:: Fundamental Group of $n$-sphere for $n \geq 2$
:: by Marco Riccardi and Artur Korni{\l}owicz
::
:: Received November 3, 2011
:: Copyright (c) 2011-2017 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 PRE_TOPC, FUNCT_1, EUCLID, RELAT_1, ARYTM_1, BORSUK_2, GRAPH_1,
ALGSTR_1, WAYBEL20, EQREL_1, NAT_1, XBOOLE_0, ARYTM_3, TOPALG_1, TOPS_2,
TOPREALB, T_0TOPSP, STRUCT_0, BORSUK_1, FUNCOP_1, SUBSET_1, TOPALG_3,
TARSKI, XREAL_0, FUNCT_2, XXREAL_0, ZFMISC_1, NUMBERS, RCOMP_1, ORDINAL2,
CARD_1, XXREAL_1, CONNSP_2, TOPMETR, MFOLD_2, PARTFUN1, METRIC_1,
CONVEX1, INCSP_1, RLTOPSP1, RVSUM_1, REAL_1, COMPLEX1, FINSEQ_1,
MEASURE5, XXREAL_2, FINSET_1, FINSEQ_2, MFOLD_1, VALUED_0, SERIES_1,
CARD_3, ORDINAL4, MEMBERED, SUPINF_2, TREAL_1, TOPREAL1, FUNCT_4,
TOPALG_6, BROUWER, XCMPLX_0;
notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, EQREL_1, RELAT_1, COMPLEX1,
FUNCOP_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, RVSUM_1, REAL_1, BINOP_1,
ORDINAL1, NUMBERS, XXREAL_0, XREAL_0, XCMPLX_0, NAT_1, FINSET_1,
STRUCT_0, PRE_TOPC, TOPS_2, BORSUK_1, CONNSP_2, TOPMETR, EUCLID,
BORSUK_2, TOPALG_1, TOPALG_3, TOPREALB, VALUED_0, METRIC_1, TOPREAL9,
TOPALG_2, RLTOPSP1, COMPTS_1, FINSEQ_1, TBSP_1, FINSEQ_2, METRIZTS,
RLVECT_1, XXREAL_1, XXREAL_2, NAT_D, BORSUK_6, MEMBERED, RCOMP_1,
BROUWER, MFOLD_0, MFOLD_1, MFOLD_2, TOPREAL1, FUNCT_4;
constructors BORSUK_6, BORSUK_3, TOPALG_3, SEQ_4, TOPREAL9, TOPREALB, MFOLD_2,
BINOP_2, FUNCSDOM, CONVEX1, MONOID_0, COMPTS_1, TBSP_1, BROUWER,
MFOLD_0, MFOLD_1, METRIZTS, NAT_D, TREAL_1, TOPREAL1, WAYBEL23, REAL_1,
NUMBERS;
registrations RELSET_1, XXREAL_0, XREAL_0, MEMBERED, STRUCT_0, BORSUK_1,
EUCLID, BORSUK_2, TOPALG_1, PRE_TOPC, ZFMISC_1, TOPGRP_1, TOPMETR,
RCOMP_3, XXREAL_2, TEX_1, TOPALG_3, FUNCT_2, FUNCT_1, FUNCOP_1, MFOLD_2,
XBOOLE_0, RELAT_1, PARTFUN1, NAT_1, MONOID_0, VALUED_0, FINSET_1,
FINSEQ_1, FINSEQ_2, NUMBERS, TOPALG_2, TOPREAL9, TBSP_1, MFOLD_0,
MFOLD_1, BORSUK_3, TOPREAL6, TOPREALB, ORDINAL1, CARD_1;
requirements REAL, ARITHM, BOOLE, SUBSET, NUMERALS;
definitions TARSKI, XBOOLE_0, STRUCT_0, T_0TOPSP, BORSUK_2, MEMBERED;
equalities STRUCT_0, TOPALG_1, RLVECT_1,BROUWER;
expansions TARSKI, XBOOLE_0, STRUCT_0, T_0TOPSP, BORSUK_2;
theorems FUNCT_2, XBOOLE_0, ZFMISC_1, TOPALG_1, TARSKI, EQREL_1, BORSUK_2,
BORSUK_6, TOPALG_3, JORDAN, TOPS_2, KNASTER, TOPREALC, PRE_TOPC, TSEP_1,
TOPREALA, XXREAL_1, XXREAL_0, TOPMETR, MFOLD_2, BINOP_1, BORSUK_1, NAT_D,
FUNCOP_1, TOPREALB, T_0TOPSP, TOPALG_2, RLTOPSP1, EUCLID, EUCLID_2,
TOPREAL9, XBOOLE_1, FUNCT_1, JORDAN16, JGRAPH_8, RELAT_1, FINSEQ_1,
FINSEQ_2, NAT_1, XREAL_1, MFOLD_1, METRIZTS, CARD_1, ORDINAL1, EUCLID_7,
PARTFUN1, TBSP_1, METRIC_1, COMPLEX1, JORDAN2C, SUBSET_1, RELSET_1,
XXREAL_2, FINSEQ_3, FINSEQ_4, VALUED_0, RFUNCT_2, TREAL_1, SYSREL,
TOPREAL1, RAMSEY_1, CHORD, XCMPLX_1, BORSUK_3, COMPTS_1, HEINE, FUNCT_4,
FUNCT_3, XREAL_0, XTUPLE_0, NUMBERS, RLVECT_1,MFOLD_0,BROUWER;
schemes NAT_1, FUNCT_1, FINSEQ_1;
begin :: Preliminaries
reserve T,U for non empty TopSpace;
reserve t for Point of T;
reserve n for Nat;
registration
let S be TopSpace, T be non empty TopSpace;
cluster constant -> continuous for Function of S,T;
correctness
proof
let f be Function of S,T;
assume
A1: f is constant;
per cases;
suppose
A2: S is empty;
for P1 being Subset of T st P1 is closed holds f"P1 is closed by A2;
hence thesis by PRE_TOPC:def 6;
end;
suppose S is not empty;
then consider y be Element of T such that
A3: rng f = {y} by A1,FUNCT_2:111;
y in rng f by A3,TARSKI:def 1;
then ex x being object st x in dom f & y = f.x by FUNCT_1:def 3;
then
A4: the_value_of f = y by A1,FUNCT_1:def 12;
f = (dom f) --> the_value_of f by A1,FUNCOP_1:80;
then f = S --> y by A4,FUNCT_2:def 1;
hence thesis;
end;
end;
end;
theorem Th1:
L[01](0,1,0,1) = id Closed-Interval-TSpace(0,1)
proof
L[01](0,1,0,1) =
(id Closed-Interval-TSpace(0,1))*(id Closed-Interval-TSpace(0,1))
by BORSUK_6:def 1,TREAL_1:10,14;
hence thesis by SYSREL:12;
end;
theorem Th2:
for r1,r2,r3,r4,r5,r6 being Real
st r1 < r2 & r3 <= r4 & r5 < r6
holds L[01](r1,r2,r3,r4)*L[01](r5,r6,r1,r2) = L[01](r5,r6,r3,r4)
proof
let r1,r2,r3,r4,r5,r6 be Real;
set f1 = L[01](r1,r2,r3,r4);
set f2 = L[01](r5,r6,r1,r2);
set f3 = L[01](r5,r6,r3,r4);
assume
A1: r1 < r2 & r3 <= r4 & r5 < r6;
A2: dom(f1*f2) = [#]Closed-Interval-TSpace(r5,r6) by FUNCT_2:def 1
.= dom f3 by FUNCT_2:def 1;
for x being object st x in dom(f1*f2) holds (f1*f2).x = f3.x
proof
let x be object;
assume
A3: x in dom(f1*f2);
then
A4: x in [#]Closed-Interval-TSpace(r5,r6);
then
A5: x in [.r5,r6.] by A1,TOPMETR:18;
reconsider r = x as Real by A3;
A6: r5 <= r & r <= r6 by A5,XXREAL_1:1;
A7: rng f2 c= [#]Closed-Interval-TSpace(r1,r2) by RELAT_1:def 19;
reconsider s = f2.x as Real;
x in dom f2 by A4,FUNCT_2:def 1;
then s in [#]Closed-Interval-TSpace(r1,r2) by A7,FUNCT_1:3;
then s in [.r1,r2.] by A1,TOPMETR:18;
then r1 <= s & s <= r2 by XXREAL_1:1;
then
A8: f1.s = (r4 - r3) / (r2 - r1) * (s - r1) + r3 by A1,BORSUK_6:35;
A9: r2 - r1 <> 0 by A1;
A10: (r4 - r3) / (r2 - r1) * s
= (r4 - r3) / (r2 - r1) * ((r2 - r1) / (r6 - r5) * (r - r5) + r1)
by A1,A6,BORSUK_6:35
.= ((r4 - r3) / (r2 - r1)) * ((r2 - r1) / (r6 - r5)) * (r - r5) +
(r4 - r3) / (r2 - r1) * r1
.= ((r4 - r3) / (r6 - r5)) * ((r2 - r1) / (r2 - r1)) * (r - r5) +
(r4 - r3) / (r2 - r1) * r1 by XCMPLX_1:85
.= ((r4 - r3) / (r6 - r5)) * 1 * (r - r5) +
(r4 - r3) / (r2 - r1) * r1 by A9,XCMPLX_1:60
.= (r4 - r3) / (r6 - r5) * (r - r5) + (r4 - r3) / (r2 - r1) * r1;
thus (f1*f2).x = f1.(f2.x) by A3,FUNCT_1:12
.= f3.x by A10,A8,A1,A6,BORSUK_6:35;
end;
hence thesis by A2,FUNCT_1:2;
end;
registration
let n be positive Nat;
cluster TOP-REAL n -> infinite;
correctness
proof
A1: the carrier of TOP-REAL n = REAL n by EUCLID:22
.= n-tuples_on REAL by EUCLID:def 1;
deffunc F(Element of n-tuples_on REAL) = $1.1;
consider f be Function such that
A2: dom f = n-tuples_on REAL &
for d be Element of n-tuples_on REAL holds f.d = F(d) from FUNCT_1:sch 4;
for y being object holds y in f.:(n-tuples_on REAL) iff y in REAL
proof
let y be object;
0+1 < n+1 by XREAL_1:6;
then
A31: 1 <= n by NAT_1:13;
then
A3: 1 in Seg n by FINSEQ_1:1;
hereby
assume y in f.:(n-tuples_on REAL);
then consider x be object such that
A4: x in dom f & x in n-tuples_on REAL & y = f.x by FUNCT_1:def 6;
reconsider x as Element of n-tuples_on REAL by A4;
A5: y = x.1 by A2,A4;
x in Funcs(Seg n,REAL) by A4,FINSEQ_2:93;
then ex f1 being Function st x = f1 & dom f1 = Seg n & rng f1 c= REAL
by FUNCT_2:def 2;
hence y in REAL by A3,A5,FUNCT_1:3;
end;
assume y in REAL;
then
A6: {y} c= REAL by ZFMISC_1:31;
set x = Seg n --> y;
A7: dom x = Seg n & rng x c= {y} by FUNCOP_1:13;
rng x c= REAL by A6;
then x in Funcs(Seg n,REAL) by A7,FUNCT_2:def 2;
then reconsider x as Element of n-tuples_on REAL by FINSEQ_2:93;
f.x = x.1 by A2 .= y by A31,FINSEQ_1:1,FUNCOP_1:7;
hence y in f.:(n-tuples_on REAL) by A2,FUNCT_1:def 6;
end;
hence thesis by A1,TARSKI:2;
end;
cluster n-locally_euclidean -> infinite for non empty TopSpace;
correctness
proof
let M be non empty TopSpace;
set TR=TOP-REAL n;
assume
A8: M is n-locally_euclidean;
consider p be object such that
A9: p in the carrier of M by XBOOLE_0:def 1;
reconsider p as Point of M by A9;
consider U be a_neighborhood of p such that
A10: M|U,Tdisk(0.TR,1) are_homeomorphic by A8,MFOLD_0:def 3;
consider f be Function of Tdisk(0.TR,1), M|U such that
A11: f is being_homeomorphism by A10, T_0TOPSP:def 1;
C: [#]Tdisk(0.TR,1) = cl_Ball(0.TR,1) by PRE_TOPC:def 5;
A12: dom f = [#]Tdisk(0.TR,1) & rng f = [#](M | U)
& f is one-to-one by A11,TOPS_2:58;
Ball(0.TR,1) is non empty ball by MFOLD_1:def 1;
then TR|Ball(0.TR,1),TR| [#]TR are_homeomorphic
by MFOLD_1:10,METRIZTS:def 1;
then consider g be Function of TR|Ball(0.TR,1), TR| [#]TR such that
B11: g is being_homeomorphism;
B12: dom g = [#](TR|Ball(0.TR,1)) & rng g = [#](TR| [#]TR)
& g is one-to-one by B11,TOPS_2:58;
Ball(0.TR,1) is infinite & Ball(0.TR,1) c= cl_Ball(0.TR,1)
by B12,PRE_TOPC:def 5,TOPREAL9:16;
then [#]Tdisk(0.TR,1) is infinite by C;
hence thesis by A12,CARD_1:59;
end;
end;
theorem Th3:
for p being Point of TOP-REAL n st p in Sphere(0.TOP-REAL n,1)
holds -p in Sphere(0.TOP-REAL n,1) \ {p}
proof
let p be Point of TOP-REAL n;
reconsider n1=n as Element of NAT by ORDINAL1:def 12;
reconsider p1=p as Point of TOP-REAL n1;
assume p in Sphere(0.TOP-REAL n,1);
then |.p1 - 0.TOP-REAL n1.| = 1 by TOPREAL9:9;
then |.p1 + -0.TOP-REAL n1.| = 1;
then |.p + (-1)*0.TOP-REAL n.| = 1 by RLVECT_1:16;
then |.p + 0.TOP-REAL n.| = 1 by RLVECT_1:10;
then
A1: |. p .| = 1 by RLVECT_1:4;
reconsider r1 = 1 as Real;
|. 0.TOP-REAL n .| <> |. p .| by A1,EUCLID_2:39;
then 0.TOP-REAL n <> (1+1)*p by RLVECT_1:11;
then 0.TOP-REAL n <> r1*p + r1*p by RLVECT_1:def 6;
then 0.TOP-REAL n <> r1*p + p by RLVECT_1:def 8;
then 0.TOP-REAL n <> p + p by RLVECT_1:def 8;
then p + -p <> p + p by RLVECT_1:5;
then
A2: not -p in {p} by TARSKI:def 1;
|. -p .| = 1 by A1,EUCLID:71;
then |.-p + 0.TOP-REAL n.| = 1 by RLVECT_1:4;
then |.-p + (-1)*0.TOP-REAL n.| = 1 by RLVECT_1:10;
then |.-p + -0.TOP-REAL n.| = 1 by RLVECT_1:16;
then |.-p1 - 0.TOP-REAL n1.| = 1;
then
-p1 in Sphere(0.TOP-REAL n1,1) by TOPREAL9:9;
hence -p in Sphere(0.TOP-REAL n,1) \ {p} by A2,XBOOLE_0:def 5;
end;
theorem Th4:
for T being non empty TopStruct, t1,t2 being Point of T
for p being Path of t1,t2
holds inf dom p = 0 & sup dom p = 1
proof
let T be non empty TopStruct;
let t1,t2 be Point of T;
let p be Path of t1,t2;
[.0,1.] = dom p by BORSUK_1:40,FUNCT_2:def 1;
hence thesis by XXREAL_2:25,29;
end;
theorem Th5:
for C1,C2 being constant Loop of t holds
C1, C2 are_homotopic
proof
let C1,C2 be constant Loop of t;
C1 = I[01] --> t by BORSUK_2:5 .= C2 by BORSUK_2:5;
hence thesis by BORSUK_2:12;
end;
theorem Th6:
for S being non empty SubSpace of T,
t1,t2 being Point of T, s1,s2 being Point of S,
A,B being Path of t1,t2, C,D being Path of s1,s2 st
s1,s2 are_connected & t1,t2 are_connected & A = C & B = D & C,D are_homotopic
holds A,B are_homotopic
proof
let S be non empty SubSpace of T;
let t1,t2 be Point of T;
let s1,s2 be Point of S;
let A,B be Path of t1,t2;
let C,D be Path of s1,s2 such that
A1: s1,s2 are_connected & t1,t2 are_connected and
A2: A = C & B = D;
given f being Function of [:I[01],I[01]:],S such that
A3: f is continuous and
A4: for t being Point of I[01] holds f.(t,0) = C.t & f.(t,1) = D.t &
f.(0,t) = s1 & f.(1,t) = s2;
reconsider g = f as Function of [:I[01],I[01]:],T by TOPREALA:7;
take g;
thus g is continuous by A3,PRE_TOPC:26;
s1 = C.0 & s2 = C.1 & t1 = A.0 & t2 = A.1 by A1,BORSUK_2:def 2;
hence thesis by A2,A4;
end;
theorem
for S being non empty SubSpace of T,
t1,t2 being Point of T, s1,s2 being Point of S,
A,B being Path of t1,t2, C,D being Path of s1,s2 st
s1,s2 are_connected & t1,t2 are_connected & A = C & B = D &
Class(EqRel(S,s1,s2),C) = Class(EqRel(S,s1,s2),D) holds
Class(EqRel(T,t1,t2),A) = Class(EqRel(T,t1,t2),B)
proof
let S be non empty SubSpace of T;
let t1,t2 be Point of T;
let s1,s2 be Point of S;
let A,B be Path of t1,t2;
let C,D be Path of s1,s2 such that
A1: s1,s2 are_connected and
A2: t1,t2 are_connected and
A3: A = C & B = D;
assume Class(EqRel(S,s1,s2),C) = Class(EqRel(S,s1,s2),D);
then C,D are_homotopic by A1,TOPALG_1:46;
then A,B are_homotopic by A1,A2,A3,Th6;
hence thesis by A2,TOPALG_1:46;
end;
theorem Th8:
for T being trivial non empty TopSpace
for t being Point of T, L being Loop of t holds
the carrier of pi_1(T,t) = { Class(EqRel(T,t),L) }
proof
let T be trivial non empty TopSpace;
let t be Point of T;
set E = EqRel(T,t);
let L be Loop of t;
thus the carrier of pi_1(T,t) c= { Class(E,L) }
proof
let x be object;
assume x in the carrier of pi_1(T,t);
then consider P being Loop of t such that
A1: x = Class(E,P) by TOPALG_1:47;
P = I[01] --> t by TOPREALC:22
.= L by TOPREALC:22;
hence x in { Class(E,L) } by A1,TARSKI:def 1;
end;
let x be object;
assume x in { Class(E,L) };
then
A2: x = Class(E,L) by TARSKI:def 1;
L in Loops t by TOPALG_1:def 1;
then x in Class E by A2,EQREL_1:def 3;
hence thesis by TOPALG_1:def 5;
end;
theorem Th9:
for p being Point of TOP-REAL n, S being Subset of TOP-REAL n
st n >= 2 & S = ([#]TOP-REAL n) \ {p}
holds (TOP-REAL n) | S is pathwise_connected
proof
let p be Point of TOP-REAL n;
let S be Subset of TOP-REAL n;
assume
A1: n >= 2;
assume
A2: S = ([#]TOP-REAL n) \ {p};
then S is infinite by A1,RAMSEY_1:4;
then reconsider T = (TOP-REAL n) | S as non empty SubSpace of TOP-REAL n;
A3:[#]T = ([#]TOP-REAL n) \ {p} by A2,PRE_TOPC:def 5;
A4: for a,b being Point of T, a1,b1 being Point of TOP-REAL n
st not p in LSeg(a1,b1) & a1=a & b1=b
holds a, b are_connected
proof
let a,b be Point of T;
let a1,b1 be Point of TOP-REAL n;
assume
A5: not p in LSeg(a1,b1);
assume
A6: a1 = a & b1 = b;
per cases;
suppose
A7: a1 <> b1;
A8: [#]((TOP-REAL n) | LSeg(a1,b1)) = LSeg(a1,b1) by PRE_TOPC:def 5;
A9: LSeg(a1,b1) c= ([#]TOP-REAL n) \ {p} by A5,ZFMISC_1:34;
reconsider Y = (TOP-REAL n) | LSeg(a1,b1) as
non empty SubSpace of T by A3,A9,A8,RLTOPSP1:68,TSEP_1:4;
LSeg(a1,b1) is_an_arc_of a1,b1 by A7,TOPREAL1:9;
then consider
h being Function of I[01], Y such that
A10: h is being_homeomorphism and
A11: h.0 = a1 & h.1 = b1 by TOPREAL1:def 1;
reconsider f = h as Function of I[01], T by A3,A9,A8,FUNCT_2:7;
take f;
thus f is continuous by A10,PRE_TOPC:26;
thus thesis by A6,A11;
end;
suppose a1 = b1; hence a, b are_connected by A6; end;
end;
for a,b being Point of T holds a, b are_connected
proof
let a,b be Point of T;
A12: the carrier of T is Subset of TOP-REAL n by TSEP_1:1;
a in the carrier of T & b in the carrier of T;
then reconsider a1 = a, b1 = b as Point of TOP-REAL n by A12;
per cases;
suppose
A13: a1 <> b1;
per cases;
suppose
A14: p in LSeg(a1,b1);
reconsider n1 = n as Element of NAT by ORDINAL1:def 12;
reconsider aa1=a1,bb1=b1 as Point of TOP-REAL n1;
consider s be Real such that
A15: 0<=s & s<=1 & p=(1-s)*aa1+s*bb1 by A14,RLTOPSP1:76;
set q1 = b1 - a1;
reconsider k = n - 1 as Nat by A1,CHORD:1;
k + 1 > 1 by A1,XXREAL_0:2;
then
A16: k >= 1 by NAT_1:13;
q1 <> 0.TOP-REAL(k+1) by A13,RLVECT_1:21;
then TPlane(q1, p) is k-manifold by MFOLD_2:30;
then [#]TPlane(q1, p) is infinite by A16;
then [#]( (TOP-REAL n) | Plane(q1, p)) is infinite by MFOLD_2:def 3;
then
A17: Plane(q1, p) is infinite;
reconsider X = Plane(q1, p) as set;
X \ {p} is infinite by A17,RAMSEY_1:4;
then consider x be object such that
A18: x in (X \ {p}) by XBOOLE_0:def 1;
A19: x in X & not x in {p} by A18,XBOOLE_0:def 5;
then x in {y where y is Point of TOP-REAL n: |( q1,y-p )| = 0 }
by MFOLD_2:def 2;
then consider c1 be Point of TOP-REAL n such that
A20: c1 = x & |( q1,c1-p )| = 0;
A21: |( q1 , q1 )| <> 0
proof
assume |( q1 , q1 )| = 0;
then q1 = 0.TOP-REAL n by EUCLID_2:41;
hence contradiction by A13,RLVECT_1:21;
end;
A22: not p in LSeg(a1,c1)
proof
assume
A23: p in LSeg(a1,c1);
reconsider cc1=c1 as Point of TOP-REAL n1;
consider r be Real such that
A24: 0<=r & r<=1 & p=(1-r)*aa1+r*cc1 by A23,RLTOPSP1:76;
A25: 1-r <> 0
proof
assume 1-r = 0;
then p = 0.TOP-REAL(n) + (1 qua Real)*c1 by A24,RLVECT_1:10
.= 0.TOP-REAL(n) + c1 by RLVECT_1:def 8
.= c1 by RLVECT_1:4;
hence contradiction by A19,A20,TARSKI:def 1;
end;
set q2 = c1 - a1;
c1-p
= c1 -(1-r)*a1 - r*c1 by A24,RLVECT_1:27
.= c1 + (-(1-r))*a1 - r*c1 by RLVECT_1:79
.= c1 + (-1+r)*a1 - r*c1
.= c1 + ((-1)*a1 + r*a1) - r*c1 by RLVECT_1:def 6
.= c1 + (-a1 + r*a1) - r*c1 by RLVECT_1:16
.= c1 + -a1 + r*a1 - r*c1 by RLVECT_1:def 3
.= q2 + r*a1 + r*(-c1) by RLVECT_1:25
.= q2 + (r*a1 + r*(-c1)) by RLVECT_1:def 3
.= q2 + r*(a1 + -c1) by RLVECT_1:def 5
.= q2 + r*(-q2) by RLVECT_1:33
.= q2 + -r*q2 by RLVECT_1:25
.= q2 + (-r)*q2 by RLVECT_1:79
.= (1 qua Real)*q2 + (-r)*q2 by RLVECT_1:def 8
.= (1+(-r))*q2 by RLVECT_1:def 6
.= (1-r)*q2;
then (1-r)*|( q1,q2 )| = 0 by A20,EUCLID_2:20;
then
A26: |( q1,q2 )| = 0 by A25,XCMPLX_1:6;
0.TOP-REAL n
= (1-r)*a1+r*c1 + -((1-s)*a1+s*b1) by A15,A24,RLVECT_1:5
.= (1-r)*a1 + -((1-s)*a1+s*b1) + r*c1 by RLVECT_1:def 3
.= (1-r)*a1 + (-(1-s)*a1 -s*b1) + r*c1 by RLVECT_1:30
.= (1-r)*a1 + -(1-s)*a1 + -s*b1 + r*c1 by RLVECT_1:def 3
.= (1-r)*a1 + (-(1-s))*a1 + -s*b1 + r*c1 by RLVECT_1:79
.= ((1-r) + (-(1-s)))*a1 + -s*b1 + r*c1 by RLVECT_1:def 6
.= (s+(-r))*a1 + -s*b1 + r*c1
.= s*a1 + (-r)*a1 + -s*b1 + r*c1 by RLVECT_1:def 6
.= s*a1 + -s*b1 + (-r)*a1 + r*c1 by RLVECT_1:def 3
.= s*a1 + s*(-b1) + (-r)*a1 + r*c1 by RLVECT_1:25
.= s*(a1 + -b1) + (-r)*a1 + r*c1 by RLVECT_1:def 5
.= s*(-(b1-a1)) + (-r)*a1 + r*c1 by RLVECT_1:33
.= s*(-q1) + ((-r)*a1 + r*c1) by RLVECT_1:def 3
.= s*(-q1) + (r*c1 + -r*a1) by RLVECT_1:79
.= s*(-q1) + (r*c1 + r*(-a1)) by RLVECT_1:25
.= s*(-q1) + r*q2 by RLVECT_1:def 5;
then
A27: 0 = |( s*(-q1) + r*q2, s*(-q1) + r*q2 )| by EUCLID_2:34
.= |( s*(-q1) , s*(-q1) )| +2*|( s*(-q1) , r*q2 )|
+|( r*q2, r*q2 )| by EUCLID_2:30;
A28: |( s*(-q1) , s*(-q1) )| = s*|( -q1 , s*(-q1) )| by EUCLID_2:19
.= s*(s*|( -q1 , -q1 )|) by EUCLID_2:20
.= s*(s*|( q1 , q1 )|) by EUCLID_2:23
.= s*s*|( q1 , q1 )|;
A29: |( r*q2 , r*q2 )| = r*|( q2 , r*q2 )| by EUCLID_2:19
.= r*(r*|( q2 , q2 )|) by EUCLID_2:20
.= r*r*|( q2 , q2 )|;
A30: |( s*(-q1) , r*q2 )| = s*|( -q1 , r*q2 )| by EUCLID_2:19
.= s*(r*|( -q1 , q2 )|) by EUCLID_2:20
.= s*(r*(-|( q1 , q2 )|)) by EUCLID_2:21
.= 0 by A26;
A31: s*s >= 0 by XREAL_1:63;
A32: r*r >= 0 by XREAL_1:63;
A33: |( q1 , q1 )| >= 0 by EUCLID_2:35;
A34: |( q2 , q2 )| >= 0 by EUCLID_2:35;
A35: s*s <> 0
proof
assume s*s = 0;
then s = 0 by XCMPLX_1:6;
then p
= (1 qua Real)*a1 + 0.TOP-REAL n by A15,RLVECT_1:10
.= (1 qua Real)*a1 by RLVECT_1:4
.= a1 by RLVECT_1:def 8;
then not p in {p} by A3,XBOOLE_0:def 5;
hence contradiction by TARSKI:def 1;
end;
thus contradiction
by A28,A29,A27,A30,A31,A32,A33,A34,A35,A21,XREAL_1:71;
end;
A36: not p in LSeg(c1,b1)
proof
assume
A37: p in LSeg(c1,b1);
reconsider cc1=c1 as Point of TOP-REAL n1;
consider r be Real such that
A38: 0<=r & r<=1 & p=(1-r)*bb1+r*cc1 by A37,RLTOPSP1:76;
A39: 1-r <> 0
proof
assume 1-r = 0;
then p = 0.TOP-REAL(n) + (1 qua Real)*c1 by A38,RLVECT_1:10
.= 0.TOP-REAL(n) + c1 by RLVECT_1:def 8
.= c1 by RLVECT_1:4;
hence contradiction by A19,A20,TARSKI:def 1;
end;
set q2 = c1 - b1;
c1-p = c1 -(1-r)*b1 - r*c1 by A38,RLVECT_1:27
.= c1 + (-(1-r))*b1 - r*c1 by RLVECT_1:79
.= c1 + (-1+r)*b1 - r*c1
.= c1 + ((-1)*b1 + r*b1) - r*c1 by RLVECT_1:def 6
.= c1 + (-b1 + r*b1) - r*c1 by RLVECT_1:16
.= c1 + -b1 + r*b1 - r*c1 by RLVECT_1:def 3
.= q2 + r*b1 + r*(-c1) by RLVECT_1:25
.= q2 + (r*b1 + r*(-c1)) by RLVECT_1:def 3
.= q2 + r*(b1 + -c1) by RLVECT_1:def 5
.= q2 + r*(-q2) by RLVECT_1:33
.= q2 + -r*q2 by RLVECT_1:25
.= q2 + (-r)*q2 by RLVECT_1:79
.= (1 qua Real)*q2 + (-r)*q2 by RLVECT_1:def 8
.= (1+(-r))*q2 by RLVECT_1:def 6
.= (1-r)*q2;
then (1-r)*|( q1,q2 )| = 0 by A20,EUCLID_2:20;
then
A40: |( q1,q2 )| = 0 by A39,XCMPLX_1:6;
A41: 0.TOP-REAL n = (1+(-r))*b1+r*c1 + -((1-s)*a1+s*b1)
by A38,A15,RLVECT_1:5
.= (1 qua Real)*b1+(-r)*b1+r*c1
+ -((1-s)*a1+s*b1) by RLVECT_1:def 6
.= b1+(-r)*b1+r*c1 + -((1-s)*a1+s*b1) by RLVECT_1:def 8
.= b1+((-r)*b1+r*c1) + -((1-s)*a1+s*b1) by RLVECT_1:def 3
.= b1+(-r*b1+r*c1) + -((1-s)*a1+s*b1) by RLVECT_1:79
.= b1+(r*(-b1)+r*c1) + -((1-s)*a1+s*b1) by RLVECT_1:25
.= b1+r*q2 + -((1-s)*a1+s*b1) by RLVECT_1:def 5
.= b1 + -((1-s)*a1+s*b1)+r*q2 by RLVECT_1:def 3
.= b1 + (-1)*(s*b1+(1-s)*a1)+r*q2 by RLVECT_1:16
.= b1 + ((-1)*(s*b1)+(-1)*((1-s)*a1))+r*q2 by RLVECT_1:def 5
.= b1 + (((-1)*s)*b1+(-1)*((1-s)*a1))+r*q2 by RLVECT_1:def 7
.= b1 + ((-s)*b1+ -((1-s)*a1))+r*q2 by RLVECT_1:16
.= b1 + (-s)*b1+ -((1-s)*a1) +r*q2 by RLVECT_1:def 3
.= (1 qua Real)*b1+(-s)*b1+ -((1-s)*a1) +r*q2 by RLVECT_1:def 8
.= (1+(-s))*b1+ -((1-s)*a1) +r*q2 by RLVECT_1:def 6
.= (1-s)*b1 +(1-s)*(-a1) +r*q2 by RLVECT_1:25
.= (1-s)*q1 + r*q2 by RLVECT_1:def 5;
set t = 1-s;
A42: 0 = |( t*q1 + r*q2, t*q1 + r*q2 )| by A41,EUCLID_2:34
.= |( t*q1 , t*q1 )| +2*|( t*q1 , r*q2 )|
+|( r*q2, r*q2 )| by EUCLID_2:30;
A43: |( t*q1 , t*q1 )| = t*|( q1 , t*q1 )| by EUCLID_2:19
.= t*(t*|( q1 , q1 )|) by EUCLID_2:20
.= t*t*|( q1 , q1 )|;
A44: |( r*q2 , r*q2 )| = r*|( q2 , r*q2 )| by EUCLID_2:19
.= r*(r*|( q2 , q2 )|) by EUCLID_2:20
.= r*r*|( q2 , q2 )|;
A45: |( t*q1 , r*q2 )| = t*|( q1 , r*q2 )| by EUCLID_2:19
.= t*(r*|( q1 , q2 )|) by EUCLID_2:20
.= 0 by A40;
A46: t*t >= 0 by XREAL_1:63;
A47: r*r >= 0 by XREAL_1:63;
A48: |( q1 , q1 )| >= 0 by EUCLID_2:35;
A49: |( q2 , q2 )| >= 0 by EUCLID_2:35;
A50: t*t <> 0
proof
assume t*t = 0;
then t = 0 by XCMPLX_1:6;
then p
= 0.TOP-REAL n + (1 qua Real)*b1 by A15,RLVECT_1:10
.= (1 qua Real)*b1 by RLVECT_1:4
.= b1 by RLVECT_1:def 8;
then not p in {p} by A3,XBOOLE_0:def 5;
hence contradiction by TARSKI:def 1;
end;
thus contradiction
by A50,A21,A43,A44,A42,A45,A46,A47,A48,A49,XREAL_1:71;
end;
reconsider c = c1 as Point of T by A20,A19,A3,XBOOLE_0:def 5;
a,c are_connected & c,b are_connected by A22,A36,A4;
hence thesis by BORSUK_6:42;
end;
suppose not p in LSeg(a1,b1); hence thesis by A4; end;
end;
suppose a1 = b1; hence a, b are_connected; end;
end;
hence thesis;
end;
theorem Th10:
for S being non empty Subset of T
st n >= 2 & S = ([#]T) \ {t} & TOP-REAL n, T are_homeomorphic
holds T | S is pathwise_connected
proof
let S be non empty Subset of T;
assume
A1: n >= 2 & S = ([#]T) \ {t} & TOP-REAL n, T are_homeomorphic;
then consider f be Function of T, TOP-REAL n such that
A2: f is being_homeomorphism by T_0TOPSP:def 1;
reconsider p = f.t as Point of TOP-REAL n;
reconsider SN = ([#]TOP-REAL n) \ {p}
as non empty Subset of TOP-REAL n by A1,RAMSEY_1:4;
A3: (TOP-REAL n) | SN is pathwise_connected by A1,Th9;
A4:dom f = [#]T & rng f = [#]TOP-REAL n by A2,TOPS_2:58;
then
A5: f"([#]TOP-REAL n) = [#]T by RELAT_1:134;
consider x be object such that
A6:f"{p} = {x} by A4,A2,FUNCT_1:74;
A7: x in f"{p} by A6,TARSKI:def 1;
then x in dom f & f.x in {p} by FUNCT_1:def 7;
then p = f.x by TARSKI:def 1;
then x = t by A2,A7,A4,FUNCT_1:def 4;
then
A8: f"SN = S by A1,A5,A6,FUNCT_1:69;
A9: dom(SN |` f) = f"SN by MFOLD_2:1 .= [#](T | f"SN) by PRE_TOPC:def 5;
rng(SN |` f) c= SN;
then rng(SN |` f) c= [#]((TOP-REAL n) | SN) by PRE_TOPC:def 5;
then reconsider g = SN|`f as Function of T | f"SN, (TOP-REAL n) | SN
by A9,FUNCT_2:2;
g is being_homeomorphism by A2,MFOLD_2:4;
then (TOP-REAL n) | SN, T | S are_homeomorphic by A8,T_0TOPSP:def 1;
hence T | S is pathwise_connected by A3,TOPALG_3:9;
end;
registration
let n be Element of NAT;
let p,q be Point of TOP-REAL n;
cluster TPlane(p,q) -> convex;
correctness
proof
set P = Plane(p,q);
for w1,w2 being Point of TOP-REAL n
st w1 in P & w2 in P holds LSeg(w1,w2) c= P
proof
let w1,w2 be Point of TOP-REAL n;
assume
A1: w1 in P & w2 in P;
reconsider n0 = n as Nat;
reconsider p0 = p, q0 = q as Point of TOP-REAL n0;
A2: P = {y where y is Point of TOP-REAL n0 : |( p0, y-q0 )| = 0}
by MFOLD_2:def 2;
consider v1 be Point of TOP-REAL n0 such that
A3: w1 = v1 & |( p0, v1-q0 )| = 0 by A1,A2;
consider v2 be Point of TOP-REAL n0 such that
A4: w2 = v2 & |( p0, v2-q0 )| = 0 by A1,A2;
for x being object st x in LSeg(w1,w2) holds x in P
proof
let x be object;
assume
A5: x in LSeg(w1,w2);
then reconsider w = x as Point of TOP-REAL n0;
reconsider r1 = 1 as Real;
consider r be Real such that
A6: 0<=r & r<=1 & w=(1-r)*w1+r*w2 by A5,RLTOPSP1:76;
A7: |( p0, (1-r)*(v1-q0) )| = (1-r)*0 by A3,EUCLID_2:20 .= 0;
A8: |( p0, r*(v2-q0) )| = r*0 by A4,EUCLID_2:20 .= 0;
A9: (1-r)*(v1-q0) + r*(v2-q0)
= (1-r)*(w1-q) + (r*w2 -r*q) by A3,A4,RLVECT_1:34
.= (1-r)*w1 - (1-r)*q + (r*w2 -r*q) by RLVECT_1:34
.= (1-r)*w1 + (-(1-r))*q + (r*w2 -r*q) by RLVECT_1:79
.= (1-r)*w1 + (r-1)*q + (r*w2 -r*q)
.= (1-r)*w1 + (r*q -r1*q) + (r*w2 -r*q) by RLVECT_1:35
.= (1-r)*w1 + ((r*q -r1*q) + (r*w2 -r*q)) by RLVECT_1:def 3
.= (1-r)*w1 + ( -r1*q + r*q + -r*q + r*w2) by RLVECT_1:def 3
.= (1-r)*w1 + ( -r1*q + (r*q - r*q) + r*w2) by RLVECT_1:def 3
.= (1-r)*w1 + ( -r1*q + 0.TOP-REAL n + r*w2) by RLVECT_1:5
.= (1-r)*w1 + ( -r1*q + r*w2) by RLVECT_1:4
.= (1-r)*w1 + r*w2 + -r1*q by RLVECT_1:def 3
.= w + -q0 by A6,RLVECT_1:def 8
.= w -q0;
0 = |( p0, (1-r)*(v1-q0) )| + |( p0, r*(v2-q0) )| by A7,A8
.= |( p0, w-q0 )| by A9,EUCLID_2:26;
hence x in P by A2;
end;
hence thesis;
end;
then P is convex Subset of TOP-REAL n by RLTOPSP1:22;
then [#]((TOP-REAL n) | P) is convex Subset of TOP-REAL n
by PRE_TOPC:def 5;
then [#]TPlane(p,q) is convex Subset of TOP-REAL n by MFOLD_2:def 3;
hence thesis by TOPALG_2:def 1;
end;
end;
begin :: Fundamental groups
definition
let T;
attr T is having_trivial_Fundamental_Group means :Def1:
for t being Point of T holds pi_1(T,t) is trivial;
end;
definition
let T;
attr T is simply_connected means
T is having_trivial_Fundamental_Group pathwise_connected;
end;
registration
cluster simply_connected ->
having_trivial_Fundamental_Group pathwise_connected
for non empty TopSpace;
coherence;
cluster having_trivial_Fundamental_Group pathwise_connected ->
simply_connected for non empty TopSpace;
coherence;
end;
theorem Th11:
T is having_trivial_Fundamental_Group implies
for t being Point of T, P,Q being Loop of t holds
P,Q are_homotopic
proof
assume
A1: T is having_trivial_Fundamental_Group;
let t be Point of T, P,Q be Loop of t;
set E = EqRel(T,t);
A2: pi_1(T,t) is trivial by A1;
Class(E,P) in the carrier of pi_1(T,t) &
Class(E,Q) in the carrier of pi_1(T,t) by TOPALG_1:47;
then Class(E,P) = Class(E,Q) by A2;
hence thesis by TOPALG_1:46;
end;
registration
let n be Nat;
cluster TOP-REAL n -> having_trivial_Fundamental_Group;
coherence;
end;
registration
cluster trivial -> having_trivial_Fundamental_Group for non empty TopSpace;
coherence
proof
let T be non empty TopSpace such that
A1: T is trivial;
let t be Point of T;
reconsider L = I[01] --> t as Loop of t by JORDAN:41;
the carrier of pi_1(T,t) = { Class(EqRel(T,t),L) } by A1,Th8;
hence thesis;
end;
end;
theorem Th12:
T is simply_connected iff
for t1,t2 being Point of T holds t1,t2 are_connected &
for P, Q being Path of t1,t2 holds
Class(EqRel(T,t1,t2),P) = Class(EqRel(T,t1,t2),Q)
proof
hereby
assume
A1: T is simply_connected;
let t1,t2 be Point of T;
thus
A2: t1,t2 are_connected by A1,BORSUK_2:def 3;
let P, Q be Path of t1,t2;
set E = EqRel(T,t1,t2);
A3: P,P+-Q+Q are_homotopic by A1,TOPALG_1:22;
set C = the constant Loop of t1;
P+-Q,C are_homotopic by A1,Th11;
then
A4: P+-Q+Q,C+Q are_homotopic by A1,BORSUK_6:76;
C+Q,Q are_homotopic by A1,BORSUK_6:83;
then P+-Q+Q,Q are_homotopic by A4,BORSUK_6:79;
then P,Q are_homotopic by A3,BORSUK_6:79;
hence Class(E,P) = Class(E,Q) by A2,TOPALG_1:46;
end;
assume
A5: for t1,t2 being Point of T holds t1,t2 are_connected &
for P, Q being Path of t1,t2 holds
Class(EqRel(T,t1,t2),P) = Class(EqRel(T,t1,t2),Q);
thus T is having_trivial_Fundamental_Group
proof
let t be Point of T;
let x, y be Element of pi_1(T,t);
(ex P being Loop of t st x = Class(EqRel(T,t),P)) &
ex P being Loop of t st y = Class(EqRel(T,t),P) by TOPALG_1:47;
hence thesis by A5;
end;
thus thesis by A5;
end;
registration
let T be having_trivial_Fundamental_Group non empty TopSpace;
let t be Point of T;
cluster pi_1(T,t) -> trivial;
coherence by Def1;
end;
theorem Th13:
for S, T being non empty TopSpace st S,T are_homeomorphic holds
S is having_trivial_Fundamental_Group implies
T is having_trivial_Fundamental_Group
proof
let S, T be non empty TopSpace;
given f being Function of S,T such that
A1: f is being_homeomorphism;
assume
A2: for s being Point of S holds pi_1(S,s) is trivial;
let t be Point of T;
rng f = [#]T by A1,TOPS_2:def 5;
then consider s being Point of S such that
A3: f.s = t by FUNCT_2:113;
A4: FundGrIso(f,s) is bijective by A1,TOPALG_3:31;
pi_1(S,s) is trivial by A2;
hence thesis by A3,A4,KNASTER:11,TOPREALC:1;
end;
theorem Th14:
for S, T being non empty TopSpace st S,T are_homeomorphic holds
S is simply_connected implies T is simply_connected by Th13,TOPALG_3:9;
theorem Th15:
for T being having_trivial_Fundamental_Group non empty TopSpace,
t being Point of T, P1,P2 being Loop of t
holds P1,P2 are_homotopic
proof
let T be having_trivial_Fundamental_Group non empty TopSpace;
let t be Point of T, P1,P2 be Loop of t;
Class(EqRel(T,t),P1) in the carrier of pi_1(T,t) &
Class(EqRel(T,t),P2) in the carrier of pi_1(T,t) by TOPALG_1:47;
then Class(EqRel(T,t),P1) = Class(EqRel(T,t),P2) by ZFMISC_1:def 10;
hence thesis by TOPALG_1:46;
end;
definition
let T, t;
let l be Loop of t;
attr l is nullhomotopic means
ex c being constant Loop of t st l,c are_homotopic;
end;
registration
let T, t;
cluster constant -> nullhomotopic for Loop of t;
coherence by BORSUK_6:88;
end;
registration
let T, t;
cluster constant for Loop of t;
existence
proof
reconsider l = I[01] --> t as constant Loop of t by JORDAN:41;
take l;
thus thesis;
end;
end;
theorem Th16:
for f being Loop of t, g being continuous Function of T,U st
f is nullhomotopic holds g*f is nullhomotopic
proof
let f be Loop of t, g be continuous Function of T,U;
given c being constant Loop of t such that
A1: f,c are_homotopic;
consider F being Function of [:I[01],I[01]:], T such that
A2: F is continuous and
A3: for s being Point of I[01] holds F.(s,0) = f.s & F.(s,1) = c.s &
F.(0,s) = t & F.(1,s) = t by A1;
reconsider d = I[01] --> g.t as constant Loop of g.t by JORDAN:41;
reconsider G = g*F as Function of [:I[01],I[01]:], U;
take d, G;
thus G is continuous by A2;
let s be Point of I[01];
reconsider j0 = 0, j1 = 1 as Point of I[01] by BORSUK_1:def 14,def 15;
set I = the carrier of I[01];
A4: the carrier of [:I[01],I[01]:] = [:I,I:] by BORSUK_1:def 2;
thus G.(s,0) = g.(F.(s,j0)) by A4,BINOP_1:18
.= g.(f.s) by A3
.= (g*f).s by FUNCT_2:15;
thus G.(s,1) = g.(F.(s,j1)) by A4,BINOP_1:18
.= g.(c.s) by A3
.= g.t by TOPALG_3:21
.= d.s by FUNCOP_1:7;
thus G.(0,s) = g.(F.(j0,s)) by A4,BINOP_1:18
.= g.t by A3;
thus G.(1,s) = g.(F.(j1,s)) by A4,BINOP_1:18
.= g.t by A3;
end;
registration
let T, U be non empty TopSpace,
t be Point of T,
f be nullhomotopic Loop of t,
g be continuous Function of T,U;
cluster g*f -> nullhomotopic for Loop of g.t;
coherence by Th16;
end;
registration
let T be having_trivial_Fundamental_Group non empty TopSpace;
let t be Point of T;
cluster -> nullhomotopic for Loop of t;
coherence
proof
let l be Loop of t;
reconsider c = I[01] --> t as constant Loop of t by JORDAN:41;
take c;
thus thesis by Th15;
end;
end;
theorem Th17:
(for t being Point of T, f being Loop of t holds f is nullhomotopic)
implies T is having_trivial_Fundamental_Group
proof
assume
A1: for t being Point of T, f being Loop of t holds f is nullhomotopic;
for t being Point of T holds pi_1(T,t) is trivial
proof
let t be Point of T;
set C = the constant Loop of t;
the carrier of pi_1(T,t) = { Class(EqRel(T,t),C) }
proof
set E = EqRel(T,t);
hereby
let x be object;
assume x in the carrier of pi_1(T,t);
then consider P be Loop of t such that
A2: x = Class(E,P) by TOPALG_1:47;
P is nullhomotopic by A1;
then consider C1 be constant Loop of t such that
A3: P, C1 are_homotopic;
C1, C are_homotopic by Th5;
then P, C are_homotopic by A3,BORSUK_6:79;
then x = Class(E,C) by A2,TOPALG_1:46;
hence x in { Class(E,C) } by TARSKI:def 1;
end;
let x be object;
assume x in { Class(E,C) };
then
A4: x = Class(E,C) by TARSKI:def 1;
C in Loops t by TOPALG_1:def 1;
then x in Class E by A4,EQREL_1:def 3;
hence thesis by TOPALG_1:def 5;
end;
hence thesis;
end;
hence thesis;
end;
registration
let n be Element of NAT;
let p,q be Point of TOP-REAL n;
cluster TPlane(p,q) -> having_trivial_Fundamental_Group;
correctness;
end;
theorem Th18:
for S being non empty SubSpace of T, s being Point of S,
f being Loop of t, g being Loop of s st
t = s & f = g & g is nullhomotopic
holds f is nullhomotopic
proof
let S be non empty SubSpace of T;
let s be Point of S;
let f be Loop of t;
let g be Loop of s such that
A1: t = s & f = g and
A2: g is nullhomotopic;
consider c be constant Loop of s such that
A3: g,c are_homotopic by A2;
c = I[01] --> s by BORSUK_2:5 .= I[01] --> t by A1;
then reconsider c as constant Loop of t by JORDAN:41;
f,c are_homotopic by A1,A3,Th6;
hence thesis;
end;
begin :: Curves
reserve T for TopStruct;
reserve f for PartFunc of R^1, T;
definition
let T,f;
attr f is parametrized-curve means :Def4:
dom f is interval Subset of REAL &
ex S being SubSpace of R^1, g being Function of S, T
st f = g & S = R^1|dom f & g is continuous;
end;
Lm1:
f = {} implies f is parametrized-curve
proof
assume
A1: f = {};
reconsider f as PartFunc of R^1, T;
dom f = {} by A1;
then
A2: dom f c= REAL;
reconsider A = {} as Subset of R^1 by XBOOLE_1:2;
reconsider S = R^1 | A as SubSpace of R^1;
reconsider g = f as Relation of [#]S, [#]T by A1;
A3: A = dom g;
reconsider g as Function of S, T;
for P1 being Subset of T st P1 is closed holds g"P1 is closed;
then g is continuous by PRE_TOPC:def 6;
hence thesis by A3,A2;
end;
registration
let T;
cluster parametrized-curve for PartFunc of R^1, T;
correctness
proof
reconsider c = {} as PartFunc of R^1, T by XBOOLE_1:2;
take c;
thus thesis by Lm1;
end;
end;
theorem
{} is parametrized-curve PartFunc of R^1, T by Lm1,XBOOLE_1:2;
definition
let T;
func Curves T -> Subset of PFuncs(REAL, [#]T) equals
{f where f is Element of PFuncs(REAL, [#]T) :
f is parametrized-curve PartFunc of R^1, T};
coherence
proof
set C = {f where f is Element of PFuncs(REAL, [#]T) :
f is parametrized-curve PartFunc of R^1, T};
for x being object st x in C holds x in PFuncs(REAL, [#]T)
proof
let x be object;
assume x in C;
then ex f be Element of PFuncs(REAL, [#]T) st
x = f & f is parametrized-curve PartFunc of R^1, T;
hence thesis;
end;
hence thesis by TARSKI:def 3;
end;
end;
registration
let T;
cluster Curves T -> non empty;
coherence
proof
reconsider c = {} as PartFunc of R^1, T by XBOOLE_1:2;
reconsider f1 = c as Element of PFuncs(REAL, [#]T)
by PARTFUN1:45,TOPMETR:17;
f1 is parametrized-curve PartFunc of R^1, T by Lm1;
then f1 in Curves T;
hence thesis;
end;
end;
definition
let T;
mode Curve of T is Element of Curves T;
end;
reserve c for Curve of T;
theorem Th20:
for f being parametrized-curve PartFunc of R^1, T holds f is Curve of T
proof
let f be parametrized-curve PartFunc of R^1, T;
reconsider f1 = f as Element of PFuncs(REAL, [#]T)
by PARTFUN1:45,TOPMETR:17;
f1 in Curves T;
hence thesis;
end;
theorem Th21:
{} is Curve of T
proof
reconsider f = {} as parametrized-curve PartFunc of R^1, T
by Lm1,XBOOLE_1:2;
f is Curve of T by Th20;
hence thesis;
end;
theorem Th22:
for t1,t2 being Point of T
for p being Path of t1,t2 st t1,t2 are_connected holds p is Curve of T
proof
let t1,t2 be Point of T;
let p be Path of t1,t2;
assume t1,t2 are_connected;
then
A1: p is continuous & p.0 = t1 & p.1 = t2 by BORSUK_2:def 2;
per cases;
suppose T is non empty;
then
A2: [#]I[01] = dom p by FUNCT_2:def 1;
then
A3: dom p c= [#]R^1 by PRE_TOPC:def 4;
then reconsider A = dom p as Subset of R^1;
A4: I[01] = R^1 | A by A2,BORSUK_1:40,TOPMETR:19,20;
rng p c= [#]T;
then reconsider c = p as PartFunc of R^1, T by A3,RELSET_1:4;
reconsider c as parametrized-curve PartFunc of R^1, T
by A2,A4,Def4,A1,BORSUK_1:40;
c is Element of Curves T by Th20;
hence thesis;
end;
suppose
A5: T is empty;
then reconsider c = p as PartFunc of R^1, T;
c = {} by A5;
then reconsider c as parametrized-curve PartFunc of R^1,T by Lm1;
c is Element of Curves T by Th20;
hence thesis;
end;
end;
theorem Th23:
c is parametrized-curve PartFunc of R^1, T
proof
c in {f where f is Element of PFuncs(REAL, [#]T) :
f is parametrized-curve PartFunc of R^1, T};
then consider f be Element of PFuncs(REAL, [#]T) such that
A1: c = f & f is parametrized-curve PartFunc of R^1, T;
thus thesis by A1;
end;
theorem Th24:
dom c c= REAL & rng c c= [#]T
proof
reconsider f = c as parametrized-curve PartFunc of R^1, T by Th23;
dom f c= [#]R^1 & rng f c= [#]T;
hence thesis by TOPMETR:17;
end;
registration
let T,c;
cluster dom c -> real-membered;
correctness
proof
let x be object;
assume
A1: x in dom c;
dom c c= REAL by Th24;
hence thesis by A1;
end;
end;
definition
let T,c;
attr c is with_first_point means :Def6:
dom c is left_end;
attr c is with_last_point means :Def7:
dom c is right_end;
end;
definition
let T,c;
attr c is with_endpoints means
c is with_first_point with_last_point;
end;
registration
let T;
cluster with_first_point with_last_point -> with_endpoints for Curve of T;
correctness;
cluster with_endpoints -> with_first_point with_last_point for Curve of T;
correctness;
end;
reserve T for non empty TopStruct;
registration
let T;
cluster with_endpoints for Curve of T;
correctness
proof
set t = the Point of T;
set f = I[01] --> t;
A1: f.0 = t & f.1 = t by BORSUK_1:def 14,def 15,FUNCOP_1:7;
set p = the Path of t, t;
t, t are_connected by A1;
then reconsider c = p as Curve of T by Th22;
take c;
A2: dom c = [.0,1.] by BORSUK_1:40,FUNCT_2:def 1;
set S = [.0,1.];
inf S = 0 by XXREAL_2:25;
then inf S in S by XXREAL_1:1;
hence dom c is left_end by A2,XXREAL_2:def 5;
sup S = 1 by XXREAL_2:29;
then sup S in S by XXREAL_1:1;
hence dom c is right_end by A2,XXREAL_2:def 6;
end;
end;
registration
let T;
let c be with_first_point Curve of T;
cluster dom c -> non empty;
correctness
proof
dom c is left_end by Def6;
hence thesis;
end;
cluster inf dom c -> real;
correctness
proof
dom c is left_end by Def6;
hence thesis;
end;
end;
registration
let T;
let c be with_last_point Curve of T;
cluster dom c -> non empty;
correctness
proof
dom c is right_end by Def7;
hence thesis;
end;
cluster sup dom c -> real;
correctness
proof
dom c is right_end by Def7;
hence thesis;
end;
end;
registration
let T;
cluster with_first_point -> non empty for Curve of T;
coherence;
cluster with_last_point -> non empty for Curve of T;
coherence;
end;
definition
let T;
let c be with_first_point Curve of T;
func the_first_point_of c -> Point of T equals
c.(inf dom c);
correctness
proof
A1: rng c c= [#]T by Th24;
dom c is left_end by Def6;
then inf dom c in dom c by XXREAL_2:def 5;
then c.(inf dom c) in rng c by FUNCT_1:3;
hence thesis by A1;
end;
end;
definition
let T;
let c be with_last_point Curve of T;
func the_last_point_of c -> Point of T equals
c.(sup dom c);
correctness
proof
A1: rng c c= [#]T by Th24;
dom c is right_end by Def7;
then sup dom c in dom c by XXREAL_2:def 6;
then c.(sup dom c) in rng c by FUNCT_1:3;
hence thesis by A1;
end;
end;
theorem Th25:
for t1,t2 being Point of T
for p being Path of t1,t2 st t1,t2 are_connected
holds p is with_endpoints Curve of T
proof
let t1,t2 be Point of T;
let p be Path of t1,t2;
assume t1,t2 are_connected;
then reconsider c = p as Curve of T by Th22;
A1: [.0,1.] = dom c by BORSUK_1:40,FUNCT_2:def 1;
0 in [.0,1.] by XXREAL_1:1;
then inf dom c in dom c by A1,Th4;
then dom c is left_end by XXREAL_2:def 5;
then
A2: c is with_first_point;
1 in [.0,1.] by XXREAL_1:1;
then sup dom c in dom c by A1,Th4;
then dom c is right_end by XXREAL_2:def 6;
then c is with_last_point;
hence thesis by A2;
end;
theorem Th26:
for c being Curve of T
for r1,r2 being Real
holds c | [.r1, r2.] is Curve of T
proof
let c be Curve of T;
let r1,r2 be Real;
reconsider f = c as parametrized-curve PartFunc of R^1, T
by Th23;
set f1 = f | [.r1, r2.];
reconsider A = dom f as interval Subset of REAL by Def4;
reconsider B = [.r1,r2.] as interval Subset of REAL;
A1: A /\ B is interval Subset of REAL;
then
A2: dom f1 is interval Subset of REAL by RELAT_1:61;
consider S be SubSpace of R^1, g be Function of S,T such that
A3: f = g & S = R^1|dom f & g is continuous by Def4;
reconsider A0 = dom f as Subset of R^1;
A4: [#]S = A0 by A3,PRE_TOPC:def 5;
reconsider K0 = (dom f)/\[.r1,r2.] as Subset of S by A4,XBOOLE_1:17;
reconsider g1 = g|K0 as Function of S|K0,T by PRE_TOPC:9;
A5: g1 is continuous by A3,TOPMETR:7;
A6: g1 = (f|dom f) | [.r1,r2.] by A3,RELAT_1:71 .= f1;
A7:(dom f)/\[.r1,r2.] = dom f1 by RELAT_1:61;
reconsider K1 = K0 as Subset of R^1|A0 by A3;
reconsider D1 = dom f1 as Subset of R^1 by A1,RELAT_1:61,TOPMETR:17;
S|K0 = R^1 | D1 by A3,A7,PRE_TOPC:7,XBOOLE_1:17;
then reconsider f1 as parametrized-curve PartFunc of R^1, T
by A2,A5,A6,Def4;
c | [.r1, r2.] = f1;
hence thesis by Th20;
end;
theorem Th27:
for c being with_endpoints Curve of T holds
dom c = [.inf dom c, sup dom c.]
proof
let c be with_endpoints Curve of T;
reconsider f = c as parametrized-curve PartFunc of R^1, T
by Th23;
dom f is interval Subset of REAL by Def4;
then reconsider A = dom c as left_end right_end interval
ext-real-membered set by Def6,Def7;
A = [. min A, max A.] by XXREAL_2:75;
hence thesis;
end;
theorem Th28:
for c being with_endpoints Curve of T st dom c = [.0,1.]
holds c is Path of the_first_point_of c,the_last_point_of c
proof
let c be with_endpoints Curve of T;
assume
A1: dom c = [.0,1.];
set t1 = the_first_point_of c,t2 = the_last_point_of c;
reconsider f = c as parametrized-curve PartFunc of R^1,T
by Th23;
consider S be SubSpace of R^1, p be Function of S, T such that
A2: f = p & S = R^1|dom f & p is continuous by Def4;
reconsider p as Function of I[01],T by A2,A1,BORSUK_1:40,FUNCT_2:def 1;
A3: S = I[01] by A2,A1,TOPMETR:19,20;
A4: p.0 = t1 by A1,A2,XXREAL_2:25;
A5: p.1 = t2 by A2,A1,XXREAL_2:29;
then t1,t2 are_connected by A2,A3,A4;
hence thesis by A3,A4,A5,A2,BORSUK_2:def 2;
end;
theorem Th29:
for c being with_endpoints Curve of T holds
c*L[01](0,1,inf dom c,sup dom c)
is Path of the_first_point_of c,the_last_point_of c
proof
let c be with_endpoints Curve of T;
set t1 = the_first_point_of c, t2 = the_last_point_of c;
reconsider c0 = c as parametrized-curve PartFunc of R^1, T
by Th23;
consider S be SubSpace of R^1, g be Function of S, T such that
A1:c0 = g & S = R^1|dom c0 & g is continuous by Def4;
reconsider S as non empty TopStruct by A1;
A2:inf dom c <= sup dom c by XXREAL_2:40;
then
A3:L[01](0,1,inf dom c,sup dom c) is continuous
Function of Closed-Interval-TSpace(0,1),
Closed-Interval-TSpace(inf dom c,sup dom c) by BORSUK_6:34;
A4:dom c0 = [.inf dom c,sup dom c.] by Th27;
then
A5:Closed-Interval-TSpace(inf dom c,sup dom c) = S by A2,A1,TOPMETR:19;
reconsider f = L[01](0,1,inf dom c,sup dom c) as Function of I[01],S
by A4,A2,A1,TOPMETR:19,20;
reconsider p = g*f as Function of I[01],T;
A6: 0 in [.0,1.] & 1 in [.0,1.] by XXREAL_1:1;
A7:dom L[01](0,1,inf dom c,sup dom c)
= the carrier of Closed-Interval-TSpace(0,1) by FUNCT_2:def 1
.= [.0,1.] by TOPMETR:18;
A8:L[01](0,1,inf dom c,sup dom c).0
= (sup dom c - inf dom c)/(1 - 0) * (0 - 0) + inf dom c
by A2,BORSUK_6:35 .= inf dom c;
A9:L[01](0,1,inf dom c,sup dom c).1
= (sup dom c - inf dom c)/(1 - 0) * (1 - 0) + inf dom c
by A2,BORSUK_6:35 .= sup dom c;
A10: p is continuous by A1,A3,A5,TOPMETR:20,TOPS_2:46;
A11: p.0 = t1 by A8,A1,A6,A7,FUNCT_1:13;
A12: p.1 = t2 by A9,A1,A6,A7,FUNCT_1:13;
then t1,t2 are_connected by A10,A11;
hence thesis by A1,A10,A11,A12,BORSUK_2:def 2;
end;
theorem
for c being with_endpoints Curve of T,
t1,t2 being Point of T
st c*L[01](0,1,inf dom c,sup dom c) is Path of t1,t2 & t1,t2 are_connected
holds t1 = the_first_point_of c & t2 = the_last_point_of c
proof
let c be with_endpoints Curve of T;
let t1,t2 be Point of T;
assume
A1: c*L[01](0,1,inf dom c,sup dom c) is Path of t1,t2 & t1,t2 are_connected;
A2: inf dom c <= sup dom c by XXREAL_2:40;
A3: 0 in [.0,1.] & 1 in [.0,1.] by XXREAL_1:1;
A4: dom L[01](0,1,inf dom c,sup dom c)
= the carrier of Closed-Interval-TSpace(0,1) by FUNCT_2:def 1
.= [.0,1.] by TOPMETR:18;
A5: L[01](0,1,inf dom c,sup dom c).0
= (sup dom c - inf dom c)/(1 - 0) * (0 - 0) + inf dom c
by A2,BORSUK_6:35 .= inf dom c;
A6:L[01](0,1,inf dom c,sup dom c).1
= (sup dom c - inf dom c)/(1 - 0) * (1 - 0) + inf dom c
by A2,BORSUK_6:35 .= sup dom c;
reconsider p = c*L[01](0,1,inf dom c,sup dom c) as Path of t1,t2 by A1;
A7: p.0 = the_first_point_of c by A5,A3,A4,FUNCT_1:13;
p.1 = the_last_point_of c by A6,A3,A4,FUNCT_1:13;
hence thesis by A7,A1,BORSUK_2:def 2;
end;
theorem Th31:
for c being with_endpoints Curve of T
holds the_first_point_of c in rng c & the_last_point_of c in rng c
proof
let c be with_endpoints Curve of T;
A1: inf dom c <= sup dom c by XXREAL_2:40;
dom c = [.inf dom c,sup dom c.] by Th27;
then inf dom c in dom c & sup dom c in dom c by A1,XXREAL_1:1;
hence thesis by FUNCT_1:3;
end;
theorem Th32:
for r1,r2 being Real
for t1,t2 being Point of T
for p1 being Path of t1,t2
st t1,t2 are_connected & r1 < r2
holds p1*L[01](r1,r2,0,1) is with_endpoints Curve of T
proof
let r1,r2 be Real;
let t1,t2 be Point of T;
let p1 be Path of t1,t2;
assume
A1: t1,t2 are_connected;
assume
A2:r1 < r2;
then
A3: L[01](r1,r2,0,1) is continuous
Function of Closed-Interval-TSpace(r1,r2),Closed-Interval-TSpace(0,1)
by BORSUK_6:34;
A4: p1 is continuous & p1.0 = t1 & p1.1 = t2 by A1,BORSUK_2:def 2;
set c = p1*L[01](r1,r2,0,1);
rng L[01](r1,r2,0,1) c= [#]Closed-Interval-TSpace(0,1) by RELAT_1:def 19;
then rng L[01](r1,r2,0,1) c= dom p1 by FUNCT_2:def 1,TOPMETR:20;
then dom c = dom L[01](r1,r2,0,1) by RELAT_1:27;
then dom c = [#]Closed-Interval-TSpace(r1,r2) by FUNCT_2:def 1;
then
A5:dom c = [.r1,r2.] by A2,TOPMETR:18;
A6:rng c c= [#]T;
then reconsider c as PartFunc of R^1, T by A5,RELSET_1:4,TOPMETR:17;
set S = R^1|dom c;
dom c = [#]S by PRE_TOPC:def 5;
then reconsider g = c as Function of S, T by A6,FUNCT_2:2;
A7: S = Closed-Interval-TSpace(r1,r2) by A2,A5,TOPMETR:19;
reconsider p2 = p1 as Function of Closed-Interval-TSpace(0,1),T
by TOPMETR:20;
g is continuous by A4,A7,A3,TOPMETR:20,TOPS_2:46;
then c is parametrized-curve by A5;
then reconsider c as Curve of T by Th20;
dom c is left_end & dom c is right_end by A2,A5,XXREAL_2:33;
then c is with_first_point & c is with_last_point;
hence thesis;
end;
theorem Th33:
for c being with_endpoints Curve of T
holds the_first_point_of c, the_last_point_of c are_connected
proof
let c be with_endpoints Curve of T;
set t1 = the_first_point_of c, t2 = the_last_point_of c;
reconsider f = c as parametrized-curve PartFunc of R^1,T
by Th23;
consider S be SubSpace of R^1, g be Function of S, T such that
A1: f = g & S = R^1|dom f & g is continuous by Def4;
set r1 = inf dom c, r2 = sup dom c;
set p = g*L[01](0,1,r1,r2);
A2: r1 <= r2 by XXREAL_2:40;
then
A3: L[01](0,1,r1,r2) is continuous
Function of Closed-Interval-TSpace(0,1),Closed-Interval-TSpace(r1,r2)
by BORSUK_6:34;
rng L[01](0,1,r1,r2) c= [#]Closed-Interval-TSpace(r1,r2) by RELAT_1:def 19;
then rng L[01](0,1,r1,r2) c= [.r1,r2.] by A2,TOPMETR:18;
then rng L[01](0,1,r1,r2) c= dom c by Th27;
then dom p = dom L[01](0,1,r1,r2) by A1,RELAT_1:27;
then
A4:dom p = [#]Closed-Interval-TSpace(0,1) by FUNCT_2:def 1;
rng p c= [#]T;
then reconsider p as Function of I[01],T by A4,FUNCT_2:2,TOPMETR:20;
dom f = [.r1,r2.] by Th27;
then S = Closed-Interval-TSpace(r1,r2) by A1,A2,TOPMETR:19;
then
A5: p is continuous by A1,A3,TOPMETR:20,TOPS_2:46;
dom p = [.0,1.] by A4,TOPMETR:18;
then
A6: 0 in dom p & 1 in dom p by XXREAL_1:1;
A7: L[01](0,1,r1,r2).0
= ((r2 - r1)/(1 - 0)) * (0 - 0) + r1 by A2,BORSUK_6:35 .= r1;
A8: L[01](0,1,r1,r2).1
= ((r2 - r1)/(1 - 0)) * (1 - 0) + r1 by A2,BORSUK_6:35 .= r2;
A9: p.0
= t1 by A1,A7,A6,FUNCT_1:12;
p.1
= t2 by A1,A8,A6,FUNCT_1:12;
hence thesis by A5,A9;
end;
definition
let T be non empty TopStruct;
let c1,c2 be with_endpoints Curve of T;
pred c1,c2 are_homotopic means
ex a,b being Point of T, p1,p2 being Path of a,b st
p1 = c1*L[01](0,1,inf dom c1,sup dom c1) &
p2 = c2*L[01](0,1,inf dom c2,sup dom c2) &
p1,p2 are_homotopic;
symmetry;
end;
definition
let T be non empty TopSpace;
let c1,c2 be with_endpoints Curve of T;
redefine pred c1,c2 are_homotopic;
reflexivity
proof
let c be with_endpoints Curve of T;
reconsider p = c*L[01](0,1,inf dom c,sup dom c) as
Path of the_first_point_of c,the_last_point_of c by Th29;
p,p are_homotopic by Th33,BORSUK_2:12;
hence thesis;
end;
symmetry;
end;
theorem Th34:
for T being non empty TopStruct,
c1,c2 being with_endpoints Curve of T,
a,b being Point of T,
p1,p2 being Path of a,b
st c1 = p1 & c2 = p2 & a,b are_connected
holds c1,c2 are_homotopic iff p1,p2 are_homotopic
proof
let T be non empty TopStruct;
let c1,c2 be with_endpoints Curve of T;
let a,b be Point of T;
let p1,p2 be Path of a,b;
assume
A1: c1 = p1 & c2 = p2;
assume
A2: a,b are_connected;
A3: 0 is Point of I[01] & 1 is Point of I[01] by BORSUK_1:40,XXREAL_1:1;
A4: inf dom c1 = 0 & sup dom c1 = 1 & inf dom c2 = 0 & sup dom c2 = 1
by A1,Th4;
A5: dom p1 = the carrier of I[01] &
dom p2 = the carrier of I[01] by FUNCT_2:def 1;
thus c1,c2 are_homotopic implies p1,p2 are_homotopic
proof
assume c1,c2 are_homotopic;
then consider aa,bb be Point of T, pp1,pp2 be Path of aa,bb such that
A6: pp1 = c1*L[01](0,1,inf dom c1,sup dom c1) &
pp2 = c2*L[01](0,1,inf dom c2,sup dom c2) &
pp1,pp2 are_homotopic;
consider f be Function of [:I[01],I[01]:], T such that
A7: f is continuous & for t being Point of I[01]
holds f.(t,0) = pp1.t & f.(t,1) = pp2.t
& f.(0,t) = aa & f.(1,t) = bb by A6;
A8: pp1 = p1 & pp2 = p2 by A1,A6,A4,A5,Th1,RELAT_1:52,TOPMETR:20;
A9: f.(0,0)=pp1.0 & f.(0,1)=pp2.0 & f.(0,0)=aa & f.(0,1)=aa by A7,A3;
A10: f.(1,0)=pp1.1 & f.(1,1)=pp2.1 & f.(1,0)=bb & f.(1,1)=bb by A7,A3;
aa = a & bb = b by A8,A9,A10,A2,BORSUK_2:def 2;
hence p1,p2 are_homotopic by A7,A8;
end;
assume
A11: p1,p2 are_homotopic;
p1 = p1*L[01](0,1,0,1) & p2 = p2*L[01](0,1,0,1)
by A5,Th1,RELAT_1:52,TOPMETR:20;
hence c1,c2 are_homotopic by A4,A1,A11;
end;
theorem Th35:
for c1,c2 being with_endpoints Curve of T
st c1, c2 are_homotopic
holds the_first_point_of c1 = the_first_point_of c2 &
the_last_point_of c1 = the_last_point_of c2
proof
let c1,c2 be with_endpoints Curve of T;
assume c1, c2 are_homotopic;
then consider a,b be Point of T, p1,p2 be Path of a,b such that
A1: p1 = c1*L[01](0,1,inf dom c1,sup dom c1) &
p2 = c2*L[01](0,1,inf dom c2,sup dom c2) &
p1,p2 are_homotopic;
A2: 0 is Point of I[01] & 1 is Point of I[01] by BORSUK_1:40,XXREAL_1:1;
consider f be Function of [:I[01],I[01]:], T such that
A3: f is continuous & for t being Point of I[01] holds f.(t,0) = p1.t &
f.(t,1) = p2.t & f.(0,t) = a & f.(1,t) = b by A1;
A4: f.(0,0)=p1.0 & f.(0,1)=p2.0 & f.(0,0)=a & f.(0,1)=a by A3,A2;
A5: f.(1,0)=p1.1 & f.(1,1)=p2.1 & f.(1,0)=b & f.(1,1)=b by A3,A2;
A6: 0 in [.0,1.] & 1 in [.0,1.] by XXREAL_1:1;
A7:dom L[01](0,1,inf dom c1,sup dom c1)
= the carrier of Closed-Interval-TSpace(0,1) by FUNCT_2:def 1
.= [.0,1.] by TOPMETR:18;
A8:dom L[01](0,1,inf dom c2,sup dom c2)
= the carrier of Closed-Interval-TSpace(0,1) by FUNCT_2:def 1
.= [.0,1.] by TOPMETR:18;
A9:inf dom c1 <= sup dom c1 by XXREAL_2:40;
A10:inf dom c2 <= sup dom c2 by XXREAL_2:40;
A11:L[01](0,1,inf dom c2,sup dom c2).0
= (sup dom c2 - inf dom c2)/(1 - 0) * (0 - 0) + inf dom c2
by A10,BORSUK_6:35 .= inf dom c2;
L[01](0,1,inf dom c1,sup dom c1).0
= (sup dom c1 - inf dom c1)/(1 - 0) * (0 - 0) + inf dom c1
by A9,BORSUK_6:35 .= inf dom c1;
then p1.0 = c1.(inf dom c1) by A1,A6,A7,FUNCT_1:13;
hence the_first_point_of c1 = the_first_point_of c2
by A4,A1,A11,A6,A8,FUNCT_1:13;
A12:L[01](0,1,inf dom c2,sup dom c2).1
= (sup dom c2 - inf dom c2)/(1 - 0) * (1 - 0) + inf dom c2
by A10,BORSUK_6:35 .= sup dom c2;
L[01](0,1,inf dom c1,sup dom c1).1
= (sup dom c1 - inf dom c1)/(1 - 0) * (1 - 0) + inf dom c1
by A9,BORSUK_6:35 .= sup dom c1;
then p1.1 = c1.(sup dom c1) by A1,A6,A7,FUNCT_1:13;
hence the_last_point_of c1 = the_last_point_of c2
by A5,A1,A12,A6,A8,FUNCT_1:13;
end;
theorem Th36:
for T being non empty TopSpace
for c1,c2 being with_endpoints Curve of T
for S being Subset of R^1
st dom c1 = dom c2 & S = dom c1 holds c1,c2 are_homotopic iff
ex f being Function of [:R^1|S,I[01]:], T, a,b being Point of T
st f is continuous &
(for t being Point of R^1|S holds f.(t,0) = c1.t & f.(t,1) = c2.t) &
(for t being Point of I[01] holds f.(inf S,t) = a & f.(sup S,t) = b)
proof
let T be non empty TopSpace;
let c1,c2 be with_endpoints Curve of T;
let S be Subset of R^1;
assume
A1: dom c1 = dom c2 & S = dom c1;
A2: inf dom c1 <= sup dom c1 by XXREAL_2:40;
A3: S = [.inf dom c1,sup dom c1.] by A1,Th27;
A4: 0 in [#]I[01] by BORSUK_1:40,XXREAL_1:1;
A5: 1 in [#]I[01] by BORSUK_1:40,XXREAL_1:1;
A6: inf S in S by A3,A2,A1,XXREAL_1:1;
A7: sup S in S by A3,A2,A1,XXREAL_1:1;
thus c1,c2 are_homotopic implies
ex f being Function of [:R^1|S,I[01]:],T, a,b being Point of T
st f is continuous &
(for t being Point of R^1|S holds f.(t,0) = c1.t & f.(t,1) = c2.t) &
(for t being Point of I[01] holds f.(inf S,t) = a & f.(sup S,t) = b)
proof
assume
A8: c1,c2 are_homotopic;
per cases;
suppose
A9: inf dom c1 < sup dom c1;
consider a,b be Point of T, p1,p2 be Path of a,b such that
A10: p1 = c1*L[01](0,1,inf dom c1,sup dom c1) &
p2 = c2*L[01](0,1,inf dom c2,sup dom c2) &
p1,p2 are_homotopic by A8;
consider f be Function of [:I[01],I[01]:], T such that
A11: f is continuous &
for t being Point of I[01] holds f.(t,0) = p1.t & f.(t,1) = p2.t
& f.(0,t) = a & f.(1,t) = b by A10;
set f1 = L[01](inf dom c1,sup dom c1,0,1);
set f2 = L[01](0,1,0,1);
reconsider S1 = R^1|S as non empty TopSpace by A1;
A12: Closed-Interval-TSpace(inf dom c1,sup dom c1) = S1
by A3,A9,TOPMETR:19;
reconsider f1 as continuous Function of S1,I[01]
by A9,A12,BORSUK_6:34,TOPMETR:20;
reconsider f2 as continuous Function of I[01],I[01]
by BORSUK_6:34,TOPMETR:20;
set h = f * [:f1,f2:];
reconsider h as Function of [:R^1|S,I[01]:],T;
take h,a,b;
thus h is continuous by A11;
A13: dom f1 = [#](R^1|S) by FUNCT_2:def 1;
A14: dom f2
= [.0,1.] by BORSUK_1:40,FUNCT_2:def 1;
A15: f2.0 = (1 - 0)/(1 - 0) * (0 - 0) + 0 by BORSUK_6:35 .= 0;
A16: f2.1 = (1 - 0)/(1 - 0) * (1 - 0) + 0 by BORSUK_6:35 .= 1;
A17: rng f1 c= [#]I[01] by RELAT_1:def 19;
A18: 0 in dom f2 by A14,XXREAL_1:1;
A19: 1 in dom f2 by A14,XXREAL_1:1;
A20: sup dom c1 - inf dom c1 <> 0 by A9;
thus for t being Point of R^1|S holds h.(t,0) = c1.t & h.(t,1) = c2.t
proof
let t be Point of R^1|S;
A21: t in dom f1 by A13;
[t,0] in [:dom f1,dom f2:] by A13,A18,ZFMISC_1:def 2;
then
A22: [t,0] in dom [:f1,f2:] by FUNCT_3:def 8;
[t,1] in [:dom f1,dom f2:] by A13,A19,ZFMISC_1:def 2;
then
A23: [t,1] in dom [:f1,f2:] by FUNCT_3:def 8;
A24: f1.t in rng f1 by A13,FUNCT_1:3;
A25: t in S by A21,A13,PRE_TOPC:def 5;
t in [#]Closed-Interval-TSpace(inf dom c1,sup dom c1) by A12;
then
A26: t in dom L[01](inf dom c1,sup dom c1,inf dom c1,sup dom c1)
by FUNCT_2:def 1;
A27: inf dom c1 <= t & t <= sup dom c1 by A25,A3,XXREAL_1:1;
A28: L[01](inf dom c1,sup dom c1,inf dom c1,sup dom c1).t
= (sup dom c1 - inf dom c1)/(sup dom c1 - inf dom c1) *
(t - inf dom c1) + inf dom c1 by A27,A9,BORSUK_6:35
.= 1 * (t - inf dom c1) + inf dom c1 by A20,XCMPLX_1:60
.= t;
thus h.(t,0) = h.([t,0]) by BINOP_1:def 1
.= f.([:f1,f2:].([t,0])) by A22,FUNCT_1:13
.= f.([:f1,f2:].(t,0)) by BINOP_1:def 1
.= f.([f1.t,f2.0]) by A13,A18,FUNCT_3:def 8
.= f.(f1.t, 0) by A15,BINOP_1:def 1
.= p1.(f1.t) by A24,A11,A17
.= (c1*L[01](0,1,inf dom c1,sup dom c1)*f1).t by A10,A13,FUNCT_1:13
.= (c1*(L[01](0,1,inf dom c1,sup dom c1)*f1)).t by RELAT_1:36
.= (c1*L[01](inf dom c1,sup dom c1,inf dom c1,sup dom c1)).t
by Th2,A9
.= c1.t by A28,A26,FUNCT_1:13;
thus h.(t,1) = h.([t,1]) by BINOP_1:def 1
.= f.([:f1,f2:].([t,1])) by A23,FUNCT_1:13
.= f.([:f1,f2:].(t,1)) by BINOP_1:def 1
.= f.([f1.t,f2.1]) by A13,A19,FUNCT_3:def 8
.= f.(f1.t, 1) by A16,BINOP_1:def 1
.= p2.(f1.t) by A24,A11,A17
.= (c2*L[01](0,1,inf dom c1,sup dom c1)*f1).t by A10,A1,A13,
FUNCT_1:13
.= (c2*(L[01](0,1,inf dom c1,sup dom c1)*f1)).t by RELAT_1:36
.= (c2*L[01](inf dom c1,sup dom c1,inf dom c1,sup dom c1)).t
by Th2,A9
.= c2.t by A28,A26,FUNCT_1:13;
end;
thus for t being Point of I[01] holds h.(inf S,t) = a & h.(sup S,t) = b
proof
let t be Point of I[01];
A29: inf S in dom f1 by A6,A13,PRE_TOPC:def 5;
A30: sup S in dom f1 by A7,A13,PRE_TOPC:def 5;
t in [#]I[01];
then
A31: t in dom f2 by FUNCT_2:def 1;
[inf S,t] in [:dom f1,dom f2:] by A31,A29,ZFMISC_1:def 2;
then
A32: [inf S,t] in dom [:f1,f2:] by FUNCT_3:def 8;
[sup S,t] in [:dom f1,dom f2:] by A31,A30,ZFMISC_1:def 2;
then
A33: [sup S,t] in dom [:f1,f2:] by FUNCT_3:def 8;
0 <= t & t <= 1 by BORSUK_1:40,XXREAL_1:1;
then
A34: f2.t = (1 - 0)/(1 - 0) * (t - 0) + 0 by BORSUK_6:35 .= t;
A35: f1.(inf S)
= (1 - 0)/(sup dom c1 - inf dom c1) *
(inf dom c1 - inf dom c1) + 0 by A1,A9,BORSUK_6:35
.= 0;
A36: f1.(sup S)
= (1 - 0)/(sup dom c1 - inf dom c1) *
(sup dom c1 - inf dom c1) + 0 by A1,A9,BORSUK_6:35
.= (sup dom c1 - inf dom c1)/(sup dom c1 - inf dom c1)*1
by XCMPLX_1:75
.= 1 by A20,XCMPLX_1:60;
thus h.(inf S,t) = h.([inf S,t]) by BINOP_1:def 1
.= f.([:f1,f2:].([inf S,t])) by A32,FUNCT_1:13
.= f.([:f1,f2:].(inf S,t)) by BINOP_1:def 1
.= f.([f1.(inf S),f2.t]) by A31,A29,FUNCT_3:def 8
.= f.(f1.(inf S), t) by A34,BINOP_1:def 1
.= a by A11,A35;
thus h.(sup S,t) = h.([sup S,t]) by BINOP_1:def 1
.= f.([:f1,f2:].([sup S,t])) by A33,FUNCT_1:13
.= f.([:f1,f2:].(sup S,t)) by BINOP_1:def 1
.= f.([f1.(sup S),f2.t]) by A31,A30,FUNCT_3:def 8
.= f.(f1.(sup S), t) by A34,BINOP_1:def 1
.= b by A11,A36;
end;
end;
suppose
not inf dom c1 < sup dom c1;
then inf dom c1 = sup dom c1 by A2,XXREAL_0:1;
then dom c1 = [.inf dom c1, inf dom c1.] by Th27;
then
A37: dom c1 = {inf dom c1} by XXREAL_1:17;
set a = the_first_point_of c1;
set f = [:R^1|S,I[01]:] --> a;
A38: for t being Point of R^1|S holds f.(t,0) = c1.t & f.(t,1) = c2.t
proof
let t be Point of R^1|S;
A39: t in [#](R^1|S) by A1,SUBSET_1:def 1;
A40: [t,0] in [:[#](R^1|S),[#]I[01]:] by A4,A1,ZFMISC_1:def 2;
A41: [t,1] in [:[#](R^1|S),[#]I[01]:] by A5,A1,ZFMISC_1:def 2;
A42: t in S by A39,PRE_TOPC:def 5;
then
A43: t = inf dom c1 by A1,A37,TARSKI:def 1;
thus f.(t,0) = f.([t,0]) by BINOP_1:def 1
.= c1.t by A43,A40,FUNCOP_1:7;
thus f.(t,1) = f.([t,1]) by BINOP_1:def 1 .= a by A41,FUNCOP_1:7
.= the_first_point_of c2 by A8,Th35 .= c2.t
by A1,A42,A37,TARSKI:def 1;
end;
for t being Point of I[01] holds f.(inf S,t) = a & f.(sup S,t) = a
proof
let t be Point of I[01];
A44: inf S in [#](R^1|S) by A6,PRE_TOPC:def 5;
A45: sup S in [#](R^1|S) by A7,PRE_TOPC:def 5;
A46: [inf S,t] in [:[#](R^1|S),[#]I[01]:] by A44,ZFMISC_1:def 2;
A47: [sup S,t] in [:[#](R^1|S),[#]I[01]:] by A45,ZFMISC_1:def 2;
thus f.(inf S,t) = f.([inf S,t]) by BINOP_1:def 1
.= a by A46,FUNCOP_1:7;
thus f.(sup S,t) = f.([sup S,t]) by BINOP_1:def 1
.= a by A47,FUNCOP_1:7;
end;
hence thesis by A38;
end;
end;
given f be Function of [:R^1|S,I[01]:], T, a,b be Point of T such that
A48: f is continuous &
(for t being Point of R^1|S holds f.(t,0) = c1.t & f.(t,1) = c2.t) &
(for t being Point of I[01] holds f.(inf S,t) = a & f.(sup S,t) = b);
A49:inf S in [#](R^1|S) by A6,PRE_TOPC:def 5;
A50:sup S in [#](R^1|S) by A7,PRE_TOPC:def 5;
A51: a = f.(inf S,0) by A4,A48
.= the_first_point_of c1 by A1,A49,A48;
b = f.(sup S,0) by A4,A48
.= the_last_point_of c1 by A1,A50,A48;
then reconsider p1 = c1*L[01](0,1,inf dom c1,sup dom c1) as Path of a,b
by A51,Th29;
A52: a = f.(inf S,1) by A5,A48
.= the_first_point_of c2 by A1,A49,A48;
b = f.(sup S,1) by A5,A48
.= the_last_point_of c2 by A1,A50,A48;
then reconsider p2 = c2*L[01](0,1,inf dom c2,sup dom c2) as Path of a,b
by A52,Th29;
set f1 = L[01](0,1,inf dom c1,sup dom c1);
set f2 = L[01](0,1,0,1);
reconsider S1 = R^1|S as non empty TopSpace by A1;
A53: Closed-Interval-TSpace(inf dom c1,sup dom c1) = S1
by A3,A2,TOPMETR:19;
reconsider f1 as continuous Function of I[01],S1
by A2,A53,BORSUK_6:34,TOPMETR:20;
reconsider f2 as continuous Function of I[01],I[01]
by BORSUK_6:34,TOPMETR:20;
set h = f * [:f1,f2:];
reconsider h as Function of [:I[01],I[01]:],T;
A54: dom f1 = [#]I[01] by FUNCT_2:def 1;
A55: dom f2
= [.0,1.] by BORSUK_1:40,FUNCT_2:def 1;
A56: f2.0 = (1 - 0)/(1 - 0) * (0 - 0) + 0 by BORSUK_6:35 .= 0;
A57: f2.1 = (1 - 0)/(1 - 0) * (1 - 0) + 0 by BORSUK_6:35 .= 1;
A58: 0 in dom f2 by A55,XXREAL_1:1;
A59: 1 in dom f2 by A55,XXREAL_1:1;
for t being Point of I[01] holds h.(t,0) = p1.t & h.(t,1) = p2.t
& h.(0,t) = a & h.(1,t) = b
proof
let t be Point of I[01];
[t,0] in [:dom f1,dom f2:] by A54,A58,ZFMISC_1:def 2;
then
A60: [t,0] in dom [:f1,f2:] by FUNCT_3:def 8;
[t,1] in [:dom f1,dom f2:] by A54,A59,ZFMISC_1:def 2;
then
A61: [t,1] in dom [:f1,f2:] by FUNCT_3:def 8;
A62: 0 in dom f1 by A54,BORSUK_1:40,XXREAL_1:1;
A63: 1 in dom f1 by A54,BORSUK_1:40,XXREAL_1:1;
[0,t] in [:dom f1,dom f2:] by A62,A55,BORSUK_1:40,ZFMISC_1:def 2;
then
A64: [0,t] in dom [:f1,f2:] by FUNCT_3:def 8;
[1,t] in [:dom f1,dom f2:] by A63,A55,BORSUK_1:40,ZFMISC_1:def 2;
then
A65: [1,t] in dom [:f1,f2:] by FUNCT_3:def 8;
A66: f1.0 = (sup dom c1 - inf dom c1)/(1 - 0) *
(0 - 0) + inf dom c1 by A2,BORSUK_6:35
.= inf S by A1;
A67: f1.1 = (sup dom c1 - inf dom c1)/(1 - 0) *
(1 - 0) + inf dom c1 by A2,BORSUK_6:35
.= sup S by A1;
0 <= t & t <= 1 by BORSUK_1:40,XXREAL_1:1;
then
A68: f2.t = (1 - 0)/(1 - 0) * (t - 0) + 0 by BORSUK_6:35 .= t;
thus h.(t,0) = h.([t,0]) by BINOP_1:def 1
.= f.([:f1,f2:].([t,0])) by A60,FUNCT_1:13
.= f.([:f1,f2:].(t,0)) by BINOP_1:def 1
.= f.([f1.t,f2.0]) by A54,A58,FUNCT_3:def 8
.= f.(f1.t, 0) by A56,BINOP_1:def 1
.= c1.(f1.t) by A48
.= p1.t by A54,FUNCT_1:13;
thus h.(t,1) = h.([t,1]) by BINOP_1:def 1
.= f.([:f1,f2:].([t,1])) by A61,FUNCT_1:13
.= f.([:f1,f2:].(t,1)) by BINOP_1:def 1
.= f.([f1.t,f2.1]) by A54,A59,FUNCT_3:def 8
.= f.(f1.t, 1) by A57,BINOP_1:def 1
.= c2.(f1.t) by A48
.= p2.t by A1,A54,FUNCT_1:13;
thus h.(0,t) = h.([0,t]) by BINOP_1:def 1
.= f.([:f1,f2:].([0,t])) by A64,FUNCT_1:13
.= f.([:f1,f2:].(0,t)) by BINOP_1:def 1
.= f.([f1.0,f2.t]) by A62,A55,BORSUK_1:40,FUNCT_3:def 8
.= f.(inf S, t) by A66,A68,BINOP_1:def 1
.= a by A48;
thus h.(1,t) = h.([1,t]) by BINOP_1:def 1
.= f.([:f1,f2:].([1,t])) by A65,FUNCT_1:13
.= f.([:f1,f2:].(1,t)) by BINOP_1:def 1
.= f.([f1.1,f2.t]) by A63,A55,BORSUK_1:40,FUNCT_3:def 8
.= f.(sup S, t) by A67,A68,BINOP_1:def 1
.= b by A48;
end;
then p1,p2 are_homotopic by A48;
hence c1,c2 are_homotopic;
end;
definition
let T be TopStruct;
let c1, c2 be Curve of T;
func c1 + c2 -> Curve of T equals :Def12:
c1 \/ c2 if c1 \/ c2 is Curve of T otherwise {};
correctness
proof
now
assume not c1 \/ c2 is Curve of T;
{} is parametrized-curve PartFunc of R^1, T by Lm1,XBOOLE_1:2;
hence {} is Curve of T by Th20;
end;
hence thesis;
end;
end;
theorem Th37:
for c being with_endpoints Curve of T
for r being Real holds
ex c1,c2 being Element of Curves T
st c = c1 + c2 & c1 = c | [.inf dom c, r.] & c2 = c | [.r, sup dom c.]
proof
let c be with_endpoints Curve of T;
let r be Real;
set c1 = c | [.inf dom c, r.];
set c2 = c | [.r, sup dom c.];
reconsider c1 as Curve of T by Th26;
reconsider c2 as Curve of T by Th26;
take c1,c2;
c1 \/ c2 = c
proof
per cases;
suppose
A1: r < inf dom c;
then [.inf dom c, r.] = {} by XXREAL_1:29;
then
A2: c1 = {};
[.inf dom c, sup dom c.] c= [.r, sup dom c.] by A1,XXREAL_1:34;
then dom c c= [.r, sup dom c.] by Th27;
hence thesis by A2,RELAT_1:68;
end;
suppose
A3: r >= inf dom c;
per cases;
suppose
A4: r > sup dom c;
then [.r, sup dom c.] = {} by XXREAL_1:29;
then
A5: c2 = {};
[.inf dom c, sup dom c.] c= [.inf dom c, r.] by A4,XXREAL_1:34;
then dom c c= [.inf dom c, r.] by Th27;
hence thesis by A5,RELAT_1:68;
end;
suppose
A6: r <= sup dom c;
dom c = [.inf dom c, sup dom c.] by Th27
.= [.inf dom c,r.] \/ [.r, sup dom c.] by A6,A3,XXREAL_1:165;
then c | dom c
= c1 \/ c2 by RELAT_1:78;
hence thesis;
end;
end;
end;
hence thesis by Def12;
end;
theorem Th38:
for T being non empty TopSpace
for c1,c2 being with_endpoints Curve of T
st sup dom c1 = inf dom c2 & the_last_point_of c1 = the_first_point_of c2
holds c1 + c2 is with_endpoints &
dom(c1 + c2) = [.inf dom c1, sup dom c2.] &
(c1+c2).inf dom c1 = the_first_point_of c1 &
(c1+c2).sup dom c2 = the_last_point_of c2
proof
let T be non empty TopSpace;
let c1,c2 be with_endpoints Curve of T;
assume
A1: sup dom c1 = inf dom c2;
assume
A2: the_last_point_of c1 = the_first_point_of c2;
set f = c1 \/ c2;
A3: dom(c1 \/ c2) = dom c1 \/ dom c2 by XTUPLE_0:23;
reconsider f1 = c1 as parametrized-curve PartFunc of R^1, T
by Th23;
A4:dom f1 is interval Subset of REAL by Def4;
reconsider f2 = c2 as parametrized-curve PartFunc of R^1, T
by Th23;
A5:dom f2 is interval Subset of REAL by Def4;
A6: dom c1 \/ dom c2 c= REAL by A4,A5,XBOOLE_1:8;
rng f1 c= [#]T & rng f2 c= [#]T;
then rng c1 \/ rng c2 c= [#]T by XBOOLE_1:8;
then
A7:rng f c= [#]T by RELAT_1:12;
A8: dom f c= [#]R^1 by A6,TOPMETR:17,XTUPLE_0:23;
reconsider S0 = dom f as Subset of R^1 by A6,TOPMETR:17,XTUPLE_0:23;
A9: inf dom c2 <= sup dom c2 by XXREAL_2:40;
A10: inf dom c1 <= sup dom c1 by XXREAL_2:40;
A11: dom f1 = [.inf dom c1, sup dom c1.] by Th27;
A12: dom f2 = [.inf dom c2, sup dom c2.] by Th27;
A13:dom f1 /\ dom f2 = {sup dom c1} by A11,A12,A1,A9,A10,XXREAL_1:418;
A14:dom f1 /\ dom f2 c= dom f by A3,XBOOLE_1:29;
set S = R^1|S0;
consider S1 be SubSpace of R^1, g1 be Function of S1, T such that
A15: f1 = g1 & S1 = R^1|dom f1 & g1 is continuous by Def4;
consider S2 be SubSpace of R^1, g2 be Function of S2, T such that
A16: f2 = g2 & S2 = R^1|dom f2 & g2 is continuous by Def4;
sup dom c1 in dom f by A13,A14,ZFMISC_1:31;
then sup dom c1 in [#]S by PRE_TOPC:def 5;
then reconsider p = sup dom c1 as Point of S;
reconsider S1,S2 as SubSpace of S by A15,A16,A3,TOPMETR:22,XBOOLE_1:7;
A17: [#]S1 \/ [#]S2 = dom f1 \/ [#]S2 by A15,PRE_TOPC:def 5
.= dom f1 \/ dom f2 by A16,PRE_TOPC:def 5
.= [#]S by A3,PRE_TOPC:def 5;
A18: [#]S1 /\ [#]S2 = dom f1 /\ [#]S2 by A15,PRE_TOPC:def 5
.= {p} by A13,A16,PRE_TOPC:def 5;
S1 = Closed-Interval-TSpace(inf dom c1,sup dom c1) by A11,A10,A15,
TOPMETR:19;
then
A19: S1 is compact by A10,HEINE:4;
S2 = Closed-Interval-TSpace(inf dom c2,sup dom c2) by A12,A9,A16,TOPMETR:19
;
then
A20: S2 is compact by A9,HEINE:4;
A21: S is T_2 by TOPMETR:2;
A22: g1+*g2 is continuous Function of S,T
by A17,A18,A19,A20,A21,A15,A16,A1,A2,COMPTS_1:20;
for x,y1,y2 being object st [x,y1] in f & [x,y2] in f holds y1 = y2
proof
let x,y1,y2 be object;
assume
A23: [x,y1] in f & [x,y2] in f;
per cases by A23,XBOOLE_0:def 3;
suppose [x,y1] in c1 & [x,y2] in c1; hence thesis by FUNCT_1:def 1; end;
suppose [x,y1] in c2 & [x,y2] in c2; hence thesis by FUNCT_1:def 1; end;
suppose
A24: [x,y1] in c1 & [x,y2] in c2;
then x in dom c1 & x in dom c2 by XTUPLE_0:def 12;
then x in (dom c1 /\ dom c2) by XBOOLE_0:def 4;
then x = p by A13,TARSKI:def 1;
then c1.p = y1 & c2.p = y2 by A24,FUNCT_1:1;
hence thesis by A1,A2;
end;
suppose
A25: [x,y1] in c2 & [x,y2] in c1;
then x in dom c2 & x in dom c1 by XTUPLE_0:def 12;
then x in (dom c2 /\ dom c1) by XBOOLE_0:def 4;
then x = p by A13,TARSKI:def 1;
then c2.p = y1 & c1.p = y2 by A25,FUNCT_1:1;
hence thesis by A1,A2;
end;
end;
then reconsider f as Function by FUNCT_1:def 1;
A26: dom f = dom g1 \/ dom g2 by A15,A16,XTUPLE_0:23;
for x being object st x in dom g1 \/ dom g2 holds
(x in dom g2 implies f.x = g2.x) &
(not x in dom g2 implies f.x = g1.x)
proof
let x be object;
assume
A27: x in dom g1 \/ dom g2;
thus x in dom g2 implies f.x = g2.x
proof
assume x in dom g2;
then [x,g2.x] in g2 by FUNCT_1:1;
then [x,g2.x] in f by A16,XBOOLE_0:def 3;
hence thesis by FUNCT_1:1;
end;
thus not x in dom g2 implies f.x = g1.x
proof
assume not x in dom g2;
then x in dom g1 by A27,XBOOLE_0:def 3;
then [x,g1.x] in g1 by FUNCT_1:1;
then [x,g1.x] in f by A15,XBOOLE_0:def 3;
hence thesis by FUNCT_1:1;
end;
end;
then
A28: f = g1+*g2 by A26,FUNCT_4:def 1;
reconsider f as PartFunc of R^1, T by A7,A8,RELSET_1:4;
dom c1 meets dom c2 by A13;
then dom f is interval Subset of REAL
by A4,A5,A3,XBOOLE_1:8,XXREAL_2:89;
then f is parametrized-curve by A22,A28;
then
A29: c1 \/ c2 is Curve of T by Th20;
then
A30: c1 + c2 = c1 \/ c2 by Def12;
A31: dom(c1 \/ c2)
= [.inf dom c1, sup dom c2.] by A3,A11,A12,A1,A9,A10,XXREAL_1:165;
A32:inf dom c1 in dom f1 by A11,A10,XXREAL_1:1;
then inf dom c1 in dom f by A3,XBOOLE_0:def 3;
then inf dom f in dom f by A31,A1,A9,A10,XXREAL_0:2,XXREAL_2:25;
then dom(c1 + c2) is left_end by A30,XXREAL_2:def 5;
then
A33: c1 + c2 is with_first_point;
A34:sup dom c2 in dom f2 by A12,A9,XXREAL_1:1;
then sup dom c2 in dom f by A3,XBOOLE_0:def 3;
then sup [.inf dom c1, sup dom c2.] in dom f
by A1,A9,A10,XXREAL_0:2,XXREAL_2:29;
then dom(c1 + c2) is right_end by A31,A30,XXREAL_2:def 6;
then
A35: c1 + c2 is with_last_point;
thus c1 + c2 is with_endpoints & dom(c1 + c2) = [.inf dom c1, sup dom c2.]
by A33,A35,A30,A3,A11,A12,A1,A9,A10,XXREAL_1:165;
A36: (c1+c2).inf dom c1 = c1.inf dom c1
proof
A37: (c1+c2).inf dom c1 = f.inf dom c1 by A29,Def12;
[inf dom c1, c1.inf dom c1] in c1 by A32,FUNCT_1:1;
then [inf dom c1, c1.inf dom c1] in f by XBOOLE_0:def 3;
hence thesis by A37,FUNCT_1:1;
end;
thus (c1+c2).inf dom c1 = the_first_point_of c1 by A36;
A38: (c1+c2).sup dom c2 = c2.sup dom c2
proof
A39: (c1+c2).sup dom c2 = f.sup dom c2 by A29,Def12;
[sup dom c2, c2.sup dom c2] in c2 by A34,FUNCT_1:1;
then [sup dom c2, c2.sup dom c2] in f by XBOOLE_0:def 3;
hence thesis by A39,FUNCT_1:1;
end;
thus (c1+c2).sup dom c2 = the_last_point_of c2 by A38;
end;
theorem Th39:
for T being non empty TopSpace
for c1,c2,c3,c4,c5,c6 being with_endpoints Curve of T
st c1,c2 are_homotopic & dom c1 = dom c2 &
c3,c4 are_homotopic & dom c3 = dom c4 &
c5 = c1 + c3 & c6 = c2 + c4
& the_last_point_of c1 = the_first_point_of c3
& sup dom c1 = inf dom c3
holds c5,c6 are_homotopic
proof
let T be non empty TopSpace;
let c1,c2,c3,c4,c5,c6 be with_endpoints Curve of T;
assume
A1: c1,c2 are_homotopic & dom c1 = dom c2;
reconsider S1 = [.inf dom c1, sup dom c1.] as non empty Subset of R^1
by Th27,TOPMETR:17;
A2: dom c1 = S1 by Th27;
then consider H1 be Function of [:R^1|S1,I[01]:], T,
a1,b1 be Point of T such that
A3: H1 is continuous &
(for t being Point of R^1|S1 holds H1.(t,0) = c1.t & H1.(t,1) = c2.t) &
(for t being Point of I[01] holds H1.(inf S1,t) = a1 & H1.(sup S1,t) = b1)
by A1,Th36;
assume
A4: c3,c4 are_homotopic & dom c3 = dom c4;
reconsider S2 = [.inf dom c3, sup dom c3.] as non empty Subset of R^1
by Th27,TOPMETR:17;
A5: dom c3 = S2 by Th27;
then consider H2 be Function of [:R^1|S2,I[01]:], T,
a2,b2 be Point of T such that
A6: H2 is continuous &
(for t being Point of R^1|S2 holds H2.(t,0) = c3.t & H2.(t,1) = c4.t) &
(for t being Point of I[01] holds H2.(inf S2,t) = a2 & H2.(sup S2,t) = b2)
by A4,Th36;
assume
A7: c5 = c1 + c3;
A8: c5 = c1 \/ c3
proof
per cases;
suppose c1 \/ c3 is Curve of T; hence thesis by A7,Def12; end;
suppose not c1 \/ c3 is Curve of T;
hence thesis by A7,Def12;
end;
end;
assume
A9: c6 = c2 + c4;
A10: c6 = c2 \/ c4
proof
per cases;
suppose c2 \/ c4 is Curve of T; hence thesis by A9,Def12; end;
suppose not c2 \/ c4 is Curve of T;
hence thesis by A9,Def12;
end;
end;
assume
A11: the_last_point_of c1 = the_first_point_of c3;
assume
A12:sup dom c1 = inf dom c3;
A13: dom c5
= dom c1 \/ dom c3 by A8,XTUPLE_0:23
.= dom c6 by A10,A1,A4,XTUPLE_0:23;
reconsider S3 = S1 \/ S2 as Subset of R^1;
A14: dom c5
= dom c1 \/ dom c3 by A8,XTUPLE_0:23
.= S3 by A5,Th27;
set T1 = [:R^1|S1,I[01]:];
set T2 = [:R^1|S2,I[01]:];
set T3 = [:R^1|S3,I[01]:];
A15:I[01] is SubSpace of I[01] by TSEP_1:2;
R^1|S1 is SubSpace of R^1|S3 by TOPMETR:4;
then
A16: T1 is SubSpace of T3 by A15,BORSUK_3:21;
R^1|S2 is SubSpace of R^1|S3 by TOPMETR:4;
then
A17: T2 is SubSpace of T3 by A15,BORSUK_3:21;
A18: [#](R^1|S1) \/ [#](R^1|S2)
= S1 \/ [#](R^1|S2) by PRE_TOPC:def 5
.= S3 by PRE_TOPC:def 5
.= [#](R^1|S3) by PRE_TOPC:def 5;
A19: [#]T1 \/ [#]T2 = [:[#](R^1|S1),[#]I[01]:] \/ [#][:R^1|S2,I[01]:]
by BORSUK_1:def 2
.= [:[#](R^1|S1),[#]I[01]:] \/ [:[#](R^1|S2),[#]I[01]:] by BORSUK_1:def 2
.= [:[#](R^1|S3),[#]I[01]:] by A18,ZFMISC_1:97
.= [#]T3 by BORSUK_1:def 2;
A20: inf dom c1 <= sup dom c1 by XXREAL_2:40;
R^1|S1 = Closed-Interval-TSpace(inf dom c1,sup dom c1)
by A20,TOPMETR:19;
then
A21:R^1|S1 is compact by A20,HEINE:4;
A22: inf dom c3 <= sup dom c3 by XXREAL_2:40;
R^1|S2 = Closed-Interval-TSpace(inf dom c3,sup dom c3)
by A22,TOPMETR:19;
then
A23:R^1|S2 is compact by A22,HEINE:4;
T3 is SubSpace of [:R^1,I[01]:] by A15,BORSUK_3:21;
then
A24: T3 is T_2 by TOPMETR:2;
for p be set st p in [#]T1 /\ [#]T2 holds H1.p = H2.p
proof
let p be set;
assume
A25: p in [#]T1 /\ [#]T2;
A26: [#]T1 /\ [#]T2 = [:[#](R^1|S1),[#]I[01]:] /\ [#][:R^1|S2,I[01]:]
by BORSUK_1:def 2
.= [:[#](R^1|S1),[#]I[01]:] /\ [:[#](R^1|S2),[#]I[01]:] by BORSUK_1:def 2
.= [:[#](R^1|S1) /\ [#](R^1|S2),[#]I[01]:] by ZFMISC_1:99;
A27: [#](R^1|S1) /\ [#](R^1|S2) = S1 /\ [#](R^1|S2) by PRE_TOPC:def 5
.= S1 /\ S2 by PRE_TOPC:def 5;
A28: S1 /\ S2
= {sup dom c1} by A22,A20,A12,XXREAL_1:418;
then consider x,y be object such that
A29: x in {sup dom c1} & y in [#]I[01] & p = [x,y] by A25,A27,A26,
ZFMISC_1:def 2;
reconsider y as Point of I[01] by A29;
A30: x = sup S1 by A2,A29,TARSKI:def 1;
A31: x = inf S2 by A5,A12,A29,TARSKI:def 1;
A32: 0 in [#]I[01] by BORSUK_1:40,XXREAL_1:1;
A33: sup S1 in [#](R^1|S1) by A30,A27,A28,A29,XBOOLE_0:def 4;
thus H1.p
= H1.(sup S1,y) by A29,A30,BINOP_1:def 1
.= b1 by A3
.= H1.(sup S1,0) by A3,A32
.= the_last_point_of c1 by A2,A3,A33
.= H2.(inf S2, 0) by A6,A31,A27,A28,A29,A11,A5
.= a2 by A32,A6
.= H2.(inf S2,y) by A6
.= H2.p by A29,A30,A2,A5,A12,BINOP_1:def 1;
end;
then consider H3 be Function of T3,T such that
A34:H3 = H1+*H2 & H3 is continuous by A16,A17,A19,A21,A23,A24,A3,A6,BORSUK_2:1;
A35:for t being Point of R^1|S3 holds H3.(t,0) = c5.t & H3.(t,1) = c6.t
proof
let t be Point of R^1|S3;
A36: 0 in [#]I[01] by BORSUK_1:40,XXREAL_1:1;
then [t,0] in [:[#](R^1|S3),[#]I[01]:] by ZFMISC_1:def 2;
then [t,0] in [#]T3;
then [t,0] in dom H3 by FUNCT_2:def 1;
then
A37: [t,0] in dom H1 \/ dom H2 by A34,FUNCT_4:def 1;
A38: 1 in [#]I[01] by BORSUK_1:40,XXREAL_1:1;
then [t,1] in [:[#](R^1|S3),[#]I[01]:] by ZFMISC_1:def 2;
then [t,1] in [#]T3;
then [t,1] in dom H3 by FUNCT_2:def 1;
then
A39: [t,1] in dom H1 \/ dom H2 by A34,FUNCT_4:def 1;
per cases;
suppose
A40: [t,0] in dom H2;
then [t,0] in [#]T2;
then [t,0] in [:[#](R^1|S2),[#]I[01]:] by BORSUK_1:def 2;
then
A41: t in [#](R^1|S2) by ZFMISC_1:87;
then
A42: t in dom c3 by A5,PRE_TOPC:def 5;
then t in dom c1 \/ dom c3 by XBOOLE_0:def 3;
then
A43: t in dom c5 by A8,XTUPLE_0:23;
[t,c3.t] in c3 by A42,FUNCT_1:1;
then
A44: [t,c3.t] in c5 by A8,XBOOLE_0:def 3;
thus H3.(t,0) = H3.([t,0]) by BINOP_1:def 1
.= H2.([t,0]) by A37,A40,A34,FUNCT_4:def 1
.= H2.(t,0) by BINOP_1:def 1
.= c3.t by A6,A41
.= c5.t by A44,A43,FUNCT_1:def 2;
[t,1] in [:[#](R^1|S2),[#]I[01]:] by A41,A38,ZFMISC_1:def 2;
then [t,1] in [#]T2;
then
A45: [t,1] in dom H2 by FUNCT_2:def 1;
t in dom c2 \/ dom c4 by A42,A4,XBOOLE_0:def 3;
then
A46: t in dom c6 by A10,XTUPLE_0:23;
[t,c4.t] in c4 by A42,A4,FUNCT_1:1;
then
A47: [t,c4.t] in c6 by A10,XBOOLE_0:def 3;
thus H3.(t,1) = H3.([t,1]) by BINOP_1:def 1
.= H2.([t,1]) by A39,A45,A34,FUNCT_4:def 1
.= H2.(t,1) by BINOP_1:def 1
.= c4.t by A6,A41
.= c6.t by A47,A46,FUNCT_1:def 2;
end;
suppose
A48: not [t,0] in dom H2;
[t,0] in dom H1 or [t,0] in dom H2 by A37,XBOOLE_0:def 3;
then [t,0] in [#]T1 by A48;
then [t,0] in [:[#](R^1|S1),[#]I[01]:] by BORSUK_1:def 2;
then
A49: t in [#](R^1|S1) by ZFMISC_1:87;
then
A50: t in dom c1 by A2,PRE_TOPC:def 5;
then t in dom c1 \/ dom c3 by XBOOLE_0:def 3;
then
A51: t in dom c5 by A8,XTUPLE_0:23;
[t,c1.t] in c1 by A50,FUNCT_1:1;
then
A52: [t,c1.t] in c5 by A8,XBOOLE_0:def 3;
thus H3.(t,0) = H3.([t,0]) by BINOP_1:def 1
.= H1.([t,0]) by A48,A37,A34,FUNCT_4:def 1
.= H1.(t,0) by BINOP_1:def 1
.= c1.t by A3,A49
.= c5.t by A52,A51,FUNCT_1:def 2;
t in dom c2 \/ dom c4 by A50,A1,XBOOLE_0:def 3;
then
A53: t in dom c6 by A10,XTUPLE_0:23;
[t,c2.t] in c2 by A50,A1,FUNCT_1:1;
then
A54: [t,c2.t] in c6 by A10,XBOOLE_0:def 3;
A55: not [t,1] in dom H2
proof
assume [t,1] in dom H2;
then [t,1] in [#]T2;
then [t,1] in [:[#](R^1|S2),[#]I[01]:] by BORSUK_1:def 2;
then t in [#](R^1|S2) by ZFMISC_1:87;
then [t,0] in [:[#](R^1|S2),[#]I[01]:] by A36,ZFMISC_1:def 2;
then [t,0] in [#]T2;
hence contradiction by A48,FUNCT_2:def 1;
end;
thus H3.(t,1) = H3.([t,1]) by BINOP_1:def 1
.= H1.([t,1]) by A55,A39,A34,FUNCT_4:def 1
.= H1.(t,1) by BINOP_1:def 1
.= c2.t by A3,A49
.= c6.t by A54,A53,FUNCT_1:def 2;
end;
end;
for t being Point of I[01] holds H3.(inf S3,t) = a1 & H3.(sup S3,t) = b2
proof
let t be Point of I[01];
A56: inf S1 = inf dom c1 by Th27
.= inf [.inf dom c1,sup dom c3.] by A22,A20,A12,XXREAL_0:2,XXREAL_2:25
.= inf S3 by A22,A20,A12,XXREAL_1:165;
inf S1 in S1 by A2,A20,XXREAL_1:1;
then inf S1 in [#](R^1|S1) by PRE_TOPC:def 5;
then [inf S1,t] in [:[#](R^1|S1),[#]I[01]:] by ZFMISC_1:def 2;
then [inf S1,t] in [#]T1;
then [inf S1,t] in dom H1 by FUNCT_2:def 1;
then
A57: [inf S3,t] in dom H1 \/ dom H2 by A56,XBOOLE_0:def 3;
thus H3.(inf S3,t) = a1
proof
per cases;
suppose
A58: [inf S3,t] in dom H2;
then [inf S3,t] in [#]T2;
then [inf S3,t] in [:[#](R^1|S2),[#]I[01]:] by BORSUK_1:def 2;
then inf S3 in [#](R^1|S2) by ZFMISC_1:87;
then inf S3 in S2 by PRE_TOPC:def 5;
then inf dom c3 <= inf S1 & inf S1 <= sup dom c3 by A56,XXREAL_1:1;
then
A59: inf dom c3 <= inf dom c1 by Th27;
A60: inf dom c3 = inf dom c1 by A59,A12,A20,XXREAL_0:1;
A61: inf S2 = inf dom c3 by Th27
.= inf S1 by A60,Th27;
A62: inf S1 = inf dom c1 by Th27
.= inf dom c3 by A59,A12,A20,XXREAL_0:1
.= sup S1 by A12,Th27;
A63: 0 in [#]I[01] by BORSUK_1:40,XXREAL_1:1;
sup dom c1 = sup S1 by Th27;
then sup S1 in S1 by A20,XXREAL_1:1;
then
A64: sup S1 in [#](R^1|S1) by PRE_TOPC:def 5;
inf dom c3 = inf S2 by Th27;
then inf S2 in S2 by A22,XXREAL_1:1;
then
A65: inf S2 in [#](R^1|S2) by PRE_TOPC:def 5;
A66: sup S1 = sup dom c1 by Th27;
A67: inf S2 = inf dom c3 by Th27;
A68: a1 = H1.(inf S1,0) by A3,A63
.= the_last_point_of c1 by A66,A62,A64,A3
.= H2.(inf S2,0) by A65,A6,A67,A11
.= a2 by A6,A63;
H3.(inf S3,t) = H3.([inf S3,t]) by BINOP_1:def 1
.= H2.([inf S3,t]) by A57,A58,A34,FUNCT_4:def 1
.= H2.(inf S2,t) by A56,A61,BINOP_1:def 1
.= a1 by A6,A68;
hence thesis;
end;
suppose
A69: not [inf S3,t] in dom H2;
H3.(inf S3,t) = H3.([inf S3,t]) by BINOP_1:def 1
.= H1.([inf S3,t]) by A57,A69,A34,FUNCT_4:def 1
.= H1.(inf S3,t) by BINOP_1:def 1
.= a1 by A56,A3;
hence thesis;
end;
end;
A70: sup S2 = sup dom c3 by Th27
.= sup [.inf dom c1,sup dom c3.] by A22,A20,A12,XXREAL_0:2,XXREAL_2:29
.= sup S3 by A22,A20,A12,XXREAL_1:165;
sup S2 in S2 by A5,A22,XXREAL_1:1;
then sup S2 in [#](R^1|S2) by PRE_TOPC:def 5;
then [sup S2,t] in [:[#](R^1|S2),[#]I[01]:] by ZFMISC_1:def 2;
then
A71: [sup S2,t] in [#]T2;
then [sup S2,t] in dom H2 by FUNCT_2:def 1;
then
A72: [sup S3,t] in dom H1 \/ dom H2 by A70,XBOOLE_0:def 3;
A73: [sup S3,t] in dom H2 by A71,A70,FUNCT_2:def 1;
H3.(sup S3,t) = H3.([sup S3,t]) by BINOP_1:def 1
.= H2.([sup S3,t]) by A72,A73,A34,FUNCT_4:def 1
.= H2.(sup S2,t) by A70,BINOP_1:def 1
.= b2 by A6;
hence H3.(sup S3,t) = b2;
end;
hence c5,c6 are_homotopic by A13,A14,A35,A34,Th36;
end;
definition
let T be TopStruct;
let f be FinSequence of Curves T;
func Partial_Sums f -> FinSequence of Curves T means :Def13:
len f = len it & f.1 = it.1 & for i being Nat
st 1<=i & i0;
reconsider q=<* f/.1 *> as FinSequence of Curves T;
A2: 0+1<=len f by A1,NAT_1:13;
then f/.1=f.1 by FINSEQ_4:15;
then
A3: q.1=f.1 by FINSEQ_1:40;
defpred P[Nat] means $1+1<=len f implies
ex g being FinSequence of Curves T st $1+1 =len g & f.1=g.1 &
(for i being Nat st 1<=i & i<$1+1 holds g.(i+1)=(g/.i)+(f/.(i+1)));
A4: for i being Nat st 1<=i & i<0+1 holds q.(i+1)=(q/.i)+(f/.(i+1));
A5: for k being Nat st P[k] holds P[k + 1]
proof
let k be Nat;
assume
A6: P[k];
now
per cases;
case
A7: k+1+1<=len f;
k+1) as FinSequence of
Curves T;
A11: Seg len g = dom g by FINSEQ_1:def 3;
A12: len g2=len g+ len (<* g/.(k+1)+(f/.(k+1+1)) *>) by FINSEQ_1:22
.= k+1+1 by A8,FINSEQ_1:40;
A13: for i being Nat st 1<=i & i< k+1+1 holds g2.(i+1)=(g2/.i)+(f
/.(i+1))
proof
let i be Nat;
assume that
A14: 1<=i and
A15: i< k+1+1;
A16: i<=k+1 by A15,NAT_1:13;
per cases by A16,XXREAL_0:1;
suppose
A17: ilen f;
hence P[k+1];
end;
end;
hence P[k + 1];
end;
len f-'1=len f -1 by A2,XREAL_1:233;
then
A25: len f-'1+1=len f;
len q=1 by FINSEQ_1:40;
then
A26: P[0] by A3,A4;
for k being Nat holds P[k] from NAT_1:sch 2(A26,A5);
hence thesis by A25;
end;
suppose
A27: len f <= 0;
take f;
thus len f = len f & f.1 = f.1;
let i be Nat;
thus thesis by A27;
end;
end;
uniqueness
proof
let g1,g2 be FinSequence of Curves T;
assume that
A28: len f=len g1 and
A29: f.1=g1.1 and
A30: for i being Nat st 1<=i & ik;
then 0+1>k;
then k=0 by NAT_1:13;
hence g1.(k+1)=g2.(k+1) by A29,A32;
end;
end;
hence P[k+1];
end;
A43: P[0];
for k being Nat holds P[k] from NAT_1:sch 2(A43,A34);
hence g1=g2 by A28,A31,FINSEQ_1:14;
end;
end;
definition
let T be TopStruct;
let f be FinSequence of Curves T;
func Sum f -> Curve of T equals :Def14:
(Partial_Sums f).len f if len f > 0 otherwise {};
coherence
proof
A1: len f = len Partial_Sums f by Def13;
now per cases;
case len f > 0;
then 0+1 <= len f by NAT_1:13;
then len f in dom Partial_Sums f by A1,FINSEQ_3:25;
then (Partial_Sums f).len f in rng Partial_Sums f by FUNCT_1:def 3;
hence (Partial_Sums f).len f is Element of Curves T;
end;
case
A2: len f <= 0;
{} is parametrized-curve PartFunc of R^1, T by Lm1,XBOOLE_1:2;
hence thesis by A2,Th20;
end;
end;
hence thesis;
end;
consistency;
end;
theorem Th40:
for c being Curve of T holds Sum <*c*> = c
proof
let c be Curve of T;
set f = <*c*>;
len f = 1 by FINSEQ_1:40;
hence Sum f = (Partial_Sums f).1 by Def14
.= f.1 by Def13 .= c by FINSEQ_1:40;
end;
Lm2:
for f1,f2 being FinSequence of Curves T holds
Partial_Sums(f1 ^ f2).(len f1) = (Partial_Sums f1).(len f1)
proof
defpred P[Nat] means
for f1,f2 being FinSequence of Curves T st len f1 = $1
holds Partial_Sums(f1 ^ f2).(len f1) = (Partial_Sums f1).(len f1);
A1: P[0]
proof
let f1,f2 be FinSequence of Curves T;
assume
A2: len f1 = 0;
then not len f1 in Seg len Partial_Sums(f1 ^ f2) by FINSEQ_1:1;
then
A3: not len f1 in dom Partial_Sums(f1 ^ f2) by FINSEQ_1:def 3;
not len f1 in Seg len Partial_Sums f1 by A2,FINSEQ_1:1;
then
A4: not len f1 in dom Partial_Sums f1 by FINSEQ_1:def 3;
thus Partial_Sums(f1 ^ f2).(len f1) = {} by A3,FUNCT_1:def 2
.= (Partial_Sums f1).(len f1) by A4,FUNCT_1:def 2;
end;
A5: for k be Nat st P[k] holds P[k + 1]
proof
let k be Nat;
assume
A6: P[k];
let f1,f2 be FinSequence of Curves T;
assume
A7: len f1 = k+1;
then consider f3 be FinSequence of Curves T,
c be Element of Curves T such that
A8: f1 = f3 ^ <*c*> by FINSEQ_2:19;
set f4 = <*c*>^f2;
A9: f1 ^ f2
= f3 ^ f4 by A8,FINSEQ_1:32;
A10: len f1 = len f3 + 1 by A8,FINSEQ_2:16;
per cases;
suppose
A11: 1 > k;
then
A12: len f3 = 0 by A10,A7,NAT_1:14;
f3 = {} by A11,A10,A7,FINSEQ_1:20;
then
A13: f1 = <*c*> by A8,FINSEQ_1:34;
thus Partial_Sums(f1 ^ f2).(len f1)
= (f1 ^ f2).1 by A12,A10,Def13
.= c by A13,FINSEQ_1:41
.= f1.1 by A13,FINSEQ_1:40
.= (Partial_Sums f1).(len f1) by A12,A10,Def13;
end;
suppose
A14: 1 <= k;
A15: k < len f1 by A7,NAT_1:16;
len(f3^f4) = k + len f4 by A10,A7,FINSEQ_1:22;
then
A16: k < len(f3^f4) by NAT_1:16;
then k in Seg len (f3 ^ f4) by A14,FINSEQ_1:1;
then k in Seg len Partial_Sums(f3 ^ f4) by Def13;
then
A17: k in dom Partial_Sums(f3 ^ f4) by FINSEQ_1:def 3;
k in Seg len f3 by A14,A10,A7,FINSEQ_1:1;
then k in Seg len Partial_Sums f3 by Def13;
then
A18: k in dom Partial_Sums f3 by FINSEQ_1:def 3;
k in Seg len f1 by A14,A15,FINSEQ_1:1;
then k in Seg len Partial_Sums f1 by Def13;
then
A19: k in dom Partial_Sums f1 by FINSEQ_1:def 3;
A20: Partial_Sums(f3 ^ f4)/.k
= Partial_Sums(f3 ^ f4).k by A17,PARTFUN1:def 6
.= (Partial_Sums f3).k by A10,A7,A6
.= (Partial_Sums f3)/.k by A18,PARTFUN1:def 6;
A21: Partial_Sums(f1)/.k
= Partial_Sums(f1).k by A19,PARTFUN1:def 6
.= (Partial_Sums f3).k by A8,A10,A7,A6
.= (Partial_Sums f3)/.k by A18,PARTFUN1:def 6;
1+1 <= k+1 by A14,XREAL_1:6;
then
A22: 1 <= k+1 by XXREAL_0:2;
0+len f1 <= len f1+len f2 by XREAL_1:6;
then k+1 <= len(f1^f2) by A7,FINSEQ_1:22;
then k+1 in Seg len(f1^f2) by A22,FINSEQ_1:1;
then
A23: k+1 in dom (f1^f2) by FINSEQ_1:def 3;
k+1 in Seg len f1 by A7,A22,FINSEQ_1:1;
then
A24: k+1 in dom f1 by FINSEQ_1:def 3;
A25: (f1^f2)/.(k+1)
= (f1^f2).(k+1) by A23,PARTFUN1:def 6
.= f1.(k+1) by A24,FINSEQ_1:def 7
.= f1/.(k+1) by A24,PARTFUN1:def 6;
thus Partial_Sums(f1 ^ f2).(len f1)
= (Partial_Sums f1)/.k + f1/.(k+1) by A7,A9,A20,A21,A25,A14,A16,Def13
.= Partial_Sums(f1).(len f1) by A14,A7,A15,Def13;
end;
end;
A26: for k being Nat holds P[k] from NAT_1:sch 2(A1,A5);
let f1,f2 be FinSequence of Curves T;
thus thesis by A26;
end;
theorem Th41:
for c being Curve of T,
f being FinSequence of Curves T holds Sum(f ^ <*c*>) = Sum f + c
proof
let c be Curve of T;
let f be FinSequence of Curves T;
per cases;
suppose
A1: len f <= 0;
A2: f = {} by A1;
reconsider c0 = {} as Curve of T by Th21;
thus Sum(f ^ <*c*>) = Sum <*c*> by A2,FINSEQ_1:34
.= c0 \/ c by Th40
.= c0 + c by Def12
.= Sum f + c by Def14,A1;
end;
suppose
A3: len f > 0;
set f1 = f ^ <*c*>;
A4: len f1 = len f + len <*c*> by FINSEQ_1:22
.= len f + 1 by FINSEQ_1:39;
A5: Sum f1 = (Partial_Sums f1).len f1 by Def14;
0 < 0 + len f by A3;
then
A6: 1 <= len f by NAT_1:19;
A7: len f < len f1 by A4,NAT_1:13;
len f in Seg len f1 by A6,A7,FINSEQ_1:1;
then len f in Seg len Partial_Sums f1 by Def13;
then len f in dom Partial_Sums f1 by FINSEQ_1:def 3;
then
A8: Partial_Sums(f1)/.(len f) = Partial_Sums(f1).(len f) by PARTFUN1:def 6
.= (Partial_Sums f).(len f) by Lm2
.= Sum f by A3,Def14;
len f1 in Seg len f1 by FINSEQ_1:3;
then len f1 in dom f1 by FINSEQ_1:def 3;
then
A9: f1/.(len f +1) = f1.(len f + 1) by A4,PARTFUN1:def 6
.= c by FINSEQ_1:42;
thus Sum(f ^ <*c*>) = Sum f + c by A8,A9,A5,A7,A4,A6,Def13;
end;
end;
theorem Th42:
for X being set, f being FinSequence of Curves T
st (for i being Nat st 1 <= i & i <= len f holds rng(f/.i) c= X)
holds rng Sum f c= X
proof
let X be set;
defpred P[Nat] means
for f being FinSequence of Curves T st len f = $1
& (for i being Nat st 1 <= i & i <= len f holds rng(f/.i) c= X)
holds rng Sum f c= X;
A1: P[0]
proof
let f be FinSequence of Curves T;
assume len f = 0;
then Sum f = {} by Def14;
then rng Sum f = {};
hence thesis;
end;
A2: for k be Nat st P[k] holds P[k + 1]
proof
let k be Nat;
assume
A3: P[k];
let f be FinSequence of Curves T;
assume
A4: len f = k+1;
then consider f1 be FinSequence of Curves T,
c be Element of Curves T such that
A5: f = f1 ^ <*c*> by FINSEQ_2:19;
assume
A6: for i being Nat st 1 <= i & i <= len f holds rng(f/.i) c= X;
A7: len f = len f1 + len<*c*> by A5,FINSEQ_1:22
.= len f1 + 1 by FINSEQ_1:39;
A8: Sum f = Sum f1 + c by A5,Th41;
per cases;
suppose not Sum f1 \/ c is Curve of T;
then Sum f = {} by A8,Def12;
then rng Sum f = {};
hence thesis;
end;
suppose Sum f1 \/ c is Curve of T;
then
A9: Sum f = Sum f1 \/ c by A8,Def12;
A10: for i being Nat st 1 <= i & i <= len f1
holds rng(f1/.i) c= X
proof
let i be Nat;
assume
A11: 1 <= i & i <= len f1;
then
A12: i+1 <= len f1 + 1 by XREAL_1:6;
i <= i+1 by NAT_1:12;
then
A13: i <= len f by A12,A7,XXREAL_0:2;
then
A14: rng(f/.i) c= X by A6,A11;
i in Seg len f by A11,A13,FINSEQ_1:1;
then i in dom f by FINSEQ_1:def 3;
then
A15: rng(f.i) c= X by A14,PARTFUN1:def 6;
i in Seg len f1 by A11,FINSEQ_1:1;
then
A16: i in dom f1 by FINSEQ_1:def 3;
then f.i = f1.i by A5,FINSEQ_1:def 7;
hence thesis by A15,A16,PARTFUN1:def 6;
end;
A17: rng Sum f1 c= X by A3,A7,A4,A10;
len f in Seg len f by A7,FINSEQ_1:3;
then
A18: len f in dom f by FINSEQ_1:def 3;
f.(len f) = c by A7,A5,FINSEQ_1:42;
then
A19: f/.(len f) = c by A18,PARTFUN1:def 6;
0 + 1 <= len f1 + 1 by XREAL_1:6;
then
A20: rng c c= X by A7,A19,A6;
rng Sum f = rng(Sum f1) \/ rng c by A9,RELAT_1:12;
hence thesis by A17,A20,XBOOLE_1:8;
end;
end;
A21: for k being Nat holds P[k] from NAT_1:sch 2(A1,A2);
let f be FinSequence of Curves T;
thus thesis by A21;
end;
theorem Th43:
for T being non empty TopSpace
for f being FinSequence of Curves T
st len f > 0 &
(for i being Nat st 1 <= i & i < len f
holds (f/.i).sup dom(f/.i) = (f/.(i+1)).inf dom(f/.(i+1))
& sup dom(f/.i) = inf dom(f/.(i+1))) &
(for i being Nat st 1 <= i & i <= len f
holds f/.i is with_endpoints)
holds ex c being with_endpoints Curve of T st
Sum f = c & dom c = [.inf dom(f/.1), sup dom(f/.(len f)).] &
the_first_point_of c = (f/.1).(inf dom(f/.1)) &
the_last_point_of c = (f/.(len f)).(sup dom(f/.(len f)))
proof
let T be non empty TopSpace;
defpred P[Nat] means
for f being FinSequence of Curves T
st len f = $1 & len f > 0 &
(for i being Nat st 1 <= i & i < len f
holds (f/.i).sup dom(f/.i) = (f/.(i+1)).inf dom(f/.(i+1))
& sup dom(f/.i) = inf dom(f/.(i+1))) &
(for i being Nat st 1 <= i & i <= len f
holds f/.i is with_endpoints)
holds ex c being with_endpoints Curve of T st
Sum f = c & dom c = [.inf dom(f/.1), sup dom(f/.(len f)).] &
the_first_point_of c = (f/.1).(inf dom(f/.1)) &
the_last_point_of c = (f/.(len f)).(sup dom(f/.(len f)));
A1: P[0];
A2: for k be Nat st P[k] holds P[k + 1]
proof
let k be Nat;
assume
A3: P[k];
let f be FinSequence of Curves T;
assume
A4: len f = k+1 & len f > 0;
consider f1 be FinSequence of Curves T,
c2 be Element of Curves T such that
A5: f = f1 ^ <*c2*> by A4,FINSEQ_2:19;
A6: len f = len f1 + len<*c2*> by A5,FINSEQ_1:22
.= len f1 + 1 by FINSEQ_1:39;
assume
A7: for i being Nat st 1 <= i & i < len f
holds (f/.i).sup dom(f/.i) = (f/.(i+1)).inf dom(f/.(i+1))
& sup dom(f/.i) = inf dom(f/.(i+1));
assume
A8: for i being Nat st 1 <= i & i <= len f
holds f/.i is with_endpoints;
A9: 1<=len f by A4,NAT_1:12;
len f in Seg len f by A4,FINSEQ_1:3;
then
A10: len f in dom f by FINSEQ_1:def 3;
c2 = f.(len f) by A5,A6,FINSEQ_1:42
.= f/.(len f) by A10,PARTFUN1:def 6;
then reconsider c2 as with_endpoints Curve of T by A9,A8;
A11: for i being Nat st 1 <= i & i < len f1
holds (f1/.i).sup dom(f1/.i) = (f1/.(i+1)).inf dom(f1/.(i+1))
& sup dom(f1/.i) = inf dom(f1/.(i+1))
proof
let i be Nat;
assume
A12: 1<=i & i < len f1;
A13: i < len f by A6,A12,NAT_1:13;
i in Seg len f1 by A12,FINSEQ_1:1;
then
A14: i in dom f1 by FINSEQ_1:def 3;
A15: dom f1 c= dom f by A5,FINSEQ_1:26;
A16: f/.i = f.i by A15,A14,PARTFUN1:def 6
.= f1.i by A5,A14,FINSEQ_1:def 7
.= f1/.i by A14,PARTFUN1:def 6;
1+1<=i+1 by A12,XREAL_1:6;
then
A17: 1 <= i+1 by XXREAL_0:2;
i+1 <= len f1 by A12,NAT_1:13;
then i+1 in Seg len f1 by A17,FINSEQ_1:1;
then
A18: i+1 in dom f1 by FINSEQ_1:def 3;
A19: f/.(i+1) = f.(i+1) by A18,A15,PARTFUN1:def 6
.= f1.(i+1) by A5,A18,FINSEQ_1:def 7
.= f1/.(i+1) by A18,PARTFUN1:def 6;
thus thesis by A16,A19,A13,A12,A7;
end;
A20: for i being Nat st 1 <= i & i <= len f1
holds f1/.i is with_endpoints
proof
let i be Nat;
assume
A21: 1<=i & i <= len f1;
A22: i <= len f by A6,A21,NAT_1:13;
i in Seg len f1 by A21,FINSEQ_1:1;
then
A23: i in dom f1 by FINSEQ_1:def 3;
A24: dom f1 c= dom f by A5,FINSEQ_1:26;
f/.i = f.i by A24,A23,PARTFUN1:def 6
.= f1.i by A5,A23,FINSEQ_1:def 7
.= f1/.i by A23,PARTFUN1:def 6;
hence thesis by A22,A21,A8;
end;
per cases;
suppose len f1 = 0;
then f1 = {};
then
A25: f = <*c2*> by A5,FINSEQ_1:34;
take c2;
1 in Seg 1 by FINSEQ_1:3;
then
A26: 1 in dom f by A25,FINSEQ_1:38;
A27: f/.1 = f.1 by A26,PARTFUN1:def 6 .= c2 by A25,FINSEQ_1:40;
A28: f/.(len f) = c2 by A27,A25,FINSEQ_1:40;
thus thesis by A25,Th40,A27,A28,Th27;
end;
suppose
A29: not len f1 = 0;
consider c1 be with_endpoints Curve of T such that
A30: Sum f1 = c1 & dom c1 = [.inf dom(f1/.1), sup dom(f1/.(len f1)).] &
the_first_point_of c1 = (f1/.1).(inf dom(f1/.1)) &
the_last_point_of c1 = (f1/.(len f1)).(sup dom(f1/.(len f1)))
by A4,A6,A11,A20,A3,A29;
set c = c1 + c2;
A31: 0+1 0 & len f1 = len f2 & Sum f1 = c1 & Sum f2 = c2 &
(for i being Nat st 1 <= i & i < len f1
holds (f1/.i).sup dom(f1/.i) = (f1/.(i+1)).inf dom(f1/.(i+1))
& sup dom(f1/.i) = inf dom(f1/.(i+1))) &
(for i being Nat st 1 <= i & i < len f2
holds (f2/.i).sup dom(f2/.i) = (f2/.(i+1)).inf dom(f2/.(i+1))
& sup dom(f2/.i) = inf dom(f2/.(i+1)) ) &
(for i being Nat st 1 <= i & i <= len f1 holds
ex c3,c4 being with_endpoints Curve of T st c3 = f1/.i & c4 = f2/.i
& c3,c4 are_homotopic & dom c3 = dom c4)
holds c1,c2 are_homotopic
proof
let T be non empty TopSpace;
defpred P[Nat] means
for f1,f2 being FinSequence of Curves T
for c1,c2 being with_endpoints Curve of T
st len f1 = $1 & len f1 > 0 & len f1 = len f2 & Sum f1 = c1 & Sum f2 = c2 &
(for i being Nat st 1 <= i & i < len f1
holds (f1/.i).sup dom(f1/.i) = (f1/.(i+1)).inf dom(f1/.(i+1))
& sup dom(f1/.i) = inf dom(f1/.(i+1))) &
(for i being Nat st 1 <= i & i < len f2
holds (f2/.i).sup dom(f2/.i) = (f2/.(i+1)).inf dom(f2/.(i+1))
& sup dom(f2/.i) = inf dom(f2/.(i+1)) ) &
(for i being Nat st 1 <= i & i <= len f1 holds
ex c3,c4 being with_endpoints Curve of T st c3 = f1/.i & c4 = f2/.i
& c3,c4 are_homotopic & dom c3 = dom c4)
holds c1,c2 are_homotopic;
A1: P[0];
A2: for k be Nat st P[k] holds P[k + 1]
proof
let k be Nat;
assume
A3: P[k];
let f1,f2 be FinSequence of Curves T;
let c1,c2 be with_endpoints Curve of T;
assume
A4: len f1 = k+1 & len f1 > 0;
consider f1a be FinSequence of Curves T,
c1b be Element of Curves T such that
A5: f1 = f1a ^ <*c1b*> by A4,FINSEQ_2:19;
A6: len f1 = len f1a + len<*c1b*> by A5,FINSEQ_1:22
.= len f1a + 1 by FINSEQ_1:39;
assume
A7: len f1 = len f2;
consider f2a be FinSequence of Curves T,
c2b be Element of Curves T such that
A8: f2 = f2a ^ <*c2b*> by A7,A4,FINSEQ_2:19;
A9: len f2 = len f2a + len<*c2b*> by A8,FINSEQ_1:22
.= len f2a + 1 by FINSEQ_1:39;
assume
A10: Sum f1 = c1 & Sum f2 = c2;
assume
A11: for i being Nat st 1 <= i & i < len f1
holds (f1/.i).sup dom(f1/.i) = (f1/.(i+1)).inf dom(f1/.(i+1))
& sup dom(f1/.i) = inf dom(f1/.(i+1));
assume
A12: for i being Nat st 1 <= i & i < len f2
holds (f2/.i).sup dom(f2/.i) = (f2/.(i+1)).inf dom(f2/.(i+1))
& sup dom(f2/.i) = inf dom(f2/.(i+1));
assume
A13: for i being Nat st 1 <= i & i <= len f1 holds
ex c3,c4 being with_endpoints Curve of T st c3 = f1/.i & c4 = f2/.i
& c3,c4 are_homotopic & dom c3 = dom c4;
A14: dom f1 = Seg len f1 by FINSEQ_1:def 3 .= dom f2 by A7,FINSEQ_1:def 3;
A15: 1 <= len f1 by A4,NAT_1:11;
then len f1 in Seg len f1 by FINSEQ_1:1;
then
A16: len f1 in dom f1 by FINSEQ_1:def 3;
then
A17: f1/.(len f1) = f1.(len f1) by PARTFUN1:def 6;
consider c1bb, c2bb be with_endpoints Curve of T such that
A18: c1bb = f1/.(len f1) & c2bb = f2/.(len f1) &
c1bb,c2bb are_homotopic & dom c1bb = dom c2bb by A13,A15;
A19: f1.(len f1) = c1b by A5,A6,FINSEQ_1:42;
A20: f2.(len f2) = c2b by A8,A9,FINSEQ_1:42;
A21: c1bb = c1b & c2bb = c2b
by A7,A16,A14,A18,A19,A20,PARTFUN1:def 6;
reconsider c1b, c2b as with_endpoints Curve of T
by A7,A20,A18,A14,A16,A17,A5,A6,FINSEQ_1:42,PARTFUN1:def 6;
per cases;
suppose
A22: len f1a > 0;
A23: for i being Nat st 1 <= i & i < len f1a
holds (f1a/.i).sup dom(f1a/.i) = (f1a/.(i+1)).inf dom(f1a/.(i+1))
& sup dom(f1a/.i) = inf dom(f1a/.(i+1))
proof
let i be Nat;
assume
A24: 1 <= i & i < len f1a;
then
A25: i + 1 < len f1a + 1 by XREAL_1:6;
i <= i + 1 by NAT_1:11;
then
A26: i < len f1 by A6,A25,XXREAL_0:2;
i in Seg len f1 by A24,A26,FINSEQ_1:1;
then
A27: i in dom f1 by FINSEQ_1:def 3;
i in Seg len f1a by A24,FINSEQ_1:1;
then
A28: i in dom f1a by FINSEQ_1:def 3;
A29: f1/.i = f1.i by A27,PARTFUN1:def 6
.= f1a.i by A28,A5,FINSEQ_1:def 7
.= f1a/.i by A28,PARTFUN1:def 6;
A30: 1 <= i+1 by NAT_1:11;
i+1 in Seg len f1 by A30,A25,A6,FINSEQ_1:1;
then
A31: i+1 in dom f1 by FINSEQ_1:def 3;
i+1 <= len f1a by A24,NAT_1:13;
then i+1 in Seg len f1a by A30,FINSEQ_1:1;
then
A32: i+1 in dom f1a by FINSEQ_1:def 3;
f1/.(i+1) = f1.(i+1) by A31,PARTFUN1:def 6
.= f1a.(i+1) by A32,A5,FINSEQ_1:def 7
.= f1a/.(i+1) by A32,PARTFUN1:def 6;
hence thesis by A26,A29,A24,A11;
end;
for i being Nat st 1 <= i & i <= len f1a
holds f1a/.i is with_endpoints
proof
let i be Nat;
assume
A33: 1 <= i & i <= len f1a;
then
A34: i + 1 <= len f1a + 1 by XREAL_1:6;
i <= i + 1 by NAT_1:11;
then
A35: i <= len f1 by A6,A34,XXREAL_0:2;
i in Seg len f1 by A33,A35,FINSEQ_1:1;
then
A36: i in dom f1 by FINSEQ_1:def 3;
i in Seg len f1a by A33,FINSEQ_1:1;
then
A37: i in dom f1a by FINSEQ_1:def 3;
A38: ex c3,c4 being with_endpoints Curve of T st c3 = f1/.i & c4 = f2/.i
& c3,c4 are_homotopic & dom c3 = dom c4 by A33,A35,A13;
f1/.i = f1.i by A36,PARTFUN1:def 6
.= f1a.i by A37,A5,FINSEQ_1:def 7
.= f1a/.i by A37,PARTFUN1:def 6;
hence thesis by A38;
end;
then consider c1a be with_endpoints Curve of T such that
A39: Sum f1a = c1a &
dom c1a = [.inf dom(f1a/.1), sup dom(f1a/.(len f1a)).] &
the_first_point_of c1a = (f1a/.1).(inf dom(f1a/.1)) &
the_last_point_of c1a = (f1a/.(len f1a)).(sup dom(f1a/.(len f1a)))
by A22,A23,Th43;
A40: for i being Nat st 1 <= i & i < len f2a
holds (f2a/.i).sup dom(f2a/.i) = (f2a/.(i+1)).inf dom(f2a/.(i+1))
& sup dom(f2a/.i) = inf dom(f2a/.(i+1))
proof
let i be Nat;
assume
A41: 1 <= i & i < len f2a;
then
A42: i + 1 < len f2a + 1 by XREAL_1:6;
i <= i + 1 by NAT_1:11;
then
A43: i < len f2 by A9,A42,XXREAL_0:2;
i in Seg len f2 by A41,A43,FINSEQ_1:1;
then
A44: i in dom f2 by FINSEQ_1:def 3;
i in Seg len f2a by A41,FINSEQ_1:1;
then
A45: i in dom f2a by FINSEQ_1:def 3;
A46: f2/.i = f2.i by A44,PARTFUN1:def 6
.= f2a.i by A45,A8,FINSEQ_1:def 7
.= f2a/.i by A45,PARTFUN1:def 6;
A47: 1 <= i+1 by NAT_1:11;
i+1 in Seg len f2 by A47,A42,A9,FINSEQ_1:1;
then
A48: i+1 in dom f2 by FINSEQ_1:def 3;
i+1 <= len f2a by A41,NAT_1:13;
then i+1 in Seg len f2a by A47,FINSEQ_1:1;
then
A49: i+1 in dom f2a by FINSEQ_1:def 3;
f2/.(i+1) = f2.(i+1) by A48,PARTFUN1:def 6
.= f2a.(i+1) by A49,A8,FINSEQ_1:def 7
.= f2a/.(i+1) by A49,PARTFUN1:def 6;
hence thesis by A43,A46,A41,A12;
end;
for i being Nat st 1 <= i & i <= len f2a
holds f2a/.i is with_endpoints
proof
let i be Nat;
assume
A50: 1 <= i & i <= len f2a;
then
A51: i + 1 <= len f2a + 1 by XREAL_1:6;
i <= i + 1 by NAT_1:11;
then
A52: i <= len f2 by A9,A51,XXREAL_0:2;
i in Seg len f2 by A50,A52,FINSEQ_1:1;
then
A53: i in dom f2 by FINSEQ_1:def 3;
i in Seg len f2a by A50,FINSEQ_1:1;
then
A54: i in dom f2a by FINSEQ_1:def 3;
A55: ex c3,c4 being with_endpoints Curve of T st c3 = f1/.i & c4 = f2/.i
& c3,c4 are_homotopic & dom c3 = dom c4 by A7,A50,A52,A13;
f2/.i = f2.i by A53,PARTFUN1:def 6
.= f2a.i by A54,A8,FINSEQ_1:def 7
.= f2a/.i by A54,PARTFUN1:def 6;
hence thesis by A55;
end;
then consider c2a be with_endpoints Curve of T such that
A56: Sum f2a = c2a &
dom c2a = [.inf dom(f2a/.1), sup dom(f2a/.(len f2a)).] &
the_first_point_of c2a = (f2a/.1).(inf dom(f2a/.1)) &
the_last_point_of c2a = (f2a/.(len f2a)).(sup dom(f2a/.(len f2a)))
by A6,A7,A9,A22,A40,Th43;
for i being Nat st 1 <= i & i <= len f1a holds
ex c3,c4 being with_endpoints Curve of T st c3 = f1a/.i & c4 = f2a/.i
& c3,c4 are_homotopic & dom c3 = dom c4
proof
let i be Nat;
assume
A57: 1 <= i & i <= len f1a;
then
A58: i + 1 <= len f1a + 1 by XREAL_1:6;
i <= i + 1 by NAT_1:11;
then
A59: i <= len f1 by A6,A58,XXREAL_0:2;
i in Seg len f1 by A57,A59,FINSEQ_1:1;
then
A60: i in dom f1 by FINSEQ_1:def 3;
i in Seg len f1a by A57,FINSEQ_1:1;
then
A61: i in dom f1a by FINSEQ_1:def 3;
i in Seg len f2 by A57,A59,A7,FINSEQ_1:1;
then
A62: i in dom f2 by FINSEQ_1:def 3;
i in Seg len f2a by A57,A6,A7,A9,FINSEQ_1:1;
then
A63: i in dom f2a by FINSEQ_1:def 3;
consider c3,c4 be with_endpoints Curve of T such that
A64: c3 = f1/.i & c4 = f2/.i & c3,c4 are_homotopic &
dom c3 = dom c4 by A57,A59,A13;
take c3,c4;
A65: f1/.i = f1.i by A60,PARTFUN1:def 6
.= f1a.i by A61,A5,FINSEQ_1:def 7
.= f1a/.i by A61,PARTFUN1:def 6;
f2/.i = f2.i by A62,PARTFUN1:def 6
.= f2a.i by A63,A8,FINSEQ_1:def 7
.= f2a/.i by A63,PARTFUN1:def 6;
hence thesis by A64,A65;
end;
then
A66: c1a, c2a are_homotopic by A3,A4,A23,A6,A22,A40,A7,A9,A39,A56;
A67: c1 = c1a + c1b by A10,A5,A39,Th41;
A68: c2 = c2a + c2b by A10,A8,A56,Th41;
A69: f1/.(len f1) = c1b by A5,A6,A17,FINSEQ_1:42;
A70: 0 + 1 < len f1a +1 by A22,XREAL_1:6;
then
A71: 1 <= len f1a & len f1a < len f1 by A6,NAT_1:13;
then len f1a in Seg len f1 by FINSEQ_1:1;
then
A72: len f1a in dom f1 by FINSEQ_1:def 3;
len f1a in Seg len f1a by A71,FINSEQ_1:1;
then
A73: len f1a in dom f1a by FINSEQ_1:def 3;
then
A74: f1a/.(len f1a) = f1a.(len f1a) by PARTFUN1:def 6
.= f1.(len f1a) by A5,A73,FINSEQ_1:def 7
.= f1/.(len f1a) by A72,PARTFUN1:def 6;
len f2a in Seg len f2 by A71,A6,A9,A7,FINSEQ_1:1;
then
A75: len f2a in dom f2 by FINSEQ_1:def 3;
len f2a in Seg len f2a by A71,A6,A7,A9,FINSEQ_1:1;
then
A76: len f2a in dom f2a by FINSEQ_1:def 3;
then
A77: f2a/.(len f2a) = f2a.(len f2a) by PARTFUN1:def 6
.= f2.(len f2a) by A8,A76,FINSEQ_1:def 7
.= f2/.(len f2a) by A75,PARTFUN1:def 6;
1 in Seg len f1 by A70,A6,FINSEQ_1:1;
then
A78: 1 in dom f1 by FINSEQ_1:def 3;
1 in Seg len f1a by A71,FINSEQ_1:1;
then
A79: 1 in dom f1a by FINSEQ_1:def 3;
then
A80: f1a/.1 = f1a.1 by PARTFUN1:def 6
.= f1.1 by A5,A79,FINSEQ_1:def 7
.= f1/.1 by A78,PARTFUN1:def 6;
1 in Seg len f2 by A70,A7,A6,FINSEQ_1:1;
then
A81: 1 in dom f2 by FINSEQ_1:def 3;
1 in Seg len f2a by A71,A6,A7,A9,FINSEQ_1:1;
then
A82: 1 in dom f2a by FINSEQ_1:def 3;
then
A83: f2a/.1 = f2a.1 by PARTFUN1:def 6
.= f2.1 by A8,A82,FINSEQ_1:def 7
.= f2/.1 by A81,PARTFUN1:def 6;
A84: ex c3,c4 being with_endpoints Curve of T st
c3 = f1/.1 & c4 = f2/.1
& c3,c4 are_homotopic & dom c3 = dom c4 by A13,A15;
A85: ex c3,c4 being with_endpoints Curve of T st
c3 = f1/.(len f1a) & c4 = f2/.(len f1a)
& c3,c4 are_homotopic & dom c3 = dom c4 by A71,A13;
A86: the_last_point_of c1a
= the_first_point_of c1b by A69,A6,A74,A11,A71,A39;
sup dom c1a = sup dom(f1/.(len f1a))
by A74,A39,XXREAL_1:29,XXREAL_2:29
.= inf dom(f1/.(len f1a+1)) by A11,A71
.= inf dom c1b by A5,A6,A17,FINSEQ_1:42;
hence c1,c2 are_homotopic
by A66,A67,A68,A18,A21,A86,Th39,A56,A84,A85,A80,A83,A6,A7,A9,A74,A77
,A39;
end;
suppose
A87: not len f1a > 0;
then f1a = {};
then f1 = <*c1b*> by A5,FINSEQ_1:34;
then
A88: Sum f1 = c1b by Th40;
f2a = {} by A87,A6,A7,A9;
then f2 = <*c2b*> by A8,FINSEQ_1:34;
hence thesis by A88,A18,A21,A10,Th40;
end;
end;
for k being Nat holds P[k] from NAT_1:sch 2(A1,A2);
hence thesis;
end;
theorem Th45:
for c being with_endpoints Curve of T
for h being FinSequence of REAL
st len h >= 2 & h.1 = inf dom c & h.len h = sup dom c & h is increasing
holds
ex f being FinSequence of Curves T
st len f = len h - 1 & c = Sum f &
(for i being Nat st 1 <= i & i <= len f
holds f/.i = c | ([.h/.i,h/.(i+1).]))
proof
defpred P[Nat] means
for c being with_endpoints Curve of T
for h being FinSequence of REAL
st len h = $1
& len h >= 2 & h.1 = inf dom c & h.len h = sup dom c & h is increasing
holds
ex f being FinSequence of Curves T
st len f = len h - 1 & c = Sum f &
(for i being Nat st 1 <= i & i <= len f
holds f/.i = c | ([.h/.i,h/.(i+1).]));
A1: P[0];
A2: for k be Nat st P[k] holds P[k + 1]
proof
let k be Nat;
assume
A3: P[k];
let c be with_endpoints Curve of T;
let h be FinSequence of REAL such that
A4: len h = k+1 & len h >= 2 & h.1 = inf dom c & h.len h = sup dom c &
h is increasing;
consider h1 be FinSequence of REAL, r be Element of REAL such that
A5: h = h1 ^ <*r*> by A4,FINSEQ_2:19;
A6: len h = len h1 + 1 by A5,FINSEQ_2:16;
reconsider r1 = h.k as Real;
consider c1,c2 be Element of Curves T such that
A7: c = c1 + c2 & c1 = c | [.inf dom c, r1.] & c2 = c | [.r1, sup dom c.]
by Th37;
A8: k < k+1 by NAT_1:13;
1 <= 1+k by NAT_1:12;
then 1 in Seg len h by A4,FINSEQ_1:1;
then
A9: 1 in dom h by FINSEQ_1:def 3;
per cases;
suppose len h1 < 2;
then len h1 < 1 + 1;
then
A10: len h1 <= 1 by NAT_1:13;
per cases;
suppose h1 = {};
then h = <*r*> by A5,FINSEQ_1:34;
then len h = 1 by FINSEQ_1:40;
hence thesis by A4;
end;
suppose h1 <> {};
then len h1 >= 1 by FINSEQ_1:20;
then
A11: len h1 = 1 by A10,XXREAL_0:1;
set f = <*c*>;
take f;
A12: len f = 1 by FINSEQ_1:40;
thus len f = len h - 1 by A11,A6,FINSEQ_1:40;
thus c = Sum f by Th40;
thus for i being Nat st 1 <= i & i <= len f
holds f/.i = c | ([.h/.i,h/.(i+1).])
proof
let i be Nat;
assume
A13: 1 <= i & i <= len f;
then
A14: i = 1 by A12,XXREAL_0:1;
i in Seg len f by A13,FINSEQ_1:1;
then
A15: i in dom f by FINSEQ_1:def 3;
A16: h/.i = inf dom c by A4,A14,A9,PARTFUN1:def 6;
len h in Seg len h by A6,FINSEQ_1:3;
then len h in dom h by FINSEQ_1:def 3;
then
A17: h/.(i+1) = sup dom c by A4,A6,A11,A14,PARTFUN1:def 6;
thus f/.i = f.i by A15,PARTFUN1:def 6
.= c | dom c by A14,FINSEQ_1:40
.= c | ([.h/.i,h/.(i+1).]) by A16,A17,Th27;
end;
end;
end;
suppose
A18: len h1 >= 2;
then
A19: 1;
take f;
A39: len f = len f1 + len <*c2*> by FINSEQ_1:22
.= len f1 + 1 by FINSEQ_1:40;
thus len f
= len h - 1 by A6,A38,A39;
thus c
= Sum f by Th41,A7,A38;
thus for i being Nat st 1 <= i & i <= len f
holds f/.i = c | ([.h/.i,h/.(i+1).])
proof
let i be Nat;
assume
A40: 1 <= i & i <= len f;
then i in Seg len f by FINSEQ_1:1;
then
A41: i in dom f by FINSEQ_1:def 3;
per cases;
suppose
A42: i = len f;
A43: h/.i = r1 by A42,A39,A38,A4,A6,A20,PARTFUN1:def 6;
1+1<=i+1 by A40,XREAL_1:6;
then 1 < len h by A42,A39,A38,A6,XXREAL_0:2;
then len h in Seg len h by FINSEQ_1:1;
then len h in dom h by FINSEQ_1:def 3;
then
A44: h/.(i+1)
= sup dom c by A4,A42,A39,A38,A6,PARTFUN1:def 6;
thus f/.i = f.i by A41,PARTFUN1:def 6
.= c | ([.h/.i,h/.(i+1).]) by A43,A44,A7,A42,A39,FINSEQ_1:42;
end;
suppose i <> len f;
then
A45: i < len f1 + 1 by A39,A40,XXREAL_0:1;
then
A46: i <= len f1 by NAT_1:13;
then i in Seg len f1 by A40,FINSEQ_1:1;
then
A47: i in dom f1 by FINSEQ_1:def 3;
i in Seg len h1 by A38,A40,A39,FINSEQ_1:1;
then
A48: i in dom h1 by FINSEQ_1:def 3;
A49: h1/.i = h1.i by A48,PARTFUN1:def 6
.= (h|dom h1).i by A5,FINSEQ_1:21
.= h.i by A48,FUNCT_1:49
.= h/.i by A48,A36,PARTFUN1:def 6;
A50: i+1 <= len h1 by A38,A45,NAT_1:13;
1 <= i+1 by NAT_1:12;
then i+1 in Seg len h1 by A50,FINSEQ_1:1;
then
A51: i+1 in dom h1 by FINSEQ_1:def 3;
A52: h1/.(i+1) = h1.(i+1) by A51,PARTFUN1:def 6
.= (h|dom h1).(i+1) by A5,FINSEQ_1:21
.= h.(i+1) by A51,FUNCT_1:49
.= h/.(i+1) by A51,A36,PARTFUN1:def 6;
A53: i+1 <= len f1+1 by A45,NAT_1:13;
h.(i+1) <= h.k
proof
per cases;
suppose i+1 = k; hence thesis; end;
suppose i+1 <> k;
then i+1 < k by A38,A6,A4,A53,XXREAL_0:1;
hence thesis by A51,A36,A20,A4,VALUED_0:def 13;
end;
end;
then
A54: h/.(i+1) <= r1 by A51,A36,PARTFUN1:def 6;
h.1 <= h.i
proof
per cases;
suppose i = 1; hence thesis; end;
suppose i <> 1;
then 1 < i by A40,XXREAL_0:1;
hence thesis by A36,A48,A9,A4,VALUED_0:def 13;
end;
end;
then
A55: inf dom c <= h/.i by A4,A36,A48,PARTFUN1:def 6;
f.i = f1.i by A47,FINSEQ_1:def 7
.= f1/.i by A47,PARTFUN1:def 6
.= (c | [.inf dom c, r1.]) | ([.h1/.i,h1/.(i+1).])
by A7,A38,A40,A46
.= c | ([.h1/.i,h1/.(i+1).])
by A55,A52,A49,A54,RELAT_1:74,XXREAL_1:34;
hence f/.i = c | ([.h/.i,h/.(i+1).])
by A41,A52,A49,PARTFUN1:def 6;
end;
end;
end;
end;
for k being Nat holds P[k] from NAT_1:sch 2(A1,A2);
hence thesis;
end;
Lm3:
for t being Point of TUnitSphere(n), f being Loop of t
st rng f <> the carrier of TUnitSphere(n) holds f is nullhomotopic
proof
let t be Point of TUnitSphere(n), f be Loop of t;
assume
A1: rng f <> the carrier of TUnitSphere(n);
for x being object holds x in rng f
implies x in the carrier of TUnitSphere(n);
then consider x be object such that
A2: x in the carrier of TUnitSphere(n) and
A3: not x in rng f by A1,TARSKI:2;
reconsider n1=n+1 as Nat;
A4: [#] Tunit_circle(n1) c= [#] TOP-REAL(n1) by PRE_TOPC:def 4;
A5: x in the carrier of Tunit_circle(n1) by A2,MFOLD_2:def 4;
then reconsider p = x as Point of TOP-REAL n1 by A4;
p in the carrier of Tcircle(0.TOP-REAL n1,1) by A5,TOPREALB:def 7;
then
A6:p in Sphere(0.TOP-REAL n1,1) by TOPREALB:9;
then -p in Sphere(0.TOP-REAL n1,1) \ {p} by Th3;
then reconsider S = (TOP-REAL n1) | (Sphere(0.TOP-REAL n1,1) \ {p})
as non empty SubSpace of TOP-REAL n1;
A7: [#]S = Sphere(0.TOP-REAL n1,1) \ {p} by PRE_TOPC:def 5;
then stereographic_projection(S,p) is being_homeomorphism
by A6,MFOLD_2:31;
then
A8: TPlane(p,0.TOP-REAL n1), S are_homeomorphic by T_0TOPSP:def 1;
A9: S is having_trivial_Fundamental_Group by A8,Th13;
Tunit_circle(n1) is SubSpace of TOP-REAL n1;
then
A10: TUnitSphere(n) is SubSpace of TOP-REAL n1 by MFOLD_2:def 4;
Sphere(0.TOP-REAL n1,1) \ {p} c= Sphere(0.TOP-REAL n1,1) by XBOOLE_1:36;
then Sphere(0.TOP-REAL n1,1) \ {p} c=
the carrier of Tcircle(0.TOP-REAL n1,1) by TOPREALB:9;
then Sphere(0.TOP-REAL n1,1) \ {p} c= the carrier of Tunit_circle(n1)
by TOPREALB:def 7;
then Sphere(0.TOP-REAL n1,1) \ {p} c= the carrier of TUnitSphere(n)
by MFOLD_2:def 4;
then reconsider S0 = S as non empty SubSpace of TUnitSphere(n)
by A7,A10,TOPMETR:3;
0 in the carrier of I[01] by BORSUK_1:43;
then
A11: 0 in dom f by FUNCT_2:def 1;
t,t are_connected;
then
A12: f is continuous & f.0 = t & f.1 = t by BORSUK_2:def 2;
then
t in rng f by A11,FUNCT_1:3;
then
A13: not t in {p} by A3,TARSKI:def 1;
A14: the carrier of TUnitSphere(n)
= the carrier of Tunit_circle(n1) by MFOLD_2:def 4
.= the carrier of Tcircle(0.TOP-REAL n1,1) by TOPREALB:def 7
.= Sphere(0.TOP-REAL n1,1) by TOPREALB:9;
reconsider t0 = t as Point of S0 by A7,A14,A13,XBOOLE_0:def 5;
dom f = the carrier of I[01] by FUNCT_2:def 1;
then reconsider f0 = f as Function of I[01],S0
by A7,A3,A14,FUNCT_2:2,ZFMISC_1:34;
A15: t0,t0 are_connected;
f0 is continuous by JORDAN16:8;
then reconsider f0 = f as Loop of t0 by A12,A15,BORSUK_2:def 2;
f0 is nullhomotopic by A9;
hence thesis by Th18;
end;
Lm4:
for t being Point of TUnitSphere(n)
for f being Loop of t
st n>=2 & rng f = the carrier of TUnitSphere(n) holds
ex g being Loop of t st
g, f are_homotopic & rng g <> the carrier of TUnitSphere(n)
proof
let t be Point of TUnitSphere(n), f be Loop of t such that
A1: n>=2 and
A2: rng f = the carrier of TUnitSphere(n);
reconsider n1 = n+1 as Element of NAT;
Tunit_circle(n1) is SubSpace of TOP-REAL n1;
then
A3: TUnitSphere(n) is SubSpace of TOP-REAL n1 by MFOLD_2:def 4;
[#] Tunit_circle(n1) c= [#] TOP-REAL(n1) by PRE_TOPC:def 4;
then
A4: rng f c= the carrier of TOP-REAL n1 by A2,MFOLD_2:def 4;
dom f = the carrier of I[01] by FUNCT_2:def 1;
then reconsider f1 = f as Function of I[01],TOP-REAL n1 by A4,FUNCT_2:2;
f1 is continuous by A3,PRE_TOPC:26;
then consider h be FinSequence of REAL such that
A5: h.1=0 & h.len h=1 & 5<=len h & rng h c= the carrier of I[01] &
h is increasing &
(for i being Nat,Q being Subset of I[01],
W being Subset of Euclid n1 st 1<=i & i p & the_last_point_of c1 <> p &
not inf dom c1 = sup dom c1
holds
ex c2 being with_endpoints Curve of TUnitSphere(n)
st c1,c2 are_homotopic & rng(c2) c= the carrier of U & dom c1 = dom c2
proof
let c1 be with_endpoints Curve of TUnitSphere(n);
assume
A36: rng(c1) c= the carrier of V;
assume
A37: the_first_point_of c1 <> p & the_last_point_of c1 <> p;
assume
A38: not inf dom c1 = sup dom c1;
set t1 = the_first_point_of c1;
set t2 = the_last_point_of c1;
reconsider p1 = c1*L[01](0,1,inf dom c1,sup dom c1) as Path of t1,t2
by Th29;
stereographic_projection(V,-p) is being_homeomorphism
by A12,A14,MFOLD_2:31;
then
A39: TPlane(-p,0.TOP-REAL n1), V are_homeomorphic by T_0TOPSP:def 1;
-p <> 0.TOP-REAL n1
proof
assume -p = 0.TOP-REAL n1;
then -p - 0.TOP-REAL n1 = 0.TOP-REAL n1 by RLVECT_1:5;
then |. 0.TOP-REAL n1 .| = 1 by A12,TOPREAL9:9;
hence contradiction by EUCLID_2:39;
end;
then
A40: TOP-REAL n, TPlane(-p,0.TOP-REAL n1) are_homeomorphic by MFOLD_2:29;
then
TOP-REAL n, V are_homeomorphic by A39,BORSUK_3:3;
then consider fh be Function of TOP-REAL n, V such that
A41: fh is being_homeomorphism;
A42: dom fh = [#]TOP-REAL n & rng fh = [#]V by A41,TOPS_2:58;
A43: [#]V is infinite by A1,A41,A42,CARD_1:59;
reconsider v = p as Point of V by A13,A14;
reconsider S = ([#]V) \ {v} as non empty Subset of V by A43,RAMSEY_1:4;
A44: V | S is pathwise_connected by A1,A40,Th10,A39,BORSUK_3:3;
A45: t1 in rng c1 by Th31;
A46: not t1 in {v} by A37,TARSKI:def 1;
A47: t2 in rng c1 by Th31;
A48: not t2 in {v} by A37,TARSKI:def 1;
t1 in S & t2 in S by A45,A46,A47,A36,A48,XBOOLE_0:def 5;
then t1 in [#](V|S) & t2 in [#](V|S) by PRE_TOPC:def 5;
then reconsider v1 = t1, v2 = t2 as Point of V|S;
A49: v1,v2 are_connected by A44;
then consider p3 be Function of I[01],V|S such that
A50: p3 is continuous & p3.0 = v1 & p3.1 = v2;
reconsider p3 as Path of v1,v2 by A50,A49,BORSUK_2:def 2;
A51: Tcircle(0.TOP-REAL n1,1) = Tunit_circle(n1) by TOPREALB:def 7
.= TUnitSphere(n) by MFOLD_2:def 4;
A52: V is SubSpace of (TOP-REAL n1) | (Sphere (0.TOP-REAL n1,1))
by TOPMETR:22,XBOOLE_1:36;
then
A53: V is SubSpace of Tcircle(0.TOP-REAL n1,1) by TOPREALB:def 6;
reconsider S0 = V as non empty SubSpace of TUnitSphere(n)
by A51,A52,TOPREALB:def 6;
reconsider s1 = t1, s2 = t2 as Point of S0 by A45,A47,A36;
A54: dom p3 = [#]I[01] by FUNCT_2:def 1;
A55: [#]S0 c= [#]TUnitSphere(n) by PRE_TOPC:def 4;
rng p3 c= [#](V|S);
then
A56: rng p3 c= S by PRE_TOPC:def 5;
then
rng p3 c= [#]S0 by XBOOLE_1:1;
then reconsider p3 as Function of I[01],TUnitSphere(n)
by A54,A55,FUNCT_2:2,XBOOLE_1:1;
V|S is SubSpace of TUnitSphere(n) by A53,A51,TSEP_1:7;
then
A57: p3 is continuous by A50,PRE_TOPC:26;
then
A58: t1,t2 are_connected by A50;
then reconsider p2 = p3 as Path of t1,t2 by A50,A57,BORSUK_2:def 2;
rng p1 c= rng c1 by RELAT_1:26;
then
A59: rng p1 c= [#]V by A36;
A60: rng p2 c= [#]V by A56,XBOOLE_1:1;
A61: s1,s2 are_connected by A58,A60,JORDAN:29;
reconsider p5=p1, p6=p2 as Path of s1,s2 by A58,A60,A59,JORDAN:29;
reconsider n0 = n as Element of NAT by ORDINAL1:def 12;
S0 is simply_connected by Th14,A39;
then Class(EqRel(S0,s1,s2),p5) = Class(EqRel(S0,s1,s2),p6) by Th12;
then p5,p6 are_homotopic by A61,TOPALG_1:46;
then
A62: p1,p2 are_homotopic by A58,A61,Th6;
set r1 = inf dom c1, r2 = sup dom c1;
A63: r1 <= r2 by XXREAL_2:40;
then
A64: r1 < r2 by A38,XXREAL_0:1;
then reconsider c2 = p2*L[01](r1,r2,0,1)
as with_endpoints Curve of TUnitSphere(n) by A58,Th32;
take c2;
rng L[01](r1,r2,0,1) c= [#]Closed-Interval-TSpace(0,1) by RELAT_1:def 19;
then rng L[01](r1,r2,0,1) c= dom p2 by FUNCT_2:def 1,TOPMETR:20;
then dom c2 = dom L[01](r1,r2,0,1) by RELAT_1:27;
then dom c2 = [#]Closed-Interval-TSpace(r1,r2) by FUNCT_2:def 1;
then
A65: dom c2 = [.r1,r2.] by A63,TOPMETR:18;
A66: c2*L[01](0,1,r1,r2)
= p2*(L[01](r1,r2,0,1)*L[01](0,1,r1,r2)) by RELAT_1:36
.= p2*(id Closed-Interval-TSpace(0,1)) by Th1,A64,Th2
.= p2 by FUNCT_2:17,TOPMETR:20;
inf dom c1 = inf dom c2 & sup dom c1 = sup dom c2 by A65,Th27;
hence c1,c2 are_homotopic by A62,A66;
A67: rng c2 c= rng p2 by RELAT_1:26;
A68: (Sphere(0.TOP-REAL n1,1) \ {p}) \ {-p} c= Sphere(0.TOP-REAL n1,1) \ {p}
by XBOOLE_1:36;
rng c2 c= ([#]V) \ {p} by A56,A67;
then rng c2 c= Sphere(0.TOP-REAL n1,1) \ ({-p} \/ {p})
by A14,XBOOLE_1:41;
then rng c2 c= (Sphere(0.TOP-REAL n1,1) \ {p}) \ {-p} by XBOOLE_1:41;
hence rng c2 c= the carrier of U by A11,A68;
thus dom c1
= dom c2 by A65,Th27;
end;
A69: for i being Nat st 1 <= i & i <= len fc1
holds i+1 in dom h & i in dom h & dom(fc1/.i) = [.h/.i,h/.(i+1).] &
h/.i < h/.(i+1)
proof
let i be Nat;
assume
A70: 1 <= i & i <= len fc1;
A71: 1 <= 1+i by NAT_1:11;
A72: i+1 <= len h - 1 + 1 by A70,A31,XREAL_1:6;
then
i+1 in Seg len h by A71,FINSEQ_1:1;
hence
A73: i+1 in dom h by FINSEQ_1:def 3;
A74: i < i+1 by NAT_1:13;
i <= len h by A72,NAT_1:13;
then i in Seg len h by A70,FINSEQ_1:1;
hence
A75: i in dom h by FINSEQ_1:def 3;
A76: h/.i = h.i by A75,PARTFUN1:def 6;
A77: h/.(i+1) = h.(i+1) by A73,PARTFUN1:def 6;
A78: 0 <= h.i
proof
per cases;
suppose i = 1; hence thesis by A5; end;
suppose not i = 1;
then
A79: 1 < i by A70,XXREAL_0:1;
1 <= len h by A72,A71,XXREAL_0:2;
then 1 in Seg len h by FINSEQ_1:1;
then 1 in dom h by FINSEQ_1:def 3;
hence thesis by A5,A75,A79,VALUED_0:def 13;
end;
end;
A80: h.(i+1) <= 1
proof
per cases;
suppose i+1 = len h; hence thesis by A5; end;
suppose not i+1 = len h;
then
A81: i+1 < len h by A72,XXREAL_0:1;
len h in Seg len h by A5,FINSEQ_1:3;
then
A82: len h in dom h by FINSEQ_1:def 3;
i+1 in Seg len h by A72,A71,FINSEQ_1:1;
then i+1 in dom h by FINSEQ_1:def 3;
hence thesis by A5,A81,A82,VALUED_0:def 13;
end;
end;
[.h.i,h.(i+1).] c= [.0,1.] by A78,A80,XXREAL_1:34;
then
A83: [.h.i,h.(i+1).] c= dom c by A30,Th27;
A84: fc1/.i = c | ([.h/.i,h/.(i+1).]) by A31,A70;
thus
dom(fc1/.i)
= [.h/.i,h/.(i+1).] by A84,A83,A76,A77,RELAT_1:62;
thus h/.i < h/.(i+1) by A77,A76,A75,A73,A74,A5,VALUED_0:def 13;
end;
A85: for i being Nat st 1 <= i & i <= len fc1
holds fc1/.i is with_endpoints &
(for c1 being with_endpoints Curve of TUnitSphere(n) st c1 = fc1/.i
ex c2 being with_endpoints Curve of TUnitSphere(n)
st c1,c2 are_homotopic & rng(c2) c= the carrier of U & dom c1 = dom c2
& dom c1 = [. h/.i, h/.(i+1) .])
proof
let i be Nat;
assume
A86: 1 <= i & i <= len fc1;
A87: fc1/.i = c | ([.h/.i,h/.(i+1).]) by A31,A86;
A88: i+1 in dom h by A69,A86;
A89: i in dom h by A69,A86;
A90: i < i+1 by NAT_1:13;
A91: h.i < h.(i+1) by A89,A88,A90,A5,VALUED_0:def 13;
A92: dom(fc1/.i) = [.h/.i,h/.(i+1).] by A69,A86;
h/.i < h/.(i+1) by A69,A86;
then dom(fc1/.i) is left_end right_end by A92,XXREAL_2:33;
then (fc1/.i) is with_first_point with_last_point;
hence (fc1/.i) is with_endpoints;
let c1 be with_endpoints Curve of TUnitSphere(n);
assume
A93: c1 = fc1/.i;
A94: dom c1 = [.inf dom c1, sup dom c1.] by Th27;
A95: inf dom c1 <= sup dom c1 by XXREAL_2:40;
A96: inf dom c1 = h/.i by A93,A94,A95,A92,XXREAL_1:66;
A97: h/.i = h.i by A89,PARTFUN1:def 6;
A98: sup dom c1 = h/.(i+1) by A93,A94,A95,A92,XXREAL_1:66;
A99: h/.(i+1) = h.(i+1) by A88,PARTFUN1:def 6;
per cases by A32,A86,A93;
suppose
A100: rng(c1) c= the carrier of U;
take c1;
thus thesis by A100,A93,A69,A86;
end;
suppose
A101: rng(c1) c= the carrier of V;
A102: rng h c= dom f by A5,FUNCT_2:def 1;
then
A103: dom f2 = dom h by RELAT_1:27;
A104: i+1 in dom f2 by A102,A88,RELAT_1:27;
A105: the_first_point_of c1 <> p
proof
assume
A106: the_first_point_of c1 = p;
inf dom c1 in dom c1 by A94,A95,XXREAL_1:1;
then c1.(inf dom c1)
= f.(h.i) by A96,A97,A93,A87,FUNCT_1:47
.= f2.i by A103,A89,FUNCT_1:12;
hence contradiction by A6,A106,A103,A89,FUNCT_1:3;
end;
A107: the_last_point_of c1 <> p
proof
assume
A108: the_last_point_of c1 = p;
sup dom c1 in dom c1 by A94,A95,XXREAL_1:1;
then c1.(sup dom c1)
= f.(h.(i+1)) by A98,A99,A93,A87,FUNCT_1:47
.= f2.(i+1) by A104,FUNCT_1:12;
hence contradiction by A6,A108,A104,FUNCT_1:3;
end;
consider c2 be with_endpoints Curve of TUnitSphere(n) such that
A109: c1,c2 are_homotopic & rng(c2) c= the carrier of U & dom c1 = dom c2
by A35,A101,A105,A107,A96,A98,A91,A97,A99;
take c2;
thus thesis by A109,A93,A69,A86;
end;
end;
defpred P[set,set] means
for i being Nat, c1 being with_endpoints Curve of TUnitSphere(n)
st i=$1 & c1 = fc1/.i holds
ex c2 being with_endpoints Curve of TUnitSphere(n) st c2 = $2 &
c2,c1 are_homotopic & rng(c2) c= the carrier of U & dom c1 = dom c2 &
dom c1 = [. h/.i, h/.(i+1) .];
A110: for k being Nat st k in Seg len fc1
ex x being Element of Curves TUnitSphere(n) st P[k,x]
proof
let k be Nat;
assume k in Seg len fc1;
then
A111: 1 <= k & k <= len fc1 by FINSEQ_1:1;
set c1 = fc1/.k;
reconsider c1 as with_endpoints Curve of TUnitSphere(n) by A111,A85;
consider c2 be with_endpoints Curve of TUnitSphere(n) such that
A112: c1,c2 are_homotopic & rng(c2) c= the carrier of U & dom c1 = dom c2 &
dom c1 = [. h/.k, h/.(k+1) .]
by A85,A111;
reconsider x = c2 as Element of Curves TUnitSphere(n);
take x;
thus thesis by A112;
end;
ex p being FinSequence of Curves TUnitSphere(n)
st dom p = Seg len fc1 &
for k being Nat st k in Seg len fc1 holds P[k,p.k]
from FINSEQ_1:sch 5(A110);
then consider fc2 be FinSequence of Curves TUnitSphere(n) such that
A113: dom fc2 = Seg len fc1 &
for k being Nat st k in Seg len fc1 holds P[k,fc2.k];
A114: len fc2 = len fc1 by A113,FINSEQ_1:def 3;
A115: 2-1 <= len h -1 by A29,XREAL_1:9;
then
A116: len fc2 > 0 by A31,A113,FINSEQ_1:def 3;
A117: for i being Nat st 1 <= i & i < len fc2
holds (fc2/.i).sup dom(fc2/.i) = (fc2/.(i+1)).inf dom(fc2/.(i+1))
& sup dom(fc2/.i) = inf dom(fc2/.(i+1))
proof
let i be Nat;
assume
A118: 1 <= i & i < len fc2;
then 1<=i & i <= len fc1 by A113,FINSEQ_1:def 3;
then
A119: i in Seg len fc1 by FINSEQ_1:1;
set ci = fc1/.i;
reconsider ci as with_endpoints Curve of TUnitSphere(n) by A118,A114,A85;
consider di be with_endpoints Curve of TUnitSphere(n) such that
A120: di = fc2.i & di,ci are_homotopic & rng(di) c= the carrier of U &
dom ci = dom di & dom ci = [. h/.i,h/.(i+1) .] by A119,A113;
1+1 <= i+1 by A118,XREAL_1:6;
then
A121: 1 <= i+1 by XXREAL_0:2;
A122: i+1 <= len fc2 by A118,NAT_1:13;
then
A123: i+1 in Seg len fc1 by A114,A121,FINSEQ_1:1;
set ci1 = fc1/.(i+1);
reconsider ci1 as with_endpoints Curve of TUnitSphere(n)
by A121,A122,A114,A85;
consider di1 be with_endpoints Curve of TUnitSphere(n) such that
A124: di1 = fc2.(i+1) & di1,ci1 are_homotopic & rng(di1) c= the carrier of U &
dom ci1 = dom di1 & dom ci1 = [. h/.(i+1),h/.((i+1)+1) .] by A123,A113;
A125: i+1 in dom fc2 by A122,A113,A114,A121,FINSEQ_1:1;
A126: h/.i < h/.(i+1) by A69,A118,A114;
A127: h/.(i+1) < h/.((i+1)+1) by A69,A121,A122,A114;
A128: dom(fc1/.i) = [.h/.i,h/.(i+1).] by A69,A118,A114;
A129: dom(fc1/.(i+1)) = [.h/.(i+1),h/.((i+1)+1).] by A69,A121,A122,A114;
A130: h/.(i+1) in [.h/.i,h/.(i+1).] by A126,XXREAL_1:1;
A131: h/.(i+1) in [.h/.(i+1),h/.((i+1)+1).] by A127,XXREAL_1:1;
A132: fc2/.i = fc2.i by A119,A113,PARTFUN1:def 6;
A133: fc2/.(i+1) = fc2.(i+1) by A125,PARTFUN1:def 6;
thus (fc2/.i).sup dom(fc2/.i) = the_last_point_of di by A120,A132
.= the_last_point_of ci by A120,Th35
.= (fc1/.i).(h/.(i+1)) by A126,A128,XXREAL_2:29
.= (c | ([.h/.i,h/.(i+1).])).(h/.(i+1)) by A31,A118,A114
.= c.(h/.(i+1)) by A130,FUNCT_1:49
.= (c | ([.h/.(i+1),h/.((i+1)+1).])).(h/.(i+1)) by A131,FUNCT_1:49
.= (fc1/.(i+1)).(h/.(i+1)) by A31,A121,A122,A114
.= the_first_point_of ci1 by A127,A129,XXREAL_2:25
.= the_first_point_of di1 by A124,Th35
.= (fc2/.(i+1)).inf dom(fc2/.(i+1)) by A124,A133;
A134: dom(fc2/.i) = [. h/.i,h/.(i+1) .]
by A120,A119,A113,PARTFUN1:def 6;
A135: dom(fc2/.(i+1))
= [. h/.(i+1),h/.(i+2) .] by A124,A125,PARTFUN1:def 6;
thus sup dom(fc2/.i) = h/.(i+1) by A134,A126,XXREAL_2:29
.= inf dom(fc2/.(i+1)) by A135,A127,XXREAL_2:25;
end;
A136: for i being Nat st 1 <= i & i <= len fc2
holds fc2/.i is with_endpoints
proof
let i be Nat;
assume
A137: 1 <= i & i <= len fc2;
then
A138: i in Seg len fc1 by A114,FINSEQ_1:1;
set ci = fc1/.i;
reconsider ci as with_endpoints Curve of TUnitSphere(n)
by A137,A114,A85;
consider di be with_endpoints Curve of TUnitSphere(n) such that
A139: di = fc2.i & di,ci are_homotopic & rng(di) c= the carrier of U &
dom ci = dom di & dom ci = [. h/.i,h/.(i+1) .] by A138,A113;
thus fc2/.i is with_endpoints
by A139,A138,A113,PARTFUN1:def 6;
end;
consider c0 be with_endpoints Curve of TUnitSphere(n) such that
A140: Sum fc2 = c0 & dom c0 = [.inf dom(fc2/.1), sup dom(fc2/.(len fc2)).] &
the_first_point_of c0 = (fc2/.1).(inf dom(fc2/.1)) &
the_last_point_of c0 = (fc2/.(len fc2)).(sup dom(fc2/.(len fc2)))
by A117,A136,A116,Th43;
A141: for i being Nat st 1 <= i & i < len fc1
holds (fc1/.i).sup dom(fc1/.i) = (fc1/.(i+1)).inf dom(fc1/.(i+1))
& sup dom(fc1/.i) = inf dom(fc1/.(i+1))
proof
let i be Nat;
assume
A142: 1 <= i & i < len fc1;
A143: i in Seg len fc1 by A142,FINSEQ_1:1;
set ci = fc1/.i;
reconsider ci as with_endpoints Curve of TUnitSphere(n) by A142,A85;
consider di be with_endpoints Curve of TUnitSphere(n) such that
A144: di = fc2.i & di,ci are_homotopic & rng(di) c= the carrier of U &
dom ci = dom di & dom ci = [. h/.i,h/.(i+1) .] by A143,A113;
1+1 <= i+1 by A142,XREAL_1:6;
then
A145: 1 <= i+1 by XXREAL_0:2;
A146: i+1 <= len fc2 by A114,A142,NAT_1:13;
then
A147: i+1 in Seg len fc1 by A114,A145,FINSEQ_1:1;
set ci1 = fc1/.(i+1);
reconsider ci1 as with_endpoints Curve of TUnitSphere(n)
by A145,A146,A114,A85;
consider di1 be with_endpoints Curve of TUnitSphere(n) such that
A148: di1 = fc2.(i+1) & di1,ci1 are_homotopic & rng(di1) c= the carrier of U &
dom ci1 = dom di1 & dom ci1 = [. h/.(i+1),h/.((i+1)+1) .] by A147,A113;
A149: h/.i < h/.(i+1) by A69,A142;
A150: h/.(i+1) < h/.((i+1)+1) by A69,A145,A146,A114;
A151: dom(fc1/.i) = [.h/.i,h/.(i+1).] by A69,A142;
A152: dom(fc1/.(i+1)) = [.h/.(i+1),h/.((i+1)+1).] by A69,A145,A146,A114;
A153: h/.(i+1) in [.h/.i,h/.(i+1).] by A149,XXREAL_1:1;
A154: h/.(i+1) in [.h/.(i+1),h/.((i+1)+1).] by A150,XXREAL_1:1;
thus (fc1/.i).sup dom(fc1/.i)
= (fc1/.i).(h/.(i+1)) by A149,A151,XXREAL_2:29
.= (c | ([.h/.i,h/.(i+1).])).(h/.(i+1)) by A31,A142
.= c.(h/.(i+1)) by A153,FUNCT_1:49
.= (c | ([.h/.(i+1),h/.((i+1)+1).])).(h/.(i+1)) by A154,FUNCT_1:49
.= (fc1/.(i+1)).(h/.(i+1)) by A31,A145,A146,A114
.= (fc1/.(i+1)).inf dom(fc1/.(i+1)) by A150,A152,XXREAL_2:25;
thus sup dom(fc1/.i) = h/.(i+1) by A144,A149,XXREAL_2:29
.= inf dom(fc1/.(i+1)) by A148,A150,XXREAL_2:25;
end;
for i being Nat st 1 <= i & i <= len fc2 holds
ex c3,c4 being with_endpoints Curve of TUnitSphere(n)
st c3 = fc2/.i & c4 = fc1/.i & c3,c4 are_homotopic & dom c3 = dom c4
proof
let i be Nat;
assume
A155: 1 <= i & i <= len fc2;
then
A156: i in Seg len fc1 by A114,FINSEQ_1:1;
set ci = fc1/.i;
reconsider ci as with_endpoints Curve of TUnitSphere(n)
by A155,A114,A85;
consider di be with_endpoints Curve of TUnitSphere(n) such that
A157: di = fc2.i & di,ci are_homotopic & rng(di) c= the carrier of U &
dom ci = dom di & dom ci = [. h/.i,h/.(i+1) .] by A156,A113;
A158: i in dom fc2 by A155,A114,A113,FINSEQ_1:1;
take di,ci;
thus thesis by A157,A158,PARTFUN1:def 6;
end;
then
A159: c0, c are_homotopic by A117,A141,A140,Th44,A114,A115,A31;
A160: dom c0 = [.0,1.]
proof
A161: 0+1 < len fc1 + 1 by A115,A31,XREAL_1:6;
A162: 1 in Seg len fc1 by A115,A31,FINSEQ_1:1;
set ci = fc1/.1;
reconsider ci as with_endpoints Curve of TUnitSphere(n) by A115,A31,A85;
consider di be with_endpoints Curve of TUnitSphere(n) such that
A163: di = fc2.1 & di,ci are_homotopic & rng(di) c= the carrier of U &
dom ci = dom di & dom ci = [. h/.1,h/.(1+1) .] by A162,A113;
1 in Seg len fc2 by A115,A31,A114,FINSEQ_1:1;
then 1 in dom fc2 by FINSEQ_1:def 3;
then
A164: dom(fc2/.1) = [.h/.1,h/.(1+1).] by A163,PARTFUN1:def 6;
A165: h/.1 < h/.(1+1) by A69,A115,A31;
1 in Seg len h by A161,A31,FINSEQ_1:1;
then
A166: 1 in dom h by FINSEQ_1:def 3;
A167: inf dom(fc2/.1) = h/.1 by A165,A164,XXREAL_2:25
.= 0 by A5,A166,PARTFUN1:def 6;
A168: len fc1 in Seg len fc1 by A115,A31,FINSEQ_1:1;
set ci1 = fc1/.(len fc1);
reconsider ci1 as with_endpoints Curve of TUnitSphere(n)
by A115,A31,A85;
consider di1 be with_endpoints Curve of TUnitSphere(n) such that
A169: di1 = fc2.(len fc1) & di1,ci1 are_homotopic &
rng(di1) c= the carrier of U &
dom ci1 = dom di1 & dom ci1 = [. h/.(len fc1),h/.(len fc1+1) .]
by A168,A113;
len fc1 in Seg len fc2 by A114,A115,A31,FINSEQ_1:1;
then len fc1 in dom fc2 by FINSEQ_1:def 3;
then
A170: dom(fc2/.len fc2) = [.h/.len fc1,h/.(len fc1+1).]
by A169,A114,PARTFUN1:def 6;
A171: h/.len fc1 < h/.(len fc1+1) by A69,A115,A31;
len h in Seg len h by A161,A31,FINSEQ_1:1;
then
A172: len h in dom h by FINSEQ_1:def 3;
A173: sup dom(fc2/.(len fc2)) = h/.(len fc1+1) by A171,A170,XXREAL_2:29
.= 1 by A5,A31,A172,PARTFUN1:def 6;
thus thesis by A140,A167,A173;
end;
for i being Nat st 1 <= i & i <= len fc2
holds rng(fc2/.i) c= the carrier of U
proof
let i be Nat;
assume
A174: 1 <= i & i <= len fc2;
then i in Seg len fc2 by FINSEQ_1:1;
then
A175: i in dom fc2 by FINSEQ_1:def 3;
A176: i in Seg len fc1 by A114,A174,FINSEQ_1:1;
reconsider c1 = fc1/.i as with_endpoints Curve of TUnitSphere(n)
by A85,A174,A114;
consider c2 be with_endpoints Curve of TUnitSphere(n) such that
A177: c2 = fc2.i & c2,c1 are_homotopic & rng(c2) c= the carrier of U &
dom c1 = dom c2 & dom c1 = [. h/.i, h/.(i+1) .] by A176,A113;
thus thesis by A175,A177,PARTFUN1:def 6;
end;
then
A178: rng c0 c= the carrier of U by A140,Th42;
A179: t,t are_connected;
A180: t
= the_first_point_of c by A30,A179,BORSUK_2:def 2
.= the_first_point_of c0 by Th35,A159;
A181: t
= the_last_point_of c by A30,A179,BORSUK_2:def 2
.= the_last_point_of c0 by Th35,A159;
reconsider f0 = c0 as Loop of t by A160,A180,A181,Th28;
A182: f0, f are_homotopic by A159,A179,Th34;
not p in rng f0
proof
assume p in rng f0;
then not p in {p} by A11,A178,XBOOLE_0:def 5;
hence contradiction by TARSKI:def 1;
end;
hence thesis by A6,A182;
end;
theorem Th46:
n >= 2 implies TUnitSphere(n) is having_trivial_Fundamental_Group
proof
assume
A1: n >= 2;
set T = TUnitSphere(n);
for t being Point of T, f being Loop of t holds f is nullhomotopic
proof
let t be Point of T, f be Loop of t;
per cases;
suppose rng f <> the carrier of TUnitSphere(n);
hence f is nullhomotopic by Lm3;
end;
suppose rng f = the carrier of TUnitSphere(n);
then consider g be Loop of t such that
A2: g, f are_homotopic and
A3: rng g <> the carrier of TUnitSphere(n) by A1,Lm4;
g is nullhomotopic by A3,Lm3;
then consider C be constant Loop of t such that
A4: g, C are_homotopic;
f, C are_homotopic by A2,A4,BORSUK_6:79;
hence f is nullhomotopic;
end;
end;
hence thesis by Th17;
end;
theorem
for n being non zero Nat, r being positive Real,
x being Point of TOP-REAL n st n >= 3 holds
Tcircle(x,r) is having_trivial_Fundamental_Group
proof
let n be non zero Nat;
let r be positive Real;
let x be Point of TOP-REAL n;
assume
A1: n >= 3;
then n-1 >= 3-1 by XREAL_1:9;
then 0 <= n-1 by XXREAL_0:2;
then
A2: n-'1+1 = n-1+1 by XREAL_0:def 2;
2+1 = 3;
then 2 <= n-'1 by A1,NAT_D:49;
then
A3: TUnitSphere(n-'1) is having_trivial_Fundamental_Group by Th46;
A4: TUnitSphere(n-'1) = Tunit_circle(n-'1+1) by MFOLD_2:def 4;
A5: Tunit_circle(n) = Tcircle(0.TOP-REAL n,1) by TOPREALB:def 7;
n in NAT by ORDINAL1:def 12;
then Tcircle(x,r), Tcircle(0.TOP-REAL n,1) are_homeomorphic by TOPREALB:20;
hence thesis by A2,A3,A4,A5,Th13;
end;