TOP-REAL 2 such that
A71: for r being Point of I[01] holds g2.r=(h4.r)*|[b,c]|+(h3.r)*|[a,c]| and
A72: g2 is continuous by A64,A66,Th13;
A73: for r being Point of I[01],s being Real st r=s holds
g2.r=(1-(2*s-1))*|[b,c]|+(2*s-1)*|[a,c]|
proof
let r be Point of I[01],s be Real;
assume
A74: r=s;
g2.r=(h4.r)*|[b,c]|+(h3.r)*|[a,c]| by A71
.=(1-((h2.r)-1))*|[b,c]|+(h3.r)*|[a,c]| by A63,A65
.=(1-((h2.r)-1))*|[b,c]|+((h2.r)-1)*|[a,c]| by A63
.=(1-(2*(h1.r)-1))*|[b,c]|+((h2.r)-1)*|[a,c]| by A59
.=(1-(2*(h1.r)-1))*|[b,c]|+(2*(h1.r)-1)*|[a,c]| by A59
.=(1-(2*s-1))*|[b,c]|+(2*(h1.r)-1)*|[a,c]| by A57,A74
.=(1-(2*s-1))*|[b,c]|+(2*s-1)*|[a,c]| by A57,A74;
hence thesis;
end;
reconsider B11=[.0,1/2.] as non empty Subset of I[01]
by A3,BORSUK_1:40,XBOOLE_1:7,XXREAL_1:1;
A75: dom (g1|B11)=dom g1 /\ B11 by RELAT_1:61
.= (the carrier of I[01]) /\ B11 by FUNCT_2:def 1
.=B11 by XBOOLE_1:28
.=the carrier of (I[01]|B11) by PRE_TOPC:8;
rng (g1|B11) c= the carrier of TOP-REAL 2;
then reconsider g11=g1|B11 as Function
of I[01]|B11,TOP-REAL 2 by A75,FUNCT_2:2;
A76: TOP-REAL 2 is SubSpace of TOP-REAL 2 by TSEP_1:2;
then
A77: g11 is continuous by A68,BORSUK_4:44;
reconsider B22=[.1/2,1.] as non empty Subset of I[01]
by A3,BORSUK_1:40,XBOOLE_1:7,XXREAL_1:1;
A78: dom (g2|B22)=dom g2 /\ B22 by RELAT_1:61
.= (the carrier of I[01]) /\ B22 by FUNCT_2:def 1
.=B22 by XBOOLE_1:28
.=the carrier of (I[01]|B22) by PRE_TOPC:8;
rng (g2|B22) c= the carrier of TOP-REAL 2;
then reconsider g22=g2|B22 as Function
of I[01]|B22,TOP-REAL 2 by A78,FUNCT_2:2;
A79: g22 is continuous by A72,A76,BORSUK_4:44;
A80: B11=[#](I[01]|B11) by PRE_TOPC:def 5;
A81: B22=[#](I[01]|B22) by PRE_TOPC:def 5;
A82: B11 is closed by Th4;
A83: B22 is closed by Th4;
A84: [#](I[01]|B11) \/ [#](I[01]|B22)=[#]I[01]
by A80,A81,BORSUK_1:40,XXREAL_1:165;
for p being object st p in ([#](I[01]|B11)) /\ ([#](I[01]|B22))
holds g11.p = g22.p
proof
let p be object;
assume
A85: p in ([#](I[01]|B11)) /\ ([#](I[01]|B22));
then
A86: p in [#](I[01]|B11) by XBOOLE_0:def 4;
A87: p in [#](I[01]|B22) by A85;
A88: p in B11 by A86,PRE_TOPC:def 5;
A89: p in B22 by A87,PRE_TOPC:def 5;
reconsider rp=p as Real by A88;
A90: rp<=1/2 by A88,XXREAL_1:1;
rp>=1/2 by A89,XXREAL_1:1;
then rp=1/2 by A90,XXREAL_0:1;
then
A91: 2*rp=1;
thus g11.p=g1.p by A88,FUNCT_1:49
.= (1-1)*|[b,d]|+(1)*|[b,c]| by A69,A88,A91
.=0.TOP-REAL 2 +1*|[b,c]| by RLVECT_1:10
.=(1-0)*|[b,c]| +(1-1)*|[a,c]| by RLVECT_1:10
.=g2.p by A73,A88,A91
.=g22.p by A89,FUNCT_1:49;
end;
then consider h being Function of I[01],TOP-REAL 2 such that
A92: h=g11+*g22 and
A93: h is continuous by A77,A79,A80,A81,A82,A83,A84,JGRAPH_2:1;
A94: dom f3=dom h by Th5;
A95: dom f3=the carrier of I[01] by Th5;
for x being object st x in dom f2 holds f3.x=h.x
proof
let x be object;
assume
A96: x in dom f2;
then reconsider rx=x as Real by A95;
A97: 0<=rx by A94,A96,BORSUK_1:40,XXREAL_1:1;
A98: rx<=1 by A94,A96,BORSUK_1:40,XXREAL_1:1;
per cases;
suppose
A99: rx<1/2;
then
A100: rx in [.0,1/2.] by A97,XXREAL_1:1;
not rx in dom g22 by A81,A99,XXREAL_1:1;
then h.rx=g11.rx by A92,FUNCT_4:11
.=g1.rx by A100,FUNCT_1:49
.=(1-(2*rx))*|[b,d]|+(2*rx)*|[b,c]| by A69,A94,A96
.=f3.rx by A32,A100;
hence thesis;
end;
suppose rx >=1/2;
then
A101: rx in [.1/2,1.] by A98,XXREAL_1:1;
then rx in [#](I[01]|B22) by PRE_TOPC:def 5;
then h.rx=g22.rx by A78,A92,FUNCT_4:13
.=g2.rx by A101,FUNCT_1:49
.=(1-(2*rx-1))*|[b,c]|+(2*rx-1)*|[a,c]| by A73,A94,A96
.=f3.rx by A35,A101;
hence thesis;
end;
end;
then
A102: f2=h by A94,FUNCT_1:2;
for x1,x2 being object st x1 in dom f3 & x2 in dom f3 & f3.x1=f3.x2
holds x1=x2
proof
let x1,x2 be object;
assume that
A103: x1 in dom f3 and
A104: x2 in dom f3 and
A105: f3.x1=f3.x2;
reconsider r1=x1 as Real by A103;
reconsider r2=x2 as Real by A104;
A106: LSeg(|[b,d]|,|[b,c]|) /\ LSeg(|[b,c]|,|[a,c]|) = {|[b,c]|} by A1,A2,Th32;
now per cases by A3,A14,A103,A104,XBOOLE_0:def 3;
case
A107: x1 in [.0,1/2.] & x2 in [.0,1/2.];
then f3.r1=(1-2*r1)*|[b,d]|+(2*r1)*|[b,c]| by A32;
then (1-2*r2)*|[b,d]|+(2*r2)*|[b,c]|= (1-2*r1)*|[b,d]|+(2*r1)*|[b,c]|
by A32,A105,A107;
then (1-2*r2)*|[b,d]|+(2*r2)*|[b,c]| -(2*r1)*|[b,c]|
= (1-2*r1)*|[b,d]| by RLVECT_4:1;
then (1-2*r2)*|[b,d]|+((2*r2)*|[b,c]| -(2*r1)*|[b,c]|)
= (1-2*r1)*|[b,d]| by RLVECT_1:def 3;
then (1-2*r2)*|[b,d]|+(2*r2-2*r1)*(|[b,c]|)
= (1-2*r1)*|[b,d]| by RLVECT_1:35;
then (2*r2-2*r1)*(|[b,c]|)+((1-2*r2)*|[b,d]|-(1-2*r1)*|[b,d]|)
= (1-2*r1)*|[b,d]|-(1-2*r1)*|[b,d]| by RLVECT_1:def 3;
then (2*r2-2*r1)*(|[b,c]|)+((1-2*r2)*|[b,d]|-(1-2*r1)*|[b,d]|)
= 0.TOP-REAL 2 by RLVECT_1:5;
then (2*r2-2*r1)*(|[b,c]|)+((1-2*r2)-(1-2*r1))*|[b,d]|
= 0.TOP-REAL 2 by RLVECT_1:35;
then (2*r2-2*r1)*(|[b,c]|)+(-(2*r2-2*r1))*|[b,d]| = 0.TOP-REAL 2;
then (2*r2-2*r1)*(|[b,c]|)+-((2*r2-2*r1)*|[b,d]|)
= 0.TOP-REAL 2 by RLVECT_1:79;
then (2*r2-2*r1)*(|[b,c]|)-((2*r2-2*r1)*|[b,d]|)
= 0.TOP-REAL 2;
then (2*r2-2*r1)*((|[b,c]|)-(|[b,d]|)) = 0.TOP-REAL 2 by RLVECT_1:34;
then (2*r2-2*r1)=0 or (|[b,c]|)-(|[b,d]|)=0.TOP-REAL 2 by RLVECT_1:11;
then (2*r2-2*r1)=0 or |[b,c]|=|[b,d]| by RLVECT_1:21;
then (2*r2-2*r1)=0 or d =|[b,c]|`2 by EUCLID:52;
hence thesis by A2,EUCLID:52;
end;
case
A108: x1 in [.0,1/2.] & x2 in [.1/2,1.];
then
A109: f3.r1=(1-2*r1)*|[b,d]|+(2*r1)*|[b,c]| by A32;
A110: 0<=r1 by A108,XXREAL_1:1;
r1<=1/2 by A108,XXREAL_1:1;
then r1*2<=1/2*2 by XREAL_1:64;
then
A111: f3.r1 in LSeg(|[b,d]|,|[b,c]|) by A109,A110;
A112: f3.r2=(1-(2*r2-1))*|[b,c]|+(2*r2-1)*|[a,c]| by A35,A108;
A113: 1/2<=r2 by A108,XXREAL_1:1;
A114: r2<=1 by A108,XXREAL_1:1;
r2*2>=1/2*2 by A113,XREAL_1:64;
then
A115: 2*r2-1>=0 by XREAL_1:48;
2*1>=2*r2 by A114,XREAL_1:64;
then 1+1-1>=2*r2-1 by XREAL_1:9;
then f3.r2 in { (1-lambda)*|[b,c]| + lambda*|[a,c]|
where lambda is Real:
0 <= lambda & lambda <= 1 } by A112,A115;
then f3.r1 in LSeg(|[b,d]|,|[b,c]|) /\ LSeg(|[b,c]|,|[a,c]|)
by A105,A111,XBOOLE_0:def 4;
then
A116: f3.r1= |[b,c]| by A106,TARSKI:def 1;
then (1-2*r1)*|[b,d]|+(2*r1)*|[b,c]|+-(|[b,c]|)=0.TOP-REAL 2
by A109,RLVECT_1:5;
then (1-2*r1)*|[b,d]|+(2*r1)*|[b,c]|+(-1)*|[b,c]|=0.TOP-REAL 2
by RLVECT_1:16;
then (1-2*r1)*|[b,d]|+((2*r1)*|[b,c]|+(-1)*|[b,c]|)=0.TOP-REAL 2
by RLVECT_1:def 3;
then (1-2*r1)*|[b,d]|+((2*r1)+(-1))*|[b,c]|=0.TOP-REAL 2 by
RLVECT_1:def 6;
then (1-2*r1)*|[b,d]|+(-(1-(2*r1)))*|[b,c]|=0.TOP-REAL 2;
then (1-2*r1)*|[b,d]|+-((1-(2*r1))*|[b,c]|)=0.TOP-REAL 2 by RLVECT_1:79
;
then (1-2*r1)*|[b,d]|-((1-(2*r1))*|[b,c]|)=0.TOP-REAL 2;
then (1-2*r1)*(|[b,d]|-(|[b,c]|))=0.TOP-REAL 2 by RLVECT_1:34;
then 1-2*r1=0 or (|[b,d]|-(|[b,c]|))=0.TOP-REAL 2 by RLVECT_1:11;
then 1-2*r1=0 or |[b,d]|=|[b,c]| by RLVECT_1:21;
then
A117: 1-2*r1=0 or d =|[b,c]|`2 by EUCLID:52;
(1-(2*r2-1))*|[b,c]|+(2*r2-1)*|[a,c]|+-(|[b,c]|)=0.TOP-REAL 2
by A105,A112,A116,RLVECT_1:5;
then
(1-(2*r2-1))*|[b,c]|+((2*r2-1))*|[a,c]|+(-1)*|[b,c]|=0.TOP-REAL 2
by RLVECT_1:16;
then (2*r2-1) *|[a,c]|+((1-(2*r2-1))*|[b,c]|+(-1)*|[b,c]|)
=0.TOP-REAL 2
by RLVECT_1:def 3;
then (2*r2-1) *|[a,c]|+((1-(2*r2-1))+(-1))*|[b,c]|=0.TOP-REAL 2
by RLVECT_1:def 6;
then (2*r2-1) *|[a,c]|+(-(2*r2-1))*|[b,c]|=0.TOP-REAL 2;
then (2*r2-1) *|[a,c]|+-((2*r2-1)*|[b,c]|)=0.TOP-REAL 2 by RLVECT_1:79;
then (2*r2-1) *|[a,c]|-((2*r2-1)*|[b,c]|)=0.TOP-REAL 2;
then (2*r2-1) *(|[a,c]|-(|[b,c]|))=0.TOP-REAL 2 by RLVECT_1:34;
then (2*r2-1)=0 or (|[a,c]|-(|[b,c]|))=0.TOP-REAL 2 by RLVECT_1:11;
then (2*r2-1)=0 or |[a,c]|=|[b,c]| by RLVECT_1:21;
then (2*r2-1)=0 or a =|[b,c]|`1 by EUCLID:52;
hence thesis by A1,A2,A117,EUCLID:52;
end;
case
A118: x1 in [.1/2,1.] & x2 in [.0,1/2.];
then
A119: f3.r2=(1-2*r2)*|[b,d]|+(2*r2)*|[b,c]| by A32;
A120: 0<=r2 by A118,XXREAL_1:1;
r2<=1/2 by A118,XXREAL_1:1;
then r2*2<=1/2*2 by XREAL_1:64;
then
A121: f3.r2 in LSeg(|[b,d]|,|[b,c]|) by A119,A120;
A122: f3.r1=(1-(2*r1-1))*|[b,c]|+(2*r1-1)*|[a,c]| by A35,A118;
A123: 1/2<=r1 by A118,XXREAL_1:1;
A124: r1<=1 by A118,XXREAL_1:1;
r1*2>=1/2*2 by A123,XREAL_1:64;
then
A125: 2*r1-1>=0 by XREAL_1:48;
2*1>=2*r1 by A124,XREAL_1:64;
then 1+1-1>=2*r1-1 by XREAL_1:9;
then f3.r1 in { (1-lambda)*|[b,c]| + lambda*|[a,c]|
where lambda is Real:
0 <= lambda & lambda <= 1 } by A122,A125;
then f3.r2 in LSeg(|[b,d]|,|[b,c]|) /\ LSeg(|[b,c]|,|[a,c]|)
by A105,A121,XBOOLE_0:def 4;
then
A126: f3.r2= |[b,c]| by A106,TARSKI:def 1;
then (1-2*r2)*|[b,d]|+(2*r2)*|[b,c]|+-(|[b,c]|)=0.TOP-REAL 2
by A119,RLVECT_1:5;
then (1-2*r2)*|[b,d]|+(2*r2)*|[b,c]|+(-1)*|[b,c]|=0.TOP-REAL 2
by RLVECT_1:16;
then (1-2*r2)*|[b,d]|+((2*r2)*|[b,c]|+(-1)*|[b,c]|)=0.TOP-REAL 2
by RLVECT_1:def 3;
then (1-2*r2)*|[b,d]|+((2*r2)+(-1))*|[b,c]|=0.TOP-REAL 2 by
RLVECT_1:def 6;
then (1-2*r2)*|[b,d]|+(-(1-(2*r2)))*|[b,c]|=0.TOP-REAL 2;
then (1-2*r2)*|[b,d]|+-((1-(2*r2))*|[b,c]|)=0.TOP-REAL 2 by RLVECT_1:79
;
then (1-2*r2)*|[b,d]|-((1-(2*r2))*|[b,c]|)=0.TOP-REAL 2;
then (1-2*r2)*(|[b,d]|-(|[b,c]|))=0.TOP-REAL 2 by RLVECT_1:34;
then 1-2*r2=0 or (|[b,d]|-(|[b,c]|))=0.TOP-REAL 2 by RLVECT_1:11;
then 1-2*r2=0 or |[b,d]|=|[b,c]| by RLVECT_1:21;
then
A127: 1-2*r2=0 or d =|[b,c]|`2 by EUCLID:52;
(1-(2*r1-1))*|[b,c]|+(2*r1-1)*|[a,c]|+-(|[b,c]|)=0.TOP-REAL 2
by A105,A122,A126,RLVECT_1:5;
then
(1-(2*r1-1))*|[b,c]|+((2*r1-1))*|[a,c]|+(-1)*|[b,c]|=0.TOP-REAL 2
by RLVECT_1:16;
then (2*r1-1) *|[a,c]|+((1-(2*r1-1))*|[b,c]|+(-1)*|[b,c]|)
=0.TOP-REAL 2
by RLVECT_1:def 3;
then (2*r1-1) *|[a,c]|+((1-(2*r1-1))+(-1))*|[b,c]|=0.TOP-REAL 2
by RLVECT_1:def 6;
then (2*r1-1) *|[a,c]|+(-(2*r1-1))*|[b,c]|=0.TOP-REAL 2;
then (2*r1-1) *|[a,c]|+-((2*r1-1)*|[b,c]|)=0.TOP-REAL 2 by RLVECT_1:79;
then (2*r1-1) *|[a,c]|-((2*r1-1)*|[b,c]|)=0.TOP-REAL 2;
then (2*r1-1) *(|[a,c]|-(|[b,c]|))=0.TOP-REAL 2 by RLVECT_1:34;
then (2*r1-1)=0 or (|[a,c]|-(|[b,c]|))=0.TOP-REAL 2 by RLVECT_1:11;
then (2*r1-1)=0 or |[a,c]|=|[b,c]| by RLVECT_1:21;
then (2*r1-1)=0 or a =|[b,c]|`1 by EUCLID:52;
hence thesis by A1,A2,A127,EUCLID:52;
end;
case
A128: x1 in [.1/2,1.] & x2 in [.1/2,1.];
then f3.r1=(1-(2*r1-1))*|[b,c]|+((2*r1-1))*|[a,c]| by A35;
then (1-(2*r2-1))*|[b,c]|+((2*r2-1))*|[a,c]|
= (1-(2*r1-1))*|[b,c]|+((2*r1-1))*|[a,c]| by A35,A105,A128;
then (1-(2*r2-1))*|[b,c]|+((2*r2-1))*|[a,c]| -((2*r1-1))*|[a,c]|
= (1-(2*r1-1))*|[b,c]| by RLVECT_4:1;
then (1-(2*r2-1))*|[b,c]|+(((2*r2-1))*|[a,c]| -((2*r1-1))*|[a,c]|)
= (1-(2*r1-1))*|[b,c]| by RLVECT_1:def 3;
then (1-(2*r2-1))*|[b,c]|+((2*r2-1)-(2*r1-1))*(|[a,c]|)
= (1-(2*r1-1))*|[b,c]| by RLVECT_1:35;
then ((2*r2-1)-(2*r1-1))*(|[a,c]|)
+((1-(2*r2-1))*|[b,c]|-(1-(2*r1-1))*|[b,c]|)
= (1-(2*r1-1))*|[b,c]|-(1-(2*r1-1))*|[b,c]| by RLVECT_1:def 3;
then ((2*r2-1)-(2*r1-1))*(|[a,c]|)
+((1-(2*r2-1))*|[b,c]|-(1-(2*r1-1))*|[b,c]|) = 0.TOP-REAL 2
by RLVECT_1:5;
then
((2*r2-1)-(2*r1-1))*(|[a,c]|)+((1-(2*r2-1))-(1-(2*r1-1)))*|[b,c]|
= 0.TOP-REAL 2 by RLVECT_1:35;
then ((2*r2-1)-(2*r1-1))*(|[a,c]|)+(-((2*r2-1)-(2*r1-1)))*|[b,c]|
= 0.TOP-REAL 2;
then ((2*r2-1)-(2*r1-1))*(|[a,c]|)+-(((2*r2-1)-(2*r1-1))*|[b,c]|)
= 0.TOP-REAL 2 by RLVECT_1:79;
then ((2*r2-1)-(2*r1-1))*(|[a,c]|)-(((2*r2-1)-(2*r1-1))*|[b,c]|)
= 0.TOP-REAL 2;
then ((2*r2-1)-(2*r1-1))*((|[a,c]|)-(|[b,c]|))
= 0.TOP-REAL 2 by RLVECT_1:34;
then ((2*r2-1)-(2*r1-1))=0 or (|[a,c]|)-(|[b,c]|)=0.TOP-REAL 2
by RLVECT_1:11;
then ((2*r2-1)-(2*r1-1))=0 or |[a,c]|=|[b,c]| by RLVECT_1:21;
then ((2*r2-1)-(2*r1-1))=0 or a =|[b,c]|`1 by EUCLID:52;
hence thesis by A1,EUCLID:52;
end;
end;
hence thesis;
end;
then
A129: f3 is one-to-one by FUNCT_1:def 4;
[#]((TOP-REAL 2)|(Lower_Arc(K))) c= rng f3
proof
let y be object;
assume y in [#]((TOP-REAL 2)|(Lower_Arc(K)));
then
A130: y in Lower_Arc(K) by PRE_TOPC:def 5;
then reconsider q=y as Point of TOP-REAL 2;
A131: Lower_Arc(K)=LSeg(|[b,d]|,|[b,c]|) \/ LSeg(|[b,c]|,|[a,c]|) by A1,A2,Th52
;
now per cases by A130,A131,XBOOLE_0:def 3;
case
A132: q in LSeg(|[b,d]|,|[b,c]|);
then
A133: 0<=((q`2)-d)/(c-d)/2 by A38;
A134: ((q`2)-d)/(c-d)/2<=1 by A38,A132;
A135: f3.(((q`2)-d)/(c-d)/2)=q by A38,A132;
((q`2)-d)/(c-d)/2 in [.0,1.] by A133,A134,XXREAL_1:1;
hence thesis by A14,A135,FUNCT_1:def 3;
end;
case
A136: q in LSeg(|[b,c]|,|[a,c]|);
then
A137: 0<=((q`1)-b)/(a-b)/2+1/2 by A47;
A138: ((q`1)-b)/(a-b)/2+1/2<=1 by A47,A136;
A139: f3.(((q`1)-b)/(a-b)/2+1/2)=q by A47,A136;
((q`1)-b)/(a-b)/2+1/2 in [.0,1.] by A137,A138,XXREAL_1:1;
hence thesis by A14,A139,FUNCT_1:def 3;
end;
end;
hence thesis;
end;
then
A140: rng f3=[#]((TOP-REAL 2)|(Lower_Arc K));
I[01] is compact by HEINE:4,TOPMETR:20;
then
A141: f3 is being_homeomorphism
by A93,A102,A129,A140,COMPTS_1:17,JGRAPH_1:45;
rng f3=Lower_Arc(K) by A140,PRE_TOPC:def 5;
hence thesis by A29,A31,A32,A35,A38,A47,A141;
end;
theorem Th55:
for a,b,c,d being Real,p1,p2 being Point of TOP-REAL 2 st a<b & c <d
& p1 in LSeg(|[a,c]|,|[a,d]|) & p2 in LSeg(|[a,c]|,|[a,d]|)
holds LE p1,p2,rectangle(a,b,c,d) iff p1`2<=p2`2
proof
let a,b,c,d be Real,p1,p2 be Point of TOP-REAL 2;
set K = rectangle(a,b,c,d);
assume that
A1: a<b and
A2: c <d and
A3: p1 in LSeg(|[a,c]|,|[a,d]|) and
A4: p2 in LSeg(|[a,c]|,|[a,d]|);
A5: K is being_simple_closed_curve by A1,A2,Th50;
A6: p1`1=a by A2,A3,Th1;
A7: c <=p1`2 by A2,A3,Th1;
A8: p2`1=a by A2,A4,Th1;
A9: E-max(K)= |[b,d]| by A1,A2,Th46;
A10: Upper_Arc(K)=LSeg(|[a,c]|,|[a,d]|) \/ LSeg(|[a,d]|,|[b,d]|) by A1,A2,Th51;
then
A11: LSeg(|[a,c]|,|[a,d]|) c= Upper_Arc(K) by XBOOLE_1:7;
A12: Upper_Arc(K) /\ Lower_Arc(K)={W-min(K),E-max(K)} by A5,JORDAN6:def 9;
A13: now
assume p2 in Lower_Arc(K);
then
A14: p2 in Upper_Arc(K) /\ Lower_Arc(K) by A4,A11,XBOOLE_0:def 4;
now
assume p2=E-max(K);
then p2`1=b by A9,EUCLID:52;
hence contradiction by A1,A4,TOPREAL3:11;
end;
hence p2=W-min(K) by A12,A14,TARSKI:def 2;
end;
thus LE p1,p2,K implies p1`2<=p2`2
proof
assume LE p1,p2,K;
then
A15: p1 in Upper_Arc(K) & p2 in Lower_Arc(K)& not p2=W-min(K) or
p1 in Upper_Arc(K) & p2 in Upper_Arc(K) &
LE p1,p2,Upper_Arc(K),W-min(K),E-max(K) or
p1 in Lower_Arc(K) & p2 in Lower_Arc(K)& not p2=W-min(K) &
LE p1,p2,Lower_Arc(K),E-max(K),W-min(K) by JORDAN6:def 10;
consider f being Function of I[01],(TOP-REAL 2)|(Upper_Arc(K)) such that
A16: f is being_homeomorphism and
A17: f.0=W-min(K) and
A18: f.1=E-max(K) and rng f=Upper_Arc(K)
and for r being Real st r in [.0,1/2.] holds f.r=(1-2*r)*|[a,c]|+(2*r)
*|[a,d]|
and for r being Real st r in [.1/2,1.] holds f.r=(1-(2*r-1))*|[a,d]|+(
2*r-1)*|[b,d]| and
A19: for p being Point of TOP-REAL 2 st p in LSeg(|[a,c]|,|[a,d]|)
holds 0<=((p`2)-c)/(d-c)/2 & ((p`2)-c)/(d-c)/2<=1 & f.(((p`2)-c)/(d-c)/2)=p
and for p being Point of TOP-REAL 2 st p in LSeg(|[a,d]|,|[b,d]|)
holds 0<=((p`1)-a)/(b-a)/2+1/2 & ((p`1)-a)/(b-a)/2+1/2<=1 & f.(((p`1)-a)/(b-a)/
2+1/2)=p
by A1,A2,Th53;
reconsider s1=((p1`2)-c)/(d-c)/2,s2=((p2`2)-c)/(d-c)/2 as Real;
A20: f.s1=p1 by A3,A19;
A21: f.s2=p2 by A4,A19;
A22: d-c >0 by A2,XREAL_1:50;
A23: s1<=1 by A3,A19;
A24: 0<=s2 by A4,A19;
s2<=1 by A4,A19;
then s1<=s2 by A13,A15,A16,A17,A18,A20,A21,A23,A24,JORDAN5C:def 3;
then ((p1`2)-c)/(d-c)/2*2<=((p2`2)-c)/(d-c)/2*2 by XREAL_1:64;
then ((p1`2)-c)/(d-c)*(d-c)<= ((p2`2)-c)/(d-c)*(d-c) by A22,XREAL_1:64;
then ((p1`2)-c)<= ((p2`2)-c)/(d-c)*(d-c) by A22,XCMPLX_1:87;
then ((p1`2)-c)<= ((p2`2)-c) by A22,XCMPLX_1:87;
then ((p1`2)-c)+c <= ((p2`2)-c)+c by XREAL_1:7;
hence thesis;
end;
thus p1`2<=p2`2 implies LE p1,p2,K
proof
assume
A25: p1`2<=p2`2;
for g being Function of I[01], (TOP-REAL 2)|Upper_Arc(K),
s1, s2 being Real st g is being_homeomorphism & g.0 = W-min(K) &
g.1 = E-max(K)
& g.s1 = p1 & 0 <= s1 & s1 <= 1 & g.s2 = p2 & 0 <= s2 & s2 <= 1
holds s1 <= s2
proof
let g be Function of I[01], (TOP-REAL 2)|Upper_Arc(K),
s1, s2 be Real;
assume that
A26: g is being_homeomorphism and
A27: g.0 = W-min(K) and g.1 = E-max(K) and
A28: g.s1 = p1 and
A29: 0 <= s1 and
A30: s1 <= 1 and
A31: g.s2 = p2 and
A32: 0 <= s2 and
A33: s2 <= 1;
A34: dom g=the carrier of I[01] by FUNCT_2:def 1;
A35: g is one-to-one by A26,TOPS_2:def 5;
A36: the
carrier of ((TOP-REAL 2)|Upper_Arc(K)) =Upper_Arc(K) by PRE_TOPC:8;
then reconsider g1=g as Function of I[01],TOP-REAL 2 by FUNCT_2:7;
g is continuous by A26,TOPS_2:def 5;
then
A37: g1 is continuous by PRE_TOPC:26;
reconsider h1=proj1 as Function of TOP-REAL 2,R^1 by TOPMETR:17;
reconsider h2=proj2 as Function of TOP-REAL 2,R^1 by TOPMETR:17;
reconsider hh1=h1 as Function of the TopStruct of TOP-REAL 2,R^1;
reconsider hh2=h2 as Function of the TopStruct of TOP-REAL 2,R^1;
A38: the TopStruct of TOP-REAL 2
= (the TopStruct of TOP-REAL 2)|([#](the TopStruct of TOP-REAL 2))
by TSEP_1:3
.= the TopStruct of ((TOP-REAL 2)|([#](TOP-REAL 2))) by PRE_TOPC:36
.= (TOP-REAL 2)|([#](TOP-REAL 2));
then (for p being Point of (TOP-REAL 2)|([#](TOP-REAL 2)) holds
hh1.p=proj1.p) implies hh1 is continuous by JGRAPH_2:29;
then
A39: (for p being Point of (TOP-REAL 2)|([#]TOP-REAL 2)holds
hh1.p=proj1.p) implies h1 is continuous by PRE_TOPC:32;
(for p being Point of (TOP-REAL 2)|([#](TOP-REAL 2)) holds
hh2.p=proj2.p) implies hh2 is continuous by A38,JGRAPH_2:30;
then (for p being Point of (TOP-REAL 2)|([#](TOP-REAL 2)) holds
hh2.p=proj2.p) implies h2 is continuous by PRE_TOPC:32;
then consider h being Function of TOP-REAL 2,R^1 such that
A40: for p being Point of TOP-REAL 2, r1,r2 being Real st hh1.p=r1 &
hh2.p=r2 holds h.p=r1+r2 and
A41: h is continuous by A39,JGRAPH_2:19;
reconsider k=h*g1 as Function of I[01],R^1;
A42: W-min K=|[a,c]| by A1,A2,Th46;
now
assume
A43: s1>s2;
A44: dom g=[.0,1.] by BORSUK_1:40,FUNCT_2:def 1;
0 in [.0,1.] by XXREAL_1:1;
then
A45: k.0=h.(W-min(K)) by A27,A44,FUNCT_1:13
.=h1.(W-min(K))+h2.(W-min(K)) by A40
.=(W-min(K))`1+proj2.(W-min(K)) by PSCOMP_1:def 5
.=(W-min(K))`1+(W-min(K))`2 by PSCOMP_1:def 6
.=a+(W-min(K))`2 by A42,EUCLID:52
.=a+c by A42,EUCLID:52;
s1 in [.0,1.] by A29,A30,XXREAL_1:1;
then
A46: k.s1=h.p1 by A28,A44,FUNCT_1:13
.=h1.p1+h2.p1 by A40
.=p1`1+proj2.p1 by PSCOMP_1:def 5
.=a+p1`2 by A6,PSCOMP_1:def 6;
A47: s2 in [.0,1.] by A32,A33,XXREAL_1:1;
then
A48: k.s2=h.p2 by A31,A44,FUNCT_1:13
.=h1.p2+h2.p2 by A40
.=p2`1+proj2.p2 by PSCOMP_1:def 5
.=a+p2`2 by A8,PSCOMP_1:def 6;
A49: k.0<=k.s1 by A7,A45,A46,XREAL_1:7;
A50: k.s1<=k.s2 by A25,A46,A48,XREAL_1:7;
A51: 0 in [.0,1.] by XXREAL_1:1;
then
A52: [.0,s2.] c= [.0,1.] by A47,XXREAL_2:def 12;
reconsider B=[.0,s2.] as Subset of I[01] by A47,A51,BORSUK_1:40
,XXREAL_2:def 12;
A53: B is connected by A32,A47,A51,BORSUK_1:40,BORSUK_4:24;
A54: 0 in B by A32,XXREAL_1:1;
A55: s2 in B by A32,XXREAL_1:1;
consider xc being Point of I[01] such that
A56: xc in B and
A57: k.xc =k.s1 by A37,A41,A49,A50,A53,A54,A55,TOPREAL5:5;
reconsider rxc=xc as Real;
A58: for x1,x2 being set st x1 in dom k & x2 in dom k &
k.x1=k.x2 holds x1=x2
proof
let x1,x2 be set;
assume that
A59: x1 in dom k and
A60: x2 in dom k and
A61: k.x1=k.x2;
reconsider r1=x1 as Point of I[01] by A59;
reconsider r2=x2 as Point of I[01] by A60;
A62: k.x1=h.(g1.x1) by A59,FUNCT_1:12
.=h1.(g1.r1)+h2.(g1.r1) by A40
.=(g1.r1)`1+proj2.(g1.r1) by PSCOMP_1:def 5
.=(g1.r1)`1+(g1.r1)`2 by PSCOMP_1:def 6;
A63: k.x2=h.(g1.x2) by A60,FUNCT_1:12
.=h1.(g1.r2)+h2.(g1.r2) by A40
.=(g1.r2)`1+proj2.(g1.r2) by PSCOMP_1:def 5
.=(g1.r2)`1+(g1.r2)`2 by PSCOMP_1:def 6;
A64: g.r1 in Upper_Arc(K) by A36;
A65: g.r2 in Upper_Arc(K) by A36;
reconsider gr1=g.r1 as Point of TOP-REAL 2 by A64;
reconsider gr2=g.r2 as Point of TOP-REAL 2 by A65;
now per cases by A10,A36,XBOOLE_0:def 3;
case
A66: g
.r1 in LSeg(|[a,c]|,|[a,d]|) & g.r2 in LSeg(|[a,c]|,|[a,d]|);
then
A67: (gr1)`1=a by A2,Th1;
(gr2)`1=a by A2,A66,Th1;
then |[(gr1)`1,(gr1)`2]|=g.r2 by A61,A62,A63,A67,EUCLID:53;
then g.r1=g.r2 by EUCLID:53;
hence thesis by A34,A35,FUNCT_1:def 4;
end;
case
A68: g
.r1 in LSeg(|[a,c]|,|[a,d]|) & g.r2 in LSeg(|[a,d]|,|[b,d]|);
then
A69: (gr1)`1=a by A2,Th1;
A70: (gr1)`2 <=d by A2,A68,Th1;
A71: (gr2)`2=d by A1,A68,Th3;
A72: a <=(gr2)`1 by A1,A68,Th3;
A73: a+(gr1)`2=(gr2)`1 +d by A1,A61,A62,A63,A68,A69,Th3;
A74: now
assume a<>gr2`1;
then a<gr2`1 by A72,XXREAL_0:1;
hence contradiction by A70,A73,XREAL_1:8;
end;
now
assume gr1`2<>d;
then d>gr1`2 by A70,XXREAL_0:1;
hence contradiction by A61,A62,A63,A69,A71,A72,XREAL_1:8;
end;
then |[(gr1)`1,(gr1)`2]|=g.r2 by A69,A71,A74,EUCLID:53;
then g.r1=g.r2 by EUCLID:53;
hence thesis by A34,A35,FUNCT_1:def 4;
end;
case
A75: g
.r1 in LSeg(|[a,d]|,|[b,d]|) & g.r2 in LSeg(|[a,c]|,|[a,d]|);
then
A76: (gr2)`1=a by A2,Th1;
A77: (gr2)`2 <=d by A2,A75,Th1;
A78: (gr1)`2=d by A1,A75,Th3;
A79: a <=(gr1)`1 by A1,A75,Th3;
A80: a+(gr2)`2=(gr1)`1 +d by A1,A61,A62,A63,A75,A76,Th3;
A81: now
assume a<>gr1`1;
then a<gr1`1 by A79,XXREAL_0:1;
hence contradiction by A77,A80,XREAL_1:8;
end;
now
assume gr2`2<>d;
then d>gr2`2 by A77,XXREAL_0:1;
hence contradiction by A61,A62,A63,A76,A78,A79,XREAL_1:8;
end;
then |[(gr2)`1,(gr2)`2]|=g.r1 by A76,A78,A81,EUCLID:53;
then g.r1=g.r2 by EUCLID:53;
hence thesis by A34,A35,FUNCT_1:def 4;
end;
case
A82: g
.r1 in LSeg(|[a,d]|,|[b,d]|) & g.r2 in LSeg(|[a,d]|,|[b,d]|);
then
A83: (gr1)`2=d by A1,Th3;
(gr2)`2=d by A1,A82,Th3;
then |[(gr1)`1,(gr1)`2]|=g.r2 by A61,A62,A63,A83,EUCLID:53;
then g.r1=g.r2 by EUCLID:53;
hence thesis by A34,A35,FUNCT_1:def 4;
end;
end;
hence thesis;
end;
A84: dom k=[.0,1.] by BORSUK_1:40,FUNCT_2:def 1;
then s1 in dom k by A29,A30,XXREAL_1:1;
then rxc=s1 by A52,A56,A57,A58,A84;
hence contradiction by A43,A56,XXREAL_1:1;
end;
hence thesis;
end;
then LE p1,p2,Upper_Arc(K),W-min(K),E-max(K) by A3,A4,A11,JORDAN5C:def 3;
hence thesis by A3,A4,A11,JORDAN6:def 10;
end;
end;
theorem Th56:
for a,b,c,d being Real,p1,p2 being Point of TOP-REAL 2
st a < b & c < d & p1 in LSeg(|[a,d]|,|[b,d]|) & p2 in LSeg(|[a,d]|,|[b,d]|)
holds LE p1,p2,rectangle(a,b,c,d) iff p1`1<=p2`1
proof
let a,b,c,d be Real,p1,p2 be Point of TOP-REAL 2;
set K = rectangle(a,b,c,d);
assume that
A1: a<b and
A2: c <d and
A3: p1 in LSeg(|[a,d]|,|[b,d]|) and
A4: p2 in LSeg(|[a,d]|,|[b,d]|);
A5: K is being_simple_closed_curve by A1,A2,Th50;
A6: p1`2=d by A1,A3,Th3;
A7: a <=p1`1 by A1,A3,Th3;
A8: p1`1 <= b by A1,A3,Th3;
A9: p2`2=d by A1,A4,Th3;
A10: W-min(K)= |[a,c]| by A1,A2,Th46;
A11: E-max(K)= |[b,d]| by A1,A2,Th46;
A12: Upper_Arc(K)=LSeg(|[a,c]|,|[a,d]|) \/ LSeg(|[a,d]|,|[b,d]|) by A1,A2,Th51;
then
A13: LSeg(|[a,d]|,|[b,d]|) c= Upper_Arc(K) by XBOOLE_1:7;
A14: Upper_Arc(K) /\ Lower_Arc(K)={W-min(K),E-max(K)} by A5,JORDAN6:def 9;
A15: now
assume p2 in Lower_Arc(K);
then
A16: p2 in Upper_Arc(K) /\ Lower_Arc(K) by A4,A13,XBOOLE_0:def 4;
now
assume p2=W-min(K);
then p2`2=c by A10,EUCLID:52;
hence contradiction by A2,A4,TOPREAL3:12;
end;
hence p2=E-max(K) by A14,A16,TARSKI:def 2;
end;
thus LE p1,p2,K implies p1`1<=p2`1
proof
assume LE p1,p2,K;
then
A17: p1 in Upper_Arc(K) & p2 in Lower_Arc(K)& not p2=W-min(K) or
p1 in Upper_Arc(K) & p2 in Upper_Arc(K) &
LE p1,p2,Upper_Arc(K),W-min(K),E-max(K) or
p1 in Lower_Arc(K) & p2 in Lower_Arc(K)& not p2=W-min(K) &
LE p1,p2,Lower_Arc(K),E-max(K),W-min(K) by JORDAN6:def 10;
now per cases;
case p2=E-max(K);
hence thesis by A8,A11,EUCLID:52;
end;
case
A18: p2<>E-max(K);
consider f being Function of I[01],(TOP-REAL 2)|(Upper_Arc(K)) such that
A19: f is being_homeomorphism and
A20: f.0=W-min(K) and
A21: f.1=E-max(K) and rng f=Upper_Arc(K)
and for r being Real st r in [.0,1/2.] holds f.r=(1-2*r)*|[a,c]|+(2*r)
*|[a,d]|
and for r being Real st r in [.1/2,1.] holds f.r=(1-(2*r-1))*|[a,d]|+(
2*r-1)*|[b,d]|
and for p being Point of TOP-REAL 2 st p in LSeg(|[a,c]|,|[a,d]|)
holds 0<=((p`2)-c)/(d-c)/2 & ((p`2)-c)/(d-c)/2<=1 & f.(((p`2)-c)/(d-c)/2)=p and
A22: for p being Point of TOP-REAL 2 st p in LSeg(|[a,d]|,|[b,d]|)
holds 0<=((p`1)-a)/(b-a)/2+1/2 & ((p`1)-a)/(b-a)/2+1/2<=1 & f.(((p`1)-a)/(b-a)/
2+1/2)=p
by A1,A2,Th53;
reconsider s1=((p1`1)-a)/(b-a)/2+1/2,s2=((p2`1)-a)/(b-a)/2+1/2 as
Real;
A23: f.s1=p1 by A3,A22;
A24: f.s2=p2 by A4,A22;
A25: b-a >0 by A1,XREAL_1:50;
A26: s1<=1 by A3,A22;
A27: 0<=s2 by A4,A22;
s2<=1 by A4,A22;
then s1<=s2 by A15,A17,A18,A19,A20,A21,A23,A24,A26,A27,JORDAN5C:def 3;
then ((p1`1)-a)/(b-a)/2<= ((p2`1)-a)/(b-a)/2 by XREAL_1:6;
then ((p1`1)-a)/(b-a)/2*2<=((p2`1)-a)/(b-a)/2*2 by XREAL_1:64;
then ((p1`1)-a)/(b-a)*(b-a)<= ((p2`1)-a)/(b-a)*(b-a) by A25,XREAL_1:64;
then ((p1`1)-a)<= ((p2`1)-a)/(b-a)*(b-a) by A25,XCMPLX_1:87;
then ((p1`1)-a)<= ((p2`1)-a) by A25,XCMPLX_1:87;
then ((p1`1)-a)+a <= ((p2`1)-a)+a by XREAL_1:7;
hence thesis;
end;
end;
hence thesis;
end;
thus p1`1<=p2`1 implies LE p1,p2,K
proof
assume
A28: p1`1<=p2`1;
for g being Function of I[01], (TOP-REAL 2)|Upper_Arc(K),
s1, s2 being Real st g is being_homeomorphism & g.0 = W-min(K) &
g.1 = E-max(K)
& g.s1 = p1 & 0 <= s1 & s1 <= 1 & g.s2 = p2 & 0 <= s2 & s2 <= 1
holds s1 <= s2
proof
let g be Function of I[01], (TOP-REAL 2)|Upper_Arc(K),
s1, s2 be Real;
assume that
A29: g is being_homeomorphism and
A30: g.0 = W-min(K) and g.1 = E-max(K) and
A31: g.s1 = p1 and
A32: 0 <= s1 and
A33: s1 <= 1 and
A34: g.s2 = p2 and
A35: 0 <= s2 and
A36: s2 <= 1;
A37: dom g=the carrier of I[01] by FUNCT_2:def 1;
A38: g is one-to-one by A29,TOPS_2:def 5;
A39: the
carrier of ((TOP-REAL 2)|Upper_Arc(K)) =Upper_Arc(K) by PRE_TOPC:8;
then reconsider g1=g as Function of I[01],TOP-REAL 2 by FUNCT_2:7;
g is continuous by A29,TOPS_2:def 5;
then
A40: g1 is continuous by PRE_TOPC:26;
reconsider h1=proj1 as Function of TOP-REAL 2,R^1 by TOPMETR:17;
reconsider h2=proj2 as Function of TOP-REAL 2,R^1 by TOPMETR:17;
reconsider hh1=h1 as Function of the TopStruct of TOP-REAL 2,R^1;
reconsider hh2=h2 as Function of the TopStruct of TOP-REAL 2,R^1;
A41: the TopStruct of TOP-REAL 2
= (the TopStruct of TOP-REAL 2)|([#](the TopStruct of TOP-REAL 2))
by TSEP_1:3
.= the TopStruct of ((TOP-REAL 2)|([#](TOP-REAL 2))) by PRE_TOPC:36
.= (TOP-REAL 2)|([#](TOP-REAL 2));
then (for p being Point of (TOP-REAL 2)|([#](TOP-REAL 2)) holds
hh1.p=proj1.p) implies hh1 is continuous by JGRAPH_2:29;
then
A42: (for p being Point of (TOP-REAL 2)|([#]TOP-REAL 2)holds
hh1.p=proj1.p) implies h1 is continuous by PRE_TOPC:32;
(for p being Point of (TOP-REAL 2)|([#](TOP-REAL 2)) holds
hh2.p=proj2.p) implies hh2 is continuous by A41,JGRAPH_2:30;
then (for p being Point of (TOP-REAL 2)|([#](TOP-REAL 2)) holds
hh2.p=proj2.p) implies h2 is continuous by PRE_TOPC:32;
then consider h being Function of TOP-REAL 2,R^1 such that
A43: for p being Point of TOP-REAL 2, r1,r2 being Real st h1.p=r1 &
h2.p=r2 holds h.p=r1+r2 and
A44: h is continuous by A42,JGRAPH_2:19;
reconsider k=h*g1 as Function of I[01],R^1;
A45: W-min K=|[a,c]| by A1,A2,Th46;
now
assume
A46: s1>s2;
A47: dom g=[.0,1.] by BORSUK_1:40,FUNCT_2:def 1;
0 in [.0,1.] by XXREAL_1:1;
then
A48: k.0=h.(W-min(K)) by A30,A47,FUNCT_1:13
.=h1.(W-min(K))+h2.(W-min(K)) by A43
.=(W-min(K))`1+proj2.(W-min(K)) by PSCOMP_1:def 5
.=(W-min(K))`1+(W-min(K))`2 by PSCOMP_1:def 6
.=(W-min(K))`1+c by A45,EUCLID:52
.=a+c by A45,EUCLID:52;
s1 in [.0,1.] by A32,A33,XXREAL_1:1;
then
A49: k.s1=h.p1 by A31,A47,FUNCT_1:13
.=h1.p1+h2.p1 by A43
.=p1`1+proj2.p1 by PSCOMP_1:def 5
.=p1`1 +d by A6,PSCOMP_1:def 6;
A50: s2 in [.0,1.] by A35,A36,XXREAL_1:1;
then
A51: k.s2=h.p2 by A34,A47,FUNCT_1:13
.=h1.p2+h2.p2 by A43
.=p2`1+proj2.p2 by PSCOMP_1:def 5
.=p2`1 +d by A9,PSCOMP_1:def 6;
A52: k.0<=k.s1 by A2,A7,A48,A49,XREAL_1:7;
A53: k.s1<=k.s2 by A28,A49,A51,XREAL_1:7;
A54: 0 in [.0,1.] by XXREAL_1:1;
then
A55: [.0,s2.] c= [.0,1.] by A50,XXREAL_2:def 12;
reconsider B=[.0,s2.] as Subset of I[01] by A50,A54,BORSUK_1:40
,XXREAL_2:def 12;
A56: B is connected by A35,A50,A54,BORSUK_1:40,BORSUK_4:24;
A57: 0 in B by A35,XXREAL_1:1;
A58: s2 in B by A35,XXREAL_1:1;
consider xc being Point of I[01] such that
A59: xc in B and
A60: k.xc =k.s1 by A40,A44,A52,A53,A56,A57,A58,TOPREAL5:5;
reconsider rxc=xc as Real;
A61: for x1,x2 being set st x1 in dom k & x2 in dom k &
k.x1=k.x2 holds x1=x2
proof
let x1,x2 be set;
assume that
A62: x1 in dom k and
A63: x2 in dom k and
A64: k.x1=k.x2;
reconsider r1=x1 as Point of I[01] by A62;
reconsider r2=x2 as Point of I[01] by A63;
A65: k.x1=h.(g1.x1) by A62,FUNCT_1:12
.=h1.(g1.r1)+h2.(g1.r1) by A43
.=(g1.r1)`1+proj2.(g1.r1) by PSCOMP_1:def 5
.=(g1.r1)`1+(g1.r1)`2 by PSCOMP_1:def 6;
A66: k.x2=h.(g1.x2) by A63,FUNCT_1:12
.=h1.(g1.r2)+h2.(g1.r2) by A43
.=(g1.r2)`1+proj2.(g1.r2) by PSCOMP_1:def 5
.=(g1.r2)`1+(g1.r2)`2 by PSCOMP_1:def 6;
A67: g.r1 in Upper_Arc(K) by A39;
A68: g.r2 in Upper_Arc(K) by A39;
reconsider gr1=g.r1 as Point of TOP-REAL 2 by A67;
reconsider gr2=g.r2 as Point of TOP-REAL 2 by A68;
now per cases by A12,A39,XBOOLE_0:def 3;
case
A69: g
.r1 in LSeg(|[a,c]|,|[a,d]|) & g.r2 in LSeg(|[a,c]|,|[a,d]|);
then
A70: (gr1)`1=a by A2,Th1;
(gr2)`1=a by A2,A69,Th1;
then |[(gr1)`1,(gr1)`2]|=g.r2 by A64,A65,A66,A70,EUCLID:53;
then g.r1=g.r2 by EUCLID:53;
hence thesis by A37,A38,FUNCT_1:def 4;
end;
case
A71: g
.r1 in LSeg(|[a,c]|,|[a,d]|) & g.r2 in LSeg(|[a,d]|,|[b,d]|);
then
A72: (gr1)`1=a by A2,Th1;
A73: (gr1)`2 <=d by A2,A71,Th1;
A74: (gr2)`2=d by A1,A71,Th3;
A75: a <=(gr2)`1 by A1,A71,Th3;
A76: a+(gr1)`2=(gr2)`1 +d by A1,A64,A65,A66,A71,A72,Th3;
A77: now
assume a<>gr2`1;
then a<gr2`1 by A75,XXREAL_0:1;
hence contradiction by A73,A76,XREAL_1:8;
end;
now
assume gr1`2<>d;
then d>gr1`2 by A73,XXREAL_0:1;
hence contradiction by A64,A65,A66,A72,A74,A75,XREAL_1:8;
end;
then |[(gr1)`1,(gr1)`2]|=g.r2 by A72,A74,A77,EUCLID:53;
then g.r1=g.r2 by EUCLID:53;
hence thesis by A37,A38,FUNCT_1:def 4;
end;
case
A78: g
.r1 in LSeg(|[a,d]|,|[b,d]|) & g.r2 in LSeg(|[a,c]|,|[a,d]|);
then
A79: (gr2)`1=a by A2,Th1;
A80: (gr2)`2 <=d by A2,A78,Th1;
A81: (gr1)`2=d by A1,A78,Th3;
A82: a <=(gr1)`1 by A1,A78,Th3;
A83: a+(gr2)`2=(gr1)`1 +d by A1,A64,A65,A66,A78,A79,Th3;
A84: now
assume a<>gr1`1;
then a<gr1`1 by A82,XXREAL_0:1;
hence contradiction by A80,A83,XREAL_1:8;
end;
now
assume gr2`2<>d;
then d>gr2`2 by A80,XXREAL_0:1;
hence contradiction by A64,A65,A66,A79,A81,A82,XREAL_1:8;
end;
then |[(gr2)`1,(gr2)`2]|=g.r1 by A79,A81,A84,EUCLID:53;
then g.r1=g.r2 by EUCLID:53;
hence thesis by A37,A38,FUNCT_1:def 4;
end;
case
A85: g
.r1 in LSeg(|[a,d]|,|[b,d]|) & g.r2 in LSeg(|[a,d]|,|[b,d]|);
then
A86: (gr1)`2=d by A1,Th3;
(gr2)`2=d by A1,A85,Th3;
then |[(gr1)`1,(gr1)`2]|=g.r2 by A64,A65,A66,A86,EUCLID:53;
then g.r1=g.r2 by EUCLID:53;
hence thesis by A37,A38,FUNCT_1:def 4;
end;
end;
hence thesis;
end;
A87: dom k=[.0,1.] by BORSUK_1:40,FUNCT_2:def 1;
then s1 in dom k by A32,A33,XXREAL_1:1;
then rxc=s1 by A55,A59,A60,A61,A87;
hence contradiction by A46,A59,XXREAL_1:1;
end;
hence thesis;
end;
then LE p1,p2,Upper_Arc(K),W-min(K),E-max(K) by A3,A4,A13,JORDAN5C:def 3;
hence thesis by A3,A4,A13,JORDAN6:def 10;
end;
end;
theorem Th57:
for a,b,c,d being Real,p1,p2 being Point of TOP-REAL 2 st a<b & c <d
& p1 in LSeg(|[b,c]|,|[b,d]|) & p2 in LSeg(|[b,c]|,|[b,d]|)
holds LE p1,p2,rectangle(a,b,c,d) iff p1`2>=p2`2
proof
let a,b,c,d be Real,p1,p2 be Point of TOP-REAL 2;
set K = rectangle(a,b,c,d);
assume that
A1: a<b and
A2: c <d and
A3: p1 in LSeg(|[b,c]|,|[b,d]|) and
A4: p2 in LSeg(|[b,c]|,|[b,d]|);
A5: K is being_simple_closed_curve by A1,A2,Th50;
A6: p1`1=b by A2,A3,Th1;
A7: p1`2 <= d by A2,A3,Th1;
A8: p2`1=b by A2,A4,Th1;
A9: p2`2 <= d by A2,A4,Th1;
A10: W-min(K)= |[a,c]| by A1,A2,Th46;
A11: E-max(K)= |[b,d]| by A1,A2,Th46;
A12: Lower_Arc(K)=LSeg(|[b,d]|,|[b,c]|) \/ LSeg(|[b,c]|,|[a,c]|) by A1,A2,Th52;
then
A13: LSeg(|[b,d]|,|[b,c]|) c= Lower_Arc(K) by XBOOLE_1:7;
A14: Upper_Arc(K) /\ Lower_Arc(K)={W-min(K),E-max(K)} by A5,JORDAN6:def 9;
A15: now
assume p1 in Upper_Arc(K);
then
A16: p1 in Upper_Arc(K) /\ Lower_Arc(K) by A3,A13,XBOOLE_0:def 4;
now
assume p1=W-min(K);
then p1`1=a by A10,EUCLID:52;
hence contradiction by A1,A3,TOPREAL3:11;
end;
hence p1=E-max(K) by A14,A16,TARSKI:def 2;
end;
thus LE p1,p2,K implies p1`2>=p2`2
proof
assume LE p1,p2,K;
then
A17: p1 in Upper_Arc(K) & p2 in Lower_Arc(K)& not p2=W-min(K) or
p1 in Upper_Arc(K) & p2 in Upper_Arc(K) &
LE p1,p2,Upper_Arc(K),W-min(K),E-max(K) or
p1 in Lower_Arc(K) & p2 in Lower_Arc(K)& not p2=W-min(K) &
LE p1,p2,Lower_Arc(K),E-max(K),W-min(K) by JORDAN6:def 10;
now per cases;
case p1=E-max(K);
hence thesis by A9,A11,EUCLID:52;
end;
case
A18: p1<>E-max(K);
consider f being Function of I[01],(TOP-REAL 2)|(Lower_Arc(K)) such that
A19: f is being_homeomorphism and
A20: f.0=E-max(K) and
A21: f.1=W-min(K) and rng f=Lower_Arc(K)
and for r being Real st r in [.0,1/2.] holds f.r=(1-2*r)*|[b,d]|+(2*r)
*|[b,c]|
and for r being Real st r in [.1/2,1.] holds f.r=(1-(2*r-1))*|[b,c]|+(
2*r-1)*|[a,c]| and
A22: for p being Point of TOP-REAL 2 st p in LSeg(|[b,d]|,|[b,c]|)
holds 0<=((p`2)-d)/(c-d)/2 & ((p`2)-d)/(c-d)/2<=1 & f.(((p`2)-d)/(c-d)/2)=p
and for p being Point of TOP-REAL 2 st p in LSeg(|[b,c]|,|[a,c]|)
holds 0<=((p`1)-b)/(a-b)/2+1/2 & ((p`1)-b)/(a-b)/2+1/2<=1 & f.(((p`1)-b)/(a-b)/
2+1/2)=p
by A1,A2,Th54;
reconsider s1=((p1`2)-d)/(c-d)/2,s2=((p2`2)-d)/(c-d)/2 as Real;
A23: f.s1=p1 by A3,A22;
A24: f.s2=p2 by A4,A22;
d-c>0 by A2,XREAL_1:50;
then
A25: -(d-c)< -0 by XREAL_1:24;
A26: s1<=1 by A3,A22;
A27: 0<=s2 by A4,A22;
s2<=1 by A4,A22;
then s1<=s2 by A15,A17,A18,A19,A20,A21,A23,A24,A26,A27,JORDAN5C:def 3;
then ((p1`2)-d)/(c-d)/2*2<=((p2`2)-d)/(c-d)/2*2 by XREAL_1:64;
then ((p1`2)-d)/(c-d)*(c-d)>= ((p2`2)-d)/(c-d)*(c-d) by A25,XREAL_1:65;
then ((p1`2)-d)>= ((p2`2)-d)/(c-d)*(c-d) by A25,XCMPLX_1:87;
then ((p1`2)-d)>= ((p2`2)-d) by A25,XCMPLX_1:87;
then ((p1`2)-d)+d >= ((p2`2)-d)+d by XREAL_1:7;
hence thesis;
end;
end;
hence thesis;
end;
thus p1`2>=p2`2 implies LE p1,p2,K
proof
assume
A28: p1`2>=p2`2;
now per cases;
case p2=W-min (K);
then p2=|[a,c]| by A1,A2,Th46;
hence contradiction by A1,A8,EUCLID:52;
end;
case
A29: p2<>W-min(K);
for g being Function of I[01], (TOP-REAL 2)|Lower_Arc(K),
s1, s2 being Real st g is being_homeomorphism
& g.0 = E-max(K) & g.1 = W-min(K) & g.s1 = p1 & 0 <= s1 & s1 <= 1
& g.s2 = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2
proof
let g be Function of I[01], (TOP-REAL 2)|Lower_Arc(K),
s1, s2 be Real;
assume that
A30: g is being_homeomorphism and
A31: g.0 = E-max(K) and g.1 = W-min(K) and
A32: g.s1 = p1 and
A33: 0 <= s1 and
A34: s1 <= 1 and
A35: g.s2 = p2 and
A36: 0 <= s2 and
A37: s2 <= 1;
A38: dom g=the carrier of I[01] by FUNCT_2:def 1;
A39: g is one-to-one by A30,TOPS_2:def 5;
A40: the carrier of ((TOP-REAL 2)|Lower_Arc(K))
=Lower_Arc(K) by PRE_TOPC:8;
then reconsider g1=g as Function of I[01],TOP-REAL 2 by FUNCT_2:7;
g is continuous by A30,TOPS_2:def 5;
then
A41: g1 is continuous by PRE_TOPC:26;
reconsider h1=proj1 as Function of TOP-REAL 2,R^1 by TOPMETR:17;
reconsider h2=proj2 as Function of TOP-REAL 2,R^1 by TOPMETR:17;
reconsider hh1=h1 as Function of the TopStruct of TOP-REAL 2,R^1;
reconsider hh2=h2 as Function of the TopStruct of TOP-REAL 2,R^1;
A42: the TopStruct of TOP-REAL 2
= (the TopStruct of TOP-REAL 2)|([#](the TopStruct of TOP-REAL 2))
by TSEP_1:3
.= the TopStruct of ((TOP-REAL 2)|([#](TOP-REAL 2))) by PRE_TOPC:36
.= (TOP-REAL 2)|([#](TOP-REAL 2));
then (for p being Point of (TOP-REAL 2)|([#](TOP-REAL 2)) holds
hh1.p=proj1.p) implies hh1 is continuous by JGRAPH_2:29;
then
A43: (for p being Point of (TOP-REAL 2)|([#]TOP-REAL 2)holds
hh1.p=proj1.p) implies h1 is continuous by PRE_TOPC:32;
(for p being Point of (TOP-REAL 2)|([#](TOP-REAL 2)) holds
hh2.p=proj2.p) implies hh2 is continuous by A42,JGRAPH_2:30;
then (for p being Point of (TOP-REAL 2)|([#](TOP-REAL 2)) holds
hh2.p=proj2.p) implies h2 is continuous by PRE_TOPC:32;
then consider h being Function of TOP-REAL 2,R^1 such that
A44: for p being Point of TOP-REAL 2, r1,r2 being Real st h1.p=r1 &
h2.p=r2 holds h.p=r1+r2 and
A45: h is continuous by A43,JGRAPH_2:19;
reconsider k=h*g1 as Function of I[01],R^1;
A46: E-max K=|[b,d]| by A1,A2,Th46;
now
assume
A47: s1>s2;
A48: dom g=[.0,1.] by BORSUK_1:40,FUNCT_2:def 1;
0 in [.0,1.] by XXREAL_1:1;
then
A49: k.0=h.(E-max(K)) by A31,A48,FUNCT_1:13
.=h1.(E-max(K))+h2.(E-max(K)) by A44
.=(E-max(K))`1+proj2.(E-max(K)) by PSCOMP_1:def 5
.=(E-max(K))`1+(E-max(K))`2 by PSCOMP_1:def 6
.=b+(E-max(K))`2 by A46,EUCLID:52
.=b+d by A46,EUCLID:52;
s1 in [.0,1.] by A33,A34,XXREAL_1:1;
then
A50: k.s1=h.p1 by A32,A48,FUNCT_1:13
.=h1.p1+h2.p1 by A44
.=p1`1+proj2.p1 by PSCOMP_1:def 5
.=b+p1`2 by A6,PSCOMP_1:def 6;
A51: s2 in [.0,1.] by A36,A37,XXREAL_1:1;
then
A52: k.s2=h.p2 by A35,A48,FUNCT_1:13
.=proj1.p2 +proj2.p2 by A44
.=p2`1+proj2.p2 by PSCOMP_1:def 5
.=b+p2`2 by A8,PSCOMP_1:def 6;
A53: k.0>=k.s1 by A7,A49,A50,XREAL_1:7;
A54: k.s1>=k.s2 by A28,A50,A52,XREAL_1:7;
A55: 0 in [.0,1.] by XXREAL_1:1;
then
A56: [.0,s2.] c= [.0,1.] by A51,XXREAL_2:def 12;
reconsider B=[.0,s2.] as Subset of I[01] by A51,A55,BORSUK_1:40
,XXREAL_2:def 12;
A57: B is connected by A36,A51,A55,BORSUK_1:40,BORSUK_4:24;
A58: 0 in B by A36,XXREAL_1:1;
A59: s2 in B by A36,XXREAL_1:1;
consider xc being Point of I[01] such that
A60: xc in B and
A61: k.xc =k.s1 by A41,A45,A53,A54,A57,A58,A59,TOPREAL5:5;
reconsider rxc=xc as Real;
A62: for x1,x2 being set st x1 in dom k & x2 in dom k &
k.x1=k.x2 holds x1=x2
proof
let x1,x2 be set;
assume that
A63: x1 in dom k and
A64: x2 in dom k and
A65: k.x1=k.x2;
reconsider r1=x1 as Point of I[01] by A63;
reconsider r2=x2 as Point of I[01] by A64;
A66: k.x1=h.(g1.x1) by A63,FUNCT_1:12
.=h1.(g1.r1)+h2.(g1.r1) by A44
.=(g1.r1)`1+proj2.(g1.r1) by PSCOMP_1:def 5
.=(g1.r1)`1+(g1.r1)`2 by PSCOMP_1:def 6;
A67: k.x2=h.(g1.x2) by A64,FUNCT_1:12
.=h1.(g1.r2)+h2.(g1.r2) by A44
.=(g1.r2)`1+proj2.(g1.r2) by PSCOMP_1:def 5
.=(g1.r2)`1+(g1.r2)`2 by PSCOMP_1:def 6;
A68: g.r1 in Lower_Arc(K) by A40;
A69: g.r2 in Lower_Arc(K) by A40;
reconsider gr1=g.r1 as Point of TOP-REAL 2 by A68;
reconsider gr2=g.r2 as Point of TOP-REAL 2 by A69;
now per cases by A12,A40,XBOOLE_0:def 3;
case
A70: g.r1 in LSeg(|[b,c]|,|[b,d]|) &
g.r2 in LSeg(|[b,c]|,|[b,d]|);
then
A71: (gr1)`1=b by A2,Th1;
(gr2)`1=b by A2,A70,Th1;
then |[(gr1)`1,(gr1)`2]|=g.r2 by A65,A66,A67,A71,EUCLID:53;
then g.r1=g.r2 by EUCLID:53;
hence thesis by A38,A39,FUNCT_1:def 4;
end;
case
A72: g.r1 in LSeg(|[b,c]|,|[b,d]|) &
g.r2 in LSeg(|[b,c]|,|[a,c]|);
then
A73: (gr1)`1=b by A2,Th1;
A74: c <=(gr1)`2 by A2,A72,Th1;
A75: (gr2)`2=c by A1,A72,Th3;
A76: (gr2)`1 <=b by A1,A72,Th3;
A77: b+(gr1)`2=(gr2)`1 +c by A1,A65,A66,A67,A72,A73,Th3;
A78: now
assume b<>gr2`1;
then b>gr2`1 by A76,XXREAL_0:1;
hence contradiction by A74,A77,XREAL_1:8;
end;
now
assume gr1`2<>c;
then c <gr1`2 by A74,XXREAL_0:1;
hence contradiction by A65,A66,A67,A73,A75,A76,XREAL_1:8;
end;
then |[(gr1)`1,(gr1)`2]|=g.r2 by A73,A75,A78,EUCLID:53;
then g.r1=g.r2 by EUCLID:53;
hence thesis by A38,A39,FUNCT_1:def 4;
end;
case
A79: g.r1 in LSeg(|[b,c]|,|[a,c]|) &
g.r2 in LSeg(|[b,c]|,|[b,d]|);
then
A80: (gr2)`1=b by A2,Th1;
A81: c <=(gr2)`2 by A2,A79,Th1;
A82: (gr1)`2=c by A1,A79,Th3;
A83: (gr1)`1 <=b by A1,A79,Th3;
A84: b+(gr2)`2=(gr1)`1 +c by A1,A65,A66,A67,A79,A80,Th3;
A85: now
assume b<>gr1`1;
then b> gr1`1 by A83,XXREAL_0:1;
hence contradiction by A81,A84,XREAL_1:8;
end;
now
assume gr2`2<> c;
then c <gr2`2 by A81,XXREAL_0:1;
hence contradiction by A65,A66,A67,A80,A82,A83,XREAL_1:8;
end;
then |[(gr2)`1,(gr2)`2]|=g.r1 by A80,A82,A85,EUCLID:53;
then g.r1=g.r2 by EUCLID:53;
hence thesis by A38,A39,FUNCT_1:def 4;
end;
case
A86: g.r1 in LSeg(|[b,c]|,|[a,c]|) &
g.r2 in LSeg(|[b,c]|,|[a,c]|);
then
A87: (gr1)`2=c by A1,Th3;
(gr2)`2=c by A1,A86,Th3;
then |[(gr1)`1,(gr1)`2]|=g.r2 by A65,A66,A67,A87,EUCLID:53;
then g.r1=g.r2 by EUCLID:53;
hence thesis by A38,A39,FUNCT_1:def 4;
end;
end;
hence thesis;
end;
A88: dom k=[.0,1.] by BORSUK_1:40,FUNCT_2:def 1;
then s1 in dom k by A33,A34,XXREAL_1:1;
then rxc=s1 by A56,A60,A61,A62,A88;
hence contradiction by A47,A60,XXREAL_1:1;
end;
hence thesis;
end;
then LE p1,p2,Lower_Arc(K),E-max(K),W-min(K) by A3,A4,A13,
JORDAN5C:def 3;
hence thesis by A3,A4,A13,A29,JORDAN6:def 10;
end;
end;
hence thesis;
end;
end;
theorem Th58:
for a,b,c,d being Real,p1,p2 being Point of TOP-REAL 2 st a<b & c <d
& p1 in LSeg(|[a,c]|,|[b,c]|) & p2 in LSeg(|[a,c]|,|[b,c]|)
holds LE p1,p2,rectangle(a,b,c,d) &
p1<>W-min rectangle(a,b,c,d) iff p1`1>=p2`1 & p2<>W-min rectangle(a,b,c,d)
proof
let a,b,c,d be Real,p1,p2 be Point of TOP-REAL 2;
set K = rectangle(a,b,c,d);
assume that
A1: a<b and
A2: c <d and
A3: p1 in LSeg(|[a,c]|,|[b,c]|) and
A4: p2 in LSeg(|[a,c]|,|[b,c]|);
A5: K is being_simple_closed_curve by A1,A2,Th50;
A6: p1`2=c by A1,A3,Th3;
A7: p1`1 <= b by A1,A3,Th3;
A8: p2`2=c by A1,A4,Th3;
A9: a <=p2`1 by A1,A4,Th3;
A10: W-min(K)= |[a,c]| by A1,A2,Th46;
A11: E-max(K)= |[b,d]| by A1,A2,Th46;
A12: Lower_Arc(K)=LSeg(|[b,d]|,|[b,c]|) \/ LSeg(|[b,c]|,|[a,c]|) by A1,A2,Th52;
then
A13: LSeg(|[b,c]|,|[a,c]|) c= Lower_Arc(K) by XBOOLE_1:7;
then
A14: p1 in Lower_Arc(K) by A3;
A15: Lower_Arc(K) c= K by A5,JORDAN6:61;
A16: Upper_Arc(K) /\ Lower_Arc(K)={W-min(K),E-max(K)} by A5,JORDAN6:def 9;
A17: now
assume p1 in Upper_Arc(K);
then p1 in Upper_Arc(K) /\ Lower_Arc(K) by A3,A13,XBOOLE_0:def 4;
then p1=W-min(K) or p1=E-max(K) by A16,TARSKI:def 2;
hence p1=W-min(K) by A2,A6,A11,EUCLID:52;
end;
thus LE p1,p2,K & p1<>W-min(K) implies p1`1>=p2`1 & p2<>W-min(K)
proof
assume that
A18: LE p1,p2,K and
A19: p1<>W-min(K);
A20: p1 in Upper_Arc(K) & p2 in Lower_Arc(K)& not p2=W-min(K) or
p1 in Upper_Arc(K) & p2 in Upper_Arc(K) &
LE p1,p2,Upper_Arc(K),W-min(K),E-max(K) or
p1 in Lower_Arc(K) & p2 in Lower_Arc(K)& not p2=W-min(K) &
LE p1,p2,Lower_Arc(K),E-max(K),W-min(K) by A18,JORDAN6:def 10;
consider f being Function of I[01],(TOP-REAL 2)|(Lower_Arc(K)) such that
A21: f is being_homeomorphism and
A22: f.0=E-max(K) and
A23: f.1=W-min(K) and rng f=Lower_Arc(K)
and for r being Real st r in [.0,1/2.] holds f.r=(1-2*r)*|[b,d]|+(2*r)
*|[b,c]|
and for r being Real st r in [.1/2,1.] holds f.r=(1-(2*r-1))*|[b,c]|+(
2*r-1)*|[a,c]|
and for p being Point of TOP-REAL 2 st p in LSeg(|[b,d]|,|[b,c]|)
holds 0<=((p`2)-d)/(c-d)/2 & ((p`2)-d)/(c-d)/2<=1 & f.(((p`2)-d)/(c-d)/2)=p
and
A24: for p being Point of TOP-REAL 2 st p in LSeg(|[b,c]|,|[a,c]|)
holds 0<=((p`1)-b)/(a-b)/2+1/2 & ((p`1)-b)/(a-b)/2+1/2<=1 & f.(((p`1)-b)/(a-b)/
2+1/2)=p
by A1,A2,Th54;
reconsider s1=((p1`1)-b)/(a-b)/2+1/2,s2=((p2`1)-b)/(a-b)/2+1/2 as Real;
A25: f.s1=p1 by A3,A24;
A26: f.s2=p2 by A4,A24;
b-a>0 by A1,XREAL_1:50;
then
A27: -(b-a)< -0 by XREAL_1:24;
A28: s1<=1 by A3,A24;
A29: 0<=s2 by A4,A24;
s2<=1 by A4,A24;
then s1<=s2 by A17,A19,A20,A21,A22,A23,A25,A26,A28,A29,JORDAN5C:def 3;
then ((p1`1)-b)/(a-b)/2<= ((p2`1)-b)/(a-b)/2 by XREAL_1:6;
then ((p1`1)-b)/(a-b)/2*2<=((p2`1)-b)/(a-b)/2*2 by XREAL_1:64;
then ((p1`1)-b)/(a-b)*(a-b)>= ((p2`1)-b)/(a-b)*(a-b) by A27,XREAL_1:65;
then ((p1`1)-b)>= ((p2`1)-b)/(a-b)*(a-b) by A27,XCMPLX_1:87;
then ((p1`1)-b)>= ((p2`1)-b) by A27,XCMPLX_1:87;
then ((p1`1)-b)+b >= ((p2`1)-b)+b by XREAL_1:7;
hence p1`1>=p2`1;
now
assume
A30: p2=W-min(K);
then LE p2,p1,K by A5,A14,A15,JORDAN7:3;
hence contradiction by A1,A2,A18,A19,A30,Th50,JORDAN6:57;
end;
hence thesis;
end;
thus p1`1>=p2`1 & p2<>W-min(K) implies LE p1,p2,K & p1<>W-min(K)
proof
assume that
A31: p1`1>=p2`1 and
A32: p2<>W-min(K);
A33: for g being Function of I[01], (TOP-REAL 2)|Lower_Arc(K),
s1, s2 being Real st g is being_homeomorphism & g.0 = E-max(K) &
g.1 = W-min(K)
& g.s1 = p1 & 0 <= s1 & s1 <= 1 & g.s2 = p2 & 0 <= s2 & s2 <= 1
holds s1 <= s2
proof
let g be Function of I[01], (TOP-REAL 2)|Lower_Arc(K),
s1, s2 be Real;
assume that
A34: g is being_homeomorphism and
A35: g.0 = E-max(K) and g.1 = W-min(K) and
A36: g.s1 = p1 and
A37: 0 <= s1 and
A38: s1 <= 1 and
A39: g.s2 = p2 and
A40: 0 <= s2 and
A41: s2 <= 1;
A42: dom g=the carrier of I[01] by FUNCT_2:def 1;
A43: g is one-to-one by A34,TOPS_2:def 5;
A44: the
carrier of ((TOP-REAL 2)|Lower_Arc(K)) =Lower_Arc(K) by PRE_TOPC:8;
then reconsider g1=g as Function of I[01],TOP-REAL 2 by FUNCT_2:7;
g is continuous by A34,TOPS_2:def 5;
then
A45: g1 is continuous by PRE_TOPC:26;
reconsider h1=proj1 as Function of TOP-REAL 2,R^1 by TOPMETR:17;
reconsider h2=proj2 as Function of TOP-REAL 2,R^1 by TOPMETR:17;
reconsider hh1=h1 as Function of the TopStruct of TOP-REAL 2,R^1;
reconsider hh2=h2 as Function of the TopStruct of TOP-REAL 2,R^1;
A46: the TopStruct of TOP-REAL 2
= (the TopStruct of TOP-REAL 2)|([#](the TopStruct of TOP-REAL 2))
by TSEP_1:3
.= the TopStruct of ((TOP-REAL 2)|([#](TOP-REAL 2))) by PRE_TOPC:36
.= (TOP-REAL 2)|([#](TOP-REAL 2));
then (for p being Point of (TOP-REAL 2)|([#](TOP-REAL 2)) holds
hh1.p=proj1.p) implies hh1 is continuous by JGRAPH_2:29;
then
A47: (for p being Point of (TOP-REAL 2)|([#]TOP-REAL 2)holds
hh1.p=proj1.p) implies h1 is continuous by PRE_TOPC:32;
(for p being Point of (TOP-REAL 2)|([#](TOP-REAL 2)) holds
hh2.p=proj2.p) implies hh2 is continuous by A46,JGRAPH_2:30;
then (for p being Point of (TOP-REAL 2)|([#](TOP-REAL 2)) holds
hh2.p=proj2.p) implies h2 is continuous by PRE_TOPC:32;
then consider h being Function of TOP-REAL 2,R^1 such that
A48: for p being Point of TOP-REAL 2, r1,r2 being Real st h1.p=r1 &
h2.p=r2 holds h.p=r1+r2 and
A49: h is continuous by A47,JGRAPH_2:19;
reconsider k=h*g1 as Function of I[01],R^1;
A50: E-max K=|[b,d]| by A1,A2,Th46;
now
assume
A51: s1>s2;
A52: dom g=[.0,1.] by BORSUK_1:40,FUNCT_2:def 1;
0 in [.0,1.] by XXREAL_1:1;
then
A53: k.0=h.(E-max(K)) by A35,A52,FUNCT_1:13
.=h1.(E-max(K))+h2.(E-max(K)) by A48
.=(E-max(K))`1+proj2.(E-max(K)) by PSCOMP_1:def 5
.=(E-max(K))`1+(E-max(K))`2 by PSCOMP_1:def 6
.=(E-max(K))`1+d by A50,EUCLID:52
.=b+d by A50,EUCLID:52;
s1 in [.0,1.] by A37,A38,XXREAL_1:1;
then
A54: k.s1=h.p1 by A36,A52,FUNCT_1:13
.=proj1.p1 +proj2.p1 by A48
.=p1`1+proj2.p1 by PSCOMP_1:def 5
.=p1`1 +c by A6,PSCOMP_1:def 6;
A55: s2 in [.0,1.] by A40,A41,XXREAL_1:1;
then
A56: k.s2=h.p2 by A39,A52,FUNCT_1:13
.=h1.p2+h2.p2 by A48
.=p2`1+proj2.p2 by PSCOMP_1:def 5
.=p2`1 +c by A8,PSCOMP_1:def 6;
A57: k.0>=k.s1 by A2,A7,A53,A54,XREAL_1:7;
A58: k.s1>=k.s2 by A31,A54,A56,XREAL_1:7;
A59: 0 in [.0,1.] by XXREAL_1:1;
then
A60: [.0,s2.] c= [.0,1.] by A55,XXREAL_2:def 12;
reconsider B=[.0,s2.] as Subset of I[01] by A55,A59,BORSUK_1:40
,XXREAL_2:def 12;
A61: B is connected by A40,A55,A59,BORSUK_1:40,BORSUK_4:24;
A62: 0 in B by A40,XXREAL_1:1;
A63: s2 in B by A40,XXREAL_1:1;
consider xc being Point of I[01] such that
A64: xc in B and
A65: k.xc =k.s1 by A45,A49,A57,A58,A61,A62,A63,TOPREAL5:5;
reconsider rxc=xc as Real;
A66: for x1,x2 being set st x1 in dom k & x2 in dom k &
k.x1=k.x2 holds x1=x2
proof
let x1,x2 be set;
assume that
A67: x1 in dom k and
A68: x2 in dom k and
A69: k.x1=k.x2;
reconsider r1=x1 as Point of I[01] by A67;
reconsider r2=x2 as Point of I[01] by A68;
A70: k.x1=h.(g1.x1) by A67,FUNCT_1:12
.=h1.(g1.r1)+h2.(g1.r1) by A48
.=(g1.r1)`1+proj2.(g1.r1) by PSCOMP_1:def 5
.=(g1.r1)`1+(g1.r1)`2 by PSCOMP_1:def 6;
A71: k.x2=h.(g1.x2) by A68,FUNCT_1:12
.=h1.(g1.r2)+h2.(g1.r2) by A48
.=(g1.r2)`1+proj2.(g1.r2) by PSCOMP_1:def 5
.=(g1.r2)`1+(g1.r2)`2 by PSCOMP_1:def 6;
A72: g.r1 in Lower_Arc(K) by A44;
A73: g.r2 in Lower_Arc(K) by A44;
reconsider gr1=g.r1 as Point of TOP-REAL 2 by A72;
reconsider gr2=g.r2 as Point of TOP-REAL 2 by A73;
now per cases by A12,A44,XBOOLE_0:def 3;
case
A74: g
.r1 in LSeg(|[b,d]|,|[b,c]|) & g.r2 in LSeg(|[b,d]|,|[b,c]|);
then
A75: (gr1)`1=b by A2,Th1;
(gr2)`1=b by A2,A74,Th1;
then |[(gr1)`1,(gr1)`2]|=g.r2 by A69,A70,A71,A75,EUCLID:53;
then g.r1=g.r2 by EUCLID:53;
hence thesis by A42,A43,FUNCT_1:def 4;
end;
case
A76: g
.r1 in LSeg(|[b,d]|,|[b,c]|) & g.r2 in LSeg(|[b,c]|,|[a,c]|);
then
A77: (gr1)`1=b by A2,Th1;
A78: c <=(gr1)`2 by A2,A76,Th1;
A79: (gr2)`2=c by A1,A76,Th3;
A80: (gr2)`1 <=b by A1,A76,Th3;
A81: b+(gr1)`2=(gr2)`1 +c by A1,A69,A70,A71,A76,A77,Th3;
A82: now
assume b<>gr2`1;
then b>gr2`1 by A80,XXREAL_0:1;
hence contradiction by A78,A81,XREAL_1:8;
end;
now
assume gr1`2<>c;
then c <gr1`2 by A78,XXREAL_0:1;
hence contradiction by A69,A70,A71,A77,A79,A80,XREAL_1:8;
end;
then |[(gr1)`1,(gr1)`2]|=g.r2 by A77,A79,A82,EUCLID:53;
then g.r1=g.r2 by EUCLID:53;
hence thesis by A42,A43,FUNCT_1:def 4;
end;
case
A83: g
.r1 in LSeg(|[b,c]|,|[a,c]|) & g.r2 in LSeg(|[b,d]|,|[b,c]|);
then
A84: (gr2)`1=b by A2,Th1;
A85: c <=(gr2)`2 by A2,A83,Th1;
A86: (gr1)`2=c by A1,A83,Th3;
A87: (gr1)`1 <=b by A1,A83,Th3;
A88: b+(gr2)`2=(gr1)`1 +c by A1,A69,A70,A71,A83,A84,Th3;
A89: now
assume b<>gr1`1;
then b>gr1`1 by A87,XXREAL_0:1;
hence contradiction by A85,A88,XREAL_1:8;
end;
now
assume gr2`2<>c;
then c <gr2`2 by A85,XXREAL_0:1;
hence contradiction by A69,A70,A71,A84,A86,A87,XREAL_1:8;
end;
then |[(gr2)`1,(gr2)`2]|=g.r1 by A84,A86,A89,EUCLID:53;
then g.r1=g.r2 by EUCLID:53;
hence thesis by A42,A43,FUNCT_1:def 4;
end;
case
A90: g
.r1 in LSeg(|[b,c]|,|[a,c]|) & g.r2 in LSeg(|[b,c]|,|[a,c]|);
then
A91: (gr1)`2=c by A1,Th3;
(gr2)`2=c by A1,A90,Th3;
then |[(gr1)`1,(gr1)`2]|=g.r2 by A69,A70,A71,A91,EUCLID:53;
then g.r1=g.r2 by EUCLID:53;
hence thesis by A42,A43,FUNCT_1:def 4;
end;
end;
hence thesis;
end;
A92: dom k=[.0,1.] by BORSUK_1:40,FUNCT_2:def 1;
then s1 in dom k by A37,A38,XXREAL_1:1;
then rxc=s1 by A60,A64,A65,A66,A92;
hence contradiction by A51,A64,XXREAL_1:1;
end;
hence thesis;
end;
A93: now
assume
A94: p1=W-min(K);
then p1`1=a by A10,EUCLID:52;
then p1`1=p2`1 by A9,A31,XXREAL_0:1;
then |[(p1)`1,(p1)`2]|=p2 by A6,A8,EUCLID:53;
hence contradiction by A32,A94,EUCLID:53;
end;
LE p1,p2,Lower_Arc(K),E-max(K),W-min(K) by A3,A4,A13,A33,JORDAN5C:def 3;
hence LE p1,p2,K by A3,A4,A13,A32,JORDAN6:def 10;
thus thesis by A93;
end;
end;
theorem Th59:
for a,b,c,d being Real,p1,p2 being Point of TOP-REAL 2 st a<b & c <d
& p1 in LSeg(|[a,c]|,|[a,d]|)
holds LE p1,p2,rectangle(a,b,c,d) iff p2 in LSeg(|[a,c]|,|[a,d]|) & p1`2<=p2`2
or p2 in LSeg(|[a,d]|,|[b,d]|) or p2 in LSeg(|[b,d]|,|[b,c]|)
or p2 in LSeg(|[b,c]|,|[a,c]|) & p2<>W-min rectangle(a,b,c,d)
proof
let a,b,c,d be Real,p1,p2 be Point of TOP-REAL 2;
set K = rectangle(a,b,c,d);
assume that
A1: a<b and
A2: c <d and
A3: p1 in LSeg(|[a,c]|,|[a,d]|);
A4: K is being_simple_closed_curve by A1,A2,Th50;
Upper_Arc(K)=LSeg(|[a,c]|,|[a,d]|) \/ LSeg(|[a,d]|,|[b,d]|) by A1,A2,Th51;
then
A5: LSeg(|[a,c]|,|[a,d]|) c= Upper_Arc(K) by XBOOLE_1:7;
A6: p1`1=a by A2,A3,Th1;
A7: c <=p1`2 by A2,A3,Th1;
A8: p1`2 <= d by A2,A3,Th1;
thus LE p1,p2,K implies p2 in LSeg(|[a,c]|,|[a,d]|) & p1`2<=p2`2
or p2 in LSeg(|[a,d]|,|[b,d]|) or p2 in LSeg(|[b,d]|,|[b,c]|)
or p2 in LSeg(|[b,c]|,|[a,c]|) & p2<>W-min(K)
proof
assume
A9: LE p1,p2,K;
then
A10: p1 in K by A4,JORDAN7:5;
A11: p2 in K by A4,A9,JORDAN7:5;
K= LSeg(|[ a,c ]|, |[ a,d ]|) \/ LSeg(|[ a,d ]|, |[ b,d ]|)
\/ (LSeg(|[ a,c ]|, |[ b,c ]|) \/ LSeg(|[ b,c ]|, |[ b,d ]|))
by SPPOL_2:def 3
.=LSeg(|[a,c]|,|[a,d]|) \/ LSeg(|[a,d]|,|[b,d]|)
\/ LSeg(|[b,d]|,|[b,c]|) \/ LSeg(|[b,c]|,|[a,c]|) by XBOOLE_1:4;
then p2 in LSeg(|[a,c]|,|[a,d]|) \/ LSeg(|[a,d]|,|[b,d]|)
\/ LSeg(|[b,d]|,|[b,c]|) or
p2 in LSeg(|[b,c]|,|[a,c]|) by A11,XBOOLE_0:def 3;
then
A12: p2 in LSeg(|[a,c]|,|[a,d]|) \/ LSeg(|[a,d]|,|[b,d]|) or
p2 in LSeg(|[b,d]|,|[b,c]|) or
p2 in LSeg(|[b,c]|,|[a,c]|) by XBOOLE_0:def 3;
now per cases by A12,XBOOLE_0:def 3;
case p2 in LSeg(|[a,c]|,|[a,d]|);
hence thesis by A1,A2,A3,A9,Th55;
end;
case p2 in LSeg(|[a,d]|,|[b,d]|);
hence thesis;
end;
case p2 in LSeg(|[b,d]|,|[b,c]|);
hence thesis;
end;
case
A13: p2 in LSeg(|[b,c]|,|[a,c]|);
now per cases;
case p2=W-min(K);
then LE p2,p1,K by A4,A10,JORDAN7:3;
hence thesis by A1,A2,A3,A9,Th50,JORDAN6:57;
end;
case p2<>W-min(K);
hence thesis by A13;
end;
end;
hence thesis;
end;
end;
hence thesis;
end;
A14: W-min(K)= |[a,c]| by A1,A2,Th46;
thus p2 in LSeg(|[a,c]|,|[a,d]|) & p1`2<=p2`2
or p2 in LSeg(|[a,d]|,|[b,d]|) or p2 in LSeg(|[b,d]|,|[b,c]|)
or p2 in LSeg(|[b,c]|,|[a,c]|) & p2<>W-min(K) implies LE p1,p2,K
proof
assume that
A15: p2 in LSeg(|[a,c]|,|[a,d]|) & p1`2<=p2`2
or p2 in LSeg(|[a,d]|,|[b,d]|) or p2 in LSeg(|[b,d]|,|[b,c]|)
or p2 in LSeg(|[b,c]|,|[a,c]|) & p2<>W-min(K);
now per cases by A15;
case p2 in LSeg(|[a,c]|,|[a,d]|) & p1`2<=p2`2;
hence thesis by A1,A2,A3,Th55;
end;
case
A16: p2 in LSeg(|[a,d]|,|[b,d]|);
then
A17: p2`2=d by A1,Th3;
A18: a <=p2`1 by A1,A16,Th3;
A19: Upper_Arc(K)=LSeg(|[a,c]|,|[a,d]|) \/ LSeg(|[a,d]|,|[b,d]|)
by A1,A2,Th51;
then
A20: p2 in Upper_Arc(K) by A16,XBOOLE_0:def 3;
A21: p1 in Upper_Arc(K) by A3,A19,XBOOLE_0:def 3;
for g being Function of I[01], (TOP-REAL 2)|Upper_Arc(K),
s1, s2 being Real st g is being_homeomorphism
& g.0 = W-min(K) & g.1 = E-max(K) & g.s1 = p1 & 0 <= s1 & s1 <= 1
& g.s2 = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2
proof
let g be Function of I[01], (TOP-REAL 2)|Upper_Arc(K),
s1, s2 be Real;
assume that
A22: g is being_homeomorphism and
A23: g.0 = W-min(K) and g.1 = E-max(K) and
A24: g.s1 = p1 and
A25: 0 <= s1 and
A26: s1 <= 1 and
A27: g.s2 = p2 and
A28: 0 <= s2 and
A29: s2 <= 1;
A30: dom g=the carrier of I[01] by FUNCT_2:def 1;
A31: g is one-to-one by A22,TOPS_2:def 5;
A32: the carrier of ((TOP-REAL 2)|Upper_Arc(K))
=Upper_Arc(K) by PRE_TOPC:8;
then reconsider g1=g as Function of I[01],TOP-REAL 2 by FUNCT_2:7;
g is continuous by A22,TOPS_2:def 5;
then
A33: g1 is continuous by PRE_TOPC:26;
reconsider h1=proj1 as Function of TOP-REAL 2,R^1 by TOPMETR:17;
reconsider h2=proj2 as Function of TOP-REAL 2,R^1 by TOPMETR:17;
reconsider hh1=h1 as Function of the TopStruct of TOP-REAL 2,R^1;
reconsider hh2=h2 as Function of the TopStruct of TOP-REAL 2,R^1;
A34: the TopStruct of TOP-REAL 2
= (the TopStruct of TOP-REAL 2)|([#](the TopStruct of TOP-REAL 2))
by TSEP_1:3
.= the TopStruct of ((TOP-REAL 2)|([#](TOP-REAL 2))) by PRE_TOPC:36
.= (TOP-REAL 2)|([#](TOP-REAL 2));
then (for p being Point of (TOP-REAL 2)|([#](TOP-REAL 2)) holds
hh1.p=proj1.p) implies hh1 is continuous by JGRAPH_2:29;
then
A35: h1 is continuous by PRE_TOPC:32;
(for p being Point of (TOP-REAL 2)|([#](TOP-REAL 2)) holds
hh2.p=proj2.p) implies hh2 is continuous by A34,JGRAPH_2:30;
then (for p being Point of (TOP-REAL 2)|([#](TOP-REAL 2)) holds
hh2.p=proj2.p) implies h2 is continuous by PRE_TOPC:32;
then consider h being Function of TOP-REAL 2,R^1 such that
A36: for p being Point of TOP-REAL 2, r1,r2 being Real st h1.p=r1 &
h2.p=r2 holds h.p=r1+r2 and
A37: h is continuous by A35,JGRAPH_2:19;
reconsider k=h*g1 as Function of I[01],R^1;
A38: W-min K=|[a,c]| by A1,A2,Th46;
now
assume
A39: s1>s2;
A40: dom g=[.0,1.] by BORSUK_1:40,FUNCT_2:def 1;
0 in [.0,1.] by XXREAL_1:1;
then
A41: k.0=h.(W-min(K)) by A23,A40,FUNCT_1:13
.=h1.(W-min(K))+h2.(W-min(K)) by A36
.=(W-min(K))`1+proj2.(W-min(K)) by PSCOMP_1:def 5
.=(W-min(K))`1+(W-min(K))`2 by PSCOMP_1:def 6
.=(W-min(K))`1+c by A38,EUCLID:52
.=a+c by A38,EUCLID:52;
s1 in [.0,1.] by A25,A26,XXREAL_1:1;
then
A42: k.s1=h.p1 by A24,A40,FUNCT_1:13
.=proj1.p1 +proj2.p1 by A36
.=p1`1+proj2.p1 by PSCOMP_1:def 5
.=a +p1`2 by A6,PSCOMP_1:def 6;
A43: s2 in [.0,1.] by A28,A29,XXREAL_1:1;
then
A44: k.s2=h.p2 by A27,A40,FUNCT_1:13
.=proj1.p2 +proj2.p2 by A36
.=p2`1+proj2.p2 by PSCOMP_1:def 5
.=p2`1 +d by A17,PSCOMP_1:def 6;
A45: k.0<=k.s1 by A7,A41,A42,XREAL_1:7;
A46: k.s1<=k.s2 by A8,A18,A42,A44,XREAL_1:7;
A47: 0 in [.0,1.] by XXREAL_1:1;
then
A48: [.0,s2.] c= [.0,1.] by A43,XXREAL_2:def 12;
reconsider B=[.0,s2.] as Subset of I[01] by A43,A47,BORSUK_1:40
,XXREAL_2:def 12;
A49: B is connected by A28,A43,A47,BORSUK_1:40,BORSUK_4:24;
A50: 0 in B by A28,XXREAL_1:1;
A51: s2 in B by A28,XXREAL_1:1;
consider xc being Point of I[01] such that
A52: xc in B and
A53: k.xc =k.s1 by A33,A37,A45,A46,A49,A50,A51,TOPREAL5:5;
reconsider rxc=xc as Real;
A54: for x1,x2 being set st x1 in dom k & x2 in dom k &
k.x1=k.x2 holds x1=x2
proof
let x1,x2 be set;
assume that
A55: x1 in dom k and
A56: x2 in dom k and
A57: k.x1=k.x2;
reconsider r1=x1 as Point of I[01] by A55;
reconsider r2=x2 as Point of I[01] by A56;
A58: k.x1=h.(g1.x1) by A55,FUNCT_1:12
.=h1.(g1.r1)+h2.(g1.r1) by A36
.=(g1.r1)`1+proj2.(g1.r1) by PSCOMP_1:def 5
.=(g1.r1)`1+(g1.r1)`2 by PSCOMP_1:def 6;
A59: k.x2=h.(g1.x2) by A56,FUNCT_1:12
.=h1.(g1.r2)+h2.(g1.r2) by A36
.=(g1.r2)`1+proj2.(g1.r2) by PSCOMP_1:def 5
.=(g1.r2)`1+(g1.r2)`2 by PSCOMP_1:def 6;
A60: g.r1 in Upper_Arc(K) by A32;
A61: g.r2 in Upper_Arc(K) by A32;
reconsider gr1=g.r1 as Point of TOP-REAL 2 by A60;
reconsider gr2=g.r2 as Point of TOP-REAL 2 by A61;
now per cases by A19,A32,XBOOLE_0:def 3;
case
A62: g.r1 in LSeg(|[a,c]|,|[a,d]|) &
g.r2 in LSeg(|[a,c]|,|[a,d]|);
then
A63: (gr1)`1=a by A2,Th1;
(gr2)`1=a by A2,A62,Th1;
then |[(gr1)`1,(gr1)`2]|=g.r2 by A57,A58,A59,A63,EUCLID:53;
then g.r1=g.r2 by EUCLID:53;
hence thesis by A30,A31,FUNCT_1:def 4;
end;
case
A64: g.r1 in LSeg(|[a,c]|,|[a,d]|) &
g.r2 in LSeg(|[a,d]|,|[b,d]|);
then
A65: (gr1)`1=a by A2,Th1;
A66: (gr1 )`2 <=d by A2,A64,Th1;
A67: (gr2)`2=d by A1,A64,Th3;
A68: a <=(gr2)`1 by A1,A64,Th3;
A69: a+(gr1)`2=(gr2)`1 +d by A1,A57,A58,A59,A64,A65,Th3;
A70: now
assume a<>gr2`1;
then a<gr2`1 by A68,XXREAL_0:1;
hence contradiction by A66,A69,XREAL_1:8;
end;
now
assume gr1`2<>d;
then d>gr1`2 by A66,XXREAL_0:1;
hence contradiction by A57,A58,A59,A65,A67,A68,XREAL_1:8;
end;
then |[(gr1)`1,(gr1)`2]|=g.r2 by A65,A67,A70,EUCLID:53;
then g.r1=g.r2 by EUCLID:53;
hence thesis by A30,A31,FUNCT_1:def 4;
end;
case
A71: g.r1 in LSeg(|[a,d]|,|[b,d]|) &
g.r2 in LSeg(|[a,c]|,|[a,d]|);
then
A72: (gr2)`1=a by A2,Th1;
A73: (gr2 )`2 <=d by A2,A71,Th1;
A74: (gr1)`2=d by A1,A71,Th3;
A75: a <=(gr1)`1 by A1,A71,Th3;
A76: a+(gr2)`2=(gr1)`1 +d by A1,A57,A58,A59,A71,A72,Th3;
A77: now
assume a<>gr1`1;
then a<gr1`1 by A75,XXREAL_0:1;
hence contradiction by A73,A76,XREAL_1:8;
end;
now
assume gr2`2<>d;
then d>gr2`2 by A73,XXREAL_0:1;
hence contradiction by A57,A58,A59,A72,A74,A75,XREAL_1:8;
end;
then |[(gr2)`1,(gr2)`2]|=g.r1 by A72,A74,A77,EUCLID:53;
then g.r1=g.r2 by EUCLID:53;
hence thesis by A30,A31,FUNCT_1:def 4;
end;
case
A78: g.r1 in LSeg(|[a,d]|,|[b,d]|) &
g.r2 in LSeg(|[a,d]|,|[b,d]|);
then
A79: (gr1)`2=d by A1,Th3;
(gr2)`2=d by A1,A78,Th3;
then |[(gr1)`1,(gr1)`2]|=g.r2 by A57,A58,A59,A79,EUCLID:53;
then g.r1=g.r2 by EUCLID:53;
hence thesis by A30,A31,FUNCT_1:def 4;
end;
end;
hence thesis;
end;
A80: dom k=[.0,1.] by BORSUK_1:40,FUNCT_2:def 1;
then s1 in dom k by A25,A26,XXREAL_1:1;
then rxc=s1 by A48,A52,A53,A54,A80;
hence contradiction by A39,A52,XXREAL_1:1;
end;
hence thesis;
end;
then LE p1,p2,Upper_Arc(K),W-min(K),E-max(K) by A20,A21,JORDAN5C:def 3;
hence thesis by A20,A21,JORDAN6:def 10;
end;
case
A81: p2 in LSeg(|[b,d]|,|[b,c]|);
then
A82: p2`1 =b by TOPREAL3:11;
Lower_Arc(K)=LSeg(|[b,d]|,|[b,c]|) \/ LSeg(|[b,c]|,|[a,c]|) by A1,A2
,Th52;
then
A83: LSeg(|[b,d]|,|[b,c]|) c= Lower_Arc(K) by XBOOLE_1:7;
p2 <> W-min(K) by A1,A14,A82,EUCLID:52;
hence thesis by A3,A5,A81,A83,JORDAN6:def 10;
end;
case
A84: p2 in LSeg(|[b,c]|,|[a,c]|) & p2<>W-min(K);
Lower_Arc(K)=LSeg(|[b,d]|,|[b,c]|) \/ LSeg(|[b,c]|,|[a,c]|) by A1,A2
,Th52;
then LSeg(|[b,c]|,|[a,c]|) c= Lower_Arc(K) by XBOOLE_1:7;
hence thesis by A3,A5,A84,JORDAN6:def 10;
end;
end;
hence thesis;
end;
end;
theorem Th60:
for a,b,c,d being Real,p1,p2 being Point of TOP-REAL 2 st a<b & c <d
& p1 in LSeg(|[a,d]|,|[b,d]|)
holds LE p1,p2,rectangle(a,b,c,d) iff p2 in LSeg(|[a,d]|,|[b,d]|) & p1`1<=p2`1
or p2 in LSeg(|[b,d]|,|[b,c]|)
or p2 in LSeg(|[b,c]|,|[a,c]|) & p2<>W-min rectangle(a,b,c,d)
proof
let a,b,c,d be Real,p1,p2 be Point of TOP-REAL 2;
set K = rectangle(a,b,c,d);
assume that
A1: a<b and
A2: c <d and
A3: p1 in LSeg(|[a,d]|,|[b,d]|);
A4: K is being_simple_closed_curve by A1,A2,Th50;
Upper_Arc(K)=LSeg(|[a,c]|,|[a,d]|) \/ LSeg(|[a,d]|,|[b,d]|) by A1,A2,Th51;
then
A5: LSeg(|[a,d]|,|[b,d]|) c= Upper_Arc(K) by XBOOLE_1:7;
A6: p1`2=d by A1,A3,Th3;
A7: a <=p1`1 by A1,A3,Th3;
thus LE p1,p2,K implies p2 in LSeg(|[a,d]|,|[b,d]|) & p1`1<=p2`1
or p2 in LSeg(|[b,d]|,|[b,c]|) or p2 in LSeg(|[b,c]|,|[a,c]|) & p2<>W-min(K)
proof
assume
A8: LE p1,p2,K;
then
A9: p1 in K by A4,JORDAN7:5;
A10: p2 in K by A4,A8,JORDAN7:5;
K= LSeg(|[ a,c ]|, |[ a,d ]|) \/ LSeg(|[ a,d ]|, |[ b,d ]|)
\/ (LSeg(|[ a,c ]|, |[ b,c ]|) \/ LSeg(|[ b,c ]|, |[ b,d ]|))
by SPPOL_2:def 3
.=LSeg(|[a,c]|,|[a,d]|) \/ LSeg(|[a,d]|,|[b,d]|)
\/ LSeg(|[b,d]|,|[b,c]|) \/ LSeg(|[b,c]|,|[a,c]|) by XBOOLE_1:4;
then p2 in LSeg(|[a,c]|,|[a,d]|) \/ LSeg(|[a,d]|,|[b,d]|)
\/ LSeg(|[b,d]|,|[b,c]|) or
p2 in LSeg(|[b,c]|,|[a,c]|) by A10,XBOOLE_0:def 3;
then
A11: p2 in LSeg(|[a,c]|,|[a,d]|) \/ LSeg(|[a,d]|,|[b,d]|) or
p2 in LSeg(|[b,d]|,|[b,c]|) or
p2 in LSeg(|[b,c]|,|[a,c]|) by XBOOLE_0:def 3;
now per cases by A11,XBOOLE_0:def 3;
case p2 in LSeg(|[a,c]|,|[a,d]|);
then LE p2,p1,K by A1,A2,A3,Th59;
hence thesis by A1,A2,A3,A8,Th50,JORDAN6:57;
end;
case p2 in LSeg(|[a,d]|,|[b,d]|);
hence thesis by A1,A2,A3,A8,Th56;
end;
case p2 in LSeg(|[b,d]|,|[b,c]|);
hence thesis;
end;
case
A12: p2 in LSeg(|[b,c]|,|[a,c]|);
now per cases;
case p2=W-min(K);
then LE p2,p1,K by A4,A9,JORDAN7:3;
hence thesis by A1,A2,A3,A8,Th50,JORDAN6:57;
end;
case p2<>W-min(K);
hence thesis by A12;
end;
end;
hence thesis;
end;
end;
hence thesis;
end;
A13: W-min(K)= |[a,c]| by A1,A2,Th46;
thus
p2 in LSeg(|[a,d]|,|[b,d]|)& p1`1<=p2`1 or p2 in LSeg(|[b,d]|,|[b,c]|)
or p2 in LSeg(|[b,c]|,|[a,c]|) & p2<>W-min(K) implies LE p1,p2,K
proof
assume that
A14: p2 in LSeg(|[a,d]|,|[b,d]|)& p1`1<=p2`1 or p2 in LSeg(|[b,d]|,|[b,c]|)
or p2 in LSeg(|[b,c]|,|[a,c]|) & p2<>W-min(K);
now per cases by A14;
case
A15: p2 in LSeg(|[a,d]|,|[b,d]|)& p1`1<=p2`1;
then
A16: p2`2=d by A1,Th3;
A17: Upper_Arc(K)=LSeg(|[a,c]|,|[a,d]|) \/ LSeg(|[a,d]|,|[b,d]|)
by A1,A2,Th51;
then
A18: p2 in Upper_Arc(K) by A15,XBOOLE_0:def 3;
A19: p1 in Upper_Arc(K) by A3,A17,XBOOLE_0:def 3;
for g being Function of I[01], (TOP-REAL 2)|Upper_Arc(K),
s1, s2 being Real st g is being_homeomorphism
& g.0 = W-min(K) & g.1 = E-max(K) & g.s1 = p1 & 0 <= s1 & s1 <= 1
& g.s2 = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2
proof
let g be Function of I[01], (TOP-REAL 2)|Upper_Arc(K),
s1, s2 be Real;
assume that
A20: g is being_homeomorphism and
A21: g.0 = W-min(K) and g.1 = E-max(K) and
A22: g.s1 = p1 and
A23: 0 <= s1 and
A24: s1 <= 1 and
A25: g.s2 = p2 and
A26: 0 <= s2 and
A27: s2 <= 1;
A28: dom g=the carrier of I[01] by FUNCT_2:def 1;
A29: g is one-to-one by A20,TOPS_2:def 5;
A30: the carrier of ((TOP-REAL 2)|Upper_Arc(K))
=Upper_Arc(K) by PRE_TOPC:8;
then reconsider g1=g as Function of I[01],TOP-REAL 2 by FUNCT_2:7;
g is continuous by A20,TOPS_2:def 5;
then
A31: g1 is continuous by PRE_TOPC:26;
reconsider h1=proj1 as Function of TOP-REAL 2,R^1 by TOPMETR:17;
reconsider h2=proj2 as Function of TOP-REAL 2,R^1 by TOPMETR:17;
reconsider hh1=h1 as Function of the TopStruct of TOP-REAL 2,R^1;
reconsider hh2=h2 as Function of the TopStruct of TOP-REAL 2,R^1;
A32: the TopStruct of TOP-REAL 2
= (the TopStruct of TOP-REAL 2)|([#](the TopStruct of TOP-REAL 2))
by TSEP_1:3
.= the TopStruct of ((TOP-REAL 2)|([#](TOP-REAL 2))) by PRE_TOPC:36
.= (TOP-REAL 2)|([#](TOP-REAL 2));
then (for p being Point of (TOP-REAL 2)|([#](TOP-REAL 2)) holds
hh1.p=proj1.p) implies hh1 is continuous by JGRAPH_2:29;
then
A33: (for p being Point of (TOP-REAL 2)|([#]TOP-REAL 2)holds
hh1.p=proj1.p) implies h1 is continuous by PRE_TOPC:32;
(for p being Point of (TOP-REAL 2)|([#](TOP-REAL 2)) holds
hh2.p=proj2.p) implies hh2 is continuous by A32,JGRAPH_2:30;
then (for p being Point of (TOP-REAL 2)|([#](TOP-REAL 2)) holds
hh2.p=proj2.p) implies h2 is continuous by PRE_TOPC:32;
then consider h being Function of TOP-REAL 2,R^1 such that
A34: for p being Point of TOP-REAL 2, r1,r2 being Real st h1.p=r1 &
h2.p=r2 holds h.p=r1+r2 and
A35: h is continuous by A33,JGRAPH_2:19;
reconsider k=h*g1 as Function of I[01],R^1;
A36: W-min K=|[a,c]| by A1,A2,Th46;
now
assume
A37: s1>s2;
A38: dom g=[.0,1.] by BORSUK_1:40,FUNCT_2:def 1;
0 in [.0,1.] by XXREAL_1:1;
then
A39: k.0=h.(W-min(K)) by A21,A38,FUNCT_1:13
.=h1.(W-min(K))+h2.(W-min(K)) by A34
.=(W-min(K))`1+proj2.(W-min(K)) by PSCOMP_1:def 5
.=(W-min(K))`1+(W-min(K))`2 by PSCOMP_1:def 6
.=(W-min(K))`1+c by A36,EUCLID:52
.=a+c by A36,EUCLID:52;
s1 in [.0,1.] by A23,A24,XXREAL_1:1;
then
A40: k.s1=h.p1 by A22,A38,FUNCT_1:13
.=proj1.p1 +proj2.p1 by A34
.=p1`1+proj2.p1 by PSCOMP_1:def 5
.=p1`1+d by A6,PSCOMP_1:def 6;
A41: s2 in [.0,1.] by A26,A27,XXREAL_1:1;
then
A42: k.s2=h.p2 by A25,A38,FUNCT_1:13
.=proj1.p2 +proj2.p2 by A34
.=p2`1+proj2.p2 by PSCOMP_1:def 5
.=p2`1 +d by A16,PSCOMP_1:def 6;
A43: k.0<=k.s1 by A2,A7,A39,A40,XREAL_1:7;
A44: k.s1<=k.s2 by A15,A40,A42,XREAL_1:7;
A45: 0 in [.0,1.] by XXREAL_1:1;
then
A46: [.0,s2.] c= [.0,1.] by A41,XXREAL_2:def 12;
reconsider B=[.0,s2.] as Subset of I[01] by A41,A45,BORSUK_1:40
,XXREAL_2:def 12;
A47: B is connected by A26,A41,A45,BORSUK_1:40,BORSUK_4:24;
A48: 0 in B by A26,XXREAL_1:1;
A49: s2 in B by A26,XXREAL_1:1;
consider xc being Point of I[01] such that
A50: xc in B and
A51: k.xc =k.s1 by A31,A35,A43,A44,A47,A48,A49,TOPREAL5:5;
reconsider rxc=xc as Real;
A52: for x1,x2 being set st x1 in dom k & x2 in dom k &
k.x1=k.x2 holds x1=x2
proof
let x1,x2 be set;
assume that
A53: x1 in dom k and
A54: x2 in dom k and
A55: k.x1=k.x2;
reconsider r1=x1 as Point of I[01] by A53;
reconsider r2=x2 as Point of I[01] by A54;
A56: k.x1=h.(g1.x1) by A53,FUNCT_1:12
.=h1.(g1.r1)+h2.(g1.r1) by A34
.=(g1.r1)`1+proj2.(g1.r1) by PSCOMP_1:def 5
.=(g1.r1)`1+(g1.r1)`2 by PSCOMP_1:def 6;
A57: k.x2=h.(g1.x2) by A54,FUNCT_1:12
.=h1.(g1.r2)+h2.(g1.r2) by A34
.=(g1.r2)`1+proj2.(g1.r2) by PSCOMP_1:def 5
.=(g1.r2)`1+(g1.r2)`2 by PSCOMP_1:def 6;
A58: g.r1 in Upper_Arc(K) by A30;
A59: g.r2 in Upper_Arc(K) by A30;
reconsider gr1=g.r1 as Point of TOP-REAL 2 by A58;
reconsider gr2=g.r2 as Point of TOP-REAL 2 by A59;
now per cases by A17,A30,XBOOLE_0:def 3;
case
A60: g.r1 in LSeg(|[a,c]|,|[a,d]|) &
g.r2 in LSeg(|[a,c]|,|[a,d]|);
then
A61: (gr1)`1=a by A2,Th1;
(gr2)`1=a by A2,A60,Th1;
then |[(gr1)`1,(gr1)`2]|=g.r2 by A55,A56,A57,A61,EUCLID:53;
then g.r1=g.r2 by EUCLID:53;
hence thesis by A28,A29,FUNCT_1:def 4;
end;
case
A62: g.r1 in LSeg(|[a,c]|,|[a,d]|) &
g.r2 in LSeg(|[a,d]|,|[b,d]|);
then
A63: (gr1)`1=a by A2,Th1;
A64: (gr1 )`2 <=d by A2,A62,Th1;
A65: (gr2)`2=d by A1,A62,Th3;
A66: a <=(gr2)`1 by A1,A62,Th3;
A67: a+(gr1)`2=(gr2)`1 +d by A1,A55,A56,A57,A62,A63,Th3;
A68: now
assume a<>gr2`1;
then a<gr2`1 by A66,XXREAL_0:1;
hence contradiction by A64,A67,XREAL_1:8;
end;
now
assume gr1`2<>d;
then d>gr1`2 by A64,XXREAL_0:1;
hence contradiction by A55,A56,A57,A63,A65,A66,XREAL_1:8;
end;
then |[(gr1)`1,(gr1)`2]|=g.r2 by A63,A65,A68,EUCLID:53;
then g.r1=g.r2 by EUCLID:53;
hence thesis by A28,A29,FUNCT_1:def 4;
end;
case
A69: g.r1 in LSeg(|[a,d]|,|[b,d]|) &
g.r2 in LSeg(|[a,c]|,|[a,d]|);
then
A70: (gr2)`1=a by A2,Th1;
A71: (gr2 )`2 <=d by A2,A69,Th1;
A72: (gr1)`2=d by A1,A69,Th3;
A73: a <=(gr1)`1 by A1,A69,Th3;
A74: a+(gr2)`2=(gr1)`1 +d by A1,A55,A56,A57,A69,A70,Th3;
A75: now
assume a<>gr1`1;
then a<gr1`1 by A73,XXREAL_0:1;
hence contradiction by A71,A74,XREAL_1:8;
end;
now
assume gr2`2<>d;
then d>gr2`2 by A71,XXREAL_0:1;
hence contradiction by A55,A56,A57,A70,A72,A73,XREAL_1:8;
end;
then |[(gr2)`1,(gr2)`2]|=g.r1 by A70,A72,A75,EUCLID:53;
then g.r1=g.r2 by EUCLID:53;
hence thesis by A28,A29,FUNCT_1:def 4;
end;
case
A76: g.r1 in LSeg(|[a,d]|,|[b,d]|) &
g.r2 in LSeg(|[a,d]|,|[b,d]|);
then
A77: (gr1)`2=d by A1,Th3;
(gr2)`2=d by A1,A76,Th3;
then |[(gr1)`1,(gr1)`2]|=g.r2 by A55,A56,A57,A77,EUCLID:53;
then g.r1=g.r2 by EUCLID:53;
hence thesis by A28,A29,FUNCT_1:def 4;
end;
end;
hence thesis;
end;
A78: dom k=[.0,1.] by BORSUK_1:40,FUNCT_2:def 1;
then s1 in dom k by A23,A24,XXREAL_1:1;
then rxc=s1 by A46,A50,A51,A52,A78;
hence contradiction by A37,A50,XXREAL_1:1;
end;
hence thesis;
end;
then LE p1,p2,Upper_Arc(K),W-min(K),E-max(K) by A18,A19,JORDAN5C:def 3;
hence thesis by A18,A19,JORDAN6:def 10;
end;
case
A79: p2 in LSeg(|[b,d]|,|[b,c]|);
then
A80: p2`1 =b by TOPREAL3:11;
Lower_Arc(K)=LSeg(|[b,d]|,|[b,c]|) \/ LSeg(|[b,c]|,|[a,c]|) by A1,A2
,Th52;
then
A81: LSeg(|[b,d]|,|[b,c]|) c= Lower_Arc(K) by XBOOLE_1:7;
p2 <> W-min(K) by A1,A13,A80,EUCLID:52;
hence thesis by A3,A5,A79,A81,JORDAN6:def 10;
end;
case
A82: p2 in LSeg(|[b,c]|,|[a,c]|) & p2<>W-min(K);
Lower_Arc(K)=LSeg(|[b,d]|,|[b,c]|) \/ LSeg(|[b,c]|,|[a,c]|) by A1,A2
,Th52;
then LSeg(|[b,c]|,|[a,c]|) c= Lower_Arc(K) by XBOOLE_1:7;
hence thesis by A3,A5,A82,JORDAN6:def 10;
end;
end;
hence thesis;
end;
end;
theorem Th61:
for a,b,c,d being Real,p1,p2 being Point of TOP-REAL 2
st a<b & c <d & p1 in LSeg(|[b,d]|,|[b,c]|)
holds LE p1,p2,rectangle(a,b,c,d) iff p2 in LSeg(|[b,d]|,|[b,c]|) & p1`2>=p2`2
or p2 in LSeg(|[b,c]|,|[a,c]|) & p2<>W-min rectangle(a,b,c,d)
proof
let a,b,c,d be Real,p1,p2 be Point of TOP-REAL 2;
set K = rectangle(a,b,c,d);
assume that
A1: a<b and
A2: c <d and
A3: p1 in LSeg(|[b,d]|,|[b,c]|);
A4: K is being_simple_closed_curve by A1,A2,Th50;
A5: p1`1=b by A2,A3,Th1;
A6: c <=p1`2 by A2,A3,Th1;
A7: p1`2 <= d by A2,A3,Th1;
thus LE p1,p2,K implies p2 in LSeg(|[b,d]|,|[b,c]|) & p1`2>=p2`2
or p2 in LSeg(|[b,c]|,|[a,c]|) & p2<>W-min(K)
proof
assume
A8: LE p1,p2,K;
then
A9: p1 in K by A4,JORDAN7:5;
A10: p2 in K by A4,A8,JORDAN7:5;
K= LSeg(|[ a,c ]|, |[ a,d ]|) \/ LSeg(|[ a,d ]|, |[ b,d ]|)
\/ (LSeg(|[ a,c ]|, |[ b,c ]|) \/ LSeg(|[ b,c ]|, |[ b,d ]|))
by SPPOL_2:def 3
.=LSeg(|[a,c]|,|[a,d]|) \/ LSeg(|[a,d]|,|[b,d]|)
\/ LSeg(|[b,d]|,|[b,c]|) \/ LSeg(|[b,c]|,|[a,c]|) by XBOOLE_1:4;
then p2 in LSeg(|[a,c]|,|[a,d]|) \/ LSeg(|[a,d]|,|[b,d]|)
\/ LSeg(|[b,d]|,|[b,c]|) or
p2 in LSeg(|[b,c]|,|[a,c]|) by A10,XBOOLE_0:def 3;
then
A11: p2 in LSeg(|[a,c]|,|[a,d]|) \/ LSeg(|[a,d]|,|[b,d]|) or
p2 in LSeg(|[b,d]|,|[b,c]|) or
p2 in LSeg(|[b,c]|,|[a,c]|) by XBOOLE_0:def 3;
now per cases by A11,XBOOLE_0:def 3;
case p2 in LSeg(|[a,c]|,|[a,d]|);
then LE p2,p1,K by A1,A2,A3,Th59;
hence thesis by A1,A2,A3,A8,Th50,JORDAN6:57;
end;
case p2 in LSeg(|[a,d]|,|[b,d]|);
then LE p2,p1,K by A1,A2,A3,Th60;
hence thesis by A1,A2,A3,A8,Th50,JORDAN6:57;
end;
case p2 in LSeg(|[b,d]|,|[b,c]|);
hence thesis by A1,A2,A3,A8,Th57;
end;
case
A12: p2 in LSeg(|[b,c]|,|[a,c]|);
now per cases;
case p2=W-min(K);
then LE p2,p1,K by A4,A9,JORDAN7:3;
hence thesis by A1,A2,A3,A8,Th50,JORDAN6:57;
end;
case p2<>W-min(K);
hence thesis by A12;
end;
end;
hence thesis;
end;
end;
hence thesis;
end;
thus
p2 in LSeg(|[b,d]|,|[b,c]|)& p1`2>=p2`2
or p2 in LSeg(|[b,c]|,|[a,c]|) & p2<>W-min(K) implies LE p1,p2,K
proof
assume that
A13: p2 in LSeg(|[b,d]|,|[b,c]|)& p1`2>=p2`2
or p2 in LSeg(|[b,c]|,|[a,c]|) & p2<>W-min(K);
now per cases by A13;
case
A14: p2 in LSeg(|[b,d]|,|[b,c]|)& p1`2>=p2`2;
then
A15: p2`1=b by A2,Th1;
W-min K=|[a,c]| by A1,A2,Th46;
then
A16: p2 <> W-min(K) by A1,A15,EUCLID:52;
A17: Lower_Arc(K)=LSeg(|[b,d]|,|[b,c]|) \/ LSeg(|[b,c]|,|[a,c]|)
by A1,A2,Th52;
then
A18: p2 in Lower_Arc(K) by A14,XBOOLE_0:def 3;
A19: p1 in Lower_Arc(K) by A3,A17,XBOOLE_0:def 3;
for g being Function of I[01], (TOP-REAL 2)|Lower_Arc(K),
s1, s2 being Real st g is being_homeomorphism
& g.0 = E-max(K) & g.1 = W-min(K) & g.s1 = p1 & 0 <= s1 & s1 <= 1
& g.s2 = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2
proof
let g be Function of I[01], (TOP-REAL 2)|Lower_Arc(K),
s1, s2 be Real;
assume that
A20: g is being_homeomorphism and
A21: g.0 = E-max(K) and g.1 = W-min(K) and
A22: g.s1 = p1 and
A23: 0 <= s1 and
A24: s1 <= 1 and
A25: g.s2 = p2 and
A26: 0 <= s2 and
A27: s2 <= 1;
A28: dom g=the carrier of I[01] by FUNCT_2:def 1;
A29: g is one-to-one by A20,TOPS_2:def 5;
A30: the carrier of ((TOP-REAL 2)|Lower_Arc(K))
=Lower_Arc(K) by PRE_TOPC:8;
then reconsider g1=g as Function of I[01],TOP-REAL 2 by FUNCT_2:7;
g is continuous by A20,TOPS_2:def 5;
then
A31: g1 is continuous by PRE_TOPC:26;
reconsider h1=proj1 as Function of TOP-REAL 2,R^1 by TOPMETR:17;
reconsider h2=proj2 as Function of TOP-REAL 2,R^1 by TOPMETR:17;
reconsider hh1=h1 as Function of the TopStruct of TOP-REAL 2,R^1;
reconsider hh2=h2 as Function of the TopStruct of TOP-REAL 2,R^1;
A32: the TopStruct of TOP-REAL 2
= (the TopStruct of TOP-REAL 2)|([#](the TopStruct of TOP-REAL 2))
by TSEP_1:3
.= the TopStruct of ((TOP-REAL 2)|([#](TOP-REAL 2))) by PRE_TOPC:36
.= (TOP-REAL 2)|([#](TOP-REAL 2));
then (for p being Point of (TOP-REAL 2)|([#](TOP-REAL 2)) holds
hh1.p=proj1.p) implies hh1 is continuous by JGRAPH_2:29;
then
A33: (for p being Point of (TOP-REAL 2)|([#]TOP-REAL 2)holds
hh1.p=proj1.p) implies h1 is continuous by PRE_TOPC:32;
(for p being Point of (TOP-REAL 2)|([#](TOP-REAL 2)) holds
hh2.p=proj2.p) implies hh2 is continuous by A32,JGRAPH_2:30;
then (for p being Point of (TOP-REAL 2)|([#](TOP-REAL 2)) holds
hh2.p=proj2.p) implies h2 is continuous by PRE_TOPC:32;
then consider h being Function of TOP-REAL 2,R^1 such that
A34: for p being Point of TOP-REAL 2, r1,r2 being Real st h1.p=r1 &
h2.p=r2 holds h.p=r1+r2 and
A35: h is continuous by A33,JGRAPH_2:19;
reconsider k=h*g1 as Function of I[01],R^1;
A36: E-max K=|[b,d]| by A1,A2,Th46;
now
assume
A37: s1>s2;
A38: dom g=[.0,1.] by BORSUK_1:40,FUNCT_2:def 1;
0 in [.0,1.] by XXREAL_1:1;
then
A39: k.0=h.(E-max(K)) by A21,A38,FUNCT_1:13
.=h1.(E-max(K))+h2.(E-max(K)) by A34
.=(E-max(K))`1+proj2.(E-max(K)) by PSCOMP_1:def 5
.=(E-max(K))`1+(E-max(K))`2 by PSCOMP_1:def 6
.=(E-max(K))`1+d by A36,EUCLID:52
.=b+d by A36,EUCLID:52;
s1 in [.0,1.] by A23,A24,XXREAL_1:1;
then
A40: k.s1=h.p1 by A22,A38,FUNCT_1:13
.=proj1.p1 +proj2.p1 by A34
.=p1`1+proj2.p1 by PSCOMP_1:def 5
.=b+p1`2 by A5,PSCOMP_1:def 6;
A41: s2 in [.0,1.] by A26,A27,XXREAL_1:1;
then
A42: k.s2=h.p2 by A25,A38,FUNCT_1:13
.=proj1.p2 +proj2.p2 by A34
.=p2`1+proj2.p2 by PSCOMP_1:def 5
.=b+p2`2 by A15,PSCOMP_1:def 6;
A43: k.0>=k.s1 by A7,A39,A40,XREAL_1:7;
A44: k.s1>=k.s2 by A14,A40,A42,XREAL_1:7;
A45: 0 in [.0,1.] by XXREAL_1:1;
then
A46: [.0,s2.] c= [.0,1.] by A41,XXREAL_2:def 12;
reconsider B=[.0,s2.] as Subset of I[01] by A41,A45,BORSUK_1:40
,XXREAL_2:def 12;
A47: B is connected by A26,A41,A45,BORSUK_1:40,BORSUK_4:24;
A48: 0 in B by A26,XXREAL_1:1;
A49: s2 in B by A26,XXREAL_1:1;
consider xc being Point of I[01] such that
A50: xc in B and
A51: k.xc =k.s1 by A31,A35,A43,A44,A47,A48,A49,TOPREAL5:5;
reconsider rxc=xc as Real;
A52: for x1,x2 being set st x1 in dom k & x2 in dom k &
k.x1=k.x2 holds x1=x2
proof
let x1,x2 be set;
assume that
A53: x1 in dom k and
A54: x2 in dom k and
A55: k.x1=k.x2;
reconsider r1=x1 as Point of I[01] by A53;
reconsider r2=x2 as Point of I[01] by A54;
A56: k.x1=h.(g1.x1) by A53,FUNCT_1:12
.=h1.(g1.r1)+h2.(g1.r1) by A34
.=(g1.r1)`1+proj2.(g1.r1) by PSCOMP_1:def 5
.=(g1.r1)`1+(g1.r1)`2 by PSCOMP_1:def 6;
A57: k.x2=h.(g1.x2) by A54,FUNCT_1:12
.=h1.(g1.r2)+h2.(g1.r2) by A34
.=(g1.r2)`1+proj2.(g1.r2) by PSCOMP_1:def 5
.=(g1.r2)`1+(g1.r2)`2 by PSCOMP_1:def 6;
A58: g.r1 in Lower_Arc(K) by A30;
A59: g.r2 in Lower_Arc(K) by A30;
reconsider gr1=g.r1 as Point of TOP-REAL 2 by A58;
reconsider gr2=g.r2 as Point of TOP-REAL 2 by A59;
now per cases by A17,A30,XBOOLE_0:def 3;
case
A60: g.r1 in LSeg(|[b,d]|,|[b,c]|) &
g.r2 in LSeg(|[b,d]|,|[b,c]|);
then
A61: (gr1)`1=b by A2,Th1;
(gr2)`1=b by A2,A60,Th1;
then |[(gr1)`1,(gr1)`2]|=g.r2 by A55,A56,A57,A61,EUCLID:53;
then g.r1=g.r2 by EUCLID:53;
hence thesis by A28,A29,FUNCT_1:def 4;
end;
case
A62: g.r1 in LSeg(|[b,d]|,|[b,c]|) &
g.r2 in LSeg(|[b,c]|,|[a,c]|);
then
A63: (gr1)`1=b by A2,Th1;
A64: c <=(gr1)`2 by A2,A62,Th1;
A65: (gr2)`2=c by A1,A62,Th3;
A66: (gr2)`1 <=b by A1,A62,Th3;
A67: b+(gr1)`2=(gr2)`1 +c by A2,A55,A56,A57,A62,A65,Th1;
A68: now
assume b<>gr2`1;
then b>gr2`1 by A66,XXREAL_0:1;
hence contradiction by A55,A56,A57,A63,A64,A65,XREAL_1:8;
end;
now
assume gr1`2<> c;
then c <gr1`2 by A64,XXREAL_0:1;
hence contradiction by A66,A67,XREAL_1:8;
end;
then |[(gr1)`1,(gr1)`2]|=g.r2 by A63,A65,A68,EUCLID:53;
then g.r1=g.r2 by EUCLID:53;
hence thesis by A28,A29,FUNCT_1:def 4;
end;
case
A69: g.r1 in LSeg(|[b,c]|,|[a,c]|) &
g.r2 in LSeg(|[b,d]|,|[b,c]|);
then
A70: (gr2)`1=b by A2,Th1;
A71: c <=(gr2)`2 by A2,A69,Th1;
A72: (gr1)`2=c by A1,A69,Th3;
A73: (gr1)`1 <=b by A1,A69,Th3;
A74: b+(gr2)`2=(gr1)`1 +c by A1,A55,A56,A57,A69,A70,Th3;
A75: now
assume b<>gr1`1;
then b>gr1`1 by A73,XXREAL_0:1;
hence contradiction by A71,A74,XREAL_1:8;
end;
now
assume gr2`2<> c;
then c < gr2`2 by A71,XXREAL_0:1;
hence contradiction by A55,A56,A57,A70,A72,A73,XREAL_1:8;
end;
then |[(gr2)`1,(gr2)`2]|=g.r1 by A70,A72,A75,EUCLID:53;
then g.r1=g.r2 by EUCLID:53;
hence thesis by A28,A29,FUNCT_1:def 4;
end;
case
A76: g.r1 in LSeg(|[b,c]|,|[a,c]|) &
g.r2 in LSeg(|[b,c]|,|[a,c]|);
then
A77: (gr1)`2=c by A1,Th3;
(gr2)`2=c by A1,A76,Th3;
then |[(gr1)`1,(gr1)`2]|=g.r2 by A55,A56,A57,A77,EUCLID:53;
then g.r1=g.r2 by EUCLID:53;
hence thesis by A28,A29,FUNCT_1:def 4;
end;
end;
hence thesis;
end;
A78: dom k=[.0,1.] by BORSUK_1:40,FUNCT_2:def 1;
then s1 in dom k by A23,A24,XXREAL_1:1;
then rxc=s1 by A46,A50,A51,A52,A78;
hence contradiction by A37,A50,XXREAL_1:1;
end;
hence thesis;
end;
then LE p1,p2,Lower_Arc(K),E-max(K),W-min(K) by A18,A19,JORDAN5C:def 3;
hence thesis by A16,A18,A19,JORDAN6:def 10;
end;
case
A79: p2 in LSeg(|[b,c]|,|[a,c]|) & p2<>W-min(K);
then
A80: p2`2=c by A1,Th3;
A81: p2`1 <= b by A1,A79,Th3;
A82: Lower_Arc(K)=LSeg(|[b,d]|,|[b,c]|) \/ LSeg(|[b,c]|,|[a,c]|)
by A1,A2,Th52;
then
A83: p2 in Lower_Arc(K) by A79,XBOOLE_0:def 3;
A84: p1 in Lower_Arc(K) by A3,A82,XBOOLE_0:def 3;
for g being Function of I[01], (TOP-REAL 2)|Lower_Arc(K),
s1, s2 being Real st g is being_homeomorphism
& g.0 = E-max(K) & g.1 = W-min(K) & g.s1 = p1 & 0 <= s1 & s1 <= 1
& g.s2 = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2
proof
let g be Function of I[01], (TOP-REAL 2)|Lower_Arc(K),
s1, s2 be Real;
assume that
A85: g is being_homeomorphism and
A86: g.0 = E-max(K) and g.1 = W-min(K) and
A87: g.s1 = p1 and
A88: 0 <= s1 and
A89: s1 <= 1 and
A90: g.s2 = p2 and
A91: 0 <= s2 and
A92: s2 <= 1;
A93: dom g=the carrier of I[01] by FUNCT_2:def 1;
A94: g is one-to-one by A85,TOPS_2:def 5;
A95: the carrier of ((TOP-REAL 2)|Lower_Arc(K))
=Lower_Arc(K) by PRE_TOPC:8;
then reconsider g1=g as Function of I[01],TOP-REAL 2 by FUNCT_2:7;
g is continuous by A85,TOPS_2:def 5;
then
A96: g1 is continuous by PRE_TOPC:26;
reconsider h1=proj1 as Function of TOP-REAL 2,R^1 by TOPMETR:17;
reconsider h2=proj2 as Function of TOP-REAL 2,R^1 by TOPMETR:17;
reconsider hh1=h1 as Function of the TopStruct of TOP-REAL 2,R^1;
reconsider hh2=h2 as Function of the TopStruct of TOP-REAL 2,R^1;
A97: the TopStruct of TOP-REAL 2
= (the TopStruct of TOP-REAL 2)|([#](the TopStruct of TOP-REAL 2))
by TSEP_1:3
.= the TopStruct of ((TOP-REAL 2)|([#](TOP-REAL 2))) by PRE_TOPC:36
.= (TOP-REAL 2)|([#](TOP-REAL 2));
then (for p being Point of (TOP-REAL 2)|([#](TOP-REAL 2)) holds
hh1.p=proj1.p) implies hh1 is continuous by JGRAPH_2:29;
then
A98: (for p being Point of (TOP-REAL 2)|([#]TOP-REAL 2)holds
hh1.p=proj1.p) implies h1 is continuous by PRE_TOPC:32;
(for p being Point of (TOP-REAL 2)|([#](TOP-REAL 2)) holds
hh2.p=proj2.p) implies hh2 is continuous by A97,JGRAPH_2:30;
then (for p being Point of (TOP-REAL 2)|([#](TOP-REAL 2)) holds
hh2.p=proj2.p) implies h2 is continuous by PRE_TOPC:32;
then consider h being Function of TOP-REAL 2,R^1 such that
A99: for p being Point of TOP-REAL 2, r1,r2 being Real st h1.p=r1 &
h2.p=r2 holds h.p=r1+r2 and
A100: h is continuous by A98,JGRAPH_2:19;
reconsider k=h*g1 as Function of I[01],R^1;
A101: E-max K=|[b,d]| by A1,A2,Th46;
now
assume
A102: s1>s2;
A103: dom g=[.0,1.] by BORSUK_1:40,FUNCT_2:def 1;
0 in [.0,1.] by XXREAL_1:1;
then
A104: k.0=h.(E-max(K)) by A86,A103,FUNCT_1:13
.=h1.(E-max(K))+h2.(E-max(K)) by A99
.=(E-max(K))`1+proj2.(E-max(K)) by PSCOMP_1:def 5
.=(E-max(K))`1+(E-max(K))`2 by PSCOMP_1:def 6
.=(E-max(K))`1+d by A101,EUCLID:52
.=b+d by A101,EUCLID:52;
s1 in [.0,1.] by A88,A89,XXREAL_1:1;
then
A105: k.s1=h.p1 by A87,A103,FUNCT_1:13
.=proj1.p1 +proj2.p1 by A99
.=p1`1+proj2.p1 by PSCOMP_1:def 5
.=b+p1`2 by A5,PSCOMP_1:def 6;
A106: s2 in [.0,1.] by A91,A92,XXREAL_1:1;
then
A107: k.s2=h.p2 by A90,A103,FUNCT_1:13
.=proj1.p2 +proj2.p2 by A99
.=p2`1+proj2.p2 by PSCOMP_1:def 5
.=p2`1+c by A80,PSCOMP_1:def 6;
A108: k.0>=k.s1 by A7,A104,A105,XREAL_1:7;
A109: k.s1>=k.s2 by A6,A81,A105,A107,XREAL_1:7;
A110: 0 in [.0,1.] by XXREAL_1:1;
then
A111: [.0,s2.] c= [.0,1.] by A106,XXREAL_2:def 12;
reconsider B=[.0,s2.] as Subset of I[01] by A106,A110,BORSUK_1:40
,XXREAL_2:def 12;
A112: B is connected by A91,A106,A110,BORSUK_1:40,BORSUK_4:24;
A113: 0 in B by A91,XXREAL_1:1;
A114: s2 in B by A91,XXREAL_1:1;
consider xc being Point of I[01] such that
A115: xc in B and
A116: k.xc =k.s1 by A96,A100,A108,A109,A112,A113,A114,TOPREAL5:5;
reconsider rxc=xc as Real;
A117: for x1,x2 being set st x1 in dom k & x2 in dom k &
k.x1=k.x2 holds x1=x2
proof
let x1,x2 be set;
assume that
A118: x1 in dom k and
A119: x2 in dom k and
A120: k.x1=k.x2;
reconsider r1=x1 as Point of I[01] by A118;
reconsider r2=x2 as Point of I[01] by A119;
A121: k.x1=h.(g1.x1) by A118,FUNCT_1:12
.=h1.(g1.r1)+h2.(g1.r1) by A99
.=(g1.r1)`1+proj2.(g1.r1) by PSCOMP_1:def 5
.=(g1.r1)`1+(g1.r1)`2 by PSCOMP_1:def 6;
A122: k.x2=h.(g1.x2) by A119,FUNCT_1:12
.=h1.(g1.r2)+h2.(g1.r2) by A99
.=(g1.r2)`1+proj2.(g1.r2) by PSCOMP_1:def 5
.=(g1.r2)`1+(g1.r2)`2 by PSCOMP_1:def 6;
A123: g.r1 in Lower_Arc(K) by A95;
A124: g.r2 in Lower_Arc(K) by A95;
reconsider gr1=g.r1 as Point of TOP-REAL 2 by A123;
reconsider gr2=g.r2 as Point of TOP-REAL 2 by A124;
now per cases by A82,A95,XBOOLE_0:def 3;
case
A125: g.r1 in LSeg(|[b,d]|,|[b,c]|) &
g.r2 in LSeg(|[b,d]|,|[b,c]|);
then
A126: (gr1)`1=b by A2,Th1;
(gr2)`1=b by A2,A125,Th1;
then |[(gr1)`1,(gr1)`2]|=g.r2 by A120,A121,A122,A126,EUCLID:53;
then g.r1=g.r2 by EUCLID:53;
hence thesis by A93,A94,FUNCT_1:def 4;
end;
case
A127: g.r1 in LSeg(|[b,d]|,|[b,c]|) &
g.r2 in LSeg(|[b,c]|,|[a,c]|);
then
A128: (gr1)`1=b by A2,Th1;
A129: c <=(gr1)`2 by A2,A127,Th1;
A130: (gr2)`2=c by A1,A127,Th3;
A131: (gr2)`1 <=b by A1,A127,Th3;
A132: b+(gr1)`2=(gr2)`1 +c by A2,A120,A121,A122,A127,A130,Th1;
A133: now
assume b<>gr2`1;
then b>gr2`1 by A131,XXREAL_0:1;
hence contradiction by A120,A121,A122,A128,A129,A130,
XREAL_1:8;
end;
now
assume gr1`2<> c;
then c <gr1`2 by A129,XXREAL_0:1;
hence contradiction by A131,A132,XREAL_1:8;
end;
then |[(gr1)`1,(gr1)`2]|=g.r2 by A128,A130,A133,EUCLID:53;
then g.r1=g.r2 by EUCLID:53;
hence thesis by A93,A94,FUNCT_1:def 4;
end;
case
A134: g.r1 in LSeg(|[b,c]|,|[a,c]|) &
g.r2 in LSeg(|[b,d]|,|[b,c]|);
then
A135: (gr2)`1=b by A2,Th1;
A136: c <=(gr2)`2 by A2,A134,Th1;
A137: (gr1)`2=c by A1,A134,Th3;
A138: (gr1)`1 <=b by A1,A134,Th3;
A139: b+(gr2)`2=(gr1)`1 +c by A1,A120,A121,A122,A134,A135,Th3;
A140: now
assume b<>gr1`1;
then b>gr1`1 by A138,XXREAL_0:1;
hence contradiction by A136,A139,XREAL_1:8;
end;
now
assume gr2`2<> c;
then c < gr2`2 by A136,XXREAL_0:1;
hence contradiction by A120,A121,A122,A135,A137,A138,XREAL_1:8;
end;
then |[(gr2)`1,(gr2)`2]|=g.r1 by A135,A137,A140,EUCLID:53;
then g.r1=g.r2 by EUCLID:53;
hence thesis by A93,A94,FUNCT_1:def 4;
end;
case
A141: g.r1 in LSeg(|[b,c]|,|[a,c]|) &
g.r2 in LSeg(|[b,c]|,|[a,c]|);
then
A142: (gr1)`2=c by A1,Th3;
(gr2)`2=c by A1,A141,Th3;
then |[(gr1)`1,(gr1)`2]|=g.r2 by A120,A121,A122,A142,EUCLID:53;
then g.r1=g.r2 by EUCLID:53;
hence thesis by A93,A94,FUNCT_1:def 4;
end;
end;
hence thesis;
end;
A143: dom k=[.0,1.] by BORSUK_1:40,FUNCT_2:def 1;
then s1 in dom k by A88,A89,XXREAL_1:1;
then rxc=s1 by A111,A115,A116,A117,A143;
hence contradiction by A102,A115,XXREAL_1:1;
end;
hence thesis;
end;
then LE p1,p2,Lower_Arc(K),E-max(K),W-min(K) by A83,A84,JORDAN5C:def 3;
hence thesis by A79,A83,A84,JORDAN6:def 10;
end;
end;
hence thesis;
end;
end;
theorem Th62:
for a,b,c,d being Real,p1,p2 being Point of TOP-REAL 2
st a<b & c < d & p1 in LSeg(|[b,c]|,|[a,c]|)& p1<>W-min rectangle(a,b,c,d)
holds LE p1,p2,rectangle(a,b,c,d) iff p2 in LSeg(|[b,c]|,|[a,c]|) &
p1`1>=p2`1 & p2<>W-min rectangle(a,b,c,d)
proof
let a,b,c,d be Real,p1,p2 be Point of TOP-REAL 2;
set K = rectangle(a,b,c,d);
assume that
A1: a<b and
A2: c <d and
A3: p1 in LSeg(|[b,c]|,|[a,c]|) and
A4: p1<>W-min(K);
A5: K is being_simple_closed_curve by A1,A2,Th50;
A6: p1`2=c by A1,A3,Th3;
A7: p1`1 <= b by A1,A3,Th3;
thus LE p1,p2,K implies
p2 in LSeg(|[b,c]|,|[a,c]|) & p1`1>=p2`1 & p2<>W-min(K)
proof
assume
A8: LE p1,p2,K;
then
A9: p2 in K by A5,JORDAN7:5;
K= LSeg(|[ a,c ]|, |[ a,d ]|) \/ LSeg(|[ a,d ]|, |[ b,d ]|)
\/ (LSeg(|[ a,c ]|, |[ b,c ]|) \/ LSeg(|[ b,c ]|, |[ b,d ]|))
by SPPOL_2:def 3
.=LSeg(|[a,c]|,|[a,d]|) \/ LSeg(|[a,d]|,|[b,d]|)
\/ LSeg(|[b,d]|,|[b,c]|) \/ LSeg(|[b,c]|,|[a,c]|) by XBOOLE_1:4;
then p2 in LSeg(|[a,c]|,|[a,d]|) \/ LSeg(|[a,d]|,|[b,d]|)
\/ LSeg(|[b,d]|,|[b,c]|) or
p2 in LSeg(|[b,c]|,|[a,c]|) by A9,XBOOLE_0:def 3;
then
A10: p2 in LSeg(|[a,c]|,|[a,d]|) \/ LSeg(|[a,d]|,|[b,d]|) or
p2 in LSeg(|[b,d]|,|[b,c]|) or
p2 in LSeg(|[b,c]|,|[a,c]|) by XBOOLE_0:def 3;
now per cases by A10,XBOOLE_0:def 3;
case p2 in LSeg(|[a,c]|,|[a,d]|);
then LE p2,p1,K by A1,A2,A3,A4,Th59;
hence thesis by A1,A2,A3,A4,A8,Th50,JORDAN6:57;
end;
case p2 in LSeg(|[a,d]|,|[b,d]|);
then LE p2,p1,K by A1,A2,A3,A4,Th60;
hence thesis by A1,A2,A3,A4,A8,Th50,JORDAN6:57;
end;
case p2 in LSeg(|[b,d]|,|[b,c]|);
then LE p2,p1,K by A1,A2,A3,A4,Th61;
hence thesis by A1,A2,A3,A4,A8,Th50,JORDAN6:57;
end;
case p2 in LSeg(|[b,c]|,|[a,c]|);
hence thesis by A1,A2,A3,A4,A8,Th58;
end;
end;
hence thesis;
end;
thus
p2 in LSeg(|[b,c]|,|[a,c]|)& p1`1>=p2`1 & p2<>W-min(K) implies LE p1,p2,K
proof
assume that
A11: p2 in LSeg(|[b,c]|,|[a,c]|) and
A12: p1`1>=p2`1 and
A13: p2<>W-min(K);
now per cases by A11,A12;
case
A14: p2 in LSeg(|[b,c]|,|[a,c]|)& p1`1>=p2`1;
then
A15: p2`2=c by A1,Th3;
A16: Lower_Arc(K)=LSeg(|[b,d]|,|[b,c]|) \/ LSeg(|[b,c]|,|[a,c]|)
by A1,A2,Th52;
then
A17: p2 in Lower_Arc(K) by A14,XBOOLE_0:def 3;
A18: p1 in Lower_Arc(K) by A3,A16,XBOOLE_0:def 3;
for g being Function of I[01], (TOP-REAL 2)|Lower_Arc(K),
s1, s2 being Real st g is being_homeomorphism
& g.0 = E-max(K) & g.1 = W-min(K) & g.s1 = p1 & 0 <= s1 & s1 <= 1
& g.s2 = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2
proof
let g be Function of I[01], (TOP-REAL 2)|Lower_Arc(K),
s1, s2 be Real;
assume that
A19: g is being_homeomorphism and
A20: g.0 = E-max(K) and g.1 = W-min(K) and
A21: g.s1 = p1 and
A22: 0 <= s1 and
A23: s1 <= 1 and
A24: g.s2 = p2 and
A25: 0 <= s2 and
A26: s2 <= 1;
A27: dom g=the carrier of I[01] by FUNCT_2:def 1;
A28: g is one-to-one by A19,TOPS_2:def 5;
A29: the carrier of ((TOP-REAL 2)|Lower_Arc(K))
=Lower_Arc(K) by PRE_TOPC:8;
then reconsider g1=g as Function of I[01],TOP-REAL 2 by FUNCT_2:7;
g is continuous by A19,TOPS_2:def 5;
then
A30: g1 is continuous by PRE_TOPC:26;
reconsider h1=proj1 as Function of TOP-REAL 2,R^1 by TOPMETR:17;
reconsider h2=proj2 as Function of TOP-REAL 2,R^1 by TOPMETR:17;
reconsider hh1=h1 as Function of the TopStruct of TOP-REAL 2,R^1;
reconsider hh2=h2 as Function of the TopStruct of TOP-REAL 2,R^1;
A31: the TopStruct of TOP-REAL 2
= (the TopStruct of TOP-REAL 2)|([#](the TopStruct of TOP-REAL 2))
by TSEP_1:3
.= the TopStruct of ((TOP-REAL 2)|([#](TOP-REAL 2))) by PRE_TOPC:36
.= (TOP-REAL 2)|([#](TOP-REAL 2));
then (for p being Point of (TOP-REAL 2)|([#](TOP-REAL 2)) holds
hh1.p=proj1.p) implies hh1 is continuous by JGRAPH_2:29;
then
A32: (for p being Point of (TOP-REAL 2)|([#]TOP-REAL 2)holds
hh1.p=proj1.p) implies h1 is continuous by PRE_TOPC:32;
(for p being Point of (TOP-REAL 2)|([#](TOP-REAL 2)) holds
hh2.p=proj2.p) implies hh2 is continuous by A31,JGRAPH_2:30;
then (for p being Point of (TOP-REAL 2)|([#](TOP-REAL 2)) holds
hh2.p=proj2.p) implies h2 is continuous by PRE_TOPC:32;
then consider h being Function of TOP-REAL 2,R^1 such that
A33: for p being Point of TOP-REAL 2, r1,r2 being Real st h1.p=r1 &
h2.p=r2 holds h.p=r1+r2 and
A34: h is continuous by A32,JGRAPH_2:19;
reconsider k=h*g1 as Function of I[01],R^1;
A35: E-max K=|[b,d]| by A1,A2,Th46;
now
assume
A36: s1>s2;
A37: dom g=[.0,1.] by BORSUK_1:40,FUNCT_2:def 1;
0 in [.0,1.] by XXREAL_1:1;
then
A38: k.0=h.(E-max(K)) by A20,A37,FUNCT_1:13
.=h1.(E-max(K))+h2.(E-max(K)) by A33
.=(E-max(K))`1+proj2.(E-max(K)) by PSCOMP_1:def 5
.=(E-max(K))`1+(E-max(K))`2 by PSCOMP_1:def 6
.=(E-max(K))`1+d by A35,EUCLID:52
.=b+d by A35,EUCLID:52;
s1 in [.0,1.] by A22,A23,XXREAL_1:1;
then
A39: k.s1=h.p1 by A21,A37,FUNCT_1:13
.=proj1.p1 +proj2.p1 by A33
.=p1`1+proj2.p1 by PSCOMP_1:def 5
.=p1`1+c by A6,PSCOMP_1:def 6;
A40: s2 in [.0,1.] by A25,A26,XXREAL_1:1;
then
A41: k.s2=h.p2 by A24,A37,FUNCT_1:13
.=proj1.p2 +proj2.p2 by A33
.=p2`1+proj2.p2 by PSCOMP_1:def 5
.=p2`1+c by A15,PSCOMP_1:def 6;
A42: k.0>=k.s1 by A2,A7,A38,A39,XREAL_1:7;
A43: k.s1>=k.s2 by A14,A39,A41,XREAL_1:7;
A44: 0 in [.0,1.] by XXREAL_1:1;
then
A45: [.0,s2.] c= [.0,1.] by A40,XXREAL_2:def 12;
reconsider B=[.0,s2.] as Subset of I[01] by A40,A44,BORSUK_1:40
,XXREAL_2:def 12;
A46: B is connected by A25,A40,A44,BORSUK_1:40,BORSUK_4:24;
A47: 0 in B by A25,XXREAL_1:1;
A48: s2 in B by A25,XXREAL_1:1;
consider xc being Point of I[01] such that
A49: xc in B and
A50: k.xc =k.s1 by A30,A34,A42,A43,A46,A47,A48,TOPREAL5:5;
reconsider rxc=xc as Real;
A51: for x1,x2 being set st x1 in dom k & x2 in dom k &
k.x1=k.x2 holds x1=x2
proof
let x1,x2 be set;
assume that
A52: x1 in dom k and
A53: x2 in dom k and
A54: k.x1=k.x2;
reconsider r1=x1 as Point of I[01] by A52;
reconsider r2=x2 as Point of I[01] by A53;
A55: k.x1=h.(g1.x1) by A52,FUNCT_1:12
.=h1.(g1.r1)+h2.(g1.r1) by A33
.=(g1.r1)`1+proj2.(g1.r1) by PSCOMP_1:def 5
.=(g1.r1)`1+(g1.r1)`2 by PSCOMP_1:def 6;
A56: k.x2=h.(g1.x2) by A53,FUNCT_1:12
.=h1.(g1.r2)+h2.(g1.r2) by A33
.=(g1.r2)`1+proj2.(g1.r2) by PSCOMP_1:def 5
.=(g1.r2)`1+(g1.r2)`2 by PSCOMP_1:def 6;
A57: g.r1 in Lower_Arc(K) by A29;
A58: g.r2 in Lower_Arc(K) by A29;
reconsider gr1=g.r1 as Point of TOP-REAL 2 by A57;
reconsider gr2=g.r2 as Point of TOP-REAL 2 by A58;
now per cases by A16,A29,XBOOLE_0:def 3;
case
A59: g.r1 in LSeg(|[b,d]|,|[b,c]|) &
g.r2 in LSeg(|[b,d]|,|[b,c]|);
then
A60: (gr1)`1=b by A2,Th1;
(gr2)`1=b by A2,A59,Th1;
then |[(gr1)`1,(gr1)`2]|=g.r2 by A54,A55,A56,A60,EUCLID:53;
then g.r1=g.r2 by EUCLID:53;
hence thesis by A27,A28,FUNCT_1:def 4;
end;
case
A61: g.r1 in LSeg(|[b,d]|,|[b,c]|) &
g.r2 in LSeg(|[b,c]|,|[a,c]|);
then
A62: (gr1)`1=b by A2,Th1;
A63: c <=(gr1)`2 by A2,A61,Th1;
A64: (gr2)`2=c by A1,A61,Th3;
A65: (gr2)`1 <=b by A1,A61,Th3;
A66: b+(gr1)`2=(gr2)`1 +c by A2,A54,A55,A56,A61,A64,Th1;
A67: now
assume b<>gr2`1;
then b>gr2`1 by A65,XXREAL_0:1;
hence contradiction by A54,A55,A56,A62,A63,A64,XREAL_1:8;
end;
now
assume gr1`2<> c;
then c <gr1`2 by A63,XXREAL_0:1;
hence contradiction by A65,A66,XREAL_1:8;
end;
then |[(gr1)`1,(gr1)`2]|=g.r2 by A62,A64,A67,EUCLID:53;
then g.r1=g.r2 by EUCLID:53;
hence thesis by A27,A28,FUNCT_1:def 4;
end;
case
A68: g.r1 in LSeg(|[b,c]|,|[a,c]|) &
g.r2 in LSeg(|[b,d]|,|[b,c]|);
then
A69: (gr2)`1=b by A2,Th1;
A70: c <=(gr2)`2 by A2,A68,Th1;
A71: (gr1)`2=c by A1,A68,Th3;
A72: (gr1)`1 <=b by A1,A68,Th3;
A73: b+(gr2)`2=(gr1)`1 +c by A1,A54,A55,A56,A68,A69,Th3;
A74: now
assume b<>gr1`1;
then b>gr1`1 by A72,XXREAL_0:1;
hence contradiction by A70,A73,XREAL_1:8;
end;
now
assume gr2`2<> c;
then c < gr2`2 by A70,XXREAL_0:1;
hence contradiction by A54,A55,A56,A69,A71,A72,XREAL_1:8;
end;
then |[(gr2)`1,(gr2)`2]|=g.r1 by A69,A71,A74,EUCLID:53;
then g.r1=g.r2 by EUCLID:53;
hence thesis by A27,A28,FUNCT_1:def 4;
end;
case
A75: g.r1 in LSeg(|[b,c]|,|[a,c]|) &
g.r2 in LSeg(|[b,c]|,|[a,c]|);
then
A76: (gr1)`2=c by A1,Th3;
(gr2)`2=c by A1,A75,Th3;
then |[(gr1)`1,(gr1)`2]|=g.r2 by A54,A55,A56,A76,EUCLID:53;
then g.r1=g.r2 by EUCLID:53;
hence thesis by A27,A28,FUNCT_1:def 4;
end;
end;
hence thesis;
end;
A77: dom k=[.0,1.] by BORSUK_1:40,FUNCT_2:def 1;
then s1 in dom k by A22,A23,XXREAL_1:1;
then rxc=s1 by A45,A49,A50,A51,A77;
hence contradiction by A36,A49,XXREAL_1:1;
end;
hence thesis;
end;
then LE p1,p2,Lower_Arc(K),E-max(K),W-min(K) by A17,A18,JORDAN5C:def 3;
hence thesis by A13,A17,A18,JORDAN6:def 10;
end;
case
A78: p2 in LSeg(|[b,c]|,|[a,c]|) & p2<>W-min(K);
then
A79: p2`2=c by A1,Th3;
A80: Lower_Arc(K)=LSeg(|[b,d]|,|[b,c]|) \/ LSeg(|[b,c]|,|[a,c]|)
by A1,A2,Th52;
then
A81: p2 in Lower_Arc(K) by A78,XBOOLE_0:def 3;
A82: p1 in Lower_Arc(K) by A3,A80,XBOOLE_0:def 3;
for g being Function of I[01], (TOP-REAL 2)|Lower_Arc(K),
s1, s2 being Real st g is being_homeomorphism
& g.0 = E-max(K) & g.1 = W-min(K) & g.s1 = p1 & 0 <= s1 & s1 <= 1
& g.s2 = p2 & 0 <= s2 & s2 <= 1 holds s1 <= s2
proof
let g be Function of I[01], (TOP-REAL 2)|Lower_Arc(K),
s1, s2 be Real;
assume that
A83: g is being_homeomorphism and
A84: g.0 = E-max(K) and g.1 = W-min(K) and
A85: g.s1 = p1 and
A86: 0 <= s1 and
A87: s1 <= 1 and
A88: g.s2 = p2 and
A89: 0 <= s2 and
A90: s2 <= 1;
A91: dom g=the carrier of I[01] by FUNCT_2:def 1;
A92: g is one-to-one by A83,TOPS_2:def 5;
A93: the carrier of ((TOP-REAL 2)|Lower_Arc(K))
=Lower_Arc(K) by PRE_TOPC:8;
then reconsider g1=g as Function of I[01],TOP-REAL 2 by FUNCT_2:7;
g is continuous by A83,TOPS_2:def 5;
then
A94: g1 is continuous by PRE_TOPC:26;
reconsider h1=proj1 as Function of TOP-REAL 2,R^1 by TOPMETR:17;
reconsider h2=proj2 as Function of TOP-REAL 2,R^1 by TOPMETR:17;
reconsider hh1=h1 as Function of the TopStruct of TOP-REAL 2,R^1;
reconsider hh2=h2 as Function of the TopStruct of TOP-REAL 2,R^1;
A95: the TopStruct of TOP-REAL 2
= (the TopStruct of TOP-REAL 2)|([#](the TopStruct of TOP-REAL 2))
by TSEP_1:3
.= the TopStruct of ((TOP-REAL 2)|([#](TOP-REAL 2))) by PRE_TOPC:36
.= (TOP-REAL 2)|([#](TOP-REAL 2));
then (for p being Point of (TOP-REAL 2)|([#](TOP-REAL 2)) holds
hh1.p=proj1.p) implies hh1 is continuous by JGRAPH_2:29;
then
A96: (for p being Point of (TOP-REAL 2)|([#]TOP-REAL 2)holds
hh1.p=proj1.p) implies h1 is continuous by PRE_TOPC:32;
(for p being Point of (TOP-REAL 2)|([#](TOP-REAL 2)) holds
hh2.p=proj2.p) implies hh2 is continuous by A95,JGRAPH_2:30;
then (for p being Point of (TOP-REAL 2)|([#](TOP-REAL 2)) holds
hh2.p=proj2.p) implies h2 is continuous by PRE_TOPC:32;
then consider h being Function of TOP-REAL 2,R^1 such that
A97: for p being Point of TOP-REAL 2, r1,r2 being Real st h1.p=r1 &
h2.p=r2 holds h.p=r1+r2 and
A98: h is continuous by A96,JGRAPH_2:19;
reconsider k=h*g1 as Function of I[01],R^1;
A99: E-max K=|[b,d]| by A1,A2,Th46;
now
assume
A100: s1>s2;
A101: dom g=[.0,1.] by BORSUK_1:40,FUNCT_2:def 1;
0 in [.0,1.] by XXREAL_1:1;
then
A102: k.0=h.(E-max(K)) by A84,A101,FUNCT_1:13
.=h1.(E-max(K))+h2.(E-max(K)) by A97
.=(E-max(K))`1+proj2.(E-max(K)) by PSCOMP_1:def 5
.=(E-max(K))`1+(E-max(K))`2 by PSCOMP_1:def 6
.=(E-max(K))`1+d by A99,EUCLID:52
.=b+d by A99,EUCLID:52;
s1 in [.0,1.] by A86,A87,XXREAL_1:1;
then
A103: k.s1=h.p1 by A85,A101,FUNCT_1:13
.=proj1.p1 +proj2.p1 by A97
.=p1`1+proj2.p1 by PSCOMP_1:def 5
.=p1`1+c by A6,PSCOMP_1:def 6;
A104: s2 in [.0,1.] by A89,A90,XXREAL_1:1;
then
A105: k.s2=h.p2 by A88,A101,FUNCT_1:13
.=proj1.p2 +proj2.p2 by A97
.=p2`1+proj2.p2 by PSCOMP_1:def 5
.=p2`1+c by A79,PSCOMP_1:def 6;
A106: k.0>=k.s1 by A2,A7,A102,A103,XREAL_1:7;
A107: k.s1>=k.s2 by A12,A103,A105,XREAL_1:7;
A108: 0 in [.0,1.] by XXREAL_1:1;
then
A109: [.0,s2.] c= [.0,1.] by A104,XXREAL_2:def 12;
reconsider B=[.0,s2.] as Subset of I[01] by A104,A108,BORSUK_1:40
,XXREAL_2:def 12;
A110: B is connected by A89,A104,A108,BORSUK_1:40,BORSUK_4:24;
A111: 0 in B by A89,XXREAL_1:1;
A112: s2 in B by A89,XXREAL_1:1;
consider xc being Point of I[01] such that
A113: xc in B and
A114: k.xc =k.s1 by A94,A98,A106,A107,A110,A111,A112,TOPREAL5:5;
reconsider rxc=xc as Real;
A115: for x1,x2 being set st x1 in dom k & x2 in dom k &
k.x1=k.x2 holds x1=x2
proof
let x1,x2 be set;
assume that
A116: x1 in dom k and
A117: x2 in dom k and
A118: k.x1=k.x2;
reconsider r1=x1 as Point of I[01] by A116;
reconsider r2=x2 as Point of I[01] by A117;
A119: k.x1=h.(g1.x1) by A116,FUNCT_1:12
.=h1.(g1.r1)+h2.(g1.r1) by A97
.=(g1.r1)`1+proj2.(g1.r1) by PSCOMP_1:def 5
.=(g1.r1)`1+(g1.r1)`2 by PSCOMP_1:def 6;
A120: k.x2=h.(g1.x2) by A117,FUNCT_1:12
.=h1.(g1.r2)+h2.(g1.r2) by A97
.=(g1.r2)`1+proj2.(g1.r2) by PSCOMP_1:def 5
.=(g1.r2)`1+(g1.r2)`2 by PSCOMP_1:def 6;
A121: g.r1 in Lower_Arc(K) by A93;
A122: g.r2 in Lower_Arc(K) by A93;
reconsider gr1=g.r1 as Point of TOP-REAL 2 by A121;
reconsider gr2=g.r2 as Point of TOP-REAL 2 by A122;
now per cases by A80,A93,XBOOLE_0:def 3;
case
A123: g.r1 in LSeg(|[b,d]|,|[b,c]|) &
g.r2 in LSeg(|[b,d]|,|[b,c]|);
then
A124: (gr1)`1=b by A2,Th1;
(gr2)`1=b by A2,A123,Th1;
then |[(gr1)`1,(gr1)`2]|=g.r2 by A118,A119,A120,A124,EUCLID:53;
then g.r1=g.r2 by EUCLID:53;
hence thesis by A91,A92,FUNCT_1:def 4;
end;
case
A125: g.r1 in LSeg(|[b,d]|,|[b,c]|) &
g.r2 in LSeg(|[b,c]|,|[a,c]|);
then
A126: (gr1)`1=b by A2,Th1;
A127: c <=(gr1)`2 by A2,A125,Th1;
A128: (gr2)`2=c by A1,A125,Th3;
A129: (gr2)`1 <=b by A1,A125,Th3;
A130: b+(gr1)`2=(gr2)`1 +c by A2,A118,A119,A120,A125,A128,Th1;
A131: now
assume b<>gr2`1;
then b>gr2`1 by A129,XXREAL_0:1;
hence contradiction by A118,A119,A120,A126,A127,A128,
XREAL_1:8;
end;
now
assume gr1`2<> c;
then c <gr1`2 by A127,XXREAL_0:1;
hence contradiction by A129,A130,XREAL_1:8;
end;
then |[(gr1)`1,(gr1)`2]|=g.r2 by A126,A128,A131,EUCLID:53;
then g.r1=g.r2 by EUCLID:53;
hence thesis by A91,A92,FUNCT_1:def 4;
end;
case
A132: g.r1 in LSeg(|[b,c]|,|[a,c]|) &
g.r2 in LSeg(|[b,d]|,|[b,c]|);
then
A133: (gr2)`1=b by A2,Th1;
A134: c <=(gr2)`2 by A2,A132,Th1;
A135: (gr1)`2=c by A1,A132,Th3;
A136: (gr1)`1 <=b by A1,A132,Th3;
A137: b+(gr2)`2=(gr1)`1 +c by A1,A118,A119,A120,A132,A133,Th3;
A138: now
assume b<>gr1`1;
then b>gr1`1 by A136,XXREAL_0:1;
hence contradiction by A134,A137,XREAL_1:8;
end;
now
assume gr2`2<> c;
then c < gr2`2 by A134,XXREAL_0:1;
hence contradiction by A118,A119,A120,A133,A135,A136,XREAL_1:8;
end;
then |[(gr2)`1,(gr2)`2]|=g.r1 by A133,A135,A138,EUCLID:53;
then g.r1=g.r2 by EUCLID:53;
hence thesis by A91,A92,FUNCT_1:def 4;
end;
case
A139: g.r1 in LSeg(|[b,c]|,|[a,c]|) &
g.r2 in LSeg(|[b,c]|,|[a,c]|);
then
A140: (gr1)`2=c by A1,Th3;
(gr2)`2=c by A1,A139,Th3;
then |[(gr1)`1,(gr1)`2]|=g.r2 by A118,A119,A120,A140,EUCLID:53;
then g.r1=g.r2 by EUCLID:53;
hence thesis by A91,A92,FUNCT_1:def 4;
end;
end;
hence thesis;
end;
A141: dom k=[.0,1.] by BORSUK_1:40,FUNCT_2:def 1;
then s1 in dom k by A86,A87,XXREAL_1:1;
then rxc=s1 by A109,A113,A114,A115,A141;
hence contradiction by A100,A113,XXREAL_1:1;
end;
hence thesis;
end;
then LE p1,p2,Lower_Arc(K),E-max(K),W-min(K) by A81,A82,JORDAN5C:def 3;
hence thesis by A78,A81,A82,JORDAN6:def 10;
end;
end;
hence thesis;
end;
end;
theorem Th63:
for x being set,a,b,c,d being Real
st x in rectangle(a,b,c,d) & a<b & c <d
holds x in LSeg(|[a,c]|,|[a,d]|) or x in LSeg(|[a,d]|,|[b,d]|)
or x in LSeg(|[b,d]|,|[b,c]|) or x in LSeg(|[b,c]|,|[a,c]|)
proof
let x be set,a,b,c,d be Real;
assume that
A1: x in rectangle(a,b,c,d) and
A2: a<b and
A3: c <d;
x in {q: q`1 = a & q`2 <= d & q`2 >= c or q`1 <= b & q`1 >= a & q`2 = d or
q`1 <= b & q`1 >= a & q`2 = c or q`1 = b & q`2 <= d & q`2 >= c}
by A1,A2,A3,SPPOL_2:54;
then consider p such that
A4: p=x and
A5: p`1 = a & p`2 <= d & p`2 >= c or p`1 <= b & p`1 >= a & p`2 = d or p
`1 <= b & p`1 >= a & p`2 = c or p`1 = b & p`2 <= d & p`2 >= c;
now per cases by A5;
case
A6: p`1=a & c <=p`2 & p`2<=d;
A7: d-c >0 by A3,XREAL_1:50;
A8: p`2 -c >=0 by A6,XREAL_1:48;
A9: d-p`2 >=0 by A6,XREAL_1:48;
reconsider r=(p`2-c)/(d-c) as Real;
A10: 1-r=(d-c)/(d-c)-(p`2-c)/(d-c) by A7,XCMPLX_1:60
.=((d-c)-(p`2 -c))/(d-c) by XCMPLX_1:120
.=(d-p`2)/(d-c);
then
A11: 1-r+r>=0+r by A7,A9,XREAL_1:7;
A12: ((1-r)*(|[a,c]|)+r*(|[a,d]|))`1
=((1-r)*(|[a,c]|))`1+(r*(|[a,d]|))`1 by TOPREAL3:2
.=(1-r)*((|[a,c]|)`1)+(r*(|[a,d]|))`1 by TOPREAL3:4
.=(1-r)*a+(r*(|[a,d]|))`1 by EUCLID:52
.=(1-r)*a+r*((|[a,d]|)`1) by TOPREAL3:4
.=(1-r)*a+r*a by EUCLID:52
.=p`1 by A6;
((1-r)*(|[a,c]|)+r*(|[a,d]|))`2
=((1-r)*(|[a,c]|))`2+(r*(|[a,d]|))`2 by TOPREAL3:2
.=(1-r)*((|[a,c]|)`2)+(r*(|[a,d]|))`2 by TOPREAL3:4
.=(1-r)*c+(r*(|[a,d]|))`2 by EUCLID:52
.=(1-r)*c+r*((|[a,d]|)`2) by TOPREAL3:4
.=(d-p`2)/(d-c)*c+(p`2-c)/(d-c)*d by A10,EUCLID:52
.=(d-p`2)*(d-c)"*c + (p`2-c)/(d-c)*d by XCMPLX_0:def 9
.=(d-c)"*((d-p`2)*c)+ (d-c)"*(p`2-c)*d by XCMPLX_0:def 9
.=(d-c)"*(d -c)*p`2
.=1*p`2 by A7,XCMPLX_0:def 7
.=p`2;
then p=|[((1-r)*(|[a,c]|)+r*(|[a,d]|))`1,
((1-r)*(|[a,c]|)+r*(|[a,d]|))`2]| by A12,EUCLID:53
.=(1-r)*(|[a,c]|)+r*(|[a,d]|) by EUCLID:53;
hence thesis by A4,A7,A8,A11;
end;
case
A13: p`2=d & a<=p`1 & p`1<=b;
A14: b-a >0 by A2,XREAL_1:50;
A15: p`1 -a >=0 by A13,XREAL_1:48;
A16: b-p`1 >=0 by A13,XREAL_1:48;
reconsider r=(p`1-a)/(b-a) as Real;
A17: 1-r=(b-a)/(b-a)-(p`1-a)/(b-a) by A14,XCMPLX_1:60
.=((b-a)-(p`1 -a))/(b-a) by XCMPLX_1:120
.=(b-p`1)/(b-a);
then
A18: 1-r+r>=0+r by A14,A16,XREAL_1:7;
A19: ((1-r)*(|[a,d]|)+r*(|[b,d]|))`1
=((1-r)*(|[a,d]|))`1+(r*(|[b,d]|))`1 by TOPREAL3:2
.=(1-r)*((|[a,d]|)`1)+(r*(|[b,d]|))`1 by TOPREAL3:4
.=(1-r)*a+(r*(|[b,d]|))`1 by EUCLID:52
.=(1-r)*a+r*((|[b,d]|)`1) by TOPREAL3:4
.=(b-p`1)/(b-a)*a+(p`1-a)/(b-a)*b by A17,EUCLID:52
.=(b-p`1)*(b-a)"*a + (p`1-a)/(b-a)*b by XCMPLX_0:def 9
.=(b-a)"*((b-p`1)*a)+ (b-a)"*(p`1-a)*b by XCMPLX_0:def 9
.=(b-a)"*(b -a)*p`1
.=1*p`1 by A14,XCMPLX_0:def 7
.=p`1;
((1-r)*(|[a,d]|)+r*(|[b,d]|))`2
=((1-r)*(|[a,d]|))`2+(r*(|[b,d]|))`2 by TOPREAL3:2
.=(1-r)*((|[a,d]|)`2)+(r*(|[b,d]|))`2 by TOPREAL3:4
.=(1-r)*d+(r*(|[b,d]|))`2 by EUCLID:52
.=(1-r)*d+r*((|[b,d]|)`2) by TOPREAL3:4
.=(1-r)*d+r*d by EUCLID:52
.=p`2 by A13;
then p=|[((1-r)*(|[a,d]|)+r*(|[b,d]|))`1,
((1-r)*(|[a,d]|)+r*(|[b,d]|))`2]| by A19,EUCLID:53
.=(1-r)*(|[a,d]|)+r*(|[b,d]|) by EUCLID:53;
hence thesis by A4,A14,A15,A18;
end;
case
A20: p`1=b & c <=p`2 & p`2<=d;
A21: d-c >0 by A3,XREAL_1:50;
A22: p`2 -c >=0 by A20,XREAL_1:48;
A23: d-p`2 >=0 by A20,XREAL_1:48;
reconsider r=(d-p`2)/(d-c) as Real;
A24: 1-r=(d-c)/(d-c)-(d-p`2)/(d-c) by A21,XCMPLX_1:60
.=((d-c)-(d-p`2))/(d-c) by XCMPLX_1:120
.=(p`2-c)/(d-c);
then
A25: 1-r+r>=0+r by A21,A22,XREAL_1:7;
A26: ((1-r)*(|[b,d]|)+r*(|[b,c]|))`1
=((1-r)*(|[b,d]|))`1+(r*(|[b,c]|))`1 by TOPREAL3:2
.=(1-r)*((|[b,d]|)`1)+(r*(|[b,c]|))`1 by TOPREAL3:4
.=(1-r)*b+(r*(|[b,c]|))`1 by EUCLID:52
.=(1-r)*b+r*((|[b,c]|)`1) by TOPREAL3:4
.=(1-r)*b+r*b by EUCLID:52
.=p`1 by A20;
((1-r)*(|[b,d]|)+r*(|[b,c]|))`2
=((1-r)*(|[b,d]|))`2+(r*(|[b,c]|))`2 by TOPREAL3:2
.=(1-r)*((|[b,d]|)`2)+(r*(|[b,c]|))`2 by TOPREAL3:4
.=(1-r)*d+(r*(|[b,c]|))`2 by EUCLID:52
.=(1-r)*d+r*((|[b,c]|)`2) by TOPREAL3:4
.=(p`2-c)/(d-c)*d+(d-p`2)/(d-c)*c by A24,EUCLID:52
.=(p`2-c)*(d-c)"*d + (d-p`2)/(d-c)*c by XCMPLX_0:def 9
.=(d-c)"*((p`2-c)*d)+ (d-c)"*(d-p`2)*c by XCMPLX_0:def 9
.=(d-c)"*(d -c)*p`2
.=1*p`2 by A21,XCMPLX_0:def 7
.=p`2;
then p=|[((1-r)*(|[b,d]|)+r*(|[b,c]|))`1,
((1-r)*(|[b,d]|)+r*(|[b,c]|))`2]| by A26,EUCLID:53
.=(1-r)*(|[b,d]|)+r*(|[b,c]|) by EUCLID:53;
hence thesis by A4,A21,A23,A25;
end;
case
A27: p`2=c & a<=p`1 & p`1<=b;
A28: b-a >0 by A2,XREAL_1:50;
A29: p`1 -a >=0 by A27,XREAL_1:48;
A30: b-p`1 >=0 by A27,XREAL_1:48;
reconsider r=(b-p`1)/(b-a) as Real;
A31: 1-r=(b-a)/(b-a)-(b-p`1)/(b-a) by A28,XCMPLX_1:60
.=((b-a)-(b-p`1))/(b-a) by XCMPLX_1:120
.=(p`1-a)/(b-a);
then
A32: 1-r+r>=0+r by A28,A29,XREAL_1:7;
A33: ((1-r)*(|[b,c]|)+r*(|[a,c]|))`1
=((1-r)*(|[b,c]|))`1+(r*(|[a,c]|))`1 by TOPREAL3:2
.=(1-r)*((|[b,c]|)`1)+(r*(|[a,c]|))`1 by TOPREAL3:4
.=(1-r)*b+(r*(|[a,c]|))`1 by EUCLID:52
.=(1-r)*b+r*((|[a,c]|)`1) by TOPREAL3:4
.=(p`1-a)/(b-a)*b+(b-p`1)/(b-a)*a by A31,EUCLID:52
.=(p`1-a)*(b-a)"*b + (b-p`1)/(b-a)*a by XCMPLX_0:def 9
.=(b-a)"*((p`1-a)*b)+ (b-a)"*(b-p`1)*a by XCMPLX_0:def 9
.=(b-a)"*(b -a)*p`1
.=1*p`1 by A28,XCMPLX_0:def 7
.=p`1;
((1-r)*(|[b,c]|)+r*(|[a,c]|))`2
=((1-r)*(|[b,c]|))`2+(r*(|[a,c]|))`2 by TOPREAL3:2
.=(1-r)*((|[b,c]|)`2)+(r*(|[a,c]|))`2 by TOPREAL3:4
.=(1-r)*c+(r*(|[a,c]|))`2 by EUCLID:52
.=(1-r)*c+r*((|[a,c]|)`2) by TOPREAL3:4
.=(1-r)*c+r*c by EUCLID:52
.=p`2 by A27;
then p=|[((1-r)*(|[b,c]|)+r*(|[a,c]|))`1,
((1-r)*(|[b,c]|)+r*(|[a,c]|))`2]| by A33,EUCLID:53
.=(1-r)*(|[b,c]|)+r*(|[a,c]|) by EUCLID:53;
hence thesis by A4,A28,A30,A32;
end;
end;
hence thesis;
end;
begin :: General Fashoda Theorem for Unit Square
theorem Th64:
for p1,p2 being Point of TOP-REAL 2
st LE p1,p2,rectangle(-1,1,-1,1) & p1 in LSeg(|[-1,-1]|,|[-1,1]|)
holds p2 in LSeg(|[-1,-1]|,|[-1,1]|)& p2`2>=p1`2 or
p2 in LSeg(|[-1,1]|,|[1,1]|) or p2 in LSeg(|[1,1]|,|[1,-1]|)
or p2 in LSeg(|[1,-1]|,|[-1,-1]|)& p2<>|[-1,-1]|
proof
let p1,p2 be Point of TOP-REAL 2;
set K = rectangle(-1,1,-1,1);
assume that
A1: LE p1,p2,K and
A2: p1 in LSeg(|[-1,-1]|,|[-1,1]|);
p2 in LSeg(|[-1,-1]|,|[-1,1]|) & p1`2<=p2`2
or p2 in LSeg(|[-1,1]|,|[1,1]|) or p2 in LSeg(|[1,1]|,|[1,-1]|)
or p2 in LSeg(|[1,-1]|,|[-1,-1]|) & p2<>W-min(K) by A1,A2,Th59;
hence thesis by Th46;
end;
theorem Th65:
for p1,p2 being Point of TOP-REAL 2,
P being non empty compact Subset of TOP-REAL 2,
f being Function of TOP-REAL 2,TOP-REAL 2 st P= circle(0,0,1) & f=Sq_Circ
& p1 in LSeg(|[-1,-1]|,|[-1,1]|)& p1`2>=0 & LE p1,p2,rectangle(-1,1,-1,1)
holds LE f.p1,f.p2,P
proof
let p1,p2 be Point of TOP-REAL 2,
P be non empty compact Subset of TOP-REAL 2,
f be Function of TOP-REAL 2,TOP-REAL 2;
set K = rectangle(-1,1,-1,1);
assume that
A1: P= circle(0,0,1) and
A2: f=Sq_Circ and
A3: p1 in LSeg(|[-1,-1]|,|[-1,1]|) and
A4: p1`2>=0 and
A5: LE p1,p2,K;
A6: K is being_simple_closed_curve by Th50;
A7: P={p: |.p.|=1} by A1,Th24;
A8: p1`1=-1 by A3,Th1;
A9: p1`2<=1 by A3,Th1;
A10: p1 in K by A5,A6,JORDAN7:5;
A11: p2 in K by A5,A6,JORDAN7:5;
A12: f.:K=P by A2,A7,Lm15,Th35,JGRAPH_3:23;
A13: dom f = the carrier of TOP-REAL 2 by FUNCT_2:def 1;
then
A14: f.p1 in P by A10,A12,FUNCT_1:def 6;
A15: f.p2 in P by A11,A12,A13,FUNCT_1:def 6;
A16: p1`1=-1 by A3,Th1;
A17: (p1`2)^2 >=0 by XREAL_1:63;
then
A18: sqrt(1+(p1`2)^2)>0 by SQUARE_1:25;
A19: p1`2<=-p1`1 by A3,A8,Th1;
p1<>0.TOP-REAL 2 by A8,EUCLID:52,54;
then
A20: f.p1= |[p1`1/sqrt(1+(p1`2/p1`1)^2),p1`2/sqrt(1+(p1`2/p1`1)^2)]|
by A2,A4,A16,A19,JGRAPH_3:def 1;
then
A21: (f.p1)`1= p1`1/sqrt(1+(p1`2/(-1))^2) by A16,EUCLID:52
.=(p1`1)/sqrt(1+(p1`2)^2);
A22: (f.p1)`2= p1`2/sqrt(1+(p1`2/(-1))^2) by A16,A20,EUCLID:52
.=(p1`2)/sqrt(1+(p1`2)^2);
A23: (f.p1)`1<0 by A16,A17,A21,SQUARE_1:25,XREAL_1:141;
A24: (f.p1)`2>=0 by A4,A18,A22;
f.p1 in {p9 where p9 is Point of TOP-REAL 2:p9 in P & p9`2>=0} by A4,A14,A18
,A22;
then
A25: f.p1 in Upper_Arc(P) by A7,JGRAPH_5:34;
now per cases by A3,A5,Th64;
case
A26: p2 in LSeg(|[-1,-1]|,|[-1,1]|)& p2`2>=p1`2;
A27: (p2`2)^2 >=0 by XREAL_1:63;
then
A28: sqrt(1+(p2`2)^2)>0 by SQUARE_1:25;
A29: p2`1=-1 by A26,Th1;
A30: -1<=p2`2 by A26,Th1;
A31: p2`2<=-p2`1 by A26,A29,Th1;
p2<>0.TOP-REAL 2 by A29,EUCLID:52,54;
then
A32: f.p2= |[p2`1/sqrt(1+(p2`2/p2`1)^2),p2`2/sqrt(1+(p2`2/p2`1)^2) ]|
by A2,A29,A30,A31,JGRAPH_3:def 1;
then
A33: (f.p2)`1= p2`1/sqrt(1+(p2`2/(-1))^2) by A29,EUCLID:52
.=(p2`1)/sqrt(1+(p2`2)^2);
A34: (f.p2)`2= p2`2/sqrt(1+(p2`2/(-1))^2) by A29,A32,EUCLID:52
.=(p2`2)/sqrt(1+(p2`2)^2);
A35: (f.p2)`1<0 by A27,A29,A33,SQUARE_1:25,XREAL_1:141;
(p1`2)*sqrt(1+(p2`2)^2)<= (p2`2)*sqrt(1+(p1`2)^2) by A4,A26,Lm3;
then (p1`2)*sqrt(1+(p2`2)^2)/sqrt(1+(p2`2)^2)
<= (p2`2)*sqrt(1+(p1`2)^2)/sqrt(1+(p2`2)^2) by A28,XREAL_1:72;
then (p1`2)
<= (p2`2)*sqrt(1+(p1`2)^2)/sqrt(1+(p2`2)^2) by A28,XCMPLX_1:89;
then (p1`2)/sqrt(1+(p1`2)^2)
<= (p2`2)*sqrt(1+(p1`2)^2)/sqrt(1+(p2`2)^2)/sqrt(1+(p1`2)^2)
by A18,XREAL_1:72;
then (p1`2)/sqrt(1+(p1`2)^2)
<= (p2`2)*sqrt(1+(p1`2)^2)/sqrt(1+(p1`2)^2)/sqrt(1+(p2`2)^2)
by XCMPLX_1:48;
then (f.p1)`2<=(f.p2)`2 by A18,A22,A34,XCMPLX_1:89;
hence thesis by A7,A14,A15,A23,A24,A35,JGRAPH_5:53;
end;
case
A36: p2 in LSeg(|[-1,1]|,|[1,1]|);
then
A37: p2`2=1 by Th3;
A38: -1<=p2`1 by A36,Th3;
A39: p2`1<=1 by A36,Th3;
(p2`1)^2 >=0 by XREAL_1:63;
then
A40: sqrt(1+(p2`1)^2)>0 by SQUARE_1:25;
p2<>0.TOP-REAL 2 by A37,EUCLID:52,54;
then
A41: f.p2= |[p2`1/sqrt(1+(p2`1/p2`2)^2),p2`2/sqrt(1+(p2`1/p2`2)^2) ]|
by A2,A37,A38,A39,JGRAPH_3:4;
then
A42: (f.p2)`1=(p2`1)/sqrt(1+(p2`1)^2) by A37,EUCLID:52;
A43: (f.p2)`2>=0 by A37,A40,A41,EUCLID:52;
-sqrt(1+(p2`1)^2)<= (p2`1)*sqrt(1+(p1`2)^2) by A4,A9,A38,A39,SQUARE_1:55;
then (p1`1)*sqrt(1+(p2`1)^2)/sqrt(1+(p2`1)^2)
<= (p2`1)*sqrt(1+(p1`2)^2)/sqrt(1+(p2`1)^2) by A8,A40,XREAL_1:72;
then (p1`1)
<= (p2`1)*sqrt(1+(p1`2)^2)/sqrt(1+(p2`1)^2) by A40,XCMPLX_1:89;
then (p1`1)/sqrt(1+(p1`2)^2)
<= (p2`1)*sqrt(1+(p1`2)^2)/sqrt(1+(p2`1)^2)/sqrt(1+(p1`2)^2)
by A18,XREAL_1:72;
then (p1`1)/sqrt(1+(p1`2)^2)
<= (p2`1)*sqrt(1+(p1`2)^2)/sqrt(1+(p1`2)^2)/sqrt(1+(p2`1)^2)
by XCMPLX_1:48;
then (f.p1)`1<=(f.p2)`1 by A18,A21,A42,XCMPLX_1:89;
hence thesis by A4,A7,A14,A15,A18,A22,A43,JGRAPH_5:54;
end;
case
A44: p2 in LSeg(|[1,1]|,|[1,-1]|);
then
A45: p2`1=1 by Th1;
A46: -1<=p2`2 by A44,Th1;
A47: p2`2<=1 by A44,Th1;
(p2`2)^2 >=0 by XREAL_1:63;
then
A48: sqrt(1+(p2`2)^2)>0 by SQUARE_1:25;
p2<>0.TOP-REAL 2 by A45,EUCLID:52,54;
then f.p2= |[p2`1/sqrt(1+(p2`2/p2`1)^2),p2`2/sqrt(1+(p2`2/p2`1)^2)]|
by A2,A45,A46,A47,JGRAPH_3:def 1;
then
A49: (f.p2)`1=(p2`1)/sqrt(1+(p2`2)^2) by A45,EUCLID:52;
(p1`1)/sqrt(1+(p1`2)^2)
<= (p2`1)*sqrt(1+(p1`2)^2)/sqrt(1+(p2`2)^2)/sqrt(1+(p1`2)^2)
by A8,A18,A45,A48,XREAL_1:72;
then (p1`1)/sqrt(1+(p1`2)^2)
<= (p2`1)*sqrt(1+(p1`2)^2)/sqrt(1+(p1`2)^2)/sqrt(1+(p2`2)^2)
by XCMPLX_1:48;
then
A50: (f.p1)`1<=(f.p2)`1 by A18,A21,A49,XCMPLX_1:89;
now per cases;
case (f.p2)`2>=0;
hence thesis by A4,A7,A14,A15,A18,A22,A50,JGRAPH_5:54;
end;
case
A51: (f.p2)`2<0;
then f.p2 in {p9 where p9 is Point of TOP-REAL 2:p9 in P & p9`2<=0}
by A15;
then
A52: f.p2 in Lower_Arc(P) by A7,JGRAPH_5:35;
W-min(P)=|[-1,0]| by A7,JGRAPH_5:29;
then f.p2 <> W-min(P) by A51,EUCLID:52;
hence thesis by A25,A52,JORDAN6:def 10;
end;
end;
hence thesis;
end;
case
A53: p2 in LSeg(|[1,-1]|,|[-1,-1]|)& p2<>|[-1,-1]|;
then
A54: p2`2=-1 by Th3;
A55: -1<=p2`1 by A53,Th3;
A56: (p2`1)^2 >=0 by XREAL_1:63;
A57: p2`1<=-p2`2 by A53,A54,Th3;
p2<>0.TOP-REAL 2 by A54,EUCLID:52,54;
then f.p2= |[p2`1/sqrt(1+(p2`1/p2`2)^2),p2`2/sqrt(1+(p2`1/p2`2)^2)]|
by A2,A54,A55,A57,JGRAPH_3:4;
then (f.p2)`2= p2`2/sqrt(1+(p2`1/(-1))^2) by A54,EUCLID:52
.=(p2`2)/sqrt(1+(p2`1)^2);
then
A58: (f.p2)`2<0 by A54,A56,SQUARE_1:25,XREAL_1:141;
then f.p2 in {p9 where p9 is Point of TOP-REAL 2:p9 in P & p9`2<=0}
by A15;
then
A59: f.p2 in Lower_Arc(P) by A7,JGRAPH_5:35;
W-min(P)=|[-1,0]| by A7,JGRAPH_5:29;
then f.p2 <> W-min(P) by A58,EUCLID:52;
hence thesis by A25,A59,JORDAN6:def 10;
end;
end;
hence thesis;
end;
theorem Th66:
for p1,p2,p3 being Point of TOP-REAL 2,
P being non empty compact Subset of TOP-REAL 2,
f being Function of TOP-REAL 2,TOP-REAL 2 st P= circle(0,0,1) & f=Sq_Circ
& p1 in LSeg(|[-1,-1]|,|[-1,1]|)& p1`2>=0 &
LE p1,p2,rectangle(-1,1,-1,1) & LE p2,p3,rectangle(-1,1,-1,1)
holds LE f.p2,f.p3,P
proof
let p1,p2,p3 be Point of TOP-REAL 2,
P be non empty compact Subset of TOP-REAL 2,
f be Function of TOP-REAL 2,TOP-REAL 2;
set K = rectangle(-1,1,-1,1);
assume that
A1: P= circle(0,0,1) and
A2: f=Sq_Circ and
A3: p1 in LSeg(|[-1,-1]|,|[-1,1]|) and
A4: p1`2>=0 and
A5: LE p1,p2,K and
A6: LE p2,p3,K;
A7: K is being_simple_closed_curve by Th50;
A8: P={p: |.p.|=1} by A1,Th24;
A9: p3 in K by A6,A7,JORDAN7:5;
A10: p2 in K by A5,A7,JORDAN7:5;
A11: f.:K=P by A2,A8,Lm15,Th35,JGRAPH_3:23;
A12: dom f = the carrier of TOP-REAL 2 by FUNCT_2:def 1;
then
A13: f.p2 in P by A10,A11,FUNCT_1:def 6;
A14: f.p3 in P by A9,A11,A12,FUNCT_1:def 6;
now per cases by A3,A5,Th64;
case p2 in LSeg(|[-1,-1]|,|[-1,1]|)& p2`2>=p1`2;
hence thesis by A1,A2,A4,A6,Th65;
end;
case
A15: p2 in LSeg(|[-1,1]|,|[1,1]|);
then
A16: p2`2=1 by Th3;
A17: -1<=p2`1 by A15,Th3;
A18: p2`1<=1 by A15,Th3;
(p2`1)^2 >=0 by XREAL_1:63;
then
A19: sqrt(1+(p2`1)^2)>0 by SQUARE_1:25;
p2<>0.TOP-REAL 2 by A16,EUCLID:52,54;
then
A20: f.p2= |[p2`1/sqrt(1+(p2`1/p2`2)^2),p2`2/sqrt(1+(p2`1/p2`2)^2) ]|
by A2,A16,A17,A18,JGRAPH_3:4;
then
A21: (f.p2)`1=(p2`1)/sqrt(1+(p2`1)^2) by A16,EUCLID:52;
A22: (f.p2)`2>=0 by A16,A19,A20,EUCLID:52;
then f.p2 in {p9 where p9 is Point of TOP-REAL 2:p9 in P & p9`2>=0}
by A13;
then
A23: f.p2 in Upper_Arc(P) by A8,JGRAPH_5:34;
now per cases by A6,A15,Th60;
case
A24: p3 in LSeg(|[-1,1]|,|[1,1]|) & p2`1<=p3`1;
then
A25: p3`2=1 by Th3;
A26: -1<=p3`1 by A24,Th3;
A27: p3`1<=1 by A24,Th3;
(p3`1)^2 >=0 by XREAL_1:63;
then
A28: sqrt(1+(p3`1)^2)>0 by SQUARE_1:25;
p3<>0.TOP-REAL 2 by A25,EUCLID:52,54;
then
A29: f
.p3= |[p3`1/sqrt(1+(p3`1/p3`2)^2),p3`2/sqrt(1+(p3`1/ p3`2)^2) ]|
by A2,A25,A26,A27,JGRAPH_3:4;
then
A30: (f.p3)`1=(p3`1)/sqrt(1+(p3`1)^2) by A25,EUCLID:52;
A31: (f.p3)`2>=0 by A25,A28,A29,EUCLID:52;
(p2`1)*sqrt(1+(p3`1)^2)<= (p3`1)*sqrt(1+(p2`1)^2)
by A24,SQUARE_1:57;
then (p2`1)*sqrt(1+(p3`1)^2)/sqrt(1+(p3`1)^2)
<= (p3`1)*sqrt(1+(p2`1)^2)/sqrt(1+(p3`1)^2) by A28,XREAL_1:72;
then (p2`1)
<= (p3`1)*sqrt(1+(p2`1)^2)/sqrt(1+(p3`1)^2) by A28,XCMPLX_1:89;
then (p2`1)/sqrt(1+(p2`1)^2)
<= (p3`1)*sqrt(1+(p2`1)^2)/sqrt(1+(p3`1)^2)/sqrt(1+(p2`1)^2)
by A19,XREAL_1:72;
then (p2`1)/sqrt(1+(p2`1)^2)
<= (p3`1)*sqrt(1+(p2`1)^2)/sqrt(1+(p2`1)^2)/sqrt(1+(p3`1)^2)
by XCMPLX_1:48;
then (f.p2)`1<=(f.p3)`1 by A19,A21,A30,XCMPLX_1:89;
hence thesis by A8,A13,A14,A22,A31,JGRAPH_5:54;
end;
case
A32: p3 in LSeg(|[1,1]|,|[1,-1]|);
then
A33: p3`1=1 by Th1;
A34: -1<=p3`2 by A32,Th1;
A35: p3`2<=1 by A32,Th1;
(p3`2)^2 >=0 by XREAL_1:63;
then
A36: sqrt(1+(p3`2)^2)>0 by SQUARE_1:25;
p3<>0.TOP-REAL 2 by A33,EUCLID:52,54;
then
f.p3= |[p3`1/sqrt(1+(p3`2/p3`1)^2),p3`2/sqrt(1+(p3`2/p3`1)^2)]|
by A2,A33,A34,A35,JGRAPH_3:def 1;
then
A37: (f.p3)`1=(p3`1)/sqrt(1+(p3`2)^2) by A33,EUCLID:52;
A38: -1<=-p2`1 by A18,XREAL_1:24;
A39: --1>=-p2`1 by A17,XREAL_1:24;
(p2`1)^2 = (-p2`1)^2;
then (--p2`1)*sqrt(1+(p3`2)^2)<= sqrt(1+(p2`1)^2)
by A34,A35,A38,A39,SQUARE_1:55;
then (p2`1)*sqrt(1+(p3`2)^2)/sqrt(1+(p3`2)^2)
<= (p3`1)*sqrt(1+(p2`1)^2)/sqrt(1+(p3`2)^2) by A33,A36,XREAL_1:72;
then (p2`1)
<= (p3`1)*sqrt(1+(p2`1)^2)/sqrt(1+(p3`2)^2) by A36,XCMPLX_1:89;
then (p2`1)/sqrt(1+(p2`1)^2)
<= (p3`1)*sqrt(1+(p2`1)^2)/sqrt(1+(p3`2)^2)/sqrt(1+(p2`1)^2)
by A19,XREAL_1:72;
then (p2`1)/sqrt(1+(p2`1)^2)
<= (p3`1)*sqrt(1+(p2`1)^2)/sqrt(1+(p2`1)^2)/sqrt(1+(p3`2)^2)
by XCMPLX_1:48;
then
A40: (f.p2)`1<=(f.p3)`1 by A19,A21,A37,XCMPLX_1:89;
now per cases;
case (f.p3)`2>=0;
hence thesis by A8,A13,A14,A22,A40,JGRAPH_5:54;
end;
case
A41: (f.p3)`2<0;
then f.p3 in {p9 where p9 is Point of TOP-REAL 2:p9 in P & p9`2<=0} by A14;
then
A42: f.p3 in Lower_Arc(P) by A8,JGRAPH_5:35;
W-min(P)=|[-1,0]| by A8,JGRAPH_5:29;
then f.p3 <> W-min(P) by A41,EUCLID:52;
hence thesis by A23,A42,JORDAN6:def 10;
end;
end;
hence thesis;
end;
case
A43: p3 in LSeg(|[1,-1]|,|[-1,-1]|) & p3<>W-min(K);
then
A44: p3`2=-1 by Th3;
A45: -1<=p3`1 by A43,Th3;
A46: (p3`1)^2 >=0 by XREAL_1:63;
A47: -p3`2>=p3`1 by A43,A44,Th3;
p3<>0.TOP-REAL 2 by A44,EUCLID:52,54;
then
f.p3= |[p3`1/sqrt(1+(p3`1/p3`2)^2),p3`2/sqrt(1+(p3`1/p3`2)^2)]|
by A2,A44,A45,A47,JGRAPH_3:4;
then (f.p3)`2= p3`2/sqrt(1+(p3`1/(-1))^2) by A44,EUCLID:52
.=(p3`2)/sqrt(1+(p3`1)^2);
then
A48: (f.p3)`2<0 by A44,A46,SQUARE_1:25,XREAL_1:141;
then f.p3 in {p9 where p9 is Point of TOP-REAL 2:p9 in P & p9`2<=0}
by A14;
then
A49: f.p3 in Lower_Arc(P) by A8,JGRAPH_5:35;
W-min(P)=|[-1,0]| by A8,JGRAPH_5:29;
then f.p3 <> W-min(P) by A48,EUCLID:52;
hence thesis by A23,A49,JORDAN6:def 10;
end;
end;
hence thesis;
end;
case
A50: p2 in LSeg(|[1,1]|,|[1,-1]|);
then
A51: p2`1=1 by Th1;
A52: -1<=p2`2 by A50,Th1;
A53: p2`2<=1 by A50,Th1;
(p2`2)^2 >=0 by XREAL_1:63;
then
A54: sqrt(1+(p2`2)^2)>0 by SQUARE_1:25;
p2<>0.TOP-REAL 2 by A51,EUCLID:52,54;
then
A55: f.p2= |[p2`1/sqrt(1+(p2`2/p2`1)^2),p2`2/sqrt(1+(p2`2/p2`1)^2) ]|
by A2,A51,A52,A53,JGRAPH_3:def 1;
then
A56: (f.p2)`1=(p2`1)/sqrt(1+(p2`2)^2) by A51,EUCLID:52;
A57: (f.p2)`2=(p2`2)/sqrt(1+(p2`2)^2) by A51,A55,EUCLID:52;
now per cases by A6,A50,Th61;
case
A58: p3 in LSeg(|[1,1]|,|[1,-1]|) & p2`2>=p3`2;
then
A59: p3`1=1 by Th1;
A60: -1<=p3`2 by A58,Th1;
A61: p3`2<=1 by A58,Th1;
(p3`2)^2 >=0 by XREAL_1:63;
then
A62: sqrt(1+(p3`2)^2)>0 by SQUARE_1:25;
p3<>0.TOP-REAL 2 by A59,EUCLID:52,54;
then
A63: f
.p3= |[p3`1/sqrt(1+(p3`2/p3`1)^2),p3`2/sqrt(1+(p3`2/ p3`1)^2) ]|
by A2,A59,A60,A61,JGRAPH_3:def 1;
then
A64: (f.p3)`2=(p3`2)/sqrt(1+(p3`2)^2) by A59,EUCLID:52;
A65: (f.p3)`1>=0 by A59,A62,A63,EUCLID:52;
(p2`2)*sqrt(1+(p3`2)^2)>= (p3`2)*sqrt(1+(p2`2)^2)
by A58,SQUARE_1:57;
then (p2`2)*sqrt(1+(p3`2)^2)/sqrt(1+(p3`2)^2)
>= (p3`2)*sqrt(1+(p2`2)^2)/sqrt(1+(p3`2)^2) by A62,XREAL_1:72;
then (p2`2)
>= (p3`2)*sqrt(1+(p2`2)^2)/sqrt(1+(p3`2)^2) by A62,XCMPLX_1:89;
then (p2`2)/sqrt(1+(p2`2)^2)
>= (p3`2)*sqrt(1+(p2`2)^2)/sqrt(1+(p3`2)^2)/sqrt(1+(p2`2)^2)
by A54,XREAL_1:72;
then (p2`2)/sqrt(1+(p2`2)^2)
>= (p3`2)*sqrt(1+(p2`2)^2)/sqrt(1+(p2`2)^2)/sqrt(1+(p3`2)^2)
by XCMPLX_1:48;
then (p2`2)/sqrt(1+(p2`2)^2) >= (p3`2)/sqrt(1+(p3`2)^2)
by A54,XCMPLX_1:89;
hence thesis by A8,A13,A14,A51,A54,A56,A57,A64,A65,JGRAPH_5:55;
end;
case
A66: p3 in LSeg(|[1,-1]|,|[-1,-1]|) & p3<>W-min(K);
then
A67: p3`2=-1 by Th3;
A68: -1<=p3`1 by A66,Th3;
A69: p3`1<=1 by A66,Th3;
A70: (p3`1)^2 >=0 by XREAL_1:63;
then
A71: sqrt(1+(p3`1)^2)>0 by SQUARE_1:25;
A72: -p3`2>=p3`1 by A66,A67,Th3;
p3<>0.TOP-REAL 2 by A67,EUCLID:52,54;
then
A73: f
.p3= |[p3`1/sqrt(1+(p3`1/p3`2)^2),p3`2/sqrt(1+(p3`1/ p3`2)^2) ]|
by A2,A67,A68,A72,JGRAPH_3:4;
then
A74: (f.p3)`1= p3`1/sqrt(1+(p3`1/(-1))^2) by A67,EUCLID:52
.=(p3`1)/sqrt(1+(p3`1)^2);
A75: (f.p3)`2= p3`2/sqrt(1+(p3`1/(-1))^2) by A67,A73,EUCLID:52
.=(p3`2)/sqrt(1+(p3`1)^2);
then
A76: (f.p3)`2<0 by A67,A70,SQUARE_1:25,XREAL_1:141;
f.p3 in {p9 where p9 is Point of TOP-REAL 2:p9 in P & p9`2<=0}
by A14,A67,A71,A75;
then
A77: f.p3 in Lower_Arc(P) by A8,JGRAPH_5:35;
W-min(P)=|[-1,0]| by A8,JGRAPH_5:29;
then
A78: f.p3 <> W-min(P) by A76,EUCLID:52;
now per cases;
case (f.p2)`2>=0;
then f.p2 in {p9 where p9 is Point of TOP-REAL 2:p9 in P & p9`2>=0} by A13;
then f.p2 in Upper_Arc(P) by A8,JGRAPH_5:34;
hence thesis by A77,A78,JORDAN6:def 10;
end;
case
A79: (f.p2)`2<0;
(p2`1)*sqrt(1+(p3`1)^2)/sqrt(1+(p3`1)^2)
>= (p3`1)*sqrt(1+(p2`2)^2)/sqrt(1+(p3`1)^2)
by A51,A52,A53,A68,A69,A71,SQUARE_1:56,XREAL_1:72;
then (p2`1)
>= (p3`1)*sqrt(1+(p2`2)^2)/sqrt(1+(p3`1)^2) by A71,XCMPLX_1:89;
then (p2`1)/sqrt(1+(p2`2)^2)
>= (p3`1)*sqrt(1+(p2`2)^2)/sqrt(1+(p3`1)^2)/sqrt(1+(p2`2)^2)
by A54,XREAL_1:72;
then (p2`1)/sqrt(1+(p2`2)^2)
>= (p3`1)*sqrt(1+(p2`2)^2)/sqrt(1+(p2`2)^2)/sqrt(1+(p3`1)^2)
by XCMPLX_1:48;
then (p2`1)/sqrt(1+(p2`2)^2) >= (p3`1)/sqrt(1+(p3`1)^2)
by A54,XCMPLX_1:89;
hence thesis by A8,A13,A14,A56,A67,A71,A74,A75,A78,A79,
JGRAPH_5:56;
end;
end;
hence thesis;
end;
end;
hence thesis;
end;
case
A80: p2 in LSeg(|[1,-1]|,|[-1,-1]|)& p2<>|[-1,-1]|;
then
A81: p2`2=-1 by Th3;
A82: -1<=p2`1 by A80,Th3;
(p2`1)^2 >=0 by XREAL_1:63;
then
A83: sqrt(1+(p2`1)^2)>0 by SQUARE_1:25;
A84: -p2`2>=p2`1 by A80,A81,Th3;
p2<>0.TOP-REAL 2 by A81,EUCLID:52,54;
then
A85: f.p2= |[p2`1/sqrt(1+(p2`1/p2`2)^2),p2`2/sqrt(1+(p2`1/p2`2)^2) ]|
by A2,A81,A82,A84,JGRAPH_3:4;
then
A86: (f.p2)`1= p2`1/sqrt(1+(p2`1/(-1))^2) by A81,EUCLID:52
.=(p2`1)/sqrt(1+(p2`1)^2);
A87: (f.p2)`2= p2`2/sqrt(1+(p2`1/(-1))^2) by A81,A85,EUCLID:52
.=(p2`2)/sqrt(1+(p2`1)^2);
A88: W-min(K)=|[-1,-1]| by Th46;
then
A89: p3 in LSeg(|[1,-1]|,|[-1,-1]|) by A6,A80,Th62;
A90: p2`1>=p3`1 by A6,A80,A88,Th62;
A91: p3`2=-1 by A89,Th3;
A92: -1<=p3`1 by A89,Th3;
A93: (p3`1)^2 >=0 by XREAL_1:63;
then
A94: sqrt(1+(p3`1)^2)>0 by SQUARE_1:25;
A95: -p3`2>=p3`1 by A89,A91,Th3;
p3<>0.TOP-REAL 2 by A91,EUCLID:52,54;
then
A96: f.p3= |[p3`1/sqrt(1+(p3`1/p3`2)^2),p3`2/sqrt(1+(p3`1/p3`2)^2) ]|
by A2,A91,A92,A95,JGRAPH_3:4;
then
A97: (f.p3)`1= p3`1/sqrt(1+(p3`1/(-1))^2) by A91,EUCLID:52
.=(p3`1)/sqrt(1+(p3`1)^2);
(f.p3)`2= p3`2/sqrt(1+(p3`1/(-1))^2) by A91,A96,EUCLID:52
.=(p3`2)/sqrt(1+(p3`1)^2);
then
A98: (f.p3)`2<0 by A91,A93,SQUARE_1:25,XREAL_1:141;
W-min(P)=|[-1,0]| by A8,JGRAPH_5:29;
then
A99: f.p3 <> W-min(P) by A98,EUCLID:52;
(p2`1)*sqrt(1+(p3`1)^2)>= (p3`1)*sqrt(1+(p2`1)^2) by A90,SQUARE_1:57;
then (p2`1)*sqrt(1+(p3`1)^2)/sqrt(1+(p3`1)^2)
>= (p3`1)*sqrt(1+(p2`1)^2)/sqrt(1+(p3`1)^2) by A94,XREAL_1:72;
then (p2`1)
>= (p3`1)*sqrt(1+(p2`1)^2)/sqrt(1+(p3`1)^2) by A94,XCMPLX_1:89;
then (p2`1)/sqrt(1+(p2`1)^2)
>= (p3`1)*sqrt(1+(p2`1)^2)/sqrt(1+(p3`1)^2)/sqrt(1+(p2`1)^2)
by A83,XREAL_1:72;
then (p2`1)/sqrt(1+(p2`1)^2)
>= (p3`1)*sqrt(1+(p2`1)^2)/sqrt(1+(p2`1)^2)/sqrt(1+(p3`1)^2)
by XCMPLX_1:48;
then (p2`1)/sqrt(1+(p2`1)^2) >= (p3`1)/sqrt(1+(p3`1)^2)
by A83,XCMPLX_1:89;
hence thesis by A8,A13,A14,A81,A83,A86,A87,A97,A98,A99,JGRAPH_5:56;
end;
end;
hence thesis;
end;
theorem Th67:
for p being Point of TOP-REAL 2, f being Function of TOP-REAL 2,TOP-REAL 2 st
f=Sq_Circ & p`1=-1 & p`2<0 holds (f.p)`1<0 & (f.p)`2<0
proof
let p be Point of TOP-REAL 2, f be Function of TOP-REAL 2,TOP-REAL 2;
assume that
A1: f=Sq_Circ and
A2: p`1=-1 and
A3: p`2<0;
now per cases;
case p=0.TOP-REAL 2;
hence contradiction by A2,EUCLID:52,54;
end;
case
A4: p<> 0.TOP-REAL 2;
now per cases;
case p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1;
then
A5: f.p=|[p`1/sqrt(1+(p`2/p`1)^2),p`2/sqrt(1+(p`2/p`1)^2)]|
by A1,A4,JGRAPH_3:def 1;
then
A6: (f.p)`1= p`1/sqrt(1+(p`2/p`1)^2) by EUCLID:52;
(f.p)`2= p`2/sqrt(1+(p`2/p`1)^2) by A5,EUCLID:52;
hence thesis by A2,A3,A6,SQUARE_1:25,XREAL_1:141;
end;
case not (p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1);
then
A7: f.p=|[p`1/sqrt(1+(p`1/p`2)^2),p`2/sqrt(1+(p`1/p`2)^2)]|
by A1,A4,JGRAPH_3:def 1;
then
A8: (f.p)`1= p`1/sqrt(1+(p`1/p`2)^2) by EUCLID:52;
(f.p)`2= p`2/sqrt(1+(p`1/p`2)^2) by A7,EUCLID:52;
hence thesis by A2,A3,A8,SQUARE_1:25,XREAL_1:141;
end;
end;
hence thesis;
end;
end;
hence thesis;
end;
theorem Th68:
for p being Point of TOP-REAL 2,
P being non empty compact Subset of TOP-REAL 2,
f being Function of TOP-REAL 2,TOP-REAL 2 st
f=Sq_Circ
holds (f.p)`1>=0 iff p`1>=0
proof
let p be Point of TOP-REAL 2, P be non empty compact Subset of TOP-REAL 2,
f be Function of TOP-REAL 2,TOP-REAL 2;
assume that
A1: f=Sq_Circ;
thus (f.p)`1>=0 implies p`1>=0
proof
assume
A2: (f.p)`1>=0;
reconsider g=(Sq_Circ") as Function
of TOP-REAL 2,TOP-REAL 2 by JGRAPH_3:29;
A3: dom f = the carrier of TOP-REAL 2 by FUNCT_2:def 1;
set q=(f.p);
now per cases;
case q=0.TOP-REAL 2;
hence (g.q)`1>=0 by A2,JGRAPH_3:28;
end;
case
A4: q<> 0.TOP-REAL 2;
now per cases;
case q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=-q`1;
then
A5: g.q=|[q`1*sqrt(1+(q`2/q`1)^2),q`2*sqrt(1+(q`2/q`1)^2)]|
by A4,JGRAPH_3:28;
(q`2/q`1)^2 >=0 by XREAL_1:63;
then sqrt(1+(q`2/q`1)^2)>0 by SQUARE_1:25;
hence (g.q)`1>=0 by A2,A5,EUCLID:52;
end;
case not (q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=-q`1);
then
A6: g.q=|[q`1*sqrt(1+(q`1/q`2)^2),q`2*sqrt(1+(q`1/q`2)^2)]|
by JGRAPH_3:28;
(q`1/q`2)^2 >=0 by XREAL_1:63;
then sqrt(1+(q`1/q`2)^2)>0 by SQUARE_1:25;
hence (g.q)`1>=0 by A2,A6,EUCLID:52;
end;
end;
hence (g.q)`1>=0;
end;
end;
hence thesis by A1,A3,FUNCT_1:34;
end;
thus p`1>=0 implies (f.p)`1>=0
proof
assume
A7: p`1>=0;
now per cases;
case p=0.TOP-REAL 2;
hence thesis by A1,A7,JGRAPH_3:def 1;
end;
case
A8: p<> 0.TOP-REAL 2;
now per cases;
case p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1;
then
A9: f.p=|[p`1/sqrt(1+(p`2/p`1)^2),p`2/sqrt(1+(p`2/p`1)^2)]|
by A1,A8,JGRAPH_3:def 1;
(p`2/p`1)^2 >=0 by XREAL_1:63;
then sqrt(1+(p`2/p`1)^2)>0 by SQUARE_1:25;
hence thesis by A7,A9,EUCLID:52;
end;
case not (p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1);
then
A10: f.p=|[p`1/sqrt(1+(p`1/p`2)^2),p`2/sqrt(1+(p`1/p`2)^2)]|
by A1,A8,JGRAPH_3:def 1;
(p`1/p`2)^2 >=0 by XREAL_1:63;
then sqrt(1+(p`1/p`2)^2)>0 by SQUARE_1:25;
hence thesis by A7,A10,EUCLID:52;
end;
end;
hence thesis;
end;
end;
hence thesis;
end;
end;
theorem Th69:
for p being Point of TOP-REAL 2,
P being non empty compact Subset of TOP-REAL 2,
f being Function of TOP-REAL 2,TOP-REAL 2 st
f=Sq_Circ
holds (f.p)`2>=0 iff p`2>=0
proof
let p be Point of TOP-REAL 2, P be non empty compact Subset of TOP-REAL 2,
f be Function of TOP-REAL 2,TOP-REAL 2;
assume
A1: f=Sq_Circ;
thus (f.p)`2>=0 implies p`2>=0
proof
assume
A2: (f.p)`2>=0;
reconsider g=(Sq_Circ") as Function
of TOP-REAL 2,TOP-REAL 2 by JGRAPH_3:29;
A3: dom f = the carrier of TOP-REAL 2 by FUNCT_2:def 1;
set q=(f.p);
now per cases;
case q=0.TOP-REAL 2;
hence (g.q)`2>=0 by A2,JGRAPH_3:28;
end;
case
A4: q<> 0.TOP-REAL 2;
now per cases;
case q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=-q`1;
then
A5: g.q=|[q`1*sqrt(1+(q`2/q`1)^2),q`2*sqrt(1+(q`2/q`1)^2)]|
by A4,JGRAPH_3:28;
(q`2/q`1)^2 >=0 by XREAL_1:63;
then sqrt(1+(q`2/q`1)^2)>0 by SQUARE_1:25;
hence (g.q)`2>=0 by A2,A5,EUCLID:52;
end;
case not (q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=-q`1);
then
A6: g.q=|[q`1*sqrt(1+(q`1/q`2)^2),q`2*sqrt(1+(q`1/q`2)^2)]|
by JGRAPH_3:28;
(q`1/q`2)^2 >=0 by XREAL_1:63;
then sqrt(1+(q`1/q`2)^2)>0 by SQUARE_1:25;
hence (g.q)`2>=0 by A2,A6,EUCLID:52;
end;
end;
hence (g.q)`2>=0;
end;
end;
hence thesis by A1,A3,FUNCT_1:34;
end;
thus p`2>=0 implies (f.p)`2>=0
proof
assume
A7: p`2>=0;
now per cases;
case p=0.TOP-REAL 2;
hence thesis by A1,A7,JGRAPH_3:def 1;
end;
case
A8: p<> 0.TOP-REAL 2;
now per cases;
case p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1;
then
A9: f.p=|[p`1/sqrt(1+(p`2/p`1)^2),p`2/sqrt(1+(p`2/p`1)^2)]|
by A1,A8,JGRAPH_3:def 1;
(p`2/p`1)^2 >=0 by XREAL_1:63;
then sqrt(1+(p`2/p`1)^2)>0 by SQUARE_1:25;
hence thesis by A7,A9,EUCLID:52;
end;
case not (p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1);
then
A10: f.p=|[p`1/sqrt(1+(p`1/p`2)^2),p`2/sqrt(1+(p`1/p`2)^2)]|
by A1,A8,JGRAPH_3:def 1;
(p`1/p`2)^2 >=0 by XREAL_1:63;
then sqrt(1+(p`1/p`2)^2)>0 by SQUARE_1:25;
hence thesis by A7,A10,EUCLID:52;
end;
end;
hence thesis;
end;
end;
hence thesis;
end;
end;
theorem Th70:
for p,q being Point of TOP-REAL 2,
f being Function of TOP-REAL 2,TOP-REAL 2 st f=Sq_Circ
& p in LSeg(|[-1,-1]|,|[-1,1]|)
& q in LSeg(|[1,-1]|,|[-1,-1]|) holds (f.p)`1<=(f.q)`1
proof
let p,q be Point of TOP-REAL 2, f be Function of TOP-REAL 2,TOP-REAL 2;
assume that
A1: f=Sq_Circ and
A2: p in LSeg(|[-1,-1]|,|[-1,1]|) and
A3: q in LSeg(|[1,-1]|,|[-1,-1]|);
A4: p`1=-1 by A2,Th1;
A5: -1<=p`2 by A2,Th1;
A6: p`2<=1 by A2,Th1;
A7: q`2=-1 by A3,Th3;
A8: -1<=q`1 by A3,Th3;
A9: q`1<=1 by A3,Th3;
A10: p<>0.TOP-REAL 2 by A4,EUCLID:52,54;
A11: q<>0.TOP-REAL 2 by A7,EUCLID:52,54;
p`2<=-p`1 by A2,A4,Th1;
then f.p= |[p`1/sqrt(1+(p`2/p`1)^2),p`2/sqrt(1+(p`2/p`1)^2)]|
by A1,A4,A5,A10,JGRAPH_3:def 1;
then
A12: (f.p)`1=(-1)/sqrt(1+(p`2/(-1))^2) by A4,EUCLID:52
.=(-1)/sqrt(1+(p`2)^2);
(p`2)^2 >=0 by XREAL_1:63;
then
A13: sqrt(1+(p`2)^2)>0 by SQUARE_1:25;
(q`1)^2 >=0 by XREAL_1:63;
then
A14: sqrt(1+(q`1)^2)>0 by SQUARE_1:25;
q`1<=-q`2 by A3,A7,Th3;
then f.q= |[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2)]|
by A1,A7,A8,A11,JGRAPH_3:4;
then
A15: (f.q)`1=q`1/sqrt(1+(q`1/(-1))^2) by A7,EUCLID:52
.=q`1/sqrt(1+(q`1)^2);
-sqrt(1+(q`1)^2)<= q`1*sqrt(1+(p`2)^2) by A5,A6,A8,A9,SQUARE_1:55;
then (-1)*sqrt(1+(q`1)^2)/sqrt(1+(q`1)^2)
<= q`1*sqrt(1+(p`2)^2)/sqrt(1+(q`1)^2) by A14,XREAL_1:72;
then (-1) <= q`1*sqrt(1+(p`2)^2)/sqrt(1+(q`1)^2) by A14,XCMPLX_1:89;
then -1<= q`1/sqrt(1+(q`1)^2)*sqrt(1+(p`2)^2) by XCMPLX_1:74;
then (-1)/sqrt(1+(p`2)^2)
<= q`1/sqrt(1+(q`1)^2)*sqrt(1+(p`2)^2)/sqrt(1+(p`2)^2) by A13,XREAL_1:72;
hence thesis by A12,A13,A15,XCMPLX_1:89;
end;
theorem Th71:
for p,q being Point of TOP-REAL 2,
f being Function of TOP-REAL 2,TOP-REAL 2 st
f=Sq_Circ & p in LSeg(|[-1,-1]|,|[-1,1]|) & q in LSeg(|[-1,-1]|,|[-1,1]|)
& p`2>=q`2 & p`2<0 holds (f.p)`2>=(f.q)`2
proof
let p,q be Point of TOP-REAL 2, f be Function of TOP-REAL 2,TOP-REAL 2;
assume that
A1: f=Sq_Circ and
A2: p in LSeg(|[-1,-1]|,|[-1,1]|) and
A3: q in LSeg(|[-1,-1]|,|[-1,1]|) and
A4: p`2>=q`2 and
A5: p`2<0;
A6: p`1=-1 by A2,Th1;
A7: -1<=p`2 by A2,Th1;
(p`2)^2 >=0 by XREAL_1:63;
then
A8: sqrt(1+(p`2)^2)>0 by SQUARE_1:25;
(q`2)^2 >=0 by XREAL_1:63;
then
A9: sqrt(1+(q`2)^2)>0 by SQUARE_1:25;
A10: p`2<=-p`1 by A5,A6;
p<>0.TOP-REAL 2 by A5,EUCLID:52,54;
then f.p= |[p`1/sqrt(1+(p`2/p`1)^2),p`2/sqrt(1+(p`2/p`1)^2)]|
by A1,A6,A7,A10,JGRAPH_3:def 1;
then
A11: (f.p)`2= p`2/sqrt(1+(p`2/(-1))^2) by A6,EUCLID:52
.=(p`2)/sqrt(1+(p`2)^2);
A12: q`1=-1 by A3,Th1;
A13: -1<=q`2 by A3,Th1;
A14: q`2<=-q`1 by A4,A5,A12;
q<>0.TOP-REAL 2 by A4,A5,EUCLID:52,54;
then f.q= |[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|
by A1,A12,A13,A14,JGRAPH_3:def 1;
then
A15: (f.q)`2= q`2/sqrt(1+(q`2/(-1))^2) by A12,EUCLID:52
.=(q`2)/sqrt(1+(q`2)^2);
(p`2)*sqrt(1+(q`2)^2)>= (q`2)*sqrt(1+(p`2)^2) by A4,A5,Lm2;
then (p`2)*sqrt(1+(q`2)^2)/sqrt(1+(q`2)^2)
>= (q`2)*sqrt(1+(p`2)^2)/sqrt(1+(q`2)^2) by A9,XREAL_1:72;
then (p`2) >= (q`2)*sqrt(1+(p`2)^2)/sqrt(1+(q`2)^2) by A9,XCMPLX_1:89;
then (p`2)/sqrt(1+(p`2)^2)
>= (q`2)*sqrt(1+(p`2)^2)/sqrt(1+(q`2)^2)/sqrt(1+(p`2)^2) by A8,XREAL_1:72;
then (p`2)/sqrt(1+(p`2)^2)
>= (q`2)*sqrt(1+(p`2)^2)/sqrt(1+(p`2)^2)/sqrt(1+(q`2)^2) by XCMPLX_1:48;
hence thesis by A8,A11,A15,XCMPLX_1:89;
end;
theorem Th72:
for p1,p2,p3,p4 being Point of TOP-REAL 2,
P being non empty compact Subset of TOP-REAL 2,
f being Function of TOP-REAL 2,TOP-REAL 2 st P= circle(0,0,1) & f=Sq_Circ
holds LE p1,p2,rectangle(-1,1,-1,1) & LE p2,p3,rectangle(-1,1,-1,1) &
LE p3,p4,rectangle(-1,1,-1,1) implies
f.p1,f.p2,f.p3,f.p4 are_in_this_order_on P
proof
let p1,p2,p3,p4 be Point of TOP-REAL 2,
P be non empty compact Subset of TOP-REAL 2,
f be Function of TOP-REAL 2,TOP-REAL 2;
set K = rectangle(-1,1,-1,1);
assume that
A1: P= circle(0,0,1) and
A2: f=Sq_Circ;
A3: K is being_simple_closed_curve by Th50;
A4: K= {p: p`1=-1 & -1 <=p`2 & p`2<=1 or p`2=1 & -1<=p`1 & p`1<=1 or
p`1=1 & -1 <=p`2 & p`2<=1 or p`2=-1 & -1<=p`1 & p`1<=1} by Lm15;
A5: P={p: |.p.|=1} by A1,Th24;
thus LE p1,p2,K & LE p2,p3,K & LE p3,p4,K implies
f.p1,f.p2,f.p3,f.p4 are_in_this_order_on P
proof
assume that
A6: LE p1,p2,K and
A7: LE p2,p3,K and
A8: LE p3,p4,K;
A9: p1 in K by A3,A6,JORDAN7:5;
A10: p2 in K by A3,A6,JORDAN7:5;
A11: p3 in K by A3,A7,JORDAN7:5;
A12: p4 in K by A3,A8,JORDAN7:5;
then
A13: ex q8 being Point of TOP-REAL 2 st ( q8=p4)&( q8`1=-1 & -1
<=q8`2 & q8`2<=1 or q8`2=1 & -1<=q8`1 & q8`1<=1 or q8 `1=1 & -1 <=q8`2 & q8`2<=
1 or q8`2=-1 & -1<=q8`1 & q8`1<=1) by A4;
A14: LE p1,p3,K by A6,A7,Th50,JORDAN6:58;
A15: LE p2,p4,K by A7,A8,Th50,JORDAN6:58;
A16: W-min(K)=|[-1,-1]| by Th46;
A17: (|[-1,0]|)`2=0 by EUCLID:52;
A18: 1/2*(|[-1,-1]|+|[-1,1]|)=1/2*(|[-1,-1]|)+1/2*(|[-1,1]|) by RLVECT_1:def 5
.= (|[1/2*(-1),1/2*(-1)]|)+1/2*(|[-1,1]|) by EUCLID:58
.= (|[1/2*(-1),1/2*(-1)]|)+(|[1/2*(-1),1/2*1]|) by EUCLID:58
.= (|[1/2*(-1)+1/2*(-1),1/2*(-1)+1/2*1]|) by EUCLID:56
.= (|[(-1),0]|);
then
A19: |[-1,0]| in LSeg(|[-1,-1]|,|[-1,1]|) by RLTOPSP1:69;
now per cases by A9,A16,Th63,RLTOPSP1:68;
case
A20: p1 in LSeg(|[-1,-1]|,|[-1,1]|);
then
A21: p1`1=-1 by Th1;
then
A22: (f.p1)`1<0 by A2,Th68;
A23: f.:K=P by A2,A5,Lm15,Th35,JGRAPH_3:23;
A24: dom f = the carrier of TOP-REAL 2 by FUNCT_2:def 1;
then
A25: f.p1 in P by A9,A23,FUNCT_1:def 6;
now per cases;
case
A26: p1`2>=0;
then
A27: LE f.p1,f.p2,P by A1,A2,A6,A20,Th65;
A28: LE f.p2,f.p3,P by A1,A2,A6,A7,A20,A26,Th66;
LE f.p3,f.p4,P by A1,A2,A8,A14,A20,A26,Th66;
hence thesis by A27,A28,JORDAN17:def 1;
end;
case
A29: p1`2<0;
now per cases;
case
A30: p2`2<0 & p2 in LSeg(|[-1,-1]|,|[-1,1]|);
then
A31: p2`1=-1 by Th1;
A32: f.p2 in P by A10,A23,A24,FUNCT_1:def 6;
A33: p1`2<=p2`2 by A6,A20,A30,Th55;
now per cases;
case
A34: p3`2<0 & p3 in LSeg(|[-1,-1]|,|[-1,1]|);
then
A35: p3`1=-1 by Th1;
A36: f.p3 in P by A11,A23,A24,FUNCT_1:def 6;
A37: p2`2<=p3`2 by A7,A30,A34,Th55;
now per cases;
case
A38: p4`2<0 & p4 in LSeg(|[-1,-1]|,|[-1,1]|);
then
A39: p4`1=-1 by Th1;
A40: (f.p2)`1<0 by A2,A30,A31,Th67;
A41: (f.p2)`2<0 by A2,A30,A31,Th67;
A42: (f.p3)`1<0 by A2,A34,A35,Th67;
A43: (f.p3)`2<0 by A2,A34,A35,Th67;
A44: (f.p4)`1<0 by A2,A38,A39,Th67;
A45: (f.p4)`2<0 by A2,A38,A39,Th67;
(f.p1)`2<=(f.p2)`2 by A2,A20,A30,A33,Th71;
then
A46: LE f.p1,f.p2,P by A5,A22,A25,A32,A40,A41,JGRAPH_5:51;
(f.p2)`2<=(f.p3)`2 by A2,A30,A34,A37,Th71;
then
A47: LE f.p2,f.p3,P by A5,A32,A36,A40,A42,A43,JGRAPH_5:51;
A48: f.p4 in P by A12,A23,A24,FUNCT_1:def 6;
p3`2<=p4`2 by A8,A34,A38,Th55;
then (f.p3)`2<=(f.p4)`2 by A2,A34,A38,Th71;
then LE f.p3,f.p4,P by A5,A36,A42,A44,A45,A48,
JGRAPH_5:51;
hence thesis by A46,A47,JORDAN17:def 1;
end;
case
A49: not(p4`2<0 & p4 in LSeg(|[-1,-1]|,|[-1,1]|));
A50: now per cases by A12,Th63;
case p4 in LSeg(|[-1,-1]|,|[-1,1]|);
hence p4 in LSeg(|[-1,-1]|,|[-1,1]|) & (|[-1,0]|)`2<=p4`2
or p4 in LSeg(|[-1,1]|,|[1,1]|) or p4 in LSeg(|[1,1]|,|[1,-1]|)
or p4 in LSeg(|[1,-1]|,|[-1,-1]|) & p4<>W-min(K) by A49,EUCLID:52;
end;
case p4 in LSeg(|[-1,1]|,|[1,1]|);
hence p4 in LSeg(|[-1,-1]|,|[-1,1]|) & (|[-1,0]|)`2<=p4`2
or p4 in LSeg(|[-1,1]|,|[1,1]|) or p4 in LSeg(|[1,1]|,|[1,-1]|)
or p4 in LSeg(|[1,-1]|,|[-1,-1]|) & p4<>W-min(K);
end;
case p4 in LSeg(|[1,1]|,|[1,-1]|);
hence p4 in LSeg(|[-1,-1]|,|[-1,1]|) & (|[-1,0]|)`2<=p4`2
or p4 in LSeg(|[-1,1]|,|[1,1]|) or p4 in LSeg(|[1,1]|,|[1,-1]|)
or p4 in LSeg(|[1,-1]|,|[-1,-1]|) & p4<>W-min(K);
end;
case
A51: p4 in LSeg(|[1,-1]|,|[-1,-1]|);
A52: W-min(K)=|[-1,-1]| by Th46;
now
assume
A53: p4= W-min(K);
then p4`2=-1 by A52,EUCLID:52;
hence contradiction by A49,A52,A53,RLTOPSP1:68;
end;
hence p4 in LSeg(|[-1,-1]|,|[-1,1]|) & (|[-1,0]|)`2<=p4`2
or p4 in LSeg(|[-1,1]|,|[1,1]|) or p4 in LSeg(|[1,1]|,|[1,-1]|)
or p4 in LSeg(|[1,-1]|,|[-1,-1]|) & p4<>W-min(K) by A51;
end;
end;
A54: (f.p2)`1<0 by A2,A30,A31,Th67;
A55: (f.p2)`2<0 by A2,A30,A31,Th67;
A56: (f.p3)`1<0 by A2,A34,A35,Th67;
A57: (f.p3)`2<0 by A2,A34,A35,Th67;
(f.p1)`2<=(f.p2)`2 by A2,A20,A30,A33,Th71;
then
A58: LE f.p1,f.p2,P by A5,A22,A25,A32,A54,A55,JGRAPH_5:51;
(f.p2)`2<=(f.p3)`2 by A2,A30,A34,A37,Th71;
then
A59: LE f.p2,f.p3,P by A5,A32,A36,A54,A56,A57,JGRAPH_5:51;
A60: now per cases;
case
A61: p4`1=-1 & p4`2<0 & p1`2<=p4`2;
now per cases by A50;
case p4 in LSeg(|[-1,-1]|,|[-1,1]|) & (|[-1,0]|)`2<=p4`2;
hence contradiction by A61,EUCLID:52;
end;
case p4 in LSeg(|[-1,1]|,|[1,1]|);
hence contradiction by A61,Th3;
end;
case p4 in LSeg(|[1,1]|,|[1,-1]|);
hence contradiction by A61,Th1;
end;
case
A62: p4
in LSeg(|[1,-1]|,|[-1,-1]|) & p4<>W-min(K);
then
A63: p4`2= -1 by Th3;
A64: W-min(K)= |[-1,-1]| by Th46;
then
A65: (W-min(K))`1=-1 by EUCLID:52;
(W-min(K))`2=-1 by A64,EUCLID:52;
hence contradiction by A61,A62,A63,A65,
TOPREAL3:6;
end;
end;
hence contradiction;
end;
case
A66: not (p4`1=-1 & p4`2<0 & p1`2<=p4`2);
A67: p4
in LSeg(|[-1,-1]|,|[-1,1]|) or p4 in LSeg(|[-1,1]|,|[1,1]|)
or p4 in LSeg(|[1,1]|,|[1,-1]|) or p4 in LSeg(|[1,-1]|,|[-1,-1]|) by A12,Th63;
now per cases by A66;
case
A68: p4`1<>-1;
A69: f.:K=P by A2,A5,Lm15,Th35,JGRAPH_3:23;
A70: dom
f = the carrier of TOP-REAL 2 by FUNCT_2:def 1;
A71: (f.p1)`2<=0 by A2,A21,A29,Th67;
A72: Upper_Arc(P)
={p where p is Point of TOP-REAL 2:p in P & p`2>=0} by A5,JGRAPH_5:34;
A73: f.p1 in P by A9,A69,A70,FUNCT_1:def 6;
Lower_Arc(P)
={p where p is Point of TOP-REAL 2:p in P & p`2<=0} by A5,JGRAPH_5:35;
then
A74: f
.p1 in Lower_Arc(P) by A71,A73;
A75: f
.(|[-1,0]|)=W-min(P) by A2,A5,Th10,JGRAPH_5:29;
A76: now
assume f.p1=W-min(P);
then p1=|[-1,0]| by A2,A70,A75,FUNCT_1:def 4;
hence contradiction by A29,EUCLID:52;
end;
now per cases by A67,A68,Th1;
case p4 in LSeg(|[-1,1]|,|[1,1]|);
then
A77: p4
`2=1 by Th3;
A78: f.p4 in P by A12,A69,A70,FUNCT_1:def 6;
(f.p4)`2>=0 by A2,A77,Th69;
then f.p4 in Upper_Arc(P) by A72,A78;
hence LE f.p4,f.p1,P by A74,A76,JORDAN6:def 10;
end;
case p4 in LSeg(|[1,1]|,|[1,-1]|);
then
A79: p4
`1=1 by Th1;
A80: f.:K=P by A2,A5,Lm15,Th35,JGRAPH_3:23;
A81: dom
f = the carrier of TOP-REAL 2 by FUNCT_2:def 1;
then
A82: f
. p4 in P by A12,A80,FUNCT_1:def 6;
A83: f.p1 in P by A9,A80,A81,FUNCT_1:def 6;
A84: ( f.p1)`1<0 by A2,A21,A29,Th67;
A85: (f.p1)`2<=0 by A2,A21,A29,Th67;
A86: f
.(|[-1,0]|)=W-min(P) by A2,A5,Th10,JGRAPH_5:29;
A87: now
assume f.p1=W-min(P);
then p1=|[-1,0]| by A2,A81,A86,FUNCT_1:def 4;
hence contradiction by A29,EUCLID:52;
end;
A88: (f.p4)`1>=0 by A2,A79,Th68;
now per cases;
case
A89: (f.p4)`2>=0;
A90: (f.p1)`2<=0 by A2,A21,A29,Th67;
A91: Upper_Arc(P)
={p where p is Point of TOP-REAL 2:p in P & p`2>=0} by A5,JGRAPH_5:34;
A92: f
.(|[-1,0]|)=W-min(P) by A2,A5,Th10,JGRAPH_5:29;
A93: now
assume f.p1=W-min(P);
then p1=|[-1,0]| by A2,A81,A92,FUNCT_1:def 4;
hence contradiction by A29,EUCLID:52;
end;
A94: f.p4 in Upper_Arc(P) by A82,A89,A91;
Lower_Arc(P)
={p where p is Point of TOP-REAL 2:p in P & p`2<=0} by A5,JGRAPH_5:35;
then f.p1 in Lower_Arc(P) by A83,A90;
hence LE f.p4,f.p1,P by A93,A94,JORDAN6:def 10;
end;
case (f.p4)`2<0;
hence LE f.p4,f.p1,P
by A5,A82,A83,A84,A85,A87,A88,
JGRAPH_5:56;
end;
end;
hence LE f.p4,f.p1,P;
end;
case
A95: p4 in LSeg(|[1,-1]|,|[-1,-1]|);
then p4`2=-1 by Th3;
then
A96: (
f .p4)`2<0 by A2,Th69;
A97: f.p4 in P by A12,A69,A70,FUNCT_1:def 6;
A98: (f.p1)`2<=0 by A2,A21,A29,Th67;
(f.p4)`1>=(f.p1)`1 by A2,A20,A95,Th70;
hence LE f.p4,f.p1,P by A5,A73,A76,A96,A97,A98,JGRAPH_5:56;
end;
end;
hence LE f.p4,f.p1,P;
end;
case
A99: p4`1=-1 & p4`2>=0;
A100: f.:K=P by A2,A5,Lm15,Th35,JGRAPH_3:23;
A101: dom
f = the carrier of TOP-REAL 2 by FUNCT_2:def 1;
then
A102: f.p4 in P by A12,A100,FUNCT_1:def 6;
A103: f.p1 in P by A9,A100,A101,FUNCT_1:def 6;
A104: (f.p4)`2>=0 by A2,A99,Th69;
A105: (f.p1)`2<=0 by A2,A21,A29,Th67;
A106: Upper_Arc(P)
={p where p is Point of TOP-REAL 2:p in P & p`2>=0} by A5,JGRAPH_5:34;
A107: f
.(|[-1,0]|)=W-min(P) by A2,A5,Th10,JGRAPH_5:29;
A108: now
assume f.p1=W-min(P);
then p1=|[-1,0]| by A2,A101,A107,
FUNCT_1:def 4;
hence contradiction by A29,EUCLID:52;
end;
A109: f.p4 in Upper_Arc(P) by A102,A104,A106;
Lower_Arc(P)
={p where p is Point of TOP-REAL 2:p in P & p`2<=0} by A5,JGRAPH_5:35;
then f.p1 in Lower_Arc(P) by A103,A105;
hence LE f.p4,f.p1,P by A108,A109,
JORDAN6:def 10;
end;
case
A110: p4`1=-1 & p4`2<0 & p1`2>p4`2;
then
A111: p4
in LSeg (|[-1,-1]|,|[-1,1]|) by A13,Th2;
A112: f.:K=P by A2,A5,Lm15,Th35,JGRAPH_3:23;
A113: dom
f = the carrier of TOP-REAL 2 by FUNCT_2:def 1;
then
A114: f.p4 in P by A12,A112,FUNCT_1:def 6;
A115: f.p1 in P by A9,A112,A113,FUNCT_1:def 6;
A116: (f.p1)`1<0 by A2,A21,A29,Th67;
A117: (f.p1)`2<0 by A2,A21,A29,Th67;
A118: (f.p4)`2<=(f.p1)`2 by A2,A20,A29,A110,A111,Th71
;
(f.p4)`1<0 by A2,A110,Th68;
hence LE f.p4,f.p1,P by A5,A114,A115,A116,A117,A118,JGRAPH_5:51;
end;
end;
hence LE f.p4,f.p1,P;
end;
end;
A119: K
={p: p`1=-1 & -1 <=p`2 & p`2<=1 or p`2=1 & -1<=p`1 & p`1<=1 or
p`1=1 & -1 <=p`2 & p`2<=1 or p`2=-1 & -1<=p`1 & p`1<=1}
by Lm15;
thus K={p: p`1=-1 & -1 <=p`2 & p`2<=1 or
p`1=1 & -1 <=p`2 & p`2<=1 or -1=p`2 & -1<=p`1 & p`1<=1 or
1=p`2 & -1<=p`1 & p`1<=1}
proof
thus
K c= {p: p`1=-1 & -1 <=p`2 & p`2<=1 or
p`1=1 & -1 <=p`2 & p`2<=1 or -1=p`2 & -1<=p`1 & p`1<=1 or
1=p`2 & -1<=p`1 & p`1<=1}
proof
let x be object;
assume x in K;
then ex p st ( p=x)&( p`1=-1 & -1 <=p`2 & p`2<=1 or p`2=1 & -1
<=p`1 & p`1<=1 or p`1=1 & -1 <=p`2 & p`2<=1 or p`2=-1 & -1<=p`1 & p`1<=1) by
A119;
hence thesis;
end;
thus
{p: p`1=-1 & -1 <=p`2 & p`2<=1 or
p`1=1 & -1 <=p`2 & p`2<=1 or -1=p`2 & -1<=p`1 & p`1<=1 or
1=p`2 & -1<=p`1 & p`1<=1} c= K
proof
let x be object;
assume x in {p: p`1=-1 & -1 <=p`2 & p`2<=1 or
p`1=1 & -1 <=p`2 & p`2<=1 or -1=p`2 & -1<=p`1 & p`1<=1 or
1=p`2 & -1<=p`1 & p`1<=1};
then ex p st ( p=x)&( p`1=-1 & -1 <=p`2 & p`2<=1 or p`1=1 & -1
<=p`2 & p`2<=1 or -1=p`2 & -1<=p`1 & p`1<=1 or 1=p`2 & -1<=p`1 & p`1<=1);
hence thesis by A119;
end;
end;
thus thesis by A58,A59,A60,JORDAN17:def 1;
end;
end;
hence thesis;
end;
case
A120: not(p3`2<0 & p3 in LSeg(|[-1,-1]|,|[-1,1]|));
A121: now per cases by A11,Th63;
case p3 in LSeg(|[-1,-1]|,|[-1,1]|);
hence p3 in LSeg(|[-1,-1]|,|[-1,1]|) & (|[-1,0]|)`2<=p3`2
or p3 in LSeg(|[-1,1]|,|[1,1]|) or p3 in LSeg(|[1,1]|,|[1,-1]|)
or p3 in LSeg(|[1,-1]|,|[-1,-1]|) & p3<>W-min(K) by A120,EUCLID:52;
end;
case p3 in LSeg(|[-1,1]|,|[1,1]|);
hence p3 in LSeg(|[-1,-1]|,|[-1,1]|) & (|[-1,0]|)`2<=p3`2
or p3 in LSeg(|[-1,1]|,|[1,1]|) or p3 in LSeg(|[1,1]|,|[1,-1]|)
or p3 in LSeg(|[1,-1]|,|[-1,-1]|) & p3<>W-min(K);
end;
case p3 in LSeg(|[1,1]|,|[1,-1]|);
hence p3 in LSeg(|[-1,-1]|,|[-1,1]|) & (|[-1,0]|)`2<=p3`2
or p3 in LSeg(|[-1,1]|,|[1,1]|) or p3 in LSeg(|[1,1]|,|[1,-1]|)
or p3 in LSeg(|[1,-1]|,|[-1,-1]|) & p3<>W-min(K);
end;
case
A122: p3 in LSeg(|[1,-1]|,|[-1,-1]|);
A123: W-min(K)=|[-1,-1]| by Th46;
now
assume
A124: p3= W-min(K);
then p3`2=-1 by A123,EUCLID:52;
hence contradiction by A120,A123,A124,RLTOPSP1:68;
end;
hence p3 in LSeg(|[-1,-1]|,|[-1,1]|) & (|[-1,0]|)`2<=p3`2
or p3 in LSeg(|[-1,1]|,|[1,1]|) or p3 in LSeg(|[1,1]|,|[1,-1]|)
or p3 in LSeg(|[1,-1]|,|[-1,-1]|) & p3<>W-min(K) by A122;
end;
end;
then
A125: LE |[-1,0]|,p3,K by A19,Th59;
A126: (f.p2)`1<0 by A2,A30,A31,Th67;
A127: (f.p2)`2<0 by A2,A30,A31,Th67;
(f.p1)`2<=(f.p2)`2 by A2,A20,A30,A33,Th71;
then
A128: LE f.p1,f.p2,P by A5,A22,A25,A32,A126,A127,JGRAPH_5:51;
A129: LE f.p3,f.p4,P by A1,A2,A8,A17,A18,A125,Th66,RLTOPSP1:69;
A130: now per cases;
case
A131: p4`1=-1 & p4`2<0 & p1`2<=p4`2;
A132: (|[-1,-1]|)`1=-1 by EUCLID:52;
A133: (|[-1,-1]|)`2=-1 by EUCLID:52;
A134: (|[-1,1]|)`1=-1 by EUCLID:52;
A135: (|[-1,1]|)`2=1 by EUCLID:52;
-1<=p4`2 by A12,Th19;
then
A136: p4
in LSeg(|[-1,-1]|,|[-1,1]|) by A131,A132,A133,A134,A135,GOBOARD7:7;
now per cases by A121;
case
A137: p3
in LSeg(|[-1,-1]|,|[-1,1]|) & (|[-1,0]|)`2<=p3`2;
then 0<=p3`2 by EUCLID:52;
hence contradiction by A8,A131,A136,A137,Th55;
end;
case
A138: p3 in LSeg(|[-1,1]|,|[1,1]|);
then LE p4,p3,K by A136,Th59;
then p3=p4 by A8,Th50,JORDAN6:57;
hence contradiction by A131,A138,Th3;
end;
case
A139: p3 in LSeg(|[1,1]|,|[1,-1]|);
then LE p4,p3,K by A136,Th59;
then p3=p4 by A8,Th50,JORDAN6:57;
hence contradiction by A131,A139,Th1;
end;
case
A140: p3 in LSeg(|[1,-1]|,|[-1,-1]|) & p3<>W-min(K);
then LE p4,p3,K by A136,Th59;
then
A141: p3=p4 by A8,Th50,JORDAN6:57;
A142: p3`2= -1 by A140,Th3;
A143: W-min(K)= |[-1,-1]| by Th46;
then
A144: (W-min(K))`1=-1 by EUCLID:52;
(W-min(K))`2=-1 by A143,EUCLID:52;
hence contradiction by A131,A140,A141,A142,A144,TOPREAL3:6;
end;
end;
hence contradiction;
end;
case
A145: not (p4`1=-1 & p4`2<0 & p1`2<=p4`2);
A146: p4
in LSeg(|[-1,-1]|,|[-1,1]|) or p4 in LSeg(|[-1,1]|,|[1,1]|)
or p4 in LSeg(|[1,1]|,|[1,-1]|) or p4 in LSeg(|[1,-1]|,|[-1,-1]|) by A12,Th63;
now per cases by A145;
case
A147: p4`1<>-1;
A148: f.:K=P by A2,A5,Lm15,Th35,JGRAPH_3:23;
A149: dom
f = the carrier of TOP-REAL 2 by FUNCT_2:def 1;
A150: (f.p1)`2<=0 by A2,A21,A29,Th67;
A151: Upper_Arc(P)
={p where p is Point of TOP-REAL 2:p in P & p`2>=0}
by A5,JGRAPH_5:34;
A152: f.p1 in P by A9,A148,A149,FUNCT_1:def 6;
Lower_Arc(P)
={p where p is Point of TOP-REAL 2:p in P & p`2<=0}
by A5,JGRAPH_5:35;
then
A153: f
.p1 in Lower_Arc( P) by A150,A152;
A154: f.(|[-1,0]|)=W-min(P) by A2,A5,Th10,JGRAPH_5:29;
A155: now
assume f.p1=W-min(P);
then p1=|[-1,0]| by A2,A149,A154,FUNCT_1:def 4;
hence contradiction by A29,EUCLID:52;
end;
now per cases by A146,A147,Th1;
case p4 in LSeg(|[-1,1]|,|[1,1]|);
then
A156: p4`2=1 by Th3;
A157: f.p4 in P by A12,A148,A149,FUNCT_1:def 6;
(f.p4)`2>=0 by A2,A156,Th69;
then f.p4 in Upper_Arc(P) by A151,A157;
hence LE f.p4,f.p1,P by A153,A155,JORDAN6:def 10;
end;
case p4 in LSeg(|[1,1]|,|[1,-1]|);
then
A158: p4`1=1 by Th1;
A159: f.:K=P by A2,A5,Lm15,Th35,JGRAPH_3:23;
A160: dom
f = the carrier of TOP-REAL 2 by FUNCT_2:def 1;
then
A161: f.p4 in P by A12,A159,FUNCT_1:def 6;
A162: f.p1 in P by A9,A159,A160,FUNCT_1:def 6;
A163: (f.p1)`1<0 by A2,A21,A29,Th67;
A164: (f.p1)`2<=0 by A2,A21,A29,Th67;
A165: f
.(|[-1,0]|)=W-min(P) by A2,A5,Th10,JGRAPH_5:29;
A166: now
assume f.p1=W-min(P);
then p1=|[-1,0]| by A2,A160,A165,FUNCT_1:def 4;
hence contradiction by A29,EUCLID:52;
end;
A167: (f.p4)`1>=0 by A2,A158,Th68;
now per cases;
case
A168: (f.p4)`2>=0;
A169: (f.p1)`2<=0 by A2,A21,A29,Th67;
A170: Upper_Arc(P)
={p where p is Point of TOP-REAL 2:p in P & p`2>=0} by A5,JGRAPH_5:34;
A171: f
.(|[-1,0]|)=W-min(P) by A2,A5,Th10,JGRAPH_5:29;
A172: now
assume f.p1=W-min(P);
then p1=|[-1,0]| by A2,A160,A171,FUNCT_1:def 4;
hence contradiction by A29,EUCLID:52;
end;
A173: f.p4 in Upper_Arc(P) by A161,A168,A170;
Lower_Arc(P)
={p where p is Point of TOP-REAL 2:p in P & p`2<=0} by A5,JGRAPH_5:35;
then f.p1 in Lower_Arc(P) by A162,A169;
hence LE f.p4,f.p1,P by A172,A173,JORDAN6:def 10;
end;
case (f.p4)`2<0;
hence LE f.p4,f.p1,P
by A5,A161,A162,A163,A164,A166,A167,
JGRAPH_5:56;
end;
end;
hence LE f.p4,f.p1,P;
end;
case
A174: p4 in LSeg(|[1,-1]|,|[-1,-1]|);
then p4`2=-1 by Th3;
then
A175: (f.p4)`2<0 by A2,Th69;
A176: f.p4 in P by A12,A148,A149,FUNCT_1:def 6;
A177: (f.p1)`2<=0 by A2,A21,A29,Th67;
(f.p4)`1>=(f.p1)`1 by A2,A20,A174,Th70;
hence LE f.p4,f.p1,P by A5,A152,A155,A175,A176,A177,JGRAPH_5:56;
end;
end;
hence LE f.p4,f.p1,P;
end;
case
A178: p4`1=-1 & p4`2>=0;
A179: f.:K=P by A2,A5,Lm15,Th35,JGRAPH_3:23;
A180: dom
f = the carrier of TOP-REAL 2 by FUNCT_2:def 1;
then
A181: f.p4 in P by A12,A179,FUNCT_1:def 6;
A182: f.p1 in P by A9,A179,A180,FUNCT_1:def 6;
A183: (f.p4)`2>=0 by A2,A178,Th69;
A184: (f.p1)`2<=0 by A2,A21,A29,Th67;
A185: Upper_Arc(P)
={p where p is Point of TOP-REAL 2:p in P & p`2>=0}
by A5,JGRAPH_5:34;
A186: f.(|[-1,0]|)=W-min(P) by A2,A5,Th10,JGRAPH_5:29;
A187: now
assume f.p1=W-min(P);
then p1=|[-1,0]| by A2,A180,A186,FUNCT_1:def 4;
hence contradiction by A29,EUCLID:52;
end;
A188: f.p4 in Upper_Arc(P) by A181,A183,A185;
Lower_Arc(P)
={p where p is Point of TOP-REAL 2:p in P & p`2<=0}
by A5,JGRAPH_5:35;
then f.p1 in Lower_Arc(P) by A182,A184;
hence LE f.p4,f.p1,P by A187,A188,JORDAN6:def 10;
end;
case
A189: p4`1=-1 & p4`2<0 & p1`2>p4`2;
then
A190: p4
in LSeg(|[-1,-1 ]|,|[-1,1]|) by A13,Th2;
A191: f.:K=P by A2,A5,Lm15,Th35,JGRAPH_3:23;
A192: dom
f = the carrier of TOP-REAL 2 by FUNCT_2:def 1;
then
A193: f.p4 in P by A12,A191,FUNCT_1:def 6;
A194: f.p1 in P by A9,A191,A192,FUNCT_1:def 6;
A195: (f.p1)`1<0 by A2,A21,A29,Th67;
A196: (f.p1)`2<0 by A2,A21,A29,Th67;
A197: (f.p4)`2<=(f.p1)`2 by A2,A20,A29,A189,A190,Th71;
(f.p4)`1<0 by A2,A189,Th68;
hence LE f.p4,f.p1,P by A5,A193,A194,A195,A196,A197,JGRAPH_5:51;
end;
end;
hence LE f.p4,f.p1,P;
end;
end;
A198: K
={p: p`1=-1 & -1 <=p`2 & p`2<=1 or p`2=1 & -1<=p`1 & p`1<=1 or
p`1=1 & -1 <=p`2 & p`2<=1 or p`2=-1 & -1<=p`1 & p`1<=1}
by Lm15;
thus K={p: p`1=-1 & -1 <=p`2 & p`2<=1 or
p`1=1 & -1 <=p`2 & p`2<=1 or -1=p`2 & -1<=p`1 & p`1<=1 or
1=p`2 & -1<=p`1 & p`1<=1}
proof
thus K c= {p: p`1=-1 & -1 <=p`2 & p`2<=1 or
p`1=1 & -1 <=p`2 & p`2<=1 or -1=p`2 & -1<=p`1 & p`1<=1 or
1=p`2 & -1<=p`1 & p`1<=1}
proof
let x be object;
assume x in K;
then ex p st ( p=x)&( p`1=-1 & -1 <=p`2 & p`2<=1 or p`2=1 & -1
<=p`1 & p`1<=1 or p`1=1 & -1 <=p`2 & p`2<=1 or p`2=-1 & -1<=p`1 & p`1<=1) by
A198;
hence thesis;
end;
thus
{p: p`1=-1 & -1 <=p`2 & p`2<=1 or
p`1=1 & -1 <=p`2 & p`2<=1 or -1=p`2 & -1<=p`1 & p`1<=1 or
1=p`2 & -1<=p`1 & p`1<=1} c= K
proof
let x be object;
assume x in {p: p`1=-1 & -1 <=p`2 & p`2<=1 or
p`1=1 & -1 <=p`2 & p`2<=1 or -1=p`2 & -1<=p`1 & p`1<=1 or
1=p`2 & -1<=p`1 & p`1<=1};
then ex p st ( p=x)&( p`1=-1 & -1 <=p`2 & p`2<=1 or p`1=1 & -1
<=p`2 & p`2<=1 or -1=p`2 & -1<=p`1 & p`1<=1 or 1=p`2 & -1<=p`1 & p`1<=1);
hence thesis by A198;
end;
end;
thus thesis by A128,A129,A130,JORDAN17:def 1;
end;
end;
hence thesis;
end;
case
A199: not(p2`2<0 & p2 in LSeg(|[-1,-1]|,|[-1,1]|));
A200: now per cases by A10,Th63;
case p2 in LSeg(|[-1,-1]|,|[-1,1]|);
hence
p2 in LSeg(|[-1,-1]|,|[-1,1]|) & (|[-1,0]|)`2<=p2`2
or p2 in LSeg(|[-1,1]|,|[1,1]|) or p2 in LSeg(|[1,1]|,|[1,-1]|)
or p2 in LSeg(|[1,-1]|,|[-1,-1]|) & p2<>W-min(K) by A199,EUCLID:52;
end;
case p2 in LSeg(|[-1,1]|,|[1,1]|);
hence
p2 in LSeg(|[-1,-1]|,|[-1,1]|) & (|[-1,0]|)`2<=p2`2
or p2 in LSeg(|[-1,1]|,|[1,1]|) or p2 in LSeg(|[1,1]|,|[1,-1]|)
or p2 in LSeg(|[1,-1]|,|[-1,-1]|) & p2<>W-min(K);
end;
case p2 in LSeg(|[1,1]|,|[1,-1]|);
hence
p2 in LSeg(|[-1,-1]|,|[-1,1]|) & (|[-1,0]|)`2<=p2`2
or p2 in LSeg(|[-1,1]|,|[1,1]|) or p2 in LSeg(|[1,1]|,|[1,-1]|)
or p2 in LSeg(|[1,-1]|,|[-1,-1]|) & p2<>W-min(K);
end;
case
A201: p2 in LSeg(|[1,-1]|,|[-1,-1]|);
A202: W-min(K)=|[-1,-1]| by Th46;
now
assume
A203: p2= W-min(K);
then p2`2=-1 by A202,EUCLID:52;
hence contradiction by A199,A202,A203,RLTOPSP1:68;
end;
hence
p2 in LSeg(|[-1,-1]|,|[-1,1]|) & (|[-1,0]|)`2<=p2`2
or p2 in LSeg(|[-1,1]|,|[1,1]|) or p2 in LSeg(|[1,1]|,|[1,-1]|)
or p2 in LSeg(|[1,-1]|,|[-1,-1]|) & p2<>W-min(K) by A201;
end;
end;
then
A204: LE |[-1,0]|,p2,K by A19,Th59;
then
A205: LE f.p2,f.p3,P by A1,A2,A7,A17,A18,Th66,RLTOPSP1:69;
LE |[-1,0]|,p3,K by A7,A204,Th50,JORDAN6:58;
then
A206: LE f.p3,f.p4,P by A1,A2,A8,A17,A18,Th66,RLTOPSP1:69;
A207: now per cases;
case
A208: p4`1=-1 & p4`2<0 & p1`2<=p4`2;
A209: (|[-1,-1]|)`1=-1 by EUCLID:52;
A210: (|[-1,-1]|)`2=-1 by EUCLID:52;
A211: (|[-1,1]|)`1=-1 by EUCLID:52;
A212: (|[-1,1]|)`2=1 by EUCLID:52;
-1<=p4`2 by A12,Th19;
then
A213: p4 in LSeg(|[-1,-1]|,|[-1,1]|) by A208,A209,A210,A211,A212,
GOBOARD7:7;
now per cases by A200;
case
A214: p2 in LSeg(|[-1,-1]|,|[-1,1]|) & (|[-1,0]|)`2<=p2`2;
then 0<=p2`2 by EUCLID:52;
hence contradiction by A15,A208,A213,A214,Th55;
end;
case
A215: p2 in LSeg(|[-1,1]|,|[1,1]|);
then LE p4,p2,K by A213,Th59;
then p2=p4 by A15,Th50,JORDAN6:57;
hence contradiction by A208,A215,Th3;
end;
case
A216: p2 in LSeg(|[1,1]|,|[1,-1]|);
then LE p4,p2,K by A213,Th59;
then p2=p4 by A15,Th50,JORDAN6:57;
hence contradiction by A208,A216,Th1;
end;
case
A217: p2 in LSeg(|[1,-1]|,|[-1,-1]|) & p2<>W-min(K);
then LE p4,p2,K by A213,Th59;
then
A218: p2=p4 by A15,Th50,JORDAN6:57;
A219: p2`2= -1 by A217,Th3;
A220: W-min(K)= |[-1,-1]| by Th46;
then
A221: (W-min(K))`1=-1 by EUCLID:52;
(W-min(K))`2=-1 by A220,EUCLID:52;
hence contradiction by A208,A217,A218,A219,A221,
TOPREAL3:6;
end;
end;
hence contradiction;
end;
case
A222: not (p4`1=-1 & p4`2<0 & p1`2<=p4`2);
A223: p4
in LSeg(|[-1,-1]|,|[-1,1]|) or p4 in LSeg(|[-1,1]|,|[1,1]|)
or p4 in LSeg(|[1,1]|,|[1,-1]|) or p4 in LSeg(|[1,-1]|,|[-1,-1]|) by A12,Th63;
now per cases by A222;
case
A224: p4`1<>-1;
A225: f.:K=P by A2,A5,Lm15,Th35,JGRAPH_3:23;
A226: dom f = the carrier of TOP-REAL 2 by FUNCT_2:def 1;
A227: (f.p1)`2<=0 by A2,A21,A29,Th67;
A228: Upper_Arc(P)
={p where p is Point of TOP-REAL 2:p in P & p`2>=0}
by A5,JGRAPH_5:34;
A229: f.p1 in P by A9,A225,A226,FUNCT_1:def 6;
Lower_Arc(P)
={p where p is Point of TOP-REAL 2:p in P & p`2<=0}
by A5,JGRAPH_5:35;
then
A230: f.p1 in Lower_Arc(P) by A227,A229;
A231: f.(|[-1,0]|)=W-min(P) by A2,A5,Th10,JGRAPH_5:29;
A232: now
assume f.p1=W-min(P);
then p1=|[-1,0]| by A2,A226,A231,FUNCT_1:def 4;
hence contradiction by A29,EUCLID:52;
end;
now per cases by A223,A224,Th1;
case p4 in LSeg(|[-1,1]|,|[1,1]|);
then
A233: p4`2=1 by Th3;
A234: f.p4 in P by A12,A225,A226,FUNCT_1:def 6;
(f.p4)`2>=0 by A2,A233,Th69;
then f.p4 in Upper_Arc(P) by A228,A234;
hence LE f.p4,f.p1,P by A230,A232,JORDAN6:def 10;
end;
case p4 in LSeg(|[1,1]|,|[1,-1]|);
then
A235: p4`1=1 by Th1;
A236: f.:K=P by A2,A5,Lm15,Th35,JGRAPH_3:23;
A237: dom
f = the carrier of TOP-REAL 2 by FUNCT_2:def 1;
then
A238: f.p4 in P by A12,A236,FUNCT_1:def 6;
A239: f.p1 in P by A9,A236,A237,FUNCT_1:def 6;
A240: (f.p1)`1<0 by A2,A21,A29,Th67;
A241: (f.p1)`2<=0 by A2,A21,A29,Th67;
A242: f.(|[-1,0]|)=W-min(P) by A2,A5,Th10,JGRAPH_5:29;
A243: now
assume f.p1=W-min(P);
then p1=|[-1,0]| by A2,A237,A242,FUNCT_1:def 4;
hence contradiction by A29,EUCLID:52;
end;
A244: (f.p4)`1>=0 by A2,A235,Th68;
now per cases;
case
A245: (f.p4)`2>=0;
A246: (f.p1)`2<=0 by A2,A21,A29,Th67;
A247: Upper_Arc(P)
={p where p is Point of TOP-REAL 2:p in P & p`2>=0} by A5,JGRAPH_5:34;
A248: f
.(|[-1,0]|)=W-min(P) by A2,A5,Th10,JGRAPH_5:29;
A249: now
assume f.p1=W-min(P);
then p1=|[-1,0]| by A2,A237,A248,FUNCT_1:def 4;
hence contradiction by A29,EUCLID:52;
end;
A250: f.p4 in Upper_Arc(P) by A238,A245,A247;
Lower_Arc(P)
={p where p is Point of TOP-REAL 2:p in P & p`2<=0} by A5,JGRAPH_5:35;
then f.p1 in Lower_Arc(P) by A239,A246;
hence LE f.p4,f.p1,P by A249,A250,JORDAN6:def 10;
end;
case (f.p4)`2<0;
hence LE f.p4,f.p1,P
by A5,A238,A239,A240,A241,A243,A244,JGRAPH_5:56
;
end;
end;
hence LE f.p4,f.p1,P;
end;
case
A251: p4 in LSeg(|[1,-1]|,|[-1,-1]|);
then p4`2=-1 by Th3;
then
A252: (f.p4)`2<0 by A2,Th69;
A253: f.p4 in P by A12,A225,A226,FUNCT_1:def 6;
A254: (f.p1)`2<=0 by A2,A21,A29,Th67;
(f.p4)`1>=(f.p1)`1 by A2,A20,A251,Th70;
hence LE f.p4,f.p1,P by A5,A229,A232,A252,A253,A254,JGRAPH_5:56;
end;
end;
hence LE f.p4,f.p1,P;
end;
case
A255: p4`1=-1 & p4`2>=0;
A256: f.:K=P by A2,A5,Lm15,Th35,JGRAPH_3:23;
A257: dom f = the carrier of TOP-REAL 2 by FUNCT_2:def 1;
then
A258: f.p4 in P by A12,A256,FUNCT_1:def 6;
A259: f.p1 in P by A9,A256,A257,FUNCT_1:def 6;
A260: (f.p4)`2>=0 by A2,A255,Th69;
A261: (f.p1)`2<=0 by A2,A21,A29,Th67;
A262: Upper_Arc(P)
={p where p is Point of TOP-REAL 2:p in P & p`2>=0}
by A5,JGRAPH_5:34;
A263: f.(|[-1,0]|)=W-min(P) by A2,A5,Th10,JGRAPH_5:29;
A264: now
assume f.p1=W-min(P);
then p1=|[-1,0]| by A2,A257,A263,FUNCT_1:def 4;
hence contradiction by A29,EUCLID:52;
end;
A265: f.p4 in Upper_Arc(P) by A258,A260,A262;
Lower_Arc(P)
={p where p is Point of TOP-REAL 2:p in P & p`2<=0}
by A5,JGRAPH_5:35;
then f.p1 in Lower_Arc(P) by A259,A261;
hence LE f.p4,f.p1,P by A264,A265,JORDAN6:def 10;
end;
case
A266: p4`1=-1 & p4`2<0 & p1`2>p4`2;
then
A267: p4
in LSeg(|[-1,-1]|,|[-1, 1]|) by A13,Th2;
A268: f.:K=P by A2,A5,Lm15,Th35,JGRAPH_3:23;
A269: dom f = the carrier of TOP-REAL 2 by FUNCT_2:def 1;
then
A270: f.p4 in P by A12,A268,FUNCT_1:def 6;
A271: f.p1 in P by A9,A268,A269,FUNCT_1:def 6;
A272: (f.p1)`1<0 by A2,A21,A29,Th67;
A273: (f.p1)`2<0 by A2,A21,A29,Th67;
A274: (f.p4)`2<=(f.p1)`2 by A2,A20,A29,A266,A267,Th71;
(f.p4)`1<0 by A2,A266,Th68;
hence LE f.p4,f.p1,P by A5,A270,A271,A272,A273,A274,JGRAPH_5:51;
end;
end;
hence LE f.p4,f.p1,P;
end;
end;
A275: K
={p: p`1=-1 & -1 <=p`2 & p`2<=1 or p`2=1 & -1<=p`1 & p`1<=1 or
p`1=1 & -1 <=p`2 & p`2<=1 or p`2=-1 & -1<=p`1 & p`1<=1}
by Lm15;
thus K={p: p`1=-1 & -1 <=p`2 & p`2<=1 or
p`1=1 & -1 <=p`2 & p`2<=1 or -1=p`2 & -1<=p`1 & p`1<=1 or
1=p`2 & -1<=p`1 & p`1<=1}
proof
thus K c= {p: p`1=-1 & -1 <=p`2 & p`2<=1 or
p`1=1 & -1 <=p`2 & p`2<=1 or -1=p`2 & -1<=p`1 & p`1<=1 or
1=p`2 & -1<=p`1 & p`1<=1}
proof
let x be object;
assume x in K;
then ex p st ( p=x)&( p`1=-1 & -1 <=p`2 & p`2<=1 or p`2=1 & -1
<=p`1 & p`1<=1 or p`1=1 & -1 <=p`2 & p`2<=1 or p`2=-1 & -1<=p`1 & p`1<=1) by
A275;
hence thesis;
end;
thus
{p: p`1=-1 & -1 <=p`2 & p`2<=1 or
p`1=1 & -1 <=p`2 & p`2<=1 or -1=p`2 & -1<=p`1 & p`1<=1 or
1=p`2 & -1<=p`1 & p`1<=1} c= K
proof
let x be object;
assume x in {p: p`1=-1 & -1 <=p`2 & p`2<=1 or
p`1=1 & -1 <=p`2 & p`2<=1 or -1=p`2 & -1<=p`1 & p`1<=1 or
1=p`2 & -1<=p`1 & p`1<=1};
then ex p st ( p=x)&( p`1=-1 & -1 <=p`2 & p`2<=1 or p`1=1 & -1
<=p`2 & p`2<=1 or -1=p`2 & -1<=p`1 & p`1<=1 or 1=p`2 & -1<=p`1 & p`1<=1);
hence thesis by A275;
end;
end;
thus thesis by A205,A206,A207,JORDAN17:def 1;
end;
end;
hence thesis;
end;
end;
hence thesis;
end;
case
A276: p1 in LSeg(|[-1,1]|,|[1,1]|);
A277: |[-1,1]| in LSeg(|[-1,1]|,|[1,1]|) by RLTOPSP1:68;
A278: (|[-1,1]|)`1=-1 by EUCLID:52;
A279: (|[-1,1]|)`2=1 by EUCLID:52;
-1 <=p1`1 by A276,Th3;
then
A280: LE |[-1,1]|,p1,K by A276,A277,A278,Th60;
then
A281: LE f.p1,f.p2,P by A1,A2,A6,A279,Th66,RLTOPSP1:68;
A282: LE |[-1,1]|,p2,K by A6,A280,Th50,JORDAN6:58;
then
A283: LE f.p2,f.p3,P by A1,A2,A7,A279,Th66,RLTOPSP1:68;
LE |[-1,1]|,p3,K by A7,A282,Th50,JORDAN6:58;
then LE f.p3,f.p4,P by A1,A2,A8,A279,Th66,RLTOPSP1:68;
hence thesis by A281,A283,JORDAN17:def 1;
end;
case
A284: p1 in LSeg(|[1,1]|,|[1,-1]|);
A285: |[-1,1]| in LSeg(|[-1,1]|,|[1,1]|) by RLTOPSP1:68;
A286: (|[-1,1]|)`1=-1 by EUCLID:52;
A287: (|[-1,1]|)`2=1 by EUCLID:52;
A288: |[1,1]| in LSeg(|[1,1]|,|[1,-1]|) by RLTOPSP1:68;
A289: |[1,1]| in LSeg(|[-1,1]|,|[1,1]|) by RLTOPSP1:68;
A290: (|[1,1]|)`1=1 by EUCLID:52;
A291: (|[1,1]|)`2=1 by EUCLID:52;
A292: LE |[-1,1]|,|[1,1]|,K by A285,A286,A289,A290,Th60;
p1`2<=1 by A284,Th1;
then LE |[1,1]|,p1,K by A284,A288,A291,Th61;
then
A293: LE |[-1,1]|,p1,K by A292,Th50,JORDAN6:58;
then
A294: LE f.p1,f.p2,P by A1,A2,A6,A287,Th66,RLTOPSP1:68;
A295: LE |[-1,1]|,p2,K by A6,A293,Th50,JORDAN6:58;
then
A296: LE f.p2,f.p3,P by A1,A2,A7,A287,Th66,RLTOPSP1:68;
LE |[-1,1]|,p3,K by A7,A295,Th50,JORDAN6:58;
then LE f.p3,f.p4,P by A1,A2,A8,A287,Th66,RLTOPSP1:68;
hence thesis by A294,A296,JORDAN17:def 1;
end;
case
A297: p1 in LSeg(|[1,-1]|,|[-1,-1]|) & p1 <> W-min(K);
A298: |[-1,1]| in LSeg(|[-1,1]|,|[1,1]|) by RLTOPSP1:68;
A299: (|[-1,1]|)`1=-1 by EUCLID:52;
A300: (|[-1,1]|)`2=1 by EUCLID:52;
A301: |[1,1]| in LSeg(|[-1,1]|,|[1,1]|) by RLTOPSP1:68;
(|[1,1]|)`1=1 by EUCLID:52;
then
A302: LE |[-1,1]|,|[1,1]|,K by A298,A299,A301,Th60;
A303: |[1,-1]| in LSeg(|[1,1]|,|[1,-1]|) by RLTOPSP1:68;
A304: |[1,-1]| in LSeg(|[1,-1]|,|[-1,-1]|) by RLTOPSP1:68;
A305: (|[1,-1]|)`1=1 by EUCLID:52;
LE |[1,1]|,|[1,-1]|,K by A301,A303,Th60;
then
A306: LE |[-1,1]|,|[1,-1]|,K by A302,Th50,JORDAN6:58;
W-min(K)=|[-1,-1]| by Th46;
then
A307: (W-min(K))`1=-1 by EUCLID:52;
p1`1<=1 by A297,Th3;
then LE |[1,-1]|,p1,K by A297,A304,A305,A307,Th62;
then
A308: LE |[-1,1]|,p1,K by A306,Th50,JORDAN6:58;
then
A309: LE f.p1,f.p2,P by A1,A2,A6,A300,Th66,RLTOPSP1:68;
A310: LE |[-1,1]|,p2,K by A6,A308,Th50,JORDAN6:58;
then
A311: LE f.p2,f.p3,P by A1,A2,A7,A300,Th66,RLTOPSP1:68;
LE |[-1,1]|,p3,K by A7,A310,Th50,JORDAN6:58;
then LE f.p3,f.p4,P by A1,A2,A8,A300,Th66,RLTOPSP1:68;
hence thesis by A309,A311,JORDAN17:def 1;
end;
end;
hence thesis;
end;
end;
theorem Th73:
for p1,p2 being Point of TOP-REAL 2,
P being non empty compact Subset of TOP-REAL 2 st
P is being_simple_closed_curve & p1 in P & p2 in P & not LE p1,p2,P
holds LE p2,p1,P
proof
let p1,p2 be Point of TOP-REAL 2,
P be non empty compact Subset of TOP-REAL 2;
assume that
A1: P is being_simple_closed_curve and
A2: p1 in P and
A3: p2 in P and
A4: not LE p1,p2,P;
A5: P=Upper_Arc(P) \/ Lower_Arc(P) by A1,JORDAN6:def 9;
A6: not p1=W-min(P) by A1,A3,A4,JORDAN7:3;
per cases by A2,A3,A5,XBOOLE_0:def 3;
suppose
A7: p1 in Upper_Arc(P) & p2 in Upper_Arc(P);
A8: Upper_Arc(P) is_an_arc_of W-min(P),E-max(P) by A1,JORDAN6:def 8;
set q1=W-min(P),q2=E-max(P);
set Q= Upper_Arc(P);
now per cases;
case
A9: p1<>p2;
now per cases by A7,A8,A9,JORDAN5C:14;
case LE p1,p2,Q,q1,q2 & not LE p2,p1,Q,q1,q2;
hence contradiction by A4,A7,JORDAN6:def 10;
end;
case LE p2,p1,Q,q1,q2 & not LE p1,p2,Q,q1,q2;
hence thesis by A7,JORDAN6:def 10;
end;
end;
hence thesis;
end;
case p1=p2;
hence thesis by A1,A2,JORDAN6:56;
end;
end;
hence thesis;
end;
suppose
A10: p1 in Upper_Arc(P) & p2 in Lower_Arc(P);
now per cases;
case p2=W-min(P);
hence thesis by A1,A2,JORDAN7:3;
end;
case p2<>W-min(P);
hence contradiction by A4,A10,JORDAN6:def 10;
end;
end;
hence thesis;
end;
suppose p1 in Lower_Arc(P) & p2 in Upper_Arc(P);
hence thesis by A6,JORDAN6:def 10;
end;
suppose
A11: p1 in Lower_Arc(P) & p2 in Lower_Arc(P);
A12: Lower_Arc(P) is_an_arc_of E-max(P),W-min(P) by A1,JORDAN6:50;
set q2=W-min(P),q1=E-max(P);
set Q= Lower_Arc(P);
now per cases;
case
A13: p1<>p2;
now per cases by A11,A12,A13,JORDAN5C:14;
case
A14: LE p1,p2,Q,q1,q2 & not LE p2,p1,Q,q1,q2;
now per cases;
case p2=W-min(P);
hence thesis by A1,A2,JORDAN7:3;
end;
case p2<>W-min(P);
hence contradiction by A4,A11,A14,JORDAN6:def 10;
end;
end;
hence thesis;
end;
case LE p2,p1,Q,q1,q2 & not LE p1,p2,Q,q1,q2;
hence thesis by A6,A11,JORDAN6:def 10;
end;
end;
hence thesis;
end;
case p1=p2;
hence thesis by A1,A2,JORDAN6:56;
end;
end;
hence thesis;
end;
end;
theorem
for p1,p2,p3 being Point of TOP-REAL 2,
P being non empty compact Subset of TOP-REAL 2 st
P is being_simple_closed_curve & p1 in P & p2 in P & p3 in P holds
LE p1,p2,P & LE p2,p3,P or LE p1,p3,P & LE p3,p2,P or
LE p2,p1,P & LE p1,p3,P or LE p2,p3,P & LE p3,p1,P or
LE p3,p1,P & LE p1,p2,P or LE p3,p2,P & LE p2,p1,P by Th73;
theorem
for p1,p2,p3 being Point of TOP-REAL 2,
P being non empty compact Subset of TOP-REAL 2 st
P is being_simple_closed_curve & p1 in P & p2 in P & p3 in P & LE p2,p3,P
holds
LE p1,p2,P or LE p2,p1,P & LE p1,p3,P or LE p3,p1,P by Th73;
theorem
for p1,p2,p3,p4 being Point of TOP-REAL 2,
P being non empty compact Subset of TOP-REAL 2 st
P is being_simple_closed_curve & p1 in P & p2 in P & p3 in P & p4 in P &
LE p2,p3,P & LE p3,p4,P holds
LE p1,p2,P or LE p2,p1,P & LE p1,p3,P or LE p3,p1,P & LE p1,p4,P or
LE p4,p1,P by Th73;
theorem Th77:
for p1,p2,p3,p4 being Point of TOP-REAL 2,
P being non empty compact Subset of TOP-REAL 2,
f being Function of TOP-REAL 2,TOP-REAL 2 st P= circle(0,0,1) & f=Sq_Circ &
LE f.p1,f.p2,P & LE f.p2,f.p3,P & LE f.p3,f.p4,P
holds p1,p2,p3,p4 are_in_this_order_on rectangle(-1,1,-1,1)
proof
let p1,p2,p3,p4 be Point of TOP-REAL 2,
P be non empty compact Subset of TOP-REAL 2,
f be Function of TOP-REAL 2,TOP-REAL 2;
set K = rectangle(-1,1,-1,1);
assume that
A1: P= circle(0,0,1) and
A2: f=Sq_Circ and
A3: LE f.p1,f.p2,P and
A4: LE f.p2,f.p3,P and
A5: LE f.p3,f.p4,P;
A6: K is being_simple_closed_curve by Th50;
A7: P={p: |.p.|=1} by A1,Th24;
then
A8: LE f.p1,f.p3,P by A3,A4,JGRAPH_3:26,JORDAN6:58;
A9: LE f.p2,f.p4,P by A4,A5,A7,JGRAPH_3:26,JORDAN6:58;
A10: f.:K=P by A2,A7,Lm15,Th35,JGRAPH_3:23;
A11: dom f = the carrier of TOP-REAL 2 by FUNCT_2:def 1;
A12: P={p: |.p.|=1} by A1,Th24;
then
A13: P is being_simple_closed_curve by JGRAPH_3:26;
then f.p1 in P by A3,JORDAN7:5;
then ex x1 being object st ( x1 in dom f)&( x1 in K)&( f.p1=f.x1)
by A10,FUNCT_1:def 6;
then
A14: p1 in K by A2,A11,FUNCT_1:def 4;
f.p2 in P by A3,A13,JORDAN7:5;
then ex x2 being object st ( x2 in dom f)&( x2 in K)&( f.p2=f.x2)
by A10,FUNCT_1:def 6;
then
A15: p2 in K by A2,A11,FUNCT_1:def 4;
f.p3 in P by A4,A13,JORDAN7:5;
then ex x3 being object st ( x3 in dom f)&( x3 in K)&( f.p3=f.x3)
by A10,FUNCT_1:def 6;
then
A16: p3 in K by A2,A11,FUNCT_1:def 4;
f.p4 in P by A5,A13,JORDAN7:5;
then ex x4 being object st ( x4 in dom f)&( x4 in K)&( f.p4=f.x4)
by A10,FUNCT_1:def 6;
then
A17: p4 in K by A2,A11,FUNCT_1:def 4;
now
assume
A18: not p1,p2,p3,p4 are_in_this_order_on K;
A19: now
assume
A20: p1,p2,p4,p3 are_in_this_order_on K;
now per cases by A20,JORDAN17:def 1;
case
A21: LE p1,p2,K & LE p2,p4,K & LE p4,p3,K;
then f.p1,f.p2,f.p4,f.p3 are_in_this_order_on P by A1,A2,Th72;
then LE f.p1,f.p2,P & LE f.p2,f.p4,P & LE f.p4,f.p3,P or
LE f.p2,f.p4,P & LE f.p4,f.p3,P & LE f.p3,f.p1,P or
LE f.p4,f.p3,P & LE f.p3,f.p1,P & LE f.p1,f.p2,P or
LE f.p3,f.p1,P & LE f.p1,f.p2,P & LE f.p2,f.p4,P by JORDAN17:def 1;
then f.p3=f.p4 or f.p3=f.p1 by A5,A8,A12,JGRAPH_3:26,JORDAN6:57;
then
A22: p3=p4 or p3=p1 by A2,A11,FUNCT_1:def 4;
LE p1,p4,K by A21,Th50,JORDAN6:58;
then p1=p4 by A18,A20,A21,A22,Th50,JORDAN6:57;
hence contradiction by A18,A20,A22;
end;
case
A23: LE p2,p4,K & LE p4,p3,K & LE p3,p1,K;
then f.p2,f.p4,f.p3,f.p1 are_in_this_order_on P by A1,A2,Th72;
then LE f.p2,f.p4,P & LE f.p4,f.p3,P & LE f.p3,f.p1,P or
LE f.p4,f.p3,P & LE f.p3,f.p1,P & LE f.p1,f.p2,P or
LE f.p3,f.p1,P & LE f.p1,f.p2,P & LE f.p2,f.p4,P or
LE f.p1,f.p2,P & LE f.p2,f.p4,P & LE f.p4,f.p3,P by JORDAN17:def 1;
then f.p3=f.p4 or LE f.p3,f.p2,P by A5,A13,JORDAN6:57,58;
then f.p3=f.p4 or f.p3=f.p2 by A4,A12,JGRAPH_3:26,JORDAN6:57;
then
A24: p3=p4 or p3=p2 by A2,A11,FUNCT_1:def 4;
then p4=p2 by A18,A20,A23,Th50,JORDAN6:57;
hence contradiction by A18,A20,A24;
end;
case
A25: LE p4,p3,K & LE p3,p1,K & LE p1,p2,K;
then f.p4,f.p3,f.p1,f.p2 are_in_this_order_on P by A1,A2,Th72;
then LE f.p4,f.p3,P & LE f.p3,f.p1,P & LE f.p1,f.p2,P or
LE f.p3,f.p1,P & LE f.p1,f.p2,P & LE f.p2,f.p4,P or
LE f.p1,f.p2,P & LE f.p2,f.p4,P & LE f.p4,f.p3,P or
LE f.p2,f.p4,P & LE f.p4,f.p3,P & LE f.p3,f.p1,P by JORDAN17:def 1;
then f.p3=f.p4 or LE f.p3,f.p2,P by A5,A13,JORDAN6:57,58;
then f.p3=f.p4 or f.p3=f.p2 by A4,A12,JGRAPH_3:26,JORDAN6:57;
then
A26: p3=p4 or p3=p2 by A2,A11,FUNCT_1:def 4;
then p3=p1 by A18,A20,A25,Th50,JORDAN6:57;
hence contradiction by A6,A18,A20,A26,JORDAN17:12;
end;
case
A27: LE p3,p1,K & LE p1,p2,K & LE p2,p4,K;
then f.p3,f.p1,f.p2,f.p4 are_in_this_order_on P by A1,A2,Th72;
then LE f.p4,f.p3,P & LE f.p3,f.p1,P & LE f.p1,f.p2,P or
LE f.p3,f.p1,P & LE f.p1,f.p2,P & LE f.p2,f.p4,P or
LE f.p1,f.p2,P & LE f.p2,f.p4,P & LE f.p4,f.p3,P or
LE f.p2,f.p4,P & LE f.p4,f.p3,P & LE f.p3,f.p1,P by JORDAN17:def 1;
then f.p3=f.p4 or LE f.p3,f.p2,P by A5,A13,JORDAN6:57,58;
then f.p3=f.p4 or f.p3=f.p2 by A4,A12,JGRAPH_3:26,JORDAN6:57;
then
A28: p3=p4 or p3=p2 by A2,A11,FUNCT_1:def 4;
then p3=p1 by A18,A20,A27,Th50,JORDAN6:57;
hence contradiction by A6,A18,A20,A28,JORDAN17:12;
end;
end;
hence contradiction;
end;
A29: now
assume
A30: p1,p3,p4,p2 are_in_this_order_on K;
now per cases by A30,JORDAN17:def 1;
case LE p1,p3,K & LE p3,p4,K & LE p4,p2,K;
then f.p1,f.p3,f.p4,f.p2 are_in_this_order_on P by A1,A2,Th72;
then LE f.p1,f.p3,P & LE f.p3,f.p4,P & LE f.p4,f.p2,P or
LE f.p3,f.p4,P & LE f.p4,f.p2,P & LE f.p2,f.p1,P or
LE f.p4,f.p2,P & LE f.p2,f.p1,P & LE f.p1,f.p3,P or
LE f.p2,f.p1,P & LE f.p1,f.p3,P & LE f.p3,f.p4,P by JORDAN17:def 1;
then f.p4=f.p2 or f.p2=f.p1 by A3,A9,A12,JGRAPH_3:26,JORDAN6:57;
then
A31: p4=p2 or p2=p1 by A2,A11,FUNCT_1:def 4;
then f.p3=f.p2 or f.p4=f.p1 by A4,A5,A6,A12,A18,A30,JGRAPH_3:26
,JORDAN17:12,JORDAN6:57;
then p3=p2 or p4=p1 by A2,A11,FUNCT_1:def 4;
hence contradiction by A6,A18,A30,A31,JORDAN17:12;
end;
case LE p3,p4,K & LE p4,p2,K & LE p2,p1,K;
then f.p3,f.p4,f.p2,f.p1 are_in_this_order_on P by A1,A2,Th72;
then
A32: LE f.p1,f.p3,P & LE f.p3,f.p4,P & LE f.p4,f.p2,P or
LE f.p3,f.p4,P & LE f.p4,f.p2,P & LE f.p2,f.p1,P or
LE f.p4,f.p2,P & LE f.p2,f.p1,P & LE f.p1,f.p3,P or
LE f.p2,f.p1,P & LE f.p1,f.p3,P & LE f.p3,f.p4,P by JORDAN17:def 1;
then f.p4=f.p2 or f.p2=f.p1 by A3,A9,A12,JGRAPH_3:26,JORDAN6:57;
then
A33: p4=p2 or p2=p1 by A2,A11,FUNCT_1:def 4;
f.p2=f.p1 or LE f.p3,f.p2,P by A3,A13,A32,JORDAN6:57,58;
then f.p2=f.p1 or f.p3=f.p2 by A4,A12,JGRAPH_3:26,JORDAN6:57;
then p2=p1 or p3=p2 by A2,A11,FUNCT_1:def 4;
hence contradiction by A6,A18,A30,A33,JORDAN17:12;
end;
case LE p4,p2,K & LE p2,p1,K & LE p1,p3,K;
then f.p4,f.p2,f.p1,f.p3 are_in_this_order_on P by A1,A2,Th72;
then
A34: LE f.p1,f.p3,P & LE f.p3,f.p4,P & LE f.p4,f.p2,P or
LE f.p3,f.p4,P & LE f.p4,f.p2,P & LE f.p2,f.p1,P or
LE f.p4,f.p2,P & LE f.p2,f.p1,P & LE f.p1,f.p3,P or
LE f.p2,f.p1,P & LE f.p1,f.p3,P & LE f.p3,f.p4,P by JORDAN17:def 1;
then f.p4=f.p2 or f.p2=f.p1 by A3,A9,A12,JGRAPH_3:26,JORDAN6:57;
then
A35: p4=p2 or p2=p1 by A2,A11,FUNCT_1:def 4;
f.p2=f.p1 or LE f.p3,f.p2,P by A3,A13,A34,JORDAN6:57,58;
then f.p2=f.p1 or f.p3=f.p2 by A4,A12,JGRAPH_3:26,JORDAN6:57;
then p2=p1 or p3=p2 by A2,A11,FUNCT_1:def 4;
hence contradiction by A6,A18,A30,A35,JORDAN17:12;
end;
case
A36: LE p2,p1,K & LE p1,p3,K & LE p3,p4,K;
then f.p2,f.p1,f.p3,f.p4 are_in_this_order_on P by A1,A2,Th72;
then LE f.p1,f.p3,P & LE f.p3,f.p4,P & LE f.p4,f.p2,P or
LE f.p3,f.p4,P & LE f.p4,f.p2,P & LE f.p2,f.p1,P or
LE f.p4,f.p2,P & LE f.p2,f.p1,P & LE f.p1,f.p3,P or
LE f.p2,f.p1,P & LE f.p1,f.p3,P & LE f.p3,f.p4,P by JORDAN17:def 1;
then f.p4=f.p2 or f.p2=f.p1 by A3,A9,A12,JGRAPH_3:26,JORDAN6:57;
then
A37: p4=p2 or p2=p1 by A2,A11,FUNCT_1:def 4;
LE p2,p3,K by A36,Th50,JORDAN6:58;
then p2=p3 by A6,A18,A30,A36,A37,JORDAN17:12,JORDAN6:57;
hence contradiction by A6,A18,A30,A37,JORDAN17:12;
end;
end;
hence contradiction;
end;
now per cases by A6,A14,A15,A16,A17,A18,JORDAN17:27;
case p1,p2,p4,p3 are_in_this_order_on K;
hence contradiction by A19;
end;
case
A38: p1,p3,p2,p4 are_in_this_order_on K;
now per cases by A38,JORDAN17:def 1;
case
A39: LE p1,p3,K & LE p3,p2,K & LE p2,p4,K;
then f.p1,f.p3,f.p2,f.p4 are_in_this_order_on P by A1,A2,Th72;
then LE f.p1,f.p3,P & LE f.p3,f.p2,P & LE f.p2,f.p4,P or
LE f.p3,f.p2,P & LE f.p2,f.p4,P & LE f.p4,f.p1,P or
LE f.p2,f.p4,P & LE f.p4,f.p1,P & LE f.p1,f.p3,P or
LE f.p4,f.p1,P & LE f.p1,f.p3,P & LE f.p3,f.p2,P
by JORDAN17:def 1;
then f.p3=f.p2 or LE f.p2,f.p1,P by A4,A13,JORDAN6:57,58;
then f.p3=f.p2 or f.p2=f.p1 by A3,A12,JGRAPH_3:26,JORDAN6:57;
then
A40: p3=p2 or p2=p1 by A2,A11,FUNCT_1:def 4;
then p3=p1 by A18,A38,A39,Th50,JORDAN6:57;
hence contradiction by A18,A38,A40;
end;
case
A41: LE p3,p2,K & LE p2,p4,K & LE p4,p1,K;
then f.p3,f.p2,f.p4,f.p1 are_in_this_order_on P by A1,A2,Th72;
then LE f.p1,f.p3,P & LE f.p3,f.p2,P & LE f.p2,f.p4,P or
LE f.p3,f.p2,P & LE f.p2,f.p4,P & LE f.p4,f.p1,P or
LE f.p2,f.p4,P & LE f.p4,f.p1,P & LE f.p1,f.p3,P or
LE f.p4,f.p1,P & LE f.p1,f.p3,P & LE f.p3,f.p2,P
by JORDAN17:def 1;
then f.p3=f.p2 or LE f.p2,f.p1,P by A4,A13,JORDAN6:57,58;
then f.p3=f.p2 or f.p2=f.p1 by A3,A12,JGRAPH_3:26,JORDAN6:57;
then
A42: p3=p2 or p2=p1 by A2,A11,FUNCT_1:def 4;
then p4=p1 by A18,A38,A41,Th50,JORDAN6:57;
hence contradiction by A6,A18,A38,A42,JORDAN17:12;
end;
case
A43: LE p2,p4,K & LE p4,p1,K & LE p1,p3,K;
then f.p2,f.p4,f.p1,f.p3 are_in_this_order_on P by A1,A2,Th72;
then LE f.p1,f.p3,P & LE f.p3,f.p2,P & LE f.p2,f.p4,P or
LE f.p3,f.p2,P & LE f.p2,f.p4,P & LE f.p4,f.p1,P or
LE f.p2,f.p4,P & LE f.p4,f.p1,P & LE f.p1,f.p3,P or
LE f.p4,f.p1,P & LE f.p1,f.p3,P & LE f.p3,f.p2,P
by JORDAN17:def 1;
then f.p3=f.p2 or LE f.p2,f.p1,P by A4,A13,JORDAN6:57,58;
then f.p3=f.p2 or f.p2=f.p1 by A3,A12,JGRAPH_3:26,JORDAN6:57;
then
A44: p3=p2 or p2=p1 by A2,A11,FUNCT_1:def 4;
then p4=p1 by A18,A38,A43,Th50,JORDAN6:57;
hence contradiction by A6,A18,A38,A44,JORDAN17:12;
end;
case
A45: LE p4,p1,K & LE p1,p3,K & LE p3,p2,K;
then f.p4,f.p1,f.p3,f.p2 are_in_this_order_on P by A1,A2,Th72;
then LE f.p1,f.p3,P & LE f.p3,f.p2,P & LE f.p2,f.p4,P or
LE f.p3,f.p2,P & LE f.p2,f.p4,P & LE f.p4,f.p1,P or
LE f.p2,f.p4,P & LE f.p4,f.p1,P & LE f.p1,f.p3,P or
LE f.p4,f.p1,P & LE f.p1,f.p3,P & LE f.p3,f.p2,P
by JORDAN17:def 1;
then f.p3=f.p2 or LE f.p2,f.p1,P by A4,A13,JORDAN6:57,58;
then f.p3=f.p2 or f.p2=f.p1 by A3,A12,JGRAPH_3:26,JORDAN6:57;
then
A46: p3=p2 or p2=p1 by A2,A11,FUNCT_1:def 4;
then p3=p1 by A18,A38,A45,Th50,JORDAN6:57;
hence contradiction by A18,A38,A46;
end;
end;
hence contradiction;
end;
case p1,p3,p4,p2 are_in_this_order_on K;
hence contradiction by A29;
end;
case
A47: p1,p4,p2,p3 are_in_this_order_on K;
now per cases by A47,JORDAN17:def 1;
case
A48: LE p1,p4,K & LE p4,p2,K & LE p2,p3,K;
then f.p1,f.p4,f.p2,f.p3 are_in_this_order_on P by A1,A2,Th72;
then
A49: LE f.p1,f.p4,P & LE f.p4,f.p2,P & LE f.p2,f.p3,P or
LE f.p4,f.p2,P & LE f.p2,f.p3,P & LE f.p3,f.p1,P or
LE f.p2,f.p3,P & LE f.p3,f.p1,P & LE f.p1,f.p4,P or
LE f.p3,f.p1,P & LE f.p1,f.p4,P & LE f.p4,f.p2,P
by JORDAN17:def 1;
then f.p4=f.p2 or LE f.p2,f.p1,P by A9,A13,JORDAN6:57,58;
then f.p4=f.p2 or f.p2=f.p1 by A3,A12,JGRAPH_3:26,JORDAN6:57;
then
A50: p4=p2 or p2=p1 by A2,A11,FUNCT_1:def 4;
then
A51: p4=p2 by A48,Th50,JORDAN6:57;
f.p3=f.p1 or LE f.p4,f.p3,P by A8,A13,A49,JORDAN6:57,58;
then f.p3=f.p1 or f.p4=f.p3 by A5,A12,JGRAPH_3:26,JORDAN6:57;
then
A52: p3=p1 or p4=p3 by A2,A11,FUNCT_1:def 4;
then p1=p2 by A18,A47,A48,A50,Th50,JORDAN6:57;
hence contradiction by A18,A47,A51,A52;
end;
case
A53: LE p4,p2,K & LE p2,p3,K & LE p3,p1,K;
then f.p4,f.p2,f.p3,f.p1 are_in_this_order_on P by A1,A2,Th72;
then
A54: LE f.p1,f.p4,P & LE f.p4,f.p2,P & LE f.p2,f.p3,P or
LE f.p4,f.p2,P & LE f.p2,f.p3,P & LE f.p3,f.p1,P or
LE f.p2,f.p3,P & LE f.p3,f.p1,P & LE f.p1,f.p4,P or
LE f.p3,f.p1,P & LE f.p1,f.p4,P & LE f.p4,f.p2,P
by JORDAN17:def 1;
then f.p4=f.p2 or LE f.p2,f.p1,P by A9,A13,JORDAN6:57,58;
then f.p4=f.p2 or f.p2=f.p1 by A3,A12,JGRAPH_3:26,JORDAN6:57;
then p4=p2 or p2=p1 by A2,A11,FUNCT_1:def 4;
then
A55: p4=p2 or p2=p1 & p3=p1 by A53,Th50,JORDAN6:57;
f.p3=f.p1 or LE f.p4,f.p3,P by A8,A13,A54,JORDAN6:57,58;
then f.p3=f.p1 or f.p4=f.p3 by A5,A12,JGRAPH_3:26,JORDAN6:57;
then p3=p1 or p4=p3 by A2,A11,FUNCT_1:def 4;
hence contradiction by A6,A29,A47,A55,JORDAN17:12;
end;
case
A56: LE p2,p3,K & LE p3,p1,K & LE p1,p4,K;
then f.p2,f.p3,f.p1,f.p4 are_in_this_order_on P by A1,A2,Th72;
then
A57: LE f.p1,f.p4,P & LE f.p4,f.p2,P & LE f.p2,f.p3,P or
LE f.p4,f.p2,P & LE f.p2,f.p3,P & LE f.p3,f.p1,P or
LE f.p2,f.p3,P & LE f.p3,f.p1,P & LE f.p1,f.p4,P or
LE f.p3,f.p1,P & LE f.p1,f.p4,P & LE f.p4,f.p2,P
by JORDAN17:def 1;
then f.p4=f.p2 or LE f.p2,f.p1,P by A9,A13,JORDAN6:57,58;
then f.p4=f.p2 or f.p2=f.p1 by A3,A12,JGRAPH_3:26,JORDAN6:57;
then p4=p2 or p2=p1 by A2,A11,FUNCT_1:def 4;
then
A58: p4=p2 or p2=p1 & p3=p1 by A56,Th50,JORDAN6:57;
f.p3=f.p1 or LE f.p4,f.p3,P by A8,A13,A57,JORDAN6:57,58;
then f.p3=f.p1 or f.p4=f.p3 by A5,A12,JGRAPH_3:26,JORDAN6:57;
then p3=p1 or p4=p3 by A2,A11,FUNCT_1:def 4;
hence contradiction by A6,A29,A47,A58,JORDAN17:12;
end;
case
A59: LE p3,p1,K & LE p1,p4,K & LE p4,p2,K;
then f.p3,f.p1,f.p4,f.p2 are_in_this_order_on P by A1,A2,Th72;
then
A60: LE f.p1,f.p4,P & LE f.p4,f.p2,P & LE f.p2,f.p3,P or
LE f.p4,f.p2,P & LE f.p2,f.p3,P & LE f.p3,f.p1,P or
LE f.p2,f.p3,P & LE f.p3,f.p1,P & LE f.p1,f.p4,P or
LE f.p3,f.p1,P & LE f.p1,f.p4,P & LE f.p4,f.p2,P
by JORDAN17:def 1;
then f.p4=f.p2 or LE f.p2,f.p1,P by A9,A13,JORDAN6:57,58;
then f.p4=f.p2 or f.p2=f.p1 by A3,A12,JGRAPH_3:26,JORDAN6:57;
then p4=p2 or p2=p1 by A2,A11,FUNCT_1:def 4;
then
A61: p4=p2 by A59,Th50,JORDAN6:57;
f.p3=f.p1 or LE f.p4,f.p3,P by A8,A13,A60,JORDAN6:57,58;
then f.p3=f.p1 or f.p4=f.p3 by A5,A12,JGRAPH_3:26,JORDAN6:57;
then p3=p1 or p4=p3 by A2,A11,FUNCT_1:def 4;
hence contradiction by A6,A29,A47,A61,JORDAN17:12;
end;
end;
hence contradiction;
end;
case
A62: p1,p4,p3,p2 are_in_this_order_on K;
now per cases by A62,JORDAN17:def 1;
case
A63: LE p1,p4,K & LE p4,p3,K & LE p3,p2,K;
then f.p1,f.p4,f.p3,f.p2 are_in_this_order_on P by A1,A2,Th72;
then
A64: LE f.p1,f.p4,P & LE f.p4,f.p3,P & LE f.p3,f.p2,P or
LE f.p4,f.p3,P & LE f.p3,f.p2,P & LE f.p2,f.p1,P or
LE f.p3,f.p2,P & LE f.p2,f.p1,P & LE f.p1,f.p4,P or
LE f.p2,f.p1,P & LE f.p1,f.p4,P & LE f.p4,f.p3,P
by JORDAN17:def 1;
then f.p3=f.p2 or f.p2=f.p1 by A3,A4,A12,JGRAPH_3:26,JORDAN6:57;
then
A65: p3=p2 or p2=p1 by A2,A11,FUNCT_1:def 4;
LE p1,p3,K by A63,Th50,JORDAN6:58;
then
A66: p3=p2 by A63,A65,Th50,JORDAN6:57;
f.p4=f.p3 or f.p2=f.p1 by A3,A5,A12,A64,JGRAPH_3:26,JORDAN6:57;
then p4=p3 or p2=p1 by A2,A11,FUNCT_1:def 4;
then p4=p3 or p2,p3,p4,p1 are_in_this_order_on K
by A6,A62,A66,JORDAN17:12;
hence contradiction by A6,A18,A62,A65,JORDAN17:12;
end;
case LE p4,p3,K & LE p3,p2,K & LE p2,p1,K;
then f.p4,f.p3,f.p2,f.p1 are_in_this_order_on P by A1,A2,Th72;
then
A67: LE f.p1,f.p4,P & LE f.p4,f.p3,P & LE f.p3,f.p2,P or
LE f.p4,f.p3,P & LE f.p3,f.p2,P & LE f.p2,f.p1,P or
LE f.p3,f.p2,P & LE f.p2,f.p1,P & LE f.p1,f.p4,P or
LE f.p2,f.p1,P & LE f.p1,f.p4,P & LE f.p4,f.p3,P
by JORDAN17:def 1;
then f.p3=f.p2 or f.p2=f.p1 by A3,A4,A12,JGRAPH_3:26,JORDAN6:57;
then
A68: p3=p2 or p2=p1 by A2,A11,FUNCT_1:def 4;
f.p4=f.p3 or f.p2=f.p1 by A3,A5,A12,A67,JGRAPH_3:26,JORDAN6:57;
then p4=p3 or p2=p1 by A2,A11,FUNCT_1:def 4;
hence contradiction by A6,A19,A62,A68,JORDAN17:12;
end;
case LE p3,p2,K & LE p2,p1,K & LE p1,p4,K;
then f.p3,f.p2,f.p1,f.p4 are_in_this_order_on P by A1,A2,Th72;
then LE f.p1,f.p4,P & LE f.p4,f.p3,P & LE f.p3,f.p2,P or
LE f.p4,f.p3,P & LE f.p3,f.p2,P & LE f.p2,f.p1,P or
LE f.p3,f.p2,P & LE f.p2,f.p1,P & LE f.p1,f.p4,P or
LE f.p2,f.p1,P & LE f.p1,f.p4,P & LE f.p4,f.p3,P
by JORDAN17:def 1;
then f.p4=f.p3 or f.p2=f.p1 by A3,A5,A12,JGRAPH_3:26,JORDAN6:57;
then p4=p3 or p2=p1 by A2,A11,FUNCT_1:def 4;
hence contradiction by A6,A19,A29,A62,JORDAN17:12;
end;
case
A69: LE p2,p1,K & LE p1,p4,K & LE p4,p3,K;
then f.p2,f.p1,f.p4,f.p3 are_in_this_order_on P by A1,A2,Th72;
then
A70: LE f.p1,f.p4,P & LE f.p4,f.p3,P & LE f.p3,f.p2,P or
LE f.p4,f.p3,P & LE f.p3,f.p2,P & LE f.p2,f.p1,P or
LE f.p3,f.p2,P & LE f.p2,f.p1,P & LE f.p1,f.p4,P or
LE f.p2,f.p1,P & LE f.p1,f.p4,P & LE f.p4,f.p3,P
by JORDAN17:def 1;
then f.p3=f.p2 or f.p2=f.p1 by A3,A4,A12,JGRAPH_3:26,JORDAN6:57;
then
A71: p3=p2 or p2=p1 by A2,A11,FUNCT_1:def 4;
LE p1,p3,K by A69,Th50,JORDAN6:58;
then
A72: p1=p2 by A69,A71,Th50,JORDAN6:57;
f.p4=f.p3 or f.p2=f.p3 by A4,A5,A12,A70,JGRAPH_3:26,JORDAN6:57;
then p4=p3 or p2=p3 by A2,A11,FUNCT_1:def 4;
hence contradiction by A6,A29,A62,A72,JORDAN17:12;
end;
end;
hence contradiction;
end;
end;
hence contradiction;
end;
hence thesis;
end;
theorem Th78:
for p1,p2,p3,p4 being Point of TOP-REAL 2,
P being non empty compact Subset of TOP-REAL 2,
f being Function of TOP-REAL 2,TOP-REAL 2 st P= circle(0,0,1) & f=Sq_Circ
holds p1,p2,p3,p4 are_in_this_order_on rectangle(-1,1,-1,1)
iff f.p1,f.p2,f.p3,f.p4 are_in_this_order_on P
proof
let p1,p2,p3,p4 be Point of TOP-REAL 2,
P be non empty compact Subset of TOP-REAL 2,
f be Function of TOP-REAL 2,TOP-REAL 2;
set K = rectangle(-1,1,-1,1);
assume that
A1: P= circle(0,0,1) and
A2: f=Sq_Circ;
A3: K is being_simple_closed_curve by Th50;
circle(0,0,1)={p5 where p5 is Point of TOP-REAL 2: |.p5.|=1} by Th24;
then
A4: P is being_simple_closed_curve by A1,JGRAPH_3:26;
thus p1,p2,p3,p4 are_in_this_order_on K implies
f.p1,f.p2,f.p3,f.p4 are_in_this_order_on P
proof
assume
A5: p1,p2,p3,p4 are_in_this_order_on K;
now per cases by A5,JORDAN17:def 1;
case LE p1,p2,K & LE p2,p3,K & LE p3,p4,K;
hence thesis by A1,A2,Th72;
end;
case LE p2,p3,K & LE p3,p4,K & LE p4,p1,K;
hence thesis by A1,A2,A4,Th72,JORDAN17:12;
end;
case LE p3,p4,K & LE p4,p1,K & LE p1,p2,K;
hence thesis by A1,A2,A4,Th72,JORDAN17:11;
end;
case LE p4,p1,K & LE p1,p2,K & LE p2,p3,K;
hence thesis by A1,A2,A4,Th72,JORDAN17:10;
end;
end;
hence thesis;
end;
thus f.p1,f.p2,f.p3,f.p4 are_in_this_order_on P implies
p1,p2,p3,p4 are_in_this_order_on K
proof
assume
A6: f.p1,f.p2,f.p3,f.p4 are_in_this_order_on P;
now per cases by A6,JORDAN17:def 1;
case LE f.p1,f.p2,P & LE f.p2,f.p3,P & LE f.p3,f.p4,P;
hence thesis by A1,A2,Th77;
end;
case LE f.p2,f.p3,P & LE f.p3,f.p4,P & LE f.p4,f.p1,P;
then p2,p3,p4,p1 are_in_this_order_on K by A1,A2,Th77;
hence thesis by A3,JORDAN17:12;
end;
case LE f.p3,f.p4,P & LE f.p4,f.p1,P & LE f.p1,f.p2,P;
then p3,p4,p1,p2 are_in_this_order_on K by A1,A2,Th77;
hence thesis by A3,JORDAN17:11;
end;
case LE f.p4,f.p1,P & LE f.p1,f.p2,P & LE f.p2,f.p3,P;
then p4,p1,p2,p3 are_in_this_order_on K by A1,A2,Th77;
hence thesis by A3,JORDAN17:10;
end;
end;
hence thesis;
end;
end;
theorem
for p1,p2,p3,p4 being Point of TOP-REAL 2
st p1,p2,p3,p4 are_in_this_order_on rectangle(-1,1,-1,1)
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=p2 & g.1=p4 &
rng f c= closed_inside_of_rectangle(-1,1,-1,1) &
rng g c= closed_inside_of_rectangle(-1,1,-1,1)
holds rng f meets rng g
proof
let p1,p2,p3,p4 be Point of TOP-REAL 2;
set K=rectangle(-1,1,-1,1), K0 = closed_inside_of_rectangle(-1,1,-1,1);
assume
A1: p1,p2,p3,p4 are_in_this_order_on K;
reconsider j = 1 as non negative Real;
reconsider P= circle(0,0,j) as non empty compact Subset of TOP-REAL 2;
A2: P={p6 where p6 is Point of TOP-REAL 2: |.p6.|=1} by Th24;
thus
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=p2 & g.1=p4 &
rng f c= K0 & rng g c= K0 holds rng f meets rng g
proof
let f,g be Function of I[01],TOP-REAL 2;
assume that
A3: f is continuous one-to-one and
A4: g is continuous one-to-one and
A5: f.0=p1 and
A6: f.1=p3 and
A7: g.0=p2 and
A8: g.1=p4 and
A9: rng f c= K0 and
A10: rng g c= K0;
reconsider s=Sq_Circ as Function of TOP-REAL 2,TOP-REAL 2;
A11: dom s=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
reconsider f1=s*f as Function of I[01],TOP-REAL 2;
reconsider g1=s*g as Function of I[01],TOP-REAL 2;
s is being_homeomorphism by JGRAPH_3:43;
then
A12: s is continuous by TOPS_2:def 5;
A13: dom f=[.0,1.] by BORSUK_1:40,FUNCT_2:def 1;
then 0 in dom f by XXREAL_1:1;
then
A14: f1.0=Sq_Circ.p1 by A5,FUNCT_1:13;
1 in dom f by A13,XXREAL_1:1;
then
A15: f1.1=Sq_Circ.p3 by A6,FUNCT_1:13;
A16: dom g=[.0,1.] by BORSUK_1:40,FUNCT_2:def 1;
then 0 in dom g by XXREAL_1:1;
then
A17: g1.0=Sq_Circ.p2 by A7,FUNCT_1:13;
1 in dom g by A16,XXREAL_1:1;
then
A18: g1.1=Sq_Circ.p4 by A8,FUNCT_1:13;
defpred P[Point of TOP-REAL 2] means |.$1.|<=1;
{p8 where p8 is Point of TOP-REAL 2: P[p8]} is Subset of TOP-REAL 2
from DOMAIN_1:sch 7;
then reconsider C0={p8 where p8 is Point of TOP-REAL 2: |.p8.|<=1} as
Subset of TOP-REAL 2;
A19: s.:K0 = C0 by Th27;
A20: rng f1 c= C0
proof
let y be object;
assume y in rng f1;
then consider x being object such that
A21: x in dom f1 and
A22: y=f1.x by FUNCT_1:def 3;
A23: x in dom f by A21,FUNCT_1:11;
A24: f.x in dom s by A21,FUNCT_1:11;
f.x in rng f by A23,FUNCT_1:3;
then s.(f.x) in s.:K0 by A9,A24,FUNCT_1:def 6;
hence thesis by A19,A21,A22,FUNCT_1:12;
end;
A25: rng g1 c= C0
proof
let y be object;
assume y in rng g1;
then consider x being object such that
A26: x in dom g1 and
A27: y=g1.x by FUNCT_1:def 3;
A28: x in dom g by A26,FUNCT_1:11;
A29: g.x in dom s by A26,FUNCT_1:11;
g.x in rng g by A28,FUNCT_1:3;
then s.(g.x) in s.:K0 by A10,A29,FUNCT_1:def 6;
hence thesis by A19,A26,A27,FUNCT_1:12;
end;
reconsider q1=s.p1,q2=s.p2,q3=s.p3,q4=s.p4 as Point of TOP-REAL 2;
q1,q2,q3,q4 are_in_this_order_on P by A1,Th78;
then rng f1 meets rng g1 by A2,A3,A4,A12,A14,A15,A17,A18,A20,A25,Th18;
then consider y being object such that
A30: y in rng f1 and
A31: y in rng g1 by XBOOLE_0:3;
consider x1 being object such that
A32: x1 in dom f1 and
A33: y=f1.x1 by A30,FUNCT_1:def 3;
consider x2 being object such that
A34: x2 in dom g1 and
A35: y=g1.x2 by A31,FUNCT_1:def 3;
dom f1 c= dom f by RELAT_1:25;
then
A36: f.x1 in rng f by A32,FUNCT_1:3;
dom g1 c= dom g by RELAT_1:25;
then
A37: g.x2 in rng g by A34,FUNCT_1:3;
y=(Sq_Circ).(f.x1) by A32,A33,FUNCT_1:12;
then
A38: Sq_Circ".y=f.x1 by A11,A36,FUNCT_1:32;
x1 in dom f by A32,FUNCT_1:11;
then
A39: f.x1 in rng f by FUNCT_1:def 3;
y=(Sq_Circ).(g.x2) by A34,A35,FUNCT_1:12;
then
A40: Sq_Circ".y=g.x2 by A11,A37,FUNCT_1:32;
x2 in dom g by A34,FUNCT_1:11;
then g.x2 in rng g by FUNCT_1:def 3;
hence thesis by A38,A39,A40,XBOOLE_0:3;
end;
end;