:: TOPALG_6 semantic presentation
begin
registration
let S be TopSpace;
let T be non empty TopSpace;
cluster Function-like constant quasi_total -> continuous for Element of bool [: the carrier of S, the carrier of T:];
correctness
coherence
for b1 being Function of S,T st b1 is constant holds
b1 is continuous ;
proof
let f be Function of S,T; ::_thesis: ( f is constant implies f is continuous )
assume A1: f is constant ; ::_thesis: f is continuous
percases ( S is empty or not S is empty ) ;
supposeA2: S is empty ; ::_thesis: f is continuous
for P1 being Subset of T st P1 is closed holds
f " P1 is closed by A2;
hence f is continuous by PRE_TOPC:def_6; ::_thesis: verum
end;
suppose not S is empty ; ::_thesis: f is continuous
then consider y being 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 set 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 f is continuous ; ::_thesis: verum
end;
end;
end;
end;
theorem Th1: :: TOPALG_6:1
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, TREAL_1:14;
hence L[01] (0,1,0,1) = id (Closed-Interval-TSpace (0,1)) by SYSREL:12; ::_thesis: verum
end;
theorem Th2: :: TOPALG_6:2
for r1, r2, r3, r4, r5, r6 being real number 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 number ; ::_thesis: ( r1 < r2 & r3 <= r4 & r5 < r6 implies (L[01] (r1,r2,r3,r4)) * (L[01] (r5,r6,r1,r2)) = L[01] (r5,r6,r3,r4) )
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 ) ; ::_thesis: (L[01] (r1,r2,r3,r4)) * (L[01] (r5,r6,r1,r2)) = L[01] (r5,r6,r3,r4)
A2: dom ((L[01] (r1,r2,r3,r4)) * (L[01] (r5,r6,r1,r2))) = [#] (Closed-Interval-TSpace (r5,r6)) by FUNCT_2:def_1
.= dom (L[01] (r5,r6,r3,r4)) by FUNCT_2:def_1 ;
for x being set st x in dom ((L[01] (r1,r2,r3,r4)) * (L[01] (r5,r6,r1,r2))) holds
((L[01] (r1,r2,r3,r4)) * (L[01] (r5,r6,r1,r2))) . x = (L[01] (r5,r6,r3,r4)) . x
proof
let x be set ; ::_thesis: ( x in dom ((L[01] (r1,r2,r3,r4)) * (L[01] (r5,r6,r1,r2))) implies ((L[01] (r1,r2,r3,r4)) * (L[01] (r5,r6,r1,r2))) . x = (L[01] (r5,r6,r3,r4)) . x )
assume A3: x in dom ((L[01] (r1,r2,r3,r4)) * (L[01] (r5,r6,r1,r2))) ; ::_thesis: ((L[01] (r1,r2,r3,r4)) * (L[01] (r5,r6,r1,r2))) . x = (L[01] (r5,r6,r3,r4)) . x
then A4: x in [#] (Closed-Interval-TSpace (r5,r6)) ;
then A5: x in [.r5,r6.] by A1, TOPMETR:18;
reconsider r = x as real number by A3;
A6: ( r5 <= r & r <= r6 ) by A5, XXREAL_1:1;
A7: rng (L[01] (r5,r6,r1,r2)) c= [#] (Closed-Interval-TSpace (r1,r2)) by RELAT_1:def_19;
reconsider s = (L[01] (r5,r6,r1,r2)) . x as real number ;
x in dom (L[01] (r5,r6,r1,r2)) by A4, FUNCT_2:def_1;
then s in rng (L[01] (r5,r6,r1,r2)) by FUNCT_1:3;
then s in [#] (Closed-Interval-TSpace (r1,r2)) by A7;
then s in [.r1,r2.] by A1, TOPMETR:18;
then ( r1 <= s & s <= r2 ) by XXREAL_1:1;
then A8: (L[01] (r1,r2,r3,r4)) . 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 ((L[01] (r1,r2,r3,r4)) * (L[01] (r5,r6,r1,r2))) . x = (L[01] (r1,r2,r3,r4)) . ((L[01] (r5,r6,r1,r2)) . x) by A3, FUNCT_1:12
.= (L[01] (r5,r6,r3,r4)) . x by A10, A8, A1, A6, BORSUK_6:35 ; ::_thesis: verum
end;
hence (L[01] (r1,r2,r3,r4)) * (L[01] (r5,r6,r1,r2)) = L[01] (r5,r6,r3,r4) by A2, FUNCT_1:2; ::_thesis: verum
end;
registration
let n be positive Nat;
cluster TOP-REAL n -> infinite ;
correctness
coherence
not TOP-REAL n is finite ;
proof
A1: the carrier of (TOP-REAL n) = REAL n by EUCLID:22
.= n -tuples_on REAL by EUCLID:def_1 ;
deffunc H1( Element of n -tuples_on REAL) -> set = n . 1;
consider f being Function such that
A2: ( dom f = n -tuples_on REAL & ( for d being Element of n -tuples_on REAL holds f . d = H1(d) ) ) from FUNCT_1:sch_4();
for y being set holds
( y in f .: (n -tuples_on REAL) iff y in REAL )
proof
let y be set ; ::_thesis: ( y in f .: (n -tuples_on REAL) iff y in REAL )
0 + 1 < n + 1 by XREAL_1:6;
then 1 <= n by NAT_1:13;
then A3: 1 in Seg n by FINSEQ_1:1;
hereby ::_thesis: ( y in REAL implies y in f .: (n -tuples_on REAL) )
assume y in f .: (n -tuples_on REAL) ; ::_thesis: y in REAL
then consider x being set such that
A4: ( x in dom f & x in n -tuples_on REAL & y = f . x ) by FUNCT_1:def_6;
reconsider x = 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;
then y in rng x by A3, A5, FUNCT_1:3;
hence y in REAL ; ::_thesis: verum
end;
assume y in REAL ; ::_thesis: y in f .: (n -tuples_on REAL)
then A6: {y} c= REAL by ZFMISC_1:31;
set x = (Seg n) --> y;
A7: ( dom ((Seg n) --> y) = Seg n & rng ((Seg n) --> y) c= {y} ) by FUNCOP_1:13;
rng ((Seg n) --> y) c= REAL by A6, XBOOLE_1:1;
then (Seg n) --> y in Funcs ((Seg n),REAL) by A7, FUNCT_2:def_2;
then reconsider x = (Seg n) --> y as Element of n -tuples_on REAL by FINSEQ_2:93;
f . x = x . 1 by A2
.= y by A3, FUNCOP_1:7 ;
hence y in f .: (n -tuples_on REAL) by A2, FUNCT_1:def_6; ::_thesis: verum
end;
hence not TOP-REAL n is finite by A1, TARSKI:1; ::_thesis: verum
end;
cluster non empty TopSpace-like n -locally_euclidean -> non empty infinite for TopStruct ;
correctness
coherence
for b1 being non empty TopSpace st b1 is n -locally_euclidean holds
b1 is infinite ;
proof
let M be non empty TopSpace; ::_thesis: ( M is n -locally_euclidean implies M is infinite )
assume A8: M is n -locally_euclidean ; ::_thesis: M is infinite
consider p being set such that
A9: p in the carrier of M by XBOOLE_0:def_1;
reconsider p = p as Point of M by A9;
consider U being a_neighborhood of p such that
A10: U, [#] (TOP-REAL n) are_homeomorphic by A8, MFOLD_1:13;
M | U,(TOP-REAL n) | ([#] (TOP-REAL n)) are_homeomorphic by A10, METRIZTS:def_1;
then consider f being Function of ((TOP-REAL n) | ([#] (TOP-REAL n))),(M | U) such that
A11: f is being_homeomorphism by T_0TOPSP:def_1;
A12: ( dom f = [#] ((TOP-REAL n) | ([#] (TOP-REAL n))) & rng f = [#] (M | U) & f is one-to-one ) by A11, TOPS_2:58;
[#] ((TOP-REAL n) | ([#] (TOP-REAL n))) is infinite by PRE_TOPC:def_5;
hence M is infinite by A12, CARD_1:59; ::_thesis: verum
end;
end;
theorem Th3: :: TOPALG_6:3
for n being Nat
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 n be Nat; ::_thesis: 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}
let p be Point of (TOP-REAL n); ::_thesis: ( p in Sphere ((0. (TOP-REAL n)),1) implies - p in (Sphere ((0. (TOP-REAL n)),1)) \ {p} )
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) ; ::_thesis: - p in (Sphere ((0. (TOP-REAL n)),1)) \ {p}
then |.(p1 - (0. (TOP-REAL n1))).| = 1 by TOPREAL9:9;
then |.(p1 + (- (0. (TOP-REAL n1)))).| = 1 by EUCLID:41;
then |.(p + ((- 1) * (0. (TOP-REAL n)))).| = 1 by EUCLID:39;
then |.(p + (0. (TOP-REAL n))).| = 1 by EUCLID:28;
then A1: |.p.| = 1 by EUCLID:27;
reconsider r1 = 1 as real number ;
|.(0. (TOP-REAL n)).| <> |.p.| by A1, EUCLID_2:39;
then 0. (TOP-REAL n) <> (1 + 1) * p by EUCLID:31;
then 0. (TOP-REAL n) <> (r1 * p) + (r1 * p) by EUCLID:33;
then 0. (TOP-REAL n) <> (r1 * p) + p by EUCLID:29;
then 0. (TOP-REAL n) <> p + p by EUCLID:29;
then p + (- p) <> p + p by EUCLID:36;
then A2: not - p in {p} by TARSKI:def_1;
|.(- p).| = 1 by A1, EUCLID:71;
then |.((- p) + (0. (TOP-REAL n))).| = 1 by EUCLID:27;
then |.((- p) + ((- 1) * (0. (TOP-REAL n)))).| = 1 by EUCLID:28;
then |.((- p) + (- (0. (TOP-REAL n)))).| = 1 by EUCLID:39;
then |.((- p1) - (0. (TOP-REAL n1))).| = 1 by EUCLID:41;
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; ::_thesis: verum
end;
theorem Th4: :: TOPALG_6:4
for T being non empty TopStruct
for 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 ; ::_thesis: for t1, t2 being Point of T
for p being Path of t1,t2 holds
( inf (dom p) = 0 & sup (dom p) = 1 )
let t1, t2 be Point of T; ::_thesis: for p being Path of t1,t2 holds
( inf (dom p) = 0 & sup (dom p) = 1 )
let p be Path of t1,t2; ::_thesis: ( inf (dom p) = 0 & sup (dom p) = 1 )
[.0,1.] = dom p by BORSUK_1:40, FUNCT_2:def_1;
hence ( inf (dom p) = 0 & sup (dom p) = 1 ) by XXREAL_2:25, XXREAL_2:29; ::_thesis: verum
end;
theorem Th5: :: TOPALG_6:5
for T being non empty TopSpace
for t being Point of T
for C1, C2 being constant Loop of t holds C1,C2 are_homotopic
proof
let T be non empty TopSpace; ::_thesis: for t being Point of T
for C1, C2 being constant Loop of t holds C1,C2 are_homotopic
let t be Point of T; ::_thesis: for C1, C2 being constant Loop of t holds C1,C2 are_homotopic
let C1, C2 be constant Loop of t; ::_thesis: C1,C2 are_homotopic
C1 = I[01] --> t by BORSUK_2:5
.= C2 by BORSUK_2:5 ;
hence C1,C2 are_homotopic by BORSUK_2:12; ::_thesis: verum
end;
theorem Th6: :: TOPALG_6:6
for T being non empty TopSpace
for S being non empty SubSpace of T
for t1, t2 being Point of T
for s1, s2 being Point of S
for A, B being Path of t1,t2
for 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 T be non empty TopSpace; ::_thesis: for S being non empty SubSpace of T
for t1, t2 being Point of T
for s1, s2 being Point of S
for A, B being Path of t1,t2
for 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
let S be non empty SubSpace of T; ::_thesis: for t1, t2 being Point of T
for s1, s2 being Point of S
for A, B being Path of t1,t2
for 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
let t1, t2 be Point of T; ::_thesis: for s1, s2 being Point of S
for A, B being Path of t1,t2
for 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
let s1, s2 be Point of S; ::_thesis: for A, B being Path of t1,t2
for 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
let A, B be Path of t1,t2; ::_thesis: for 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
let C, D be Path of s1,s2; ::_thesis: ( s1,s2 are_connected & t1,t2 are_connected & A = C & B = D & C,D are_homotopic implies A,B are_homotopic )
assume that
A1: ( s1,s2 are_connected & t1,t2 are_connected ) and
A2: ( A = C & B = D ) ; ::_thesis: ( not C,D are_homotopic or A,B are_homotopic )
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 ) ; :: according to BORSUK_2:def_7 ::_thesis: A,B are_homotopic
reconsider g = f as Function of [:I[01],I[01]:],T by TOPREALA:7;
take g ; :: according to BORSUK_2:def_7 ::_thesis: ( g is continuous & ( for b1 being Element of the carrier of I[01] holds
( g . (b1,0) = A . b1 & g . (b1,1) = B . b1 & g . (0,b1) = t1 & g . (1,b1) = t2 ) ) )
thus g is continuous by A3, PRE_TOPC:26; ::_thesis: for b1 being Element of the carrier of I[01] holds
( g . (b1,0) = A . b1 & g . (b1,1) = B . b1 & g . (0,b1) = t1 & g . (1,b1) = t2 )
( s1 = C . 0 & s2 = C . 1 & t1 = A . 0 & t2 = A . 1 ) by A1, BORSUK_2:def_2;
hence for b1 being Element of the carrier of I[01] holds
( g . (b1,0) = A . b1 & g . (b1,1) = B . b1 & g . (0,b1) = t1 & g . (1,b1) = t2 ) by A2, A4; ::_thesis: verum
end;
theorem :: TOPALG_6:7
for T being non empty TopSpace
for S being non empty SubSpace of T
for t1, t2 being Point of T
for s1, s2 being Point of S
for A, B being Path of t1,t2
for 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 T be non empty TopSpace; ::_thesis: for S being non empty SubSpace of T
for t1, t2 being Point of T
for s1, s2 being Point of S
for A, B being Path of t1,t2
for 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)
let S be non empty SubSpace of T; ::_thesis: for t1, t2 being Point of T
for s1, s2 being Point of S
for A, B being Path of t1,t2
for 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)
let t1, t2 be Point of T; ::_thesis: for s1, s2 being Point of S
for A, B being Path of t1,t2
for 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)
let s1, s2 be Point of S; ::_thesis: for A, B being Path of t1,t2
for 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)
let A, B be Path of t1,t2; ::_thesis: for 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)
let C, D be Path of s1,s2; ::_thesis: ( s1,s2 are_connected & t1,t2 are_connected & A = C & B = D & Class ((EqRel (S,s1,s2)),C) = Class ((EqRel (S,s1,s2)),D) implies Class ((EqRel (T,t1,t2)),A) = Class ((EqRel (T,t1,t2)),B) )
assume that
A1: s1,s2 are_connected and
A2: t1,t2 are_connected and
A3: ( A = C & B = D ) ; ::_thesis: ( not Class ((EqRel (S,s1,s2)),C) = Class ((EqRel (S,s1,s2)),D) or Class ((EqRel (T,t1,t2)),A) = Class ((EqRel (T,t1,t2)),B) )
assume Class ((EqRel (S,s1,s2)),C) = Class ((EqRel (S,s1,s2)),D) ; ::_thesis: Class ((EqRel (T,t1,t2)),A) = Class ((EqRel (T,t1,t2)),B)
then C,D are_homotopic by A1, TOPALG_1:46;
then A,B are_homotopic by A1, A2, A3, Th6;
hence Class ((EqRel (T,t1,t2)),A) = Class ((EqRel (T,t1,t2)),B) by A2, TOPALG_1:46; ::_thesis: verum
end;
theorem Th8: :: TOPALG_6:8
for T being non empty trivial TopSpace
for t being Point of T
for L being Loop of t holds the carrier of (pi_1 (T,t)) = {(Class ((EqRel (T,t)),L))}
proof
let T be non empty trivial TopSpace; ::_thesis: for t being Point of T
for L being Loop of t holds the carrier of (pi_1 (T,t)) = {(Class ((EqRel (T,t)),L))}
let t be Point of T; ::_thesis: for L being Loop of t holds the carrier of (pi_1 (T,t)) = {(Class ((EqRel (T,t)),L))}
set E = EqRel (T,t);
let L be Loop of t; ::_thesis: the carrier of (pi_1 (T,t)) = {(Class ((EqRel (T,t)),L))}
thus the carrier of (pi_1 (T,t)) c= {(Class ((EqRel (T,t)),L))} :: according to XBOOLE_0:def_10 ::_thesis: {(Class ((EqRel (T,t)),L))} c= the carrier of (pi_1 (T,t))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of (pi_1 (T,t)) or x in {(Class ((EqRel (T,t)),L))} )
assume x in the carrier of (pi_1 (T,t)) ; ::_thesis: x in {(Class ((EqRel (T,t)),L))}
then consider P being Loop of t such that
A1: x = Class ((EqRel (T,t)),P) by TOPALG_1:47;
P = I[01] --> t by TOPREALC:22
.= L by TOPREALC:22 ;
hence x in {(Class ((EqRel (T,t)),L))} by A1, TARSKI:def_1; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(Class ((EqRel (T,t)),L))} or x in the carrier of (pi_1 (T,t)) )
assume x in {(Class ((EqRel (T,t)),L))} ; ::_thesis: x in the carrier of (pi_1 (T,t))
then A2: x = Class ((EqRel (T,t)),L) by TARSKI:def_1;
L in Loops t by TOPALG_1:def_1;
then x in Class (EqRel (T,t)) by A2, EQREL_1:def_3;
hence x in the carrier of (pi_1 (T,t)) by TOPALG_1:def_5; ::_thesis: verum
end;
theorem Th9: :: TOPALG_6:9
for n being Nat
for p being Point of (TOP-REAL n)
for 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 n be Nat; ::_thesis: for p being Point of (TOP-REAL n)
for S being Subset of (TOP-REAL n) st n >= 2 & S = ([#] (TOP-REAL n)) \ {p} holds
(TOP-REAL n) | S is pathwise_connected
let p be Point of (TOP-REAL n); ::_thesis: for S being Subset of (TOP-REAL n) st n >= 2 & S = ([#] (TOP-REAL n)) \ {p} holds
(TOP-REAL n) | S is pathwise_connected
let S be Subset of (TOP-REAL n); ::_thesis: ( n >= 2 & S = ([#] (TOP-REAL n)) \ {p} implies (TOP-REAL n) | S is pathwise_connected )
assume A1: n >= 2 ; ::_thesis: ( not S = ([#] (TOP-REAL n)) \ {p} or (TOP-REAL n) | S is pathwise_connected )
assume A2: S = ([#] (TOP-REAL n)) \ {p} ; ::_thesis: (TOP-REAL n) | S is pathwise_connected
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
for 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; ::_thesis: for a1, b1 being Point of (TOP-REAL n) st not p in LSeg (a1,b1) & a1 = a & b1 = b holds
a,b are_connected
let a1, b1 be Point of (TOP-REAL n); ::_thesis: ( not p in LSeg (a1,b1) & a1 = a & b1 = b implies a,b are_connected )
assume A5: not p in LSeg (a1,b1) ; ::_thesis: ( not a1 = a or not b1 = b or a,b are_connected )
assume A6: ( a1 = a & b1 = b ) ; ::_thesis: a,b are_connected
percases ( a1 <> b1 or a1 = b1 ) ;
supposeA7: a1 <> b1 ; ::_thesis: a,b are_connected
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 ; :: according to BORSUK_2:def_1 ::_thesis: ( f is continuous & f . 0 = a & f . 1 = b )
thus f is continuous by A10, PRE_TOPC:26; ::_thesis: ( f . 0 = a & f . 1 = b )
thus ( f . 0 = a & f . 1 = b ) by A6, A11; ::_thesis: verum
end;
suppose a1 = b1 ; ::_thesis: a,b are_connected
hence a,b are_connected by A6; ::_thesis: verum
end;
end;
end;
for a, b being Point of T holds a,b are_connected
proof
let a, b be Point of T; ::_thesis: a,b are_connected
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;
percases ( a1 <> b1 or a1 = b1 ) ;
supposeA13: a1 <> b1 ; ::_thesis: a,b are_connected
percases ( p in LSeg (a1,b1) or not p in LSeg (a1,b1) ) ;
supposeA14: p in LSeg (a1,b1) ; ::_thesis: a,b are_connected
reconsider n1 = n as Element of NAT by ORDINAL1:def_12;
reconsider aa1 = a1, bb1 = b1 as Point of (TOP-REAL n1) ;
consider s being Real such that
A15: ( 0 <= s & s <= 1 & p = ((1 - s) * aa1) + (s * bb1) ) by A14, JGRAPH_1:35;
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;
b1 - a1 <> 0. (TOP-REAL (k + 1)) by A13, EUCLID:43;
then TPlane ((b1 - a1),p) is k -manifold by MFOLD_2:30;
then [#] (TPlane ((b1 - a1),p)) is infinite by A16;
then [#] ((TOP-REAL n) | (Plane ((b1 - a1),p))) is infinite by MFOLD_2:def_3;
then A17: Plane ((b1 - a1),p) is infinite ;
reconsider X = Plane ((b1 - a1),p) as set ;
X \ {p} is infinite by A17, RAMSEY_1:4;
then consider x being set 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) : |((b1 - a1),(y - p))| = 0 } by MFOLD_2:def_2;
then consider c1 being Point of (TOP-REAL n) such that
A20: ( c1 = x & |((b1 - a1),(c1 - p))| = 0 ) ;
A21: |((b1 - a1),(b1 - a1))| <> 0
proof
assume |((b1 - a1),(b1 - a1))| = 0 ; ::_thesis: contradiction
then b1 - a1 = 0. (TOP-REAL n) by EUCLID_2:41;
hence contradiction by A13, EUCLID:43; ::_thesis: verum
end;
A22: not p in LSeg (a1,c1)
proof
assume A23: p in LSeg (a1,c1) ; ::_thesis: contradiction
reconsider cc1 = c1 as Point of (TOP-REAL n1) ;
consider r being Real such that
A24: ( 0 <= r & r <= 1 & p = ((1 - r) * aa1) + (r * cc1) ) by A23, JGRAPH_1:35;
A25: 1 - r <> 0
proof
assume 1 - r = 0 ; ::_thesis: contradiction
then p = (0. (TOP-REAL n)) + (1 * c1) by A24, EUCLID:29
.= (0. (TOP-REAL n)) + c1 by EUCLID:29
.= c1 by EUCLID:27 ;
hence contradiction by A19, A20, TARSKI:def_1; ::_thesis: verum
end;
set q2 = c1 - a1;
c1 - p = (c1 - ((1 - r) * a1)) - (r * c1) by A24, EUCLID:46
.= (c1 + (- ((1 - r) * a1))) - (r * c1) by EUCLID:41
.= (c1 + ((- (1 - r)) * a1)) - (r * c1) by EUCLID:40
.= (c1 + (((- 1) + r) * a1)) - (r * c1)
.= (c1 + (((- 1) * a1) + (r * a1))) - (r * c1) by EUCLID:33
.= (c1 + ((- a1) + (r * a1))) - (r * c1) by EUCLID:39
.= ((c1 + (- a1)) + (r * a1)) - (r * c1) by EUCLID:26
.= ((c1 - a1) + (r * a1)) - (r * c1) by EUCLID:41
.= ((c1 - a1) + (r * a1)) + (- (r * c1)) by EUCLID:41
.= ((c1 - a1) + (r * a1)) + (r * (- c1)) by EUCLID:40
.= (c1 - a1) + ((r * a1) + (r * (- c1))) by EUCLID:26
.= (c1 - a1) + (r * (a1 + (- c1))) by EUCLID:32
.= (c1 - a1) + (r * (- (c1 - a1))) by EUCLID:44
.= (c1 - a1) + (- (r * (c1 - a1))) by EUCLID:40
.= (c1 - a1) + ((- r) * (c1 - a1)) by EUCLID:40
.= (1 * (c1 - a1)) + ((- r) * (c1 - a1)) by EUCLID:29
.= (1 + (- r)) * (c1 - a1) by EUCLID:33
.= (1 - r) * (c1 - a1) ;
then (1 - r) * |((b1 - a1),(c1 - a1))| = 0 by A20, EUCLID_2:20;
then A26: |((b1 - a1),(c1 - a1))| = 0 by A25, XCMPLX_1:6;
0. (TOP-REAL n) = (((1 - r) * a1) + (r * c1)) + (- (((1 - s) * a1) + (s * b1))) by A15, A24, EUCLID:36
.= (((1 - r) * a1) + (- (((1 - s) * a1) + (s * b1)))) + (r * c1) by EUCLID:26
.= (((1 - r) * a1) + ((- ((1 - s) * a1)) - (s * b1))) + (r * c1) by EUCLID:38
.= (((1 - r) * a1) + ((- ((1 - s) * a1)) + (- (s * b1)))) + (r * c1) by EUCLID:41
.= ((((1 - r) * a1) + (- ((1 - s) * a1))) + (- (s * b1))) + (r * c1) by EUCLID:26
.= ((((1 - r) * a1) + ((- (1 - s)) * a1)) + (- (s * b1))) + (r * c1) by EUCLID:40
.= ((((1 - r) + (- (1 - s))) * a1) + (- (s * b1))) + (r * c1) by EUCLID:33
.= (((s + (- r)) * a1) + (- (s * b1))) + (r * c1)
.= (((s * a1) + ((- r) * a1)) + (- (s * b1))) + (r * c1) by EUCLID:33
.= (((s * a1) + (- (s * b1))) + ((- r) * a1)) + (r * c1) by EUCLID:26
.= (((s * a1) + (s * (- b1))) + ((- r) * a1)) + (r * c1) by EUCLID:40
.= ((s * (a1 + (- b1))) + ((- r) * a1)) + (r * c1) by EUCLID:32
.= ((s * (- (b1 - a1))) + ((- r) * a1)) + (r * c1) by EUCLID:44
.= (s * (- (b1 - a1))) + (((- r) * a1) + (r * c1)) by EUCLID:26
.= (s * (- (b1 - a1))) + ((r * c1) + (- (r * a1))) by EUCLID:40
.= (s * (- (b1 - a1))) + ((r * c1) + (r * (- a1))) by EUCLID:40
.= (s * (- (b1 - a1))) + (r * (c1 + (- a1))) by EUCLID:32
.= (s * (- (b1 - a1))) + (r * (c1 - a1)) by EUCLID:41 ;
then A27: 0 = |(((s * (- (b1 - a1))) + (r * (c1 - a1))),((s * (- (b1 - a1))) + (r * (c1 - a1))))| by EUCLID_2:34
.= (|((s * (- (b1 - a1))),(s * (- (b1 - a1))))| + (2 * |((s * (- (b1 - a1))),(r * (c1 - a1)))|)) + |((r * (c1 - a1)),(r * (c1 - a1)))| by EUCLID_2:30 ;
A28: |((s * (- (b1 - a1))),(s * (- (b1 - a1))))| = s * |((- (b1 - a1)),(s * (- (b1 - a1))))| by EUCLID_2:19
.= s * (s * |((- (b1 - a1)),(- (b1 - a1)))|) by EUCLID_2:20
.= s * (s * |((b1 - a1),(b1 - a1))|) by EUCLID_2:23
.= (s * s) * |((b1 - a1),(b1 - a1))| ;
A29: |((r * (c1 - a1)),(r * (c1 - a1)))| = r * |((c1 - a1),(r * (c1 - a1)))| by EUCLID_2:19
.= r * (r * |((c1 - a1),(c1 - a1))|) by EUCLID_2:20
.= (r * r) * |((c1 - a1),(c1 - a1))| ;
A30: |((s * (- (b1 - a1))),(r * (c1 - a1)))| = s * |((- (b1 - a1)),(r * (c1 - a1)))| by EUCLID_2:19
.= s * (r * |((- (b1 - a1)),(c1 - a1))|) by EUCLID_2:20
.= s * (r * (- |((b1 - a1),(c1 - a1))|)) by EUCLID_2:21
.= 0 by A26 ;
A31: s * s >= 0 by XREAL_1:63;
A32: r * r >= 0 by XREAL_1:63;
A33: |((b1 - a1),(b1 - a1))| >= 0 by EUCLID_2:35;
A34: |((c1 - a1),(c1 - a1))| >= 0 by EUCLID_2:35;
A35: s * s <> 0
proof
assume s * s = 0 ; ::_thesis: contradiction
then s = 0 by XCMPLX_1:6;
then p = (1 * a1) + (0. (TOP-REAL n)) by A15, EUCLID:29
.= 1 * a1 by EUCLID:27
.= a1 by EUCLID:29 ;
then not p in {p} by A3, XBOOLE_0:def_5;
hence contradiction by TARSKI:def_1; ::_thesis: verum
end;
thus contradiction by A28, A29, A27, A30, A31, A32, A33, A34, A35, A21, XREAL_1:71; ::_thesis: verum
end;
A36: not p in LSeg (c1,b1)
proof
assume A37: p in LSeg (c1,b1) ; ::_thesis: contradiction
reconsider cc1 = c1 as Point of (TOP-REAL n1) ;
consider r being Real such that
A38: ( 0 <= r & r <= 1 & p = ((1 - r) * bb1) + (r * cc1) ) by A37, JGRAPH_1:35;
A39: 1 - r <> 0
proof
assume 1 - r = 0 ; ::_thesis: contradiction
then p = (0. (TOP-REAL n)) + (1 * c1) by A38, EUCLID:29
.= (0. (TOP-REAL n)) + c1 by EUCLID:29
.= c1 by EUCLID:27 ;
hence contradiction by A19, A20, TARSKI:def_1; ::_thesis: verum
end;
set q2 = c1 - b1;
c1 - p = (c1 - ((1 - r) * b1)) - (r * c1) by A38, EUCLID:46
.= (c1 + (- ((1 - r) * b1))) - (r * c1) by EUCLID:41
.= (c1 + ((- (1 - r)) * b1)) - (r * c1) by EUCLID:40
.= (c1 + (((- 1) + r) * b1)) - (r * c1)
.= (c1 + (((- 1) * b1) + (r * b1))) - (r * c1) by EUCLID:33
.= (c1 + ((- b1) + (r * b1))) - (r * c1) by EUCLID:39
.= ((c1 + (- b1)) + (r * b1)) - (r * c1) by EUCLID:26
.= ((c1 - b1) + (r * b1)) - (r * c1) by EUCLID:41
.= ((c1 - b1) + (r * b1)) + (- (r * c1)) by EUCLID:41
.= ((c1 - b1) + (r * b1)) + (r * (- c1)) by EUCLID:40
.= (c1 - b1) + ((r * b1) + (r * (- c1))) by EUCLID:26
.= (c1 - b1) + (r * (b1 + (- c1))) by EUCLID:32
.= (c1 - b1) + (r * (- (c1 - b1))) by EUCLID:44
.= (c1 - b1) + (- (r * (c1 - b1))) by EUCLID:40
.= (c1 - b1) + ((- r) * (c1 - b1)) by EUCLID:40
.= (1 * (c1 - b1)) + ((- r) * (c1 - b1)) by EUCLID:29
.= (1 + (- r)) * (c1 - b1) by EUCLID:33
.= (1 - r) * (c1 - b1) ;
then (1 - r) * |((b1 - a1),(c1 - b1))| = 0 by A20, EUCLID_2:20;
then A40: |((b1 - a1),(c1 - b1))| = 0 by A39, XCMPLX_1:6;
A41: 0. (TOP-REAL n) = (((1 + (- r)) * b1) + (r * c1)) + (- (((1 - s) * a1) + (s * b1))) by A38, A15, EUCLID:36
.= (((1 * b1) + ((- r) * b1)) + (r * c1)) + (- (((1 - s) * a1) + (s * b1))) by EUCLID:33
.= ((b1 + ((- r) * b1)) + (r * c1)) + (- (((1 - s) * a1) + (s * b1))) by EUCLID:29
.= (b1 + (((- r) * b1) + (r * c1))) + (- (((1 - s) * a1) + (s * b1))) by EUCLID:26
.= (b1 + ((- (r * b1)) + (r * c1))) + (- (((1 - s) * a1) + (s * b1))) by EUCLID:40
.= (b1 + ((r * (- b1)) + (r * c1))) + (- (((1 - s) * a1) + (s * b1))) by EUCLID:40
.= (b1 + (r * ((- b1) + c1))) + (- (((1 - s) * a1) + (s * b1))) by EUCLID:32
.= (b1 + (r * (c1 - b1))) + (- (((1 - s) * a1) + (s * b1))) by EUCLID:41
.= (b1 + (- (((1 - s) * a1) + (s * b1)))) + (r * (c1 - b1)) by EUCLID:26
.= (b1 + ((- 1) * ((s * b1) + ((1 - s) * a1)))) + (r * (c1 - b1)) by EUCLID:39
.= (b1 + (((- 1) * (s * b1)) + ((- 1) * ((1 - s) * a1)))) + (r * (c1 - b1)) by EUCLID:32
.= (b1 + ((((- 1) * s) * b1) + ((- 1) * ((1 - s) * a1)))) + (r * (c1 - b1)) by EUCLID:30
.= (b1 + (((- s) * b1) + (- ((1 - s) * a1)))) + (r * (c1 - b1)) by EUCLID:39
.= ((b1 + ((- s) * b1)) + (- ((1 - s) * a1))) + (r * (c1 - b1)) by EUCLID:26
.= (((1 * b1) + ((- s) * b1)) + (- ((1 - s) * a1))) + (r * (c1 - b1)) by EUCLID:29
.= (((1 + (- s)) * b1) + (- ((1 - s) * a1))) + (r * (c1 - b1)) by EUCLID:33
.= (((1 - s) * b1) + ((1 - s) * (- a1))) + (r * (c1 - b1)) by EUCLID:40
.= ((1 - s) * (b1 + (- a1))) + (r * (c1 - b1)) by EUCLID:32
.= ((1 - s) * (b1 - a1)) + (r * (c1 - b1)) by EUCLID:41 ;
set t = 1 - s;
A42: 0 = |((((1 - s) * (b1 - a1)) + (r * (c1 - b1))),(((1 - s) * (b1 - a1)) + (r * (c1 - b1))))| by A41, EUCLID_2:34
.= (|(((1 - s) * (b1 - a1)),((1 - s) * (b1 - a1)))| + (2 * |(((1 - s) * (b1 - a1)),(r * (c1 - b1)))|)) + |((r * (c1 - b1)),(r * (c1 - b1)))| by EUCLID_2:30 ;
A43: |(((1 - s) * (b1 - a1)),((1 - s) * (b1 - a1)))| = (1 - s) * |((b1 - a1),((1 - s) * (b1 - a1)))| by EUCLID_2:19
.= (1 - s) * ((1 - s) * |((b1 - a1),(b1 - a1))|) by EUCLID_2:20
.= ((1 - s) * (1 - s)) * |((b1 - a1),(b1 - a1))| ;
A44: |((r * (c1 - b1)),(r * (c1 - b1)))| = r * |((c1 - b1),(r * (c1 - b1)))| by EUCLID_2:19
.= r * (r * |((c1 - b1),(c1 - b1))|) by EUCLID_2:20
.= (r * r) * |((c1 - b1),(c1 - b1))| ;
A45: |(((1 - s) * (b1 - a1)),(r * (c1 - b1)))| = (1 - s) * |((b1 - a1),(r * (c1 - b1)))| by EUCLID_2:19
.= (1 - s) * (r * |((b1 - a1),(c1 - b1))|) by EUCLID_2:20
.= 0 by A40 ;
A46: (1 - s) * (1 - s) >= 0 by XREAL_1:63;
A47: r * r >= 0 by XREAL_1:63;
A48: |((b1 - a1),(b1 - a1))| >= 0 by EUCLID_2:35;
A49: |((c1 - b1),(c1 - b1))| >= 0 by EUCLID_2:35;
A50: (1 - s) * (1 - s) <> 0
proof
assume (1 - s) * (1 - s) = 0 ; ::_thesis: contradiction
then 1 - s = 0 by XCMPLX_1:6;
then p = (0. (TOP-REAL n)) + (1 * b1) by A15, EUCLID:29
.= 1 * b1 by EUCLID:27
.= b1 by EUCLID:29 ;
then not p in {p} by A3, XBOOLE_0:def_5;
hence contradiction by TARSKI:def_1; ::_thesis: verum
end;
thus contradiction by A50, A21, A43, A44, A42, A45, A46, A47, A48, A49, XREAL_1:71; ::_thesis: verum
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 a,b are_connected by BORSUK_6:42; ::_thesis: verum
end;
suppose not p in LSeg (a1,b1) ; ::_thesis: a,b are_connected
hence a,b are_connected by A4; ::_thesis: verum
end;
end;
end;
suppose a1 = b1 ; ::_thesis: a,b are_connected
hence a,b are_connected ; ::_thesis: verum
end;
end;
end;
hence (TOP-REAL n) | S is pathwise_connected by BORSUK_2:def_3; ::_thesis: verum
end;
theorem Th10: :: TOPALG_6:10
for T being non empty TopSpace
for t being Point of T
for n being Nat
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 T be non empty TopSpace; ::_thesis: for t being Point of T
for n being Nat
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
let t be Point of T; ::_thesis: for n being Nat
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
let n be Nat; ::_thesis: 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
let S be non empty Subset of T; ::_thesis: ( n >= 2 & S = ([#] T) \ {t} & TOP-REAL n,T are_homeomorphic implies T | S is pathwise_connected )
assume A1: ( n >= 2 & S = ([#] T) \ {t} & TOP-REAL n,T are_homeomorphic ) ; ::_thesis: T | S is pathwise_connected
then consider f being 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 being set 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; ::_thesis: verum
end;
registration
let n be Element of NAT ;
let p, q be Point of (TOP-REAL n);
cluster TPlane (p,q) -> convex ;
correctness
coherence
TPlane (p,q) is convex ;
proof
set P = Plane (p,q);
for w1, w2 being Point of (TOP-REAL n) st w1 in Plane (p,q) & w2 in Plane (p,q) holds
LSeg (w1,w2) c= Plane (p,q)
proof
let w1, w2 be Point of (TOP-REAL n); ::_thesis: ( w1 in Plane (p,q) & w2 in Plane (p,q) implies LSeg (w1,w2) c= Plane (p,q) )
assume A1: ( w1 in Plane (p,q) & w2 in Plane (p,q) ) ; ::_thesis: LSeg (w1,w2) c= Plane (p,q)
reconsider n0 = n as Nat ;
reconsider p0 = p, q0 = q as Point of (TOP-REAL n0) ;
A2: Plane (p,q) = { y where y is Point of (TOP-REAL n0) : |(p0,(y - q0))| = 0 } by MFOLD_2:def_2;
consider v1 being Point of (TOP-REAL n0) such that
A3: ( w1 = v1 & |(p0,(v1 - q0))| = 0 ) by A1, A2;
consider v2 being Point of (TOP-REAL n0) such that
A4: ( w2 = v2 & |(p0,(v2 - q0))| = 0 ) by A1, A2;
for x being set st x in LSeg (w1,w2) holds
x in Plane (p,q)
proof
let x be set ; ::_thesis: ( x in LSeg (w1,w2) implies x in Plane (p,q) )
assume A5: x in LSeg (w1,w2) ; ::_thesis: x in Plane (p,q)
then reconsider w = x as Point of (TOP-REAL n0) ;
reconsider r1 = 1 as real number ;
consider r being Real such that
A6: ( 0 <= r & r <= 1 & w = ((1 - r) * w1) + (r * w2) ) by A5, JGRAPH_1:35;
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, EUCLID:49
.= (((1 - r) * w1) - ((1 - r) * q)) + ((r * w2) - (r * q)) by EUCLID:49
.= (((1 - r) * w1) + (- ((1 - r) * q))) + ((r * w2) - (r * q)) by EUCLID:41
.= (((1 - r) * w1) + ((- (1 - r)) * q)) + ((r * w2) - (r * q)) by EUCLID:40
.= (((1 - r) * w1) + ((r - 1) * q)) + ((r * w2) - (r * q))
.= (((1 - r) * w1) + ((r * q) - (r1 * q))) + ((r * w2) - (r * q)) by EUCLID:50
.= ((1 - r) * w1) + (((r * q) - (r1 * q)) + ((r * w2) - (r * q))) by EUCLID:26
.= ((1 - r) * w1) + (((r * q) - (r1 * q)) + ((r * w2) + (- (r * q)))) by EUCLID:41
.= ((1 - r) * w1) + (((r * q) + (- (r1 * q))) + ((r * w2) + (- (r * q)))) by EUCLID:41
.= ((1 - r) * w1) + ((((- (r1 * q)) + (r * q)) + (- (r * q))) + (r * w2)) by EUCLID:26
.= ((1 - r) * w1) + (((- (r1 * q)) + ((r * q) + (- (r * q)))) + (r * w2)) by EUCLID:26
.= ((1 - r) * w1) + (((- (r1 * q)) + ((r * q) - (r * q))) + (r * w2)) by EUCLID:41
.= ((1 - r) * w1) + (((- (r1 * q)) + (0. (TOP-REAL n))) + (r * w2)) by EUCLID:42
.= ((1 - r) * w1) + ((- (r1 * q)) + (r * w2)) by EUCLID:27
.= (((1 - r) * w1) + (r * w2)) + (- (r1 * q)) by EUCLID:26
.= w + (- q0) by A6, EUCLID:29
.= w - q0 by EUCLID:41 ;
0 = |(p0,((1 - r) * (v1 - q0)))| + |(p0,(r * (v2 - q0)))| by A7, A8
.= |(p0,(w - q0))| by A9, EUCLID_2:26 ;
hence x in Plane (p,q) by A2; ::_thesis: verum
end;
hence LSeg (w1,w2) c= Plane (p,q) by TARSKI:def_3; ::_thesis: verum
end;
then Plane (p,q) is convex Subset of (TOP-REAL n) by RLTOPSP1:22;
then [#] ((TOP-REAL n) | (Plane (p,q))) 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 TPlane (p,q) is convex by TOPALG_2:def_1; ::_thesis: verum
end;
end;
begin
definition
let T be non empty TopSpace;
attrT is having_trivial_Fundamental_Group means :Def1: :: TOPALG_6:def 1
for t being Point of T holds pi_1 (T,t) is trivial ;
end;
:: deftheorem Def1 defines having_trivial_Fundamental_Group TOPALG_6:def_1_:_
for T being non empty TopSpace holds
( T is having_trivial_Fundamental_Group iff for t being Point of T holds pi_1 (T,t) is trivial );
definition
let T be non empty TopSpace;
attrT is simply_connected means :Def2: :: TOPALG_6:def 2
( T is having_trivial_Fundamental_Group & T is pathwise_connected );
end;
:: deftheorem Def2 defines simply_connected TOPALG_6:def_2_:_
for T being non empty TopSpace holds
( T is simply_connected iff ( T is having_trivial_Fundamental_Group & T is pathwise_connected ) );
registration
cluster non empty TopSpace-like simply_connected -> non empty pathwise_connected having_trivial_Fundamental_Group for TopStruct ;
coherence
for b1 being non empty TopSpace st b1 is simply_connected holds
( b1 is having_trivial_Fundamental_Group & b1 is pathwise_connected ) by Def2;
cluster non empty TopSpace-like pathwise_connected having_trivial_Fundamental_Group -> non empty simply_connected for TopStruct ;
coherence
for b1 being non empty TopSpace st b1 is having_trivial_Fundamental_Group & b1 is pathwise_connected holds
b1 is simply_connected by Def2;
end;
theorem Th11: :: TOPALG_6:11
for T being non empty TopSpace st T is having_trivial_Fundamental_Group holds
for t being Point of T
for P, Q being Loop of t holds P,Q are_homotopic
proof
let T be non empty TopSpace; ::_thesis: ( T is having_trivial_Fundamental_Group implies for t being Point of T
for P, Q being Loop of t holds P,Q are_homotopic )
assume A1: T is having_trivial_Fundamental_Group ; ::_thesis: for t being Point of T
for P, Q being Loop of t holds P,Q are_homotopic
let t be Point of T; ::_thesis: for P, Q being Loop of t holds P,Q are_homotopic
let P, Q be Loop of t; ::_thesis: P,Q are_homotopic
set E = EqRel (T,t);
A2: pi_1 (T,t) is trivial by A1, Def1;
( Class ((EqRel (T,t)),P) in the carrier of (pi_1 (T,t)) & Class ((EqRel (T,t)),Q) in the carrier of (pi_1 (T,t)) ) by TOPALG_1:47;
then Class ((EqRel (T,t)),P) = Class ((EqRel (T,t)),Q) by A2, ZFMISC_1:def_10;
hence P,Q are_homotopic by TOPALG_1:46; ::_thesis: verum
end;
registration
let n be Nat;
cluster TOP-REAL n -> having_trivial_Fundamental_Group ;
coherence
TOP-REAL n is having_trivial_Fundamental_Group
proof
let o be Point of (TOP-REAL n); :: according to TOPALG_6:def_1 ::_thesis: pi_1 ((TOP-REAL n),o) is trivial
thus pi_1 ((TOP-REAL n),o) is trivial ; ::_thesis: verum
end;
end;
registration
cluster non empty trivial TopSpace-like -> non empty having_trivial_Fundamental_Group for TopStruct ;
coherence
for b1 being non empty TopSpace st b1 is trivial holds
b1 is having_trivial_Fundamental_Group
proof
let T be non empty TopSpace; ::_thesis: ( T is trivial implies T is having_trivial_Fundamental_Group )
assume A1: T is trivial ; ::_thesis: T is having_trivial_Fundamental_Group
let t be Point of T; :: according to TOPALG_6:def_1 ::_thesis: pi_1 (T,t) is trivial
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 pi_1 (T,t) is trivial ; ::_thesis: verum
end;
end;
theorem Th12: :: TOPALG_6:12
for T being non empty TopSpace holds
( 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
let T be non empty TopSpace; ::_thesis: ( 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) ) ) )
hereby ::_thesis: ( ( 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) ) ) ) implies T is simply_connected )
assume A1: T is simply_connected ; ::_thesis: 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) ) )
let t1, t2 be Point of T; ::_thesis: ( 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 A2: t1,t2 are_connected by A1, BORSUK_2:def_3; ::_thesis: for P, Q being Path of t1,t2 holds Class ((EqRel (T,t1,t2)),P) = Class ((EqRel (T,t1,t2)),Q)
let P, Q be Path of t1,t2; ::_thesis: Class ((EqRel (T,t1,t2)),P) = Class ((EqRel (T,t1,t2)),Q)
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), the constant Loop of t1 are_homotopic by A1, Th11;
then A4: (P + (- Q)) + Q, the constant Loop of t1 + Q are_homotopic by A1, BORSUK_6:76;
the constant Loop of t1 + 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 ((EqRel (T,t1,t2)),P) = Class ((EqRel (T,t1,t2)),Q) by A2, TOPALG_1:46; ::_thesis: verum
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) ) ) ; ::_thesis: T is simply_connected
thus T is having_trivial_Fundamental_Group :: according to TOPALG_6:def_2 ::_thesis: T is pathwise_connected
proof
let t be Point of T; :: according to TOPALG_6:def_1 ::_thesis: pi_1 (T,t) is trivial
let x, y be Element of (pi_1 (T,t)); :: according to STRUCT_0:def_10 ::_thesis: x = y
( 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 x = y by A5; ::_thesis: verum
end;
thus T is pathwise_connected by A5, BORSUK_2:def_3; ::_thesis: verum
end;
registration
let T be non empty having_trivial_Fundamental_Group TopSpace;
let t be Point of T;
cluster FundamentalGroup (T,t) -> trivial ;
coherence
pi_1 (T,t) is trivial by Def1;
end;
theorem Th13: :: TOPALG_6:13
for S, T being non empty TopSpace st S,T are_homeomorphic & S is having_trivial_Fundamental_Group holds
T is having_trivial_Fundamental_Group
proof
let S, T be non empty TopSpace; ::_thesis: ( S,T are_homeomorphic & S is having_trivial_Fundamental_Group implies T is having_trivial_Fundamental_Group )
given f being Function of S,T such that A1: f is being_homeomorphism ; :: according to T_0TOPSP:def_1 ::_thesis: ( not S is having_trivial_Fundamental_Group or T is having_trivial_Fundamental_Group )
assume A2: for s being Point of S holds pi_1 (S,s) is trivial ; :: according to TOPALG_6:def_1 ::_thesis: T is having_trivial_Fundamental_Group
let t be Point of T; :: according to TOPALG_6:def_1 ::_thesis: pi_1 (T,t) is trivial
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 pi_1 (T,t) is trivial by A3, A4, KNASTER:11, TOPREALC:1; ::_thesis: verum
end;
theorem Th14: :: TOPALG_6:14
for S, T being non empty TopSpace st S,T are_homeomorphic & S is simply_connected holds
T is simply_connected
proof
let S, T be non empty TopSpace; ::_thesis: ( S,T are_homeomorphic & S is simply_connected implies T is simply_connected )
assume ( S,T are_homeomorphic & S is simply_connected ) ; ::_thesis: T is simply_connected
hence ( T is having_trivial_Fundamental_Group & T is pathwise_connected ) by Th13, TOPALG_3:9; :: according to TOPALG_6:def_2 ::_thesis: verum
end;
theorem Th15: :: TOPALG_6:15
for T being non empty having_trivial_Fundamental_Group TopSpace
for t being Point of T
for P1, P2 being Loop of t holds P1,P2 are_homotopic
proof
let T be non empty having_trivial_Fundamental_Group TopSpace; ::_thesis: for t being Point of T
for P1, P2 being Loop of t holds P1,P2 are_homotopic
let t be Point of T; ::_thesis: for P1, P2 being Loop of t holds P1,P2 are_homotopic
let P1, P2 be Loop of t; ::_thesis: P1,P2 are_homotopic
( 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 P1,P2 are_homotopic by TOPALG_1:46; ::_thesis: verum
end;
definition
let T be non empty TopSpace;
let t be Point of T;
let l be Loop of t;
attrl is nullhomotopic means :Def3: :: TOPALG_6:def 3
ex c being constant Loop of t st l,c are_homotopic ;
end;
:: deftheorem Def3 defines nullhomotopic TOPALG_6:def_3_:_
for T being non empty TopSpace
for t being Point of T
for l being Loop of t holds
( l is nullhomotopic iff ex c being constant Loop of t st l,c are_homotopic );
registration
let T be non empty TopSpace;
let t be Point of T;
cluster constant -> nullhomotopic for Path of t,t;
coherence
for b1 being Loop of t st b1 is constant holds
b1 is nullhomotopic
proof
let l be Loop of t; ::_thesis: ( l is constant implies l is nullhomotopic )
assume l is constant ; ::_thesis: l is nullhomotopic
then reconsider l = l as constant Loop of t ;
take l ; :: according to TOPALG_6:def_3 ::_thesis: l,l are_homotopic
thus l,l are_homotopic by BORSUK_6:88; ::_thesis: verum
end;
end;
registration
let T be non empty TopSpace;
let t be Point of T;
cluster non empty Relation-like the carrier of I[01] -defined the carrier of T -valued Function-like constant total quasi_total continuous for Path of t,t;
existence
ex b1 being Loop of t st b1 is constant
proof
reconsider l = I[01] --> t as constant Loop of t by JORDAN:41;
take l ; ::_thesis: l is constant
thus l is constant ; ::_thesis: verum
end;
end;
theorem Th16: :: TOPALG_6:16
for T, U being non empty TopSpace
for t being Point of T
for f being Loop of t
for g being continuous Function of T,U st f is nullhomotopic holds
g * f is nullhomotopic
proof
let T, U be non empty TopSpace; ::_thesis: for t being Point of T
for f being Loop of t
for g being continuous Function of T,U st f is nullhomotopic holds
g * f is nullhomotopic
let t be Point of T; ::_thesis: for f being Loop of t
for g being continuous Function of T,U st f is nullhomotopic holds
g * f is nullhomotopic
let f be Loop of t; ::_thesis: for g being continuous Function of T,U st f is nullhomotopic holds
g * f is nullhomotopic
let g be continuous Function of T,U; ::_thesis: ( f is nullhomotopic implies g * f is nullhomotopic )
given c being constant Loop of t such that A1: f,c are_homotopic ; :: according to TOPALG_6:def_3 ::_thesis: g * f is nullhomotopic
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, BORSUK_2:def_7;
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 ; :: according to TOPALG_6:def_3 ::_thesis: g * f,d are_homotopic
take G ; :: according to BORSUK_2:def_7 ::_thesis: ( G is continuous & ( for b1 being Element of the carrier of I[01] holds
( G . (b1,0) = (g * f) . b1 & G . (b1,1) = d . b1 & G . (0,b1) = g . t & G . (1,b1) = g . t ) ) )
thus G is continuous by A2; ::_thesis: for b1 being Element of the carrier of I[01] holds
( G . (b1,0) = (g * f) . b1 & G . (b1,1) = d . b1 & G . (0,b1) = g . t & G . (1,b1) = g . t )
let s be Point of I[01]; ::_thesis: ( G . (s,0) = (g * f) . s & G . (s,1) = d . s & G . (0,s) = g . t & G . (1,s) = g . t )
reconsider j0 = 0 , j1 = 1 as Point of I[01] by BORSUK_1:def_14, BORSUK_1:def_15;
set I = the carrier of I[01];
A4: the carrier of [:I[01],I[01]:] = [: the carrier of I[01], the carrier of I[01]:] 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 ; ::_thesis: ( G . (s,1) = d . s & G . (0,s) = g . t & G . (1,s) = g . t )
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 ; ::_thesis: ( G . (0,s) = g . t & G . (1,s) = g . t )
thus G . (0,s) = g . (F . (j0,s)) by A4, BINOP_1:18
.= g . t by A3 ; ::_thesis: G . (1,s) = g . t
thus G . (1,s) = g . (F . (j1,s)) by A4, BINOP_1:18
.= g . t by A3 ; ::_thesis: verum
end;
registration
let T, U be non empty TopSpace;
let t be Point of T;
let f be nullhomotopic Loop of t;
let g be continuous Function of T,U;
clusterf * g -> nullhomotopic for Loop of g . t;
coherence
for b1 being Loop of g . t st b1 = g * f holds
b1 is nullhomotopic by Th16;
end;
registration
let T be non empty having_trivial_Fundamental_Group TopSpace;
let t be Point of T;
cluster -> nullhomotopic for Path of t,t;
coherence
for b1 being Loop of t holds b1 is nullhomotopic
proof
let l be Loop of t; ::_thesis: l is nullhomotopic
reconsider c = I[01] --> t as constant Loop of t by JORDAN:41;
take c ; :: according to TOPALG_6:def_3 ::_thesis: l,c are_homotopic
thus l,c are_homotopic by Th15; ::_thesis: verum
end;
end;
theorem Th17: :: TOPALG_6:17
for T being non empty TopSpace st ( for t being Point of T
for f being Loop of t holds f is nullhomotopic ) holds
T is having_trivial_Fundamental_Group
proof
let T be non empty TopSpace; ::_thesis: ( ( for t being Point of T
for f being Loop of t holds f is nullhomotopic ) implies T is having_trivial_Fundamental_Group )
assume A1: for t being Point of T
for f being Loop of t holds f is nullhomotopic ; ::_thesis: T is having_trivial_Fundamental_Group
for t being Point of T holds pi_1 (T,t) is trivial
proof
let t be Point of T; ::_thesis: pi_1 (T,t) is trivial
set C = the constant Loop of t;
the carrier of (pi_1 (T,t)) = {(Class ((EqRel (T,t)), the constant Loop of t))}
proof
set E = EqRel (T,t);
hereby :: according to TARSKI:def_3,XBOOLE_0:def_10 ::_thesis: {(Class ((EqRel (T,t)), the constant Loop of t))} c= the carrier of (pi_1 (T,t))
let x be set ; ::_thesis: ( x in the carrier of (pi_1 (T,t)) implies x in {(Class ((EqRel (T,t)), the constant Loop of t))} )
assume x in the carrier of (pi_1 (T,t)) ; ::_thesis: x in {(Class ((EqRel (T,t)), the constant Loop of t))}
then consider P being Loop of t such that
A2: x = Class ((EqRel (T,t)),P) by TOPALG_1:47;
P is nullhomotopic by A1;
then consider C1 being constant Loop of t such that
A3: P,C1 are_homotopic by Def3;
C1, the constant Loop of t are_homotopic by Th5;
then P, the constant Loop of t are_homotopic by A3, BORSUK_6:79;
then x = Class ((EqRel (T,t)), the constant Loop of t) by A2, TOPALG_1:46;
hence x in {(Class ((EqRel (T,t)), the constant Loop of t))} by TARSKI:def_1; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(Class ((EqRel (T,t)), the constant Loop of t))} or x in the carrier of (pi_1 (T,t)) )
assume x in {(Class ((EqRel (T,t)), the constant Loop of t))} ; ::_thesis: x in the carrier of (pi_1 (T,t))
then A4: x = Class ((EqRel (T,t)), the constant Loop of t) by TARSKI:def_1;
the constant Loop of t in Loops t by TOPALG_1:def_1;
then x in Class (EqRel (T,t)) by A4, EQREL_1:def_3;
hence x in the carrier of (pi_1 (T,t)) by TOPALG_1:def_5; ::_thesis: verum
end;
hence pi_1 (T,t) is trivial ; ::_thesis: verum
end;
hence T is having_trivial_Fundamental_Group by Def1; ::_thesis: verum
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
coherence
TPlane (p,q) is having_trivial_Fundamental_Group ;
proof
for t being Point of (TPlane (p,q)) holds pi_1 ((TPlane (p,q)),t) is trivial ;
hence TPlane (p,q) is having_trivial_Fundamental_Group by Def1; ::_thesis: verum
end;
end;
theorem Th18: :: TOPALG_6:18
for T being non empty TopSpace
for t being Point of T
for S being non empty SubSpace of T
for s being Point of S
for f being Loop of t
for g being Loop of s st t = s & f = g & g is nullhomotopic holds
f is nullhomotopic
proof
let T be non empty TopSpace; ::_thesis: for t being Point of T
for S being non empty SubSpace of T
for s being Point of S
for f being Loop of t
for g being Loop of s st t = s & f = g & g is nullhomotopic holds
f is nullhomotopic
let t be Point of T; ::_thesis: for S being non empty SubSpace of T
for s being Point of S
for f being Loop of t
for g being Loop of s st t = s & f = g & g is nullhomotopic holds
f is nullhomotopic
let S be non empty SubSpace of T; ::_thesis: for s being Point of S
for f being Loop of t
for g being Loop of s st t = s & f = g & g is nullhomotopic holds
f is nullhomotopic
let s be Point of S; ::_thesis: for f being Loop of t
for g being Loop of s st t = s & f = g & g is nullhomotopic holds
f is nullhomotopic
let f be Loop of t; ::_thesis: for g being Loop of s st t = s & f = g & g is nullhomotopic holds
f is nullhomotopic
let g be Loop of s; ::_thesis: ( t = s & f = g & g is nullhomotopic implies f is nullhomotopic )
assume that
A1: ( t = s & f = g ) and
A2: g is nullhomotopic ; ::_thesis: f is nullhomotopic
consider c being constant Loop of s such that
A3: g,c are_homotopic by A2, Def3;
c = I[01] --> s by BORSUK_2:5
.= I[01] --> t by A1 ;
then reconsider c = c as constant Loop of t by JORDAN:41;
f,c are_homotopic by A1, A3, Th6;
hence f is nullhomotopic by Def3; ::_thesis: verum
end;
begin
definition
let T be TopStruct ;
let f be PartFunc of R^1,T;
attrf is parametrized-curve means :Def4: :: TOPALG_6:def 4
( dom f is interval Subset of REAL & ex S being SubSpace of R^1 ex g being Function of S,T st
( f = g & S = R^1 | (dom f) & g is continuous ) );
end;
:: deftheorem Def4 defines parametrized-curve TOPALG_6:def_4_:_
for T being TopStruct
for f being PartFunc of R^1,T holds
( f is parametrized-curve iff ( dom f is interval Subset of REAL & ex S being SubSpace of R^1 ex g being Function of S,T st
( f = g & S = R^1 | (dom f) & g is continuous ) ) );
Lm1: for T being TopStruct
for f being PartFunc of R^1,T st f = {} holds
f is parametrized-curve
proof
let T be TopStruct ; ::_thesis: for f being PartFunc of R^1,T st f = {} holds
f is parametrized-curve
let f be PartFunc of R^1,T; ::_thesis: ( f = {} implies f is parametrized-curve )
assume A1: f = {} ; ::_thesis: f is parametrized-curve
reconsider f = f as PartFunc of R^1,T ;
dom f = {} by A1;
then A2: dom f c= REAL by XBOOLE_1:2;
reconsider A = {} as Subset of R^1 by XBOOLE_1:2;
reconsider S = R^1 | A as SubSpace of R^1 ;
{} c= [:([#] S),([#] T):] ;
then reconsider g = f as Relation of ([#] S),([#] T) by A1;
A3: A = dom g ;
reconsider g = 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 f is parametrized-curve by A3, A2, Def4; ::_thesis: verum
end;
registration
let T be TopStruct ;
cluster Relation-like the carrier of R^1 -defined the carrier of T -valued Function-like parametrized-curve for Element of bool [: the carrier of R^1, the carrier of T:];
correctness
existence
ex b1 being PartFunc of R^1,T st b1 is parametrized-curve ;
proof
reconsider c = {} as PartFunc of R^1,T by XBOOLE_1:2;
take c ; ::_thesis: c is parametrized-curve
thus c is parametrized-curve by Lm1; ::_thesis: verum
end;
end;
theorem :: TOPALG_6:19
for T being TopStruct holds {} is parametrized-curve PartFunc of R^1,T by Lm1, XBOOLE_1:2;
definition
let T be TopStruct ;
func Curves T -> Subset of (PFuncs (REAL,([#] T))) equals :: TOPALG_6:def 5
{ f where f is Element of PFuncs (REAL,([#] T)) : f is parametrized-curve PartFunc of R^1,T } ;
coherence
{ f where f is Element of PFuncs (REAL,([#] T)) : f is parametrized-curve PartFunc of R^1,T } is Subset of (PFuncs (REAL,([#] T)))
proof
set C = { f where f is Element of PFuncs (REAL,([#] T)) : f is parametrized-curve PartFunc of R^1,T } ;
for x being set st x in { f where f is Element of PFuncs (REAL,([#] T)) : f is parametrized-curve PartFunc of R^1,T } holds
x in PFuncs (REAL,([#] T))
proof
let x be set ; ::_thesis: ( x in { f where f is Element of PFuncs (REAL,([#] T)) : f is parametrized-curve PartFunc of R^1,T } implies x in PFuncs (REAL,([#] T)) )
assume x in { f where f is Element of PFuncs (REAL,([#] T)) : f is parametrized-curve PartFunc of R^1,T } ; ::_thesis: x in PFuncs (REAL,([#] T))
then ex f being Element of PFuncs (REAL,([#] T)) st
( x = f & f is parametrized-curve PartFunc of R^1,T ) ;
hence x in PFuncs (REAL,([#] T)) ; ::_thesis: verum
end;
hence { f where f is Element of PFuncs (REAL,([#] T)) : f is parametrized-curve PartFunc of R^1,T } is Subset of (PFuncs (REAL,([#] T))) by TARSKI:def_3; ::_thesis: verum
end;
end;
:: deftheorem defines Curves TOPALG_6:def_5_:_
for T being TopStruct holds Curves T = { f where f is Element of PFuncs (REAL,([#] T)) : f is parametrized-curve PartFunc of R^1,T } ;
registration
let T be TopStruct ;
cluster Curves T -> non empty ;
coherence
not Curves T is empty
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 not Curves T is empty ; ::_thesis: verum
end;
end;
definition
let T be TopStruct ;
mode Curve of T is Element of Curves T;
correctness
;
end;
theorem Th20: :: TOPALG_6:20
for T being TopStruct
for f being parametrized-curve PartFunc of R^1,T holds f is Curve of T
proof
let T be TopStruct ; ::_thesis: for f being parametrized-curve PartFunc of R^1,T holds f is Curve of T
let f be parametrized-curve PartFunc of R^1,T; ::_thesis: f is Curve of T
reconsider f1 = f as Element of PFuncs (REAL,([#] T)) by PARTFUN1:45, TOPMETR:17;
f1 in Curves T ;
hence f is Curve of T ; ::_thesis: verum
end;
theorem Th21: :: TOPALG_6:21
for T being TopStruct holds {} is Curve of T
proof
let T be TopStruct ; ::_thesis: {} is Curve of T
reconsider f = {} as parametrized-curve PartFunc of R^1,T by Lm1, XBOOLE_1:2;
f is Curve of T by Th20;
hence {} is Curve of T ; ::_thesis: verum
end;
theorem Th22: :: TOPALG_6:22
for T being TopStruct
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 T be TopStruct ; ::_thesis: 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
let t1, t2 be Point of T; ::_thesis: for p being Path of t1,t2 st t1,t2 are_connected holds
p is Curve of T
let p be Path of t1,t2; ::_thesis: ( t1,t2 are_connected implies p is Curve of T )
assume t1,t2 are_connected ; ::_thesis: p is Curve of T
then A1: ( p is continuous & p . 0 = t1 & p . 1 = t2 ) by BORSUK_2:def_2;
percases ( not T is empty or T is empty ) ;
suppose not T is empty ; ::_thesis: p is Curve of T
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, TOPMETR:20;
rng p c= [#] T ;
then reconsider c = p as PartFunc of R^1,T by A3, RELSET_1:4;
reconsider c = 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 p is Curve of T ; ::_thesis: verum
end;
supposeA5: T is empty ; ::_thesis: p is Curve of T
then reconsider c = p as PartFunc of R^1,T ;
c = {} by A5;
then reconsider c = c as parametrized-curve PartFunc of R^1,T by Lm1;
c is Element of Curves T by Th20;
hence p is Curve of T ; ::_thesis: verum
end;
end;
end;
theorem Th23: :: TOPALG_6:23
for T being TopStruct
for c being Curve of T holds c is parametrized-curve PartFunc of R^1,T
proof
let T be TopStruct ; ::_thesis: for c being Curve of T holds c is parametrized-curve PartFunc of R^1,T
let c be Curve of T; ::_thesis: c is parametrized-curve PartFunc of R^1,T
c in { f where f is Element of PFuncs (REAL,([#] T)) : f is parametrized-curve PartFunc of R^1,T } ;
then consider f being Element of PFuncs (REAL,([#] T)) such that
A1: ( c = f & f is parametrized-curve PartFunc of R^1,T ) ;
thus c is parametrized-curve PartFunc of R^1,T by A1; ::_thesis: verum
end;
theorem Th24: :: TOPALG_6:24
for T being TopStruct
for c being Curve of T holds
( dom c c= REAL & rng c c= [#] T )
proof
let T be TopStruct ; ::_thesis: for c being Curve of T holds
( dom c c= REAL & rng c c= [#] T )
let c be Curve of T; ::_thesis: ( dom c c= REAL & rng c c= [#] T )
reconsider f = c as parametrized-curve PartFunc of R^1,T by Th23;
( dom f c= [#] R^1 & rng f c= [#] T ) ;
hence ( dom c c= REAL & rng c c= [#] T ) by TOPMETR:17; ::_thesis: verum
end;
registration
let T be TopStruct ;
let c be Curve of T;
cluster dom c -> real-membered ;
correctness
coherence
dom c is real-membered ;
proof
for x being set st x in dom c holds
x is real
proof
let x be set ; ::_thesis: ( x in dom c implies x is real )
assume A1: x in dom c ; ::_thesis: x is real
dom c c= REAL by Th24;
hence x is real by A1; ::_thesis: verum
end;
hence dom c is real-membered by MEMBERED:def_3; ::_thesis: verum
end;
end;
definition
let T be TopStruct ;
let c be Curve of T;
attrc is with_first_point means :Def6: :: TOPALG_6:def 6
dom c is left_end ;
attrc is with_last_point means :Def7: :: TOPALG_6:def 7
dom c is right_end ;
end;
:: deftheorem Def6 defines with_first_point TOPALG_6:def_6_:_
for T being TopStruct
for c being Curve of T holds
( c is with_first_point iff dom c is left_end );
:: deftheorem Def7 defines with_last_point TOPALG_6:def_7_:_
for T being TopStruct
for c being Curve of T holds
( c is with_last_point iff dom c is right_end );
definition
let T be TopStruct ;
let c be Curve of T;
attrc is with_endpoints means :Def8: :: TOPALG_6:def 8
( c is with_first_point & c is with_last_point );
end;
:: deftheorem Def8 defines with_endpoints TOPALG_6:def_8_:_
for T being TopStruct
for c being Curve of T holds
( c is with_endpoints iff ( c is with_first_point & c is with_last_point ) );
registration
let T be TopStruct ;
cluster with_first_point with_last_point -> with_endpoints for Element of Curves T;
correctness
coherence
for b1 being Curve of T st b1 is with_first_point & b1 is with_last_point holds
b1 is with_endpoints ;
by Def8;
cluster with_endpoints -> with_first_point with_last_point for Element of Curves T;
correctness
coherence
for b1 being Curve of T st b1 is with_endpoints holds
( b1 is with_first_point & b1 is with_last_point );
by Def8;
end;
registration
let T be non empty TopStruct ;
cluster Relation-like Function-like with_endpoints for Element of Curves T;
correctness
existence
ex b1 being Curve of T st b1 is with_endpoints ;
proof
set t = the Point of T;
set f = I[01] --> the Point of T;
A1: ( (I[01] --> the Point of T) . 0 = the Point of T & (I[01] --> the Point of T) . 1 = the Point of T ) by BORSUK_1:def_14, BORSUK_1:def_15, FUNCOP_1:7;
set p = the Path of the Point of T, the Point of T;
the Point of T, the Point of T are_connected by A1, BORSUK_2:def_1;
then reconsider c = the Path of the Point of T, the Point of T as Curve of T by Th22;
take c ; ::_thesis: c is with_endpoints
A2: dom c = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1;
set S = [.0,1.];
inf [.0,1.] = 0 by XXREAL_2:25;
then inf [.0,1.] in [.0,1.] by XXREAL_1:1;
hence dom c is left_end by A2, XXREAL_2:def_5; :: according to TOPALG_6:def_6,TOPALG_6:def_8 ::_thesis: c is with_last_point
sup [.0,1.] = 1 by XXREAL_2:29;
then sup [.0,1.] in [.0,1.] by XXREAL_1:1;
hence dom c is right_end by A2, XXREAL_2:def_6; :: according to TOPALG_6:def_7 ::_thesis: verum
end;
end;
registration
let T be non empty TopStruct ;
let c be with_first_point Curve of T;
cluster dom c -> non empty ;
correctness
coherence
not dom c is empty ;
proof
dom c is left_end by Def6;
hence not dom c is empty ; ::_thesis: verum
end;
cluster inf (dom c) -> real ;
correctness
coherence
inf (dom c) is real ;
proof
dom c is left_end by Def6;
hence inf (dom c) is real ; ::_thesis: verum
end;
end;
registration
let T be non empty TopStruct ;
let c be with_last_point Curve of T;
cluster dom c -> non empty ;
correctness
coherence
not dom c is empty ;
proof
dom c is right_end by Def7;
hence not dom c is empty ; ::_thesis: verum
end;
cluster sup (dom c) -> real ;
correctness
coherence
sup (dom c) is real ;
proof
dom c is right_end by Def7;
hence sup (dom c) is real ; ::_thesis: verum
end;
end;
registration
let T be non empty TopStruct ;
cluster with_first_point -> non empty for Element of Curves T;
coherence
for b1 being Curve of T st b1 is with_first_point holds
not b1 is empty
proof
let C be Curve of T; ::_thesis: ( C is with_first_point implies not C is empty )
assume ( C is with_first_point & C is empty ) ; ::_thesis: contradiction
then reconsider c = {} as with_first_point Curve of T ;
dom c is left_end ;
hence contradiction ; ::_thesis: verum
end;
cluster with_last_point -> non empty for Element of Curves T;
coherence
for b1 being Curve of T st b1 is with_last_point holds
not b1 is empty
proof
let C be Curve of T; ::_thesis: ( C is with_last_point implies not C is empty )
assume ( C is with_last_point & C is empty ) ; ::_thesis: contradiction
then reconsider c = {} as with_last_point Curve of T ;
dom c is right_end ;
hence contradiction ; ::_thesis: verum
end;
end;
definition
let T be non empty TopStruct ;
let c be with_first_point Curve of T;
func the_first_point_of c -> Point of T equals :: TOPALG_6:def 9
c . (inf (dom c));
correctness
coherence
c . (inf (dom c)) is Point of T;
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 c . (inf (dom c)) is Point of T by A1; ::_thesis: verum
end;
end;
:: deftheorem defines the_first_point_of TOPALG_6:def_9_:_
for T being non empty TopStruct
for c being with_first_point Curve of T holds the_first_point_of c = c . (inf (dom c));
definition
let T be non empty TopStruct ;
let c be with_last_point Curve of T;
func the_last_point_of c -> Point of T equals :: TOPALG_6:def 10
c . (sup (dom c));
correctness
coherence
c . (sup (dom c)) is Point of T;
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 c . (sup (dom c)) is Point of T by A1; ::_thesis: verum
end;
end;
:: deftheorem defines the_last_point_of TOPALG_6:def_10_:_
for T being non empty TopStruct
for c being with_last_point Curve of T holds the_last_point_of c = c . (sup (dom c));
theorem Th25: :: TOPALG_6:25
for T being non empty TopStruct
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 T be non empty TopStruct ; ::_thesis: 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
let t1, t2 be Point of T; ::_thesis: for p being Path of t1,t2 st t1,t2 are_connected holds
p is with_endpoints Curve of T
let p be Path of t1,t2; ::_thesis: ( t1,t2 are_connected implies p is with_endpoints Curve of T )
assume t1,t2 are_connected ; ::_thesis: p is with_endpoints Curve of T
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 by Def6;
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 by Def7;
hence p is with_endpoints Curve of T by A2; ::_thesis: verum
end;
theorem Th26: :: TOPALG_6:26
for T being non empty TopStruct
for c being Curve of T
for r1, r2 being real number holds c | [.r1,r2.] is Curve of T
proof
let T be non empty TopStruct ; ::_thesis: for c being Curve of T
for r1, r2 being real number holds c | [.r1,r2.] is Curve of T
let c be Curve of T; ::_thesis: for r1, r2 being real number holds c | [.r1,r2.] is Curve of T
let r1, r2 be real number ; ::_thesis: c | [.r1,r2.] is Curve of T
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 (f | [.r1,r2.]) is interval Subset of REAL by RELAT_1:61;
consider S being SubSpace of R^1 , g being 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
.= f | [.r1,r2.] ;
A7: (dom f) /\ [.r1,r2.] = dom (f | [.r1,r2.]) by RELAT_1:61;
reconsider K1 = K0 as Subset of (R^1 | A0) by A3;
reconsider D1 = dom (f | [.r1,r2.]) 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 = f | [.r1,r2.] as parametrized-curve PartFunc of R^1,T by A2, A5, A6, Def4;
c | [.r1,r2.] = f1 ;
hence c | [.r1,r2.] is Curve of T by Th20; ::_thesis: verum
end;
theorem Th27: :: TOPALG_6:27
for T being non empty TopStruct
for c being with_endpoints Curve of T holds dom c = [.(inf (dom c)),(sup (dom c)).]
proof
let T be non empty TopStruct ; ::_thesis: for c being with_endpoints Curve of T holds dom c = [.(inf (dom c)),(sup (dom c)).]
let c be with_endpoints Curve of T; ::_thesis: dom c = [.(inf (dom c)),(sup (dom c)).]
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 ext-real-membered left_end right_end interval set by Def6, Def7;
A = [.(min A),(max A).] by XXREAL_2:75;
hence dom c = [.(inf (dom c)),(sup (dom c)).] ; ::_thesis: verum
end;
theorem Th28: :: TOPALG_6:28
for T being non empty TopStruct
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 T be non empty TopStruct ; ::_thesis: 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
let c be with_endpoints Curve of T; ::_thesis: ( dom c = [.0,1.] implies c is Path of the_first_point_of c, the_last_point_of c )
assume A1: dom c = [.0,1.] ; ::_thesis: c is Path of the_first_point_of c, the_last_point_of c
set t1 = the_first_point_of c;
set t2 = the_last_point_of c;
reconsider f = c as parametrized-curve PartFunc of R^1,T by Th23;
consider S being SubSpace of R^1 , p being Function of S,T such that
A2: ( f = p & S = R^1 | (dom f) & p is continuous ) by Def4;
reconsider p = 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, TOPMETR:20;
A4: p . 0 = the_first_point_of c by A1, A2, XXREAL_2:25;
A5: p . 1 = the_last_point_of c by A2, A1, XXREAL_2:29;
then the_first_point_of c, the_last_point_of c are_connected by A2, A3, A4, BORSUK_2:def_1;
hence c is Path of the_first_point_of c, the_last_point_of c by A3, A4, A5, A2, BORSUK_2:def_2; ::_thesis: verum
end;
theorem Th29: :: TOPALG_6:29
for T being non empty TopStruct
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 T be non empty TopStruct ; ::_thesis: 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
let c be with_endpoints Curve of T; ::_thesis: c * (L[01] (0,1,(inf (dom c)),(sup (dom c)))) is Path of the_first_point_of c, the_last_point_of c
set t1 = the_first_point_of c;
set t2 = the_last_point_of c;
reconsider c0 = c as parametrized-curve PartFunc of R^1,T by Th23;
consider S being SubSpace of R^1 , g being Function of S,T such that
A1: ( c0 = g & S = R^1 | (dom c0) & g is continuous ) by Def4;
reconsider S = 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, TOPMETR: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 = the_first_point_of c by A8, A1, A6, A7, FUNCT_1:13;
A12: p . 1 = the_last_point_of c by A9, A1, A6, A7, FUNCT_1:13;
then the_first_point_of c, the_last_point_of c are_connected by A10, A11, BORSUK_2:def_1;
hence c * (L[01] (0,1,(inf (dom c)),(sup (dom c)))) is Path of the_first_point_of c, the_last_point_of c by A1, A10, A11, A12, BORSUK_2:def_2; ::_thesis: verum
end;
theorem :: TOPALG_6:30
for T being non empty TopStruct
for c being with_endpoints Curve of T
for 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 T be non empty TopStruct ; ::_thesis: for c being with_endpoints Curve of T
for 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 )
let c be with_endpoints Curve of T; ::_thesis: for 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 )
let t1, t2 be Point of T; ::_thesis: ( c * (L[01] (0,1,(inf (dom c)),(sup (dom c)))) is Path of t1,t2 & t1,t2 are_connected implies ( t1 = the_first_point_of c & t2 = the_last_point_of c ) )
assume A1: ( c * (L[01] (0,1,(inf (dom c)),(sup (dom c)))) is Path of t1,t2 & t1,t2 are_connected ) ; ::_thesis: ( t1 = the_first_point_of c & t2 = the_last_point_of c )
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 ( t1 = the_first_point_of c & t2 = the_last_point_of c ) by A7, A1, BORSUK_2:def_2; ::_thesis: verum
end;
theorem Th31: :: TOPALG_6:31
for T being non empty TopStruct
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 T be non empty TopStruct ; ::_thesis: 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 )
let c be with_endpoints Curve of T; ::_thesis: ( the_first_point_of c in rng c & the_last_point_of c in rng c )
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 ( the_first_point_of c in rng c & the_last_point_of c in rng c ) by FUNCT_1:3; ::_thesis: verum
end;
theorem Th32: :: TOPALG_6:32
for T being non empty TopStruct
for r1, r2 being real number
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 T be non empty TopStruct ; ::_thesis: for r1, r2 being real number
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
let r1, r2 be real number ; ::_thesis: 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
let t1, t2 be Point of T; ::_thesis: 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
let p1 be Path of t1,t2; ::_thesis: ( t1,t2 are_connected & r1 < r2 implies p1 * (L[01] (r1,r2,0,1)) is with_endpoints Curve of T )
assume A1: t1,t2 are_connected ; ::_thesis: ( not r1 < r2 or p1 * (L[01] (r1,r2,0,1)) is with_endpoints Curve of T )
assume A2: r1 < r2 ; ::_thesis: p1 * (L[01] (r1,r2,0,1)) is with_endpoints Curve of T
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 (p1 * (L[01] (r1,r2,0,1))) = dom (L[01] (r1,r2,0,1)) by RELAT_1:27;
then dom (p1 * (L[01] (r1,r2,0,1))) = [#] (Closed-Interval-TSpace (r1,r2)) by FUNCT_2:def_1;
then A5: dom (p1 * (L[01] (r1,r2,0,1))) = [.r1,r2.] by A2, TOPMETR:18;
A6: rng (p1 * (L[01] (r1,r2,0,1))) c= [#] T ;
then reconsider c = p1 * (L[01] (r1,r2,0,1)) as PartFunc of R^1,T by A5, RELSET_1:4, TOPMETR:17;
set S = R^1 | (dom c);
dom c = [#] (R^1 | (dom c)) by PRE_TOPC:def_5;
then reconsider g = c as Function of (R^1 | (dom c)),T by A6, FUNCT_2:2;
A7: R^1 | (dom c) = 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, Def4;
then reconsider c = 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 ) by Def6, Def7;
hence p1 * (L[01] (r1,r2,0,1)) is with_endpoints Curve of T ; ::_thesis: verum
end;
theorem Th33: :: TOPALG_6:33
for T being non empty TopStruct
for c being with_endpoints Curve of T holds the_first_point_of c, the_last_point_of c are_connected
proof
let T be non empty TopStruct ; ::_thesis: for c being with_endpoints Curve of T holds the_first_point_of c, the_last_point_of c are_connected
let c be with_endpoints Curve of T; ::_thesis: the_first_point_of c, the_last_point_of c are_connected
set t1 = the_first_point_of c;
set t2 = the_last_point_of c;
reconsider f = c as parametrized-curve PartFunc of R^1,T by Th23;
consider S being SubSpace of R^1 , g being Function of S,T such that
A1: ( f = g & S = R^1 | (dom f) & g is continuous ) by Def4;
set r1 = inf (dom c);
set r2 = sup (dom c);
set p = g * (L[01] (0,1,(inf (dom c)),(sup (dom c))));
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;
rng (L[01] (0,1,(inf (dom c)),(sup (dom c)))) c= [#] (Closed-Interval-TSpace ((inf (dom c)),(sup (dom c)))) by RELAT_1:def_19;
then rng (L[01] (0,1,(inf (dom c)),(sup (dom c)))) c= [.(inf (dom c)),(sup (dom c)).] by A2, TOPMETR:18;
then rng (L[01] (0,1,(inf (dom c)),(sup (dom c)))) c= dom c by Th27;
then dom (g * (L[01] (0,1,(inf (dom c)),(sup (dom c))))) = dom (L[01] (0,1,(inf (dom c)),(sup (dom c)))) by A1, RELAT_1:27;
then A4: dom (g * (L[01] (0,1,(inf (dom c)),(sup (dom c))))) = [#] (Closed-Interval-TSpace (0,1)) by FUNCT_2:def_1;
rng (g * (L[01] (0,1,(inf (dom c)),(sup (dom c))))) c= [#] T ;
then reconsider p = g * (L[01] (0,1,(inf (dom c)),(sup (dom c)))) as Function of I[01],T by A4, FUNCT_2:2, TOPMETR:20;
dom f = [.(inf (dom c)),(sup (dom c)).] by Th27;
then S = Closed-Interval-TSpace ((inf (dom c)),(sup (dom c))) 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,(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) ;
A8: (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) ;
A9: p . 0 = the_first_point_of c by A1, A7, A6, FUNCT_1:12;
p . 1 = the_last_point_of c by A1, A8, A6, FUNCT_1:12;
hence the_first_point_of c, the_last_point_of c are_connected by A5, A9, BORSUK_2:def_1; ::_thesis: verum
end;
definition
let T be non empty TopStruct ;
let c1, c2 be with_endpoints Curve of T;
predc1,c2 are_homotopic means :Def11: :: TOPALG_6:def 11
ex a, b being Point of T ex 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
for c1, c2 being with_endpoints Curve of T st ex a, b being Point of T ex 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 ) holds
ex a, b being Point of T ex p1, p2 being Path of a,b st
( p1 = c2 * (L[01] (0,1,(inf (dom c2)),(sup (dom c2)))) & p2 = c1 * (L[01] (0,1,(inf (dom c1)),(sup (dom c1)))) & p1,p2 are_homotopic ) ;
end;
:: deftheorem Def11 defines are_homotopic TOPALG_6:def_11_:_
for T being non empty TopStruct
for c1, c2 being with_endpoints Curve of T holds
( c1,c2 are_homotopic iff ex a, b being Point of T ex 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 ) );
definition
let T be non empty TopSpace;
let c1, c2 be with_endpoints Curve of T;
:: original: are_homotopic
redefine predc1,c2 are_homotopic ;
reflexivity
for c1 being with_endpoints Curve of T holds (T,b1,b1)
proof
let c be with_endpoints Curve of T; ::_thesis: (T,c,c)
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 (T,c,c) by Def11; ::_thesis: verum
end;
symmetry
for c1, c2 being with_endpoints Curve of T st (T,b1,b2) holds
(T,b2,b1) ;
end;
theorem Th34: :: TOPALG_6:34
for T being non empty TopStruct
for c1, c2 being with_endpoints Curve of T
for a, b being Point of T
for 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 ; ::_thesis: for c1, c2 being with_endpoints Curve of T
for a, b being Point of T
for 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 )
let c1, c2 be with_endpoints Curve of T; ::_thesis: for a, b being Point of T
for 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 )
let a, b be Point of T; ::_thesis: for 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 )
let p1, p2 be Path of a,b; ::_thesis: ( c1 = p1 & c2 = p2 & a,b are_connected implies ( c1,c2 are_homotopic iff p1,p2 are_homotopic ) )
assume A1: ( c1 = p1 & c2 = p2 ) ; ::_thesis: ( not a,b are_connected or ( c1,c2 are_homotopic iff p1,p2 are_homotopic ) )
assume A2: a,b are_connected ; ::_thesis: ( c1,c2 are_homotopic iff p1,p2 are_homotopic )
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 ) ::_thesis: ( p1,p2 are_homotopic implies c1,c2 are_homotopic )
proof
assume c1,c2 are_homotopic ; ::_thesis: p1,p2 are_homotopic
then consider aa, bb being Point of T, pp1, pp2 being 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 ) by Def11;
consider f being 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, BORSUK_2:def_7;
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, BORSUK_2:def_7; ::_thesis: verum
end;
assume A11: p1,p2 are_homotopic ; ::_thesis: c1,c2 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, Def11; ::_thesis: verum
end;
theorem Th35: :: TOPALG_6:35
for T being non empty TopStruct
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 T be non empty TopStruct ; ::_thesis: 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 )
let c1, c2 be with_endpoints Curve of T; ::_thesis: ( c1,c2 are_homotopic implies ( the_first_point_of c1 = the_first_point_of c2 & the_last_point_of c1 = the_last_point_of c2 ) )
assume c1,c2 are_homotopic ; ::_thesis: ( the_first_point_of c1 = the_first_point_of c2 & the_last_point_of c1 = the_last_point_of c2 )
then consider a, b being Point of T, p1, p2 being 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 ) by Def11;
A2: ( 0 is Point of I[01] & 1 is Point of I[01] ) by BORSUK_1:40, XXREAL_1:1;
consider f being 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, BORSUK_2:def_7;
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; ::_thesis: the_last_point_of c1 = the_last_point_of c2
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; ::_thesis: verum
end;
theorem Th36: :: TOPALG_6:36
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 ex 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; ::_thesis: 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 ex 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 ) ) ) )
let c1, c2 be with_endpoints Curve of T; ::_thesis: 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 ex 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 ) ) ) )
let S be Subset of R^1; ::_thesis: ( dom c1 = dom c2 & S = dom c1 implies ( c1,c2 are_homotopic iff ex f being Function of [:(R^1 | S),I[01]:],T ex 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 ) ) ) ) )
assume A1: ( dom c1 = dom c2 & S = dom c1 ) ; ::_thesis: ( c1,c2 are_homotopic iff ex f being Function of [:(R^1 | S),I[01]:],T ex 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 ) ) ) )
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 ex 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 ) ) ) ) ::_thesis: ( ex f being Function of [:(R^1 | S),I[01]:],T ex 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 ) ) ) implies c1,c2 are_homotopic )
proof
assume A8: c1,c2 are_homotopic ; ::_thesis: ex f being Function of [:(R^1 | S),I[01]:],T ex 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 ) ) )
percases ( inf (dom c1) < sup (dom c1) or not inf (dom c1) < sup (dom c1) ) ;
supposeA9: inf (dom c1) < sup (dom c1) ; ::_thesis: ex f being Function of [:(R^1 | S),I[01]:],T ex 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 ) ) )
consider a, b being Point of T, p1, p2 being 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, Def11;
consider f being 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, BORSUK_2:def_7;
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 = L[01] ((inf (dom c1)),(sup (dom c1)),0,1) as continuous Function of S1,I[01] by A9, A12, BORSUK_6:34, TOPMETR:20;
reconsider f2 = L[01] (0,1,0,1) as continuous Function of I[01],I[01] by BORSUK_6:34, TOPMETR:20;
set h = f * [:f1,f2:];
reconsider h = f * [:f1,f2:] as Function of [:(R^1 | S),I[01]:],T ;
take h ; ::_thesis: ex a, b being Point of T st
( h is continuous & ( for t being Point of (R^1 | S) holds
( h . (t,0) = c1 . t & h . (t,1) = c2 . t ) ) & ( for t being Point of I[01] holds
( h . ((inf S),t) = a & h . ((sup S),t) = b ) ) )
take a ; ::_thesis: ex b being Point of T st
( h is continuous & ( for t being Point of (R^1 | S) holds
( h . (t,0) = c1 . t & h . (t,1) = c2 . t ) ) & ( for t being Point of I[01] holds
( h . ((inf S),t) = a & h . ((sup S),t) = b ) ) )
take b ; ::_thesis: ( h is continuous & ( for t being Point of (R^1 | S) holds
( h . (t,0) = c1 . t & h . (t,1) = c2 . t ) ) & ( for t being Point of I[01] holds
( h . ((inf S),t) = a & h . ((sup S),t) = b ) ) )
thus h is continuous by A11; ::_thesis: ( ( for t being Point of (R^1 | S) holds
( h . (t,0) = c1 . t & h . (t,1) = c2 . t ) ) & ( for t being Point of I[01] holds
( h . ((inf S),t) = a & h . ((sup S),t) = b ) ) )
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 ) ::_thesis: for t being Point of I[01] holds
( h . ((inf S),t) = a & h . ((sup S),t) = b )
proof
let t be Point of (R^1 | S); ::_thesis: ( h . (t,0) = c1 . t & h . (t,1) = c2 . t )
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 ; ::_thesis: h . (t,1) = c2 . t
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 ; ::_thesis: verum
end;
thus for t being Point of I[01] holds
( h . ((inf S),t) = a & h . ((sup S),t) = b ) ::_thesis: verum
proof
let t be Point of I[01]; ::_thesis: ( h . ((inf S),t) = a & h . ((sup S),t) = b )
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 ; ::_thesis: h . ((sup S),t) = b
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 ; ::_thesis: verum
end;
end;
suppose not inf (dom c1) < sup (dom c1) ; ::_thesis: ex f being Function of [:(R^1 | S),I[01]:],T ex 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 ) ) )
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]:] --> (the_first_point_of c1);
A38: for t being Point of (R^1 | S) holds
( ([:(R^1 | S),I[01]:] --> (the_first_point_of c1)) . (t,0) = c1 . t & ([:(R^1 | S),I[01]:] --> (the_first_point_of c1)) . (t,1) = c2 . t )
proof
let t be Point of (R^1 | S); ::_thesis: ( ([:(R^1 | S),I[01]:] --> (the_first_point_of c1)) . (t,0) = c1 . t & ([:(R^1 | S),I[01]:] --> (the_first_point_of c1)) . (t,1) = c2 . t )
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 ([:(R^1 | S),I[01]:] --> (the_first_point_of c1)) . (t,0) = ([:(R^1 | S),I[01]:] --> (the_first_point_of c1)) . [t,0] by BINOP_1:def_1
.= c1 . t by A43, A40, FUNCOP_1:7 ; ::_thesis: ([:(R^1 | S),I[01]:] --> (the_first_point_of c1)) . (t,1) = c2 . t
thus ([:(R^1 | S),I[01]:] --> (the_first_point_of c1)) . (t,1) = ([:(R^1 | S),I[01]:] --> (the_first_point_of c1)) . [t,1] by BINOP_1:def_1
.= the_first_point_of c1 by A41, FUNCOP_1:7
.= the_first_point_of c2 by A8, Th35
.= c2 . t by A1, A42, A37, TARSKI:def_1 ; ::_thesis: verum
end;
for t being Point of I[01] holds
( ([:(R^1 | S),I[01]:] --> (the_first_point_of c1)) . ((inf S),t) = the_first_point_of c1 & ([:(R^1 | S),I[01]:] --> (the_first_point_of c1)) . ((sup S),t) = the_first_point_of c1 )
proof
let t be Point of I[01]; ::_thesis: ( ([:(R^1 | S),I[01]:] --> (the_first_point_of c1)) . ((inf S),t) = the_first_point_of c1 & ([:(R^1 | S),I[01]:] --> (the_first_point_of c1)) . ((sup S),t) = the_first_point_of c1 )
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 ([:(R^1 | S),I[01]:] --> (the_first_point_of c1)) . ((inf S),t) = ([:(R^1 | S),I[01]:] --> (the_first_point_of c1)) . [(inf S),t] by BINOP_1:def_1
.= the_first_point_of c1 by A46, FUNCOP_1:7 ; ::_thesis: ([:(R^1 | S),I[01]:] --> (the_first_point_of c1)) . ((sup S),t) = the_first_point_of c1
thus ([:(R^1 | S),I[01]:] --> (the_first_point_of c1)) . ((sup S),t) = ([:(R^1 | S),I[01]:] --> (the_first_point_of c1)) . [(sup S),t] by BINOP_1:def_1
.= the_first_point_of c1 by A47, FUNCOP_1:7 ; ::_thesis: verum
end;
hence ex f being Function of [:(R^1 | S),I[01]:],T ex 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 ) ) ) by A38; ::_thesis: verum
end;
end;
end;
given f being Function of [:(R^1 | S),I[01]:],T, a, b being 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 ) ) ) ; ::_thesis: c1,c2 are_homotopic
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 = L[01] (0,1,(inf (dom c1)),(sup (dom c1))) as continuous Function of I[01],S1 by A2, A53, BORSUK_6:34, TOPMETR:20;
reconsider f2 = L[01] (0,1,0,1) as continuous Function of I[01],I[01] by BORSUK_6:34, TOPMETR:20;
set h = f * [:f1,f2:];
reconsider h = f * [:f1,f2:] 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]; ::_thesis: ( h . (t,0) = p1 . t & h . (t,1) = p2 . t & h . (0,t) = a & h . (1,t) = b )
[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 ; ::_thesis: ( h . (t,1) = p2 . t & h . (0,t) = a & h . (1,t) = b )
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 ; ::_thesis: ( h . (0,t) = a & h . (1,t) = b )
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 ; ::_thesis: h . (1,t) = b
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 ; ::_thesis: verum
end;
then p1,p2 are_homotopic by A48, BORSUK_2:def_7;
hence c1,c2 are_homotopic by Def11; ::_thesis: verum
end;
definition
let T be TopStruct ;
let c1, c2 be Curve of T;
funcc1 + c2 -> Curve of T equals :Def12: :: TOPALG_6:def 12
c1 \/ c2 if c1 \/ c2 is Curve of T
otherwise {} ;
correctness
coherence
( ( c1 \/ c2 is Curve of T implies c1 \/ c2 is Curve of T ) & ( c1 \/ c2 is not Curve of T implies {} is Curve of T ) );
consistency
for b1 being Curve of T holds verum;
proof
now__::_thesis:_(_c1_\/_c2_is_not_Curve_of_T_implies_{}_is_Curve_of_T_)
assume c1 \/ c2 is not Curve of T ; ::_thesis: {} is Curve of T
{} is parametrized-curve PartFunc of R^1,T by Lm1, XBOOLE_1:2;
hence {} is Curve of T by Th20; ::_thesis: verum
end;
hence ( ( c1 \/ c2 is Curve of T implies c1 \/ c2 is Curve of T ) & ( c1 \/ c2 is not Curve of T implies {} is Curve of T ) & ( for b1 being Curve of T holds verum ) ) ; ::_thesis: verum
end;
end;
:: deftheorem Def12 defines + TOPALG_6:def_12_:_
for T being TopStruct
for c1, c2 being Curve of T holds
( ( c1 \/ c2 is Curve of T implies c1 + c2 = c1 \/ c2 ) & ( c1 \/ c2 is not Curve of T implies c1 + c2 = {} ) );
theorem Th37: :: TOPALG_6:37
for T being non empty TopStruct
for c being with_endpoints Curve of T
for r being real number 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 T be non empty TopStruct ; ::_thesis: for c being with_endpoints Curve of T
for r being real number ex c1, c2 being Element of Curves T st
( c = c1 + c2 & c1 = c | [.(inf (dom c)),r.] & c2 = c | [.r,(sup (dom c)).] )
let c be with_endpoints Curve of T; ::_thesis: for r being real number ex c1, c2 being Element of Curves T st
( c = c1 + c2 & c1 = c | [.(inf (dom c)),r.] & c2 = c | [.r,(sup (dom c)).] )
let r be real number ; ::_thesis: ex c1, c2 being Element of Curves T st
( c = c1 + c2 & c1 = c | [.(inf (dom c)),r.] & c2 = c | [.r,(sup (dom c)).] )
set c1 = c | [.(inf (dom c)),r.];
set c2 = c | [.r,(sup (dom c)).];
reconsider c1 = c | [.(inf (dom c)),r.] as Curve of T by Th26;
reconsider c2 = c | [.r,(sup (dom c)).] as Curve of T by Th26;
take c1 ; ::_thesis: ex c2 being Element of Curves T st
( c = c1 + c2 & c1 = c | [.(inf (dom c)),r.] & c2 = c | [.r,(sup (dom c)).] )
take c2 ; ::_thesis: ( c = c1 + c2 & c1 = c | [.(inf (dom c)),r.] & c2 = c | [.r,(sup (dom c)).] )
c1 \/ c2 = c
proof
percases ( r < inf (dom c) or r >= inf (dom c) ) ;
supposeA1: r < inf (dom c) ; ::_thesis: c1 \/ c2 = 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 c1 \/ c2 = c by A2, RELAT_1:68; ::_thesis: verum
end;
supposeA3: r >= inf (dom c) ; ::_thesis: c1 \/ c2 = c
percases ( r > sup (dom c) or r <= sup (dom c) ) ;
supposeA4: r > sup (dom c) ; ::_thesis: c1 \/ c2 = 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 c1 \/ c2 = c by A5, RELAT_1:68; ::_thesis: verum
end;
supposeA6: r <= sup (dom c) ; ::_thesis: c1 \/ c2 = 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 c1 \/ c2 = c ; ::_thesis: verum
end;
end;
end;
end;
end;
hence ( c = c1 + c2 & c1 = c | [.(inf (dom c)),r.] & c2 = c | [.r,(sup (dom c)).] ) by Def12; ::_thesis: verum
end;
theorem Th38: :: TOPALG_6:38
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; ::_thesis: 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 )
let c1, c2 be with_endpoints Curve of T; ::_thesis: ( sup (dom c1) = inf (dom c2) & the_last_point_of c1 = the_first_point_of c2 implies ( 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 ) )
assume A1: sup (dom c1) = inf (dom c2) ; ::_thesis: ( not the_last_point_of c1 = the_first_point_of c2 or ( 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 ) )
assume A2: the_last_point_of c1 = the_first_point_of c2 ; ::_thesis: ( 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 )
set f = c1 \/ c2;
A3: dom (c1 \/ c2) = (dom c1) \/ (dom c2) by RELAT_1:1;
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 (c1 \/ c2) c= [#] T by RELAT_1:12;
A8: dom (c1 \/ c2) c= [#] R^1 by A6, RELAT_1:1, TOPMETR:17;
reconsider S0 = dom (c1 \/ c2) as Subset of R^1 by A6, RELAT_1:1, TOPMETR:17;
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 (c1 \/ c2) by A3, XBOOLE_1:29;
set S = R^1 | S0;
consider S1 being SubSpace of R^1 , g1 being Function of S1,T such that
A15: ( f1 = g1 & S1 = R^1 | (dom f1) & g1 is continuous ) by Def4;
consider S2 being SubSpace of R^1 , g2 being Function of S2,T such that
A16: ( f2 = g2 & S2 = R^1 | (dom f2) & g2 is continuous ) by Def4;
sup (dom c1) in dom (c1 \/ c2) by A13, A14, ZFMISC_1:31;
then sup (dom c1) in [#] (R^1 | S0) by PRE_TOPC:def_5;
then reconsider p = sup (dom c1) as Point of (R^1 | S0) ;
reconsider S1 = S1, S2 = S2 as SubSpace of R^1 | S0 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
.= [#] (R^1 | S0) 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: R^1 | S0 is T_2 by TOPMETR:2;
A22: g1 +* g2 is continuous Function of (R^1 | S0),T by A17, A18, A19, A20, A21, A15, A16, A1, A2, COMPTS_1:20;
for x, y1, y2 being set st [x,y1] in c1 \/ c2 & [x,y2] in c1 \/ c2 holds
y1 = y2
proof
let x, y1, y2 be set ; ::_thesis: ( [x,y1] in c1 \/ c2 & [x,y2] in c1 \/ c2 implies y1 = y2 )
assume A23: ( [x,y1] in c1 \/ c2 & [x,y2] in c1 \/ c2 ) ; ::_thesis: y1 = y2
percases ( ( [x,y1] in c1 & [x,y2] in c1 ) or ( [x,y1] in c2 & [x,y2] in c2 ) or ( [x,y1] in c1 & [x,y2] in c2 ) or ( [x,y1] in c2 & [x,y2] in c1 ) ) by A23, XBOOLE_0:def_3;
suppose ( [x,y1] in c1 & [x,y2] in c1 ) ; ::_thesis: y1 = y2
hence y1 = y2 by FUNCT_1:def_1; ::_thesis: verum
end;
suppose ( [x,y1] in c2 & [x,y2] in c2 ) ; ::_thesis: y1 = y2
hence y1 = y2 by FUNCT_1:def_1; ::_thesis: verum
end;
supposeA24: ( [x,y1] in c1 & [x,y2] in c2 ) ; ::_thesis: y1 = y2
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 y1 = y2 by A1, A2; ::_thesis: verum
end;
supposeA25: ( [x,y1] in c2 & [x,y2] in c1 ) ; ::_thesis: y1 = y2
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 y1 = y2 by A1, A2; ::_thesis: verum
end;
end;
end;
then reconsider f = c1 \/ c2 as Function by FUNCT_1:def_1;
A26: dom f = (dom g1) \/ (dom g2) by A15, A16, RELAT_1:1;
for x being set 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 set ; ::_thesis: ( x in (dom g1) \/ (dom g2) implies ( ( x in dom g2 implies f . x = g2 . x ) & ( not x in dom g2 implies f . x = g1 . x ) ) )
assume A27: x in (dom g1) \/ (dom g2) ; ::_thesis: ( ( x in dom g2 implies f . x = g2 . x ) & ( not x in dom g2 implies f . x = g1 . x ) )
thus ( x in dom g2 implies f . x = g2 . x ) ::_thesis: ( not x in dom g2 implies f . x = g1 . x )
proof
assume x in dom g2 ; ::_thesis: f . x = g2 . x
then [x,(g2 . x)] in g2 by FUNCT_1:1;
then [x,(g2 . x)] in f by A16, XBOOLE_0:def_3;
hence f . x = g2 . x by FUNCT_1:1; ::_thesis: verum
end;
thus ( not x in dom g2 implies f . x = g1 . x ) ::_thesis: verum
proof
assume not x in dom g2 ; ::_thesis: f . x = g1 . x
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 f . x = g1 . x by FUNCT_1:1; ::_thesis: verum
end;
end;
then A28: f = g1 +* g2 by A26, FUNCT_4:def_1;
reconsider f = f as PartFunc of R^1,T by A7, A8, RELSET_1:4;
dom c1 meets dom c2 by A13, XBOOLE_0:def_7;
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, Def4;
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 by Def6;
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 by Def7;
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; ::_thesis: ( (c1 + c2) . (inf (dom c1)) = the_first_point_of c1 & (c1 + c2) . (sup (dom c2)) = the_last_point_of c2 )
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 (c1 + c2) . (inf (dom c1)) = c1 . (inf (dom c1)) by A37, FUNCT_1:1; ::_thesis: verum
end;
thus (c1 + c2) . (inf (dom c1)) = the_first_point_of c1 by A36; ::_thesis: (c1 + c2) . (sup (dom c2)) = the_last_point_of c2
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 (c1 + c2) . (sup (dom c2)) = c2 . (sup (dom c2)) by A39, FUNCT_1:1; ::_thesis: verum
end;
thus (c1 + c2) . (sup (dom c2)) = the_last_point_of c2 by A38; ::_thesis: verum
end;
theorem Th39: :: TOPALG_6:39
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; ::_thesis: 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
let c1, c2, c3, c4, c5, c6 be with_endpoints Curve of T; ::_thesis: ( 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) implies c5,c6 are_homotopic )
assume A1: ( c1,c2 are_homotopic & dom c1 = dom c2 ) ; ::_thesis: ( not c3,c4 are_homotopic or not dom c3 = dom c4 or not c5 = c1 + c3 or not c6 = c2 + c4 or not the_last_point_of c1 = the_first_point_of c3 or not sup (dom c1) = inf (dom c3) or c5,c6 are_homotopic )
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 being Function of [:(R^1 | S1),I[01]:],T, a1, b1 being 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 ) ; ::_thesis: ( not c5 = c1 + c3 or not c6 = c2 + c4 or not the_last_point_of c1 = the_first_point_of c3 or not sup (dom c1) = inf (dom c3) or c5,c6 are_homotopic )
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 being Function of [:(R^1 | S2),I[01]:],T, a2, b2 being 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 ; ::_thesis: ( not c6 = c2 + c4 or not the_last_point_of c1 = the_first_point_of c3 or not sup (dom c1) = inf (dom c3) or c5,c6 are_homotopic )
A8: c5 = c1 \/ c3
proof
percases ( c1 \/ c3 is Curve of T or not c1 \/ c3 is Curve of T ) ;
suppose c1 \/ c3 is Curve of T ; ::_thesis: c5 = c1 \/ c3
hence c5 = c1 \/ c3 by A7, Def12; ::_thesis: verum
end;
suppose c1 \/ c3 is not Curve of T ; ::_thesis: c5 = c1 \/ c3
hence c5 = c1 \/ c3 by A7, Def12; ::_thesis: verum
end;
end;
end;
assume A9: c6 = c2 + c4 ; ::_thesis: ( not the_last_point_of c1 = the_first_point_of c3 or not sup (dom c1) = inf (dom c3) or c5,c6 are_homotopic )
A10: c6 = c2 \/ c4
proof
percases ( c2 \/ c4 is Curve of T or not c2 \/ c4 is Curve of T ) ;
suppose c2 \/ c4 is Curve of T ; ::_thesis: c6 = c2 \/ c4
hence c6 = c2 \/ c4 by A9, Def12; ::_thesis: verum
end;
suppose c2 \/ c4 is not Curve of T ; ::_thesis: c6 = c2 \/ c4
hence c6 = c2 \/ c4 by A9, Def12; ::_thesis: verum
end;
end;
end;
assume A11: the_last_point_of c1 = the_first_point_of c3 ; ::_thesis: ( not sup (dom c1) = inf (dom c3) or c5,c6 are_homotopic )
assume A12: sup (dom c1) = inf (dom c3) ; ::_thesis: c5,c6 are_homotopic
A13: dom c5 = (dom c1) \/ (dom c3) by A8, RELAT_1:1
.= dom c6 by A10, A1, A4, RELAT_1:1 ;
reconsider S3 = S1 \/ S2 as Subset of R^1 ;
A14: dom c5 = (dom c1) \/ (dom c3) by A8, RELAT_1:1
.= 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: [:(R^1 | S1),I[01]:] is SubSpace of [:(R^1 | S3),I[01]:] by A15, BORSUK_3:21;
R^1 | S2 is SubSpace of R^1 | S3 by TOPMETR:4;
then A17: [:(R^1 | S2),I[01]:] is SubSpace of [:(R^1 | S3),I[01]:] 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: ([#] [:(R^1 | S1),I[01]:]) \/ ([#] [:(R^1 | S2),I[01]:]) = [:([#] (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
.= [#] [:(R^1 | S3),I[01]:] 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;
[:(R^1 | S3),I[01]:] is SubSpace of [:R^1,I[01]:] by A15, BORSUK_3:21;
then A24: [:(R^1 | S3),I[01]:] is T_2 by TOPMETR:2;
for p being set st p in ([#] [:(R^1 | S1),I[01]:]) /\ ([#] [:(R^1 | S2),I[01]:]) holds
H1 . p = H2 . p
proof
let p be set ; ::_thesis: ( p in ([#] [:(R^1 | S1),I[01]:]) /\ ([#] [:(R^1 | S2),I[01]:]) implies H1 . p = H2 . p )
assume A25: p in ([#] [:(R^1 | S1),I[01]:]) /\ ([#] [:(R^1 | S2),I[01]:]) ; ::_thesis: H1 . p = H2 . p
A26: ([#] [:(R^1 | S1),I[01]:]) /\ ([#] [:(R^1 | S2),I[01]:]) = [:([#] (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 being set 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 = 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 ; ::_thesis: verum
end;
then consider H3 being Function of [:(R^1 | S3),I[01]:],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); ::_thesis: ( H3 . (t,0) = c5 . t & H3 . (t,1) = c6 . t )
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 [#] [:(R^1 | S3),I[01]:] ;
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 [#] [:(R^1 | S3),I[01]:] ;
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;
percases ( [t,0] in dom H2 or not [t,0] in dom H2 ) ;
supposeA40: [t,0] in dom H2 ; ::_thesis: ( H3 . (t,0) = c5 . t & H3 . (t,1) = c6 . t )
then [t,0] in [#] [:(R^1 | S2),I[01]:] ;
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, RELAT_1:1;
[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 ; ::_thesis: H3 . (t,1) = c6 . t
[t,1] in [:([#] (R^1 | S2)),([#] I[01]):] by A41, A38, ZFMISC_1:def_2;
then [t,1] in [#] [:(R^1 | S2),I[01]:] ;
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, RELAT_1:1;
[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 ; ::_thesis: verum
end;
supposeA48: not [t,0] in dom H2 ; ::_thesis: ( H3 . (t,0) = c5 . t & H3 . (t,1) = c6 . t )
( [t,0] in dom H1 or [t,0] in dom H2 ) by A37, XBOOLE_0:def_3;
then [t,0] in [#] [:(R^1 | S1),I[01]:] 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, RELAT_1:1;
[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 ; ::_thesis: H3 . (t,1) = c6 . t
t in (dom c2) \/ (dom c4) by A50, A1, XBOOLE_0:def_3;
then A53: t in dom c6 by A10, RELAT_1:1;
[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 ; ::_thesis: contradiction
then [t,1] in [#] [:(R^1 | S2),I[01]:] ;
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 [#] [:(R^1 | S2),I[01]:] ;
hence contradiction by A48, FUNCT_2:def_1; ::_thesis: verum
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 ; ::_thesis: verum
end;
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]; ::_thesis: ( H3 . ((inf S3),t) = a1 & H3 . ((sup S3),t) = b2 )
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 [#] [:(R^1 | S1),I[01]:] ;
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 ::_thesis: H3 . ((sup S3),t) = b2
proof
percases ( [(inf S3),t] in dom H2 or not [(inf S3),t] in dom H2 ) ;
supposeA58: [(inf S3),t] in dom H2 ; ::_thesis: H3 . ((inf S3),t) = a1
then [(inf S3),t] in [#] [:(R^1 | S2),I[01]:] ;
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 H3 . ((inf S3),t) = a1 ; ::_thesis: verum
end;
supposeA69: not [(inf S3),t] in dom H2 ; ::_thesis: H3 . ((inf S3),t) = a1
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 H3 . ((inf S3),t) = a1 ; ::_thesis: verum
end;
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 [#] [:(R^1 | S2),I[01]:] ;
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 ; ::_thesis: verum
end;
hence c5,c6 are_homotopic by A13, A14, A35, A34, Th36; ::_thesis: verum
end;
definition
let T be TopStruct ;
let f be FinSequence of Curves T;
func Partial_Sums f -> FinSequence of Curves T means :Def13: :: TOPALG_6:def 13
( len f = len it & f . 1 = it . 1 & ( for i being Nat st 1 <= i & i < len f holds
it . (i + 1) = (it /. i) + (f /. (i + 1)) ) );
existence
ex b1 being FinSequence of Curves T st
( len f = len b1 & f . 1 = b1 . 1 & ( for i being Nat st 1 <= i & i < len f holds
b1 . (i + 1) = (b1 /. i) + (f /. (i + 1)) ) )
proof
percases ( len f > 0 or len f <= 0 ) ;
supposeA1: len f > 0 ; ::_thesis: ex b1 being FinSequence of Curves T st
( len f = len b1 & f . 1 = b1 . 1 & ( for i being Nat st 1 <= i & i < len f holds
b1 . (i + 1) = (b1 /. i) + (f /. (i + 1)) ) )
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 S1[ 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 S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A6: S1[k] ; ::_thesis: S1[k + 1]
now__::_thesis:_(_(_(k_+_1)_+_1_<=_len_f_&_S1[k_+_1]_)_or_(_(k_+_1)_+_1_>_len_f_&_S1[k_+_1]_)_)
percases ( (k + 1) + 1 <= len f or (k + 1) + 1 > len f ) ;
caseA7: (k + 1) + 1 <= len f ; ::_thesis: S1[k + 1]
k + 1 < (k + 1) + 1 by XREAL_1:29;
then consider g being FinSequence of Curves T such that
A8: k + 1 = len g and
A9: f . 1 = g . 1 and
A10: for i being Nat st 1 <= i & i < k + 1 holds
g . (i + 1) = (g /. i) + (f /. (i + 1)) by A6, A7, XXREAL_0:2;
reconsider g2 = g ^ <*((g /. (k + 1)) + (f /. ((k + 1) + 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; ::_thesis: ( 1 <= i & i < (k + 1) + 1 implies g2 . (i + 1) = (g2 /. i) + (f /. (i + 1)) )
assume that
A14: 1 <= i and
A15: i < (k + 1) + 1 ; ::_thesis: g2 . (i + 1) = (g2 /. i) + (f /. (i + 1))
A16: i <= k + 1 by A15, NAT_1:13;
percases ( i < k + 1 or i = k + 1 ) by A16, XXREAL_0:1;
supposeA17: i < k + 1 ; ::_thesis: g2 . (i + 1) = (g2 /. i) + (f /. (i + 1))
A18: 1 <= i + 1 by NAT_1:12;
i + 1 <= k + 1 by A17, NAT_1:13;
then i + 1 in Seg (len g) by A8, A18, FINSEQ_1:1;
then A19: g2 . (i + 1) = g . (i + 1) by A11, FINSEQ_1:def_7;
i in Seg (len g) by A8, A14, A16, FINSEQ_1:1;
then A20: g2 . i = g . i by A11, FINSEQ_1:def_7;
A21: g /. i = g . i by A8, A14, A17, FINSEQ_4:15;
g2 /. i = g2 . i by A12, A14, A15, FINSEQ_4:15;
hence g2 . (i + 1) = (g2 /. i) + (f /. (i + 1)) by A10, A14, A17, A19, A20, A21; ::_thesis: verum
end;
supposeA22: i = k + 1 ; ::_thesis: g2 . (i + 1) = (g2 /. i) + (f /. (i + 1))
A23: g2 /. i = g2 . i by A12, A14, A15, FINSEQ_4:15;
i in Seg (len g) by A8, A14, A16, FINSEQ_1:1;
then A24: g . i = g2 . i by A11, FINSEQ_1:def_7;
g /. i = g . i by A8, A14, A16, FINSEQ_4:15;
hence g2 . (i + 1) = (g2 /. i) + (f /. (i + 1)) by A8, A22, A24, A23, FINSEQ_1:42; ::_thesis: verum
end;
end;
end;
1 <= k + 1 by NAT_1:11;
then 1 in Seg (len g) by A8, FINSEQ_1:1;
then g2 . 1 = f . 1 by A9, A11, FINSEQ_1:def_7;
hence S1[k + 1] by A12, A13; ::_thesis: verum
end;
case (k + 1) + 1 > len f ; ::_thesis: S1[k + 1]
hence S1[k + 1] ; ::_thesis: verum
end;
end;
end;
hence S1[k + 1] ; ::_thesis: verum
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: S1[ 0 ] by A3, A4;
for k being Nat holds S1[k] from NAT_1:sch_2(A26, A5);
hence ex b1 being FinSequence of Curves T st
( len f = len b1 & f . 1 = b1 . 1 & ( for i being Nat st 1 <= i & i < len f holds
b1 . (i + 1) = (b1 /. i) + (f /. (i + 1)) ) ) by A25; ::_thesis: verum
end;
supposeA27: len f <= 0 ; ::_thesis: ex b1 being FinSequence of Curves T st
( len f = len b1 & f . 1 = b1 . 1 & ( for i being Nat st 1 <= i & i < len f holds
b1 . (i + 1) = (b1 /. i) + (f /. (i + 1)) ) )
take f ; ::_thesis: ( len f = len f & f . 1 = f . 1 & ( for i being Nat st 1 <= i & i < len f holds
f . (i + 1) = (f /. i) + (f /. (i + 1)) ) )
thus ( len f = len f & f . 1 = f . 1 ) ; ::_thesis: for i being Nat st 1 <= i & i < len f holds
f . (i + 1) = (f /. i) + (f /. (i + 1))
let i be Nat; ::_thesis: ( 1 <= i & i < len f implies f . (i + 1) = (f /. i) + (f /. (i + 1)) )
thus ( 1 <= i & i < len f implies f . (i + 1) = (f /. i) + (f /. (i + 1)) ) by A27; ::_thesis: verum
end;
end;
end;
uniqueness
for b1, b2 being FinSequence of Curves T st len f = len b1 & f . 1 = b1 . 1 & ( for i being Nat st 1 <= i & i < len f holds
b1 . (i + 1) = (b1 /. i) + (f /. (i + 1)) ) & len f = len b2 & f . 1 = b2 . 1 & ( for i being Nat st 1 <= i & i < len f holds
b2 . (i + 1) = (b2 /. i) + (f /. (i + 1)) ) holds
b1 = b2
proof
let g1, g2 be FinSequence of Curves T; ::_thesis: ( len f = len g1 & f . 1 = g1 . 1 & ( for i being Nat st 1 <= i & i < len f holds
g1 . (i + 1) = (g1 /. i) + (f /. (i + 1)) ) & len f = len g2 & f . 1 = g2 . 1 & ( for i being Nat st 1 <= i & i < len f holds
g2 . (i + 1) = (g2 /. i) + (f /. (i + 1)) ) implies g1 = g2 )
assume that
A28: len f = len g1 and
A29: f . 1 = g1 . 1 and
A30: for i being Nat st 1 <= i & i < len f holds
g1 . (i + 1) = (g1 /. i) + (f /. (i + 1)) ; ::_thesis: ( not len f = len g2 or not f . 1 = g2 . 1 or ex i being Nat st
( 1 <= i & i < len f & not g2 . (i + 1) = (g2 /. i) + (f /. (i + 1)) ) or g1 = g2 )
defpred S1[ Nat] means ( 1 <= $1 & $1 <= len f implies g1 . $1 = g2 . $1 );
assume that
A31: len f = len g2 and
A32: f . 1 = g2 . 1 and
A33: for i being Nat st 1 <= i & i < len f holds
g2 . (i + 1) = (g2 /. i) + (f /. (i + 1)) ; ::_thesis: g1 = g2
A34: for k being Nat st S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A35: S1[k] ; ::_thesis: S1[k + 1]
( 1 <= k + 1 & k + 1 <= len f implies g1 . (k + 1) = g2 . (k + 1) )
proof
assume that
1 <= k + 1 and
A36: k + 1 <= len f ; ::_thesis: g1 . (k + 1) = g2 . (k + 1)
A37: k < k + 1 by XREAL_1:29;
then A38: k < len f by A36, XXREAL_0:2;
percases ( 1 <= k or 1 > k ) ;
supposeA39: 1 <= k ; ::_thesis: g1 . (k + 1) = g2 . (k + 1)
then A40: g2 . (k + 1) = (g2 /. k) + (f /. (k + 1)) by A33, A38;
A41: k <= len g2 by A31, A36, A37, XXREAL_0:2;
A42: g1 /. k = g1 . k by A28, A38, A39, FINSEQ_4:15;
g1 . (k + 1) = (g1 /. k) + (f /. (k + 1)) by A30, A38, A39;
hence g1 . (k + 1) = g2 . (k + 1) by A35, A36, A37, A39, A40, A42, A41, FINSEQ_4:15, XXREAL_0:2; ::_thesis: verum
end;
suppose 1 > k ; ::_thesis: g1 . (k + 1) = g2 . (k + 1)
then 0 + 1 > k ;
then k = 0 by NAT_1:13;
hence g1 . (k + 1) = g2 . (k + 1) by A29, A32; ::_thesis: verum
end;
end;
end;
hence S1[k + 1] ; ::_thesis: verum
end;
A43: S1[ 0 ] ;
for k being Nat holds S1[k] from NAT_1:sch_2(A43, A34);
hence g1 = g2 by A28, A31, FINSEQ_1:14; ::_thesis: verum
end;
end;
:: deftheorem Def13 defines Partial_Sums TOPALG_6:def_13_:_
for T being TopStruct
for f, b3 being FinSequence of Curves T holds
( b3 = Partial_Sums f iff ( len f = len b3 & f . 1 = b3 . 1 & ( for i being Nat st 1 <= i & i < len f holds
b3 . (i + 1) = (b3 /. i) + (f /. (i + 1)) ) ) );
definition
let T be TopStruct ;
let f be FinSequence of Curves T;
func Sum f -> Curve of T equals :Def14: :: TOPALG_6:def 14
(Partial_Sums f) . (len f) if len f > 0
otherwise {} ;
coherence
( ( len f > 0 implies (Partial_Sums f) . (len f) is Curve of T ) & ( not len f > 0 implies {} is Curve of T ) )
proof
A1: len f = len (Partial_Sums f) by Def13;
now__::_thesis:_(_(_len_f_>_0_&_(Partial_Sums_f)_._(len_f)_is_Element_of_Curves_T_)_or_(_len_f_<=_0_&_(_len_f_>_0_implies_(Partial_Sums_f)_._(len_f)_is_Curve_of_T_)_&_(_not_len_f_>_0_implies_{}_is_Curve_of_T_)_)_)
percases ( len f > 0 or len f <= 0 ) ;
case len f > 0 ; ::_thesis: (Partial_Sums f) . (len f) is Element of Curves T
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 ; ::_thesis: verum
end;
caseA2: len f <= 0 ; ::_thesis: ( ( len f > 0 implies (Partial_Sums f) . (len f) is Curve of T ) & ( not len f > 0 implies {} is Curve of T ) )
{} is parametrized-curve PartFunc of R^1,T by Lm1, XBOOLE_1:2;
hence ( ( len f > 0 implies (Partial_Sums f) . (len f) is Curve of T ) & ( not len f > 0 implies {} is Curve of T ) ) by A2, Th20; ::_thesis: verum
end;
end;
end;
hence ( ( len f > 0 implies (Partial_Sums f) . (len f) is Curve of T ) & ( not len f > 0 implies {} is Curve of T ) ) ; ::_thesis: verum
end;
consistency
for b1 being Curve of T holds verum ;
end;
:: deftheorem Def14 defines Sum TOPALG_6:def_14_:_
for T being TopStruct
for f being FinSequence of Curves T holds
( ( len f > 0 implies Sum f = (Partial_Sums f) . (len f) ) & ( not len f > 0 implies Sum f = {} ) );
theorem Th40: :: TOPALG_6:40
for T being non empty TopStruct
for c being Curve of T holds Sum <*c*> = c
proof
let T be non empty TopStruct ; ::_thesis: for c being Curve of T holds Sum <*c*> = c
let c be Curve of T; ::_thesis: Sum <*c*> = c
set f = <*c*>;
len <*c*> = 1 by FINSEQ_1:40;
hence Sum <*c*> = (Partial_Sums <*c*>) . 1 by Def14
.= <*c*> . 1 by Def13
.= c by FINSEQ_1:40 ;
::_thesis: verum
end;
Lm2: for T being non empty TopStruct
for f1, f2 being FinSequence of Curves T holds (Partial_Sums (f1 ^ f2)) . (len f1) = (Partial_Sums f1) . (len f1)
proof
let T be non empty TopStruct ; ::_thesis: for f1, f2 being FinSequence of Curves T holds (Partial_Sums (f1 ^ f2)) . (len f1) = (Partial_Sums f1) . (len f1)
defpred S1[ 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: S1[ 0 ]
proof
let f1, f2 be FinSequence of Curves T; ::_thesis: ( len f1 = 0 implies (Partial_Sums (f1 ^ f2)) . (len f1) = (Partial_Sums f1) . (len f1) )
assume A2: len f1 = 0 ; ::_thesis: (Partial_Sums (f1 ^ f2)) . (len f1) = (Partial_Sums f1) . (len f1)
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 ; ::_thesis: verum
end;
A5: for k being Nat st S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A6: S1[k] ; ::_thesis: S1[k + 1]
let f1, f2 be FinSequence of Curves T; ::_thesis: ( len f1 = k + 1 implies (Partial_Sums (f1 ^ f2)) . (len f1) = (Partial_Sums f1) . (len f1) )
assume A7: len f1 = k + 1 ; ::_thesis: (Partial_Sums (f1 ^ f2)) . (len f1) = (Partial_Sums f1) . (len f1)
then consider f3 being FinSequence of Curves T, c being Element of Curves T such that
A8: f1 = f3 ^ <*c*> by FINSEQ_2:19;
set f4 = <*c*> ^ f2;
A9: f1 ^ f2 = f3 ^ (<*c*> ^ f2) by A8, FINSEQ_1:32;
A10: len f1 = (len f3) + 1 by A8, FINSEQ_2:16;
percases ( 1 > k or 1 <= k ) ;
supposeA11: 1 > k ; ::_thesis: (Partial_Sums (f1 ^ f2)) . (len f1) = (Partial_Sums f1) . (len f1)
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 ; ::_thesis: verum
end;
supposeA14: 1 <= k ; ::_thesis: (Partial_Sums (f1 ^ f2)) . (len f1) = (Partial_Sums f1) . (len f1)
A15: k < len f1 by A7, NAT_1:16;
A16: len (<*c*> ^ f2) = (len <*c*>) + (len f2) by FINSEQ_1:22
.= 1 + (len f2) by FINSEQ_1:39 ;
len (f3 ^ (<*c*> ^ f2)) = k + (len (<*c*> ^ f2)) by A10, A7, FINSEQ_1:22;
then A17: k < len (f3 ^ (<*c*> ^ f2)) by A16, NAT_1:16;
then k in Seg (len (f3 ^ (<*c*> ^ f2))) by A14, FINSEQ_1:1;
then k in Seg (len (Partial_Sums (f3 ^ (<*c*> ^ f2)))) by Def13;
then A18: k in dom (Partial_Sums (f3 ^ (<*c*> ^ f2))) 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 A19: 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 A20: k in dom (Partial_Sums f1) by FINSEQ_1:def_3;
A21: (Partial_Sums (f3 ^ (<*c*> ^ f2))) /. k = (Partial_Sums (f3 ^ (<*c*> ^ f2))) . k by A18, PARTFUN1:def_6
.= (Partial_Sums f3) . k by A10, A7, A6
.= (Partial_Sums f3) /. k by A19, PARTFUN1:def_6 ;
A22: (Partial_Sums f1) /. k = (Partial_Sums f1) . k by A20, PARTFUN1:def_6
.= (Partial_Sums f3) . k by A8, A10, A7, A6
.= (Partial_Sums f3) /. k by A19, PARTFUN1:def_6 ;
1 + 1 <= k + 1 by A14, XREAL_1:6;
then A23: 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 A23, FINSEQ_1:1;
then A24: k + 1 in dom (f1 ^ f2) by FINSEQ_1:def_3;
k + 1 in Seg (len f1) by A7, A23, FINSEQ_1:1;
then A25: k + 1 in dom f1 by FINSEQ_1:def_3;
A26: (f1 ^ f2) /. (k + 1) = (f1 ^ f2) . (k + 1) by A24, PARTFUN1:def_6
.= f1 . (k + 1) by A25, FINSEQ_1:def_7
.= f1 /. (k + 1) by A25, PARTFUN1:def_6 ;
thus (Partial_Sums (f1 ^ f2)) . (len f1) = ((Partial_Sums f1) /. k) + (f1 /. (k + 1)) by A7, A9, A21, A22, A26, A14, A17, Def13
.= (Partial_Sums f1) . (len f1) by A14, A7, A15, Def13 ; ::_thesis: verum
end;
end;
end;
A27: for k being Nat holds S1[k] from NAT_1:sch_2(A1, A5);
let f1, f2 be FinSequence of Curves T; ::_thesis: (Partial_Sums (f1 ^ f2)) . (len f1) = (Partial_Sums f1) . (len f1)
thus (Partial_Sums (f1 ^ f2)) . (len f1) = (Partial_Sums f1) . (len f1) by A27; ::_thesis: verum
end;
theorem Th41: :: TOPALG_6:41
for T being non empty TopStruct
for c being Curve of T
for f being FinSequence of Curves T holds Sum (f ^ <*c*>) = (Sum f) + c
proof
let T be non empty TopStruct ; ::_thesis: for c being Curve of T
for f being FinSequence of Curves T holds Sum (f ^ <*c*>) = (Sum f) + c
let c be Curve of T; ::_thesis: for f being FinSequence of Curves T holds Sum (f ^ <*c*>) = (Sum f) + c
let f be FinSequence of Curves T; ::_thesis: Sum (f ^ <*c*>) = (Sum f) + c
percases ( len f <= 0 or len f > 0 ) ;
supposeA1: len f <= 0 ; ::_thesis: Sum (f ^ <*c*>) = (Sum f) + c
A2: f = {} by A1, FINSEQ_1:20;
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 ; ::_thesis: verum
end;
supposeA3: len f > 0 ; ::_thesis: Sum (f ^ <*c*>) = (Sum f) + c
set f1 = f ^ <*c*>;
A4: len (f ^ <*c*>) = (len f) + (len <*c*>) by FINSEQ_1:22
.= (len f) + 1 by FINSEQ_1:39 ;
A5: Sum (f ^ <*c*>) = (Partial_Sums (f ^ <*c*>)) . (len (f ^ <*c*>)) by A4, Def14;
0 < 0 + (len f) by A3;
then A6: 1 <= len f by NAT_1:19;
A7: len f < len (f ^ <*c*>) by A4, NAT_1:13;
len f in Seg (len (f ^ <*c*>)) by A6, A7, FINSEQ_1:1;
then len f in Seg (len (Partial_Sums (f ^ <*c*>))) by Def13;
then len f in dom (Partial_Sums (f ^ <*c*>)) by FINSEQ_1:def_3;
then A8: (Partial_Sums (f ^ <*c*>)) /. (len f) = (Partial_Sums (f ^ <*c*>)) . (len f) by PARTFUN1:def_6
.= (Partial_Sums f) . (len f) by Lm2
.= Sum f by A3, Def14 ;
len (f ^ <*c*>) in Seg (len (f ^ <*c*>)) by A4, FINSEQ_1:3;
then len (f ^ <*c*>) in dom (f ^ <*c*>) by FINSEQ_1:def_3;
then A9: (f ^ <*c*>) /. ((len f) + 1) = (f ^ <*c*>) . ((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; ::_thesis: verum
end;
end;
end;
theorem Th42: :: TOPALG_6:42
for T being non empty TopStruct
for X being set
for 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 T be non empty TopStruct ; ::_thesis: for X being set
for 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
let X be set ; ::_thesis: for 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
defpred S1[ 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: S1[ 0 ]
proof
let f be FinSequence of Curves T; ::_thesis: ( len f = 0 & ( for i being Nat st 1 <= i & i <= len f holds
rng (f /. i) c= X ) implies rng (Sum f) c= X )
assume len f = 0 ; ::_thesis: ( ex i being Nat st
( 1 <= i & i <= len f & not rng (f /. i) c= X ) or rng (Sum f) c= X )
then Sum f = {} by Def14;
then rng (Sum f) = {} ;
hence ( ex i being Nat st
( 1 <= i & i <= len f & not rng (f /. i) c= X ) or rng (Sum f) c= X ) by XBOOLE_1:2; ::_thesis: verum
end;
A2: for k being Nat st S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A3: S1[k] ; ::_thesis: S1[k + 1]
let f be FinSequence of Curves T; ::_thesis: ( len f = k + 1 & ( for i being Nat st 1 <= i & i <= len f holds
rng (f /. i) c= X ) implies rng (Sum f) c= X )
assume A4: len f = k + 1 ; ::_thesis: ( ex i being Nat st
( 1 <= i & i <= len f & not rng (f /. i) c= X ) or rng (Sum f) c= X )
then consider f1 being FinSequence of Curves T, c being 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 ; ::_thesis: rng (Sum f) 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;
percases ( not (Sum f1) \/ c is Curve of T or (Sum f1) \/ c is Curve of T ) ;
suppose (Sum f1) \/ c is not Curve of T ; ::_thesis: rng (Sum f) c= X
then Sum f = {} by A8, Def12;
then rng (Sum f) = {} ;
hence rng (Sum f) c= X by XBOOLE_1:2; ::_thesis: verum
end;
suppose (Sum f1) \/ c is Curve of T ; ::_thesis: rng (Sum f) c= X
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; ::_thesis: ( 1 <= i & i <= len f1 implies rng (f1 /. i) c= X )
assume A11: ( 1 <= i & i <= len f1 ) ; ::_thesis: rng (f1 /. i) c= X
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 rng (f1 /. i) c= X by A15, A16, PARTFUN1:def_6; ::_thesis: verum
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 rng (Sum f) c= X by A17, A20, XBOOLE_1:8; ::_thesis: verum
end;
end;
end;
A21: for k being Nat holds S1[k] from NAT_1:sch_2(A1, A2);
let f be FinSequence of Curves T; ::_thesis: ( ( for i being Nat st 1 <= i & i <= len f holds
rng (f /. i) c= X ) implies rng (Sum f) c= X )
thus ( ( for i being Nat st 1 <= i & i <= len f holds
rng (f /. i) c= X ) implies rng (Sum f) c= X ) by A21; ::_thesis: verum
end;
theorem Th43: :: TOPALG_6:43
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; ::_thesis: 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)))) )
defpred S1[ 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: S1[ 0 ] ;
A2: for k being Nat st S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A3: S1[k] ; ::_thesis: S1[k + 1]
let f be FinSequence of Curves T; ::_thesis: ( len f = k + 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 ) implies 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)))) ) )
assume A4: ( len f = k + 1 & len f > 0 ) ; ::_thesis: ( ex i being Nat st
( 1 <= i & i < len f & not ( (f /. i) . (sup (dom (f /. i))) = (f /. (i + 1)) . (inf (dom (f /. (i + 1)))) & sup (dom (f /. i)) = inf (dom (f /. (i + 1))) ) ) or ex i being Nat st
( 1 <= i & i <= len f & not f /. i is with_endpoints ) or 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)))) ) )
consider f1 being FinSequence of Curves T, c2 being 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))) ) ; ::_thesis: ( ex i being Nat st
( 1 <= i & i <= len f & not f /. i is with_endpoints ) or 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)))) ) )
assume A8: for i being Nat st 1 <= i & i <= len f holds
f /. i is with_endpoints ; ::_thesis: 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)))) )
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 = 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; ::_thesis: ( 1 <= i & i < len f1 implies ( (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: ( 1 <= i & i < len f1 ) ; ::_thesis: ( (f1 /. i) . (sup (dom (f1 /. i))) = (f1 /. (i + 1)) . (inf (dom (f1 /. (i + 1)))) & sup (dom (f1 /. i)) = inf (dom (f1 /. (i + 1))) )
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 ( (f1 /. i) . (sup (dom (f1 /. i))) = (f1 /. (i + 1)) . (inf (dom (f1 /. (i + 1)))) & sup (dom (f1 /. i)) = inf (dom (f1 /. (i + 1))) ) by A16, A19, A13, A12, A7; ::_thesis: verum
end;
A20: for i being Nat st 1 <= i & i <= len f1 holds
f1 /. i is with_endpoints
proof
let i be Nat; ::_thesis: ( 1 <= i & i <= len f1 implies f1 /. i is with_endpoints )
assume A21: ( 1 <= i & i <= len f1 ) ; ::_thesis: f1 /. i is with_endpoints
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 f1 /. i is with_endpoints by A22, A21, A8; ::_thesis: verum
end;
percases ( len f1 = 0 or not len f1 = 0 ) ;
suppose len f1 = 0 ; ::_thesis: 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)))) )
then f1 = {} by FINSEQ_1:20;
then A25: f = <*c2*> by A5, FINSEQ_1:34;
take c2 ; ::_thesis: ( Sum f = c2 & dom c2 = [.(inf (dom (f /. 1))),(sup (dom (f /. (len f)))).] & the_first_point_of c2 = (f /. 1) . (inf (dom (f /. 1))) & the_last_point_of c2 = (f /. (len f)) . (sup (dom (f /. (len f)))) )
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 ( Sum f = c2 & dom c2 = [.(inf (dom (f /. 1))),(sup (dom (f /. (len f)))).] & the_first_point_of c2 = (f /. 1) . (inf (dom (f /. 1))) & the_last_point_of c2 = (f /. (len f)) . (sup (dom (f /. (len f)))) ) by A25, Th40, A27, A28, Th27; ::_thesis: verum
end;
supposeA29: not len f1 = 0 ; ::_thesis: 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)))) )
consider c1 being 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 < (len f1) + 1 by A29, XREAL_1:6;
then A32: 1 <= len f1 by NAT_1:13;
A33: len f1 < len f by A6, NAT_1:13;
then A34: ( (f /. (len f1)) . (sup (dom (f /. (len f1)))) = (f /. ((len f1) + 1)) . (inf (dom (f /. ((len f1) + 1)))) & sup (dom (f /. (len f1))) = inf (dom (f /. ((len f1) + 1))) ) by A32, A7;
(len f1) + 1 in Seg (len f) by A6, A31, FINSEQ_1:1;
then A35: (len f1) + 1 in dom f by FINSEQ_1:def_3;
A36: f /. ((len f1) + 1) = f . ((len f1) + 1) by A35, PARTFUN1:def_6
.= c2 by A5, FINSEQ_1:42 ;
A37: inf (dom (f1 /. 1)) <= sup (dom (f1 /. (len f1))) by A30, XXREAL_1:29;
A38: dom f1 c= dom f by A5, FINSEQ_1:26;
len f1 in Seg (len f1) by A29, FINSEQ_1:3;
then A39: len f1 in dom f1 by FINSEQ_1:def_3;
A40: f1 /. (len f1) = f1 . (len f1) by A39, PARTFUN1:def_6
.= f . (len f1) by A5, A39, FINSEQ_1:def_7
.= f /. (len f1) by A39, A38, PARTFUN1:def_6 ;
A41: sup (dom c1) = inf (dom c2) by A36, A34, A40, A30, XXREAL_1:29, XXREAL_2:29;
A42: the_last_point_of c1 = the_first_point_of c2 by A36, A30, A40, A33, A32, A7;
A43: ( 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 ) by A41, A42, Th38;
1 in Seg (len f1) by A32, FINSEQ_1:1;
then A44: 1 in dom f1 by FINSEQ_1:def_3;
A45: f1 /. 1 = f1 . 1 by A44, PARTFUN1:def_6
.= f . 1 by A44, A5, FINSEQ_1:def_7
.= f /. 1 by A44, A38, PARTFUN1:def_6 ;
reconsider c = c1 + c2 as with_endpoints Curve of T by A41, A42, Th38;
take c ; ::_thesis: ( 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)))) )
inf (dom c1) <= sup (dom c2) by A43, XXREAL_1:29;
hence ( 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)))) ) by A43, A45, A37, A30, A5, Th41, A36, A6, XXREAL_2:25, XXREAL_2:29; ::_thesis: verum
end;
end;
end;
for k being Nat holds S1[k] from NAT_1:sch_2(A1, A2);
hence 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)))) ) ; ::_thesis: verum
end;
theorem Th44: :: TOPALG_6:44
for T being non empty TopSpace
for f1, f2 being FinSequence of Curves T
for c1, c2 being with_endpoints Curve of T st 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
proof
let T be non empty TopSpace; ::_thesis: for f1, f2 being FinSequence of Curves T
for c1, c2 being with_endpoints Curve of T st 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
defpred S1[ 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: S1[ 0 ] ;
A2: for k being Nat st S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A3: S1[k] ; ::_thesis: S1[k + 1]
let f1, f2 be FinSequence of Curves T; ::_thesis: for c1, c2 being with_endpoints Curve of T st len f1 = k + 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
let c1, c2 be with_endpoints Curve of T; ::_thesis: ( len f1 = k + 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 ) ) implies c1,c2 are_homotopic )
assume A4: ( len f1 = k + 1 & len f1 > 0 ) ; ::_thesis: ( not len f1 = len f2 or not Sum f1 = c1 or not Sum f2 = c2 or ex i being Nat st
( 1 <= i & i < len f1 & not ( (f1 /. i) . (sup (dom (f1 /. i))) = (f1 /. (i + 1)) . (inf (dom (f1 /. (i + 1)))) & sup (dom (f1 /. i)) = inf (dom (f1 /. (i + 1))) ) ) or ex i being Nat st
( 1 <= i & i < len f2 & not ( (f2 /. i) . (sup (dom (f2 /. i))) = (f2 /. (i + 1)) . (inf (dom (f2 /. (i + 1)))) & sup (dom (f2 /. i)) = inf (dom (f2 /. (i + 1))) ) ) or ex i being Nat st
( 1 <= i & i <= len f1 & ( for c3, c4 being with_endpoints Curve of T holds
( not c3 = f1 /. i or not c4 = f2 /. i or not c3,c4 are_homotopic or not dom c3 = dom c4 ) ) ) or c1,c2 are_homotopic )
consider f1a being FinSequence of Curves T, c1b being 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 ; ::_thesis: ( not Sum f1 = c1 or not Sum f2 = c2 or ex i being Nat st
( 1 <= i & i < len f1 & not ( (f1 /. i) . (sup (dom (f1 /. i))) = (f1 /. (i + 1)) . (inf (dom (f1 /. (i + 1)))) & sup (dom (f1 /. i)) = inf (dom (f1 /. (i + 1))) ) ) or ex i being Nat st
( 1 <= i & i < len f2 & not ( (f2 /. i) . (sup (dom (f2 /. i))) = (f2 /. (i + 1)) . (inf (dom (f2 /. (i + 1)))) & sup (dom (f2 /. i)) = inf (dom (f2 /. (i + 1))) ) ) or ex i being Nat st
( 1 <= i & i <= len f1 & ( for c3, c4 being with_endpoints Curve of T holds
( not c3 = f1 /. i or not c4 = f2 /. i or not c3,c4 are_homotopic or not dom c3 = dom c4 ) ) ) or c1,c2 are_homotopic )
consider f2a being FinSequence of Curves T, c2b being 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 ) ; ::_thesis: ( ex i being Nat st
( 1 <= i & i < len f1 & not ( (f1 /. i) . (sup (dom (f1 /. i))) = (f1 /. (i + 1)) . (inf (dom (f1 /. (i + 1)))) & sup (dom (f1 /. i)) = inf (dom (f1 /. (i + 1))) ) ) or ex i being Nat st
( 1 <= i & i < len f2 & not ( (f2 /. i) . (sup (dom (f2 /. i))) = (f2 /. (i + 1)) . (inf (dom (f2 /. (i + 1)))) & sup (dom (f2 /. i)) = inf (dom (f2 /. (i + 1))) ) ) or ex i being Nat st
( 1 <= i & i <= len f1 & ( for c3, c4 being with_endpoints Curve of T holds
( not c3 = f1 /. i or not c4 = f2 /. i or not c3,c4 are_homotopic or not dom c3 = dom c4 ) ) ) or c1,c2 are_homotopic )
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))) ) ; ::_thesis: ( ex i being Nat st
( 1 <= i & i < len f2 & not ( (f2 /. i) . (sup (dom (f2 /. i))) = (f2 /. (i + 1)) . (inf (dom (f2 /. (i + 1)))) & sup (dom (f2 /. i)) = inf (dom (f2 /. (i + 1))) ) ) or ex i being Nat st
( 1 <= i & i <= len f1 & ( for c3, c4 being with_endpoints Curve of T holds
( not c3 = f1 /. i or not c4 = f2 /. i or not c3,c4 are_homotopic or not dom c3 = dom c4 ) ) ) or c1,c2 are_homotopic )
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))) ) ; ::_thesis: ( ex i being Nat st
( 1 <= i & i <= len f1 & ( for c3, c4 being with_endpoints Curve of T holds
( not c3 = f1 /. i or not c4 = f2 /. i or not c3,c4 are_homotopic or not dom c3 = dom c4 ) ) ) or c1,c2 are_homotopic )
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 ) ; ::_thesis: c1,c2 are_homotopic
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 being 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 = c1b, c2b = c2b as with_endpoints Curve of T by A7, A20, A18, A14, A16, A17, A5, A6, FINSEQ_1:42, PARTFUN1:def_6;
percases ( len f1a > 0 or not len f1a > 0 ) ;
supposeA22: len f1a > 0 ; ::_thesis: c1,c2 are_homotopic
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; ::_thesis: ( 1 <= i & i < len f1a implies ( (f1a /. i) . (sup (dom (f1a /. i))) = (f1a /. (i + 1)) . (inf (dom (f1a /. (i + 1)))) & sup (dom (f1a /. i)) = inf (dom (f1a /. (i + 1))) ) )
assume A24: ( 1 <= i & i < len f1a ) ; ::_thesis: ( (f1a /. i) . (sup (dom (f1a /. i))) = (f1a /. (i + 1)) . (inf (dom (f1a /. (i + 1)))) & sup (dom (f1a /. i)) = inf (dom (f1a /. (i + 1))) )
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 ( (f1a /. i) . (sup (dom (f1a /. i))) = (f1a /. (i + 1)) . (inf (dom (f1a /. (i + 1)))) & sup (dom (f1a /. i)) = inf (dom (f1a /. (i + 1))) ) by A26, A29, A24, A11; ::_thesis: verum
end;
for i being Nat st 1 <= i & i <= len f1a holds
f1a /. i is with_endpoints
proof
let i be Nat; ::_thesis: ( 1 <= i & i <= len f1a implies f1a /. i is with_endpoints )
assume A33: ( 1 <= i & i <= len f1a ) ; ::_thesis: f1a /. i is with_endpoints
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 f1a /. i is with_endpoints by A38; ::_thesis: verum
end;
then consider c1a being 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; ::_thesis: ( 1 <= i & i < len f2a implies ( (f2a /. i) . (sup (dom (f2a /. i))) = (f2a /. (i + 1)) . (inf (dom (f2a /. (i + 1)))) & sup (dom (f2a /. i)) = inf (dom (f2a /. (i + 1))) ) )
assume A41: ( 1 <= i & i < len f2a ) ; ::_thesis: ( (f2a /. i) . (sup (dom (f2a /. i))) = (f2a /. (i + 1)) . (inf (dom (f2a /. (i + 1)))) & sup (dom (f2a /. i)) = inf (dom (f2a /. (i + 1))) )
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 ( (f2a /. i) . (sup (dom (f2a /. i))) = (f2a /. (i + 1)) . (inf (dom (f2a /. (i + 1)))) & sup (dom (f2a /. i)) = inf (dom (f2a /. (i + 1))) ) by A43, A46, A41, A12; ::_thesis: verum
end;
for i being Nat st 1 <= i & i <= len f2a holds
f2a /. i is with_endpoints
proof
let i be Nat; ::_thesis: ( 1 <= i & i <= len f2a implies f2a /. i is with_endpoints )
assume A50: ( 1 <= i & i <= len f2a ) ; ::_thesis: f2a /. i is with_endpoints
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 f2a /. i is with_endpoints by A55; ::_thesis: verum
end;
then consider c2a being 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; ::_thesis: ( 1 <= i & i <= len f1a implies ex c3, c4 being with_endpoints Curve of T st
( c3 = f1a /. i & c4 = f2a /. i & c3,c4 are_homotopic & dom c3 = dom c4 ) )
assume A57: ( 1 <= i & i <= len f1a ) ; ::_thesis: ex c3, c4 being with_endpoints Curve of T st
( c3 = f1a /. i & c4 = f2a /. i & c3,c4 are_homotopic & dom c3 = dom c4 )
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 being 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 ; ::_thesis: ex c4 being with_endpoints Curve of T st
( c3 = f1a /. i & c4 = f2a /. i & c3,c4 are_homotopic & dom c3 = dom c4 )
take c4 ; ::_thesis: ( c3 = f1a /. i & c4 = f2a /. i & c3,c4 are_homotopic & dom c3 = dom 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 ( c3 = f1a /. i & c4 = f2a /. i & c3,c4 are_homotopic & dom c3 = dom c4 ) by A64, A65; ::_thesis: verum
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; ::_thesis: verum
end;
supposeA87: not len f1a > 0 ; ::_thesis: c1,c2 are_homotopic
then f1a = {} by FINSEQ_1:20;
then f1 = <*c1b*> by A5, FINSEQ_1:34;
then A88: Sum f1 = c1b by Th40;
f2a = {} by A87, A6, A7, A9, FINSEQ_1:20;
then f2 = <*c2b*> by A8, FINSEQ_1:34;
hence c1,c2 are_homotopic by A88, A18, A21, A10, Th40; ::_thesis: verum
end;
end;
end;
for k being Nat holds S1[k] from NAT_1:sch_2(A1, A2);
hence for f1, f2 being FinSequence of Curves T
for c1, c2 being with_endpoints Curve of T st 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 ; ::_thesis: verum
end;
theorem Th45: :: TOPALG_6:45
for T being non empty TopStruct
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
let T be non empty TopStruct ; ::_thesis: 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)).] ) )
defpred S1[ 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: S1[ 0 ] ;
A2: for k being Nat st S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A3: S1[k] ; ::_thesis: S1[k + 1]
let c be with_endpoints Curve of T; ::_thesis: for h being FinSequence of REAL st len h = k + 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)).] ) )
let h be FinSequence of REAL ; ::_thesis: ( len h = k + 1 & len h >= 2 & h . 1 = inf (dom c) & h . (len h) = sup (dom c) & h is increasing implies 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)).] ) ) )
assume A4: ( len h = k + 1 & len h >= 2 & h . 1 = inf (dom c) & h . (len h) = sup (dom c) & h is increasing ) ; ::_thesis: 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)).] ) )
consider h1 being FinSequence of REAL , r being 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 number ;
consider c1, c2 being 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;
percases ( len h1 < 2 or len h1 >= 2 ) ;
suppose len h1 < 2 ; ::_thesis: 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)).] ) )
then len h1 < 1 + 1 ;
then A10: len h1 <= 1 by NAT_1:13;
percases ( h1 = {} or h1 <> {} ) ;
suppose h1 = {} ; ::_thesis: 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)).] ) )
then h = <*r*> by A5, FINSEQ_1:34;
then len h = 1 by FINSEQ_1:40;
hence 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)).] ) ) by A4; ::_thesis: verum
end;
suppose h1 <> {} ; ::_thesis: 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)).] ) )
then len h1 >= 1 by FINSEQ_1:20;
then A11: len h1 = 1 by A10, XXREAL_0:1;
set f = <*c*>;
take <*c*> ; ::_thesis: ( len <*c*> = (len h) - 1 & c = Sum <*c*> & ( for i being Nat st 1 <= i & i <= len <*c*> holds
<*c*> /. i = c | [.(h /. i),(h /. (i + 1)).] ) )
A12: len <*c*> = 1 by FINSEQ_1:40;
thus len <*c*> = (len h) - 1 by A11, A6, FINSEQ_1:40; ::_thesis: ( c = Sum <*c*> & ( for i being Nat st 1 <= i & i <= len <*c*> holds
<*c*> /. i = c | [.(h /. i),(h /. (i + 1)).] ) )
thus c = Sum <*c*> by Th40; ::_thesis: for i being Nat st 1 <= i & i <= len <*c*> holds
<*c*> /. i = c | [.(h /. i),(h /. (i + 1)).]
thus for i being Nat st 1 <= i & i <= len <*c*> holds
<*c*> /. i = c | [.(h /. i),(h /. (i + 1)).] ::_thesis: verum
proof
let i be Nat; ::_thesis: ( 1 <= i & i <= len <*c*> implies <*c*> /. i = c | [.(h /. i),(h /. (i + 1)).] )
assume A13: ( 1 <= i & i <= len <*c*> ) ; ::_thesis: <*c*> /. i = c | [.(h /. i),(h /. (i + 1)).]
then A14: i = 1 by A12, XXREAL_0:1;
i in Seg (len <*c*>) by A13, FINSEQ_1:1;
then A15: i in dom <*c*> 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 <*c*> /. i = <*c*> . i by A15, PARTFUN1:def_6
.= c by A14, FINSEQ_1:40
.= c | (dom c)
.= c | [.(h /. i),(h /. (i + 1)).] by A16, A17, Th27 ; ::_thesis: verum
end;
end;
end;
end;
supposeA18: len h1 >= 2 ; ::_thesis: 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)).] ) )
then A19: 1 < k by A4, A6, XXREAL_0:2;
then k in Seg (len h) by A4, A8, FINSEQ_1:1;
then A20: k in dom h by FINSEQ_1:def_3;
k + 1 in Seg (len h) by A4, FINSEQ_1:4;
then A21: k + 1 in dom h by FINSEQ_1:def_3;
h . k <= h . (k + 1) by A8, A20, A21, A4, VALUED_0:def_13;
then [.(inf (dom c)),r1.] c= [.(inf (dom c)),(sup (dom c)).] by A4, XXREAL_1:34;
then A22: [.(inf (dom c)),r1.] c= dom c by Th27;
A23: dom c1 = (dom c) /\ [.(inf (dom c)),r1.] by A7, RELAT_1:61
.= [.(inf (dom c)),r1.] by A22, XBOOLE_1:28 ;
A24: inf (dom c) <= r1 by A4, A19, A9, A20, VALUED_0:def_13;
then A25: r1 = sup (dom c1) by A23, XXREAL_2:29;
A26: inf (dom c1) = inf (dom c) by A24, A23, XXREAL_2:25;
then inf (dom c1) in [.(inf (dom c)),r1.] by A24, XXREAL_1:1;
then dom c1 is left_end by A23, XXREAL_2:def_5;
then A27: c1 is with_first_point by Def6;
r1 in [.(inf (dom c)),r1.] by A24, XXREAL_1:1;
then dom c1 is right_end by A25, A23, XXREAL_2:def_6;
then A28: c1 is with_last_point by Def7;
reconsider c1 = c1 as with_endpoints Curve of T by A27, A28;
A29: h1 = h | (dom h1) by A5, FINSEQ_1:21;
1 in Seg k by A19, FINSEQ_1:1;
then A30: 1 in dom h1 by A4, A6, FINSEQ_1:def_3;
k in Seg k by A19, FINSEQ_1:1;
then A31: len h1 in dom h1 by A4, A6, FINSEQ_1:def_3;
A32: h1 . 1 = inf (dom c1) by A4, A26, A29, A30, FUNCT_1:49;
A33: h1 . (len h1) = h . k by A6, A4, A29, A31, FUNCT_1:49
.= sup (dom c1) by A24, A23, XXREAL_2:29 ;
A34: dom h c= REAL by XBOOLE_1:1;
rng h c= REAL ;
then reconsider h0 = h as PartFunc of REAL,REAL by A34, RELSET_1:4;
A35: h0 | (dom h0) is increasing by A4;
len h1 <= len h by A6, NAT_1:19;
then Seg (len h1) c= Seg (len h) by FINSEQ_1:5;
then Seg (len h1) c= dom h by FINSEQ_1:def_3;
then A36: dom h1 c= dom h by FINSEQ_1:def_3;
then A37: h1 is increasing by A29, A35, RFUNCT_2:28;
consider f1 being FinSequence of Curves T such that
A38: ( len f1 = (len h1) - 1 & c1 = Sum f1 & ( for i being Nat st 1 <= i & i <= len f1 holds
f1 /. i = c1 | [.(h1 /. i),(h1 /. (i + 1)).] ) ) by A3, A4, A6, A18, A32, A33, A37;
set f = f1 ^ <*c2*>;
take f1 ^ <*c2*> ; ::_thesis: ( len (f1 ^ <*c2*>) = (len h) - 1 & c = Sum (f1 ^ <*c2*>) & ( for i being Nat st 1 <= i & i <= len (f1 ^ <*c2*>) holds
(f1 ^ <*c2*>) /. i = c | [.(h /. i),(h /. (i + 1)).] ) )
A39: len (f1 ^ <*c2*>) = (len f1) + (len <*c2*>) by FINSEQ_1:22
.= (len f1) + 1 by FINSEQ_1:40 ;
thus len (f1 ^ <*c2*>) = (len h) - 1 by A6, A38, A39; ::_thesis: ( c = Sum (f1 ^ <*c2*>) & ( for i being Nat st 1 <= i & i <= len (f1 ^ <*c2*>) holds
(f1 ^ <*c2*>) /. i = c | [.(h /. i),(h /. (i + 1)).] ) )
thus c = Sum (f1 ^ <*c2*>) by Th41, A7, A38; ::_thesis: for i being Nat st 1 <= i & i <= len (f1 ^ <*c2*>) holds
(f1 ^ <*c2*>) /. i = c | [.(h /. i),(h /. (i + 1)).]
thus for i being Nat st 1 <= i & i <= len (f1 ^ <*c2*>) holds
(f1 ^ <*c2*>) /. i = c | [.(h /. i),(h /. (i + 1)).] ::_thesis: verum
proof
let i be Nat; ::_thesis: ( 1 <= i & i <= len (f1 ^ <*c2*>) implies (f1 ^ <*c2*>) /. i = c | [.(h /. i),(h /. (i + 1)).] )
assume A40: ( 1 <= i & i <= len (f1 ^ <*c2*>) ) ; ::_thesis: (f1 ^ <*c2*>) /. i = c | [.(h /. i),(h /. (i + 1)).]
then i in Seg (len (f1 ^ <*c2*>)) by FINSEQ_1:1;
then A41: i in dom (f1 ^ <*c2*>) by FINSEQ_1:def_3;
percases ( i = len (f1 ^ <*c2*>) or i <> len (f1 ^ <*c2*>) ) ;
supposeA42: i = len (f1 ^ <*c2*>) ; ::_thesis: (f1 ^ <*c2*>) /. i = c | [.(h /. i),(h /. (i + 1)).]
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 (f1 ^ <*c2*>) /. i = (f1 ^ <*c2*>) . i by A41, PARTFUN1:def_6
.= c | [.(h /. i),(h /. (i + 1)).] by A43, A44, A7, A42, A39, FINSEQ_1:42 ; ::_thesis: verum
end;
suppose i <> len (f1 ^ <*c2*>) ; ::_thesis: (f1 ^ <*c2*>) /. i = c | [.(h /. i),(h /. (i + 1)).]
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
percases ( i + 1 = k or i + 1 <> k ) ;
suppose i + 1 = k ; ::_thesis: h . (i + 1) <= h . k
hence h . (i + 1) <= h . k ; ::_thesis: verum
end;
suppose i + 1 <> k ; ::_thesis: h . (i + 1) <= h . k
then i + 1 < k by A38, A6, A4, A53, XXREAL_0:1;
hence h . (i + 1) <= h . k by A51, A36, A20, A4, VALUED_0:def_13; ::_thesis: verum
end;
end;
end;
then A54: h /. (i + 1) <= r1 by A51, A36, PARTFUN1:def_6;
h . 1 <= h . i
proof
percases ( i = 1 or i <> 1 ) ;
suppose i = 1 ; ::_thesis: h . 1 <= h . i
hence h . 1 <= h . i ; ::_thesis: verum
end;
suppose i <> 1 ; ::_thesis: h . 1 <= h . i
then 1 < i by A40, XXREAL_0:1;
hence h . 1 <= h . i by A36, A48, A9, A4, VALUED_0:def_13; ::_thesis: verum
end;
end;
end;
then A55: inf (dom c) <= h /. i by A4, A36, A48, PARTFUN1:def_6;
(f1 ^ <*c2*>) . 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 (f1 ^ <*c2*>) /. i = c | [.(h /. i),(h /. (i + 1)).] by A41, A52, A49, PARTFUN1:def_6; ::_thesis: verum
end;
end;
end;
end;
end;
end;
for k being Nat holds S1[k] from NAT_1:sch_2(A1, A2);
hence 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)).] ) ) ; ::_thesis: verum
end;
Lm3: for n being Nat
for t being Point of (TUnitSphere n)
for f being Loop of t st rng f <> the carrier of (TUnitSphere n) holds
f is nullhomotopic
proof
let n be Nat; ::_thesis: for t being Point of (TUnitSphere n)
for f being Loop of t st rng f <> the carrier of (TUnitSphere n) holds
f is nullhomotopic
let t be Point of (TUnitSphere n); ::_thesis: for f being Loop of t st rng f <> the carrier of (TUnitSphere n) holds
f is nullhomotopic
let f be Loop of t; ::_thesis: ( rng f <> the carrier of (TUnitSphere n) implies f is nullhomotopic )
assume A1: rng f <> the carrier of (TUnitSphere n) ; ::_thesis: f is nullhomotopic
for x being set st x in rng f holds
x in the carrier of (TUnitSphere n) ;
then consider x being set such that
A2: x in the carrier of (TUnitSphere n) and
A3: not x in rng f by A1, TARSKI:1;
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 f is nullhomotopic by Th18; ::_thesis: verum
end;
Lm4: for n being Nat
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 n be Nat; ::_thesis: 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) )
let t be Point of (TUnitSphere n); ::_thesis: 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) )
let f be Loop of t; ::_thesis: ( n >= 2 & rng f = the carrier of (TUnitSphere n) implies ex g being Loop of t st
( g,f are_homotopic & rng g <> the carrier of (TUnitSphere n) ) )
assume that
A1: n >= 2 and
A2: rng f = the carrier of (TUnitSphere n) ; ::_thesis: ex g being Loop of t st
( g,f are_homotopic & rng g <> 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 being 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 Element of NAT
for Q being Subset of I[01]
for W being Subset of (Euclid n1) st 1 <= i & i < len h & Q = [.(h /. i),(h /. (i + 1)).] & W = f1 .: Q holds
diameter W < 1 ) ) by JGRAPH_8:1;
set f2 = f * h;
for x being set st x in rng (f * h) holds
x in the carrier of (TUnitSphere n) ;
then consider x being set such that
A6: ( x in the carrier of (TUnitSphere n) & not x in rng (f * h) ) by A1, TARSKI:1;
A7: [#] (Tunit_circle n1) c= [#] (TOP-REAL n1) by PRE_TOPC:def_4;
A8: x in the carrier of (Tunit_circle n1) by A6, MFOLD_2:def_4;
then reconsider p = x as Point of (TOP-REAL n1) by A7;
p in the carrier of (Tcircle ((0. (TOP-REAL n1)),1)) by A8, TOPREALB:def_7;
then A9: p in Sphere ((0. (TOP-REAL n1)),1) by TOPREALB:9;
then A10: - p in (Sphere ((0. (TOP-REAL n1)),1)) \ {p} by Th3;
then reconsider U = (TOP-REAL n1) | ((Sphere ((0. (TOP-REAL n1)),1)) \ {p}) as non empty SubSpace of TOP-REAL n1 ;
A11: [#] U = (Sphere ((0. (TOP-REAL n1)),1)) \ {p} by PRE_TOPC:def_5;
A12: - p in Sphere ((0. (TOP-REAL n1)),1) by A10, XBOOLE_0:def_5;
then A13: - (- p) in (Sphere ((0. (TOP-REAL n1)),1)) \ {(- p)} by Th3;
then reconsider V = (TOP-REAL n1) | ((Sphere ((0. (TOP-REAL n1)),1)) \ {(- p)}) as non empty SubSpace of TOP-REAL n1 ;
A14: [#] V = (Sphere ((0. (TOP-REAL n1)),1)) \ {(- p)} by PRE_TOPC:def_5;
A15: for i being Element of NAT st 1 <= i & i < len h & not f .: [.(h /. i),(h /. (i + 1)).] c= the carrier of U holds
f .: [.(h /. i),(h /. (i + 1)).] c= the carrier of V
proof
let i be Element of NAT ; ::_thesis: ( 1 <= i & i < len h & not f .: [.(h /. i),(h /. (i + 1)).] c= the carrier of U implies f .: [.(h /. i),(h /. (i + 1)).] c= the carrier of V )
assume A16: ( 1 <= i & i < len h ) ; ::_thesis: ( f .: [.(h /. i),(h /. (i + 1)).] c= the carrier of U or f .: [.(h /. i),(h /. (i + 1)).] c= the carrier of V )
i in Seg (len h) by A16, FINSEQ_1:1;
then A17: i in dom h by FINSEQ_1:def_3;
reconsider h1 = h as real-valued FinSequence ;
reconsider i1 = i + 1 as Nat ;
A18: ( i1 <= len h & 1 <= i1 ) by A16, NAT_1:13;
then i1 in Seg (len h) by FINSEQ_1:1;
then A19: i1 in dom h by FINSEQ_1:def_3;
h1 . (i + 1) <= 1 by A5, A18, EUCLID_7:7;
then A20: h /. (i + 1) <= 1 by A19, PARTFUN1:def_6;
h1 . 1 <= h1 . i by A5, A16, EUCLID_7:7;
then 0 <= h /. i by A5, A17, PARTFUN1:def_6;
then reconsider Q = [.(h /. i),(h /. (i + 1)).] as Subset of I[01] by A20, BORSUK_1:40, XXREAL_1:34;
f .: Q c= the carrier of (TUnitSphere n) ;
then f .: Q c= [#] (Tunit_circle n1) by MFOLD_2:def_4;
then f .: Q c= the carrier of (Tcircle ((0. (TOP-REAL n1)),1)) by TOPREALB:def_7;
then A21: f .: Q c= Sphere ((0. (TOP-REAL n1)),1) by TOPREALB:9;
reconsider W = f1 .: Q as Subset of (Euclid n1) by EUCLID:67;
A22: diameter W < 1 by A16, A5;
Sphere ((0. (TOP-REAL n1)),1) is bounded Subset of (Euclid n1) by JORDAN2C:11;
then A23: W is bounded by A21, TBSP_1:14;
( not p in f .: Q or not - p in f .: Q )
proof
assume A24: ( p in f .: Q & - p in f .: Q ) ; ::_thesis: contradiction
reconsider p1 = p, p2 = - p as Point of (Euclid n1) by EUCLID:67;
A25: dist (p1,p2) <= diameter W by A24, A23, TBSP_1:def_8;
A26: Euclid n1 = MetrStruct(# (REAL n1),(Pitag_dist n1) #) by EUCLID:def_7;
reconsider p3 = p1, p4 = p2 as Element of REAL n1 by A26;
reconsider r1 = 1 as real number ;
dist (p1,p2) = the distance of (Euclid n1) . (p1,p2) by METRIC_1:def_1
.= |.(p3 - p4).| by A26, EUCLID:def_6 ;
then |.(p - (- p)).| < 1 by A25, A22, XXREAL_0:2;
then |.(p + (- (- p))).| < 1 by EUCLID:41;
then |.((r1 * p) + p).| < 1 by EUCLID:29;
then |.((r1 * p) + (r1 * p)).| < 1 by EUCLID:29;
then |.((1 + 1) * p).| < 1 by EUCLID:33;
then A27: (abs 2) * |.p.| < 1 by EUCLID:11;
|.(p - (0. (TOP-REAL n1))).| = 1 by A9, TOPREAL9:9;
then |.(p + (- (0. (TOP-REAL n1)))).| = 1 by EUCLID:41;
then |.(p + ((- 1) * (0. (TOP-REAL n1)))).| = 1 by EUCLID:39;
then |.(p + (0. (TOP-REAL n1))).| = 1 by EUCLID:28;
then A28: |.p.| = 1 by EUCLID:27;
|.2.| = 2 by COMPLEX1:43;
hence contradiction by A28, A27; ::_thesis: verum
end;
hence ( f .: [.(h /. i),(h /. (i + 1)).] c= the carrier of U or f .: [.(h /. i),(h /. (i + 1)).] c= the carrier of V ) by A14, A11, A21, ZFMISC_1:34; ::_thesis: verum
end;
( f is Path of t,t & t,t are_connected ) ;
then reconsider c = f as with_endpoints Curve of (TUnitSphere n) by Th25;
A29: 2 <= len h by A5, XXREAL_0:2;
A30: ( inf (dom f) = 0 & sup (dom f) = 1 ) by Th4;
then consider fc1 being FinSequence of Curves (TUnitSphere n) such that
A31: ( len fc1 = (len h) - 1 & c = Sum fc1 & ( for i being Nat st 1 <= i & i <= len fc1 holds
fc1 /. i = c | [.(h /. i),(h /. (i + 1)).] ) ) by A5, A29, Th45;
A32: for i being Nat st 1 <= i & i <= len fc1 & not rng (fc1 /. i) c= the carrier of U holds
rng (fc1 /. i) c= the carrier of V
proof
let i be Nat; ::_thesis: ( 1 <= i & i <= len fc1 & not rng (fc1 /. i) c= the carrier of U implies rng (fc1 /. i) c= the carrier of V )
assume A33: ( 1 <= i & i <= len fc1 ) ; ::_thesis: ( rng (fc1 /. i) c= the carrier of U or rng (fc1 /. i) c= the carrier of V )
then A34: i < ((len h) - 1) + 1 by A31, NAT_1:13;
reconsider i0 = i as Element of NAT by ORDINAL1:def_12;
f .: [.(h /. i0),(h /. (i0 + 1)).] = rng (f | [.(h /. i0),(h /. (i0 + 1)).]) by RELAT_1:115
.= rng (fc1 /. i) by A33, A31 ;
hence ( rng (fc1 /. i) c= the carrier of U or rng (fc1 /. i) c= the carrier of V ) by A33, A34, A15; ::_thesis: verum
end;
A35: for c1 being with_endpoints Curve of (TUnitSphere n) st rng c1 c= the carrier of V & the_first_point_of c1 <> 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); ::_thesis: ( rng c1 c= the carrier of V & the_first_point_of c1 <> p & the_last_point_of c1 <> p & not inf (dom c1) = sup (dom c1) implies 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 ) )
assume A36: rng c1 c= the carrier of V ; ::_thesis: ( not the_first_point_of c1 <> p or not the_last_point_of c1 <> p or inf (dom c1) = sup (dom c1) or 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 ) )
assume A37: ( the_first_point_of c1 <> p & the_last_point_of c1 <> p ) ; ::_thesis: ( inf (dom c1) = sup (dom c1) or 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 ) )
assume A38: not inf (dom c1) = sup (dom c1) ; ::_thesis: 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 )
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 the_first_point_of c1, the_last_point_of c1 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) ; ::_thesis: contradiction
then (- p) - (0. (TOP-REAL n1)) = 0. (TOP-REAL n1) by EUCLID:42;
then |.(0. (TOP-REAL n1)).| = 1 by A12, TOPREAL9:9;
hence contradiction by EUCLID_2:39; ::_thesis: verum
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 being Function of (TOP-REAL n),V such that
A41: fh is being_homeomorphism by T_0TOPSP:def_1;
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: the_first_point_of c1 in rng c1 by Th31;
A46: not the_first_point_of c1 in {v} by A37, TARSKI:def_1;
A47: the_last_point_of c1 in rng c1 by Th31;
A48: not the_last_point_of c1 in {v} by A37, TARSKI:def_1;
( the_first_point_of c1 in S & the_last_point_of c1 in S ) by A45, A46, A47, A36, A48, XBOOLE_0:def_5;
then ( the_first_point_of c1 in [#] (V | S) & the_last_point_of c1 in [#] (V | S) ) by PRE_TOPC:def_5;
then reconsider v1 = the_first_point_of c1, v2 = the_last_point_of c1 as Point of (V | S) ;
A49: v1,v2 are_connected by A44, BORSUK_2:def_3;
then consider p3 being Function of I[01],(V | S) such that
A50: ( p3 is continuous & p3 . 0 = v1 & p3 . 1 = v2 ) by BORSUK_2:def_1;
reconsider p3 = 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 = the_first_point_of c1, s2 = the_last_point_of c1 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 = 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: the_first_point_of c1, the_last_point_of c1 are_connected by A50, BORSUK_2:def_1;
then reconsider p2 = p3 as Path of the_first_point_of c1, the_last_point_of c1 by A50, A57, BORSUK_2:def_2;
rng p1 c= rng c1 by RELAT_1:26;
then A59: rng p1 c= [#] V by A36, XBOOLE_1:1;
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);
set r2 = sup (dom c1);
A63: inf (dom c1) <= sup (dom c1) by XXREAL_2:40;
then A64: inf (dom c1) < sup (dom c1) by A38, XXREAL_0:1;
then reconsider c2 = p2 * (L[01] ((inf (dom c1)),(sup (dom c1)),0,1)) as with_endpoints Curve of (TUnitSphere n) by A58, Th32;
take c2 ; ::_thesis: ( c1,c2 are_homotopic & rng c2 c= the carrier of U & dom c1 = dom c2 )
rng (L[01] ((inf (dom c1)),(sup (dom c1)),0,1)) c= [#] (Closed-Interval-TSpace (0,1)) by RELAT_1:def_19;
then rng (L[01] ((inf (dom c1)),(sup (dom c1)),0,1)) c= dom p2 by FUNCT_2:def_1, TOPMETR:20;
then dom c2 = dom (L[01] ((inf (dom c1)),(sup (dom c1)),0,1)) by RELAT_1:27;
then dom c2 = [#] (Closed-Interval-TSpace ((inf (dom c1)),(sup (dom c1)))) by FUNCT_2:def_1;
then A65: dom c2 = [.(inf (dom c1)),(sup (dom c1)).] by A63, TOPMETR:18;
A66: c2 * (L[01] (0,1,(inf (dom c1)),(sup (dom c1)))) = p2 * ((L[01] ((inf (dom c1)),(sup (dom c1)),0,1)) * (L[01] (0,1,(inf (dom c1)),(sup (dom c1))))) 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, Def11; ::_thesis: ( rng c2 c= the carrier of U & dom c1 = dom c2 )
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, XBOOLE_1:1;
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, XBOOLE_1:1; ::_thesis: dom c1 = dom c2
thus dom c1 = dom c2 by A65, Th27; ::_thesis: verum
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; ::_thesis: ( 1 <= i & i <= len fc1 implies ( i + 1 in dom h & i in dom h & dom (fc1 /. i) = [.(h /. i),(h /. (i + 1)).] & h /. i < h /. (i + 1) ) )
assume A70: ( 1 <= i & i <= len fc1 ) ; ::_thesis: ( i + 1 in dom h & i in dom h & dom (fc1 /. i) = [.(h /. i),(h /. (i + 1)).] & h /. i < h /. (i + 1) )
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; ::_thesis: ( i in dom h & dom (fc1 /. i) = [.(h /. i),(h /. (i + 1)).] & h /. i < h /. (i + 1) )
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; ::_thesis: ( dom (fc1 /. i) = [.(h /. i),(h /. (i + 1)).] & h /. i < h /. (i + 1) )
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
percases ( i = 1 or not i = 1 ) ;
suppose i = 1 ; ::_thesis: 0 <= h . i
hence 0 <= h . i by A5; ::_thesis: verum
end;
suppose not i = 1 ; ::_thesis: 0 <= h . i
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 0 <= h . i by A5, A75, A79, VALUED_0:def_13; ::_thesis: verum
end;
end;
end;
A80: h . (i + 1) <= 1
proof
percases ( i + 1 = len h or not i + 1 = len h ) ;
suppose i + 1 = len h ; ::_thesis: h . (i + 1) <= 1
hence h . (i + 1) <= 1 by A5; ::_thesis: verum
end;
suppose not i + 1 = len h ; ::_thesis: h . (i + 1) <= 1
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 h . (i + 1) <= 1 by A5, A81, A82, VALUED_0:def_13; ::_thesis: verum
end;
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; ::_thesis: h /. i < h /. (i + 1)
thus h /. i < h /. (i + 1) by A77, A76, A75, A73, A74, A5, VALUED_0:def_13; ::_thesis: verum
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 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 & dom c1 = [.(h /. i),(h /. (i + 1)).] ) ) )
proof
let i be Nat; ::_thesis: ( 1 <= i & i <= len fc1 implies ( fc1 /. i is with_endpoints & ( for c1 being with_endpoints Curve of (TUnitSphere n) st c1 = fc1 /. i 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 & dom c1 = [.(h /. i),(h /. (i + 1)).] ) ) ) )
assume A86: ( 1 <= i & i <= len fc1 ) ; ::_thesis: ( fc1 /. i is with_endpoints & ( for c1 being with_endpoints Curve of (TUnitSphere n) st c1 = fc1 /. i 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 & dom c1 = [.(h /. i),(h /. (i + 1)).] ) ) )
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 & dom (fc1 /. i) is right_end ) by A92, XXREAL_2:33;
then ( fc1 /. i is with_first_point & fc1 /. i is with_last_point ) by Def7, Def6;
hence fc1 /. i is with_endpoints ; ::_thesis: for c1 being with_endpoints Curve of (TUnitSphere n) st c1 = fc1 /. i 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 & dom c1 = [.(h /. i),(h /. (i + 1)).] )
let c1 be with_endpoints Curve of (TUnitSphere n); ::_thesis: ( c1 = fc1 /. i implies 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)).] ) )
assume A93: c1 = fc1 /. i ; ::_thesis: 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)).] )
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;
percases ( rng c1 c= the carrier of U or rng c1 c= the carrier of V ) by A32, A86, A93;
supposeA100: rng c1 c= the carrier of U ; ::_thesis: 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)).] )
take c1 ; ::_thesis: ( c1,c1 are_homotopic & rng c1 c= the carrier of U & dom c1 = dom c1 & dom c1 = [.(h /. i),(h /. (i + 1)).] )
thus ( c1,c1 are_homotopic & rng c1 c= the carrier of U & dom c1 = dom c1 & dom c1 = [.(h /. i),(h /. (i + 1)).] ) by A100, A93, A69, A86; ::_thesis: verum
end;
supposeA101: rng c1 c= the carrier of V ; ::_thesis: 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)).] )
A102: rng h c= dom f by A5, FUNCT_2:def_1;
then A103: dom (f * h) = dom h by RELAT_1:27;
A104: i + 1 in dom (f * h) by A102, A88, RELAT_1:27;
A105: the_first_point_of c1 <> p
proof
assume A106: the_first_point_of c1 = p ; ::_thesis: contradiction
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
.= (f * h) . i by A103, A89, FUNCT_1:12 ;
hence contradiction by A6, A106, A103, A89, FUNCT_1:3; ::_thesis: verum
end;
A107: the_last_point_of c1 <> p
proof
assume A108: the_last_point_of c1 = p ; ::_thesis: contradiction
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
.= (f * h) . (i + 1) by A104, FUNCT_1:12 ;
hence contradiction by A6, A108, A104, FUNCT_1:3; ::_thesis: verum
end;
consider c2 being 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 ; ::_thesis: ( c1,c2 are_homotopic & rng c2 c= the carrier of U & dom c1 = dom c2 & dom c1 = [.(h /. i),(h /. (i + 1)).] )
thus ( c1,c2 are_homotopic & rng c2 c= the carrier of U & dom c1 = dom c2 & dom c1 = [.(h /. i),(h /. (i + 1)).] ) by A109, A93, A69, A86; ::_thesis: verum
end;
end;
end;
defpred S1[ set , set ] means for i being Nat
for 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) holds
ex x being Element of Curves (TUnitSphere n) st S1[k,x]
proof
let k be Nat; ::_thesis: ( k in Seg (len fc1) implies ex x being Element of Curves (TUnitSphere n) st S1[k,x] )
assume k in Seg (len fc1) ; ::_thesis: ex x being Element of Curves (TUnitSphere n) st S1[k,x]
then A111: ( 1 <= k & k <= len fc1 ) by FINSEQ_1:1;
set c1 = fc1 /. k;
reconsider c1 = fc1 /. k as with_endpoints Curve of (TUnitSphere n) by A111, A85;
consider c2 being 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 ; ::_thesis: S1[k,x]
thus S1[k,x] by A112; ::_thesis: verum
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
S1[k,p . k] ) ) from FINSEQ_1:sch_5(A110);
then consider fc2 being FinSequence of Curves (TUnitSphere n) such that
A113: ( dom fc2 = Seg (len fc1) & ( for k being Nat st k in Seg (len fc1) holds
S1[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; ::_thesis: ( 1 <= i & i < len fc2 implies ( (fc2 /. i) . (sup (dom (fc2 /. i))) = (fc2 /. (i + 1)) . (inf (dom (fc2 /. (i + 1)))) & sup (dom (fc2 /. i)) = inf (dom (fc2 /. (i + 1))) ) )
assume A118: ( 1 <= i & i < len fc2 ) ; ::_thesis: ( (fc2 /. i) . (sup (dom (fc2 /. i))) = (fc2 /. (i + 1)) . (inf (dom (fc2 /. (i + 1)))) & sup (dom (fc2 /. i)) = inf (dom (fc2 /. (i + 1))) )
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 = fc1 /. i as with_endpoints Curve of (TUnitSphere n) by A118, A114, A85;
consider di being 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 = fc1 /. (i + 1) as with_endpoints Curve of (TUnitSphere n) by A121, A122, A114, A85;
consider di1 being 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 ; ::_thesis: sup (dom (fc2 /. i)) = inf (dom (fc2 /. (i + 1)))
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 ; ::_thesis: verum
end;
A136: for i being Nat st 1 <= i & i <= len fc2 holds
fc2 /. i is with_endpoints
proof
let i be Nat; ::_thesis: ( 1 <= i & i <= len fc2 implies fc2 /. i is with_endpoints )
assume A137: ( 1 <= i & i <= len fc2 ) ; ::_thesis: fc2 /. i is with_endpoints
then A138: i in Seg (len fc1) by A114, FINSEQ_1:1;
set ci = fc1 /. i;
reconsider ci = fc1 /. i as with_endpoints Curve of (TUnitSphere n) by A137, A114, A85;
consider di being 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; ::_thesis: verum
end;
consider c0 being 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; ::_thesis: ( 1 <= i & i < len fc1 implies ( (fc1 /. i) . (sup (dom (fc1 /. i))) = (fc1 /. (i + 1)) . (inf (dom (fc1 /. (i + 1)))) & sup (dom (fc1 /. i)) = inf (dom (fc1 /. (i + 1))) ) )
assume A142: ( 1 <= i & i < len fc1 ) ; ::_thesis: ( (fc1 /. i) . (sup (dom (fc1 /. i))) = (fc1 /. (i + 1)) . (inf (dom (fc1 /. (i + 1)))) & sup (dom (fc1 /. i)) = inf (dom (fc1 /. (i + 1))) )
A143: i in Seg (len fc1) by A142, FINSEQ_1:1;
set ci = fc1 /. i;
reconsider ci = fc1 /. i as with_endpoints Curve of (TUnitSphere n) by A142, A85;
consider di being 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 = fc1 /. (i + 1) as with_endpoints Curve of (TUnitSphere n) by A145, A146, A114, A85;
consider di1 being 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 ; ::_thesis: sup (dom (fc1 /. i)) = inf (dom (fc1 /. (i + 1)))
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 ; ::_thesis: verum
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; ::_thesis: ( 1 <= i & i <= len fc2 implies 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 ) )
assume A155: ( 1 <= i & i <= len fc2 ) ; ::_thesis: 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 )
then A156: i in Seg (len fc1) by A114, FINSEQ_1:1;
set ci = fc1 /. i;
reconsider ci = fc1 /. i as with_endpoints Curve of (TUnitSphere n) by A155, A114, A85;
consider di being 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 ; ::_thesis: ex c4 being with_endpoints Curve of (TUnitSphere n) st
( di = fc2 /. i & c4 = fc1 /. i & di,c4 are_homotopic & dom di = dom c4 )
take ci ; ::_thesis: ( di = fc2 /. i & ci = fc1 /. i & di,ci are_homotopic & dom di = dom ci )
thus ( di = fc2 /. i & ci = fc1 /. i & di,ci are_homotopic & dom di = dom ci ) by A157, A158, PARTFUN1:def_6; ::_thesis: verum
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 = fc1 /. 1 as with_endpoints Curve of (TUnitSphere n) by A115, A31, A85;
consider di being 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 = fc1 /. (len fc1) as with_endpoints Curve of (TUnitSphere n) by A115, A31, A85;
consider di1 being 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 dom c0 = [.0,1.] by A140, A167, A173; ::_thesis: verum
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; ::_thesis: ( 1 <= i & i <= len fc2 implies rng (fc2 /. i) c= the carrier of U )
assume A174: ( 1 <= i & i <= len fc2 ) ; ::_thesis: rng (fc2 /. i) c= the carrier of U
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 being 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 rng (fc2 /. i) c= the carrier of U by A175, A177, PARTFUN1:def_6; ::_thesis: verum
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 ; ::_thesis: contradiction
then not p in {p} by A11, A178, XBOOLE_0:def_5;
hence contradiction by TARSKI:def_1; ::_thesis: verum
end;
hence ex g being Loop of t st
( g,f are_homotopic & rng g <> the carrier of (TUnitSphere n) ) by A6, A182; ::_thesis: verum
end;
theorem Th46: :: TOPALG_6:46
for n being Nat st n >= 2 holds
TUnitSphere n is having_trivial_Fundamental_Group
proof
let n be Nat; ::_thesis: ( n >= 2 implies TUnitSphere n is having_trivial_Fundamental_Group )
assume A1: n >= 2 ; ::_thesis: TUnitSphere n is having_trivial_Fundamental_Group
set T = TUnitSphere n;
for t being Point of (TUnitSphere n)
for f being Loop of t holds f is nullhomotopic
proof
let t be Point of (TUnitSphere n); ::_thesis: for f being Loop of t holds f is nullhomotopic
let f be Loop of t; ::_thesis: f is nullhomotopic
percases ( rng f <> the carrier of (TUnitSphere n) or rng f = the carrier of (TUnitSphere n) ) ;
suppose rng f <> the carrier of (TUnitSphere n) ; ::_thesis: f is nullhomotopic
hence f is nullhomotopic by Lm3; ::_thesis: verum
end;
suppose rng f = the carrier of (TUnitSphere n) ; ::_thesis: f is nullhomotopic
then consider g being 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 being constant Loop of t such that
A4: g,C are_homotopic by Def3;
f,C are_homotopic by A2, A4, BORSUK_6:79;
hence f is nullhomotopic by Def3; ::_thesis: verum
end;
end;
end;
hence TUnitSphere n is having_trivial_Fundamental_Group by Th17; ::_thesis: verum
end;
theorem :: TOPALG_6:47
for n being non empty Nat
for r being real positive number
for x being Point of (TOP-REAL n) st n >= 3 holds
Tcircle (x,r) is having_trivial_Fundamental_Group
proof
let n be non empty Nat; ::_thesis: for r being real positive number
for x being Point of (TOP-REAL n) st n >= 3 holds
Tcircle (x,r) is having_trivial_Fundamental_Group
let r be real positive number ; ::_thesis: for x being Point of (TOP-REAL n) st n >= 3 holds
Tcircle (x,r) is having_trivial_Fundamental_Group
let x be Point of (TOP-REAL n); ::_thesis: ( n >= 3 implies Tcircle (x,r) is having_trivial_Fundamental_Group )
assume A1: n >= 3 ; ::_thesis: Tcircle (x,r) is having_trivial_Fundamental_Group
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 Tcircle (x,r) is having_trivial_Fundamental_Group by A2, A3, A4, A5, Th13; ::_thesis: verum
end;