:: Jordan Curve Theorem
:: by Artur Korni{\l}owicz
::
:: Received September 15, 2005
:: Copyright (c) 2005-2026 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, SUBSET_1, PRE_TOPC, EUCLID, TOPREAL2, RELAT_1, RCOMP_1,
JORDAN21, TARSKI, XBOOLE_0, RELAT_2, METRIC_1, XXREAL_0, CARD_1, ARYTM_3,
SUPINF_2, ARYTM_1, MCART_1, SPPOL_1, RLTOPSP1, PSCOMP_1, JORDAN6,
JORDAN2C, CONVEX1, TOPREAL1, STRUCT_0, TOPMETR, VALUED_1, TREAL_1,
ZFMISC_1, REAL_1, FUNCOP_1, ORDINAL2, COMPLEX1, BROUWER, TOPREALB,
XCMPLX_0, TOPS_1, PCOMPS_1, XXREAL_2, FUNCT_1, TOPS_2, FUNCT_2, GRAPH_1,
BORSUK_2, BORSUK_1, XXREAL_1, JGRAPH_6, JORDAN5C, JORDAN3, SQUARE_1,
PARTFUN3, PARTFUN1, TOPREALA, SETFAM_1, PROB_1, ABIAN, CONNSP_1,
CONNSP_2, TOPREAL4, FUNCT_4, TOPGRP_1, JORDAN24, CONNSP_3, JORDAN1,
JORDAN, MEASURE5, SEQ_4, NAT_1, FUNCT_7;
notations TARSKI, XBOOLE_0, XTUPLE_0, SUBSET_1, ZFMISC_1, ORDINAL1, SQUARE_1,
MCART_1, RELAT_1, VALUED_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2,
FUNCT_3, FUNCT_4, FUNCOP_1, NUMBERS, XCMPLX_0, XXREAL_0, XXREAL_2,
XREAL_0, COMPLEX1, REAL_1, RCOMP_1, DOMAIN_1, STRUCT_0, PRE_TOPC, TOPS_1,
BORSUK_1, TOPS_2, COMPTS_1, TREAL_1, CONNSP_1, CONNSP_2, CONNSP_3,
METRIC_1, TBSP_1, RLVECT_1, PCOMPS_1, RLTOPSP1, EUCLID, BORSUK_2,
TOPREAL1, TOPREAL2, TOPMETR, SPPOL_1, JORDAN1, JORDAN2C, JORDAN5C,
TOPREAL4, JORDAN6, JGRAPH_6, TOPREAL6, TOPREAL9, BORSUK_6, TOPREALA,
JORDAN21, TOPGRP_1, JORDAN24, ABIAN, PARTFUN3, TOPREALB, BROUWER,
PSCOMP_1;
constructors FUNCT_4, SQUARE_1, COMPLEX1, ABIAN, TOPS_1, CONNSP_1, COMPTS_1,
TBSP_1, TSEP_1, TOPREAL1, TREAL_1, TOPREAL4, SPPOL_1, JORDAN1, SPPOL_2,
CONNSP_3, WAYBEL_3, JORDAN5C, JORDAN6, TOPGRP_1, JORDAN2C, TOPREAL6,
JORDAN21, JGRAPH_6, TOPREAL9, TOPREALA, PARTFUN3, BROUWER, JORDAN24,
BORSUK_6, SEQ_4, FUNCSDOM, CONVEX1, BINOP_2, PSCOMP_1, XTUPLE_0, REAL_1;
registrations XBOOLE_0, SUBSET_1, FUNCT_1, RELSET_1, FUNCT_2, NUMBERS,
XXREAL_0, XREAL_0, SQUARE_1, NAT_1, MEMBERED, STRUCT_0, PRE_TOPC, TOPS_1,
COMPTS_1, METRIC_1, BORSUK_1, TEX_2, EUCLID, TOPMETR, TOPREAL1, TOPREAL2,
SPPOL_1, JORDAN1, SPPOL_2, PSCOMP_1, BORSUK_2, WAYBEL_2, WAYBEL_3,
JORDAN5A, JORDAN6, TOPGRP_1, YELLOW13, JORDAN2C, TOPREAL6, JORDAN21,
TOPREAL9, TOPREALA, TOPREALB, RCOMP_3, PARTFUN3, TOPALG_5, BROUWER,
JGRAPH_8, VALUED_0, XXREAL_2, RELAT_1, MEASURE6, PARTFUN4, VALUED_1,
XTUPLE_0, ORDINAL1, CARD_1;
requirements NUMERALS, BOOLE, SUBSET, ARITHM, REAL;
definitions TARSKI, XBOOLE_0, BORSUK_2, JORDAN1, SPPOL_1, JORDAN2C, JORDAN21,
JORDAN24, TOPREAL2, BORSUK_1, PSCOMP_1, ABIAN, XXREAL_0;
equalities JORDAN2C, JORDAN21, PSCOMP_1, SQUARE_1, JGRAPH_6, TOPS_1, SUBSET_1,
TOPREALA, STRUCT_0, RELAT_1, ALGSTR_0, RLTOPSP1;
expansions TARSKI, XBOOLE_0, BORSUK_2, JORDAN1, SPPOL_1, JORDAN2C, JORDAN24,
CONNSP_1, ABIAN;
theorems JORDAN6, PRE_TOPC, XBOOLE_0, EUCLID, TARSKI, SQUARE_1, CONNSP_1,
JORDAN2C, JGRAPH_6, XREAL_1, JORDAN1K, TOPREAL6, BORSUK_2, JORDAN21,
TOPREAL1, TOPREAL3, SPPOL_1, SUBSET_1, TOPMETR, TOPREAL9, TOPS_1,
SPRECT_3, TOPRNS_1, TOPS_2, NAT_1, JORDAN1, TOPREALA, BORSUK_1, FUNCT_2,
RELAT_1, GOBOARD7, FUNCT_1, BORSUK_6, XBOOLE_1, ZFMISC_1, CONNSP_3,
TOPALG_2, GOBOARD9, SPPOL_2, PSCOMP_1, TOPREAL4, TSEP_1, TOPALG_3,
JGRAPH_8, COMPTS_1, SEQ_4, FUNCOP_1, WEIERSTR, JORDAN5C, JORDAN16,
XREAL_0, XCMPLX_1, SPRECT_1, JORDAN24, TOPGRP_1, TOPREAL2, RCOMP_1,
BORSUK_4, BROUWER, ENUMSET1, METRIC_1, PCOMPS_1, TOPREALB, ABSVALUE,
MCART_1, BORSUK_5, METRIC_6, COMPLEX1, JGRAPH_2, SEQ_2, TIETZE, CONNSP_2,
GRCAT_1, FUNCT_4, TREAL_1, JGRAPH_1, PARTFUN3, RFUNCT_1, YELLOW12,
XCMPLX_0, FUNCT_3, JORDAN5B, TOPALG_5, GOBOARD6, XXREAL_0, VALUED_1,
XXREAL_1, RLTOPSP1, ORDINAL1, JORDAN5A, RLVECT_1;
schemes FUNCT_2;
begin :: Preliminaries
:: I would like to thank Professor Yatsuka Nakamura
:: for including me to the team working on the formalization
:: of the Jordan Curve Theorem. Especially, I am very grateful
:: to Professor Nakamura for inviting me to Shinshu University, Nagano
:: to work on the project together.
:: I am also thankful to Professor Andrzej Trybulec for his
:: continual help and fruitful discussions during the formalization.
reserve a, b, c, d, r, s for Real,
n for Element of NAT,
p, p1, p2 for Point of TOP-REAL 2,
x, y for Point of TOP-REAL n,
C for Simple_closed_curve,
A, B, P for Subset of TOP-REAL 2,
U, V for Subset of (TOP-REAL 2)|C`,
D for compact with_the_max_arc Subset of TOP-REAL 2;
set T2 = TOP-REAL 2;
Lm1: for A, B, C, Z being set st A c= Z & B c= Z & C c= Z holds
A \/ B \/ C c= Z
proof
let A, B, C, Z be set;
assume that
A1: A c= Z and
A2: B c= Z;
A \/ B c= Z by A1,A2,XBOOLE_1:8;
hence thesis by XBOOLE_1:8;
end;
Lm2: for A, B, C, D, Z being set st A c= Z & B c= Z & C c= Z & D c= Z holds
A \/ B \/ C \/ D c= Z
proof
let A, B, C, D, Z be set;
assume that
A1: A c= Z and
A2: B c= Z and
A3: C c= Z;
A \/ B \/ C c= Z by A1,A2,A3,Lm1;
hence thesis by XBOOLE_1:8;
end;
Lm3: for A, B, C, D, Z being set st A misses Z & B misses Z &
C misses Z & D misses Z holds A \/ B \/ C \/ D misses Z
proof
let A, B, C, D, Z be set;
assume that
A1: A misses Z and
A2: B misses Z and
A3: C misses Z;
A \/ B \/ C misses Z by A1,A2,A3,XBOOLE_1:114;
hence thesis by XBOOLE_1:70;
end;
registration
let M be symmetric triangle Reflexive MetrStruct, x, y be Point of M;
cluster dist(x,y) -> non negative;
coherence by METRIC_1:5;
end;
registration
let n be Element of NAT, x, y be Point of TOP-REAL n;
cluster dist(x,y) -> non negative;
coherence
proof
ex p, q being Point of Euclid n st p = x & q = y & dist(x,y) = dist(p,q)
by TOPREAL6:def 1;
hence 0 <= dist(x,y);
end;
end;
theorem Th1:
for p1, p2 being Point of TOP-REAL n st p1 <> p2 holds 1/2*(p1+p2) <> p1
proof
let p1, p2 be Point of TOP-REAL n;
set r = 1/2;
assume that
A1: p1 <> p2 and
A2: r*(p1+p2) = p1;
r*(p1+p2) = r*p1+r*p2 by RLVECT_1:def 5;
then 0.TOP-REAL n = p1-(r*p1+r*p2) by A2,RLVECT_1:5
.= p1-r*p1-r*p2 by RLVECT_1:27
.= 1 * p1-r*p1-r*p2 by RLVECT_1:def 8
.= (1-r)*p1-r*p2 by RLVECT_1:35
.= r*(p1-p2) by RLVECT_1:34;
then p1-p2 = 0.TOP-REAL n by RLVECT_1:11;
hence thesis by A1,RLVECT_1:21;
end;
theorem Th2:
p1`2 < p2`2 implies p1`2 < (1/2*(p1+p2))`2
proof
assume
A1: p1`2 < p2`2;
(1/2*(p1+p2))`2 = 1/2*((p1+p2)`2) by TOPREAL3:4
.= 1/2*(p1`2+p2`2) by TOPREAL3:2
.= (p1`2+p2`2)/2;
hence thesis by A1,XREAL_1:226;
end;
theorem Th3:
p1`2 < p2`2 implies (1/2*(p1+p2))`2 < p2`2
proof
assume
A1: p1`2 < p2`2;
(1/2*(p1+p2))`2 = 1/2*((p1+p2)`2) by TOPREAL3:4
.= 1/2*(p1`2+p2`2) by TOPREAL3:2
.= (p1`2+p2`2)/2;
hence thesis by A1,XREAL_1:226;
end;
theorem Th4:
for A being vertical Subset of TOP-REAL 2 holds A /\ B is vertical
proof
let A be vertical Subset of TOP-REAL 2;
let p, q be Point of T2;
assume that
A1: p in A /\ B and
A2: q in A /\ B;
A3: p in A by A1,XBOOLE_0:def 4;
q in A by A2,XBOOLE_0:def 4;
hence thesis by A3,SPPOL_1:def 3;
end;
theorem
for A being horizontal Subset of TOP-REAL 2 holds A /\ B is horizontal
proof
let A be horizontal Subset of TOP-REAL 2;
let p, q be Point of T2;
assume that
A1: p in A /\ B and
A2: q in A /\ B;
A3: p in A by A1,XBOOLE_0:def 4;
q in A by A2,XBOOLE_0:def 4;
hence thesis by A3,SPPOL_1:def 2;
end;
theorem
p in LSeg(p1,p2) & LSeg(p1,p2) is vertical implies LSeg(p,p2) is vertical
proof
assume
A1: p in LSeg(p1,p2);
assume
A2: LSeg(p1,p2) is vertical;
then
A3: p1`1 = p2`1 by SPPOL_1:16;
p1 in LSeg(p1,p2) by RLTOPSP1:68;
then p`1 = p1`1 by A1,A2;
hence thesis by A3,SPPOL_1:16;
end;
theorem
p in LSeg(p1,p2) & LSeg(p1,p2) is horizontal implies LSeg(p,p2) is horizontal
proof
assume
A1: p in LSeg(p1,p2);
assume
A2: LSeg(p1,p2) is horizontal;
then
A3: p1`2 = p2`2 by SPPOL_1:15;
p1 in LSeg(p1,p2) by RLTOPSP1:68;
then p`2 = p1`2 by A1,A2;
hence thesis by A3,SPPOL_1:15;
end;
registration
:: LSeg(NW-corner C,NE-corner C) is horizontal; ::SPRECT_3:29
let P be Subset of TOP-REAL 2;
cluster LSeg(SW-corner P,SE-corner P) -> horizontal;
coherence
proof
(SW-corner P)`2 = S-bound P by EUCLID:52
.= (SE-corner P)`2 by EUCLID:52;
hence thesis by SPPOL_1:15;
end;
cluster LSeg(NW-corner P,SW-corner P) -> vertical;
coherence
proof
(NW-corner P)`1 = W-bound P by EUCLID:52
.= (SW-corner P)`1 by EUCLID:52;
hence thesis by SPPOL_1:16;
end;
cluster LSeg(NE-corner P,SE-corner P) -> vertical;
coherence
proof
(NE-corner P)`1 = E-bound P by EUCLID:52
.= (SE-corner P)`1 by EUCLID:52;
hence thesis by SPPOL_1:16;
end;
end;
registration
let P be Subset of TOP-REAL 2;
cluster LSeg(SE-corner P,SW-corner P) -> horizontal;
coherence;
cluster LSeg(SW-corner P,NW-corner P) -> vertical;
coherence;
cluster LSeg(SE-corner P,NE-corner P) -> vertical;
coherence;
end;
registration
cluster vertical non empty compact -> with_the_max_arc for
Subset of TOP-REAL 2;
coherence
proof
let A be Subset of TOP-REAL 2;
assume
A1: A is vertical non empty compact;
then
A2: W-bound A = E-bound A by SPRECT_1:15;
A3: E-min A in A by A1,SPRECT_1:14;
(E-min A)`1 = E-bound A by EUCLID:52;
then E-min A in Vertical_Line((W-bound A+E-bound A)/2) by A2,JORDAN6:31;
hence A meets Vertical_Line((W-bound A+E-bound A)/2) by A3,XBOOLE_0:3;
end;
end;
theorem Th8:
p1`1 <= r & r <= p2`1 implies LSeg(p1,p2) meets Vertical_Line(r)
proof
assume that
A1: p1`1 <= r and
A2: r <= p2`1;
set a = p1`1, b = p2`1;
set l = (r-a) / (b-a);
set k = (1-l)*p1+l*p2;
A3: a-a <= r-a by A1,XREAL_1:9;
A4: r-a <= b-a by A2,XREAL_1:9;
then l <= 1 by A3,XREAL_1:183;
then
A5: k in LSeg(p1,p2) by A3,A4;
per cases;
suppose a <> b;
then
A6: b-a <> 0;
k`1 = (1-l)*a+l*b by TOPREAL9:41
.= a+l*(b-a)
.= a+(r-a) by A6,XCMPLX_1:87;
then k in Vertical_Line(r) by JORDAN6:31;
hence thesis by A5,XBOOLE_0:3;
end;
suppose
A7: a = b;
A8: p1 in LSeg(p1,p2) by RLTOPSP1:68;
a = r by A1,A2,A7,XXREAL_0:1;
then p1 in Vertical_Line(r) by JORDAN6:31;
hence thesis by A8,XBOOLE_0:3;
end;
end;
theorem
p1`2 <= r & r <= p2`2 implies LSeg(p1,p2) meets Horizontal_Line(r)
proof
assume that
A1: p1`2 <= r and
A2: r <= p2`2;
set a = p1`2, b = p2`2;
set l = (r-a) / (b-a);
set k = (1-l)*p1+l*p2;
A3: a-a <= r-a by A1,XREAL_1:9;
A4: r-a <= b-a by A2,XREAL_1:9;
then l <= 1 by A3,XREAL_1:183;
then
A5: k in LSeg(p1,p2) by A3,A4;
per cases;
suppose a <> b;
then
A6: b-a <> 0;
k`2 = (1-l)*a+l*b by TOPREAL9:42
.= a+l*(b-a)
.= a+(r-a) by A6,XCMPLX_1:87;
then k in Horizontal_Line(r) by JORDAN6:32;
hence thesis by A5,XBOOLE_0:3;
end;
suppose
A7: a = b;
A8: p1 in LSeg(p1,p2) by RLTOPSP1:68;
a = r by A1,A2,A7,XXREAL_0:1;
then p1 in Horizontal_Line(r) by JORDAN6:32;
hence thesis by A8,XBOOLE_0:3;
end;
end;
registration
let n;
cluster empty -> bounded for Subset of TOP-REAL n;
coherence;
cluster non bounded -> non empty for Subset of TOP-REAL n;
coherence;
end;
registration
let n be non zero Nat;
cluster open closed non bounded convex for Subset of TOP-REAL n;
existence
proof
take [#]TOP-REAL n;
reconsider n as Element of NAT by ORDINAL1:def 12;
n >= 1 by NAT_1:14;
then [#]TOP-REAL n is not bounded by JORDAN2C:35;
hence thesis;
end;
end;
theorem Th10:
for C being compact Subset of TOP-REAL 2 holds
north_halfline UMP C \ {UMP C} misses C
proof
let C be compact Subset of TOP-REAL 2;
set p = UMP C;
set L = north_halfline p;
set w = (W-bound C + E-bound C) / 2;
assume L \ {p} meets C;
then consider x being object such that
A1: x in L \ {p} and
A2: x in C by XBOOLE_0:3;
A3: x in L by A1,ZFMISC_1:56;
A4: x <> p by A1,ZFMISC_1:56;
reconsider x as Point of T2 by A1;
A5: x`1 = p`1 by A3,TOPREAL1:def 10;
A6: x`2 >= p`2 by A3,TOPREAL1:def 10;
x`2 <> p`2 by A4,A5,TOPREAL3:6;
then
A7: x`2 > p`2 by A6,XXREAL_0:1;
x`1 = w by A5,EUCLID:52;
then x in Vertical_Line w by JORDAN6:31;
then x in C /\ Vertical_Line w by A2,XBOOLE_0:def 4;
hence thesis by A7,JORDAN21:28;
end;
theorem Th11:
for C being compact Subset of TOP-REAL 2 holds
south_halfline LMP C \ {LMP C} misses C
proof
let C be compact Subset of TOP-REAL 2;
set p = LMP C;
set L = south_halfline p;
set w = (W-bound C + E-bound C) / 2;
assume L \ {p} meets C;
then consider x being object such that
A1: x in L \ {p} and
A2: x in C by XBOOLE_0:3;
A3: x in L by A1,ZFMISC_1:56;
A4: x <> p by A1,ZFMISC_1:56;
reconsider x as Point of T2 by A1;
A5: x`1 = p`1 by A3,TOPREAL1:def 12;
A6: x`2 <= p`2 by A3,TOPREAL1:def 12;
x`2 <> p`2 by A4,A5,TOPREAL3:6;
then
A7: x`2 < p`2 by A6,XXREAL_0:1;
x`1 = w by A5,EUCLID:52;
then x in Vertical_Line w by JORDAN6:31;
then x in C /\ Vertical_Line w by A2,XBOOLE_0:def 4;
hence thesis by A7,JORDAN21:29;
end;
theorem Th12:
for C being compact Subset of TOP-REAL 2 holds
north_halfline UMP C \ {UMP C} c= UBD C
proof
let C be compact Subset of TOP-REAL 2;
set A = north_halfline UMP C \ {UMP C};
reconsider A as non bounded Subset of T2 by JORDAN2C:122,TOPREAL6:90;
A is convex by JORDAN21:6;
hence thesis by Th10,JORDAN2C:125;
end;
theorem Th13:
for C being compact Subset of TOP-REAL 2 holds
south_halfline LMP C \ {LMP C} c= UBD C
proof
let C be compact Subset of TOP-REAL 2;
set A = south_halfline LMP C \ {LMP C};
reconsider A as non bounded Subset of T2 by JORDAN2C:123,TOPREAL6:90;
A is convex by JORDAN21:7;
hence thesis by Th11,JORDAN2C:125;
end;
theorem Th14:
A is_inside_component_of B implies UBD B misses A
proof
assume A is_inside_component_of B;
then A c= BDD B by JORDAN2C:22;
hence thesis by JORDAN2C:24,XBOOLE_1:63;
end;
theorem
A is_outside_component_of B implies BDD B misses A
proof
assume
A1: A is_outside_component_of B;
BDD B misses UBD B by JORDAN2C:24;
hence thesis by A1,JORDAN2C:23,XBOOLE_1:63;
end;
Lm4: p in C implies {p} misses U
proof
assume
A1: p in C;
A2: U is Subset of T2 by PRE_TOPC:11;
the carrier of T2|C` = C` by PRE_TOPC:8;
then U misses C by A2,SUBSET_1:23;
then not p in U by A1,XBOOLE_0:3;
hence thesis by ZFMISC_1:50;
end;
set C0 = Closed-Interval-TSpace(0,1);
set C1 = Closed-Interval-TSpace(-1,1);
set l0 = (#)(-1,1);
set l1 = (-1,1)(#);
set h1 = L[01](l0,l1);
Lm5: the carrier of [:T2,T2:] = [:the carrier of T2, the carrier of T2:]
by BORSUK_1:def 2;
Lm6: now
let T be non empty TopSpace;
let a be Element of REAL;
set c = the carrier of T;
set f = c --> a;
thus f is continuous
proof
A1: dom f = c by FUNCT_2:def 1;
A2: rng f = {a} by FUNCOP_1:8;
let Y be Subset of REAL;
assume Y is closed;
per cases;
suppose a in Y;
then
A3: rng f c= Y by A2,ZFMISC_1:31;
f"Y = f"(rng f /\ Y) by RELAT_1:133
.= f"rng f by A3,XBOOLE_1:28
.= [#]T by A1,RELAT_1:134;
hence thesis;
end;
suppose not a in Y;
then
A4: rng f misses Y by A2,ZFMISC_1:50;
f"Y = f"(rng f /\ Y) by RELAT_1:133
.= f"{} by A4
.= {}T;
hence thesis;
end;
end;
end;
theorem Th16:
for n being Nat
for r being positive Real
for a being Point of TOP-REAL n holds a in Ball(a,r)
proof let n be Nat;
let r be positive Real;
let a be Point of TOP-REAL n;
|. a-a .| = 0 by TOPRNS_1:28;
hence thesis by TOPREAL9:7;
end;
theorem Th17:
for r being non negative Real
for p being Point of TOP-REAL n holds p is Point of Tdisk(p,r)
proof
let r be non negative Real;
let p be Point of TOP-REAL n;
A1: the carrier of Tdisk(p,r) = cl_Ball(p,r) by BROUWER:3;
|. p-p .| = 0 by TOPRNS_1:28;
hence thesis by A1,TOPREAL9:8;
end;
registration
let r be positive Real;
let n be non zero Element of NAT;
let p, q be Point of TOP-REAL n;
cluster cl_Ball(p,r) \ {q} -> non empty;
coherence
proof
A1: the carrier of Tcircle(p,r) = Sphere(p,r) by TOPREALB:9;
A2: the carrier of Tdisk(p,r) = cl_Ball(p,r) by BROUWER:3;
A3: Sphere(p,r) c= cl_Ball(p,r) by TOPREAL9:17;
set a = the Point of Tcircle(p,r);
A4: a in Sphere(p,r) by A1;
per cases;
suppose
A5: a = q;
A6: p is Point of Tdisk(p,r) by Th17;
|. p-p .| <> r by TOPRNS_1:28;
then p <> q by A1,A5,TOPREAL9:9;
hence thesis by A2,A6,ZFMISC_1:56;
end;
suppose a <> q;
hence thesis by A3,A4,ZFMISC_1:56;
end;
end;
end;
theorem Th18:
r <= s implies Ball(x,r) c= Ball(x,s)
proof
reconsider xe = x as Point of Euclid n by TOPREAL3:8;
A1: Ball(x,r) = Ball(xe,r) by TOPREAL9:13;
Ball(x,s) = Ball(xe,s) by TOPREAL9:13;
hence thesis by A1,PCOMPS_1:1;
end;
theorem Th19:
cl_Ball(x,r) \ Ball(x,r) = Sphere(x,r)
proof
thus cl_Ball(x,r) \ Ball(x,r) c= Sphere(x,r)
proof
let a be object;
assume
A1: a in cl_Ball(x,r) \ Ball(x,r);
then reconsider a as Point of TOP-REAL n;
A2: a in cl_Ball(x,r) by A1,XBOOLE_0:def 5;
A3: not a in Ball(x,r) by A1,XBOOLE_0:def 5;
A4: |. a-x .| <= r by A2,TOPREAL9:8;
|. a-x .| >= r by A3,TOPREAL9:7;
then |. a-x .| = r by A4,XXREAL_0:1;
hence thesis by TOPREAL9:9;
end;
let a be object;
assume
A5: a in Sphere(x,r);
then reconsider a as Point of TOP-REAL n;
A6: |. a-x .| = r by A5,TOPREAL9:9;
then
A7: a in cl_Ball(x,r) by TOPREAL9:8;
not a in Ball(x,r) by A6,TOPREAL9:7;
hence thesis by A7,XBOOLE_0:def 5;
end;
theorem Th20:
y in Sphere(x,r) implies LSeg(x,y) \ {x,y} c= Ball(x,r)
proof
assume
A1: y in Sphere(x,r);
per cases;
suppose
A2: r = 0;
reconsider xe = x as Point of Euclid n by TOPREAL3:8;
Sphere(x,r) = Sphere(xe,r) by TOPREAL9:15;
then Sphere(x,r) = {x} by A2,TOPREAL6:54;
then
A3: x = y by A1,TARSKI:def 1;
A4: LSeg(x,x) = {x} by RLTOPSP1:70;
A5: {x,x} = {x} by ENUMSET1:29;
{x} \ {x} = {} by XBOOLE_1:37;
hence thesis by A3,A4,A5;
end;
suppose
A6: r <> 0;
let k be object;
assume
A7: k in LSeg(x,y) \ {x,y};
then k in LSeg(x,y) by XBOOLE_0:def 5;
then consider l being Real such that
A8: k = (1-l)*x + l*y and
A9: 0 <= l and
A10: l <= 1;
reconsider k as Point of TOP-REAL n by A8;
not k in {x,y} by A7,XBOOLE_0:def 5;
then k <> y by TARSKI:def 2;
then l <> 1 by A8,TOPREAL9:4;
then
A11: l < 1 by A10,XXREAL_0:1;
k-x = (1-l)*x - x + l*y by A8,RLVECT_1:def 3
.= 1 * x - l*x - x + l*y by RLVECT_1:35
.= x - l*x - x + l*y by RLVECT_1:def 8
.= x +- l*x +- x + l*y
.= x +- x +- l*x + l*y by RLVECT_1:def 3
.= x - x - l*x + l*y
.= 0.TOP-REAL n - l*x + l*y by RLVECT_1:5
.= l*y - l*x by RLVECT_1:4
.= l*(y-x) by RLVECT_1:34;
then
A12: |. k-x .| = |.l.| * |. y-x .| by TOPRNS_1:7
.= l*|. y-x .| by A9,ABSVALUE:def 1
.= l*r by A1,TOPREAL9:9;
0 <= r by A1;
then l*r < 1 * r by A6,A11,XREAL_1:68;
hence thesis by A12,TOPREAL9:7;
end;
end;
theorem Th21:
r < s implies cl_Ball(x,r) c= Ball(x,s)
proof
assume
A1: r < s;
let a be object;
assume
A2: a in cl_Ball(x,r);
then reconsider a as Point of TOP-REAL n;
|. a-x .| <= r by A2,TOPREAL9:8;
then |. a-x .| < s by A1,XXREAL_0:2;
hence thesis by TOPREAL9:7;
end;
theorem Th22:
r < s implies Sphere(x,r) c= Ball(x,s)
proof
assume r < s;
then
A1: cl_Ball(x,r) c= Ball(x,s) by Th21;
Sphere(x,r) c= cl_Ball(x,r) by TOPREAL9:17;
hence thesis by A1;
end;
theorem Th23:
for r being non zero Real holds Cl Ball(x,r) = cl_Ball(x,r)
proof
let r be non zero Real;
thus Cl Ball(x,r) c= cl_Ball(x,r) by TOPREAL9:16,TOPS_1:5;
per cases;
suppose
Ball(x,r) is empty;
then r < 0;
hence thesis;
end;
suppose
A1: Ball(x,r) is non empty;
let a be object;
assume
A2: a in cl_Ball(x,r);
then reconsider a as Point of TOP-REAL n;
reconsider ae = a as Point of Euclid n by TOPREAL3:8;
A3: 0 < r by A1;
for s being Real st 0 < s & s < r holds Ball(ae,s) meets Ball(x,r)
proof
let s be Real such that
A4: 0 < s and
A5: s < r;
now
A6: Ball(x,r) \/ Sphere(x,r) = cl_Ball(x,r) by TOPREAL9:18;
per cases by A2,A6,XBOOLE_0:def 3;
suppose
A7: a in Ball(x,r);
|.a-a.| = 0 by TOPRNS_1:28;
then a in Ball(a,s) by A4,TOPREAL9:7;
hence Ball(a,s) meets Ball(x,r) by A7,XBOOLE_0:3;
end;
suppose
A8: a in Sphere(x,r);
then
A9: |. a-x .| = r by TOPREAL9:9;
|. x-x .| = 0 by TOPRNS_1:28;
then
A10: x in Ball(x,r) by A3,TOPREAL9:7;
set z = s/(2*r);
set q = (1-z)*a+z*x;
1 * r < 2*r by A3,XREAL_1:68;
then s < 2*r by A5,XXREAL_0:2;
then
A11: z < 1 by A4,XREAL_1:189;
0 < 2*r by A3,XREAL_1:129;
then
A12: 0 < z by A4,XREAL_1:139;
A13: q in LSeg(a,x) by A3,A4,A11;
Ball(x,r) misses Sphere(x,r) by TOPREAL9:19;
then
A14: a <> x by A8,A10,XBOOLE_0:3;
then
A15: q <> a by A12,TOPREAL9:4;
q <> x by A11,A14,TOPREAL9:4;
then not q in {a,x} by A15,TARSKI:def 2;
then
A16: q in LSeg(a,x) \ {a,x} by A13,XBOOLE_0:def 5;
A17: LSeg(a,x) \ {a,x} c= Ball(x,r) by A8,Th20;
q-a = (1-z)*a - a + z*x by RLVECT_1:def 3
.= 1 * a - z*a - a + z*x by RLVECT_1:35
.= a - z*a - a + z*x by RLVECT_1:def 8
.= a +- z*a +- a + z*x
.= a +- a +- z*a + z*x by RLVECT_1:def 3
.= a - a - z*a + z*x
.= 0.TOP-REAL n - z*a + z*x by RLVECT_1:5
.= z*x - z*a by RLVECT_1:4
.= z*(x-a) by RLVECT_1:34;
then |.q-a.| = |.z.| * |.x-a.| by TOPRNS_1:7
.= z*|.x-a.| by A3,A4,ABSVALUE:def 1
.= z*|. a-x .| by TOPRNS_1:27
.= s/2 by A9,XCMPLX_1:92;
then
A18: q in Sphere(a,s/2) by TOPREAL9:9;
s/2 < s/1 by A4,XREAL_1:76;
then Sphere(a,s/2) c= Ball(a,s) by Th22;
hence Ball(a,s) meets Ball(x,r) by A16,A17,A18,XBOOLE_0:3;
end;
end;
hence thesis by TOPREAL9:13;
end;
hence thesis by A3,GOBOARD6:93;
end;
end;
theorem Th24:
for r being non zero Real holds Fr Ball(x,r) = Sphere(x,r)
proof
let r be non zero Real;
set P = Ball(x,r);
thus Fr P = Cl P \ P by TOPS_1:42
.= cl_Ball(x,r) \ P by Th23
.= Sphere(x,r) by Th19;
end;
registration
let n be non zero Element of NAT;
cluster bounded -> proper for Subset of TOP-REAL n;
coherence
proof
[#]TOP-REAL n is not bounded by JORDAN2C:35,NAT_1:14;
hence thesis by SUBSET_1:def 6;
end;
end;
registration
let n;
cluster non empty closed convex bounded for Subset of TOP-REAL n;
existence
proof
take cl_Ball(0.TOP-REAL n,1);
thus thesis;
end;
cluster non empty open convex bounded for Subset of TOP-REAL n;
existence
proof
take Ball(0.TOP-REAL n,1);
thus thesis;
end;
end;
registration
let n be Element of NAT;
let A be bounded Subset of TOP-REAL n;
cluster Cl A -> bounded;
coherence by TOPREAL6:63;
end;
registration
let n be Element of NAT;
let A be bounded Subset of TOP-REAL n;
cluster Fr A -> bounded;
coherence by TOPREAL6:89;
end;
theorem Th25:
for A being closed Subset of TOP-REAL n, p being Point of TOP-REAL n st
not p in A ex r being positive Real st Ball(p,r) misses A
proof
let A be closed Subset of TOP-REAL n, p be Point of TOP-REAL n;
assume not p in A;
then
A1: p in A` by SUBSET_1:29;
reconsider e = p as Point of Euclid n by TOPREAL3:8;
A2: the TopStruct of TOP-REAL n = TopSpaceMetr Euclid n by EUCLID:def 8;
then reconsider AA = A` as Subset of TopSpaceMetr Euclid n;
AA is open by A2,PRE_TOPC:30;
then consider r being Real such that
A3: r > 0 and
A4: Ball(e,r) c= A` by A1,TOPMETR:15;
reconsider r as positive Real by A3;
take r;
Ball(p,r) = Ball(e,r) by TOPREAL9:13;
hence thesis by A4,SUBSET_1:23;
end;
theorem Th26:
for A being bounded Subset of TOP-REAL n, a being Point of TOP-REAL n
ex r being positive Real st A c= Ball(a,r)
proof
let A be bounded Subset of TOP-REAL n;
let a be Point of TOP-REAL n;
reconsider C = A as bounded Subset of Euclid n by JORDAN2C:11;
consider r being Real, x being Element of Euclid n such that
A1: 0 < r and
A2: C c= Ball(x,r) by METRIC_6:def 3;
reconsider r as positive Real by A1;
reconsider x1 = x as Point of TOP-REAL n by TOPREAL3:8;
take s = r+|.x1-a.|;
let p be object;
assume
A3: p in A;
then reconsider p1 = p as Point of TOP-REAL n;
p = p1;
then reconsider p as Point of Euclid n by TOPREAL3:8;
A4: dist(p,x) < r by A2,A3,METRIC_1:11;
A5: |.p1-x1.| = dist(p,x) by SPPOL_1:39;
A6: |.p1-a.| <= |.p1-x1.| + |.x1-a.| by TOPRNS_1:34;
|.p1-x1.| + |.x1-a.| < s by A4,A5,XREAL_1:6;
then |.p1-a.| < s by A6,XXREAL_0:2;
hence thesis by TOPREAL9:7;
end;
theorem
for S, T being TopStruct, f being Function
of S, T st f is being_homeomorphism holds f is onto;
registration
let T be non empty T_2 TopSpace;
cluster -> T_2 for non empty SubSpace of T;
coherence;
end;
registration
let p, r;
cluster Tdisk(p,r) -> closed;
coherence
proof
let A be Subset of T2;
assume A = the carrier of Tdisk(p,r);
then A = cl_Ball(p,r) by BROUWER:3;
hence thesis;
end;
end;
registration
let p, r;
cluster Tdisk(p,r) -> compact;
coherence
proof
set D = Tdisk(p,r);
reconsider Q = [#]D as Subset of T2 by TSEP_1:1;
[#]D = cl_Ball(p,r) by BROUWER:3;
then Q is compact by TOPREAL6:79;
then [#]D is compact by COMPTS_1:2;
hence thesis by COMPTS_1:1;
end;
end;
begin :: Paths
theorem
for T being non empty TopSpace, a, b being Point of T
for f being Path of a,b st a,b are_connected holds rng f is connected
proof
let T be non empty TopSpace, a, b be Point of T;
let f be Path of a,b such that
A1: a,b are_connected;
A2: dom f = the carrier of I[01] by FUNCT_2:def 1;
reconsider A = [.0,1.] as interval Subset of R^1 by TOPMETR:17;
reconsider B = A as Subset of I[01] by BORSUK_1:40;
A3: B is connected by CONNSP_1:23;
A4: f is continuous by A1,BORSUK_2:def 2;
f.:B = rng f by A2,BORSUK_1:40,RELAT_1:113;
hence thesis by A3,A4,TOPS_2:61;
end;
theorem Th29:
for X being non empty TopSpace, Y being non empty SubSpace of X,
x1, x2 being Point of X, y1, y2 being Point of Y,
f being Path of x1,x2 st x1 = y1 & x2 = y2 & x1,x2 are_connected &
rng f c= the carrier of Y holds y1,y2 are_connected & f is Path of y1,y2
proof
let X be non empty TopSpace, Y be non empty SubSpace of X,
x1, x2 be Point of X, y1, y2 be Point of Y, f be Path of x1,x2 such that
A1: x1 = y1 and
A2: x2 = y2 and
A3: x1, x2 are_connected;
assume rng f c= the carrier of Y;
then reconsider g = f as Function of I[01], Y by FUNCT_2:6;
A4: f is continuous by A3,BORSUK_2:def 2;
A5: f.0 = y1 & f.1 = y2 by A1,A2,A3,BORSUK_2:def 2;
A6: g is continuous by A4,PRE_TOPC:27;
thus
ex f being Function of I[01], Y st f is continuous & f.0 = y1 & f.1 = y2
proof
take g;
thus g is continuous by A4,PRE_TOPC:27;
thus thesis by A1,A2,A3,BORSUK_2:def 2;
end;
y1, y2 are_connected by A5,A6;
hence thesis by A5,A6,BORSUK_2:def 2;
end;
theorem Th30:
for X being pathwise_connected non empty TopSpace,
Y being non empty SubSpace of X,
x1, x2 being Point of X, y1, y2 being Point of Y,
f being Path of x1,x2 st x1 = y1 & x2 = y2 & rng f c= the carrier of Y holds
y1,y2 are_connected & f is Path of y1,y2
proof
let X be pathwise_connected non empty TopSpace, Y be non empty SubSpace of X,
x1, x2 be Point of X, y1, y2 be Point of Y;
x1,x2 are_connected by BORSUK_2:def 3;
hence thesis by Th29;
end;
Lm7: for T being non empty TopSpace, a, b being Point of T
for f being Path of a,b st a,b are_connected holds rng f c= rng -f
proof
let T be non empty TopSpace;
let a, b be Point of T;
let f be Path of a,b;
assume
A1: a,b are_connected;
let y be object;
assume y in rng f;
then consider x being object such that
A2: x in dom f and
A3: f.x = y by FUNCT_1:def 3;
reconsider x as Point of I[01] by A2;
A4: dom -f = the carrier of I[01] by FUNCT_2:def 1;
A5: 1-x is Point of I[01] by JORDAN5B:4;
then (-f).(1-x) = f.(1-(1-x)) by A1,BORSUK_2:def 6;
hence thesis by A3,A4,A5,FUNCT_1:def 3;
end;
theorem Th31:
for T being non empty TopSpace, a, b being Point of T
for f being Path of a,b st a,b are_connected holds rng f = rng -f
proof
let T be non empty TopSpace;
let a, b be Point of T;
let f be Path of a,b;
assume
A1: a,b are_connected;
hence rng f c= rng -f by Lm7;
f = --f by A1,BORSUK_6:43;
hence thesis by A1,Lm7;
end;
theorem Th32:
for T being pathwise_connected non empty TopSpace, a, b being Point of T
for f being Path of a,b holds rng f = rng -f
by Th31,BORSUK_2:def 3;
theorem Th33:
for T being non empty TopSpace, a, b, c being Point of T
for f being Path of a,b, g being Path of b,c st
a,b are_connected & b,c are_connected holds rng f c= rng(f+g)
proof
let T be non empty TopSpace;
let a, b, c be Point of T;
let f be Path of a,b;
let g be Path of b,c;
assume that
A1: a,b are_connected and
A2: b,c are_connected;
let y be object;
assume y in rng f;
then consider x being object such that
A3: x in dom f and
A4: f.x = y by FUNCT_1:def 3;
A5: dom(f+g) = the carrier of I[01] by FUNCT_2:def 1;
reconsider x as Point of I[01] by A3;
1/2*x = x/2;
then
A6: x/2 is Point of I[01] by BORSUK_6:6;
x <= 1 by BORSUK_1:43;
then x/2 <= 1/2 by XREAL_1:72;
then (f+g).(x/2) = f.(2*(x/2)) by A1,A2,A6,BORSUK_2:def 5;
hence thesis by A4,A5,A6,FUNCT_1:def 3;
end;
theorem
for T being pathwise_connected non empty TopSpace, a, b, c being Point of T
for f being Path of a,b, g being Path of b,c holds rng f c= rng(f+g)
proof
let T be pathwise_connected non empty TopSpace;
let a, b, c be Point of T;
let f be Path of a,b;
let g be Path of b,c;
A1: a,b are_connected by BORSUK_2:def 3;
b,c are_connected by BORSUK_2:def 3;
hence thesis by A1,Th33;
end;
theorem Th35:
for T being non empty TopSpace, a, b, c being Point of T
for f being Path of b,c, g being Path of a,b st
a,b are_connected & b,c are_connected holds rng f c= rng(g+f)
proof
let T be non empty TopSpace;
let a, b, c be Point of T;
let f be Path of b,c;
let g be Path of a,b;
assume that
A1: a,b are_connected and
A2: b,c are_connected;
let y be object;
assume y in rng f;
then consider x being object such that
A3: x in dom f and
A4: f.x = y by FUNCT_1:def 3;
A5: dom(g+f) = the carrier of I[01] by FUNCT_2:def 1;
reconsider x as Point of I[01] by A3;
A6: 0 <= x by BORSUK_1:43;
then
A7: 0+1/2 <= x/2+1/2 by XREAL_1:6;
x <= 1 by BORSUK_1:43;
then x+1 <= 1+1 by XREAL_1:6;
then (x+1)/2 <= 2/2 by XREAL_1:72;
then
A8: x/2+1/2 is Point of I[01] by A6,BORSUK_1:43;
then (g+f).(x/2+1/2) = f.(2*(x/2+1/2)-1) by A1,A2,A7,BORSUK_2:def 5;
hence thesis by A4,A5,A8,FUNCT_1:def 3;
end;
theorem
for T being pathwise_connected non empty TopSpace, a, b, c being Point of T
for f being Path of b,c, g being Path of a,b holds rng f c= rng(g+f)
proof
let T be pathwise_connected non empty TopSpace;
let a, b, c be Point of T;
let f be Path of b,c;
let g be Path of a,b;
A1: a,b are_connected by BORSUK_2:def 3;
b,c are_connected by BORSUK_2:def 3;
hence thesis by A1,Th35;
end;
theorem Th37:
for T being non empty TopSpace, a, b, c being Point of T
for f being Path of a,b, g being Path of b,c st
a,b are_connected & b,c are_connected holds rng(f+g) = rng f \/ rng g
proof
let T be non empty TopSpace;
let a, b, c be Point of T;
let f be Path of a,b;
let g be Path of b,c;
assume that
A1: a,b are_connected and
A2: b,c are_connected;
thus rng(f+g) c= rng f \/ rng g
proof
let y be object;
assume y in rng(f+g);
then consider x being object such that
A3: x in dom(f+g) and
A4: y = (f+g).x by FUNCT_1:def 3;
reconsider x as Point of I[01] by A3;
per cases;
suppose
A5: x <= 1/2;
then
A6: (f+g).x = f.(2*x) by A1,A2,BORSUK_2:def 5;
A7: rng f c= rng f \/ rng g by XBOOLE_1:7;
A8: dom f = the carrier of I[01] by FUNCT_2:def 1;
2*x is Point of I[01] by A5,BORSUK_6:3;
then y in rng f by A4,A6,A8,FUNCT_1:def 3;
hence thesis by A7;
end;
suppose
A9: 1/2 <= x;
then
A10: (f+g).x = g.(2*x-1) by A1,A2,BORSUK_2:def 5;
A11: rng g c= rng f \/ rng g by XBOOLE_1:7;
A12: dom g = the carrier of I[01] by FUNCT_2:def 1;
2*x-1 is Point of I[01] by A9,BORSUK_6:4;
then y in rng g by A4,A10,A12,FUNCT_1:def 3;
hence thesis by A11;
end;
end;
A13: rng f c= rng(f+g) by A1,A2,Th33;
rng g c= rng(f+g) by A1,A2,Th35;
hence thesis by A13,XBOOLE_1:8;
end;
theorem
for T being pathwise_connected non empty TopSpace, a, b, c being Point of T
for f being Path of a,b, g being Path of b,c holds rng(f+g) = rng f \/ rng g
proof
let T be pathwise_connected non empty TopSpace;
let a, b, c be Point of T;
let f be Path of a,b;
let g be Path of b,c;
A1: a,b are_connected by BORSUK_2:def 3;
b,c are_connected by BORSUK_2:def 3;
hence thesis by A1,Th37;
end;
theorem Th39:
for T being non empty TopSpace, a, b, c, d being Point of T
for f being Path of a,b, g being Path of b,c, h being Path of c,d st
a,b are_connected & b,c are_connected & c,d are_connected holds
rng(f+g+h) = rng f \/ rng g \/ rng h
proof
let T be non empty TopSpace;
let a, b, c, d be Point of T;
let f be Path of a,b;
let g be Path of b,c;
let h be Path of c,d;
assume that
A1: a,b are_connected and
A2: b,c are_connected and
A3: c,d are_connected;
a,c are_connected by A1,A2,BORSUK_6:42;
hence rng(f+g+h) = rng(f+g) \/ rng h by A3,Th37
.= rng f \/ rng g \/ rng h by A1,A2,Th37;
end;
theorem Th40:
for T being pathwise_connected non empty TopSpace,
a, b, c, d being Point of T
for f being Path of a,b, g being Path of b,c, h being Path of c,d
holds rng(f+g+h) = rng f \/ rng g \/ rng h
proof
let T be pathwise_connected non empty TopSpace;
let a, b, c, d be Point of T;
let f be Path of a,b;
let g be Path of b,c;
let h be Path of c,d;
A1: a,b are_connected by BORSUK_2:def 3;
A2: b,c are_connected by BORSUK_2:def 3;
c,d are_connected by BORSUK_2:def 3;
hence thesis by A1,A2,Th39;
end;
Lm8: for T being non empty TopSpace, a, b, c, d, e being Point of T
for f being Path of a,b, g being Path of b,c,
h being Path of c,d, i being Path of d,e st
a,b are_connected & b,c are_connected &
c,d are_connected & d,e are_connected holds
rng(f+g+h+i) = rng f \/ rng g \/ rng h \/ rng i
proof
let T be non empty TopSpace;
let a, b, c, d, e be Point of T;
let f be Path of a,b;
let g be Path of b,c;
let h be Path of c,d;
let i be Path of d,e;
assume that
A1: a,b are_connected and
A2: b,c are_connected and
A3: c,d are_connected and
A4: d,e are_connected;
a,c are_connected by A1,A2,BORSUK_6:42;
then a,d are_connected by A3,BORSUK_6:42;
hence rng(f+g+h+i) = rng(f+g+h) \/ rng i by A4,Th37
.= rng f \/ rng g \/ rng h \/ rng i by A1,A2,A3,Th39;
end;
Lm9: for T being pathwise_connected non empty TopSpace,
a, b, c, d, e being Point of T for f being Path of a,b, g being Path of b,c,
h being Path of c,d, i being Path of d,e
holds rng(f+g+h+i) = rng f \/ rng g \/ rng h \/ rng i
proof
let T be pathwise_connected non empty TopSpace;
let a, b, c, d, e be Point of T;
let f be Path of a,b;
let g be Path of b,c;
let h be Path of c,d;
let i be Path of d,e;
A1: a,b are_connected by BORSUK_2:def 3;
A2: b,c are_connected by BORSUK_2:def 3;
A3: c,d are_connected by BORSUK_2:def 3;
d,e are_connected by BORSUK_2:def 3;
hence thesis by A1,A2,A3,Lm8;
end;
Lm10: for T being non empty TopSpace, a, b, c, d, e, z being Point of T
for f being Path of a,b, g being Path of b,c,
h being Path of c,d, i being Path of d,e, j being Path of e,z st
a,b are_connected & b,c are_connected & c,d are_connected &
d,e are_connected & e,z are_connected holds
rng(f+g+h+i+j) = rng f \/ rng g \/ rng h \/ rng i \/ rng j
proof
let T be non empty TopSpace;
let a, b, c, d, e, z be Point of T;
let f be Path of a,b;
let g be Path of b,c;
let h be Path of c,d;
let i be Path of d,e;
let j be Path of e,z;
assume that
A1: a,b are_connected and
A2: b,c are_connected and
A3: c,d are_connected and
A4: d,e are_connected and
A5: e,z are_connected;
a,c are_connected by A1,A2,BORSUK_6:42;
then a,d are_connected by A3,BORSUK_6:42;
then a,e are_connected by A4,BORSUK_6:42;
hence rng(f+g+h+i+j) = rng(f+g+h+i) \/ rng j by A5,Th37
.= rng f \/ rng g \/ rng h \/ rng i \/ rng j by A1,A2,A3,A4,Lm8;
end;
Lm11: for T being pathwise_connected non empty TopSpace,
a, b, c, d, e, z being Point of T
for f being Path of a,b, g being Path of b,c,
h being Path of c,d, i being Path of d,e, j being Path of e,z
holds rng(f+g+h+i+j) = rng f \/ rng g \/ rng h \/ rng i \/ rng j
proof
let T be pathwise_connected non empty TopSpace;
let a, b, c, d, e, z be Point of T;
let f be Path of a,b;
let g be Path of b,c;
let h be Path of c,d;
let i be Path of d,e;
let j be Path of e,z;
A1: a,b are_connected by BORSUK_2:def 3;
A2: b,c are_connected by BORSUK_2:def 3;
A3: c,d are_connected by BORSUK_2:def 3;
A4: d,e are_connected by BORSUK_2:def 3;
e,z are_connected by BORSUK_2:def 3;
hence thesis by A1,A2,A3,A4,Lm10;
end;
theorem Th41:
for T being non empty TopSpace, a being Point of T holds
I[01] --> a is Path of a,a
proof
let T be non empty TopSpace, a be Point of T;
thus a,a are_connected;
thus thesis by BORSUK_1:def 14,def 15,TOPALG_3:4;
end;
theorem Th42:
for p1, p2 being Point of TOP-REAL n, P being Subset of TOP-REAL n
holds P is_an_arc_of p1,p2 implies
ex F being Path of p1,p2, f being Function of I[01], (TOP-REAL n)|P st
rng f = P & F = f
proof
let p1, p2 be Point of TOP-REAL n, P be Subset of TOP-REAL n;
assume
A1: P is_an_arc_of p1,p2;
then reconsider P1 = P as non empty Subset of TOP-REAL n by TOPREAL1:1;
consider h being Function of I[01], (TOP-REAL n)|P such that
A2: h is being_homeomorphism and
A3: h.0 = p1 and
A4: h.1 = p2 by A1,TOPREAL1:def 1;
h is Function of I[01], (TOP-REAL n)|P1;
then reconsider h1 = h as Function of I[01],TOP-REAL n by TOPREALA:7;
h1 is continuous by A2,PRE_TOPC:26;
then reconsider f = h as Path of p1,p2 by A3,A4,BORSUK_2:def 4;
take f, h;
thus rng h = [#]((TOP-REAL n)|P) by A2,TOPS_2:def 5
.= P by PRE_TOPC:8;
thus thesis;
end;
theorem Th43:
for p1, p2 being Point of TOP-REAL n
ex F being Path of p1,p2, f being Function
of I[01], (TOP-REAL n)|LSeg(p1,p2) st rng f = LSeg(p1,p2) & F = f
proof
let p1, p2 be Point of TOP-REAL n;
per cases;
suppose
A1: p1 = p2;
then reconsider g = I[01] --> p1 as Path of p1,p2 by Th41;
take g;
A2: LSeg(p1,p2) = {p1} by A1,RLTOPSP1:70;
A3: rng g = {p1} by FUNCOP_1:8;
the carrier of (TOP-REAL n)|LSeg(p1,p2) = LSeg(p1,p2) by PRE_TOPC:8;
then reconsider f = g as Function of I[01],(TOP-REAL n)|LSeg(p1,p2)
by A2,A3,FUNCT_2:6;
take f;
thus thesis by A1,A3,RLTOPSP1:70;
end;
suppose p1 <> p2;
hence thesis by Th42,TOPREAL1:9;
end;
end;
theorem Th44:
for p1,p2,q1,q2 being Point of TOP-REAL 2
st P is_an_arc_of p1,p2 & q1 in P & q2 in P &
q1 <> p1 & q1 <> p2 & q2 <> p1 & q2 <> p2
ex f being Path of q1,q2 st rng f c= P & rng f misses {p1,p2}
proof
let p1,p2,q1,q2 be Point of TOP-REAL 2 such that
A1: P is_an_arc_of p1,p2 and
A2: q1 in P and
A3: q2 in P and
A4: q1 <> p1 and
A5: q1 <> p2 and
A6: q2 <> p1 and
A7: q2 <> p2;
per cases;
suppose q1 = q2;
then reconsider f = I[01] --> q1 as Path of q1,q2 by Th41;
take f;
A8: rng f = {q1} by FUNCOP_1:8;
thus rng f c= P
by A2,A8,TARSKI:def 1;
A9: not p1 in {q1} by A4,TARSKI:def 1;
not p2 in {q1} by A5,TARSKI:def 1;
hence thesis by A8,A9,ZFMISC_1:51;
end;
suppose q1 <> q2;
then consider Q being non empty Subset of T2 such that
A10: Q is_an_arc_of q1,q2 and
A11: Q c= P and
A12: Q misses {p1,p2} by A1,A2,A3,A4,A5,A6,A7,JORDAN16:23;
consider g being Path of q1,q2, f being Function of I[01], T2|Q such that
A13: rng f = Q and
A14: g = f by A10,Th42;
reconsider h = f as Function of I[01],T2 by TOPREALA:7;
the carrier of T2|Q = Q by PRE_TOPC:8;
then reconsider z1 = q1, z2 = q2 as Point of T2|Q by A10,TOPREAL1:1;
A15: z1,z2 are_connected
proof
take f;
thus f is continuous by A14,PRE_TOPC:27;
thus thesis by A14,BORSUK_2:def 4;
end;
A16: f is continuous by A14,PRE_TOPC:27;
f.0 =z1 & f.1=z2 by A14,BORSUK_2:def 4;
then f is Path of z1,z2 by A15,A16,BORSUK_2:def 2;
then reconsider h as Path of q1,q2 by A15,TOPALG_2:1;
take h;
thus thesis by A11,A12,A13;
end;
end;
begin :: Rectangles
theorem Th45:
a <= b & c <= d implies
rectangle(a,b,c,d) c= closed_inside_of_rectangle(a,b,c,d)
proof
assume that
A1: a <= b and
A2: c <= d;
let x be object;
assume x in rectangle(a,b,c,d);
then x in {p: 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} by A1,A2,SPPOL_2:54;
then ex p st x = p & (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);
hence thesis by A1,A2;
end;
theorem Th46:
inside_of_rectangle(a,b,c,d) c= closed_inside_of_rectangle(a,b,c,d)
proof
let x be object;
assume x in inside_of_rectangle(a,b,c,d);
then ex p st x = p & a < p`1 & p`1 < b & c < p`2 & p`2 < d;
hence thesis;
end;
theorem Th47:
closed_inside_of_rectangle(a,b,c,d) = outside_of_rectangle(a,b,c,d)`
proof
set R = closed_inside_of_rectangle(a,b,c,d);
set O = outside_of_rectangle(a,b,c,d);
thus R c= O`
proof
let x be object;
assume x in R;
then consider p such that
A1: x = p and
A2: a <= p`1 and
A3: p`1 <= b and
A4: c <= p`2 and
A5: p`2 <= d;
now
assume p in O;
then ex p1 st p1 = p &
not(a <= p1`1 & p1`1 <= b & c <= p1`2 & p1`2 <= d);
hence contradiction by A2,A3,A4,A5;
end;
hence thesis by A1,SUBSET_1:29;
end;
let x be object;
assume
A6: x in O`;
then
A7: not x in O by XBOOLE_0:def 5;
reconsider x as Point of T2 by A6;
A8: a <= x`1 by A7;
A9: x`1 <= b by A7;
A10: c <= x`2 by A7;
x`2 <= d by A7;
hence thesis by A8,A9,A10;
end;
registration
let a, b, c, d be Real;
cluster closed_inside_of_rectangle(a,b,c,d) -> closed;
coherence
proof
set P2 = outside_of_rectangle(a,b,c,d);
reconsider P2 as open Subset of T2 by JORDAN1:34;
P2` is closed;
hence thesis by Th47;
end;
end;
theorem Th48:
closed_inside_of_rectangle(a,b,c,d) misses outside_of_rectangle(a,b,c,d)
proof
set R = closed_inside_of_rectangle(a,b,c,d);
set P2 = outside_of_rectangle(a,b,c,d);
assume R meets P2;
then consider x being object such that
A1: x in R and
A2: x in P2 by XBOOLE_0:3;
A3: ex p st x = p & a <= p`1 & p`1 <= b & c <= p`2 & p`2 <= d by A1;
ex p st x = p & not (a <= p`1 & p`1 <= b & c <= p`2 & p`2 <= d) by A2;
hence thesis by A3;
end;
theorem Th49:
closed_inside_of_rectangle(a,b,c,d) /\ inside_of_rectangle(a,b,c,d)
= inside_of_rectangle(a,b,c,d)
proof
set R = closed_inside_of_rectangle(a,b,c,d);
set P1 = inside_of_rectangle(a,b,c,d);
thus R /\ P1 c= P1 by XBOOLE_1:17;
P1 /\ P1 c= P1 /\ R by Th46,XBOOLE_1:26;
hence thesis;
end;
theorem Th50:
a < b & c < d implies
Int closed_inside_of_rectangle(a,b,c,d) = inside_of_rectangle(a,b,c,d)
proof
assume that
A1: a < b and
A2: c < d;
set P = rectangle(a,b,c,d);
set R = closed_inside_of_rectangle(a,b,c,d);
set P1 = inside_of_rectangle(a,b,c,d);
set P2 = outside_of_rectangle(a,b,c,d);
A3: P = {p where p is Point of T2: 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} by A1,A2,SPPOL_2:54;
A4: R misses P2 by Th48;
thus Int R = (Cl P2``)` by Th47
.= (P2 \/ P)` by A1,A2,A3,JORDAN1:44
.= P2` /\ P` by XBOOLE_1:53
.= R /\ P` by Th47
.= R /\ (P1 \/ P2) by A1,A2,A3,JORDAN1:36
.= R /\ P1 \/ R /\ P2 by XBOOLE_1:23
.= R /\ P1 \/ {} by A4
.= P1 by Th49;
end;
theorem Th51:
a <= b & c <= d implies
closed_inside_of_rectangle(a,b,c,d) \ inside_of_rectangle(a,b,c,d)
= rectangle(a,b,c,d)
proof
assume that
A1: a <= b and
A2: c <= d;
set R = rectangle(a,b,c,d);
set P = closed_inside_of_rectangle(a,b,c,d);
set P1 = inside_of_rectangle(a,b,c,d);
A3: R = {p where p is Point of T2: 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} by A1,A2,SPPOL_2:54;
thus P \ P1 c= R
proof
let x be object;
assume
A4: x in P \ P1;
then
A5: not x in P1 by XBOOLE_0:def 5;
x in P by A4,XBOOLE_0:def 5;
then consider p such that
A6: x = p and
A7: a <= p`1 and
A8: p`1 <= b and
A9: c <= p`2 and
A10: p`2 <= d;
not (a < p`1 & p`1 < b & c < p`2 & p`2 < d) by A5,A6;
then 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 by A7,A8,A9,A10,XXREAL_0:1;
hence thesis by A3,A6;
end;
let x be object;
assume
A11: x in R;
then
A12: ex p st p = x & (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) by A3;
A13: R c= P by A1,A2,Th45;
now
assume x in P1;
then ex p st x = p & a < p`1 & p`1 < b & c < p`2 & p`2 < d;
hence contradiction by A12;
end;
hence thesis by A11,A13,XBOOLE_0:def 5;
end;
theorem Th52:
a < b & c < d implies
Fr closed_inside_of_rectangle(a,b,c,d) = rectangle(a,b,c,d)
proof
assume that
A1: a < b and
A2: c < d;
set P = closed_inside_of_rectangle(a,b,c,d);
thus Fr P = P \ Int P by TOPS_1:43
.= P \ inside_of_rectangle(a,b,c,d) by A1,A2,Th50
.= rectangle(a,b,c,d) by A1,A2,Th51;
end;
theorem
a <= b & c <= d implies W-bound closed_inside_of_rectangle(a,b,c,d) = a
proof
assume that
A1: a <= b and
A2: c <= d;
set X = closed_inside_of_rectangle(a,b,c,d);
reconsider Z = (proj1|X).:the carrier of (T2|X) as Subset of REAL;
A3: X = the carrier of (T2|X) by PRE_TOPC:8;
A4: |[a,c]| in X by A1,A2,TOPREALA:31;
A5: 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
A6: p0 in the carrier of T2|X and p0 in the carrier of T2|X and
A7: p = (proj1|X).p0 by FUNCT_2:64;
ex p1 st p0 = p1 & a <= p1`1 & p1`1 <= b & c <= p1`2 & p1`2 <= d by A3,A6;
hence thesis by A3,A6,A7,PSCOMP_1:22;
end;
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;
A9: |[a,c]|`1 = a by EUCLID:52;
(proj1|X). |[a,c]| = |[a,c]|`1 by A1,A2,PSCOMP_1:22,TOPREALA:31;
hence thesis by A3,A4,A8,A9,FUNCT_2:35;
end;
hence thesis by A4,A5,SEQ_4:44;
end;
theorem
a <= b & c <= d implies S-bound closed_inside_of_rectangle(a,b,c,d) = c
proof
assume that
A1: a <= b and
A2: c <= d;
set X = closed_inside_of_rectangle(a,b,c,d);
reconsider Z = (proj2|X).:the carrier of (T2|X) as Subset of REAL;
A3: X = the carrier of (T2|X) by PRE_TOPC:8;
A4: |[a,c]| in X by A1,A2,TOPREALA:31;
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 T2|X and p0 in the carrier of T2|X and
A7: p = (proj2|X).p0 by FUNCT_2:64;
ex p1 st p0 = p1 & a <= p1`1 & p1`1 <= b & c <= p1`2 & p1`2 <= d by A3,A6;
hence thesis by A3,A6,A7,PSCOMP_1:23;
end;
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;
A9: |[a,c]|`2 = c by EUCLID:52;
(proj2|X). |[a,c]| = |[a,c]|`2 by A1,A2,PSCOMP_1:23,TOPREALA:31;
hence thesis by A3,A4,A8,A9,FUNCT_2:35;
end;
hence thesis by A4,A5,SEQ_4:44;
end;
theorem
a <= b & c <= d implies E-bound closed_inside_of_rectangle(a,b,c,d) = b
proof
assume that
A1: a <= b and
A2: c <= d;
set X = closed_inside_of_rectangle(a,b,c,d);
reconsider Z = (proj1|X).:the carrier of (T2|X) as Subset of REAL;
A3: X = the carrier of (T2|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 T2|X and p0 in the carrier of T2|X and
A6: p = (proj1|X).p0 by FUNCT_2:64;
ex p1 st p0 = p1 & a <= p1`1 & p1`1 <= b & c <= p1`2 & p1`2 <= d by A3,A5;
hence thesis by 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 being Real st p in Z holds p <= q;
A9: |[b,d]|`1 = b by EUCLID:52;
|[b,d]|`2 = d by EUCLID:52;
then
A10: |[b,d]| in X by A1,A2,A9;
then (proj1|X). |[b,d]| = |[b,d]|`1 by PSCOMP_1:22;
hence thesis by A3,A8,A9,A10,FUNCT_2:35;
end;
|[a,c]| in X by A1,A2,TOPREALA:31;
hence thesis by A4,A7,SEQ_4:46;
end;
theorem
a <= b & c <= d implies N-bound closed_inside_of_rectangle(a,b,c,d) = d
proof
assume that
A1: a <= b and
A2: c <= d;
set X = closed_inside_of_rectangle(a,b,c,d);
reconsider Z = (proj2|X).:the carrier of (T2|X) as Subset of REAL;
A3: X = the carrier of (T2|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 T2|X and p0 in the carrier of T2|X and
A6: p = (proj2|X).p0 by FUNCT_2:64;
ex p1 st p0 = p1 & a <= p1`1 & p1`1 <= b & c <= p1`2 & p1`2 <= d by A3,A5;
hence thesis by 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 being Real st p in Z holds p <= q;
A9: |[b,d]|`1 = b by EUCLID:52;
A10: |[b,d]|`2 = d by EUCLID:52;
then
A11: |[b,d]| in X by A1,A2,A9;
then (proj2|X). |[b,d]| = |[b,d]|`2 by PSCOMP_1:23;
hence thesis by A3,A8,A10,A11,FUNCT_2:35;
end;
|[a,c]| in X by A1,A2,TOPREALA:31;
hence thesis by A4,A7,SEQ_4:46;
end;
theorem Th57:
a < b & c < d & p1 in closed_inside_of_rectangle(a,b,c,d) &
not p2 in closed_inside_of_rectangle(a,b,c,d) & P is_an_arc_of p1,p2 implies
Segment(P,p1,p2,p1,First_Point(P,p1,p2,rectangle(a,b,c,d))) c=
closed_inside_of_rectangle(a,b,c,d)
proof
set R = closed_inside_of_rectangle(a,b,c,d);
set dR = rectangle(a,b,c,d);
set n = First_Point(P,p1,p2,dR);
assume that
A1: a < b and
A2: c < d and
A3: p1 in R and
A4: not p2 in R and
A5: P is_an_arc_of p1,p2;
let x be object;
assume that
A6: x in Segment(P,p1,p2,p1,n) and
A7: not x in R;
reconsider x as Point of T2 by A6;
A8: Fr R = dR by A1,A2,Th52;
p1 in P by A5,TOPREAL1:1;
then
A9: P meets R by A3,XBOOLE_0:3;
p2 in P by A5,TOPREAL1:1;
then P \ R <> {}T2 by A4,XBOOLE_0:def 5;
then
A10: P meets dR by A5,A8,A9,CONNSP_1:22,JORDAN6:10;
A11: P is closed by A5,JORDAN6:11;
then
A12: P /\ dR is closed;
A13: n in P /\ dR by A5,A10,A11,JORDAN5C:def 1;
per cases;
suppose x = n;
then
A14: x in dR by A13,XBOOLE_0:def 4;
dR c= R by A1,A2,Th45;
hence thesis by A7,A14;
end;
suppose
A15: x <> n;
reconsider P as non empty Subset of T2 by A5,TOPREAL1:1;
consider f being Function of I[01], T2|P such that
A16: f is being_homeomorphism and
A17: f.0 = p1 and
A18: f.1 = p2 by A5,TOPREAL1:def 1;
A19: rng f = [#](T2|P) by A16,TOPS_2:def 5
.= P by PRE_TOPC:def 5;
n in P by A13,XBOOLE_0:def 4;
then consider na being object such that
A20: na in dom f and
A21: f.na = n by A19,FUNCT_1:def 3;
reconsider na as Real by A20;
A22: 0 <= na by A20,BORSUK_1:43;
A23: na <= 1 by A20,BORSUK_1:43;
A24: Segment(P,p1,p2,p1,n) c= P by JORDAN16:2;
then consider xa being object such that
A25: xa in dom f and
A26: f.xa = x by A6,A19,FUNCT_1:def 3;
reconsider xa as Real by A25;
A27: 0 <= xa by A25,BORSUK_1:43;
A28: xa <= 1 by A25,BORSUK_1:43;
A29: Segment(P,p1,p2,p1,x) is_an_arc_of p1,x by A3,A5,A6,A7,A24,JORDAN16:24;
then p1 in Segment(P,p1,p2,p1,x) by TOPREAL1:1;
then
A30: Segment(P,p1,p2,p1,x) meets R by A3,XBOOLE_0:3;
x in Segment(P,p1,p2,p1,x) by A29,TOPREAL1:1;
then Segment(P,p1,p2,p1,x) \ R <> {}T2 by A7,XBOOLE_0:def 5;
then Segment(P,p1,p2,p1,x) meets Fr R by A29,A30,CONNSP_1:22,JORDAN6:10;
then consider z being object such that
A31: z in Segment(P,p1,p2,p1,x) and
A32: z in dR by A8,XBOOLE_0:3;
reconsider z as Point of T2 by A31;
Segment(P,p1,p2,p1,x) = {p: LE p1,p,P,p1,p2 & LE p,x,P,p1,p2}
by JORDAN6:26;
then
A33: ex zz being Point of T2 st ( zz = z)&( LE p1,zz,P,p1,p2)&(
LE zz,x,P,p1,p2) by A31;
Segment(P,p1,p2,p1,x) c= P by JORDAN16:2;
then consider za being object such that
A34: za in dom f and
A35: f.za = z by A19,A31,FUNCT_1:def 3;
reconsider za as Real by A34;
A36: 0 <= za by A34,BORSUK_1:43;
A37: za <= 1 by A34,BORSUK_1:43;
A38: na <= za by A5,A10,A12,A16,A17,A18,A21,A23,A32,A35,A36,JORDAN5C:def 1;
A39: za <= xa by A16,A17,A18,A26,A27,A28,A33,A35,A37,JORDAN5C:def 3;
Segment(P,p1,p2,p1,n) = {p: LE p1,p,P,p1,p2 & LE p,n,P,p1,p2}
by JORDAN6:26;
then ex xx being Point of T2 st ( xx = x)&( LE p1,xx,P,p1,p2)&(
LE xx,n,P,p1,p2) by A6;
then xa <= na by A16,A17,A18,A21,A22,A23,A26,A28,JORDAN5C:def 3;
then xa < na by A15,A21,A26,XXREAL_0:1;
hence thesis by A38,A39,XXREAL_0:2;
end;
end;
begin :: Some useful functions
definition
let S, T be non empty TopSpace, x be Point of [:S,T:];
redefine func x`1 -> Element of S; :: redefines `1 (XTUPLE_0, definition)
coherence
proof
the carrier of [:S,T:] = [:the carrier of S, the carrier of T:]
by BORSUK_1:def 2;
hence thesis by MCART_1:10;
end;
redefine func x`2 -> Element of T; :: redefines `2 (XTUPLE_0, definition)
coherence
proof
the carrier of [:S,T:] = [:the carrier of S, the carrier of T:]
by BORSUK_1:def 2;
hence thesis by MCART_1:10;
end;
end;
definition
let o be Point of TOP-REAL 2;
func diffX2_1(o) -> RealMap of [:TOP-REAL 2,TOP-REAL 2:] means
:Def1:
for x being Point of [:TOP-REAL 2,TOP-REAL 2:] holds it.x = x`2`1 - o`1;
existence
proof
deffunc F(Point of [:T2,T2:]) = In($1`2`1 - o`1,REAL);
consider xo being RealMap of [:T2,T2:] such that
A1: for x being Point of [:T2,T2:] holds xo.x = F(x) from FUNCT_2:sch 4;
take xo;
let x be Point of [:TOP-REAL 2,TOP-REAL 2:];
xo.x = F(x) by A1;
hence thesis;
end;
uniqueness
proof
let f, g be RealMap of [:T2,T2:] such that
A2: for x being Point of [:T2,T2:] holds f.x = x`2`1 - o`1 and
A3: for x being Point of [:T2,T2:] holds g.x = x`2`1 - o`1;
now
let x be Point of [:T2,T2:];
thus f.x = x`2`1 - o`1 by A2
.= g.x by A3;
end;
hence thesis by FUNCT_2:63;
end;
func diffX2_2(o) -> RealMap of [:TOP-REAL 2,TOP-REAL 2:] means
:Def2:
for x being Point of [:TOP-REAL 2,TOP-REAL 2:] holds it.x = x`2`2 - o`2;
existence
proof
deffunc F(Point of [:T2,T2:]) = In($1`2`2 - o`2,REAL);
consider xo being RealMap of [:T2,T2:] such that
A4: for x being Point of [:T2,T2:] holds xo.x = F(x) from FUNCT_2:sch 4;
take xo;
let x be Point of [:TOP-REAL 2,TOP-REAL 2:];
xo.x = F(x) by A4;
hence thesis;
end;
uniqueness
proof
let f, g be RealMap of [:T2,T2:] such that
A5: for x being Point of [:T2,T2:] holds f.x = x`2`2 - o`2 and
A6: for x being Point of [:T2,T2:] holds g.x = x`2`2 - o`2;
now
let x be Point of [:T2,T2:];
thus f.x = x`2`2 - o`2 by A5
.= g.x by A6;
end;
hence thesis by FUNCT_2:63;
end;
end;
definition
func diffX1_X2_1 -> RealMap of [:TOP-REAL 2, TOP-REAL 2:] means
:Def3:
for x being Point of [:TOP-REAL 2, TOP-REAL 2:] holds it.x = x`1`1 - x`2`1;
existence
proof
deffunc F(Point of [:T2,T2:]) = In($1`1`1 - $1`2`1,REAL);
consider xo being RealMap of [:T2,T2:] such that
A1: for x being Point of [:T2,T2:] holds xo.x = F(x) from FUNCT_2:sch 4;
take xo;
let x be Point of [:TOP-REAL 2, TOP-REAL 2:];
xo.x = F(x) by A1;
hence thesis;
end;
uniqueness
proof
let f, g be RealMap of [:T2,T2:] such that
A2: for x being Point of [:T2,T2:] holds f.x = x`1`1 - x`2`1 and
A3: for x being Point of [:T2,T2:] holds g.x = x`1`1 - x`2`1;
now
let x be Point of [:T2,T2:];
thus f.x = x`1`1 - x`2`1 by A2
.= g.x by A3;
end;
hence thesis by FUNCT_2:63;
end;
func diffX1_X2_2 -> RealMap of [:TOP-REAL 2, TOP-REAL 2:] means
:Def4:
for x being Point of [:TOP-REAL 2, TOP-REAL 2:] holds it.x = x`1`2 - x`2`2;
existence
proof
deffunc F(Point of [:T2,T2:]) = In($1`1`2 - $1`2`2,REAL);
consider xo being RealMap of [:T2,T2:] such that
A4: for x being Point of [:T2,T2:] holds xo.x = F(x) from FUNCT_2:sch 4;
take xo;
let x be Point of [:TOP-REAL 2, TOP-REAL 2:];
xo.x = F(x) by A4;
hence thesis;
end;
uniqueness
proof
let f, g be RealMap of [:T2,T2:] such that
A5: for x being Point of [:T2,T2:] holds f.x = x`1`2 - x`2`2 and
A6: for x being Point of [:T2,T2:] holds g.x = x`1`2 - x`2`2;
now
let x be Point of [:T2,T2:];
thus f.x = x`1`2 - x`2`2 by A5
.= g.x by A6;
end;
hence thesis by FUNCT_2:63;
end;
func Proj2_1 -> RealMap of [:TOP-REAL 2, TOP-REAL 2:] means
:Def5:
for x being Point of [:TOP-REAL 2, TOP-REAL 2:] holds it.x = x`2`1;
existence
proof
deffunc F(Point of [:T2,T2:]) = In($1`2`1,REAL);
consider xo being RealMap of [:T2,T2:] such that
A7: for x being Point of [:T2,T2:] holds xo.x = F(x) from FUNCT_2:sch 4;
take xo;
let x be Point of [:TOP-REAL 2, TOP-REAL 2:];
xo.x = F(x) by A7;
hence thesis;
end;
uniqueness
proof
let f, g be RealMap of [:T2,T2:] such that
A8: for x being Point of [:T2,T2:] holds f.x = x`2`1 and
A9: for x being Point of [:T2,T2:] holds g.x = x`2`1;
now
let x be Point of [:T2,T2:];
thus f.x = x`2`1 by A8
.= g.x by A9;
end;
hence thesis by FUNCT_2:63;
end;
func Proj2_2 -> RealMap of [:TOP-REAL 2, TOP-REAL 2:] means
:Def6:
for x being Point of [:TOP-REAL 2, TOP-REAL 2:] holds it.x = x`2`2;
existence
proof
deffunc F(Point of [:T2,T2:]) = In($1`2`2,REAL);
consider xo being RealMap of [:T2,T2:] such that
A10: for x being Point of [:T2,T2:] holds xo.x = F(x) from FUNCT_2:sch 4;
take xo;
let x be Point of [:TOP-REAL 2, TOP-REAL 2:];
xo.x = F(x) by A10;
hence thesis;
end;
uniqueness
proof
let f, g be RealMap of [:T2,T2:] such that
A11: for x being Point of [:T2,T2:] holds f.x = x`2`2 and
A12: for x being Point of [:T2,T2:] holds g.x = x`2`2;
now
let x be Point of [:T2,T2:];
thus f.x = x`2`2 by A11
.= g.x by A12;
end;
hence thesis by FUNCT_2:63;
end;
end;
theorem Th58:
for o being Point of TOP-REAL 2 holds
diffX2_1(o) is continuous Function of [:TOP-REAL 2, TOP-REAL 2:], R^1
proof
let o be Point of TOP-REAL 2;
reconsider Xo = diffX2_1(o) as Function of [:T2,T2:],R^1 by TOPMETR:17;
for p being Point of [:T2,T2:], V being Subset of R^1
st Xo.p in V & V is open holds
ex W being Subset of [:T2,T2:] st p in W & W is open & Xo.:W c= V
proof
let p be Point of [:T2,T2:], V be Subset of R^1 such that
A1: Xo.p in V and
A2: V is open;
A3: Xo.p = p`2`1 - o`1 by Def1;
set r = p`2`1 - o`1;
reconsider V1 = V as open Subset of REAL by A2,BORSUK_5:39,TOPMETR:17;
consider g being Real such that
A4: 0 < g and
A5: ].r-g,r+g.[ c= V1 by A1,A3,RCOMP_1:19;
reconsider g as Element of REAL by XREAL_0:def 1;
set W2 = {|[x,y]| where x, y is Real:
p`2`1-g < x & x < p`2`1+g};
W2 c= the carrier of T2
proof
let a be object;
assume a in W2;
then ex x, y being Real st
a = |[x,y]| & p`2`1-g < x & x < p`2`1+g;
hence thesis;
end;
then reconsider W2 as Subset of T2;
take [:[#]T2,W2:];
A6: p`2 = |[p`2`1,p`2`2]| by EUCLID:53;
A7: p = [p`1,p`2] by Lm5,MCART_1:21;
A8: p`2`1-g < p`2`1-0 by A4,XREAL_1:15;
p`2`1+0 < p`2`1+g by A4,XREAL_1:6;
then p`2 in W2 by A6,A8;
hence p in [:[#]T2,W2:] by A7,ZFMISC_1:def 2;
W2 is open by PSCOMP_1:19;
hence [:[#]T2,W2:] is open by BORSUK_1:6;
let b be object;
assume b in Xo.:[:[#]T2,W2:];
then consider a being Point of [:T2,T2:] such that
A9: a in [:[#]T2,W2:] and
A10: Xo.a = b by FUNCT_2:65;
A11: a = [a`1,a`2] by Lm5,MCART_1:21;
A12: (diffX2_1(o)).a = a`2`1 - o`1 by Def1;
a`2 in W2 by A9,A11,ZFMISC_1:87;
then consider x2, y2 being Real such that
A13: a`2 = |[x2,y2]| and
A14: p`2`1-g < x2 and
A15: x2 < p`2`1+g;
A16: a`2`1 = x2 by A13,EUCLID:52;
then
A17: p`2`1 - g - o`1 < a`2`1 - o`1 by A14,XREAL_1:9;
a`2`1 - o`1 < p`2`1 + g - o`1 by A15,A16,XREAL_1:9;
then a`2`1 - o`1 in ].r-g,r+g.[ by A17,XXREAL_1:4;
hence thesis by A5,A10,A12;
end;
hence thesis by JGRAPH_2:10;
end;
theorem Th59:
for o being Point of TOP-REAL 2 holds
diffX2_2(o) is continuous Function of [:TOP-REAL 2, TOP-REAL 2:], R^1
proof
let o be Point of TOP-REAL 2;
reconsider Yo = diffX2_2(o) as Function of [:T2,T2:],R^1 by TOPMETR:17;
for p being Point of [:T2,T2:], V being Subset of R^1
st Yo.p in V & V is open holds
ex W being Subset of [:T2,T2:] st p in W & W is open & Yo.:W c= V
proof
let p be Point of [:T2,T2:], V be Subset of R^1 such that
A1: Yo.p in V and
A2: V is open;
A3: p = [p`1,p`2] by Lm5,MCART_1:21;
A4: Yo.p = p`2`2 - o`2 by Def2;
set r = p`2`2 - o`2;
reconsider V1 = V as open Subset of REAL by A2,BORSUK_5:39,TOPMETR:17;
consider g being Real such that
A5: 0 < g and
A6: ].r-g,r+g.[ c= V1 by A1,A4,RCOMP_1:19;
reconsider g as Element of REAL by XREAL_0:def 1;
set W2 = {|[x,y]| where x, y is Real:
p`2`2-g < y & y < p`2`2+g};
W2 c= the carrier of T2
proof
let a be object;
assume a in W2;
then ex x, y being Real st
a = |[x,y]| & p`2`2-g < y & y < p`2`2+g;
hence thesis;
end;
then reconsider W2 as Subset of T2;
take [:[#]T2,W2:];
A7: p`2 = |[p`2`1,p`2`2]| by EUCLID:53;
A8: p`2`2-g < p`2`2-0 by A5,XREAL_1:15;
p`2`2+0 < p`2`2+g by A5,XREAL_1:6;
then p`2 in W2 by A7,A8;
hence p in [:[#]T2,W2:] by A3,ZFMISC_1:def 2;
W2 is open by PSCOMP_1:21;
hence [:[#]T2,W2:] is open by BORSUK_1:6;
let b be object;
assume b in Yo.:[:[#]T2,W2:];
then consider a being Point of [:T2,T2:] such that
A9: a in [:[#]T2,W2:] and
A10: Yo.a = b by FUNCT_2:65;
A11: a = [a`1,a`2] by Lm5,MCART_1:21;
A12: (diffX2_2(o)).a = a`2`2 - o`2 by Def2;
a`2 in W2 by A9,A11,ZFMISC_1:87;
then consider x2, y2 being Real such that
A13: a`2 = |[x2,y2]| and
A14: p`2`2-g < y2 and
A15: y2 < p`2`2+g;
A16: a`2`2 = y2 by A13,EUCLID:52;
then
A17: p`2`2 - g - o`2 < a`2`2 - o`2 by A14,XREAL_1:9;
a`2`2 - o`2 < p`2`2 + g - o`2 by A15,A16,XREAL_1:9;
then a`2`2 - o`2 in ].r-g,r+g.[ by A17,XXREAL_1:4;
hence thesis by A6,A10,A12;
end;
hence thesis by JGRAPH_2:10;
end;
theorem Th60:
diffX1_X2_1 is continuous Function of [:TOP-REAL 2, TOP-REAL 2:], R^1
proof
reconsider Dx = diffX1_X2_1 as Function of [:T2,T2:],R^1 by TOPMETR:17;
for p being Point of [:T2,T2:], V being Subset of R^1
st Dx.p in V & V is open holds
ex W being Subset of [:T2,T2:] st p in W & W is open & Dx.:W c= V
proof
let p be Point of [:T2,T2:], V be Subset of R^1 such that
A1: Dx.p in V and
A2: V is open;
A3: p = [p`1,p`2] by Lm5,MCART_1:21;
A4: diffX1_X2_1.p = p`1`1 - p`2`1 by Def3;
set r = p`1`1 - p`2`1;
reconsider V1 = V as open Subset of REAL by A2,BORSUK_5:39,TOPMETR:17;
consider g being Real such that
A5: 0 < g and
A6: ].r-g,r+g.[ c= V1 by A1,A4,RCOMP_1:19;
reconsider g as Element of REAL by XREAL_0:def 1;
set W1 = {|[x,y]| where x, y is Real:
p`1`1-g/2 < x & x < p`1`1+g/2};
set W2 = {|[x,y]| where x, y is Real:
p`2`1-g/2 < x & x < p`2`1+g/2};
W1 c= the carrier of T2
proof
let a be object;
assume a in W1;
then ex x, y being Real st
a = |[x,y]| & p`1`1-g/2 < x & x < p`1`1+g/2;
hence thesis;
end;
then reconsider W1 as Subset of T2;
W2 c= the carrier of T2
proof
let a be object;
assume a in W2;
then ex x, y being Real st
a = |[x,y]| & p`2`1-g/2 < x & x < p`2`1+g/2;
hence thesis;
end;
then reconsider W2 as Subset of T2;
take [:W1,W2:];
A7: p`1 = |[p`1`1,p`1`2]| by EUCLID:53;
A8: 0/2 < g/2 by A5,XREAL_1:74;
then
A9: p`1`1-g/2 < p`1`1-0 by XREAL_1:15;
p`1`1+0 < p`1`1+g/2 by A8,XREAL_1:6;
then
A10: p`1 in W1 by A7,A9;
A11: p`2 = |[p`2`1,p`2`2]| by EUCLID:53;
A12: p`2`1-g/2 < p`2`1-0 by A8,XREAL_1:15;
p`2`1+0 < p`2`1+g/2 by A8,XREAL_1:6;
then p`2 in W2 by A11,A12;
hence p in [:W1,W2:] by A3,A10,ZFMISC_1:def 2;
A13: W1 is open by PSCOMP_1:19;
W2 is open by PSCOMP_1:19;
hence [:W1,W2:] is open by A13,BORSUK_1:6;
let b be object;
assume b in Dx.:[:W1,W2:];
then consider a being Point of [:T2,T2:] such that
A14: a in [:W1,W2:] and
A15: Dx.a = b by FUNCT_2:65;
A16: a = [a`1,a`2] by Lm5,MCART_1:21;
A17: diffX1_X2_1.a = a`1`1 - a`2`1 by Def3;
a`1 in W1 by A14,A16,ZFMISC_1:87;
then consider x1, y1 being Real such that
A18: a`1 = |[x1,y1]| and
A19: p`1`1-g/2 < x1 and
A20: x1 < p`1`1+g/2;
A21: a`1`1 = x1 by A18,EUCLID:52;
A22: p`1`1-g/2+g/2 < x1+g/2 by A19,XREAL_1:6;
A23: p`1`1-x1 > p`1`1-(p`1`1+g/2) by A20,XREAL_1:15;
A24: p`1`1-x1 < x1+g/2-x1 by A22,XREAL_1:9;
p`1`1-x1 > -g/2 by A23;
then
A25: |.p`1`1-x1.| < g/2 by A24,SEQ_2:1;
a`2 in W2 by A14,A16,ZFMISC_1:87;
then consider x2, y2 being Real such that
A26: a`2 = |[x2,y2]| and
A27: p`2`1-g/2 < x2 and
A28: x2 < p`2`1+g/2;
A29: a`2`1 = x2 by A26,EUCLID:52;
A30: p`2`1-g/2+g/2 < x2+g/2 by A27,XREAL_1:6;
A31: p`2`1-x2 > p`2`1-(p`2`1+g/2) by A28,XREAL_1:15;
A32: p`2`1-x2 < x2+g/2-x2 by A30,XREAL_1:9;
p`2`1-x2 > -g/2 by A31;
then |.p`2`1-x2.| < g/2 by A32,SEQ_2:1;
then
A33: |.p`1`1-x1.|+|.p`2`1-x2.| < g/2+g/2 by A25,XREAL_1:8;
|.p`1`1-x1-(p`2`1-x2).| <= |.p`1`1-x1.|+|.p`2`1-x2.| by COMPLEX1:57;
then |.-(p`1`1-x1-(p`2`1-x2)).| <= |.p`1`1-x1.|+|.p`2`1-x2.|
by COMPLEX1:52;
then |.x1-x2-r.| < g by A33,XXREAL_0:2;
then a`1`1 - a`2`1 in ].r-g,r+g.[ by A21,A29,RCOMP_1:1;
hence thesis by A6,A15,A17;
end;
hence thesis by JGRAPH_2:10;
end;
theorem Th61:
diffX1_X2_2 is continuous Function of [:TOP-REAL 2, TOP-REAL 2:], R^1
proof
reconsider Dy = diffX1_X2_2 as Function of [:T2,T2:],R^1 by TOPMETR:17;
for p being Point of [:T2,T2:], V being Subset of R^1
st Dy.p in V & V is open holds
ex W being Subset of [:T2,T2:] st p in W & W is open & Dy.:W c= V
proof
let p be Point of [:T2,T2:], V be Subset of R^1 such that
A1: Dy.p in V and
A2: V is open;
A3: p = [p`1,p`2] by Lm5,MCART_1:21;
A4: diffX1_X2_2.p = p`1`2 - p`2`2 by Def4;
set r = p`1`2 - p`2`2;
reconsider V1 = V as open Subset of REAL by A2,BORSUK_5:39,TOPMETR:17;
consider g being Real such that
A5: 0 < g and
A6: ].r-g,r+g.[ c= V1 by A1,A4,RCOMP_1:19;
reconsider g as Element of REAL by XREAL_0:def 1;
set W1 = {|[x,y]| where x, y is Real:
p`1`2-g/2 < y & y < p`1`2+g/2};
set W2 = {|[x,y]| where x, y is Real:
p`2`2-g/2 < y & y < p`2`2+g/2};
W1 c= the carrier of T2
proof
let a be object;
assume a in W1;
then ex x, y being Real st
a = |[x,y]| & p`1`2-g/2 < y & y < p`1`2+g/2;
hence thesis;
end;
then reconsider W1 as Subset of T2;
W2 c= the carrier of T2
proof
let a be object;
assume a in W2;
then ex x, y being Real st
a = |[x,y]| & p`2`2-g/2 < y & y < p`2`2+g/2;
hence thesis;
end;
then reconsider W2 as Subset of T2;
take [:W1,W2:];
A7: p`1 = |[p`1`1,p`1`2]| by EUCLID:53;
A8: 0/2 < g/2 by A5,XREAL_1:74;
then
A9: p`1`2-g/2 < p`1`2-0 by XREAL_1:15;
p`1`2+0 < p`1`2+g/2 by A8,XREAL_1:6;
then
A10: p`1 in W1 by A7,A9;
A11: p`2 = |[p`2`1,p`2`2]| by EUCLID:53;
A12: p`2`2-g/2 < p`2`2-0 by A8,XREAL_1:15;
p`2`2+0 < p`2`2+g/2 by A8,XREAL_1:6;
then p`2 in W2 by A11,A12;
hence p in [:W1,W2:] by A3,A10,ZFMISC_1:def 2;
A13: W1 is open by PSCOMP_1:21;
W2 is open by PSCOMP_1:21;
hence [:W1,W2:] is open by A13,BORSUK_1:6;
let b be object;
assume b in Dy.:[:W1,W2:];
then consider a being Point of [:T2,T2:] such that
A14: a in [:W1,W2:] and
A15: Dy.a = b by FUNCT_2:65;
A16: a = [a`1,a`2] by Lm5,MCART_1:21;
A17: diffX1_X2_2.a = a`1`2 - a`2`2 by Def4;
a`1 in W1 by A14,A16,ZFMISC_1:87;
then consider x1, y1 being Real such that
A18: a`1 = |[x1,y1]| and
A19: p`1`2-g/2 < y1 and
A20: y1 < p`1`2+g/2;
A21: a`1`2 = y1 by A18,EUCLID:52;
A22: p`1`2-g/2+g/2 < y1+g/2 by A19,XREAL_1:6;
A23: p`1`2-y1 > p`1`2-(p`1`2+g/2) by A20,XREAL_1:15;
A24: p`1`2-y1 < y1+g/2-y1 by A22,XREAL_1:9;
p`1`2-y1 > -g/2 by A23;
then
A25: |.p`1`2-y1.| < g/2 by A24,SEQ_2:1;
a`2 in W2 by A14,A16,ZFMISC_1:87;
then consider x2, y2 being Real such that
A26: a`2 = |[x2,y2]| and
A27: p`2`2-g/2 < y2 and
A28: y2 < p`2`2+g/2;
A29: a`2`2 = y2 by A26,EUCLID:52;
A30: p`2`2-g/2+g/2 < y2+g/2 by A27,XREAL_1:6;
A31: p`2`2-y2 > p`2`2-(p`2`2+g/2) by A28,XREAL_1:15;
A32: p`2`2-y2 < y2+g/2-y2 by A30,XREAL_1:9;
p`2`2-y2 > -g/2 by A31;
then |.p`2`2-y2.| < g/2 by A32,SEQ_2:1;
then
A33: |.p`1`2-y1.|+|.p`2`2-y2.| < g/2+g/2 by A25,XREAL_1:8;
|.p`1`2-y1-(p`2`2-y2).| <= |.p`1`2-y1.|+|.p`2`2-y2.| by COMPLEX1:57;
then |.-(p`1`2-y1-(p`2`2-y2)).| <= |.p`1`2-y1.|+|.p`2`2-y2.|
by COMPLEX1:52;
then |.y1-y2-r.| < g by A33,XXREAL_0:2;
then a`1`2 - a`2`2 in ].r-g,r+g.[ by A21,A29,RCOMP_1:1;
hence thesis by A6,A15,A17;
end;
hence thesis by JGRAPH_2:10;
end;
theorem Th62:
Proj2_1 is continuous Function of [:TOP-REAL 2, TOP-REAL 2:], R^1
proof
reconsider fX2 = Proj2_1 as Function of [:T2,T2:],R^1 by TOPMETR:17;
for p being Point of [:T2,T2:], V being Subset of R^1
st fX2.p in V & V is open holds
ex W being Subset of [:T2,T2:] st p in W & W is open & fX2.:W c= V
proof
let p be Point of [:T2,T2:], V be Subset of R^1 such that
A1: fX2.p in V and
A2: V is open;
A3: p = [p`1,p`2] by Lm5,MCART_1:21;
A4: fX2.p = p`2`1 by Def5;
reconsider V1 = V as open Subset of REAL by A2,BORSUK_5:39,TOPMETR:17;
consider g being Real such that
A5: 0 < g and
A6: ].p`2`1-g,p`2`1+g.[ c= V1 by A1,A4,RCOMP_1:19;
reconsider g as Element of REAL by XREAL_0:def 1;
set W1 = {|[x,y]| where x, y is Real:
p`2`1-g < x & x < p`2`1+g};
W1 c= the carrier of T2
proof
let a be object;
assume a in W1;
then ex x, y being Real st
a = |[x,y]| & p`2`1-g < x & x < p`2`1+g;
hence thesis;
end;
then reconsider W1 as Subset of T2;
take [:[#]T2,W1:];
A7: p`2 = |[p`2`1,p`2`2]| by EUCLID:53;
A8: p`2`1-g < p`2`1-0 by A5,XREAL_1:15;
p`2`1+0 < p`2`1+g by A5,XREAL_1:6;
then p`2 in W1 by A7,A8;
hence p in [:[#]T2,W1:] by A3,ZFMISC_1:def 2;
W1 is open by PSCOMP_1:19;
hence [:[#]T2,W1:] is open by BORSUK_1:6;
let b be object;
assume b in fX2.:[:[#]T2,W1:];
then consider a being Point of [:T2,T2:] such that
A9: a in [:[#]T2,W1:] and
A10: fX2.a = b by FUNCT_2:65;
A11: a = [a`1,a`2] by Lm5,MCART_1:21;
A12: fX2.a = a`2`1 by Def5;
a`2 in W1 by A9,A11,ZFMISC_1:87;
then consider x1, y1 being Real such that
A13: a`2 = |[x1,y1]| and
A14: p`2`1-g < x1 and
A15: x1 < p`2`1+g;
A16: a`2`1 = x1 by A13,EUCLID:52;
A17: p`2`1-g+g < x1+g by A14,XREAL_1:6;
A18: p`2`1-x1 > p`2`1-(p`2`1+g) by A15,XREAL_1:15;
A19: p`2`1-x1 < x1+g-x1 by A17,XREAL_1:9;
p`2`1-x1 > -g by A18;
then |.p`2`1-x1.| < g by A19,SEQ_2:1;
then |.-(p`2`1-x1).| < g by COMPLEX1:52;
then |.x1-p`2`1.| < g;
then a`2`1 in ].p`2`1-g,p`2`1+g.[ by A16,RCOMP_1:1;
hence thesis by A6,A10,A12;
end;
hence thesis by JGRAPH_2:10;
end;
theorem Th63:
Proj2_2 is continuous Function of [:TOP-REAL 2, TOP-REAL 2:], R^1
proof
reconsider fY2 = Proj2_2 as Function of [:T2,T2:],R^1 by TOPMETR:17;
for p being Point of [:T2,T2:], V being Subset of R^1
st fY2.p in V & V is open holds
ex W being Subset of [:T2,T2:] st p in W & W is open & fY2.:W c= V
proof
let p be Point of [:T2,T2:], V be Subset of R^1 such that
A1: fY2.p in V and
A2: V is open;
A3: p = [p`1,p`2] by Lm5,MCART_1:21;
A4: fY2.p = p`2`2 by Def6;
reconsider V1 = V as open Subset of REAL by A2,BORSUK_5:39,TOPMETR:17;
consider g being Real such that
A5: 0 < g and
A6: ].p`2`2-g,p`2`2+g.[ c= V1 by A1,A4,RCOMP_1:19;
reconsider g as Element of REAL by XREAL_0:def 1;
set W1 = {|[x,y]| where x, y is Real:
p`2`2-g < y & y < p`2`2+g};
W1 c= the carrier of T2
proof
let a be object;
assume a in W1;
then ex x, y being Real st
a = |[x,y]| & p`2`2-g < y & y < p`2`2+g;
hence thesis;
end;
then reconsider W1 as Subset of T2;
take [:[#]T2,W1:];
A7: p`2 = |[p`2`1,p`2`2]| by EUCLID:53;
A8: p`2`2-g < p`2`2-0 by A5,XREAL_1:15;
p`2`2+0 < p`2`2+g by A5,XREAL_1:6;
then p`2 in W1 by A7,A8;
hence p in [:[#]T2,W1:] by A3,ZFMISC_1:def 2;
W1 is open by PSCOMP_1:21;
hence [:[#]T2,W1:] is open by BORSUK_1:6;
let b be object;
assume b in fY2.:[:[#]T2,W1:];
then consider a being Point of [:T2,T2:] such that
A9: a in [:[#]T2,W1:] and
A10: fY2.a = b by FUNCT_2:65;
A11: a = [a`1,a`2] by Lm5,MCART_1:21;
A12: fY2.a = a`2`2 by Def6;
a`2 in W1 by A9,A11,ZFMISC_1:87;
then consider x1, y1 being Real such that
A13: a`2 = |[x1,y1]| and
A14: p`2`2-g < y1 and
A15: y1 < p`2`2+g;
A16: a`2`2 = y1 by A13,EUCLID:52;
A17: p`2`2-g+g < y1+g by A14,XREAL_1:6;
A18: p`2`2-y1 > p`2`2-(p`2`2+g) by A15,XREAL_1:15;
A19: p`2`2-y1 < y1+g-y1 by A17,XREAL_1:9;
p`2`2-y1 > -g by A18;
then |.p`2`2-y1.| < g by A19,SEQ_2:1;
then |.-(p`2`2-y1).| < g by COMPLEX1:52;
then |.y1-p`2`2.| < g;
then a`2`2 in ].p`2`2-g,p`2`2+g.[ by A16,RCOMP_1:1;
hence thesis by A6,A10,A12;
end;
hence thesis by JGRAPH_2:10;
end;
registration
let o be Point of TOP-REAL 2;
cluster diffX2_1(o) -> continuous;
coherence
proof
diffX2_1(o) is continuous Function of [:T2,T2:],R^1 by Th58;
hence thesis by JORDAN5A:27;
end;
cluster diffX2_2(o) -> continuous;
coherence
proof
diffX2_2(o) is continuous Function of [:T2,T2:],R^1 by Th59;
hence thesis by JORDAN5A:27;
end;
end;
registration
cluster diffX1_X2_1 -> continuous;
coherence by Th60,JORDAN5A:27;
cluster diffX1_X2_2 -> continuous;
coherence by Th61,JORDAN5A:27;
cluster Proj2_1 -> continuous;
coherence by Th62,JORDAN5A:27;
cluster Proj2_2 -> continuous;
coherence by Th63,JORDAN5A:27;
end;
definition
let n be non zero Element of NAT, o, p be Point of TOP-REAL n,
r be positive Real such that
A1: p is Point of Tdisk(o,r);
set X = (TOP-REAL n)|(cl_Ball(o,r)\{p});
func DiskProj(o,r,p) -> Function
of (TOP-REAL n)|(cl_Ball(o,r)\{p}), Tcircle(o,r) means
:Def7:
for x being Point of (TOP-REAL n)|(cl_Ball(o,r)\{p})
ex y being Point of TOP-REAL n st x = y & it.x = HC(p,y,o,r);
existence
proof
A2: the carrier of X = cl_Ball(o,r)\{p} by PRE_TOPC:8;
defpred P[object,object] means
ex z being Point of TOP-REAL n st $1 = z & $2 = HC(p,z,o,r);
A3: for x being object st x in the carrier of X
ex y being object st y in the carrier of Tcircle(o,r) & P[x,y]
proof
let x be object such that
A4: x in the carrier of X;
reconsider z = x as Point of TOP-REAL n by A4,PRE_TOPC:25;
z in cl_Ball(o,r) by A2,A4,XBOOLE_0:def 5;
then
A5: z is Point of Tdisk(o,r) by BROUWER:3;
p <> z by A2,A4,ZFMISC_1:56;
then HC(p,z,o,r) is Point of Tcircle(o,r) by A1,A5,BROUWER:6;
hence thesis;
end;
consider f being Function of the carrier of X, the carrier of Tcircle(o,r)
such that
A6: for x being object st x in the carrier of X holds P[x,f.x]
from FUNCT_2:sch 1(A3);
reconsider f as Function of X, Tcircle(o,r);
take f;
let x be Point of X;
thus thesis by A6;
end;
uniqueness
proof
let f, g be Function of X, Tcircle(o,r) such that
A7: for x being Point of X
ex y being Point of TOP-REAL n st x = y & f.x = HC(p,y,o,r) and
A8: for x being Point of X
ex y being Point of TOP-REAL n st x = y & g.x = HC(p,y,o,r);
now
let x be object such that
A9: x in the carrier of X;
A10: ex y being Point of TOP-REAL n st x = y & f.x = HC(p,y,o,r) by A7,A9;
ex y being Point of TOP-REAL n st x = y & g.x = HC(p,y,o,r) by A8,A9;
hence f.x = g.x by A10;
end;
hence thesis by FUNCT_2:12;
end;
end;
theorem Th64:
for o, p being Point of TOP-REAL 2,
r being positive Real st p is Point of Tdisk(o,r) holds
DiskProj(o,r,p) is continuous
proof
let o, p be Point of TOP-REAL 2;
let r be positive Real such that
A1: p is Point of Tdisk(o,r);
set D = Tdisk(o,r);
set cB = cl_Ball(o,r);
set Bp = cB \ {p};
set OK = [:Bp,{p}:];
set D1 = T2|Bp;
set D2 = T2|{p};
set S1 = Tcircle(o,r);
A2: p in {p} by TARSKI:def 1;
A3: the carrier of D = cl_Ball(o,r) by BROUWER:3;
A4: the carrier of D1 = Bp by PRE_TOPC:8;
A5: the carrier of D2 = {p} by PRE_TOPC:8;
set TD = [:T2,T2:] | OK;
set gg = DiskProj(o,r,p);
set xo = diffX2_1(o);
set yo = diffX2_2(o);
set dx = diffX1_X2_1;
set dy = diffX1_X2_2;
set fx2 = Proj2_1;
set fy2 = Proj2_2;
reconsider rr = r^2 as Element of REAL by XREAL_0:def 1;
set f1 = (the carrier of [:T2,T2:]) --> rr;
reconsider f1 as continuous RealMap of [:T2,T2:] by Lm6;
set Zf1 = f1 | OK;
set Zfx2 = fx2 | OK;
set Zfy2 = fy2 | OK;
set Zdx = dx | OK;
set Zdy = dy | OK;
set Zxo = xo | OK;
set Zyo = yo | OK;
set xx = Zxo(#)Zdx;
set yy = Zyo(#)Zdy;
set m = Zdx(#)Zdx + Zdy(#)Zdy;
A6: the carrier of TD = OK by PRE_TOPC:8;
A7: for y being Point of D1, z being Point of D2 holds Zdx. [y,z] = dx. [y,z]
proof
let y be Point of D1;
let z be Point of D2;
[y,z] in OK by A4,A5,ZFMISC_1:def 2;
hence thesis by FUNCT_1:49;
end;
A8: for y being Point of D1, z being Point of D2 holds Zdy. [y,z] = dy. [y,z]
proof
let y be Point of D1;
let z be Point of D2;
[y,z] in OK by A4,A5,ZFMISC_1:def 2;
hence thesis by FUNCT_1:49;
end;
A9: for
y being Point of D1, z being Point of D2 holds Zfx2. [y,z] = fx2. [y,z]
proof
let y be Point of D1;
let z be Point of D2;
[y,z] in OK by A4,A5,ZFMISC_1:def 2;
hence thesis by FUNCT_1:49;
end;
A10: for
y being Point of D1, z being Point of D2 holds Zfy2. [y,z] = fy2. [y,z]
proof
let y be Point of D1;
let z be Point of D2;
[y,z] in OK by A4,A5,ZFMISC_1:def 2;
hence thesis by FUNCT_1:49;
end;
A11: for
y being Point of D1, z being Point of D2 holds Zf1. [y,z] = f1. [y,z]
proof
let y be Point of D1;
let z be Point of D2;
[y,z] in OK by A4,A5,ZFMISC_1:def 2;
hence thesis by FUNCT_1:49;
end;
A12: for
y being Point of D1, z being Point of D2 holds Zxo. [y,z] = xo. [y,z]
proof
let y be Point of D1;
let z be Point of D2;
[y,z] in OK by A4,A5,ZFMISC_1:def 2;
hence thesis by FUNCT_1:49;
end;
A13: for
y being Point of D1, z being Point of D2 holds Zyo. [y,z] = yo. [y,z]
proof
let y be Point of D1;
let z be Point of D2;
[y,z] in OK by A4,A5,ZFMISC_1:def 2;
hence thesis by FUNCT_1:49;
end;
now
let b be Real;
assume b in rng m;
then consider a being object such that
A14: a in dom m and
A15: m.a = b by FUNCT_1:def 3;
consider y, z being object such that
A16: y in Bp and
A17: z in {p} and
A18: a = [y,z] by A14,ZFMISC_1:def 2;
A19: z = p by A17,TARSKI:def 1;
reconsider y, z as Point of T2 by A16,A17;
A20: y <> z by A16,A19,ZFMISC_1:56;
A21: dx. [y,z] = [y,z]`1`1 - [y,z]`2`1 by Def3;
A22: dy. [y,z] = [y,z]`1`2 - [y,z]`2`2 by Def4;
set r1 = y`1-z`1;
set r2 = y`2-z`2;
A23: Zdx. [y,z] = dx. [y,z] by A4,A5,A7,A16,A17;
A24: Zdy. [y,z] = dy. [y,z] by A4,A5,A8,A16,A17;
dom m c= the carrier of TD by RELAT_1:def 18;
then
a in the carrier of TD by A14;
then
A25: m. [y,z] = (Zdx(#)Zdx). [y,z] + (Zdy(#)Zdy). [y,z] by A18,VALUED_1:1
.= Zdx. [y,z] * Zdx. [y,z] + (Zdy(#)Zdy). [y,z] by VALUED_1:5
.= r1^2+r2^2 by A21,A22,A23,A24,VALUED_1:5;
now
assume
A26: r1^2+r2^2 = 0;
then
A27: r1 = 0 by COMPLEX1:1;
r2 = 0 by A26,COMPLEX1:1;
hence contradiction by A20,A27,TOPREAL3:6;
end;
hence 0 < b by A15,A18,A25;
end;
then reconsider m as positive-yielding continuous RealMap of TD
by PARTFUN3:def 1;
set p1 = (xx+yy)(#)(xx+yy);
set p2 = Zxo(#)Zxo + Zyo(#)Zyo - Zf1;
A28: dom p2 = the carrier of TD by FUNCT_2:def 1;
now
let b be Real;
assume b in rng p2;
then consider a being object such that
A29: a in dom p2 and
A30: p2.a = b by FUNCT_1:def 3;
consider y, z being object such that
A31: y in Bp and
A32: z in {p} and
A33: a = [y,z] by A29,ZFMISC_1:def 2;
reconsider y, z as Point of T2 by A31,A32;
set r3 = z`1-o`1, r4 = z`2-o`2;
A34: Zf1. [y,z] = f1. [y,z] by A4,A5,A11,A31,A32;
A35: Zxo. [y,z] = xo. [y,z] by A4,A5,A12,A31,A32;
A36: Zyo. [y,z] = yo. [y,z] by A4,A5,A13,A31,A32;
A37: xo. [y,z] = [y,z]`2`1 - o`1 by Def1;
A38: yo. [y,z] = [y,z]`2`2 - o`2 by Def2;
dom p2 c= the carrier of TD by RELAT_1:def 18;
then
A39: a in the carrier of TD by A29;
A40: p2. [y,z] = (Zxo(#)Zxo + Zyo(#)Zyo). [y,z] - Zf1. [y,z]
by A29,A33,VALUED_1:13
.= (Zxo(#)Zxo + Zyo(#)Zyo). [y,z] - r^2 by A34,FUNCOP_1:7
.= (Zxo(#)Zxo). [y,z] + (Zyo(#)Zyo). [y,z] - r^2
by A33,A39,VALUED_1:1
.= Zxo. [y,z] * Zxo. [y,z] + (Zyo(#)Zyo). [y,z] - r^2 by VALUED_1:5
.= r3^2+r4^2-r^2 by A35,A36,A37,A38,VALUED_1:5;
z = p by A32,TARSKI:def 1;
then |. z-o .| <= r by A1,A3,TOPREAL9:8;
then
A41: |. z-o .|^2 <= r^2 by SQUARE_1:15;
|. z-o .|^2 = ((z-o)`1)^2+((z-o)`2)^2 by JGRAPH_1:29
.= r3^2+((z-o)`2)^2 by TOPREAL3:3
.= r3^2+r4^2 by TOPREAL3:3;
then r3^2+r4^2-r^2 <= r^2-r^2 by A41,XREAL_1:9;
hence 0 >= b by A30,A33,A40;
end;
then reconsider p2 as nonpositive-yielding continuous RealMap of TD
by PARTFUN3:def 3;
set pp = p1 - m(#)p2;
set k = (-(xx+yy) + sqrt(pp)) / m;
set x3 = Zfx2 + k(#)Zdx;
set y3 = Zfy2 + k(#)Zdy;
reconsider X3 = x3, Y3 = y3 as Function of TD,R^1 by TOPMETR:17;
set F = <:X3,Y3:>;
set R = R2Homeomorphism;
A42: for x being Point of D1 holds gg.x = (R*F). [x,p]
proof
let x be Point of D1;
consider y being Point of T2 such that
A43: x = y and
A44: gg.x = HC(p,y,o,r) by A1,Def7;
A45: x <> p by A4,ZFMISC_1:56;
A46: [y,p] in OK by A2,A4,A43,ZFMISC_1:def 2;
set r1 = y`1-p`1, r2 = y`2-p`2, r3 = p`1-o`1, r4 = p`2-o`2;
set l = (-(r3*r1+r4*r2)+sqrt((r3*r1+r4*r2)^2-(r1^2+r2^2)*(r3^2+r4^2-r^2)))
/ (r1^2+r2^2);
A47: fx2. [y,p] = [y,p]`2`1 by Def5;
A48: fy2. [y,p] = [y,p]`2`2 by Def6;
A49: dx. [y,p] = [y,p]`1`1 - [y,p]`2`1 by Def3;
A50: dy. [y,p] = [y,p]`1`2 - [y,p]`2`2 by Def4;
A51: xo. [y,p] = [y,p]`2`1 - o`1 by Def1;
A52: yo. [y,p] = [y,p]`2`2 - o`2 by Def2;
A53: dom X3 = the carrier of TD by FUNCT_2:def 1;
A54: dom Y3 = the carrier of TD by FUNCT_2:def 1;
A55: dom pp = the carrier of TD by FUNCT_2:def 1;
A56: p is Point of D2 by A5,TARSKI:def 1;
then
A57: Zdx. [y,p] = dx. [y,p] by A7,A43;
A58: Zdy. [y,p] = dy. [y,p] by A8,A43,A56;
A59: Zf1. [y,p] = f1. [y,p] by A11,A43,A56;
A60: Zxo. [y,p] = xo. [y,p] by A12,A43,A56;
A61: Zyo. [y,p] = yo. [y,p] by A13,A43,A56;
A62: m. [y,p] = (Zdx(#)Zdx). [y,p] + (Zdy(#)Zdy). [y,p] by A6,A46,VALUED_1:1
.= Zdx. [y,p] * Zdx. [y,p] + (Zdy(#)Zdy). [y,p] by VALUED_1:5
.= r1^2+r2^2 by A49,A50,A57,A58,VALUED_1:5;
A63: xx. [y,p] = Zxo. [y,p] * Zdx. [y,p] by VALUED_1:5;
A64: yy. [y,p] = Zyo. [y,p] * Zdy. [y,p] by VALUED_1:5;
A65: (xx+yy). [y,p] = xx. [y,p] + yy. [y,p] by A6,A46,VALUED_1:1;
then
A66: p1. [y,p] = (r3*r1+r4*r2)^2
by A49,A50,A51,A52,A57,A58,A60,A61,A63,A64,VALUED_1:5;
A67: p2. [y,p] = (Zxo(#)Zxo + Zyo(#)Zyo). [y,p] - Zf1. [y,p]
by A6,A28,A46,VALUED_1:13
.= (Zxo(#)Zxo + Zyo(#)Zyo). [y,p] - r^2 by A59,FUNCOP_1:7
.= (Zxo(#)Zxo). [y,p] + (Zyo(#)Zyo). [y,p] - r^2 by A6,A46,VALUED_1:1
.= Zxo. [y,p] * Zxo. [y,p] + (Zyo(#)Zyo). [y,p] - r^2 by VALUED_1:5
.= r3^2+r4^2-r^2 by A51,A52,A60,A61,VALUED_1:5;
dom sqrt pp = the carrier of TD by FUNCT_2:def 1;
then
A68: sqrt(pp). [y,p] = sqrt(pp. [y,p]) by A6,A46,PARTFUN3:def 5
.= sqrt(p1. [y,p] - (m(#)p2). [y,p]) by A6,A46,A55,VALUED_1:13
.= sqrt((r3*r1+r4*r2)^2-(r1^2+r2^2)*(r3^2+r4^2-r^2))
by A62,A66,A67,VALUED_1:5;
dom k = the carrier of TD by FUNCT_2:def 1;
then
A69: k. [y,p] = (-(xx+yy) + sqrt(pp)). [y,p] * (m. [y,p])" by A6,A46,
RFUNCT_1:def 1
.= (-(xx+yy) + sqrt(pp)). [y,p] / m. [y,p] by XCMPLX_0:def 9
.= ((-(xx+yy)). [y,p] + sqrt(pp). [y,p]) / (r1^2+r2^2)
by A6,A46,A62,VALUED_1:1
.= l by A49,A50,A51,A52,A57,A58,A60,A61,A63,A64,A65,A68,VALUED_1:8;
A70: X3. [y,p] = Zfx2. [y,p] + (k(#)Zdx). [y,p] by A6,A46,VALUED_1:1
.= p`1 + (k(#)Zdx). [y,p] by A9,A43,A47,A56
.= p`1+l*r1 by A49,A57,A69,VALUED_1:5;
A71: Y3. [y,p] = Zfy2. [y,p] + (k(#)Zdy). [y,p] by A6,A46,VALUED_1:1
.= p`2 + (k(#)Zdy). [y,p] by A10,A43,A48,A56
.= p`2+l*r2 by A50,A58,A69,VALUED_1:5;
A72: y in Bp by A4,A43;
Bp c= cB by XBOOLE_1:36;
hence gg.x = |[ p`1+l*r1, p`2+l*r2 ]| by A1,A3,A43,A44,A45,A72,BROUWER:8
.= R. [X3. [y,p], Y3. [y,p]] by A70,A71,TOPREALA:def 2
.= R.(F. [y,p]) by A6,A46,A53,A54,FUNCT_3:49
.= (R*F). [x,p] by A6,A43,A46,FUNCT_2:15;
end;
A73: X3 is continuous by JORDAN5A:27;
Y3 is continuous by JORDAN5A:27;
then reconsider F as continuous Function of TD,[:R^1,R^1:]
by A73,YELLOW12:41;
for pp being Point of D1, V being Subset of S1
st gg.pp in V & V is open holds
ex W being Subset of D1 st pp in W & W is open & gg.:W c= V
proof
let pp be Point of D1, V be Subset of S1 such that
A74: gg.pp in V and
A75: V is open;
reconsider p1 = pp, fp = p as Point of T2 by PRE_TOPC:25;
A76: [pp,p] in OK by A2,A4,ZFMISC_1:def 2;
consider V1 being Subset of T2 such that
A77: V1 is open and
A78: V1 /\ [#]S1 = V by A75,TOPS_2:24;
A79: gg.pp = (R*F). [pp,p] by A42;
R" is being_homeomorphism by TOPREALA:34,TOPS_2:56;
then
A80: R" .:V1 is open by A77,TOPGRP_1:25;
A81: dom F = the carrier of [:T2,T2:] | OK by FUNCT_2:def 1;
A82: dom R = the carrier of [:R^1,R^1:] by FUNCT_2:def 1;
then
A83: rng F c= dom R;
then
A84: dom (R*F) = dom F by RELAT_1:27;
A85: rng R = [#]T2 by TOPREALA:34,TOPS_2:def 5;
A86: R"*(R*F) = R"*R*F by RELAT_1:36
.= id dom R*F by A85,TOPREALA:34,TOPS_2:52;
dom id dom R = dom R;
then
A87: dom (id dom R*F) = dom F by A83,RELAT_1:27;
for x being object st x in dom F holds (id dom R*F).x = F.x
proof
let x be object such that
A88: x in dom F;
A89: F.x in rng F by A88,FUNCT_1:def 3;
thus (id dom R*F).x = id dom R.(F.x) by A88,FUNCT_1:13
.= F.x by A82,A89,FUNCT_1:18;
end;
then
A90: id dom R*F = F by A87,FUNCT_1:2;
(R*F). [p1,fp] in V1 by A74,A78,A79,XBOOLE_0:def 4;
then R" .((R*F). [p1,fp]) in R" .:V1 by FUNCT_2:35;
then (R"*(R*F)). [p1,fp] in R" .:V1 by A6,A76,A81,A84,FUNCT_1:13;
then consider W being Subset of TD such that
A91: [p1,fp] in W and
A92: W is open and
A93: F.:W c= R" .:V1 by A6,A76,A80,A86,A90,JGRAPH_2:10;
consider WW being Subset of [:T2,T2:] such that
A94: WW is open and
A95: WW /\ [#]TD = W by A92,TOPS_2:24;
consider SF being Subset-Family of [:T2,T2:] such that
A96: WW = union SF and
A97: for e being set st e in SF
ex X1 being Subset of T2, Y1 being Subset of T2 st
e = [:X1,Y1:] & X1 is open & Y1 is open by A94,BORSUK_1:5;
[p1,fp] in WW by A91,A95,XBOOLE_0:def 4;
then consider Z being set such that
A98: [p1,fp] in Z and
A99: Z in SF by A96,TARSKI:def 4;
consider X1, Y1 being Subset of T2 such that
A100: Z = [:X1,Y1:] and
A101: X1 is open and Y1 is open by A97,A99;
set ZZ = Z /\ [#]TD;
reconsider XX = X1 /\ [#]D1 as open Subset of D1 by A101,TOPS_2:24;
take XX;
pp in X1 by A98,A100,ZFMISC_1:87;
hence pp in XX by XBOOLE_0:def 4;
thus XX is open;
let b be object;
assume b in gg.:XX;
then consider a being Point of D1 such that
A102: a in XX and
A103: b = gg.a by FUNCT_2:65;
reconsider a1 = a, fa = fp as Point of T2 by PRE_TOPC:25;
A104: a in X1 by A102,XBOOLE_0:def 4;
A105: [a,p] in OK by A2,A4,ZFMISC_1:def 2;
fa in Y1 by A98,A100,ZFMISC_1:87;
then [a,fa] in Z by A100,A104,ZFMISC_1:def 2;
then [a,fa] in ZZ by A6,A105,XBOOLE_0:def 4;
then
A106: F. [a1,fa] in F.:ZZ by FUNCT_2:35;
A107: R qua Function" = R" by TOPREALA:34,TOPS_2:def 4;
A108: dom(R") = [#]T2 by A85,TOPREALA:34,TOPS_2:49;
Z c= WW by A96,A99,ZFMISC_1:74;
then ZZ c= WW /\ [#]TD by XBOOLE_1:27;
then F.:ZZ c= F.:W by A95,RELAT_1:123;
then F. [a1,fa] in F.:W by A106;
then R.(F. [a1,fa]) in R.:(R" .:V1) by A93,FUNCT_2:35;
then (R*F). [a1,fa] in R.:(R" .:V1) by A6,A105,FUNCT_2:15;
then (R*F). [a1,fa] in V1 by A107,A108,PARTFUN3:1,TOPREALA:34;
then gg.a in V1 by A42;
hence thesis by A78,A103,XBOOLE_0:def 4;
end;
hence thesis by JGRAPH_2:10;
end;
theorem Th65:
for n being non zero Element of NAT, o, p being Point of TOP-REAL n,
r being positive Real st p in Ball(o,r) holds
DiskProj(o,r,p)|Sphere(o,r) = id Sphere(o,r)
proof
let n be non zero Element of NAT;
let o, p be Point of TOP-REAL n;
let r be positive Real;
assume
A1: p in Ball(o,r);
A2: the carrier of Tdisk(o,r) = cl_Ball(o,r) by BROUWER:3;
A3: the carrier of (TOP-REAL n)|(cl_Ball(o,r)\{p}) = cl_Ball(o,r)\{p}
by PRE_TOPC:8;
A4: dom DiskProj(o,r,p) = the carrier of (TOP-REAL n)|(cl_Ball(o,r)\{p})
by FUNCT_2:def 1;
A5: Sphere(o,r) misses Ball(o,r) by TOPREAL9:19;
A6: Sphere(o,r) c= cl_Ball(o,r) by TOPREAL9:17;
A7: Ball(o,r) c= cl_Ball(o,r) by TOPREAL9:16;
A8: Sphere(o,r) c= cl_Ball(o,r)\{p}
proof
let a be object;
assume
A9: a in Sphere(o,r);
then a <> p by A1,A5,XBOOLE_0:3;
hence thesis by A6,A9,ZFMISC_1:56;
end;
then
A10: dom(DiskProj(o,r,p)|Sphere(o,r)) = Sphere(o,r) by A3,A4,RELAT_1:62;
A11: dom id Sphere(o,r) = Sphere(o,r);
now
let x be object;
assume
A12: x in dom(DiskProj(o,r,p)|Sphere(o,r));
then x in dom DiskProj(o,r,p) by RELAT_1:57;
then consider y being Point of TOP-REAL n such that
A13: x = y and
A14: (DiskProj(o,r,p)).x = HC(p,y,o,r) by A1,A2,A7,Def7;
y in halfline(p,y) by TOPREAL9:28;
then
A15: x in halfline(p,y) /\ Sphere(o,r) by A12,A13,XBOOLE_0:def 4;
A16: x <> p by A1,A5,A12,XBOOLE_0:3;
thus (DiskProj(o,r,p)|Sphere(o,r)).x = (DiskProj(o,r,p)).x
by A12,FUNCT_1:47
.= x by A1,A2,A6,A7,A10,A12,A13,A14,A15,A16,BROUWER:def 3
.= (id Sphere(o,r)).x by A12,FUNCT_1:18;
end;
hence thesis by A3,A4,A8,A11,FUNCT_1:2,RELAT_1:62;
end;
definition
let n be non zero Element of NAT, o, p be Point of TOP-REAL n,
r be positive Real such that
A1: p in Ball(o,r);
set X = Tcircle(o,r);
func RotateCircle(o,r,p) -> Function of Tcircle(o,r), Tcircle(o,r) means
:Def8:
for x being Point of Tcircle(o,r)
ex y being Point of TOP-REAL n st x = y & it.x = HC(y,p,o,r);
existence
proof
A2: the carrier of X = Sphere(o,r) by TOPREALB:9;
defpred P[object,object] means
ex z being Point of TOP-REAL n st $1 = z & $2 = HC(z,p,o,r);
A3: for x being object st x in the carrier of X
ex y being object st y in the carrier of X & P[x,y]
proof
let x be object such that
A4: x in the carrier of X;
reconsider z = x as Point of TOP-REAL n by A4,PRE_TOPC:25;
Sphere(o,r) c= cl_Ball(o,r) by TOPREAL9:17;
then
A5: z is Point of Tdisk(o,r) by A2,A4,BROUWER:3;
Ball(o,r) c= cl_Ball(o,r) by TOPREAL9:16;
then
A6: p is Point of Tdisk(o,r) by A1,BROUWER:3;
Ball(o,r) misses Sphere(o,r) by TOPREAL9:19;
then p <> z by A1,A2,A4,XBOOLE_0:3;
then HC(z,p,o,r) is Point of X by A5,A6,BROUWER:6;
hence thesis;
end;
consider f being Function of the carrier of X, the carrier of X such that
A7: for x being object st x in the carrier of X holds P[x,f.x]
from FUNCT_2:sch 1(A3);
reconsider f as Function of X, X;
take f;
let x be Point of X;
thus thesis by A7;
end;
uniqueness
proof
let f, g be Function of X, X such that
A8: for x being Point of X
ex y being Point of TOP-REAL n st x = y & f.x = HC(y,p,o,r) and
A9: for x being Point of X
ex y being Point of TOP-REAL n st x = y & g.x = HC(y,p,o,r);
now
let x be object such that
A10: x in the carrier of X;
A11: ex y being Point of TOP-REAL n st x = y & f.x = HC(y,p,o,r) by A8,A10;
ex y being Point of TOP-REAL n st x = y & g.x = HC(y,p,o,r) by A9,A10;
hence f.x = g.x by A11;
end;
hence thesis by FUNCT_2:12;
end;
end;
theorem Th66:
for o, p being Point of TOP-REAL 2,
r being positive Real st p in Ball(o,r) holds
RotateCircle(o,r,p) is continuous
proof
let o, p be Point of TOP-REAL 2;
let r be positive Real such that
A1: p in Ball(o,r);
set D = Tdisk(o,r);
set cB = cl_Ball(o,r);
set Bp = Sphere(o,r);
set OK = [:{p},Bp:];
set D1 = T2|{p};
set D2 = T2|Bp;
set S1 = Tcircle(o,r);
A2: D2 = S1 by TOPREALB:def 6;
A3: Ball(o,r) misses Sphere(o,r) by TOPREAL9:19;
A4: p in {p} by TARSKI:def 1;
A5: Bp c= cB by TOPREAL9:17;
A6: Ball(o,r) c= cB by TOPREAL9:16;
A7: the carrier of D = cB by BROUWER:3;
A8: the carrier of D1 = {p} by PRE_TOPC:8;
A9: the carrier of D2 = Bp by PRE_TOPC:8;
set TD = [:T2,T2:] | OK;
set gg = RotateCircle(o,r,p);
set xo = diffX2_1(o);
set yo = diffX2_2(o);
set dx = diffX1_X2_1;
set dy = diffX1_X2_2;
set fx2 = Proj2_1;
set fy2 = Proj2_2;
reconsider rr = r^2 as Element of REAL by XREAL_0:def 1;
set f1 = (the carrier of [:T2,T2:]) --> rr;
reconsider f1 as continuous RealMap of [:T2,T2:] by Lm6;
set Zf1 = f1 | OK;
set Zfx2 = fx2 | OK;
set Zfy2 = fy2 | OK;
set Zdx = dx | OK;
set Zdy = dy | OK;
set Zxo = xo | OK;
set Zyo = yo | OK;
set xx = Zxo(#)Zdx;
set yy = Zyo(#)Zdy;
set m = Zdx(#)Zdx + Zdy(#)Zdy;
A10: the carrier of TD = OK by PRE_TOPC:8;
A11: for
y being Point of D1, z being Point of D2 holds Zdx. [y,z] = dx. [y,z]
proof
let y be Point of D1;
let z be Point of D2;
[y,z] in OK by A8,A9,ZFMISC_1:def 2;
hence thesis by FUNCT_1:49;
end;
A12: for
y being Point of D1, z being Point of D2 holds Zdy. [y,z] = dy. [y,z]
proof
let y be Point of D1;
let z be Point of D2;
[y,z] in OK by A8,A9,ZFMISC_1:def 2;
hence thesis by FUNCT_1:49;
end;
A13: for
y being Point of D1, z being Point of D2 holds Zfx2. [y,z] = fx2. [y,z]
proof
let y be Point of D1;
let z be Point of D2;
[y,z] in OK by A8,A9,ZFMISC_1:def 2;
hence thesis by FUNCT_1:49;
end;
A14: for
y being Point of D1, z being Point of D2 holds Zfy2. [y,z] = fy2. [y,z]
proof
let y be Point of D1;
let z be Point of D2;
[y,z] in OK by A8,A9,ZFMISC_1:def 2;
hence thesis by FUNCT_1:49;
end;
A15: for
y being Point of D1, z being Point of D2 holds Zf1. [y,z] = f1. [y,z]
proof
let y be Point of D1;
let z be Point of D2;
[y,z] in OK by A8,A9,ZFMISC_1:def 2;
hence thesis by FUNCT_1:49;
end;
A16: for
y being Point of D1, z being Point of D2 holds Zxo. [y,z] = xo. [y,z]
proof
let y be Point of D1;
let z be Point of D2;
[y,z] in OK by A8,A9,ZFMISC_1:def 2;
hence thesis by FUNCT_1:49;
end;
A17: for
y being Point of D1, z being Point of D2 holds Zyo. [y,z] = yo. [y,z]
proof
let y be Point of D1;
let z be Point of D2;
[y,z] in OK by A8,A9,ZFMISC_1:def 2;
hence thesis by FUNCT_1:49;
end;
now
let b be Real;
assume b in rng m;
then consider a being object such that
A18: a in dom m and
A19: m.a = b by FUNCT_1:def 3;
consider y, z being object such that
A20: y in {p} and
A21: z in Bp and
A22: a = [y,z] by A18,ZFMISC_1:def 2;
A23: y = p by A20,TARSKI:def 1;
reconsider y, z as Point of T2 by A20,A21;
A24: y <> z by A1,A3,A21,A23,XBOOLE_0:3;
A25: dx. [y,z] = [y,z]`1`1 - [y,z]`2`1 by Def3;
A26: dy. [y,z] = [y,z]`1`2 - [y,z]`2`2 by Def4;
set r1 = y`1-z`1;
set r2 = y`2-z`2;
A27: Zdx. [y,z] = dx. [y,z] by A8,A9,A11,A20,A21;
A28: Zdy. [y,z] = dy. [y,z] by A8,A9,A12,A20,A21;
dom m c= the carrier of TD by RELAT_1:def 18;
then
a in the carrier of TD by A18;
then
A29: m. [y,z] = (Zdx(#)Zdx). [y,z] + (Zdy(#)Zdy). [y,z] by A22,VALUED_1:1
.= Zdx. [y,z] * Zdx. [y,z] + (Zdy(#)Zdy). [y,z] by VALUED_1:5
.= r1^2+r2^2 by A25,A26,A27,A28,VALUED_1:5;
now
assume
A30: r1^2+r2^2 = 0;
then
A31: r1 = 0 by COMPLEX1:1;
r2 = 0 by A30,COMPLEX1:1;
hence contradiction by A24,A31,TOPREAL3:6;
end;
hence 0 < b by A19,A22,A29;
end;
then reconsider m as positive-yielding continuous RealMap of TD
by PARTFUN3:def 1;
set p1 = (xx+yy)(#)(xx+yy);
set p2 = Zxo(#)Zxo + Zyo(#)Zyo - Zf1;
A32: dom p2 = the carrier of TD by FUNCT_2:def 1;
now
let b be Real;
assume b in rng p2;
then consider a being object such that
A33: a in dom p2 and
A34: p2.a = b by FUNCT_1:def 3;
consider y, z being object such that
A35: y in {p} and
A36: z in Bp and
A37: a = [y,z] by A33,ZFMISC_1:def 2;
reconsider y, z as Point of T2 by A35,A36;
set r3 = z`1-o`1, r4 = z`2-o`2;
A38: Zf1. [y,z] = f1. [y,z] by A8,A9,A15,A35,A36;
A39: Zxo. [y,z] = xo. [y,z] by A8,A9,A16,A35,A36;
A40: Zyo. [y,z] = yo. [y,z] by A8,A9,A17,A35,A36;
A41: xo. [y,z] = [y,z]`2`1 - o`1 by Def1;
A42: yo. [y,z] = [y,z]`2`2 - o`2 by Def2;
dom p2 c= the carrier of TD by RELAT_1:def 18;
then
A43: a in the carrier of TD by A33;
A44: p2. [y,z] = (Zxo(#)Zxo + Zyo(#)Zyo). [y,z] - Zf1. [y,z]
by A33,A37,VALUED_1:13
.= (Zxo(#)Zxo + Zyo(#)Zyo). [y,z] - r^2 by A38,FUNCOP_1:7
.= (Zxo(#)Zxo). [y,z] + (Zyo(#)Zyo). [y,z] - r^2 by A37,A43,VALUED_1:1
.= Zxo. [y,z] * Zxo. [y,z] + (Zyo(#)Zyo). [y,z] - r^2 by VALUED_1:5
.= r3^2+r4^2-r^2 by A39,A40,A41,A42,VALUED_1:5;
|. z-o .| <= r by A5,A36,TOPREAL9:8;
then
A45: |. z-o .|^2 <= r^2 by SQUARE_1:15;
|. z-o .|^2 = ((z-o)`1)^2+((z-o)`2)^2 by JGRAPH_1:29
.= r3^2+((z-o)`2)^2 by TOPREAL3:3
.= r3^2+r4^2 by TOPREAL3:3;
then r3^2+r4^2-r^2 <= r^2-r^2 by A45,XREAL_1:9;
hence 0 >= b by A34,A37,A44;
end;
then reconsider p2 as nonpositive-yielding continuous RealMap of TD
by PARTFUN3:def 3;
set pp = p1 - m(#)p2;
set k = (-(xx+yy) + sqrt(pp)) / m;
set x3 = Zfx2 + k(#)Zdx;
set y3 = Zfy2 + k(#)Zdy;
reconsider X3 = x3, Y3 = y3 as Function of TD,R^1 by TOPMETR:17;
set F = <:X3,Y3:>;
set R = R2Homeomorphism;
A46: for x being Point of D2 holds gg.x = (R*F). [p,x]
proof
let x be Point of D2;
consider y being Point of T2 such that
A47: x = y and
A48: gg.x = HC(y,p,o,r) by A1,A2,Def8;
A49: x <> p by A1,A3,A9,XBOOLE_0:3;
A50: [p,y] in OK by A4,A9,A47,ZFMISC_1:def 2;
set r1 = p`1-y`1, r2 = p`2-y`2, r3 = y`1-o`1, r4 = y`2-o`2;
set l = (-(r3*r1+r4*r2)+sqrt((r3*r1+r4*r2)^2-(r1^2+r2^2)*(r3^2+r4^2-r^2)))
/ (r1^2+r2^2);
A51: fx2. [p,y] = [p,y]`2`1 by Def5;
A52: fy2. [p,y] = [p,y]`2`2 by Def6;
A53: dx. [p,y] = [p,y]`1`1 - [p,y]`2`1 by Def3;
A54: dy. [p,y] = [p,y]`1`2 - [p,y]`2`2 by Def4;
A55: xo. [p,y] = [p,y]`2`1 - o`1 by Def1;
A56: yo. [p,y] = [p,y]`2`2 - o`2 by Def2;
A57: dom X3 = the carrier of TD by FUNCT_2:def 1;
A58: dom Y3 = the carrier of TD by FUNCT_2:def 1;
A59: dom pp = the carrier of TD by FUNCT_2:def 1;
A60: p is Point of D1 by A8,TARSKI:def 1;
then
A61: Zdx. [p,y] = dx. [p,y] by A11,A47;
A62: Zdy. [p,y] = dy. [p,y] by A12,A47,A60;
A63: Zf1. [p,y] = f1. [p,y] by A15,A47,A60;
A64: Zxo. [p,y] = xo. [p,y] by A16,A47,A60;
A65: Zyo. [p,y] = yo. [p,y] by A17,A47,A60;
A66: m. [p,y] = (Zdx(#)Zdx). [p,y] + (Zdy(#)Zdy). [p,y] by A10,A50,VALUED_1:1
.= Zdx. [p,y] * Zdx. [p,y] + (Zdy(#)Zdy). [p,y] by VALUED_1:5
.= r1^2+r2^2 by A53,A54,A61,A62,VALUED_1:5;
A67: xx. [p,y] = Zxo. [p,y] * Zdx. [p,y] by VALUED_1:5;
A68: yy. [p,y] = Zyo. [p,y] * Zdy. [p,y] by VALUED_1:5;
A69: (xx+yy). [p,y] = xx. [p,y] + yy. [p,y] by A10,A50,VALUED_1:1;
then
A70: p1. [p,y] = (r3*r1+r4*r2)^2
by A53,A54,A55,A56,A61,A62,A64,A65,A67,A68,VALUED_1:5;
A71: p2. [p,y] = (Zxo(#)Zxo + Zyo(#)Zyo). [p,y] - Zf1. [p,y]
by A10,A32,A50,VALUED_1:13
.= (Zxo(#)Zxo + Zyo(#)Zyo). [p,y] - r^2 by A63,FUNCOP_1:7
.= (Zxo(#)Zxo). [p,y] + (Zyo(#)Zyo). [p,y] - r^2 by A10,A50,VALUED_1:1
.= Zxo. [p,y] * Zxo. [p,y] + (Zyo(#)Zyo). [p,y] - r^2 by VALUED_1:5
.= r3^2+r4^2-r^2 by A55,A56,A64,A65,VALUED_1:5;
dom sqrt pp = the carrier of TD by FUNCT_2:def 1;
then
A72: sqrt(pp). [p,y] = sqrt(pp. [p,y]) by A10,A50,PARTFUN3:def 5
.= sqrt(p1. [p,y] - (m(#)p2). [p,y]) by A10,A50,A59,VALUED_1:13
.= sqrt((r3*r1+r4*r2)^2-(r1^2+r2^2)*(r3^2+r4^2-r^2))
by A66,A70,A71,VALUED_1:5;
dom k = the carrier of TD by FUNCT_2:def 1;
then
A73: k. [p,y] = (-(xx+yy) + sqrt(pp)). [p,y] * (m. [p,y])" by A10,A50,
RFUNCT_1:def 1
.= (-(xx+yy) + sqrt(pp)). [p,y] / m. [p,y] by XCMPLX_0:def 9
.= ((-(xx+yy)). [p,y] + sqrt(pp). [p,y]) / (r1^2+r2^2)
by A10,A50,A66,VALUED_1:1
.= l by A53,A54,A55,A56,A61,A62,A64,A65,A67,A68,A69,A72,VALUED_1:8;
A74: X3. [p,y] = Zfx2. [p,y] + (k(#)Zdx). [p,y] by A10,A50,VALUED_1:1
.= y`1 + (k(#)Zdx). [p,y] by A13,A47,A51,A60
.= y`1+l*r1 by A53,A61,A73,VALUED_1:5;
A75: Y3. [p,y] = Zfy2. [p,y] + (k(#)Zdy). [p,y] by A10,A50,VALUED_1:1
.= y`2 + (k(#)Zdy). [p,y] by A14,A47,A52,A60
.= y`2+l*r2 by A54,A62,A73,VALUED_1:5;
y in the carrier of D2 by A47;
hence gg.x = |[ y`1+l*r1, y`2+l*r2 ]| by A1,A5,A6,A7,A9,A47,A48,A49,
BROUWER:8
.= R. [X3. [p,y], Y3. [p,y]] by A74,A75,TOPREALA:def 2
.= R.(F. [p,y]) by A10,A50,A57,A58,FUNCT_3:49
.= (R*F). [p,x] by A10,A47,A50,FUNCT_2:15;
end;
A76: X3 is continuous by JORDAN5A:27;
Y3 is continuous by JORDAN5A:27;
then reconsider F as continuous Function of TD,[:R^1,R^1:] by A76,YELLOW12:41
;
for pp being Point of D2, V being Subset of S1
st gg.pp in V & V is open holds
ex W being Subset of D2 st pp in W & W is open & gg.:W c= V
proof
let pp be Point of D2, V be Subset of S1 such that
A77: gg.pp in V and
A78: V is open;
reconsider p1 = pp, fp = p as Point of T2 by PRE_TOPC:25;
A79: [p,pp] in OK by A4,A9,ZFMISC_1:def 2;
consider V1 being Subset of T2 such that
A80: V1 is open and
A81: V1 /\ [#]S1 = V by A78,TOPS_2:24;
A82: gg.pp = (R*F). [p,pp] by A46;
R" is being_homeomorphism by TOPREALA:34,TOPS_2:56;
then
A83: R" .:V1 is open by A80,TOPGRP_1:25;
A84: dom F = the carrier of [:T2,T2:] | OK by FUNCT_2:def 1;
A85: dom R = the carrier of [:R^1,R^1:] by FUNCT_2:def 1;
then
A86: rng F c= dom R;
then
A87: dom (R*F) = dom F by RELAT_1:27;
A88: rng R = [#]T2 by TOPREALA:34,TOPS_2:def 5;
A89: R"*(R*F) = R"*R*F by RELAT_1:36
.= id dom R*F by A88,TOPREALA:34,TOPS_2:52;
dom id dom R = dom R;
then
A90: dom (id dom R*F) = dom F by A86,RELAT_1:27;
for x being object st x in dom F holds (id dom R*F).x = F.x
proof
let x be object such that
A91: x in dom F;
A92: F.x in rng F by A91,FUNCT_1:def 3;
thus (id dom R*F).x = id dom R.(F.x) by A91,FUNCT_1:13
.= F.x by A85,A92,FUNCT_1:18;
end;
then
A93: id dom R*F = F by A90,FUNCT_1:2;
(R*F). [fp,p1] in V1 by A77,A81,A82,XBOOLE_0:def 4;
then R" .((R*F). [fp,p1]) in R" .:V1 by FUNCT_2:35;
then (R"*(R*F)). [fp,p1] in R" .:V1 by A10,A79,A84,A87,FUNCT_1:13;
then consider W being Subset of TD such that
A94: [fp,p1] in W and
A95: W is open and
A96: F.:W c= R" .:V1 by A10,A79,A83,A89,A93,JGRAPH_2:10;
consider WW being Subset of [:T2,T2:] such that
A97: WW is open and
A98: WW /\ [#]TD = W by A95,TOPS_2:24;
consider SF being Subset-Family of [:T2,T2:] such that
A99: WW = union SF and
A100: for e being set st e in SF
ex X1 being Subset of T2, Y1 being Subset of T2 st
e = [:X1,Y1:] & X1 is open & Y1 is open by A97,BORSUK_1:5;
[fp,p1] in WW by A94,A98,XBOOLE_0:def 4;
then consider Z being set such that
A101: [fp,p1] in Z and
A102: Z in SF by A99,TARSKI:def 4;
consider X1, Y1 being Subset of T2 such that
A103: Z = [:X1,Y1:] and X1 is open and
A104: Y1 is open by A100,A102;
set ZZ = Z /\ [#]TD;
reconsider XX = Y1 /\ [#]D2 as open Subset of D2 by A104,TOPS_2:24;
take XX;
pp in Y1 by A101,A103,ZFMISC_1:87;
hence pp in XX by XBOOLE_0:def 4;
thus XX is open;
let b be object;
assume b in gg.:XX;
then consider a being Point of D2 such that
A105: a in XX and
A106: b = gg.a by A2,FUNCT_2:65;
reconsider a1 = a, fa = fp as Point of T2 by PRE_TOPC:25;
A107: a in Y1 by A105,XBOOLE_0:def 4;
A108: [p,a] in OK by A4,A9,ZFMISC_1:def 2;
fa in X1 by A101,A103,ZFMISC_1:87;
then [fa,a] in Z by A103,A107,ZFMISC_1:def 2;
then [fa,a] in ZZ by A10,A108,XBOOLE_0:def 4;
then
A109: F. [fa,a1] in F.:ZZ by FUNCT_2:35;
A110: R qua Function" = R" by TOPREALA:34,TOPS_2:def 4;
A111: dom(R") = [#]T2 by A88,TOPREALA:34,TOPS_2:49;
A112: gg.a1 in the carrier of S1 by A2,FUNCT_2:5;
Z c= WW by A99,A102,ZFMISC_1:74;
then ZZ c= WW /\ [#]TD by XBOOLE_1:27;
then F.:ZZ c= F.:W by A98,RELAT_1:123;
then F. [fa,a1] in F.:W by A109;
then R.(F. [fa,a1]) in R.:(R" .:V1) by A96,FUNCT_2:35;
then (R*F). [fa,a1] in R.:(R" .:V1) by A10,A108,FUNCT_2:15;
then (R*F). [fa,a1] in V1 by A110,A111,PARTFUN3:1,TOPREALA:34;
then gg.a in V1 by A46;
hence thesis by A81,A106,A112,XBOOLE_0:def 4;
end;
hence thesis by A2,JGRAPH_2:10;
end;
theorem Th67:
for n being non zero Element of NAT for o, p being Point of TOP-REAL n,
r being positive Real st p in Ball(o,r)
holds RotateCircle(o,r,p) is without_fixpoints
proof
let n be non zero Element of NAT;
let o, p be Point of TOP-REAL n;
let r be positive Real;
assume
A1: p in Ball(o,r);
set f = RotateCircle(o,r,p);
let x be object;
assume
A2: x in dom f;
set S = Tcircle(o,r);
A3: dom f = the carrier of S by FUNCT_2:def 1;
consider y being Point of TOP-REAL n such that
A4: x = y and
A5: f.x = HC(y,p,o,r) by A1,A2,Def8;
A6: the carrier of S = Sphere(o,r) by TOPREALB:9;
Sphere(o,r) c= cl_Ball(o,r) by TOPREAL9:17;
then
A7: y is Point of Tdisk(o,r) by A2,A3,A4,A6,BROUWER:3;
Ball(o,r) c= cl_Ball(o,r) by TOPREAL9:16;
then
A8: p is Point of Tdisk(o,r) by A1,BROUWER:3;
Ball(o,r) misses Sphere(o,r) by TOPREAL9:19;
then y <> p by A1,A2,A4,A6,XBOOLE_0:3;
hence thesis by A4,A5,A7,A8,BROUWER:def 3;
end;
begin :: Jordan's Curve Theorem
theorem Th68:
U = P & U is a_component &
V is a_component & U <> V implies Cl P misses V
proof
assume that
A1: U = P and
A2: U is a_component and
A3: V is a_component and
A4: U <> V;
assume Cl P meets V;
then
A5: ex x being object st x in Cl P & x in V by XBOOLE_0:3;
the carrier of T2|C` = C` by PRE_TOPC:8;
then reconsider V1 = V as Subset of T2 by XBOOLE_1:1;
reconsider T2C = T2|C` as non empty SubSpace of T2;
T2C is locally_connected by JORDAN2C:81;
then V is open by A3,CONNSP_2:15;
then V1 is open by TSEP_1:17;
then P meets V1 by A5,PRE_TOPC:def 7;
hence thesis by A1,A2,A3,A4,CONNSP_1:35;
end;
theorem Th69:
U is a_component implies
(TOP-REAL 2)|C`|U is pathwise_connected
proof
set T = T2|C`;
assume
A1: U is a_component;
let a, b be Point of T|U;
A2: the carrier of T|U = U by PRE_TOPC:8;
A3: U <> {}T by A1,CONNSP_1:32;
per cases;
suppose
A4: a = b;
reconsider TU = T|U as non empty TopSpace by A3;
reconsider a as Point of TU;
reconsider f = I[01] --> a as Function of I[01],T|U;
take f;
thus thesis by A4,BORSUK_1:def 14,def 15,TOPALG_3:4;
end;
suppose
A5: a <> b;
A6: T|U is SubSpace of T2 by TSEP_1:7;
then reconsider a1 = a, b1 = b as Point of T2 by A3,PRE_TOPC:25;
reconsider V = U as Subset of T2 by PRE_TOPC:11;
V is_a_component_of C` by A1;
then
A7: V is open by SPRECT_3:8;
U is connected by A1;
then V is connected by CONNSP_1:23;
then consider P being Subset of T2 such that
A8: P is_S-P_arc_joining a1,b1 and
A9: P c= V by A2,A3,A5,A7,TOPREAL4:29;
A10: a1 in P by A8,TOPREAL4:3;
P is_an_arc_of a1,b1 by A8,TOPREAL4:2;
then consider g being Function of I[01], T2|P such that
A11: g is being_homeomorphism and
A12: g.0 = a and
A13: g.1 = b by TOPREAL1:def 1;
A14: the carrier of T2|P = P by PRE_TOPC:8;
then reconsider f = g as Function of I[01], T|U by A2,A9,A10,FUNCT_2:7;
take f;
T2|P is SubSpace of T|U by A2,A6,A9,A14,TSEP_1:4;
hence f is continuous by A11,PRE_TOPC:26;
thus thesis by A12,A13;
end;
end;
::Th5_6: ex r ...
Lm12: for r being non negative Real st A is_an_arc_of p1,p2 &
A is Subset of Tdisk(p,r) ex f being Function
of Tdisk(p,r), (TOP-REAL 2)|A st f is continuous & f|A = id A
proof
let r be non negative Real;
set D = Tdisk(p,r);
assume that
A1: A is_an_arc_of p1,p2 and
A2: A is Subset of D;
reconsider A1 = A as non empty Subset of D by A1,A2,TOPREAL1:1;
reconsider A2 = A as non empty Subset of T2 by A1,TOPREAL1:1;
set TA = T2|A2;
consider h being Function of I[01],TA such that
A3: h is being_homeomorphism and h.0 = p1 and h.1 = p2 by A1,TOPREAL1:def 1;
A4: h1 is being_homeomorphism by TREAL_1:17;
reconsider hh = h as Function of C0,TA by TOPMETR:20;
A5: TA = D|A1 by TOPALG_5:4;
then reconsider f = h1*hh" as Function of D|A1,C1;
A is closed by A1,JORDAN6:11;
then
A6: A1 is closed by TSEP_1:12;
hh" is continuous by A3,TOPMETR:20,TOPS_2:def 5;
then consider g being continuous Function of D,C1 such that
A7: g|A1 = f by A4,A5,A6,TIETZE:23;
reconsider R = hh*h1"*g as Function of D,(TOP-REAL 2)|A;
take R;
h1" is continuous by A4,TOPS_2:def 5;
hence R is continuous by A3,TOPMETR:20;
A8: the carrier of TA = A1 by PRE_TOPC:8;
A9: dom R = the carrier of D by FUNCT_2:def 1;
A10: dom id A = A;
now
let a be object;
assume
A11: a in dom(R|A);
then
A12: a in dom R by RELAT_1:57;
A13: dom g = the carrier of D by FUNCT_2:def 1;
A14: dom(h1*hh") = the carrier of TA by FUNCT_2:def 1;
A15: hh*h1"*(h1*hh") = hh*h1"*h1*hh" by RELAT_1:36
.= hh*(h1"*h1)*hh" by RELAT_1:36
.= hh*id C0*hh" by A4,GRCAT_1:41
.= hh*hh" by FUNCT_2:17
.= id TA by A3,GRCAT_1:41;
thus (R|A).a = R.a by A11,FUNCT_1:49
.= (hh*h1").(g.a) by A13,A12,FUNCT_1:13
.= (hh*h1").((h1*hh").a) by A7,A11,FUNCT_1:49
.= (id A).a by A8,A11,A14,A15,FUNCT_1:13;
end;
hence thesis by A2,A9,A10,FUNCT_1:2,RELAT_1:62;
end;
::Th5_6: ex f ...
Lm13: for r being positive Real st A is_an_arc_of p1,p2 & A c= C &
C c= Ball(p,r) & p in U & Cl P /\ P` c= A & P c= Ball(p,r)
for f being Function of Tdisk(p,r), (TOP-REAL 2)|A st f is continuous &
f|A = id A &
:: komponenty
U = P & U is a_component & B = cl_Ball(p,r) \ {p}
ex g being Function of Tdisk(p,r), (TOP-REAL 2)|B st g is continuous &
for x being Point of Tdisk(p,r) holds (x in Cl P implies g.x = f.x) &
(x in P` implies g.x = x)
proof
let r be positive Real;
set D = Tdisk(p,r);
assume that
A1: A is_an_arc_of p1,p2 and
A2: A c= C and
A3: C c= Ball(p,r) and
A4: p in U and
A5: Cl P /\ P` c= A and
A6: P c= Ball(p,r);
let f be Function of D, T2|A;
assume that
A7: f is continuous and
A8: f|A = id A and
A9: U = P and
A10: U is a_component and
A11: B = cl_Ball(p,r) \ {p};
reconsider B1 = B as non empty Subset of T2 by A11;
reconsider T2B1 = T2|B1 as non empty SubSpace of T2;
A12: the carrier of T2|C` = C` by PRE_TOPC:8;
A13: the carrier of T2|A = A by PRE_TOPC:8;
A14: the carrier of D = cl_Ball(p,r) by BROUWER:3;
A15: Ball(p,r) c= cl_Ball(p,r) by TOPREAL9:16;
A16: A <> {} by A1,TOPREAL1:1;
reconsider A1 = A as non empty Subset of T2 by A1,TOPREAL1:1;
A17: not p in C by A4,A12,XBOOLE_0:def 5;
|. p-p .| = 0 by TOPRNS_1:28;
then
A18: p in [#]D by A14,TOPREAL9:8;
A19: P c= Cl P by PRE_TOPC:18;
then reconsider F1 = (Cl P) /\ [#]D as non empty Subset of D
by A4,A9,A18,XBOOLE_0:def 4;
A20: Sphere(p,r) c= cl_Ball(p,r) by TOPREAL9:17;
A21: Ball(p,r) misses Sphere(p,r) by TOPREAL9:19;
consider e being Point of T2 such that
A22: e in Sphere(p,r) by SUBSET_1:4;
not e in P by A6,A21,A22,XBOOLE_0:3;
then e in P` by SUBSET_1:29;
then reconsider F3 = P` /\ [#]D as non empty Subset of D
by A14,A20,A22,XBOOLE_0:def 4;
reconsider T1 = D|F1 as non empty SubSpace of D;
reconsider T3 = D|F3 as non empty SubSpace of D;
A23: the carrier of T1 = F1 by PRE_TOPC:8;
A24: the carrier of T3 = F3 by PRE_TOPC:8;
A25: the carrier of T2B1 = B1 by PRE_TOPC:8;
A26: A c= B
proof
let a be object;
assume a in A;
then
A27: a in C by A2;
then a in Ball(p,r) by A3;
hence thesis by A11,A15,A17,A27,ZFMISC_1:56;
end;
A28: F3 c= B
proof
let a be object;
assume
A29: a in F3;
then a in P` by XBOOLE_0:def 4;
then not a in P by XBOOLE_0:def 5;
hence thesis by A4,A9,A11,A14,A29,ZFMISC_1:56;
end;
f|F1 is Function of F1,A by A13,A16,FUNCT_2:32;
then reconsider f1 = f|F1 as Function
of T1,T2B1 by A16,A23,A25,A26,FUNCT_2:7;
reconsider g1 = id F3 as Function of T3,T2B1 by A24,A25,A28,FUNCT_2:7;
A30: F1 = [#]T1 by PRE_TOPC:8;
A31: F3 = [#]T3 by PRE_TOPC:8;
A32: [#]T1 \/ [#]T3 = [#]D
proof
thus [#]T1 \/ [#]T3 c= [#]D by A30,A31,XBOOLE_1:8;
let p be object;
assume
A33: p in [#]D;
per cases;
suppose p in P;
then p in F1 by A19,A33,XBOOLE_0:def 4;
hence thesis by A30,XBOOLE_0:def 3;
end;
suppose not p in P;
then p in P` by A14,A33,SUBSET_1:29;
then p in F3 by A33,XBOOLE_0:def 4;
hence thesis by A31,XBOOLE_0:def 3;
end;
end;
reconsider DT = [#]D as closed Subset of T2 by BORSUK_1:def 11,TSEP_1:1;
DT /\ Cl P is closed;
then
A34: F1 is closed by TSEP_1:8;
P is_a_component_of C`
by A9,A10;
then P is open by SPRECT_3:8;
then DT /\ P` is closed;
then
A35: F3 is closed by TSEP_1:8;
reconsider f2 = f|F1 as Function of T1,T2|A1 by A23,FUNCT_2:32;
A36: T2|A1 is SubSpace of T2B1 by A13,A25,A26,TSEP_1:4;
T3 is SubSpace of T2 by TSEP_1:7;
then
A37: T3 is SubSpace of T2B1 by A24,A25,A28,TSEP_1:4;
f2 is continuous by A7,TOPMETR:7;
then
A38: f1 is continuous by A36,PRE_TOPC:26;
reconsider g2 = id F3 as Function of T3,T3 by A24;
g2 = id T3 by PRE_TOPC:8;
then
A39: g1 is continuous by A37,PRE_TOPC:26;
A40: for x being set st x in Cl P & x in P` holds f.x = x
proof
let x be set;
assume that
A41: x in Cl P and
A42: x in P`;
A43: x in Cl P /\ P` by A41,A42,XBOOLE_0:def 4;
then (id A).x = x by A5,FUNCT_1:18;
hence thesis by A5,A8,A43,FUNCT_1:49;
end;
for x being object st x in [#]T1 /\ [#]T3 holds f1.x = g1.x
proof
let x be object;
assume
A44: x in [#]T1 /\ [#]T3;
then
A45: x in [#]T1 by XBOOLE_0:def 4;
then
A46: x in Cl P by A30,XBOOLE_0:def 4;
x in P` by A31,A44,XBOOLE_0:def 4;
then
A47: f.x = x by A40,A46;
thus f1.x = f.x by A30,A45,FUNCT_1:49
.= g1.x by A31,A44,A47,FUNCT_1:18;
end;
then consider g being Function of D,T2|B such that
A48: g = f1+*g1 and
A49: g is continuous by A30,A31,A32,A34,A35,A38,A39,JGRAPH_2:1;
take g;
thus g is continuous by A49;
let x be Point of D;
A50: dom g1 = the carrier of T3 by FUNCT_2:def 1;
hereby
assume
A51: x in Cl P;
then
A52: x in F1 by XBOOLE_0:def 4;
per cases;
suppose not x in dom g1;
hence g.x = f1.x by A48,FUNCT_4:11
.= f.x by A52,FUNCT_1:49;
end;
suppose
A53: x in dom g1;
then
A54: x in P` by XBOOLE_0:def 4;
thus g.x = g1.x by A48,A53,FUNCT_4:13
.= x by A53,FUNCT_1:18
.= f.x by A40,A51,A54;
end;
end;
assume x in P`;
then
A55: x in F3 by XBOOLE_0:def 4;
hence g.x = g1.x by A48,A50,FUNCT_4:13
.= x by A55,FUNCT_1:18;
end;
Lm14: for A being non empty Subset of T2 st U <> V
for r being positive Real st A c= C &
C c= Ball(p,r) & p in V & Cl P /\ P` c= A & Ball(p,r) meets P
for f being Function of Tdisk(p,r), (TOP-REAL 2)|A st f is continuous &
f|A = id A &
:: komponenty
U = P & U is a_component &
V is a_component & B = cl_Ball(p,r) \ {p}
ex g being Function of Tdisk(p,r), (TOP-REAL 2)|B st g is continuous &
for x being Point of Tdisk(p,r) holds (x in Cl P implies g.x = x) &
(x in P` implies g.x = f.x)
proof
let A be non empty Subset of T2 such that
A1: U <> V;
let r be positive Real;
set D = Tdisk(p,r);
assume that
A2: A c= C and
A3: C c= Ball(p,r) and
A4: p in V and
A5: Cl P /\ P` c= A and
A6: Ball(p,r) meets P;
let f be Function of D, T2|A;
assume that
A7: f is continuous and
A8: f|A = id A and
A9: U = P and
A10: U is a_component and
A11: V is a_component and
A12: B = cl_Ball(p,r) \ {p};
reconsider B1 = B as non empty Subset of T2 by A12;
reconsider T2B1 = T2|B1 as non empty SubSpace of T2;
A13: the carrier of T2|C` = C` by PRE_TOPC:8;
A14: the carrier of T2|A = A by PRE_TOPC:8;
A15: the carrier of D = cl_Ball(p,r) by BROUWER:3;
A16: Ball(p,r) c= cl_Ball(p,r) by TOPREAL9:16;
A17: not p in C by A4,A13,XBOOLE_0:def 5;
|. p-p .| = 0 by TOPRNS_1:28;
then
A18: p in [#]D by A15,TOPREAL9:8;
A19: P c= Cl P by PRE_TOPC:18;
ex j being object st j in Ball(p,r) & j in P by A6,XBOOLE_0:3;
then reconsider F1 = (Cl P) /\ [#]D as non empty Subset of D
by A15,A16,A19,XBOOLE_0:def 4;
not p in P by A1,A10,A11,CONNSP_1:35,A4,A9,XBOOLE_0:3;
then p in P` by SUBSET_1:29;
then reconsider F3 = P` /\ [#]D as non empty Subset of D
by A18,XBOOLE_0:def 4;
set T1 = D|F1;
set T3 = D|F3;
A20: the carrier of T1 = F1 by PRE_TOPC:8;
A21: the carrier of T3 = F3 by PRE_TOPC:8;
A22: the carrier of T2|B1 = B1 by PRE_TOPC:8;
A23: A c= B
proof
let a be object;
assume a in A;
then
A24: a in C by A2;
then a in Ball(p,r) by A3;
hence thesis by A12,A16,A17,A24,ZFMISC_1:56;
end;
A25: F1 c= B
proof
let a be object;
assume
A26: a in F1;
then
A27: a in Cl P by XBOOLE_0:def 4;
not p in Cl P by A4,XBOOLE_0:3,A1,A9,A10,A11,Th68;
hence thesis by A12,A15,A26,A27,ZFMISC_1:56;
end;
then reconsider f1 = id F1 as Function of T1,T2B1 by A20,A22,FUNCT_2:7;
f|F3 is Function of F3,A by A14;
then reconsider g1 = f|F3 as Function of T3,T2B1 by A21,A22,A23,FUNCT_2:7;
A28: F1 = [#]T1 by PRE_TOPC:8;
A29: F3 = [#]T3 by PRE_TOPC:8;
A30: [#]T1 \/ [#]T3 = [#]D
proof
thus [#]T1 \/ [#]T3 c= [#]D by A28,A29,XBOOLE_1:8;
let p be object;
assume
A31: p in [#]D;
per cases;
suppose p in P;
then p in F1 by A19,A31,XBOOLE_0:def 4;
hence thesis by A28,XBOOLE_0:def 3;
end;
suppose not p in P;
then p in P` by A15,A31,SUBSET_1:29;
then p in F3 by A31,XBOOLE_0:def 4;
hence thesis by A29,XBOOLE_0:def 3;
end;
end;
reconsider DT = [#]D as closed Subset of T2 by BORSUK_1:def 11,TSEP_1:1;
DT /\ Cl P is closed;
then
A32: F1 is closed by TSEP_1:8;
P is_a_component_of C`
by A9,A10;
then P is open by SPRECT_3:8;
then DT /\ P` is closed;
then
A33: F3 is closed by TSEP_1:8;
A34: id T1 = id F1 by PRE_TOPC:8;
T1 is SubSpace of T2 by TSEP_1:7;
then T1 is SubSpace of T2B1 by A20,A22,A25,TSEP_1:4;
then
A35: f1 is continuous by A34,PRE_TOPC:26;
A36: T2|A is SubSpace of T2B1 by A14,A22,A23,TSEP_1:4;
reconsider g2 = g1 as Function of T3,T2|A by A21;
g2 is continuous by A7,TOPMETR:7;
then
A37: g1 is continuous by A36,PRE_TOPC:26;
A38: for x being set st x in Cl P & x in P` holds f.x = x
proof
let x be set;
assume that
A39: x in Cl P and
A40: x in P`;
A41: x in Cl P /\ P` by A39,A40,XBOOLE_0:def 4;
then (id A).x = x by A5,FUNCT_1:18;
hence thesis by A5,A8,A41,FUNCT_1:49;
end;
for x being object st x in [#]T1 /\ [#]T3 holds f1.x = g1.x
proof
let x be object;
assume
A42: x in [#]T1 /\ [#]T3;
then
A43: x in [#]T1 by XBOOLE_0:def 4;
then
A44: x in Cl P by A28,XBOOLE_0:def 4;
x in P` by A29,A42,XBOOLE_0:def 4;
then
A45: f.x = x by A38,A44;
thus f1.x = x by A28,A43,FUNCT_1:18
.= g1.x by A29,A42,A45,FUNCT_1:49;
end;
then consider g being Function of D,T2|B such that
A46: g = f1+*g1 and
A47: g is continuous by A28,A29,A30,A32,A33,A35,A37,JGRAPH_2:1;
take g;
thus g is continuous by A47;
let x be Point of D;
A48: dom g1 = the carrier of T3 by FUNCT_2:def 1;
hereby
assume
A49: x in Cl P;
then
A50: x in F1 by XBOOLE_0:def 4;
per cases;
suppose not x in dom g1;
hence g.x = f1.x by A46,FUNCT_4:11
.= x by A50,FUNCT_1:18;
end;
suppose
A51: x in dom g1;
then
A52: x in P` by A21,XBOOLE_0:def 4;
thus g.x = g1.x by A46,A51,FUNCT_4:13
.= f.x by A21,A51,FUNCT_1:49
.= x by A38,A49,A52;
end;
end;
assume x in P`;
then
A53: x in F3 by XBOOLE_0:def 4;
hence g.x = g1.x by A21,A46,A48,FUNCT_4:13
.= f.x by A53,FUNCT_1:49;
end;
Lm15: BDD C is non empty &
U = P & U is a_component implies C = Fr P
proof
assume that
A1: BDD C is non empty and
A2: U = P and
A3: U is a_component and
A4: C <> Fr P;
A5: the carrier of T2|C` = C` by PRE_TOPC:8;
reconsider T2C = T2|C` as non empty SubSpace of T2;
A6: T2C is locally_connected by JORDAN2C:81;
then U is open by A3,CONNSP_2:15;
then reconsider P as open Subset of T2 by A2,TSEP_1:17;
A7: Fr P = Cl P /\ P` by PRE_TOPC:22;
set Z = {X where X is Subset of T2|C`: X is a_component & X <> U};
set V = union Z;
A8: V \/ U \/ C = the carrier of T2
proof
A9: V c= the carrier of T2
proof
let a be object;
assume a in V;
then consider A being set such that
A10: a in A and
A11: A in Z by TARSKI:def 4;
ex X being Subset of T2|C` st X = A & X is a_component &
X <> U by A11;
hence thesis by A5,A10,TARSKI:def 3;
end;
U c= the carrier of T2 by A5,XBOOLE_1:1;
then V \/ U c= the carrier of T2 by A9,XBOOLE_1:8;
hence V \/ U \/ C c= the carrier of T2 by XBOOLE_1:8;
let a be object;
assume
A12: a in the carrier of T2;
per cases;
suppose a in C;
hence thesis by XBOOLE_0:def 3;
end;
suppose not a in C;
then reconsider a as Point of T2|C` by A5,A12,SUBSET_1:29;
A13: a in Component_of a by CONNSP_1:38;
per cases;
suppose Component_of a = U;
then a in V \/ U by A13,XBOOLE_0:def 3;
hence thesis by XBOOLE_0:def 3;
end;
suppose
A14: Component_of a <> U;
Component_of a is a_component by CONNSP_1:40;
then Component_of a in Z by A14;
then a in V by A13,TARSKI:def 4;
then a in V \/ U by XBOOLE_0:def 3;
hence thesis by XBOOLE_0:def 3;
end;
end;
end;
A15: P misses P` by XBOOLE_1:79;
Fr P c= C
proof
let a be object;
assume
A16: a in Fr P;
then
A17: a in Cl P by XBOOLE_0:def 4;
A18: a in P` by A7,A16,XBOOLE_0:def 4;
assume not a in C;
then a in V \/ U by A8,A16,XBOOLE_0:def 3;
then a in V or a in U by XBOOLE_0:def 3;
then consider O being set such that
A19: a in O and
A20: O in Z by A2,A15,A18,TARSKI:def 4,XBOOLE_0:3;
consider X being Subset of T2|C` such that
A21: X = O and
A22: X is a_component and
A23: X <> U by A20;
Cl P misses X by A2,A3,A22,A23,Th68;
hence thesis by A17,A19,A21,XBOOLE_0:3;
end;
then Fr P c< C by A4;
then consider p1, p2, A such that
A24: A is_an_arc_of p1,p2 and
A25: Fr P c= A and
A26: A c= C by BORSUK_4:59;
A27: U <> {}(T2|C`) by A3,CONNSP_1:32;
per cases;
suppose P is bounded;
then reconsider P as bounded Subset of T2;
consider p being object such that
A28: p in U by A27,XBOOLE_0:def 1;
reconsider p as Point of T2 by A2,A28;
A29: P \/ C is bounded by TOPREAL6:67;
then reconsider PC = P \/ C as bounded Subset of Euclid 2
by JORDAN2C:11;
consider r being positive Real such that
A30: PC c= Ball(p,r) by A29,Th26;
C c= PC by XBOOLE_1:7;
then
A31: C c= Ball(p,r) by A30;
set D = Tdisk(p,r);
set S = Tcircle(p,r);
set B = cl_Ball(p,r) \ {p};
A32: the carrier of S = Sphere(p,r) by TOPREALB:9;
A33: the carrier of D = cl_Ball(p,r) by BROUWER:3;
A34: Sphere(p,r) c= cl_Ball(p,r) by TOPREAL9:17;
A35: Ball(p,r) misses Sphere(p,r) by TOPREAL9:19;
A36: Ball(p,r) c= cl_Ball(p,r) by TOPREAL9:16;
A c= Ball(p,r) by A26,A31;
then A is Subset of D by A33,A36,XBOOLE_1:1;
then consider R being Function of D, T2|A such that
A37: R is continuous and
A38: R|A = id A by A24,Lm12;
P c= PC by XBOOLE_1:7;
then
A39: P c= Ball(p,r) by A30;
then consider f being Function of D, T2|B such that
A40: f is continuous and
A41: for x being Point of D holds (x in Cl P implies f.x = R.x) &
(x in P` implies f.x = x) by A2,A3,A7,A24,A25,A26,A28,A31,A37,A38,Lm13;
set g = DiskProj(p,r,p);
set h = RotateCircle(p,r,p);
A42: S is SubSpace of D by A32,A33,A34,TSEP_1:4;
reconsider F = h*(g*f) as Function of D,D by A32,A33,A34,FUNCT_2:7;
p is Point of D by Th17;
then
A43: g is continuous by Th64;
|. p-p .| = 0 by TOPRNS_1:28;
then
A44: p in Ball(p,r) by TOPREAL9:7;
then h is continuous by Th66;
then
A45: F is continuous by A40,A42,A43,PRE_TOPC:26;
now
let x be object;
per cases;
suppose
A46: x in dom F;
A47: Ball(p,r) \/ Sphere(p,r) = cl_Ball(p,r) by TOPREAL9:18;
now per cases by A33,A46,A47,XBOOLE_0:def 3;
suppose
A48: x in Ball(p,r);
F.x in the carrier of S by A46,FUNCT_2:5;
hence F.x <> x by A32,A35,A48,XBOOLE_0:3;
end;
suppose
A49: x in Sphere(p,r);
A50: dom f = the carrier of D by FUNCT_2:def 1;
not x in P by A35,A39,A49,XBOOLE_0:3;
then
A51: x in P` by A49,SUBSET_1:29;
A52: g|Sphere(p,r) = id Sphere(p,r) by A44,Th65;
h is without_fixpoints by A44,Th67;
then
A53: not x is_a_fixpoint_of h;
A54: dom h = the carrier of S by FUNCT_2:def 1;
F.x = h.((g*f).x) by A46,FUNCT_1:12
.= h.(g.(f.x)) by A33,A34,A49,A50,FUNCT_1:13
.= h.(g.x) by A33,A34,A41,A49,A51
.= h.((id Sphere(p,r)).x) by A49,A52,FUNCT_1:49
.= h.x by A49,FUNCT_1:18;
hence F.x <> x by A32,A49,A53,A54;
end;
end;
hence not x is_a_fixpoint_of F;
end;
suppose not x in dom F;
hence not x is_a_fixpoint_of F;
end;
end;
then not F is with_fixpoint;
hence thesis by A45,BROUWER:14;
end;
suppose
A55: not P is bounded;
consider p being object such that
A56: p in BDD C by A1;
consider Z being set such that
A57: p in Z and
A58: Z in {B where B is Subset of T2: B is_inside_component_of C}
by A56,TARSKI:def 4;
consider P1 being Subset of T2 such that
A59: Z = P1 and
A60: P1 is_inside_component_of C by A58;
consider U1 being Subset of T2|C` such that
A61: U1 = P1 and
A62: U1 is a_component and
U1 is bounded Subset of Euclid 2 by A60,JORDAN2C:13;
U1 is open by A6,A62,CONNSP_2:15;
then reconsider P1 as non empty open bounded Subset of T2
by A57,A59,A60,A61,TSEP_1:17;
reconsider p as Point of T2 by A57,A59;
A63: p in P1 by A57,A59;
A64: P1 \/ C is bounded by TOPREAL6:67;
then reconsider PC = P1 \/ C as bounded Subset of Euclid 2
by JORDAN2C:11;
consider rv being positive Real such that
A65: PC c= Ball(p,rv) by A64,Th26;
not P c= Ball(p,rv) by A55,RLTOPSP1:42;
then consider u being object such that
A66: u in P and
A67: not u in Ball(p,rv);
reconsider u as Point of T2 by A66;
set r = |.u-p.|;
P misses P1 by A2,A3,A55,A61,A62,CONNSP_1:35;
then p <> u by A57,A59,A66,XBOOLE_0:3;
then reconsider r as non zero non negative Real by TOPRNS_1:28;
A68: r >= rv by A67,TOPREAL9:7;
then Ball(p,rv) c= Ball(p,r) by Th18;
then
A69: PC c= Ball(p,r) by A65;
A70: Fr Ball(p,r) = Sphere(p,r) by Th24;
u in Sphere(p,r) by TOPREAL9:9;
then
A71: P meets Ball(p,r) by A66,A70,TOPS_1:28;
A72: C c= PC by XBOOLE_1:7;
then
A73: C c= Ball(p,r) by A69;
set D = Tdisk(p,r);
set S = Tcircle(p,r);
set B = cl_Ball(p,r) \ {p};
A74: the carrier of S = Sphere(p,r) by TOPREALB:9;
A75: the carrier of D = cl_Ball(p,r) by BROUWER:3;
A76: Sphere(p,r) c= cl_Ball(p,r) by TOPREAL9:17;
A77: Ball(p,r) misses Sphere(p,r) by TOPREAL9:19;
A78: Ball(p,r) c= cl_Ball(p,r) by TOPREAL9:16;
A c= Ball(p,r) by A26,A73;
then A is Subset of D by A75,A78,XBOOLE_1:1;
then consider R being Function of D, T2|A such that
A79: R is continuous and
A80: R|A = id A by A24,Lm12;
p1 in A by A24,TOPREAL1:1;
then consider f being Function of D, T2|B such that
A81: f is continuous and
A82: for x being Point of D holds (x in Cl P implies f.x = x) &
(x in P` implies f.x = R.x)
by A2,A3,A7,A25,A26,A55,A61,A62,A63,A71,A73,A79,A80,Lm14;
set g = DiskProj(p,r,p);
set h = RotateCircle(p,r,p);
A83: S is SubSpace of D by A74,A75,A76,TSEP_1:4;
reconsider F = h*(g*f) as Function of D,D by A74,A75,A76,FUNCT_2:7;
p is Point of D by Th17;
then
A84: g is continuous by Th64;
|. p-p .| = 0 by TOPRNS_1:28;
then
A85: p in Ball(p,r) by TOPREAL9:7;
then h is continuous by Th66;
then
A86: F is continuous by A81,A83,A84,PRE_TOPC:26;
now
let x be object;
per cases;
suppose
A87: x in dom F;
A88: Ball(p,r) \/ Sphere(p,r) = cl_Ball(p,r) by TOPREAL9:18;
now per cases by A75,A87,A88,XBOOLE_0:def 3;
suppose
A89: x in Ball(p,r);
F.x in the carrier of S by A87,FUNCT_2:5;
hence F.x <> x by A74,A77,A89,XBOOLE_0:3;
end;
suppose
A90: x in Sphere(p,r);
A91: dom f = the carrier of D by FUNCT_2:def 1;
A92: P c= Cl P by PRE_TOPC:18;
set SS = Sphere(p,r);
SS c= C`
proof
let a be object;
assume
A93: a in SS;
assume not a in C`;
then
A94: a in C by A93,SUBSET_1:29;
reconsider a as Point of T2 by A93;
a in PC by A72,A94;
then |.a-p.| < rv by A65,TOPREAL9:7;
hence contradiction by A68,A93,TOPREAL9:9;
end;
then reconsider SS as Subset of T2|C` by PRE_TOPC:8;
A95: u in SS by TOPREAL9:9;
SS is connected by CONNSP_1:23;
then SS misses U or SS c= U by A3,CONNSP_1:36;
then
A96: x in P by A2,A66,A90,A95,XBOOLE_0:3;
A97: g|Sphere(p,r) = id Sphere(p,r) by A85,Th65;
h is without_fixpoints by A85,Th67;
then
A98: not x is_a_fixpoint_of h;
A99: dom h = the carrier of S by FUNCT_2:def 1;
F.x = h.((g*f).x) by A87,FUNCT_1:12
.= h.(g.(f.x)) by A75,A76,A90,A91,FUNCT_1:13
.= h.(g.x) by A75,A76,A82,A90,A92,A96
.= h.((id Sphere(p,r)).x) by A90,A97,FUNCT_1:49
.= h.x by A90,FUNCT_1:18;
hence F.x <> x by A74,A90,A98,A99;
end;
end;
hence not x is_a_fixpoint_of F;
end;
suppose not x in dom F;
hence not x is_a_fixpoint_of F;
end;
end;
then F is without_fixpoints;
hence thesis by A86,BROUWER:14;
end;
end;
set rp = 1;
set rl = -rp;
set rg = 3;
set rd = -rg;
set a = |[rl,0]|;
set b = |[rp,0]|;
set c = |[0,rg]|;
set d = |[0,rd]|;
set lg = |[rl,rg]|;
set pg = |[rp,rg]|;
set ld = |[rl,rd]|;
set pd = |[rp,rd]|;
set R = closed_inside_of_rectangle(rl,rp,rd,rg);
set dR = rectangle(rl,rp,rd,rg);
set TR = Trectangle(rl,rp,rd,rg);
Lm16: a`1 = rl by EUCLID:52;
Lm17: b`1 = rp by EUCLID:52;
Lm18: a`2 = 0 by EUCLID:52;
Lm19: b`2 = 0 by EUCLID:52;
Lm20: c`1 = 0 by EUCLID:52;
Lm21: c`2 = rg by EUCLID:52;
Lm22: d`1 = 0 by EUCLID:52;
Lm23: d`2 = rd by EUCLID:52;
Lm24: lg`1 = rl by EUCLID:52;
Lm25: lg`2 = rg by EUCLID:52;
Lm26: ld`1 = rl by EUCLID:52;
Lm27: ld`2 = rd by EUCLID:52;
Lm28: pg`1 = rp by EUCLID:52;
Lm29: pg`2 = rg by EUCLID:52;
Lm30: pd`1 = rp by EUCLID:52;
Lm31: pd`2 = rd by EUCLID:52;
Lm32: ld = |[ld`1,ld`2]| by EUCLID:53;
Lm33: lg = |[lg`1,lg`2]| by EUCLID:53;
Lm34: pd = |[pd`1,pd`2]| by EUCLID:53;
Lm35: pg = |[pg`1,pg`2]| by EUCLID:53;
Lm36: dR = (LSeg(ld,lg) \/ LSeg(lg,pg)) \/ (LSeg(pg,pd) \/ LSeg(pd,ld))
by SPPOL_2:def 3;
Lm37: LSeg(ld,lg) c= LSeg(ld,lg) \/ LSeg(lg,pg) by XBOOLE_1:7;
LSeg(ld,lg) \/ LSeg(lg,pg) c= dR by Lm36,XBOOLE_1:7;
then
Lm38: LSeg(ld,lg) c= dR by Lm37;
Lm39: LSeg(lg,pg) c= LSeg(ld,lg) \/ LSeg(lg,pg) by XBOOLE_1:7;
LSeg(ld,lg) \/ LSeg(lg,pg) c= dR by Lm36,XBOOLE_1:7;
then
Lm40: LSeg(lg,pg) c= dR by Lm39;
Lm41: LSeg(pg,pd) c= LSeg(pg,pd) \/ LSeg(pd,ld) by XBOOLE_1:7;
LSeg(pg,pd) \/ LSeg(pd,ld) c= dR by Lm36,XBOOLE_1:7;
then
Lm42: LSeg(pg,pd) c= dR by Lm41;
Lm43: LSeg(pd,ld) c= LSeg(pg,pd) \/ LSeg(pd,ld) by XBOOLE_1:7;
LSeg(pg,pd) \/ LSeg(pd,ld) c= dR by Lm36,XBOOLE_1:7;
then
Lm44: LSeg(pd,ld) c= dR by Lm43;
Lm45: LSeg(ld,lg) is vertical by Lm24,Lm26,SPPOL_1:16;
Lm46: LSeg(pd,pg) is vertical by Lm28,Lm30,SPPOL_1:16;
Lm47: LSeg(a,lg) is vertical by Lm16,Lm24,SPPOL_1:16;
Lm48: LSeg(a,ld) is vertical by Lm16,Lm26,SPPOL_1:16;
Lm49: LSeg(b,pg) is vertical by Lm17,Lm28,SPPOL_1:16;
Lm50: LSeg(b,pd) is vertical by Lm17,Lm30,SPPOL_1:16;
Lm51: LSeg(ld,d) is horizontal by Lm23,Lm27,SPPOL_1:15;
Lm52: LSeg(pd,d) is horizontal by Lm23,Lm31,SPPOL_1:15;
Lm53: LSeg(lg,c) is horizontal by Lm21,Lm25,SPPOL_1:15;
Lm54: LSeg(pg,c) is horizontal by Lm21,Lm29,SPPOL_1:15;
Lm55: LSeg(lg,pg) is horizontal by Lm25,Lm29,SPPOL_1:15;
Lm56: LSeg(ld,pd) is horizontal by Lm27,Lm31,SPPOL_1:15;
Lm57: LSeg(a,lg) c= LSeg(ld,lg)
by Lm16,Lm18,Lm25,Lm26,Lm27,Lm45,Lm47,GOBOARD7:63;
Lm58: LSeg(a,ld) c= LSeg(ld,lg) by Lm18,Lm25,Lm26,Lm27,Lm45,Lm48,GOBOARD7:63;
Lm59: LSeg(b,pg) c= LSeg(pd,pg)
by Lm17,Lm19,Lm29,Lm30,Lm31,Lm46,Lm49,GOBOARD7:63;
Lm60: LSeg(b,pd) c= LSeg(pd,pg) by Lm19,Lm29,Lm30,Lm31,Lm46,Lm50,GOBOARD7:63;
Lm61: dR = {p where p is Point of T2: p`1 = rl & p`2 <= rg & p`2 >= rd or
p`1 <= rp & p`1 >= rl & p`2 = rg or p`1 <= rp & p`1 >= rl & p`2 = rd or
p`1 = rp & p`2 <= rg & p`2 >= rd} by SPPOL_2:54;
then
Lm62: c in dR by Lm20,Lm21;
Lm63: d in dR by Lm22,Lm23,Lm61;
Lm64: (2+1)^2 = 4 + 4 + 1;
then
Lm65: sqrt(9) = 3 by SQUARE_1:def 2;
Lm66: dist(a,b) = sqrt ((a`1-b`1)^2 + (a`2-b`2)^2) by TOPREAL6:92
.= --2 by Lm16,Lm17,Lm18,Lm19,SQUARE_1:23;
theorem Th70:
for h being Homeomorphism of TOP-REAL 2 holds
h.:C is being_simple_closed_curve
proof
let h be Homeomorphism of T2;
consider f being Function of T2|R^2-unit_square, T2|C such that
A1: f is being_homeomorphism by TOPREAL2:def 1;
reconsider g = h|C as Function of T2|C,T2|(h.:C) by JORDAN24:12;
take g*f;
g is being_homeomorphism by JORDAN24:14;
hence thesis by A1,TOPS_2:57;
end;
theorem Th71:
|[-1,0]|,|[1,0]| realize-max-dist-in P implies
P c= closed_inside_of_rectangle(-1,1,-3,3)
proof
assume that
A1: a in P and
A2: b in P and
A3: for x, y being Point of TOP-REAL 2 st x in P & y in P holds
dist(a,b) >= dist(x,y);
let p be object;
assume
A4: p in P;
then reconsider p as Point of TOP-REAL 2;
A5: dist(a,p) = sqrt((rl-p`1)^2 + (0-p`2)^2) by Lm16,Lm18,TOPREAL6:92
.= sqrt((rl-p`1)^2 + p`2^2);
A6: now
assume 9 < p`2^2;
then 0+9 < (rl-p`1)^2+p`2^2 by XREAL_1:8;
then 3 < sqrt((rl-p`1)^2+p`2^2) by Lm65,SQUARE_1:27;
then 2 < sqrt((rl-p`1)^2+p`2^2) by XXREAL_0:2;
hence contradiction by A1,A3,A4,A5,Lm66;
end;
A7: now
assume
A8: rl > p`1;
then LSeg(p,b) meets Vertical_Line(rl) by Lm17,Th8;
then consider x being object such that
A9: x in LSeg(p,b) and
A10: x in Vertical_Line(rl) by XBOOLE_0:3;
reconsider x as Point of T2 by A9;
A11: x`1 = rl by A10,JORDAN6:31;
A12: dist(p,b) = dist(p,x)+dist(x,b) by A9,JORDAN1K:29;
A13: dist(x,b) = sqrt((x`1-b`1)^2 + (x`2-b`2)^2) by TOPREAL6:92
.= sqrt((-2)^2 + (x`2-0)^2) by A11,Lm17,EUCLID:52
.= sqrt(4 + x`2^2);
now
assume dist(x,b) < dist(a,b);
then 4 + x`2^2 < 4 + 0 by A13,Lm66,SQUARE_1:20,26;
hence contradiction by XREAL_1:6;
end;
then dist(p,b) + 0 > dist(a,b) + 0 by A8,A11,A12,JORDAN1K:22,XREAL_1:8;
hence contradiction by A2,A3,A4;
end;
A14: now
assume
A15: p`1 > rp;
then LSeg(p,a) meets Vertical_Line(rp) by Lm16,Th8;
then consider x being object such that
A16: x in LSeg(p,a) and
A17: x in Vertical_Line(rp) by XBOOLE_0:3;
reconsider x as Point of T2 by A16;
A18: x`1 = rp by A17,JORDAN6:31;
A19: dist(p,a) = dist(p,x)+dist(x,a) by A16,JORDAN1K:29;
A20: dist(x,a) = sqrt((x`1-a`1)^2 + (x`2-a`2)^2) by TOPREAL6:92
.= sqrt(4 + x`2^2) by A18,Lm16,Lm18;
now
assume dist(x,a) < dist(a,b);
then 4 + x`2^2 < 4 + 0 by A20,Lm66,SQUARE_1:20,26;
hence contradiction by XREAL_1:6;
end;
then dist(p,a) + 0 > dist(a,b) + 0 by A15,A18,A19,JORDAN1K:22,XREAL_1:8;
hence contradiction by A1,A3,A4;
end;
A21: now
assume rd > p`2;
then p`2^2 > rd^2 by SQUARE_1:44;
hence contradiction by A6;
end;
rg >= p`2 by A6,Lm64,SQUARE_1:16;
hence thesis by A7,A14,A21;
end;
Lm67: dR c= R by Th45;
Lm68: lg`2 = lg`2;
Lm69: lg`1 <= c`1 by Lm24,EUCLID:52;
c`1 <= pg`1 by Lm28,EUCLID:52;
then LSeg(lg,c) c= LSeg(lg,pg) by Lm53,Lm55,Lm68,Lm69,GOBOARD7:64;
then
Lm70: LSeg(lg,c) c= dR by Lm40;
LSeg(pg,c) c= LSeg(lg,pg) by Lm20,Lm21,Lm24,Lm25,Lm28,Lm54,Lm55,GOBOARD7:64;
then
Lm71: LSeg(pg,c) c= dR by Lm40;
Lm72: ld`2 = ld`2;
Lm73: ld`1 <= d`1 by Lm26,EUCLID:52;
d`1 <= pd`1 by Lm30,EUCLID:52;
then LSeg(ld,d) c= LSeg(ld,pd) by Lm51,Lm56,Lm72,Lm73,GOBOARD7:64;
then
Lm74: LSeg(ld,d) c= dR by Lm44;
LSeg(pd,d) c= LSeg(ld,pd) by Lm22,Lm23,Lm26,Lm27,Lm30,Lm52,Lm56,GOBOARD7:64;
then
Lm75: LSeg(pd,d) c= dR by Lm44;
Lm76: 0 <= p`2 & p in dR implies p in LSeg(a,lg) or p in LSeg(lg,c) or
p in LSeg(c,pg) or p in LSeg(pg,b)
proof
assume
A1: 0 <= p`2;
assume p in dR;
then consider p1 such that
A2: p1 = p and
A3: p1`1 = rl & p1`2 <= rg & p1`2 >= rd or
p1`1 <= rp & p1`1 >= rl & p1`2 = rg or
p1`1 <= rp & p1`1 >= rl & p1`2 = rd or
p1`1 = rp & p1`2 <= rg & p1`2 >= rd by Lm61;
per cases by A3;
suppose p1`1 = rl & p1`2 <= rg & p1`2 >= rd;
hence thesis by A1,A2,Lm16,Lm18,Lm24,Lm25,GOBOARD7:7;
end;
suppose
A4: p1`1 <= rp & p1`1 >= rl & p1`2 = rg;
per cases;
suppose p1`1 <= c`1;
hence thesis by A2,A4,Lm21,Lm24,Lm25,GOBOARD7:8;
end;
suppose c`1 <= p1`1;
hence thesis by A2,A4,Lm21,Lm28,Lm29,GOBOARD7:8;
end;
end;
suppose p1`1 <= rp & p1`1 >= rl & p1`2 = rd;
hence thesis by A1,A2;
end;
suppose p1`1 = rp & p1`2 <= rg & p1`2 >= rd;
hence thesis by A1,A2,Lm17,Lm19,Lm28,Lm29,GOBOARD7:7;
end;
end;
Lm77: p`2 <= 0 & p in dR implies p in LSeg(a,ld) or p in LSeg(ld,d) or
p in LSeg(d,pd) or p in LSeg(pd,b)
proof
assume
A1: p`2 <= 0;
assume p in dR;
then consider p1 such that
A2: p1 = p and
A3: p1`1 = rl & p1`2 <= rg & p1`2 >= rd or
p1`1 <= rp & p1`1 >= rl & p1`2 = rg or
p1`1 <= rp & p1`1 >= rl & p1`2 = rd or
p1`1 = rp & p1`2 <= rg & p1`2 >= rd by Lm61;
per cases by A3;
suppose p1`1 = rl & p1`2 <= rg & p1`2 >= rd;
hence thesis by A1,A2,Lm16,Lm18,Lm26,Lm27,GOBOARD7:7;
end;
suppose p1`1 <= rp & p1`1 >= rl & p1`2 = rg;
hence thesis by A1,A2;
end;
suppose
A4: p1`1 <= rp & p1`1 >= rl & p1`2 = rd;
per cases;
suppose p1`1 <= d`1;
hence thesis by A2,A4,Lm23,Lm26,Lm27,GOBOARD7:8;
end;
suppose d`1 <= p1`1;
hence thesis by A2,A4,Lm23,Lm30,Lm31,GOBOARD7:8;
end;
end;
suppose p1`1 = rp & p1`2 <= rg & p1`2 >= rd;
hence thesis by A1,A2,Lm17,Lm19,Lm30,Lm31,GOBOARD7:7;
end;
end;
theorem Th72:
|[-1,0]|,|[1,0]| realize-max-dist-in P implies
P misses LSeg(|[-1,3]|,|[1,3]|)
proof
assume
A1: a,b realize-max-dist-in P;
assume P meets LSeg(lg,pg);
then consider x being object such that
A2: x in P and
A3: x in LSeg(lg,pg) by XBOOLE_0:3;
reconsider x as Point of T2 by A2;
lg in LSeg(lg,pg) by RLTOPSP1:68;
then
A4: x`2 = rg by A3,Lm25,Lm55;
A5: dist(a,x) = sqrt ((a`1-x`1)^2 + (a`2-x`2)^2) by TOPREAL6:92
.= sqrt ((rl-x`1)^2 + rg^2) by A4,Lm18,EUCLID:52;
0+4 < (rl-x`1)^2+9 by XREAL_1:8;
then 2 < dist(a,x) by A5,SQUARE_1:20,27;
hence thesis by A1,A2,Lm66;
end;
theorem Th73:
|[-1,0]|,|[1,0]| realize-max-dist-in P implies
P misses LSeg(|[-1,-3]|,|[1,-3]|)
proof
assume
A1: a,b realize-max-dist-in P;
assume P meets LSeg(ld,pd);
then consider x being object such that
A2: x in P and
A3: x in LSeg(ld,pd) by XBOOLE_0:3;
reconsider x as Point of T2 by A2;
ld in LSeg(ld,pd) by RLTOPSP1:68;
then
A4: x`2 = rd by A3,Lm27,Lm56;
A5: dist(a,x) = sqrt ((a`1-x`1)^2 + (a`2-x`2)^2) by TOPREAL6:92
.= sqrt ((rl-x`1)^2 + (-rd)^2) by A4,Lm18,EUCLID:52;
0+4 < (rl-x`1)^2+9 by XREAL_1:8;
then 2 < dist(a,x) by A5,SQUARE_1:20,27;
hence thesis by A1,A2,Lm66;
end;
theorem Th74:
|[-1,0]|,|[1,0]| realize-max-dist-in P implies
P /\ rectangle(-1,1,-3,3) = {|[-1,0]|,|[1,0]|}
proof
assume
A1: a,b realize-max-dist-in P;
then
A2: a in P;
A3: b in P by A1;
thus P /\ dR c= {a,b}
proof
let x be object;
assume
A4: x in P /\ dR;
then
A5: x in P by XBOOLE_0:def 4;
x in dR by A4,XBOOLE_0:def 4;
then
A6: x in LSeg(ld,lg) \/ LSeg(lg,pg) or
x in LSeg(pg,pd) \/ LSeg(pd,ld) by Lm36,XBOOLE_0:def 3;
reconsider x as Point of T2 by A4;
per cases by A6,XBOOLE_0:def 3;
suppose
A7: x in LSeg(ld,lg);
ld in LSeg(ld,lg) by RLTOPSP1:68;
then
A8: x`1 = rl by A7,Lm26,Lm45;
per cases;
suppose x`2 = 0;
then x = a by A8,Lm16,Lm18,TOPREAL3:6;
hence thesis by TARSKI:def 2;
end;
suppose x`2 <> 0;
then
A9: x`2^2 > 0 by SQUARE_1:12;
A10: dist(b,x) = sqrt ((rp-rl)^2 + (0-x`2)^2) by A8,Lm17,Lm19,TOPREAL6:92
.= sqrt (4 + x`2^2);
0+4 < x`2^2+4 by A9,XREAL_1:6;
then 2 < sqrt(x`2^2+4) by SQUARE_1:20,27;
hence thesis by A1,A5,A10,Lm66;
end;
end;
suppose x in LSeg(lg,pg);
then LSeg(lg,pg) meets P by A5,XBOOLE_0:3;
hence thesis by A1,Th72;
end;
suppose
A11: x in LSeg(pg,pd);
pd in LSeg(pd,pg) by RLTOPSP1:68;
then
A12: x`1 = rp by A11,Lm30,Lm46;
per cases;
suppose x`2 = 0;
then x = b by A12,Lm17,Lm19,TOPREAL3:6;
hence thesis by TARSKI:def 2;
end;
suppose x`2 <> 0;
then
A13: x`2^2 > 0 by SQUARE_1:12;
A14: dist(x,a) = sqrt ((x`1-a`1)^2 + (x`2-a`2)^2) by TOPREAL6:92
.= sqrt (4 + x`2^2) by A12,Lm16,Lm18;
0+4 < x`2^2+4 by A13,XREAL_1:6;
then 2 < sqrt(x`2^2+4) by SQUARE_1:20,27;
hence thesis by A1,A5,A14,Lm66;
end;
end;
suppose x in LSeg(pd,ld);
then LSeg(pd,ld) meets P by A5,XBOOLE_0:3;
hence thesis by A1,Th73;
end;
end;
let x be object;
assume x in {a,b};
then
A15: x = a or x = b by TARSKI:def 2;
A16: a in dR by Lm16,Lm18,Lm61;
b in dR by Lm17,Lm19,Lm61;
hence thesis by A2,A3,A15,A16,XBOOLE_0:def 4;
end;
Lm78: |[-1,0]|,|[1,0]| realize-max-dist-in C implies LSeg(lg,c) misses C
proof
assume a,b realize-max-dist-in C;
then
A1: C /\ dR = {a,b} by Th74;
assume LSeg(lg,c) meets C;
then consider q being object such that
A2: q in LSeg(lg,c) and
A3: q in C by XBOOLE_0:3;
reconsider q as Point of T2 by A3;
q in dR /\ C by A2,A3,Lm70,XBOOLE_0:def 4;
then q = a or q = b by A1,TARSKI:def 2;
hence contradiction by A2,Lm18,Lm19,TOPREAL3:12;
end;
Lm79: |[-1,0]|,|[1,0]| realize-max-dist-in C implies LSeg(pg,c) misses C
proof
assume a,b realize-max-dist-in C;
then
A1: C /\ dR = {a,b} by Th74;
assume LSeg(pg,c) meets C;
then consider q being object such that
A2: q in LSeg(pg,c) and
A3: q in C by XBOOLE_0:3;
reconsider q as Point of T2 by A3;
q in dR /\ C by A2,A3,Lm71,XBOOLE_0:def 4;
then q = a or q = b by A1,TARSKI:def 2;
hence contradiction by A2,Lm18,Lm19,TOPREAL3:12;
end;
Lm80: |[-1,0]|,|[1,0]| realize-max-dist-in C implies LSeg(ld,d) misses C
proof
assume a,b realize-max-dist-in C;
then
A1: C /\ dR = {a,b} by Th74;
assume LSeg(ld,d) meets C;
then consider q being object such that
A2: q in LSeg(ld,d) and
A3: q in C by XBOOLE_0:3;
reconsider q as Point of T2 by A3;
q in dR /\ C by A2,A3,Lm74,XBOOLE_0:def 4;
then q = a or q = b by A1,TARSKI:def 2;
hence contradiction by A2,Lm18,Lm19,TOPREAL3:12;
end;
Lm81: |[-1,0]|,|[1,0]| realize-max-dist-in C implies LSeg(pd,d) misses C
proof
assume a,b realize-max-dist-in C;
then
A1: C /\ dR = {a,b} by Th74;
assume LSeg(pd,d) meets C;
then consider q being object such that
A2: q in LSeg(pd,d) and
A3: q in C by XBOOLE_0:3;
reconsider q as Point of T2 by A3;
q in dR /\ C by A2,A3,Lm75,XBOOLE_0:def 4;
then q = a or q = b by A1,TARSKI:def 2;
hence contradiction by A2,Lm18,Lm19,TOPREAL3:12;
end;
Lm82: |[
-1,0]|,|[1,0]| realize-max-dist-in C & p in C` & p in LSeg(a,lg) implies
LSeg(p,lg) misses C
proof
assume that
A1: a,b realize-max-dist-in C and
A2: p in C` and
A3: p in LSeg(a,lg);
A4: C /\ dR = {a,b} by A1,Th74;
assume LSeg(p,lg) meets C;
then consider q being object such that
A5: q in LSeg(p,lg) and
A6: q in C by XBOOLE_0:3;
reconsider q as Point of T2 by A6;
lg in LSeg(a,lg) by RLTOPSP1:68;
then
A7: p`1 = lg`1 by A3,Lm47;
A8: p`2 <= lg`2 by A3,Lm25,JGRAPH_6:1;
A9: LSeg(p,lg) is vertical by A7,SPPOL_1:16;
a`2 <= p`2 by A3,Lm18,JGRAPH_6:1;
then LSeg(p,lg) c= LSeg(ld,lg)
by A7,A8,A9,Lm18,Lm24,Lm26,Lm27,Lm45,GOBOARD7:63;
then LSeg(p,lg) c= dR by Lm38;
then q in dR /\ C by A5,A6,XBOOLE_0:def 4;
then
A10: q = a or q = b by A4,TARSKI:def 2;
a in LSeg(a,lg) by RLTOPSP1:68;
then
A11: a`1 = p`1 by A3,Lm47;
A12: a in C by A1;
not p in C by A2,XBOOLE_0:def 5;
then a`2 <> p`2 by A11,A12,TOPREAL3:6;
then
A13: a`2 < p`2 by A3,Lm18,JGRAPH_6:1;
p = |[p`1,p`2]| by EUCLID:53;
hence contradiction by A5,A7,A8,A10,A13,Lm17,Lm24,Lm33,JGRAPH_6:1;
end;
Lm83: |[
-1,0]|,|[1,0]| realize-max-dist-in C & p in C` & p in LSeg(b,pg) implies
LSeg(p,pg) misses C
proof
assume that
A1: a,b realize-max-dist-in C and
A2: p in C` and
A3: p in LSeg(b,pg);
A4: C /\ dR = {a,b} by A1,Th74;
assume LSeg(p,pg) meets C;
then consider q being object such that
A5: q in LSeg(p,pg) and
A6: q in C by XBOOLE_0:3;
reconsider q as Point of T2 by A6;
pg in LSeg(b,pg) by RLTOPSP1:68;
then
A7: p`1 = pg`1 by A3,Lm49;
A8: p`2 <= pg`2 by A3,Lm29,JGRAPH_6:1;
A9: LSeg(p,pg) is vertical by A7,SPPOL_1:16;
b`2 <= p`2 by A3,Lm19,JGRAPH_6:1;
then LSeg(p,pg) c= LSeg(pd,pg)
by A7,A8,A9,Lm19,Lm28,Lm30,Lm31,Lm46,GOBOARD7:63;
then LSeg(p,pg) c= dR by Lm42;
then q in dR /\ C by A5,A6,XBOOLE_0:def 4;
then
A10: q = a or q = b by A4,TARSKI:def 2;
b in LSeg(b,pg) by RLTOPSP1:68;
then
A11: b`1 = p`1 by A3,Lm49;
A12: b in C by A1;
not p in C by A2,XBOOLE_0:def 5;
then b`2 <> p`2 by A11,A12,TOPREAL3:6;
then
A13: b`2 < p`2 by A3,Lm19,JGRAPH_6:1;
p = |[p`1,p`2]| by EUCLID:53;
hence contradiction by A5,A7,A8,A10,A13,Lm16,Lm28,Lm35,JGRAPH_6:1;
end;
Lm84: |[
-1,0]|,|[1,0]| realize-max-dist-in C & p in C` & p in LSeg(a,ld) implies
LSeg(p,ld) misses C
proof
assume that
A1: a,b realize-max-dist-in C and
A2: p in C` and
A3: p in LSeg(a,ld);
A4: C /\ dR = {a,b} by A1,Th74;
assume LSeg(p,ld) meets C;
then consider q being object such that
A5: q in LSeg(p,ld) and
A6: q in C by XBOOLE_0:3;
reconsider q as Point of T2 by A6;
ld in LSeg(a,ld) by RLTOPSP1:68;
then
A7: p`1 = ld`1 by A3,Lm48;
A8: ld`2 <= p`2 by A3,Lm27,JGRAPH_6:1;
A9: LSeg(p,ld) is vertical by A7,SPPOL_1:16;
p`2 <= a`2 by A3,Lm18,JGRAPH_6:1;
then LSeg(p,ld) c= LSeg(ld,lg) by A7,A8,A9,Lm18,Lm25,Lm45,GOBOARD7:63;
then LSeg(p,ld) c= dR by Lm38;
then q in dR /\ C by A5,A6,XBOOLE_0:def 4;
then
A10: q = a or q = b by A4,TARSKI:def 2;
a in LSeg(a,ld) by RLTOPSP1:68;
then
A11: a`1 = p`1 by A3,Lm48;
A12: a in C by A1;
not p in C by A2,XBOOLE_0:def 5;
then a`2 <> p`2 by A11,A12,TOPREAL3:6;
then
A13: p`2 < a`2 by A3,Lm18,JGRAPH_6:1;
p = |[p`1,p`2]| by EUCLID:53;
hence contradiction by A5,A7,A8,A10,A13,Lm17,Lm26,Lm32,JGRAPH_6:1;
end;
Lm85: |[
-1,0]|,|[1,0]| realize-max-dist-in C & p in C` & p in LSeg(b,pd) implies
LSeg(p,pd) misses C
proof
assume that
A1: a,b realize-max-dist-in C and
A2: p in C` and
A3: p in LSeg(b,pd);
A4: C /\ dR = {a,b} by A1,Th74;
assume LSeg(p,pd) meets C;
then consider q being object such that
A5: q in LSeg(p,pd) and
A6: q in C by XBOOLE_0:3;
reconsider q as Point of T2 by A6;
pd in LSeg(b,pd) by RLTOPSP1:68;
then
A7: p`1 = pd`1 by A3,Lm50;
A8: pd`2 <= p`2 by A3,Lm31,JGRAPH_6:1;
A9: LSeg(p,pd) is vertical by A7,SPPOL_1:16;
p`2 <= b`2 by A3,Lm19,JGRAPH_6:1;
then LSeg(p,pd) c= LSeg(pd,pg) by A7,A8,A9,Lm19,Lm29,Lm46,GOBOARD7:63;
then LSeg(p,pd) c= dR by Lm42;
then q in dR /\ C by A5,A6,XBOOLE_0:def 4;
then
A10: q = a or q = b by A4,TARSKI:def 2;
b in LSeg(b,pd) by RLTOPSP1:68;
then
A11: b`1 = p`1 by A3,Lm50;
A12: b in C by A1;
not p in C by A2,XBOOLE_0:def 5;
then b`2 <> p`2 by A11,A12,TOPREAL3:6;
then
A13: p`2 < b`2 by A3,Lm19,JGRAPH_6:1;
p = |[p`1,p`2]| by EUCLID:53;
hence contradiction by A5,A7,A8,A10,A13,Lm16,Lm30,Lm34,JGRAPH_6:1;
end;
Lm86: |[0,r]| in rectangle(rl,rp,rd,rg) implies r = rd or r = rg
proof
assume |[0,r]| in dR;
then ex p st p = |[0,r]| & (p`1 = rl & p`2 <= rg & p`2 >= rd or
p`1 <= rp & p`1 >= rl & p`2 = rg or p`1 <= rp & p`1 >= rl & p`2 = rd or
p`1 = rp & p`2 <= rg & p`2 >= rd) by Lm61;
hence thesis by EUCLID:52;
end;
theorem Th75:
|[-1,0]|,|[1,0]| realize-max-dist-in P implies W-bound P = -1
proof
assume
A1: a,b realize-max-dist-in P;
then
A2: P c= R by Th71;
A3: P = the carrier of (T2|P) by PRE_TOPC:8;
A4: a in P by A1;
reconsider P as non empty Subset of T2 by A1;
reconsider Z = (proj1|P).:the carrier of (T2|P) as Subset of REAL;
A5: for p be Real st p in Z holds p >= rl
proof
let p be Real;
assume p in Z;
then consider p0 being object such that
A6: p0 in the carrier of T2|P and p0 in the carrier of T2|P and
A7: p = (proj1|P).p0 by FUNCT_2:64;
p0 in R by A2,A3,A6;
then ex p1 st p0 = p1 & rl <= p1`1 & p1`1 <= rp & rd <= p1`2 & p1`2 <= rg;
hence thesis by A3,A6,A7,PSCOMP_1:22;
end;
for q being Real st
for p being Real st p in Z holds p >= q holds rl >= q
proof
let q be Real such that
A8: for p being Real st p in Z holds p >= q;
(proj1|P).a = a`1 by A4,PSCOMP_1:22;
hence thesis by A3,A4,A8,Lm16,FUNCT_2:35;
end;
hence thesis by A5,SEQ_4:44;
end;
theorem Th76:
|[-1,0]|,|[1,0]| realize-max-dist-in P implies E-bound P = 1
proof
assume
A1: a,b realize-max-dist-in P;
then
A2: P c= R by Th71;
A3: b in P by A1;
reconsider P as non empty Subset of T2 by A1;
reconsider Z = (proj1|P).:the carrier of (T2|P) as Subset of REAL;
A4: P = the carrier of (T2|P) by PRE_TOPC:8;
A5: for p be Real st p in Z holds p <= rp
proof
let p be Real;
assume p in Z;
then consider p0 being object such that
A6: p0 in the carrier of T2|P and p0 in the carrier of T2|P and
A7: p = (proj1|P).p0 by FUNCT_2:64;
p0 in R by A2,A4,A6;
then ex p1 st p0 = p1 & rl <= p1`1 & p1`1 <= rp & rd <= p1`2 & p1`2 <= rg;
hence thesis by A4,A6,A7,PSCOMP_1:22;
end;
for q being Real st
for p being Real st p in Z holds p <= q holds rp <= q
proof
let q be Real such that
A8: for p being Real st p in Z holds p <= q;
(proj1|P).b = b`1 by A3,PSCOMP_1:22;
hence thesis by A3,A4,A8,Lm17,FUNCT_2:35;
end;
hence thesis by A5,SEQ_4:46;
end;
theorem Th77:
for P being compact Subset of TOP-REAL 2 holds
|[-1,0]|,|[1,0]| realize-max-dist-in P implies W-most P = {|[-1,0]|}
proof
let P be compact Subset of T2;
assume
A1: a,b realize-max-dist-in P;
then
A2: P c= R by Th71;
set L = LSeg(SW-corner P, NW-corner P);
A3: a in P by A1;
A4: (SW-corner P)`1 = |[rl,S-bound P]|`1 by A1,Th75
.= rl by EUCLID:52;
A5: (NW-corner P)`1 = |[rl,N-bound P]|`1 by A1,Th75
.= rl by EUCLID:52;
thus W-most P c= {a}
proof
let x be object;
assume
A6: x in W-most P;
then
A7: x in P by XBOOLE_0:def 4;
reconsider x as Point of T2 by A6;
A8: x in L by A6,XBOOLE_0:def 4;
SW-corner P in L by RLTOPSP1:68;
then
A9: x`1 = rl by A4,A8,SPPOL_1:def 3;
x in R by A2,A7;
then ex p st x = p & rl <= p`1 & p`1 <= rp & rd <= p`2 & p`2 <= rg;
then x in dR by A9,Lm61;
then x in P /\ dR by A7,XBOOLE_0:def 4;
then x in {a,b} by A1,Th74;
then x = a or x = b by TARSKI:def 2;
hence thesis by A9,EUCLID:52,TARSKI:def 1;
end;
let x be object;
assume x in {a};
then
A10: x = a by TARSKI:def 1;
A11: (SW-corner P)`2 = S-bound P by EUCLID:52;
A12: (NW-corner P)`2 = N-bound P by EUCLID:52;
A13: (SW-corner P)`2 <= a`2 by A3,A11,PSCOMP_1:24;
a`2 <= (NW-corner P)`2 by A3,A12,PSCOMP_1:24;
then a in L by A4,A5,A13,Lm16,GOBOARD7:7;
hence thesis by A3,A10,XBOOLE_0:def 4;
end;
theorem Th78:
for P being compact Subset of TOP-REAL 2 holds
|[-1,0]|,|[1,0]| realize-max-dist-in P implies E-most P = {|[1,0]|}
proof
let P be compact Subset of T2;
assume
A1: a,b realize-max-dist-in P;
then
A2: P c= R by Th71;
set L = LSeg(SE-corner P, NE-corner P);
A3: b in P by A1;
A4: (SE-corner P)`1 = |[rp,S-bound P]|`1 by A1,Th76
.= rp by EUCLID:52;
A5: (NE-corner P)`1 = |[rp,N-bound P]|`1 by A1,Th76
.= rp by EUCLID:52;
thus E-most P c= {b}
proof
let x be object;
assume
A6: x in E-most P;
then
A7: x in P by XBOOLE_0:def 4;
reconsider x as Point of T2 by A6;
A8: x in L by A6,XBOOLE_0:def 4;
SE-corner P in L by RLTOPSP1:68;
then
A9: x`1 = rp by A4,A8,SPPOL_1:def 3;
x in R by A2,A7;
then ex p st x = p & rl <= p`1 & p`1 <= rp & rd <= p`2 & p`2 <= rg;
then x in dR by A9,Lm61;
then x in P /\ dR by A7,XBOOLE_0:def 4;
then x in {a,b} by A1,Th74;
then x = a or x = b by TARSKI:def 2;
hence thesis by A9,EUCLID:52,TARSKI:def 1;
end;
let x be object;
assume x in {b};
then
A10: x = b by TARSKI:def 1;
A11: (SE-corner P)`2 = S-bound P by EUCLID:52;
A12: (NE-corner P)`2 = N-bound P by EUCLID:52;
A13: (SE-corner P)`2 <= b`2 by A3,A11,PSCOMP_1:24;
b`2 <= (NE-corner P)`2 by A3,A12,PSCOMP_1:24;
then b in L by A4,A5,A13,Lm17,GOBOARD7:7;
hence thesis by A3,A10,XBOOLE_0:def 4;
end;
theorem Th79:
for P being compact Subset of TOP-REAL 2 holds
|[-1,0]|,|[1,0]| realize-max-dist-in P implies
W-min P = |[-1,0]| & W-max P = |[-1,0]|
proof
let P be compact Subset of T2;
set M = W-most P;
assume
A1: a,b realize-max-dist-in P;
then
A2: M = {a} by Th77;
set f = proj2|M;
A3: dom f = the carrier of (T2|M) by FUNCT_2:def 1;
A4: the carrier of (T2|M) = M by PRE_TOPC:8;
A5: a in {a} by TARSKI:def 1;
A6: f.:the carrier of (T2|M) = Im(f,a) by A1,A4,Th77
.= {f.a} by A2,A3,A4,A5,FUNCT_1:59
.= {proj2.a} by A2,A5,FUNCT_1:49
.= {a`2} by PSCOMP_1:def 6;
then
A7: lower_bound (proj2|M) = a`2 by SEQ_4:9;
A8: upper_bound (proj2|M) = a`2 by A6,SEQ_4:9;
a = |[a`1,a`2]| by EUCLID:53;
hence thesis by A1,A7,A8,Lm16,Th75;
end;
theorem Th80:
for P being compact Subset of TOP-REAL 2 holds
|[-1,0]|,|[1,0]| realize-max-dist-in P implies
E-min P = |[1,0]| & E-max P = |[1,0]|
proof
let P be compact Subset of T2;
set M = E-most P;
assume
A1: a,b realize-max-dist-in P;
then
A2: M = {b} by Th78;
set f = proj2|M;
A3: dom f = the carrier of (T2|M) by FUNCT_2:def 1;
A4: the carrier of (T2|M) = M by PRE_TOPC:8;
A5: b in {b} by TARSKI:def 1;
A6: f.:the carrier of (T2|M) = Im(f,b) by A1,A4,Th78
.= {f.b} by A2,A3,A4,A5,FUNCT_1:59
.= {proj2.b} by A2,A5,FUNCT_1:49
.= {b`2} by PSCOMP_1:def 6;
then
A7: lower_bound (proj2|M) = b`2 by SEQ_4:9;
A8: upper_bound (proj2|M) = b`2 by A6,SEQ_4:9;
b = |[b`1,b`2]| by EUCLID:53;
hence thesis by A1,A7,A8,Lm17,Th76;
end;
Lm87: |[-1,0]|,|[1,0]| realize-max-dist-in P implies
c`1 = (W-bound P + E-bound P) / 2
proof
assume
A1: a,b realize-max-dist-in P;
then
A2: W-bound P = rl by Th75;
E-bound P = rp by A1,Th76;
hence thesis by A2,EUCLID:52;
end;
Lm88: |[-1,0]|,|[1,0]| realize-max-dist-in P implies
d`1 = (W-bound P + E-bound P) / 2
proof
assume
A1: a,b realize-max-dist-in P;
then
A2: W-bound P = rl by Th75;
E-bound P = rp by A1,Th76;
hence thesis by A2,EUCLID:52;
end;
theorem Th81:
|[-1,0]|,|[1,0]| realize-max-dist-in P implies
LSeg(|[0,3]|,UMP P) is vertical
proof
assume a,b realize-max-dist-in P;
then c`1 = (W-bound P + E-bound P) / 2 by Lm87
.= (UMP P)`1 by EUCLID:52;
hence thesis by SPPOL_1:16;
end;
theorem Th82:
|[-1,0]|,|[1,0]| realize-max-dist-in P implies
LSeg(LMP P,|[0,-3]|) is vertical
proof
assume a,b realize-max-dist-in P;
then d`1 = (W-bound P + E-bound P) / 2 by Lm88
.= (LMP P)`1 by EUCLID:52;
hence thesis by SPPOL_1:16;
end;
theorem Th83:
|[-1,0]|,|[1,0]| realize-max-dist-in P & p in P implies p`2 < 3
proof
assume that
A1: a,b realize-max-dist-in P and
A2: p in P;
A3: P /\ dR = {a,b} by A1,Th74;
P c= R by A1,Th71;
then p in R by A2;
then
A4: ex p1 st p1 = p & rl <= p1`1 & p1`1 <= rp & rd <= p1`2 & p1`2 <= rg;
now
assume
A5: p`2 = c`2;
then p in LSeg(lg,pg) by A4,Lm21,Lm24,Lm25,Lm28,Lm29,GOBOARD7:8;
then p in P /\ dR by A2,Lm40,XBOOLE_0:def 4;
hence contradiction by A3,A5,Lm18,Lm19,Lm21,TARSKI:def 2;
end;
hence thesis by A4,Lm21,XXREAL_0:1;
end;
theorem Th84:
|[-1,0]|,|[1,0]| realize-max-dist-in P & p in P implies -3 < p`2
proof
assume that
A1: a,b realize-max-dist-in P and
A2: p in P;
A3: P /\ dR = {a,b} by A1,Th74;
P c= R by A1,Th71;
then p in R by A2;
then
A4: ex p1 st p1 = p & rl <= p1`1 & p1`1 <= rp & rd <= p1`2 & p1`2 <= rg;
now
assume
A5: p`2 = d`2;
then p in LSeg(ld,pd) by A4,Lm23,Lm26,Lm27,Lm30,Lm31,GOBOARD7:8;
then p in P /\ dR by A2,Lm44,XBOOLE_0:def 4;
then p = a or p = b by A3,TARSKI:def 2;
hence contradiction by A5,Lm23,EUCLID:52;
end;
hence thesis by A4,Lm23,XXREAL_0:1;
end;
theorem Th85:
|[-1,0]|,|[1,0]| realize-max-dist-in D & p in LSeg(|[0,3]|,UMP D) implies
(UMP D)`2 <= p`2
proof
set x = UMP D;
assume that
A1: a,b realize-max-dist-in D and
A2: p in LSeg(c,x);
A3: x in LSeg(c,x) by RLTOPSP1:68;
A4: LSeg(c,x) is vertical by A1,Th81;
A5: c = |[c`1,c`2]| by EUCLID:53;
A6: x = |[x`1,x`2]| by EUCLID:53;
c in LSeg(c,x) by RLTOPSP1:68;
then
A7: c`1 = x`1 by A3,A4;
x`2 <= c`2 by A1,Lm21,Th83,JORDAN21:30;
hence thesis by A2,A5,A6,A7,JGRAPH_6:1;
end;
theorem Th86:
|[-1,0]|,|[1,0]| realize-max-dist-in D & p in LSeg(LMP D,|[0,-3]|) implies
p`2 <= (LMP D)`2
proof
set x = LMP D;
assume that
A1: a,b realize-max-dist-in D and
A2: p in LSeg(x,d);
A3: x in LSeg(x,d) by RLTOPSP1:68;
A4: LSeg(x,d) is vertical by A1,Th82;
A5: d = |[d`1,d`2]| by EUCLID:53;
A6: x = |[x`1,x`2]| by EUCLID:53;
d in LSeg(x,d) by RLTOPSP1:68;
then
A7: d`1 = x`1 by A3,A4;
d`2 <= x`2 by A1,Lm23,Th84,JORDAN21:31;
hence thesis by A2,A5,A6,A7,JGRAPH_6:1;
end;
theorem Th87:
|[-1,0]|,|[1,0]| realize-max-dist-in D implies
LSeg(|[0,3]|,UMP D) c= north_halfline UMP D
proof
set p = UMP D;
assume
A1: a,b realize-max-dist-in D;
let x be object;
assume
A2: x in LSeg(c,p);
then reconsider x as Point of T2;
A3: p in LSeg(c,p) by RLTOPSP1:68;
LSeg(c,p) is vertical by A1,Th81;
then
A4: x`1 = p`1 by A2,A3;
p`2 <= x`2 by A1,A2,Th85;
hence thesis by A4,TOPREAL1:def 10;
end;
theorem Th88:
|[-1,0]|,|[1,0]| realize-max-dist-in D implies
LSeg(LMP D,|[0,-3]|) c= south_halfline LMP D
proof
set p = LMP D;
assume
A1: a,b realize-max-dist-in D;
let x be object;
assume
A2: x in LSeg(p,d);
then reconsider x as Point of T2;
A3: p in LSeg(p,d) by RLTOPSP1:68;
A4: LSeg(p,d) is vertical by A1,Th82;
then
A5: x`1 = p`1 by A2,A3;
A6: d = |[d`1,d`2]| by EUCLID:53;
A7: p = |[p`1,p`2]| by EUCLID:53;
d in LSeg(p,d) by RLTOPSP1:68;
then
A8: d`1 = p`1 by A3,A4;
d`2 <= p`2 by A1,Lm23,Th84,JORDAN21:31;
then x`2 <= p`2 by A2,A6,A7,A8,JGRAPH_6:1;
hence thesis by A5,TOPREAL1:def 12;
end;
theorem Th89:
|[-1,0]|,|[1,0]| realize-max-dist-in C & P is_inside_component_of C implies
LSeg(|[0,3]|,UMP C) misses P
proof
set m = UMP C;
set L = LSeg(c,m);
assume that
A1: a,b realize-max-dist-in C and
A2: P is_inside_component_of C;
A3: ex VP being Subset of T2|C` st ( VP = P)&( VP
is a_component)&( VP is bounded Subset of Euclid 2) by A2,JORDAN2C:13;
m in L by RLTOPSP1:68;
then {m} c= L by ZFMISC_1:31;
then
A4: L = L \ {m} \/ {m} by XBOOLE_1:45;
A5: L \ {m} c= north_halfline m \ {m} by A1,Th87,XBOOLE_1:33;
north_halfline m \ {m} c= UBD C by Th12;
then L \ {m} c= UBD C by A5;
then
A6: L \ {m} misses P by A2,Th14,XBOOLE_1:63;
{m} misses P by A3,Lm4,JORDAN21:30;
hence thesis by A4,A6,XBOOLE_1:70;
end;
theorem Th90:
|[-1,0]|,|[1,0]| realize-max-dist-in C & P is_inside_component_of C implies
LSeg(LMP C,|[0,-3]|) misses P
proof
set m = LMP C;
set L = LSeg(m,d);
assume that
A1: a,b realize-max-dist-in C and
A2: P is_inside_component_of C;
A3: ex VP being Subset of T2|C` st ( VP = P)&( VP
is a_component)&( VP is bounded Subset of Euclid 2) by A2,JORDAN2C:13;
m in L by RLTOPSP1:68;
then {m} c= L by ZFMISC_1:31;
then
A4: L = L \ {m} \/ {m} by XBOOLE_1:45;
A5: L \ {m} c= south_halfline m \ {m} by A1,Th88,XBOOLE_1:33;
south_halfline m \ {m} c= UBD C by Th13;
then L \ {m} c= UBD C by A5;
then
A6: L \ {m} misses P by A2,Th14,XBOOLE_1:63;
{m} misses P by A3,Lm4,JORDAN21:31;
hence thesis by A4,A6,XBOOLE_1:70;
end;
theorem Th91:
|[-1,0]|,|[1,0]| realize-max-dist-in D implies
LSeg(|[0,3]|,UMP D) /\ D = {UMP D}
proof
assume
A1: a,b realize-max-dist-in D;
set m = UMP D;
set w = (W-bound D + E-bound D) / 2;
A2: c`1 = w by A1,Lm87;
A3: m`1 = w by EUCLID:52;
A4: m in LSeg(c,m) by RLTOPSP1:68;
A5: m in D by JORDAN21:30;
thus LSeg(c,m) /\ D c= {m}
proof
let x be object;
assume
A6: x in LSeg(c,m) /\ D;
then
A7: x in LSeg(c,m) by XBOOLE_0:def 4;
A8: x in D by A6,XBOOLE_0:def 4;
reconsider x as Point of T2 by A6;
LSeg(c,m) is vertical by A2,A3,SPPOL_1:16;
then
A9: x`1 = m`1 by A4,A7;
then x in Vertical_Line w by A3,JORDAN6:31;
then x in D /\ Vertical_Line w by A8,XBOOLE_0:def 4;
then
A10: x`2 <= m`2 by JORDAN21:28;
m`2 <= x`2 by A1,A7,Th85;
then x`2 = m`2 by A10,XXREAL_0:1;
then x = m by A9,TOPREAL3:6;
hence thesis by TARSKI:def 1;
end;
let x be object;
assume x in {m};
then x = m by TARSKI:def 1;
hence thesis by A4,A5,XBOOLE_0:def 4;
end;
theorem
|[-1,0]|,|[1,0]| realize-max-dist-in D implies
LSeg(|[0,-3]|,LMP D) /\ D = {LMP D}
proof
assume
A1: a,b realize-max-dist-in D;
set m = LMP D;
set w = (W-bound D + E-bound D) / 2;
A2: d`1 = w by A1,Lm88;
A3: m`1 = w by EUCLID:52;
A4: m in LSeg(d,m) by RLTOPSP1:68;
A5: m in D by JORDAN21:31;
thus LSeg(d,m) /\ D c= {m}
proof
let x be object;
assume
A6: x in LSeg(d,m) /\ D;
then
A7: x in LSeg(d,m) by XBOOLE_0:def 4;
A8: x in D by A6,XBOOLE_0:def 4;
reconsider x as Point of T2 by A6;
LSeg(d,m) is vertical by A2,A3,SPPOL_1:16;
then
A9: x`1 = m`1 by A4,A7;
then x in Vertical_Line w by A3,JORDAN6:31;
then x in D /\ Vertical_Line w by A8,XBOOLE_0:def 4;
then
A10: m`2 <= x`2 by JORDAN21:29;
x`2 <= m`2 by A1,A7,Th86;
then x`2 = m`2 by A10,XXREAL_0:1;
then x = m by A9,TOPREAL3:6;
hence thesis by TARSKI:def 1;
end;
let x be object;
assume x in {m};
then x = m by TARSKI:def 1;
hence thesis by A4,A5,XBOOLE_0:def 4;
end;
theorem Th93:
P is compact &
|[-1,0]|,|[1,0]| realize-max-dist-in P & A is_inside_component_of P implies
A c= closed_inside_of_rectangle(-1,1,-3,3)
proof
assume that
A1: P is compact and
A2: |[-1,0]|,|[1,0]| realize-max-dist-in P and
A3: A is_inside_component_of P;
let x be object;
assume that
A4: x in A and
A5: not x in R;
P c= R by A2,Th71;
then
A6: R` c= P` by SUBSET_1:12;
reconsider x as Point of T2 by A4;
A7: not (rl <= x`1 & x`1 <= rp & rd <= x`2 & x`2 <= rg) by A5;
per cases;
suppose
A8: 0 <= x`1;
set E = east_halfline(x);
E c= R`
proof
let e be object;
assume
A9: e in E;
then reconsider e as Point of T2;
A10: e`1 >= x`1 by A9,TOPREAL1:def 11;
now
assume e in R;
then ex p st e = p & rl <= p`1 & p`1 <= rp & rd <= p`2 & p`2 <= rg;
hence contradiction by A7,A8,A9,A10,TOPREAL1:def 11,XXREAL_0:2;
end;
hence thesis by SUBSET_1:29;
end;
then E c= P` by A6;
then E misses P by SUBSET_1:23;
then
A11: E c= UBD P by A1,JORDAN2C:127;
x in E by TOPREAL1:38;
then A meets UBD P by A4,A11,XBOOLE_0:3;
hence thesis by A3,Th14;
end;
suppose
A12: x`1 < 0;
set E = west_halfline(x);
E c= R`
proof
let e be object;
assume
A13: e in E;
then reconsider e as Point of T2;
A14: e`1 <= x`1 by A13,TOPREAL1:def 13;
now
assume e in R;
then ex p st e = p & rl <= p`1 & p`1 <= rp & rd <= p`2 & p`2 <= rg;
hence contradiction by A7,A12,A13,A14,TOPREAL1:def 13,XXREAL_0:2;
end;
hence thesis by SUBSET_1:29;
end;
then E c= P` by A6;
then E misses P by SUBSET_1:23;
then
A15: E c= UBD P by A1,JORDAN2C:126;
x in E by TOPREAL1:38;
then A meets UBD P by A4,A15,XBOOLE_0:3;
hence thesis by A3,Th14;
end;
end;
Lm89: p in R implies R c= Ball(p,10)
proof
assume p in R;
then consider p1 such that
A1: p1 = p and
A2: rl <= p1`1 and
A3: p1`1 <= rp and
A4: rd <= p1`2 and
A5: p1`2 <= rg;
let x be object;
assume
A6: x in R;
then reconsider x as Point of T2;
consider p2 such that
A7: p2 = x and
A8: rl <= p2`1 and
A9: p2`1 <= rp and
A10: rd <= p2`2 and
A11: p2`2 <= rg by A6;
A12: ex s, t being Point of Euclid 2 st
s = p1 & t = p2 & dist(p1,p2) = dist(s,t) by TOPREAL6:def 1;
dist(p1,p2) <= (rp-rl) + (rg-rd) by A2,A3,A4,A5,A8,A9,A10,A11,TOPREAL6:95;
then dist(p1,p2) < 10 by XXREAL_0:2;
then |.x-p.| < 10 by A1,A7,A12,SPPOL_1:39;
hence thesis by TOPREAL9:7;
end;
theorem
|[-1,0]|,|[1,0]| realize-max-dist-in C implies LSeg(|[0,3]|,|[0,-3]|) meets C
proof
assume
A1: a,b realize-max-dist-in C;
set Jc = Upper_Arc C;
consider Pf being Path of c,d, f being Function
of I[01], T2|LSeg(c,d) such that
A2: rng f = LSeg(c,d) and
A3: Pf = f by Th43;
A4: a = W-min C by A1,Th79;
b = E-max C by A1,Th80;
then Jc is_an_arc_of a,b by A4,JORDAN6:def 8;
then consider Pg being Path of a,b, g being Function
of I[01], T2|Jc such that
A5: rng g = Jc and
A6: Pg = g by Th42;
A7: Jc c= C by JORDAN6:61;
A8: C c= R by A1,Th71;
A9: a in C by A1;
A10: b in C by A1;
A11: the carrier of TR = R by PRE_TOPC:8;
reconsider AR = a, BR = b, CR = c, DR = d
as Point of TR by A8,A9,A10,Lm62,Lm63,Lm67,PRE_TOPC:8;
rng Pg c= the carrier of TR by A5,A6,A7,A8,A11;
then reconsider h = Pg as Path of AR,BR by Th30;
LSeg(c,d) c= R by Lm62,Lm63,Lm67,JORDAN1:def 1;
then reconsider v = Pf as Path of CR,DR by A2,A3,A11,Th30;
consider s, t being Point of I[01] such that
A12: h.s = v.t by Lm16,Lm17,Lm21,Lm23,JGRAPH_8:6;
A13: dom h = the carrier of I[01] by FUNCT_2:def 1;
dom v = the carrier of I[01] by FUNCT_2:def 1;
then
A14: v.t in rng Pf by FUNCT_1:def 3;
h.s in rng Pg by A13,FUNCT_1:def 3;
hence thesis by A2,A3,A5,A6,A7,A12,A14,XBOOLE_0:3;
end;
Lm90: |[-1,0]|,|[1,0]| realize-max-dist-in C implies
ex Jc, Jd being compact with_the_max_arc Subset of T2 st
Jc is_an_arc_of |[-1,0]|,|[1,0]| & Jd is_an_arc_of |[-1,0]|,|[1,0]| &
C = Jc \/ Jd & Jc /\ Jd = {|[-1,0]|,|[1,0]|} & UMP C in Jc & LMP C in Jd &
W-bound C = W-bound Jc & E-bound C = E-bound Jc
proof
assume
A1: a,b realize-max-dist-in C;
set U = Upper_Arc C;
set L = Lower_Arc C;
A2: U \/ L = C by JORDAN6:def 9;
A3: UMP C in C by JORDAN21:30;
LMP C in C by JORDAN21:31;
then
A4: LMP C in U or LMP C in L by A2,XBOOLE_0:def 3;
A5: W-min C = a by A1,Th79;
A6: E-max C = b by A1,Th80;
per cases by A2,A3,XBOOLE_0:def 3;
suppose
A7: UMP C in U;
take U, L;
thus thesis by A4,A5,A6,A7,JORDAN21:17,18,50,JORDAN6:50;
end;
suppose
A8: UMP C in L;
take L, U;
thus thesis by A4,A5,A6,A8,JORDAN21:19,20,49,JORDAN6:50;
end;
end;
theorem Th95:
|[-1,0]|,|[1,0]| realize-max-dist-in C implies
for Jc, Jd being compact with_the_max_arc Subset of TOP-REAL 2 st
Jc is_an_arc_of |[-1,0]|,|[1,0]| & Jd is_an_arc_of |[-1,0]|,|[1,0]| &
C = Jc \/ Jd & Jc /\ Jd = {|[-1,0]|,|[1,0]|} & UMP C in Jc & LMP C in Jd &
W-bound C = W-bound Jc & E-bound C = E-bound Jc
for Ux being Subset of TOP-REAL 2 st Ux = Component_of Down ((1/2) *
((UMP (LSeg(LMP Jc,|[0,-3]|) /\ Jd)) + LMP Jc), C`)
holds Ux is_inside_component_of C & for V being Subset of TOP-REAL 2 st
V is_inside_component_of C holds V = Ux
proof
set m = UMP C;
set j = LMP C;
assume
A1: a,b realize-max-dist-in C;
let Jc, Jd being compact with_the_max_arc Subset of T2 such that
A2: Jc is_an_arc_of a,b and
A3: Jd is_an_arc_of a,b and
A4: C = Jc \/ Jd and
A5: Jc /\ Jd = {a,b} and
A6: UMP C in Jc and
A7: LMP C in Jd and
A8: W-bound C = W-bound Jc and
A9: E-bound C = E-bound Jc;
set l = LMP Jc;
set LJ = LSeg(l,d) /\ Jd;
set k = UMP LJ;
set x = (1/2)*(k+l);
set w = (W-bound C + E-bound C) / 2;
let Ux be Subset of TOP-REAL 2 such that
A10: Ux = Component_of Down (x,C`);
A11: C c= R by A1,Th71;
A12: W-bound C = rl by A1,Th75;
A13: E-bound C = rp by A1,Th76;
A14: a in C by A1;
A15: b in C by A1;
A16: m in C by JORDAN21:30;
A17: l in Jc by JORDAN21:31;
A18: Jd c= C by A4,XBOOLE_1:7;
A19: Jc c= C by A4,XBOOLE_1:7;
then
A20: l in C by A17;
A21: m`2 < c`2 by A1,Lm21,Th83,JORDAN21:30;
A22: l`1 = 0 by A8,A9,A12,A13,EUCLID:52;
A23: c`1 = w by A1,Lm87;
A24: m`1 = w by EUCLID:52;
A25: m <> a by A12,A13,Lm16,EUCLID:52;
A26: m <> b by A12,A13,Lm17,EUCLID:52;
A27: l <> a by A8,A9,A12,A13,Lm16,EUCLID:52;
A28: l <> b by A8,A9,A12,A13,Lm17,EUCLID:52;
then consider Pml being Path of m,l such that
A29: rng Pml c= Jc and
A30: rng Pml misses {a,b} by A2,A6,A17,A25,A26,A27,Th44;
set ml = rng Pml;
A31: ml c= C by A19,A29;
A32: j in C by A7,A18;
A33: LSeg(l,d) is vertical by A22,Lm22,SPPOL_1:16;
A34: d`2 <= j`2 by A1,A7,A18,Lm23,Th84;
A35: j`1 = 0 by A12,A13,EUCLID:52;
l in Vertical_Line w by A12,A13,A22,JORDAN6:31;
then
A36: l in C /\ Vertical_Line w by A17,A19,XBOOLE_0:def 4;
then j`2 <= l`2 by JORDAN21:29;
then j in LSeg(l,d) by A22,A34,A35,Lm22,GOBOARD7:7;
then
A37: LJ is non empty by A7,XBOOLE_0:def 4;
A38: LJ is vertical by A33,Th4;
then
A39: k in LJ by A37,JORDAN21:30;
then
A40: k in LSeg(l,d) by XBOOLE_0:def 4;
A41: k in Jd by A39,XBOOLE_0:def 4;
then
A42: k in C by A18;
A43: d in LSeg(l,d) by RLTOPSP1:68;
then
A44: k`1 = 0 by A33,A40,Lm22;
then
A45: k <> a by EUCLID:52;
A46: k <> b by A44,EUCLID:52;
A47: j <> a by A35,EUCLID:52;
j <> b by A35,EUCLID:52;
then consider Pkj being Path of k,j such that
A48: rng Pkj c= Jd and
A49: rng Pkj misses {a,b} by A3,A7,A41,A45,A46,A47,Th44;
set kj = rng Pkj;
A50: kj c= C by A18,A48;
A51: x in LSeg(k,l) by RLTOPSP1:69;
A52: Component_of Down(x,C`) is a_component by CONNSP_1:40;
A53: the carrier of T2|C` = C` by PRE_TOPC:8;
A54: LSeg(l,k) is vertical by A22,A44,SPPOL_1:16;
A55: k in LSeg(l,k) by RLTOPSP1:68;
A56: l = |[l`1,l`2]| by EUCLID:53;
A57: k = |[k`1,k`2]| by EUCLID:53;
A58: d = |[d`1,d`2]| by EUCLID:53;
d`2 <= l`2 by A1,A17,A19,Lm23,Th84;
then
A59: k`2 <= l`2 by A22,A40,A56,A58,Lm22,JGRAPH_6:1;
A60: a <> k by A44,EUCLID:52;
b <> k by A44,EUCLID:52;
then not k in {a,b} by A60,TARSKI:def 2;
then
A61: k <> l by A5,A17,A41,XBOOLE_0:def 4;
then k`2 <> l`2 by A22,A44,TOPREAL3:6;
then
A62: k`2 < l`2 by A59,XXREAL_0:1;
k in Vertical_Line w by A12,A13,A44,JORDAN6:31;
then k in C /\ Vertical_Line w by A18,A41,XBOOLE_0:def 4;
then j`2 <= k`2 by JORDAN21:29;
then d`2 <= k`2 by A1,A7,A18,Lm23,Th84,XXREAL_0:2;
then
A63: LSeg(l,k) c= LSeg(l,d) by A33,A44,A54,A59,Lm22,GOBOARD7:63;
A64: LSeg(l,k) \ {l,k} c= C`
proof
let q be object;
assume that
A65: q in LSeg(l,k) \ {l,k} and
A66: not q in C`;
A67: q in LSeg(l,k) by A65,XBOOLE_0:def 5;
reconsider q as Point of T2 by A65;
A68: q in C by A66,SUBSET_1:29;
A69: q`1 = w by A12,A13,A44,A54,A55,A67;
then
A70: q in Vertical_Line w by JORDAN6:31;
per cases by A4,A68,XBOOLE_0:def 3;
suppose q in Jc;
then q in Jc /\ Vertical_Line w by A70,XBOOLE_0:def 4;
then
A71: l`2 <= q`2 by A8,A9,JORDAN21:29;
q`2 <= l`2 by A22,A44,A56,A57,A59,A67,JGRAPH_6:1;
then l`2 = q`2 by A71,XXREAL_0:1;
then l = q by A12,A13,A22,A69,TOPREAL3:6;
then q in {l,k} by TARSKI:def 2;
hence contradiction by A65,XBOOLE_0:def 5;
end;
suppose q in Jd;
then
A72: q in LJ by A63,A67,XBOOLE_0:def 4;
A73: q`1 = d`1 by A33,A43,A63,A67;
A74: W-bound LSeg(l,d) <= W-bound LJ by A72,PSCOMP_1:69,XBOOLE_1:17;
A75: E-bound LJ <= E-bound LSeg(l,d) by A72,PSCOMP_1:67,XBOOLE_1:17;
A76: W-bound LJ = E-bound LJ by A37,A38,SPRECT_1:15;
A77: W-bound LSeg(l,d) = d`1 by A22,Lm22,SPRECT_1:54;
then W-bound LSeg(l,d) = W-bound LJ by A22,A74,A75,A76,Lm22,SPRECT_1:57;
then q in Vertical_Line ((W-bound LJ + E-bound LJ) / 2)
by A73,A76,A77,JORDAN6:31;
then q in LJ /\ Vertical_Line ((W-bound LJ + E-bound LJ) / 2)
by A72,XBOOLE_0:def 4;
then
A78: q`2 <= k`2 by JORDAN21:28;
k`2 <= q`2 by A22,A44,A56,A57,A59,A67,JGRAPH_6:1;
then k`2 = q`2 by A78,XXREAL_0:1;
then k = q by A12,A13,A44,A69,TOPREAL3:6;
then q in {l,k} by TARSKI:def 2;
hence contradiction by A65,XBOOLE_0:def 5;
end;
end;
then reconsider X = LSeg(l,k) \ {l,k} as Subset of T2|C` by PRE_TOPC:8;
now
assume x in {l,k};
then x = l or x = k by TARSKI:def 2;
hence contradiction by A61,Th1;
end;
then
A79: x in LSeg(l,k) \ {l,k} by A51,XBOOLE_0:def 5;
then Component_of(x,C`) = Component_of Down(x,C`) by A64,CONNSP_3:27;
then
A80: x in Component_of Down(x,C`) by A64,A79,CONNSP_3:26;
then
A81: X meets Ux by A10,A79,XBOOLE_0:3;
LSeg(l,k) \ {l,k} is convex by JORDAN1:46;
then X is connected by CONNSP_1:23;
then
A82: X c= Component_of Down(x,C`) by A10,A52,A81,CONNSP_1:36;
A83: LSeg(l,k) c= R by A11,A20,A42,JORDAN1:def 1;
A84: the carrier of TR = R by PRE_TOPC:8;
reconsider AR = a, BR = b, CR = c, DR = d
as Point of TR by A11,A14,A15,Lm62,Lm63,Lm67,PRE_TOPC:8;
consider Pcm being Path of c,m, fcm being Function of I[01], T2|LSeg(c,m)
such that
A85: rng fcm = LSeg(c,m) and
A86: Pcm = fcm by Th43;
A87: LSeg(c,m) c= R by A11,A16,Lm62,Lm67,JORDAN1:def 1;
A88: ml c= R by A11,A31;
thus Ux is_inside_component_of C
proof
thus
A89: Ux is_a_component_of C` by A10,A52;
assume not Ux is bounded;
then not Ux c= Ball(x,10) by RLTOPSP1:42;
then consider u being object such that
A90: u in Ux and
A91: not u in Ball(x,10);
A92: R c= Ball(x,10) by A51,A83,Lm89;
reconsider u as Point of T2 by A90;
A93: Ux is open by A89,SPRECT_3:8;
Component_of Down(x,C`) is connected by A52;
then
A94: Ux is connected by A10,CONNSP_1:23;
x in Ball(x,10) by Th16;
then consider P1 being Subset of T2 such that
A95: P1 is_S-P_arc_joining x,u and
A96: P1 c= Ux by A10,A80,A90,A91,A93,A94,TOPREAL4:29;
A97: P1 is_an_arc_of x,u by A95,TOPREAL4:2;
reconsider P2 = P1 as Subset of T2|C` by A10,A96,XBOOLE_1:1;
A98: P2 c= Component_of Down(x,C`) by A10,A96;
A99: P2 misses C by A53,SUBSET_1:23;
then
A100: P2 misses Jc by A4,XBOOLE_1:7,63;
A101: P2 misses Jd by A4,A99,XBOOLE_1:7,63;
A102: x`1 = 1/2*((k+l)`1) by TOPREAL3:4
.= 1/2*(k`1+l`1) by TOPREAL3:2
.= 0 by A22,A44;
then
A103: LSeg(d,x) is vertical by Lm22,SPPOL_1:16;
A104: x = |[x`1,x`2]| by EUCLID:53;
A105: x`2 < l`2 by A62,Th3;
A106: k`2 < x`2 by A62,Th2;
then
A107: d`2 <= x`2 by A1,A18,A41,Lm23,Th84,XXREAL_0:2;
d`1 = d`1;
then
A108: LSeg(d,x) c= LSeg(d,l) by A33,A103,A105,A107,GOBOARD7:63;
A109: LSeg(d,x) misses Jc
proof
assume not thesis;
then consider q being object such that
A110: q in LSeg(d,x) and
A111: q in Jc by XBOOLE_0:3;
reconsider q as Point of T2 by A110;
q`2 <= x`2 by A58,A102,A104,A107,A110,Lm22,JGRAPH_6:1;
then
A112: q`2 < l`2 by A105,XXREAL_0:2;
q`1 = 0 by A33,A43,A108,A110,Lm22;
then q in Vertical_Line w by A12,A13,JORDAN6:31;
then q in Jc /\ Vertical_Line w by A111,XBOOLE_0:def 4;
hence contradiction by A8,A9,A112,JORDAN21:29;
end;
set n = First_Point(P1,x,u,dR);
A113: not u in R by A91,A92;
A114: Fr R = dR by Th52;
u in P1 by A97,TOPREAL1:1;
then
A115: P1 \ R <> {}T2 by A113,XBOOLE_0:def 5;
x in P1 by A97,TOPREAL1:1;
then P1 meets R by A51,A83,XBOOLE_0:3;
then
A116: P1 meets dR by A97,A114,A115,CONNSP_1:22,JORDAN6:10;
P1 is closed by A95,JORDAN6:11,TOPREAL4:2;
then
A117: n in P1 /\ dR by A97,A116,JORDAN5C:def 1;
then
A118: n in dR by XBOOLE_0:def 4;
A119: n in P1 by A117,XBOOLE_0:def 4;
set alpha = Segment(P1,x,u,x,n);
A120: rd < k`2 by A1,A18,A41,Th84;
l`2 <= m`2 by A36,JORDAN21:28;
then x`2 < m`2 by A105,XXREAL_0:2;
then not x in dR by A21,A102,A104,A106,A120,Lm86;
then
A121: alpha is_an_arc_of x,n by A95,A118,A119,JORDAN16:24,TOPREAL4:2;
A122: alpha misses Jc by A100,JORDAN16:2,XBOOLE_1:63;
A123: alpha misses Jd by A101,JORDAN16:2,XBOOLE_1:63;
consider Pdx being Path of d,x,
fdx being Function of I[01], T2|LSeg(d,x) such that
A124: rng fdx = LSeg(d,x) and
A125: Pdx = fdx by Th43;
consider PJc being Path of a,b, fJc being Function
of I[01], T2|Jc such that
A126: rng fJc = Jc and
A127: PJc = fJc by A2,Th42;
consider PJd being Path of a,b, fJd being Function
of I[01], T2|Jd such that
A128: rng fJd = Jd and
A129: PJd = fJd by A3,Th42;
consider Palpha being Path of x,n,
falpha being Function of I[01], T2|alpha such that
A130: rng falpha = alpha and
A131: Palpha = falpha by A121,Th42;
n in R by A118,Lm67;
then
A132: ex p st p = n & rl <= p`1 & p`1 <= rp & rd <= p`2 & p`2 <= rg;
rng PJc c= the carrier of TR by A11,A19,A84,A126,A127;
then reconsider h = PJc as Path of AR,BR by Th30;
rng PJd c= the carrier of TR by A11,A18,A84,A128,A129;
then reconsider H = PJd as Path of AR,BR by Th30;
A133: LSeg(d,x) c= R by A51,A83,Lm63,Lm67,JORDAN1:def 1;
A134: alpha c= R by A51,A83,A95,A113,Th57,TOPREAL4:2;
A135: ld in LSeg(ld,lg) by RLTOPSP1:68;
A136: pd in LSeg(pd,pg) by RLTOPSP1:68;
LSeg(lg,c) misses C by A1,Lm78;
then
A137: LSeg(lg,c) misses Jc by A4,XBOOLE_1:7,63;
A138: LSeg(lg,c) c= R by Lm67,Lm70;
A139: LSeg(pg,c) c= R by Lm67,Lm71;
LSeg(pg,c) misses C by A1,Lm79;
then
A140: LSeg(pg,c) misses Jc by A4,XBOOLE_1:7,63;
consider Plx being Path of l,x, flx being Function of I[01], T2|LSeg(l,x)
such that
A141: rng flx = LSeg(l,x) and
A142: Plx = flx by Th43;
set PCX = Pcm + Pml + Plx;
A143: rng PCX = rng Pcm \/ rng Pml \/ rng Plx by Th40;
A144: ml misses Jd
proof
assume ml meets Jd;
then consider q being object such that
A145: q in ml and
A146: q in Jd by XBOOLE_0:3;
q in {a,b} by A5,A29,A145,A146,XBOOLE_0:def 4;
hence contradiction by A30,A145,XBOOLE_0:3;
end;
A147: LSeg(c,m) /\ C = {m} by A1,Th91;
A148: LSeg(c,m) misses Jd
proof
assume LSeg(c,m) meets Jd;
then consider q being object such that
A149: q in LSeg(c,m) and
A150: q in Jd by XBOOLE_0:3;
q in {m} by A18,A147,A149,A150,XBOOLE_0:def 4;
then q = m by TARSKI:def 1;
then m in {a,b} by A5,A6,A150,XBOOLE_0:def 4;
hence contradiction by A25,A26,TARSKI:def 2;
end;
LSeg(l,x) is vertical by A22,A102,SPPOL_1:16;
then
A151: LSeg(l,x) c= LSeg(l,k) by A44,A54,A102,A105,A106,GOBOARD7:63;
l in LSeg(l,x) by RLTOPSP1:68;
then {l} c= LSeg(l,x) by ZFMISC_1:31;
then
A152: LSeg(l,x) = LSeg(l,x) \ {l} \/ {l} by XBOOLE_1:45;
LSeg(l,x) \ {l} c= LSeg(l,k) \ {l,k}
proof
let q be object;
assume
A153: q in LSeg(l,x) \ {l};
then
A154: q in LSeg(l,x) by ZFMISC_1:56;
A155: q <> l by A153,ZFMISC_1:56;
q <> k by A22,A56,A102,A104,A105,A106,A154,JGRAPH_6:1;
then not q in {l,k} by A155,TARSKI:def 2;
hence thesis by A151,A154,XBOOLE_0:def 5;
end;
then LSeg(l,x) \ {l} c= C` by A64;
then LSeg(l,x) \ {l} misses C by SUBSET_1:23;
then
A156: LSeg(l,x) \ {l} misses Jd by A4,XBOOLE_1:7,63;
{l} misses Jd
proof
assume {l} meets Jd;
then l in Jd by ZFMISC_1:50;
then l in {a,b} by A5,A17,XBOOLE_0:def 4;
hence thesis by A27,A28,TARSKI:def 2;
end;
then LSeg(l,x) misses Jd by A152,A156,XBOOLE_1:70;
then
A157: rng PCX misses Jd by A85,A86,A141,A142,A143,A144,A148,XBOOLE_1:114;
LSeg(l,x) c= R by A83,A151;
then
A158: rng PCX c= R by A85,A86,A87,A88,A141,A142,A143,Lm1;
LSeg(ld,d) misses C by A1,Lm80;
then
A159: LSeg(ld,d) misses Jd by A4,XBOOLE_1:7,63;
LSeg(pd,d) misses C by A1,Lm81;
then
A160: LSeg(pd,d) misses Jd by A4,XBOOLE_1:7,63;
per cases;
suppose
A161: n`2 < 0;
per cases by A118,A161,Lm77;
suppose
A162: n in LSeg(a,ld);
consider Pnld being Path of n,ld,
fnld being Function of I[01], T2|LSeg(n,ld) such that
A163: rng fnld = LSeg(n,ld) and
A164: Pnld = fnld by Th43;
consider Pldd being Path of ld,d,
fldd being Function of I[01], T2|LSeg(ld,d) such that
A165: rng fldd = LSeg(ld,d) and
A166: Pldd = fldd by Th43;
A167: ld`1 = n`1 by A135,A162,Lm45,Lm58;
then LSeg(n,ld) is vertical by SPPOL_1:16;
then LSeg(n,ld) c= LSeg(ld,lg) by A132,A167,Lm25,Lm27,Lm45,GOBOARD7:63;
then
A168: LSeg(n,ld) c= dR by Lm38;
set K1 = PCX + Palpha + Pnld + Pldd;
LSeg(n,ld) misses C by A1,A53,A98,A119,A162,Lm84;
then
A169: LSeg(n,ld) misses Jd by A4,XBOOLE_1:7,63;
A170: rng K1 = rng PCX \/ rng Palpha \/ rng Pnld \/ rng Pldd by Lm9;
then
A171: rng PJd misses rng K1 by A123,A128,A129,A130,A131,A157,A159,A163,A164
,A165,A166,A169,Lm3;
A172: LSeg(ld,d) c= R by Lm67,Lm74;
LSeg(n,ld) c= R by A168,Lm67;
then rng K1 c= the carrier of TR
by A84,A130,A131,A134,A158,A163,A164,A165,A166,A170,A172,Lm2;
then reconsider v = K1 as Path of CR,DR by Th30;
consider s, t being Point of I[01] such that
A173: H.s = v.t by Lm16,Lm17,Lm21,Lm23,JGRAPH_8:6;
A174: dom H = the carrier of I[01] by FUNCT_2:def 1;
A175: dom v = the carrier of I[01] by FUNCT_2:def 1;
A176: H.s in rng PJd by A174,FUNCT_1:def 3;
v.t in rng K1 by A175,FUNCT_1:def 3;
hence contradiction by A171,A173,A176,XBOOLE_0:3;
end;
suppose
A177: n in LSeg(ld,d);
consider Pnd being Path of n,d,
fnd being Function of I[01], T2|LSeg(n,d) such that
A178: rng fnd = LSeg(n,d) and
A179: Pnd = fnd by Th43;
set K1 = PCX + Palpha + Pnd;
ld in LSeg(ld,d) by RLTOPSP1:68;
then
A180: ld`2 = n`2 by A177,Lm51;
then
A181: LSeg(n,d) is horizontal by Lm23,Lm27,SPPOL_1:15;
A182: ld`1 <= n`1 by A177,Lm26,JGRAPH_6:3;
n`1 <= d`1 by A177,Lm22,JGRAPH_6:3;
then
A183: LSeg(n,d) c= LSeg(ld,d) by A180,A181,A182,Lm51,GOBOARD7:64;
then
A184: LSeg(n,d) c= dR by Lm74;
LSeg(n,d) misses C by A1,A183,Lm80,XBOOLE_1:63;
then
A185: LSeg(n,d) misses Jd by A4,XBOOLE_1:7,63;
A186: rng K1 = rng PCX \/ rng Palpha \/ rng Pnd by Th40;
then
A187: rng K1 misses Jd by A123,A130,A131,A157,A178,A179,A185,XBOOLE_1:114;
LSeg(n,d) c= R by A184,Lm67;
then rng K1 c= the carrier of TR by A84,A130,A131,A134,A158,A178,A179
,A186,Lm1;
then reconsider v = K1 as Path of CR,DR by Th30;
consider s, t being Point of I[01] such that
A188: H.s = v.t by Lm16,Lm17,Lm21,Lm23,JGRAPH_8:6;
A189: dom H = the carrier of I[01] by FUNCT_2:def 1;
A190: dom v = the carrier of I[01] by FUNCT_2:def 1;
A191: H.s in rng PJd by A189,FUNCT_1:def 3;
v.t in rng K1 by A190,FUNCT_1:def 3;
hence contradiction by A128,A129,A187,A188,A191,XBOOLE_0:3;
end;
suppose
A192: n in LSeg(d,pd);
consider Pnd being Path of n,d,
fnd being Function of I[01], T2|LSeg(n,d) such that
A193: rng fnd = LSeg(n,d) and
A194: Pnd = fnd by Th43;
set K1 = PCX + Palpha + Pnd;
pd in LSeg(pd,d) by RLTOPSP1:68;
then pd`2 = n`2 by A192,Lm52;
then
A195: LSeg(n,d) is horizontal by Lm23,Lm31,SPPOL_1:15;
A196: d`2 = d`2;
A197: d`1 <= n`1 by A192,Lm22,JGRAPH_6:3;
n`1 <= pd`1 by A192,Lm30,JGRAPH_6:3;
then
A198: LSeg(n,d) c= LSeg(pd,d) by A195,A196,A197,Lm52,GOBOARD7:64;
then
A199: LSeg(n,d) c= dR by Lm75;
LSeg(n,d) misses C by A1,A198,Lm81,XBOOLE_1:63;
then
A200: LSeg(n,d) misses Jd by A4,XBOOLE_1:7,63;
A201: rng K1 = rng PCX \/ rng Palpha \/ rng Pnd by Th40;
then
A202: rng K1 misses Jd by A123,A130,A131,A157,A193,A194,A200,XBOOLE_1:114;
LSeg(n,d) c= R by A199,Lm67;
then rng K1 c= the carrier of TR by A84,A130,A131,A134,A158,A193,A194
,A201,Lm1;
then reconsider v = K1 as Path of CR,DR by Th30;
consider s, t being Point of I[01] such that
A203: H.s = v.t by Lm16,Lm17,Lm21,Lm23,JGRAPH_8:6;
A204: dom H = the carrier of I[01] by FUNCT_2:def 1;
A205: dom v = the carrier of I[01] by FUNCT_2:def 1;
A206: H.s in rng PJd by A204,FUNCT_1:def 3;
v.t in rng K1 by A205,FUNCT_1:def 3;
hence contradiction by A128,A129,A202,A203,A206,XBOOLE_0:3;
end;
suppose
A207: n in LSeg(pd,b);
consider Pnpd being Path of n,pd,
fnpd being Function of I[01], T2|LSeg(n,pd) such that
A208: rng fnpd = LSeg(n,pd) and
A209: Pnpd = fnpd by Th43;
consider Ppdd being Path of pd,d,
fpdd being Function of I[01], T2|LSeg(pd,d) such that
A210: rng fpdd = LSeg(pd,d) and
A211: Ppdd = fpdd by Th43;
A212: pd`1 = n`1 by A136,A207,Lm46,Lm60;
then LSeg(n,pd) is vertical by SPPOL_1:16;
then LSeg(n,pd) c= LSeg(pd,pg) by A132,A212,Lm29,Lm31,Lm46,GOBOARD7:63;
then
A213: LSeg(n,pd) c= dR by Lm42;
set K1 = PCX + Palpha + Pnpd + Ppdd;
LSeg(n,pd) misses C by A1,A53,A98,A119,A207,Lm85;
then
A214: LSeg(n,pd) misses Jd by A4,XBOOLE_1:7,63;
A215: rng K1 = rng PCX \/ rng Palpha \/ rng Pnpd \/ rng Ppdd by Lm9;
then
A216: rng PJd misses rng K1 by A123,A128,A129,A130,A131,A157,A160,A208,A209
,A210,A211,A214,Lm3;
A217: LSeg(pd,d) c= R by Lm67,Lm75;
LSeg(n,pd) c= R by A213,Lm67;
then rng K1 c= the carrier of TR
by A84,A130,A131,A134,A158,A208,A209,A210,A211,A215,A217,Lm2;
then reconsider v = K1 as Path of CR,DR by Th30;
consider s, t being Point of I[01] such that
A218: H.s = v.t by Lm16,Lm17,Lm21,Lm23,JGRAPH_8:6;
A219: dom H = the carrier of I[01] by FUNCT_2:def 1;
A220: dom v = the carrier of I[01] by FUNCT_2:def 1;
A221: H.s in rng PJd by A219,FUNCT_1:def 3;
v.t in rng K1 by A220,FUNCT_1:def 3;
hence contradiction by A216,A218,A221,XBOOLE_0:3;
end;
end;
suppose
A222: n`2 >= 0;
per cases by A118,A222,Lm76;
suppose
A223: n in LSeg(a,lg);
consider Pnlg being Path of n,lg,
fnlg being Function of I[01], T2|LSeg(n,lg) such that
A224: rng fnlg = LSeg(n,lg) and
A225: Pnlg = fnlg by Th43;
consider Plgc being Path of lg,c,
flgc being Function of I[01], T2|LSeg(lg,c) such that
A226: rng flgc = LSeg(lg,c) and
A227: Plgc = flgc by Th43;
A228: ld`1 = n`1 by A135,A223,Lm45,Lm57;
then LSeg(n,lg) is vertical by Lm24,Lm26,SPPOL_1:16;
then LSeg(n,lg) c= LSeg(ld,lg) by A132,A228,Lm25,Lm27,Lm45,GOBOARD7:63;
then
A229: LSeg(n,lg) c= dR by Lm38;
set K1 = Pdx + Palpha + Pnlg + Plgc;
LSeg(n,lg) misses C by A1,A53,A98,A119,A223,Lm82;
then
A230: LSeg(n,lg) misses Jc by A4,XBOOLE_1:7,63;
A231: rng K1 = rng Pdx \/ rng Palpha \/ rng Pnlg \/ rng Plgc by Lm9;
then
A232: rng K1 misses Jc by A109,A122,A124,A125,A130,A131,A137,A224,A225,A226
,A227,A230,Lm3;
A233: rng K1 = rng -K1 by Th32;
LSeg(n,lg) c= R by A229,Lm67;
then rng K1 c= the carrier of TR
by A84,A124,A125,A130,A131,A133,A134,A138,A224,A225,A226,A227,A231,Lm2;
then reconsider v = -K1 as Path of CR,DR by A233,Th30;
consider s, t being Point of I[01] such that
A234: h.s = v.t by Lm16,Lm17,Lm21,Lm23,JGRAPH_8:6;
A235: dom h = the carrier of I[01] by FUNCT_2:def 1;
A236: dom v = the carrier of I[01] by FUNCT_2:def 1;
A237: h.s in rng PJc by A235,FUNCT_1:def 3;
v.t in rng -K1 by A236,FUNCT_1:def 3;
hence contradiction by A126,A127,A232,A233,A234,A237,XBOOLE_0:3;
end;
suppose
A238: n in LSeg(lg,c);
consider Pnc being Path of n,c,
fnc being Function of I[01], T2|LSeg(n,c) such that
A239: rng fnc = LSeg(n,c) and
A240: Pnc = fnc by Th43;
set K1 = Pdx + Palpha + Pnc;
lg in LSeg(lg,c) by RLTOPSP1:68;
then
A241: lg`2 = n`2 by A238,Lm53;
then
A242: LSeg(n,c) is horizontal by Lm21,Lm25,SPPOL_1:15;
A243: lg`1 <= n`1 by A238,Lm24,JGRAPH_6:3;
n`1 <= c`1 by A238,Lm20,JGRAPH_6:3;
then
A244: LSeg(n,c) c= LSeg(lg,c) by A241,A242,A243,Lm53,GOBOARD7:64;
then
A245: LSeg(n,c) c= dR by Lm70;
LSeg(n,c) misses C by A1,A244,Lm78,XBOOLE_1:63;
then
A246: LSeg(n,c) misses Jc by A4,XBOOLE_1:7,63;
A247: rng K1 = rng Pdx \/ rng Palpha \/ rng Pnc by Th40;
then
A248: rng K1 misses Jc by A109,A122,A124,A125,A130,A131,A239,A240,A246,
XBOOLE_1:114;
A249: rng K1 = rng -K1 by Th32;
LSeg(n,c) c= R by A245,Lm67;
then rng K1 c= the carrier of TR
by A84,A124,A125,A130,A131,A133,A134,A239,A240,A247,Lm1;
then reconsider v = -K1 as Path of CR,DR by A249,Th30;
consider s, t being Point of I[01] such that
A250: h.s = v.t by Lm16,Lm17,Lm21,Lm23,JGRAPH_8:6;
A251: dom h = the carrier of I[01] by FUNCT_2:def 1;
A252: dom v = the carrier of I[01] by FUNCT_2:def 1;
A253: h.s in rng PJc by A251,FUNCT_1:def 3;
v.t in rng -K1 by A252,FUNCT_1:def 3;
hence contradiction by A126,A127,A248,A249,A250,A253,XBOOLE_0:3;
end;
suppose
A254: n in LSeg(c,pg);
consider Pnc being Path of n,c,
fnc being Function of I[01], T2|LSeg(n,c) such that
A255: rng fnc = LSeg(n,c) and
A256: Pnc = fnc by Th43;
set K1 = Pdx + Palpha + Pnc;
pg in LSeg(pg,c) by RLTOPSP1:68;
then pg`2 = n`2 by A254,Lm54;
then
A257: LSeg(n,c) is horizontal by Lm21,Lm29,SPPOL_1:15;
A258: c`2 = c`2;
A259: c`1 <= n`1 by A254,Lm20,JGRAPH_6:3;
n`1 <= pg`1 by A254,Lm28,JGRAPH_6:3;
then
A260: LSeg(c,n) c= LSeg(c,pg) by A257,A258,A259,Lm54,GOBOARD7:64;
then
A261: LSeg(n,c) c= dR by Lm71;
LSeg(n,c) misses C by A1,A260,Lm79,XBOOLE_1:63;
then
A262: LSeg(n,c) misses Jc by A4,XBOOLE_1:7,63;
A263: rng K1 = rng Pdx \/ rng Palpha \/ rng Pnc by Th40;
then
A264: rng K1 misses Jc by A109,A122,A124,A125,A130,A131,A255,A256,A262,
XBOOLE_1:114;
A265: rng K1 = rng -K1 by Th32;
LSeg(n,c) c= R by A261,Lm67;
then rng K1 c= the carrier of TR
by A84,A124,A125,A130,A131,A133,A134,A255,A256,A263,Lm1;
then reconsider v = -K1 as Path of CR,DR by A265,Th30;
consider s, t being Point of I[01] such that
A266: h.s = v.t by Lm16,Lm17,Lm21,Lm23,JGRAPH_8:6;
A267: dom h = the carrier of I[01] by FUNCT_2:def 1;
A268: dom v = the carrier of I[01] by FUNCT_2:def 1;
A269: h.s in rng PJc by A267,FUNCT_1:def 3;
v.t in rng -K1 by A268,FUNCT_1:def 3;
hence contradiction by A126,A127,A264,A265,A266,A269,XBOOLE_0:3;
end;
suppose
A270: n in LSeg(pg,b);
consider Pnpg being Path of n,pg,
fnpg being Function of I[01], T2|LSeg(n,pg) such that
A271: rng fnpg = LSeg(n,pg) and
A272: Pnpg = fnpg by Th43;
consider Ppgc being Path of pg,c,
fpgc being Function of I[01], T2|LSeg(pg,c) such that
A273: rng fpgc = LSeg(pg,c) and
A274: Ppgc = fpgc by Th43;
A275: pd`1 = n`1 by A136,A270,Lm46,Lm59;
then LSeg(n,pg) is vertical by Lm28,Lm30,SPPOL_1:16;
then LSeg(n,pg) c= LSeg(pd,pg) by A132,A275,Lm29,Lm31,Lm46,GOBOARD7:63;
then
A276: LSeg(n,pg) c= dR by Lm42;
set K1 = Pdx + Palpha + Pnpg + Ppgc;
LSeg(n,pg) misses C by A1,A53,A98,A119,A270,Lm83;
then
A277: LSeg(n,pg) misses Jc by A4,XBOOLE_1:7,63;
A278: rng K1 = rng Pdx \/ rng Palpha \/ rng Pnpg \/ rng Ppgc by Lm9;
then
A279: rng K1 misses Jc by A109,A122,A124,A125,A130,A131,A140,A271,A272,A273
,A274,A277,Lm3;
A280: rng K1 = rng -K1 by Th32;
LSeg(n,pg) c= R by A276,Lm67;
then rng K1 c= the carrier of TR
by A84,A124,A125,A130,A131,A133,A134,A139,A271,A272,A273,A274,A278,Lm2;
then reconsider v = -K1 as Path of CR,DR by A280,Th30;
consider s, t being Point of I[01] such that
A281: h.s = v.t by Lm16,Lm17,Lm21,Lm23,JGRAPH_8:6;
A282: dom h = the carrier of I[01] by FUNCT_2:def 1;
A283: dom v = the carrier of I[01] by FUNCT_2:def 1;
A284: h.s in rng PJc by A282,FUNCT_1:def 3;
v.t in rng -K1 by A283,FUNCT_1:def 3;
hence contradiction by A126,A127,A279,A280,A281,A284,XBOOLE_0:3;
end;
end;
end;
:: uniqueness
let V be Subset of T2;
assume
A285: V is_inside_component_of C;
assume
A286: V <> Ux;
consider VP being Subset of T2|C` such that
A287: VP = V and
A288: VP is a_component and
VP is bounded Subset of Euclid 2 by A285,JORDAN2C:13;
reconsider T2C = T2|C` as non empty SubSpace of T2;
VP <> {}(T2|C`) by A288,CONNSP_1:32;
then reconsider VP as non empty Subset of T2C;
A289: V misses C by A53,A287,SUBSET_1:23;
consider Pjd being Path of j,d,
fjd being Function of I[01], T2|LSeg(j,d) such that
A290: rng fjd = LSeg(j,d) and
A291: Pjd = fjd by Th43;
consider Plk being Path of l,k,
flk being Function of I[01], T2|LSeg(l,k) such that
A292: rng flk = LSeg(l,k) and
A293: Plk = flk by Th43;
set beta = Pcm + Pml + Plk + Pkj + Pjd;
A294: rng beta = rng Pcm \/ rng Pml \/ rng Plk \/ rng Pkj \/ rng Pjd by Lm11;
dom beta = [#]I[01] by FUNCT_2:def 1;
then beta.:dom beta is compact by WEIERSTR:8;
then
A295: rng beta is closed by RELAT_1:113;
A296: ml misses V by A19,A29,A289,XBOOLE_1:1,63;
{l,k} c= LSeg(l,k)
proof
let x be object;
assume x in {l,k};
then x = l or x = k by TARSKI:def 2;
hence thesis by RLTOPSP1:68;
end;
then
A297: LSeg(l,k) = LSeg(l,k) \ {l,k} \/ {l,k} by XBOOLE_1:45;
A298: LSeg(l,k) \ {l,k} misses V
proof
assume not thesis;
then ex q being object st ( q in LSeg(l,k) \ {l,k})&( q in V)
by XBOOLE_0:3;
then V meets Ux by A10,A82,XBOOLE_0:3;
hence contradiction by A10,A52,A286,A287,A288,CONNSP_1:35;
end;
A299: not l in V by A17,A19,A289,XBOOLE_0:3;
not k in V by A18,A41,A289,XBOOLE_0:3;
then {l,k} misses V by A299,ZFMISC_1:51;
then
A300: LSeg(l,k) misses V by A297,A298,XBOOLE_1:70;
A301: kj misses V by A50,A289,XBOOLE_1:63;
A302: LSeg(j,d) misses V by A1,A285,Th90;
LSeg(c,m) misses V by A1,A285,Th89;
then LSeg(c,m) \/ ml \/ LSeg(l,k) misses V by A296,A300,XBOOLE_1:114;
then
A303: rng beta misses V by A85,A86,A290,A291,A292,A293,A294,A301,A302,
XBOOLE_1:114;
A304: m = |[m`1,m`2]| by EUCLID:53;
A305: c = |[c`1,c`2]| by EUCLID:53;
A306: j = |[j`1,j`2]| by EUCLID:53;
A307: not a in LSeg(c,m) by A12,A13,A21,A23,A24,A304,A305,Lm16,JGRAPH_6:1;
not a in ml by A30,ZFMISC_1:49;
then
A308: not a in LSeg(c,m) \/ ml by A307,XBOOLE_0:def 3;
not a in LSeg(l,k) by A22,A44,A56,A57,A59,Lm16,JGRAPH_6:1;
then
A309: not a in LSeg(c,m) \/ ml \/ LSeg(l,k) by A308,XBOOLE_0:def 3;
not a in kj by A49,ZFMISC_1:49;
then
A310: not a in LSeg(c,m) \/ ml \/ LSeg(l,k) \/ kj by A309,XBOOLE_0:def 3;
not a in LSeg(j,d) by A34,A35,A58,A306,Lm16,Lm22,JGRAPH_6:1;
then not a in rng beta by A85,A86,A290,A291,A292,A293,A294,A310,
XBOOLE_0:def 3;
then consider ra being positive Real such that
A311: Ball(a,ra) misses rng beta by A295,Th25;
A312: not b in LSeg(c,m) by A12,A13,A21,A23,A24,A304,A305,Lm17,JGRAPH_6:1;
not b in ml by A30,ZFMISC_1:49;
then
A313: not b in LSeg(c,m) \/ ml by A312,XBOOLE_0:def 3;
not b in LSeg(l,k) by A22,A44,A56,A57,A59,Lm17,JGRAPH_6:1;
then
A314: not b in LSeg(c,m) \/ ml \/ LSeg(l,k) by A313,XBOOLE_0:def 3;
not b in kj by A49,ZFMISC_1:49;
then
A315: not b in LSeg(c,m) \/ ml \/ LSeg(l,k) \/ kj by A314,XBOOLE_0:def 3;
not b in LSeg(j,d) by A34,A35,A58,A306,Lm17,Lm22,JGRAPH_6:1;
then not b in rng beta by A85,A86,A290,A291,A292,A293,A294,A315,
XBOOLE_0:def 3;
then consider rb being positive Real such that
A316: Ball(b,rb) misses rng beta by A295,Th25;
set A = Ball(a,ra), B = Ball(b,rb);
A317: a in A by Th16;
A318: b in B by Th16;
VP is non empty;
then consider t being object such that
A319: t in V by A287;
V in {W where W is Subset of T2: W is_inside_component_of C} by A285;
then t in BDD C by A319,TARSKI:def 4;
then
A320: C = Fr V by A287,A288,Lm15;
then a in Cl V by A14,XBOOLE_0:def 4;
then A meets V by A317,PRE_TOPC:def 7;
then consider u being object such that
A321: u in A and
A322: u in V by XBOOLE_0:3;
b in Cl V by A15,A320,XBOOLE_0:def 4;
then B meets V by A318,PRE_TOPC:def 7;
then consider v being object such that
A323: v in B and
A324: v in V by XBOOLE_0:3;
reconsider u, v as Point of T2 by A321,A323;
A325: the carrier of T2C|VP = VP by PRE_TOPC:8;
reconsider u1 = u, v1 = v as Point of T2C|VP by A287,A322,A324,PRE_TOPC:8;
T2C|VP is pathwise_connected by A288,Th69;
then
A326: u1,v1 are_connected;
then consider fuv being Function of I[01], T2C|VP such that
A327: fuv is continuous and
A328: fuv.0 = u1 and
A329: fuv.1 = v1;
A330: T2C|VP = T2|V by A287,GOBOARD9:2;
fuv is Path of u1,v1 by A326,A327,A328,A329,BORSUK_2:def 2;
then reconsider uv = fuv as Path of u,v by A326,A330,TOPALG_2:1;
A331: rng fuv c= the carrier of T2C|VP;
then
A332: rng uv misses rng beta by A287,A303,A325,XBOOLE_1:63;
consider au being Path of a,u,
fau being Function of I[01], T2|LSeg(a,u) such that
A333: rng fau = LSeg(a,u) and
A334: au = fau by Th43;
consider vb being Path of v,b,
fvb being Function of I[01], T2|LSeg(v,b) such that
A335: rng fvb = LSeg(v,b) and
A336: vb = fvb by Th43;
set AB = au + uv + vb;
A337: rng AB = rng au \/ rng uv \/ rng vb by Th40;
a in A by Th16;
then LSeg(a,u) c= A by A321,JORDAN1:def 1;
then
A338: LSeg(a,u) misses rng beta by A311,XBOOLE_1:63;
b in B by Th16;
then LSeg(v,b) c= B by A323,JORDAN1:def 1;
then LSeg(v,b) misses rng beta by A316,XBOOLE_1:63;
then
A339: rng AB misses rng beta by A332,A333,A334,A335,A336,A337,A338,XBOOLE_1:114
;
A340: a,b are_connected by BORSUK_2:def 3;
A341: V c= R by A1,A285,Th93;
then
A342: LSeg(a,u) c= R by A11,A14,A322,JORDAN1:def 1;
A343: LSeg(v,b) c= R by A11,A15,A324,A341,JORDAN1:def 1;
rng uv c= R by A287,A325,A331,A341;
then LSeg(a,u) \/ rng uv c= R by A342,XBOOLE_1:8;
then rng AB c= the carrier of TR by A84,A333,A334,A335,A336,A337,A343,
XBOOLE_1:8;
then reconsider h = AB as Path of AR,BR by A340,Th29;
A344: c,d are_connected by BORSUK_2:def 3;
LSeg(c,m) \/ ml c= R by A87,A88,XBOOLE_1:8;
then
A345: LSeg(c,m) \/ ml \/ LSeg(l,k) c= R by A83,XBOOLE_1:8;
kj c= R by A11,A50;
then
A346: LSeg(c,m) \/ ml \/ LSeg(l,k) \/ kj c= R by A345,XBOOLE_1:8;
LSeg(j,d) c= R by A11,A32,Lm63,Lm67,JORDAN1:def 1;
then rng beta c= the carrier of TR by A84,A85,A86,A290,A291,A292,A293,A294
,A346,XBOOLE_1:8;
then reconsider v = beta as Path of CR,DR by A344,Th29;
consider s, t being Point of I[01] such that
A347: h.s = v.t by Lm16,Lm17,Lm21,Lm23,JGRAPH_8:6;
A348: dom h = the carrier of I[01] by FUNCT_2:def 1;
A349: dom v = the carrier of I[01] by FUNCT_2:def 1;
A350: h.s in rng AB by A348,FUNCT_1:def 3;
v.t in rng beta by A349,FUNCT_1:def 3;
hence contradiction by A339,A347,A350,XBOOLE_0:3;
end;
theorem Th96:
|[-1,0]|,|[1,0]| realize-max-dist-in C implies
for Jc, Jd being compact with_the_max_arc Subset of TOP-REAL 2 st
Jc is_an_arc_of |[-1,0]|,|[1,0]| & Jd is_an_arc_of |[-1,0]|,|[1,0]| &
C = Jc \/ Jd & Jc /\ Jd = {|[-1,0]|,|[1,0]|} & UMP C in Jc & LMP C in Jd &
W-bound C = W-bound Jc & E-bound C = E-bound Jc holds
BDD C = Component_of Down ((1/2) *
((UMP (LSeg(LMP Jc,|[0,-3]|) /\ Jd)) + LMP Jc), C`)
proof
assume
A1: a,b realize-max-dist-in C;
let Jc, Jd being compact with_the_max_arc Subset of T2 such that
A2: Jc is_an_arc_of a,b and
A3: Jd is_an_arc_of a,b and
A4: C = Jc \/ Jd and
A5: Jc /\ Jd = {a,b} and
A6: UMP C in Jc and
A7: LMP C in Jd and
A8: W-bound C = W-bound Jc and
A9: E-bound C = E-bound Jc;
reconsider
Ux = Component_of Down((1/2) * ((UMP (LSeg(LMP Jc,d) /\ Jd)) + LMP Jc),C`)
as Subset of T2 by PRE_TOPC:11;
Ux = BDD C
proof
Ux is_inside_component_of C by A1,A2,A3,A4,A5,A6,A7,A8,A9,Th95;
hence Ux c= BDD C by JORDAN2C:22;
set F = {B where B is Subset of T2: B is_inside_component_of C};
let q be object;
assume q in BDD C;
then consider Z being set such that
A10: q in Z and
A11: Z in F by TARSKI:def 4;
ex B being Subset of T2 st Z = B & B is_inside_component_of C by A11;
hence thesis by A1,A2,A3,A4,A5,A6,A7,A8,A9,A10,Th95;
end;
hence thesis;
end;
Lm91: |[-1,0]|,|[1,0]| realize-max-dist-in C implies C is Jordan
proof
assume
A1: a,b realize-max-dist-in C;
then consider Jc, Jd being compact with_the_max_arc Subset of T2 such that
A2: Jc is_an_arc_of a,b and
A3: Jd is_an_arc_of a,b and
A4: C = Jc \/ Jd and
A5: Jc /\ Jd = {a,b} and
A6: UMP C in Jc and
A7: LMP C in Jd and
A8: W-bound C = W-bound Jc and
A9: E-bound C = E-bound Jc by Lm90;
set l = LMP Jc;
set LJ = LSeg(l,d) /\ Jd;
set k = UMP LJ;
set x = (1/2)*(k+l);
A10: Component_of Down(x,C`) is a_component by CONNSP_1:40;
A11: Component_of Down(x,C`) = BDD C by A1,A2,A3,A4,A5,A6,A7,A8,A9,Th96;
thus C` <> {};
take A1 = UBD C, A2 = BDD C;
thus C` = A1 \/ A2 by JORDAN2C:27;
thus A1 misses A2 by JORDAN2C:24;
A12: Component_of Down(x,C`) <> {}(T2|C`) by A10,CONNSP_1:32;
A1 is_a_component_of C` by JORDAN2C:124;
then
A13: ex B1 being Subset of T2|C` st B1 = A1 & B1 is a_component;
then
A14: C = Fr A1 by A11,A12,Lm15
.= Cl A1 /\ Cl A1`;
A15: C = Fr A2 by A10,A11,A12,Lm15
.= Cl A2 /\ Cl A2`;
A2 c= C` by JORDAN2C:25;
then C misses A2 by SUBSET_1:23;
then
A16: C c= Cl A2 \ A2 by A15,XBOOLE_1:17,86;
A17: A1 misses A2 by JORDAN2C:24;
then A2 c= A1` by SUBSET_1:23;
then
A18: Cl A2 c= A1` by TOPS_1:5;
A1 \/ A2 = C` by JORDAN2C:27;
then A1 \/ A2 misses C by SUBSET_1:23;
then C misses A1 by XBOOLE_1:70;
then
A19: A2 \/ C misses A1 by A17,XBOOLE_1:70;
A2 \/ A1 = C` by JORDAN2C:27;
then (A2 \/ A1)` misses C` by SUBSET_1:23;
then (A2 \/ A1)` /\ C` = {};
then (A2 \/ A1 \/ C)` = {} by XBOOLE_1:53;
then ((A2 \/ C) \/ A1)` = {} by XBOOLE_1:4;
then (A2 \/ C)` /\ A1` = {} by XBOOLE_1:53;
then (A2 \/ C)` misses A1`;
then Cl A2 c= A2 \/ C by A18,A19,SUBSET_1:25;
then
A20: Cl A2 \ A2 c= C by XBOOLE_1:43;
A1 c= C` by JORDAN2C:26;
then C misses A1 by SUBSET_1:23;
then
A21: C c= Cl A1 \ A1 by A14,XBOOLE_1:17,86;
A1 c= A2` by A17,SUBSET_1:23;
then
A22: Cl A1 c= A2` by TOPS_1:5;
A2 \/ A1 = C` by JORDAN2C:27;
then A2 \/ A1 misses C by SUBSET_1:23;
then C misses A2 by XBOOLE_1:70;
then
A23: A1 \/ C misses A2 by A17,XBOOLE_1:70;
A1 \/ A2 = C` by JORDAN2C:27;
then (A1 \/ A2)` misses C` by SUBSET_1:23;
then (A1 \/ A2)` /\ C` = {};
then (A1 \/ A2 \/ C)` = {} by XBOOLE_1:53;
then ((A1 \/ C) \/ A2)` = {} by XBOOLE_1:4;
then (A1 \/ C)` /\ A2` = {} by XBOOLE_1:53;
then (A1 \/ C)` misses A2`;
then Cl A1 c= A1 \/ C by A22,A23,SUBSET_1:25;
then Cl A1 \ A1 c= C by XBOOLE_1:43;
hence (Cl A1) \ A1 = C by A21
.= (Cl A2) \ A2 by A16,A20;
thus thesis by A11,A13,CONNSP_1:40;
end;
Lm92: C is Jordan
proof
consider f being Homeomorphism of T2 such that
A1: a,b realize-max-dist-in f.:C by JORDAN24:7;
A2: f" is Homeomorphism of T2 by TOPGRP_1:30;
f.:C is Simple_closed_curve by Th70;
then f.:C is Jordan by A1,Lm91;
then
A3: f".:(f.:C) is Jordan by A2,JORDAN24:16;
A4: f" = f qua Function" by TOPS_2:def 4;
dom f = the carrier of T2 by FUNCT_2:def 1;
hence thesis by A3,A4,FUNCT_1:107;
end;
registration :: do not remove it
let C;
cluster BDD C -> non empty;
coherence
proof
C is Jordan by Lm92;
then BDD C is_inside_component_of C by JORDAN2C:108;
then BDD C is_a_component_of C`;
then
ex B1 being Subset of T2|C` st B1 = BDD C & B1 is a_component;
then BDD C <> {}(T2|C`) by CONNSP_1:32;
hence thesis;
end;
end;
theorem
U = P & U is a_component implies C = Fr P
proof
BDD C is non empty;
hence thesis by Lm15;
end;
theorem :: Jordan's Curve Theorem
for C being Simple_closed_curve ex A1, A2 being Subset of TOP-REAL 2 st
C` = A1 \/ A2 & A1 misses A2 & (Cl A1) \ A1 = (Cl A2) \ A2 &
for C1, C2 being Subset of (TOP-REAL 2)|C` st C1 = A1 & C2 = A2 holds
C1 is a_component & C2 is a_component
proof
let C;
C is Jordan by Lm92;
hence thesis;
end;
::$N Jordan Curve Theorem
::$L https://en.wikipedia.org/wiki/Jordan_curve_theorem Jordan Curve Theorem
theorem
for C being Simple_closed_curve holds C is Jordan by Lm92;