:: JORDAN2C semantic presentation
begin
registration
let n be Nat;
cluster TOP-REAL n -> add-continuous Mult-continuous ;
coherence
( TOP-REAL n is add-continuous & TOP-REAL n is Mult-continuous )
proof
set T = TOP-REAL n;
set E = Euclid n;
set TE = TopSpaceMetr (Euclid n);
A1: TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def_8;
A2: n in NAT by ORDINAL1:def_12;
thus TOP-REAL n is add-continuous ::_thesis: TOP-REAL n is Mult-continuous
proof
let x1, x2 be Point of (TOP-REAL n); :: according to RLTOPSP1:def_8 ::_thesis: for b1 being Element of bool the carrier of (TOP-REAL n) holds
( not b1 is open or not K258((TOP-REAL n),x1,x2) in b1 or ex b2, b3 being Element of bool the carrier of (TOP-REAL n) st
( b2 is open & b3 is open & x1 in b2 & x2 in b3 & b2 + b3 c= b1 ) )
let V be Subset of (TOP-REAL n); ::_thesis: ( not V is open or not K258((TOP-REAL n),x1,x2) in V or ex b1, b2 being Element of bool the carrier of (TOP-REAL n) st
( b1 is open & b2 is open & x1 in b1 & x2 in b2 & b1 + b2 c= V ) )
assume that
A3: V is open and
A4: x1 + x2 in V ; ::_thesis: ex b1, b2 being Element of bool the carrier of (TOP-REAL n) st
( b1 is open & b2 is open & x1 in b1 & x2 in b2 & b1 + b2 c= V )
reconsider X1 = x1, X2 = x2, X12 = x1 + x2 as Point of (Euclid n) by A1, TOPMETR:12;
reconsider v = V as Subset of (TopSpaceMetr (Euclid n)) by A1;
V in the topology of (TOP-REAL n) by A3, PRE_TOPC:def_2;
then v is open by A1, PRE_TOPC:def_2;
then consider r being real number such that
A5: r > 0 and
A6: Ball (X12,r) c= v by A4, TOPMETR:15;
set r2 = r / 2;
reconsider B1 = Ball (X1,(r / 2)), B2 = Ball (X2,(r / 2)) as Subset of (TOP-REAL n) by A1, TOPMETR:12;
take B1 ; ::_thesis: ex b1 being Element of bool the carrier of (TOP-REAL n) st
( B1 is open & b1 is open & x1 in B1 & x2 in b1 & B1 + b1 c= V )
take B2 ; ::_thesis: ( B1 is open & B2 is open & x1 in B1 & x2 in B2 & B1 + B2 c= V )
thus ( B1 is open & B2 is open & x1 in B1 & x2 in B2 ) by A5, GOBOARD6:1, GOBOARD6:3, XREAL_1:215; ::_thesis: B1 + B2 c= V
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in B1 + B2 or x in V )
assume x in B1 + B2 ; ::_thesis: x in V
then x in { (b1 + b2) where b1, b2 is Element of (TOP-REAL n) : ( b1 in B1 & b2 in B2 ) } by RUSUB_4:def_9;
then consider b1, b2 being Element of (TOP-REAL n) such that
A7: x = b1 + b2 and
A8: b1 in B1 and
A9: b2 in B2 ;
reconsider e1 = b1, e2 = b2, e12 = b1 + b2 as Point of (Euclid n) by A1, TOPMETR:12;
reconsider y1 = x1, y2 = x2, c1 = b1, c2 = b2 as Element of REAL n by EUCLID:22;
dist (X2,e2) < r / 2 by A9, METRIC_1:11;
then A10: |.(y2 - c2).| < r / 2 by A2, SPPOL_1:5;
dist (X1,e1) < r / 2 by A8, METRIC_1:11;
then |.(y1 - c1).| < r / 2 by A2, SPPOL_1:5;
then A11: |.(y1 - c1).| + |.(y2 - c2).| < (r / 2) + (r / 2) by A10, XREAL_1:8;
A12: (y1 + y2) - (c1 + c2) = (y1 + y2) + (- (c2 + c1))
.= (y1 + y2) + ((- c2) + (- c1)) by RVSUM_1:26
.= ((y1 + y2) + (- c2)) + (- c1) by RVSUM_1:15
.= ((y2 + (- c2)) + y1) + (- c1) by RVSUM_1:15
.= (y2 + (- c2)) + (y1 + (- c1)) by RVSUM_1:15
.= (y2 - c2) + (y1 + (- c1))
.= (y2 - c2) + (y1 - c1) ;
A13: dist (X12,e12) = |.((y1 - c1) + (y2 - c2)).| by A2, A12, SPPOL_1:5;
|.((y1 - c1) + (y2 - c2)).| <= |.(y1 - c1).| + |.(y2 - c2).| by EUCLID:12;
then dist (X12,e12) < r by A11, A13, XXREAL_0:2;
then e12 in Ball (X12,r) by METRIC_1:11;
hence x in V by A6, A7; ::_thesis: verum
end;
let a be Real; :: according to RLTOPSP1:def_9 ::_thesis: for b1 being Element of the carrier of (TOP-REAL n)
for b2 being Element of bool the carrier of (TOP-REAL n) holds
( not b2 is open or not a * b1 in b2 or ex b3 being Element of REAL ex b4 being Element of bool the carrier of (TOP-REAL n) st
( b4 is open & b1 in b4 & ( for b5 being Element of REAL holds
( b3 <= abs (b5 - a) or b5 * b4 c= b2 ) ) ) )
let x be Point of (TOP-REAL n); ::_thesis: for b1 being Element of bool the carrier of (TOP-REAL n) holds
( not b1 is open or not a * x in b1 or ex b2 being Element of REAL ex b3 being Element of bool the carrier of (TOP-REAL n) st
( b3 is open & x in b3 & ( for b4 being Element of REAL holds
( b2 <= abs (b4 - a) or b4 * b3 c= b1 ) ) ) )
let V be Subset of (TOP-REAL n); ::_thesis: ( not V is open or not a * x in V or ex b1 being Element of REAL ex b2 being Element of bool the carrier of (TOP-REAL n) st
( b2 is open & x in b2 & ( for b3 being Element of REAL holds
( b1 <= abs (b3 - a) or b3 * b2 c= V ) ) ) )
assume that
A14: V is open and
A15: a * x in V ; ::_thesis: ex b1 being Element of REAL ex b2 being Element of bool the carrier of (TOP-REAL n) st
( b2 is open & x in b2 & ( for b3 being Element of REAL holds
( b1 <= abs (b3 - a) or b3 * b2 c= V ) ) )
reconsider X = x, AX = a * x as Point of (Euclid n) by A1, TOPMETR:12;
reconsider v = V as Subset of (TopSpaceMetr (Euclid n)) by A1;
V in the topology of (TOP-REAL n) by A14, PRE_TOPC:def_2;
then v is open by A1, PRE_TOPC:def_2;
then consider r being real number such that
A16: r > 0 and
A17: Ball (AX,r) c= v by A15, TOPMETR:15;
set r2 = r / 2;
A18: r / 2 > 0 by A16, XREAL_1:215;
then A19: (r / 2) / 2 > 0 by XREAL_1:215;
ex m being positive Real st (abs a) * m < r / 2
proof
percases ( abs a = 0 or abs a > 0 ) by COMPLEX1:46;
suppose abs a = 0 ; ::_thesis: ex m being positive Real st (abs a) * m < r / 2
then (abs a) * 1 < r / 2 by A16, XREAL_1:215;
hence ex m being positive Real st (abs a) * m < r / 2 ; ::_thesis: verum
end;
supposeA20: abs a > 0 ; ::_thesis: ex m being positive Real st (abs a) * m < r / 2
then reconsider m = ((r / 2) / 2) / (abs a) as positive Real by A19, XREAL_1:139;
take m ; ::_thesis: (abs a) * m < r / 2
(r / 2) / 2 < r / 2 by A16, XREAL_1:215, XREAL_1:216;
hence (abs a) * m < r / 2 by A20, XCMPLX_1:87; ::_thesis: verum
end;
end;
end;
then consider m being positive Real such that
A21: (abs a) * m < r / 2 ;
reconsider B = Ball (X,m) as Subset of (TOP-REAL n) by A1, TOPMETR:12;
reconsider nr = (r / 2) / (|.x.| + m) as positive Real by A18, XREAL_1:139;
take nr ; ::_thesis: ex b1 being Element of bool the carrier of (TOP-REAL n) st
( b1 is open & x in b1 & ( for b2 being Element of REAL holds
( nr <= abs (b2 - a) or b2 * b1 c= V ) ) )
take B ; ::_thesis: ( B is open & x in B & ( for b1 being Element of REAL holds
( nr <= abs (b1 - a) or b1 * B c= V ) ) )
thus ( B is open & x in B ) by GOBOARD6:1, GOBOARD6:3; ::_thesis: for b1 being Element of REAL holds
( nr <= abs (b1 - a) or b1 * B c= V )
let s be Real; ::_thesis: ( nr <= abs (s - a) or s * B c= V )
assume A22: abs (s - a) < nr ; ::_thesis: s * B c= V
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in s * B or z in V )
assume z in s * B ; ::_thesis: z in V
then consider b being Element of (TOP-REAL n) such that
A23: z = s * b and
A24: b in B ;
reconsider e = b, se = s * b as Point of (Euclid n) by A1, TOPMETR:12;
reconsider y = x, c = b as Element of REAL n by EUCLID:22;
reconsider Y = y, C = c as Element of n -tuples_on REAL ;
c = C - (n |-> 0) by RVSUM_1:32
.= C - (Y - Y) by RVSUM_1:37
.= (C - Y) + Y by RVSUM_1:41 ;
then A25: |.c.| <= |.(c - y).| + |.y.| by EUCLID:12;
A26: dist (X,e) < m by A24, METRIC_1:11;
then |.(c - y).| < m by A2, SPPOL_1:5;
then |.(c - y).| + |.y.| <= m + |.x.| by XREAL_1:6;
then |.c.| <= m + |.x.| by A25, XXREAL_0:2;
then A27: nr * |.c.| <= nr * (m + |.x.|) by XREAL_1:64;
(a * y) + (- (a * c)) = (a * y) + ((- 1) * (a * c))
.= (a * y) + (((- 1) * a) * c) by RVSUM_1:49
.= (a * y) + (a * ((- 1) * c)) by RVSUM_1:49
.= a * (y + ((- 1) * c)) by RVSUM_1:51
.= a * (y + (- c))
.= a * (y - c) ;
then A28: |.((a * y) + (- (a * c))).| = (abs a) * |.(y - c).| by EUCLID:11;
( abs a >= 0 & |.(y - c).| = dist (X,e) ) by A2, COMPLEX1:46, SPPOL_1:5;
then |.((a * y) + (- (a * c))).| <= (abs a) * m by A26, A28, XREAL_1:64;
then A29: |.((a * y) + (- (a * c))).| < r / 2 by A21, XXREAL_0:2;
(a * c) + (- (s * c)) = (a * c) + ((- 1) * (s * c))
.= (a * c) + (((- 1) * s) * c) by RVSUM_1:49
.= (a + ((- 1) * s)) * c by RVSUM_1:50 ;
then |.((a * c) + (- (s * c))).| = (abs (a - s)) * |.c.| by EUCLID:11
.= (abs (- (a - s))) * |.c.| by COMPLEX1:52 ;
then ( nr * (|.x.| + m) = r / 2 & |.((a * c) + (- (s * c))).| <= nr * |.c.| ) by A22, XCMPLX_1:87, XREAL_1:64;
then |.((a * c) + (- (s * c))).| <= r / 2 by A27, XXREAL_0:2;
then A30: ( |.(((a * y) + (- (a * c))) + ((a * c) + (- (s * c)))).| <= |.((a * y) + (- (a * c))).| + |.((a * c) + (- (s * c))).| & |.((a * y) + (- (a * c))).| + |.((a * c) + (- (s * c))).| < (r / 2) + (r / 2) ) by A29, EUCLID:12, XREAL_1:8;
(a * y) - (s * c) = ((a * Y) - (n |-> 0)) - (s * C) by RVSUM_1:32
.= ((a * y) - ((a * C) - (a * C))) - (s * c) by RVSUM_1:37
.= (((a * y) - (a * C)) + (a * C)) - (s * c) by RVSUM_1:41
.= (((a * y) - (a * C)) + (a * C)) + (- (s * c))
.= ((a * y) - (a * C)) + ((a * c) + (- (s * c))) by RVSUM_1:15
.= ((a * y) + (- (a * c))) + ((a * c) + (- (s * c))) ;
then dist (AX,se) = |.(((a * y) + (- (a * c))) + ((a * c) + (- (s * c)))).| by A2, SPPOL_1:5;
then dist (AX,se) < r by A30, XXREAL_0:2;
then se in Ball (AX,r) by METRIC_1:11;
hence z in V by A17, A23; ::_thesis: verum
end;
end;
begin
theorem :: JORDAN2C:1
canceled;
theorem :: JORDAN2C:2
canceled;
theorem :: JORDAN2C:3
canceled;
theorem :: JORDAN2C:4
canceled;
theorem :: JORDAN2C:5
canceled;
theorem Th6: :: JORDAN2C:6
for r, s being Real
for f being increasing FinSequence of REAL st rng f = {r,s} & len f = 2 & r <= s holds
( f . 1 = r & f . 2 = s )
proof
let r, s be Real; ::_thesis: for f being increasing FinSequence of REAL st rng f = {r,s} & len f = 2 & r <= s holds
( f . 1 = r & f . 2 = s )
let f be increasing FinSequence of REAL ; ::_thesis: ( rng f = {r,s} & len f = 2 & r <= s implies ( f . 1 = r & f . 2 = s ) )
assume that
A1: rng f = {r,s} and
A2: len f = 2 and
A3: r <= s ; ::_thesis: ( f . 1 = r & f . 2 = s )
now__::_thesis:_(_f_._1_=_s_&_f_._2_=_r_implies_(_f_._1_=_r_&_f_._2_=_s_)_)
A4: 2 in dom f by A2, FINSEQ_3:25;
A5: 1 in dom f by A2, FINSEQ_3:25;
assume ( f . 1 = s & f . 2 = r ) ; ::_thesis: ( f . 1 = r & f . 2 = s )
hence ( f . 1 = r & f . 2 = s ) by A3, A5, A4, SEQM_3:def_1; ::_thesis: verum
end;
hence ( f . 1 = r & f . 2 = s ) by A1, A2, FINSEQ_3:151; ::_thesis: verum
end;
theorem :: JORDAN2C:7
canceled;
theorem :: JORDAN2C:8
for n being Element of NAT
for q being Point of (TOP-REAL n) holds abs |.q.| = |.q.| by ABSVALUE:def_1;
theorem Th9: :: JORDAN2C:9
for n being Element of NAT
for q1, q2 being Point of (TOP-REAL n) holds abs (|.q1.| - |.q2.|) <= |.(q1 - q2).|
proof
let n be Element of NAT ; ::_thesis: for q1, q2 being Point of (TOP-REAL n) holds abs (|.q1.| - |.q2.|) <= |.(q1 - q2).|
let q1, q2 be Point of (TOP-REAL n); ::_thesis: abs (|.q1.| - |.q2.|) <= |.(q1 - q2).|
percases ( |.q1.| >= |.q2.| or |.q1.| < |.q2.| ) ;
suppose |.q1.| >= |.q2.| ; ::_thesis: abs (|.q1.| - |.q2.|) <= |.(q1 - q2).|
then |.q1.| - |.q2.| >= 0 by XREAL_1:48;
then |.q1.| - |.q2.| = abs (|.q1.| - |.q2.|) by ABSVALUE:def_1;
hence abs (|.q1.| - |.q2.|) <= |.(q1 - q2).| by TOPRNS_1:32; ::_thesis: verum
end;
supposeA1: |.q1.| < |.q2.| ; ::_thesis: abs (|.q1.| - |.q2.|) <= |.(q1 - q2).|
A2: |.q2.| - |.q1.| <= |.(q2 - q1).| by TOPRNS_1:32;
|.q2.| - |.q1.| > 0 by A1, XREAL_1:50;
then abs (|.q2.| - |.q1.|) <= |.(q2 - q1).| by A2, ABSVALUE:def_1;
then abs (|.q2.| - |.q1.|) <= |.(q1 - q2).| by TOPRNS_1:27;
hence abs (|.q1.| - |.q2.|) <= |.(q1 - q2).| by UNIFORM1:11; ::_thesis: verum
end;
end;
end;
theorem Th10: :: JORDAN2C:10
for r being Real holds |.|[r]|.| = abs r
proof
let r be Real; ::_thesis: |.|[r]|.| = abs r
set p = |[r]|;
reconsider w = |[r]| as Element of REAL 1 by EUCLID:22;
sqr w = <*(r ^2)*> by RVSUM_1:55;
then |.|[r]|.| = sqrt (r ^2) by FINSOP_1:11
.= abs r by COMPLEX1:72 ;
hence |.|[r]|.| = abs r ; ::_thesis: verum
end;
Lm1: for n being Nat
for r being Real st r > 0 holds
for x, y, z being Element of (Euclid n) st x = 0* n holds
for p being Element of (TOP-REAL n) st p = y & r * p = z holds
r * (dist (x,y)) = dist (x,z)
proof
let n be Nat; ::_thesis: for r being Real st r > 0 holds
for x, y, z being Element of (Euclid n) st x = 0* n holds
for p being Element of (TOP-REAL n) st p = y & r * p = z holds
r * (dist (x,y)) = dist (x,z)
let r be Real; ::_thesis: ( r > 0 implies for x, y, z being Element of (Euclid n) st x = 0* n holds
for p being Element of (TOP-REAL n) st p = y & r * p = z holds
r * (dist (x,y)) = dist (x,z) )
assume A1: r > 0 ; ::_thesis: for x, y, z being Element of (Euclid n) st x = 0* n holds
for p being Element of (TOP-REAL n) st p = y & r * p = z holds
r * (dist (x,y)) = dist (x,z)
let x, y, z be Element of (Euclid n); ::_thesis: ( x = 0* n implies for p being Element of (TOP-REAL n) st p = y & r * p = z holds
r * (dist (x,y)) = dist (x,z) )
assume A2: x = 0* n ; ::_thesis: for p being Element of (TOP-REAL n) st p = y & r * p = z holds
r * (dist (x,y)) = dist (x,z)
let p be Element of (TOP-REAL n); ::_thesis: ( p = y & r * p = z implies r * (dist (x,y)) = dist (x,z) )
assume that
A3: p = y and
A4: r * p = z ; ::_thesis: r * (dist (x,y)) = dist (x,z)
reconsider x1 = x, y1 = y as Element of REAL n ;
A5: dist (x,z) = (Pitag_dist n) . (x,z)
.= |.(x1 - (r * y1)).| by A3, A4, EUCLID:def_6 ;
A6: r * x1 = n |-> (0 * r) by A2, RVSUM_1:48
.= x1 by A2 ;
dist (x,y) = (Pitag_dist n) . (x,y)
.= |.(x1 - y1).| by EUCLID:def_6 ;
hence r * (dist (x,y)) = (abs r) * |.(x1 - y1).| by A1, ABSVALUE:def_1
.= |.(r * (x1 - y1)).| by EUCLID:11
.= |.((r * x1) + (r * (- y1))).| by RVSUM_1:51
.= |.((r * x1) + (((- 1) * r) * y1)).| by RVSUM_1:49
.= dist (x,z) by A5, A6, RVSUM_1:49 ;
::_thesis: verum
end;
Lm2: for n being Nat
for r, s being Real st r > 0 holds
for x being Element of (Euclid n) st x = 0* n holds
for A being Subset of (TOP-REAL n) st A = Ball (x,s) holds
r * A = Ball (x,(r * s))
proof
let n be Nat; ::_thesis: for r, s being Real st r > 0 holds
for x being Element of (Euclid n) st x = 0* n holds
for A being Subset of (TOP-REAL n) st A = Ball (x,s) holds
r * A = Ball (x,(r * s))
let r, s be Real; ::_thesis: ( r > 0 implies for x being Element of (Euclid n) st x = 0* n holds
for A being Subset of (TOP-REAL n) st A = Ball (x,s) holds
r * A = Ball (x,(r * s)) )
assume A1: r > 0 ; ::_thesis: for x being Element of (Euclid n) st x = 0* n holds
for A being Subset of (TOP-REAL n) st A = Ball (x,s) holds
r * A = Ball (x,(r * s))
let x be Element of (Euclid n); ::_thesis: ( x = 0* n implies for A being Subset of (TOP-REAL n) st A = Ball (x,s) holds
r * A = Ball (x,(r * s)) )
assume A2: x = 0* n ; ::_thesis: for A being Subset of (TOP-REAL n) st A = Ball (x,s) holds
r * A = Ball (x,(r * s))
let A be Subset of (TOP-REAL n); ::_thesis: ( A = Ball (x,s) implies r * A = Ball (x,(r * s)) )
assume A3: A = Ball (x,s) ; ::_thesis: r * A = Ball (x,(r * s))
thus r * A c= Ball (x,(r * s)) :: according to XBOOLE_0:def_10 ::_thesis: Ball (x,(r * s)) c= r * A
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in r * A or y in Ball (x,(r * s)) )
assume y in r * A ; ::_thesis: y in Ball (x,(r * s))
then consider v being Element of (TOP-REAL n) such that
A4: y = r * v and
A5: v in A ;
v in { q where q is Element of (Euclid n) : dist (x,q) < s } by A5, A3, METRIC_1:def_14;
then consider q being Element of (Euclid n) such that
A6: v = q and
A7: dist (x,q) < s ;
reconsider p = y as Element of (Euclid n) by A4, EUCLID:67;
r * (dist (x,q)) = dist (x,p) by A1, A2, A6, A4, Lm1;
then dist (x,p) < r * s by A7, A1, XREAL_1:68;
then y in { e where e is Element of (Euclid n) : dist (x,e) < r * s } ;
hence y in Ball (x,(r * s)) by METRIC_1:def_14; ::_thesis: verum
end;
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in Ball (x,(r * s)) or y in r * A )
assume y in Ball (x,(r * s)) ; ::_thesis: y in r * A
then y in { q where q is Element of (Euclid n) : dist (x,q) < r * s } by METRIC_1:def_14;
then consider z being Element of (Euclid n) such that
A8: y = z and
A9: dist (x,z) < r * s ;
reconsider q = z as Element of (TOP-REAL n) by EUCLID:67;
set p = (r ") * q;
A10: y = 1 * q by A8, RVSUM_1:52
.= ((r ") * r) * q by A1, XCMPLX_0:def_7
.= r * ((r ") * q) by RVSUM_1:49 ;
reconsider f = (r ") * q as Element of (Euclid n) by EUCLID:67;
A11: dist (x,f) = (r ") * (dist (x,z)) by A1, A2, Lm1;
s = 1 * s
.= ((r ") * ((r ") ")) * s by A1, XCMPLX_0:def_7
.= (r ") * (r * s) ;
then dist (x,f) < s by A9, A11, A1, XREAL_1:68;
then (r ") * q in { e where e is Element of (Euclid n) : dist (x,e) < s } ;
then (r ") * q in A by A3, METRIC_1:def_14;
hence y in r * A by A10; ::_thesis: verum
end;
Lm3: for n being Nat
for r, s, t being Real st 0 < s & s <= t holds
for x being Element of (Euclid n) st x = 0* n holds
for BA being Subset of (TOP-REAL n) st BA = Ball (x,r) holds
s * BA c= t * BA
proof
let n be Nat; ::_thesis: for r, s, t being Real st 0 < s & s <= t holds
for x being Element of (Euclid n) st x = 0* n holds
for BA being Subset of (TOP-REAL n) st BA = Ball (x,r) holds
s * BA c= t * BA
let r, s, t be Real; ::_thesis: ( 0 < s & s <= t implies for x being Element of (Euclid n) st x = 0* n holds
for BA being Subset of (TOP-REAL n) st BA = Ball (x,r) holds
s * BA c= t * BA )
assume that
A1: 0 < s and
A2: s <= t ; ::_thesis: for x being Element of (Euclid n) st x = 0* n holds
for BA being Subset of (TOP-REAL n) st BA = Ball (x,r) holds
s * BA c= t * BA
let x be Element of (Euclid n); ::_thesis: ( x = 0* n implies for BA being Subset of (TOP-REAL n) st BA = Ball (x,r) holds
s * BA c= t * BA )
assume A3: x = 0* n ; ::_thesis: for BA being Subset of (TOP-REAL n) st BA = Ball (x,r) holds
s * BA c= t * BA
let BA be Subset of (TOP-REAL n); ::_thesis: ( BA = Ball (x,r) implies s * BA c= t * BA )
assume A4: BA = Ball (x,r) ; ::_thesis: s * BA c= t * BA
let e be set ; :: according to TARSKI:def_3 ::_thesis: ( not e in s * BA or e in t * BA )
assume e in s * BA ; ::_thesis: e in t * BA
then consider w being Element of (TOP-REAL n) such that
A5: e = s * w and
A6: w in BA ;
w in { q where q is Element of (Euclid n) : dist (x,q) < r } by A6, A4, METRIC_1:def_14;
then consider q being Element of (Euclid n) such that
A7: w = q and
A8: dist (x,q) < r ;
set p = (s / t) * w;
A9: e = s * w by A5
.= (t * (s / t)) * w by A1, A2, XCMPLX_1:87
.= t * ((s / t) * w) by RVSUM_1:49
.= t * ((s / t) * w) ;
reconsider y = (s / t) * w as Element of (Euclid n) by EUCLID:67;
A10: dist (x,y) = (s / t) * (dist (x,q)) by A3, A7, Lm1, A1, A2, XREAL_1:139;
s / t <= 1 by A1, A2, XREAL_1:183;
then dist (x,y) <= dist (x,q) by A10, METRIC_1:5, XREAL_1:153;
then dist (x,y) < r by A8, XXREAL_0:2;
then (s / t) * w in { f where f is Element of (Euclid n) : dist (x,f) < r } ;
then (s / t) * w in BA by A4, METRIC_1:def_14;
hence e in t * BA by A9; ::_thesis: verum
end;
theorem Th11: :: JORDAN2C:11
for n being Nat
for A being Subset of (TOP-REAL n) holds
( A is bounded iff A is bounded Subset of (Euclid n) )
proof
let n be Nat; ::_thesis: for A being Subset of (TOP-REAL n) holds
( A is bounded iff A is bounded Subset of (Euclid n) )
let A be Subset of (TOP-REAL n); ::_thesis: ( A is bounded iff A is bounded Subset of (Euclid n) )
reconsider z = 0* n as Element of (Euclid n) ;
thus ( A is bounded implies A is bounded Subset of (Euclid n) ) ::_thesis: ( A is bounded Subset of (Euclid n) implies A is bounded )
proof
assume A1: A is bounded ; ::_thesis: A is bounded Subset of (Euclid n)
reconsider B = A as Subset of (Euclid n) by EUCLID:67;
z = 0. (TOP-REAL n) by EUCLID:70;
then reconsider V = Ball (z,1) as a_neighborhood of 0. (TOP-REAL n) by GOBOARD6:2;
consider s being Real such that
A2: s > 0 and
A3: for t being Real st t > s holds
A c= t * V by A1, RLTOPSP1:def_12;
set r = s + 1;
0 < s + 1 by A2;
then (s + 1) * V = Ball (z,((s + 1) * 1)) by Lm2;
then B c= Ball (z,(s + 1)) by A3, XREAL_1:29;
hence A is bounded Subset of (Euclid n) by A2, METRIC_6:def_3; ::_thesis: verum
end;
assume A4: A is bounded Subset of (Euclid n) ; ::_thesis: A is bounded
then reconsider B = A as Subset of (Euclid n) ;
consider r1 being Real such that
A5: 0 < r1 and
A6: B c= Ball (z,r1) by A4, METRIC_6:29;
let V be a_neighborhood of 0. (TOP-REAL n); :: according to RLTOPSP1:def_12 ::_thesis: ex b1 being Element of REAL st
( not b1 <= 0 & ( for b2 being Element of REAL holds
( b2 <= b1 or A c= b2 * V ) ) )
0. (TOP-REAL n) = 0* n by EUCLID:70;
then z in Int V by CONNSP_2:def_1;
then consider r2 being real number such that
A7: r2 > 0 and
A8: Ball (z,r2) c= V by GOBOARD6:5;
reconsider r2 = r2 as Real by XREAL_0:def_1;
take s = r1 / r2; ::_thesis: ( not s <= 0 & ( for b1 being Element of REAL holds
( b1 <= s or A c= b1 * V ) ) )
thus A9: s > 0 by A5, A7, XREAL_1:139; ::_thesis: for b1 being Element of REAL holds
( b1 <= s or A c= b1 * V )
let t be Real; ::_thesis: ( t <= s or A c= t * V )
reconsider BA = Ball (z,r2) as Subset of (TOP-REAL n) by EUCLID:67;
s * r2 = r1 by A7, XCMPLX_1:87;
then A10: A c= s * BA by A6, A9, Lm2;
assume t > s ; ::_thesis: A c= t * V
then s * BA c= t * BA by A9, Lm3;
then A11: A c= t * BA by A10, XBOOLE_1:1;
t * BA c= t * V by A8, CONVEX1:39;
hence A c= t * V by A11, XBOOLE_1:1; ::_thesis: verum
end;
theorem :: JORDAN2C:12
for n being Element of NAT
for A, B being Subset of (TOP-REAL n) st B is bounded & A c= B holds
A is bounded by RLTOPSP1:42;
definition
canceled;
let n be Nat;
let A, B be Subset of (TOP-REAL n);
predB is_inside_component_of A means :Def2: :: JORDAN2C:def 2
( B is_a_component_of A ` & B is bounded );
end;
:: deftheorem JORDAN2C:def_1_:_
canceled;
:: deftheorem Def2 defines is_inside_component_of JORDAN2C:def_2_:_
for n being Nat
for A, B being Subset of (TOP-REAL n) holds
( B is_inside_component_of A iff ( B is_a_component_of A ` & B is bounded ) );
registration
let M be non empty MetrStruct ;
cluster bounded for Element of bool the carrier of M;
existence
ex b1 being Subset of M st b1 is bounded
proof
take {} M ; ::_thesis: {} M is bounded
take 1 ; :: according to TBSP_1:def_7 ::_thesis: ( not 1 <= 0 & ( for b1, b2 being Element of the carrier of M holds
( not b1 in {} M or not b2 in {} M or dist (b1,b2) <= 1 ) ) )
thus ( not 1 <= 0 & ( for b1, b2 being Element of the carrier of M holds
( not b1 in {} M or not b2 in {} M or dist (b1,b2) <= 1 ) ) ) ; ::_thesis: verum
end;
end;
theorem Th13: :: JORDAN2C:13
for n being Element of NAT
for A, B being Subset of (TOP-REAL n) holds
( B is_inside_component_of A iff ex C being Subset of ((TOP-REAL n) | (A `)) st
( C = B & C is a_component & C is bounded Subset of (Euclid n) ) )
proof
let n be Element of NAT ; ::_thesis: for A, B being Subset of (TOP-REAL n) holds
( B is_inside_component_of A iff ex C being Subset of ((TOP-REAL n) | (A `)) st
( C = B & C is a_component & C is bounded Subset of (Euclid n) ) )
let A, B be Subset of (TOP-REAL n); ::_thesis: ( B is_inside_component_of A iff ex C being Subset of ((TOP-REAL n) | (A `)) st
( C = B & C is a_component & C is bounded Subset of (Euclid n) ) )
A1: ( B is_a_component_of A ` iff ex C being Subset of ((TOP-REAL n) | (A `)) st
( C = B & C is a_component ) ) by CONNSP_1:def_6;
A2: ( B is_inside_component_of A iff ( B is_a_component_of A ` & B is bounded ) ) by Def2;
thus ( B is_inside_component_of A implies ex C being Subset of ((TOP-REAL n) | (A `)) st
( C = B & C is a_component & C is bounded Subset of (Euclid n) ) ) ::_thesis: ( ex C being Subset of ((TOP-REAL n) | (A `)) st
( C = B & C is a_component & C is bounded Subset of (Euclid n) ) implies B is_inside_component_of A )
proof
assume A3: B is_inside_component_of A ; ::_thesis: ex C being Subset of ((TOP-REAL n) | (A `)) st
( C = B & C is a_component & C is bounded Subset of (Euclid n) )
then B is bounded Subset of (Euclid n) by A2, Th11;
hence ex C being Subset of ((TOP-REAL n) | (A `)) st
( C = B & C is a_component & C is bounded Subset of (Euclid n) ) by A1, A3, Def2; ::_thesis: verum
end;
given C being Subset of ((TOP-REAL n) | (A `)) such that A4: ( C = B & C is a_component & C is bounded Subset of (Euclid n) ) ; ::_thesis: B is_inside_component_of A
( B is bounded & B is_a_component_of A ` ) by A4, Th11, CONNSP_1:def_6;
hence B is_inside_component_of A by Def2; ::_thesis: verum
end;
definition
let n be Nat;
let A, B be Subset of (TOP-REAL n);
predB is_outside_component_of A means :Def3: :: JORDAN2C:def 3
( B is_a_component_of A ` & not B is bounded );
end;
:: deftheorem Def3 defines is_outside_component_of JORDAN2C:def_3_:_
for n being Nat
for A, B being Subset of (TOP-REAL n) holds
( B is_outside_component_of A iff ( B is_a_component_of A ` & not B is bounded ) );
theorem Th14: :: JORDAN2C:14
for n being Element of NAT
for A, B being Subset of (TOP-REAL n) holds
( B is_outside_component_of A iff ex C being Subset of ((TOP-REAL n) | (A `)) st
( C = B & C is a_component & C is not bounded Subset of (Euclid n) ) )
proof
let n be Element of NAT ; ::_thesis: for A, B being Subset of (TOP-REAL n) holds
( B is_outside_component_of A iff ex C being Subset of ((TOP-REAL n) | (A `)) st
( C = B & C is a_component & C is not bounded Subset of (Euclid n) ) )
let A, B be Subset of (TOP-REAL n); ::_thesis: ( B is_outside_component_of A iff ex C being Subset of ((TOP-REAL n) | (A `)) st
( C = B & C is a_component & C is not bounded Subset of (Euclid n) ) )
A1: ( B is_a_component_of A ` iff ex C being Subset of ((TOP-REAL n) | (A `)) st
( C = B & C is a_component ) ) by CONNSP_1:def_6;
A2: ( B is_outside_component_of A iff ( B is_a_component_of A ` & not B is bounded ) ) by Def3;
thus ( B is_outside_component_of A implies ex C being Subset of ((TOP-REAL n) | (A `)) st
( C = B & C is a_component & C is not bounded Subset of (Euclid n) ) ) ::_thesis: ( ex C being Subset of ((TOP-REAL n) | (A `)) st
( C = B & C is a_component & C is not bounded Subset of (Euclid n) ) implies B is_outside_component_of A )
proof
reconsider D2 = B as Subset of (Euclid n) by TOPREAL3:8;
assume A3: B is_outside_component_of A ; ::_thesis: ex C being Subset of ((TOP-REAL n) | (A `)) st
( C = B & C is a_component & C is not bounded Subset of (Euclid n) )
then consider C being Subset of ((TOP-REAL n) | (A `)) such that
A4: C = B and
A5: C is a_component by A1, Def3;
now__::_thesis:_ex_D_being_Subset_of_(Euclid_n)_st_
(_D_=_C_&_not_D_is_bounded_)
assume for D being Subset of (Euclid n) st D = C holds
D is bounded ; ::_thesis: contradiction
then D2 is bounded by A4;
hence contradiction by A2, A3, Th11; ::_thesis: verum
end;
hence ex C being Subset of ((TOP-REAL n) | (A `)) st
( C = B & C is a_component & C is not bounded Subset of (Euclid n) ) by A4, A5; ::_thesis: verum
end;
given C being Subset of ((TOP-REAL n) | (A `)) such that A6: ( C = B & C is a_component & C is not bounded Subset of (Euclid n) ) ; ::_thesis: B is_outside_component_of A
( not B is bounded & B is_a_component_of A ` ) by A6, Th11, CONNSP_1:def_6;
hence B is_outside_component_of A by Def3; ::_thesis: verum
end;
theorem :: JORDAN2C:15
for n being Element of NAT
for A, B being Subset of (TOP-REAL n) st B is_inside_component_of A holds
B c= A `
proof
let n be Element of NAT ; ::_thesis: for A, B being Subset of (TOP-REAL n) st B is_inside_component_of A holds
B c= A `
let A, B be Subset of (TOP-REAL n); ::_thesis: ( B is_inside_component_of A implies B c= A ` )
assume B is_inside_component_of A ; ::_thesis: B c= A `
then B is_a_component_of A ` by Def2;
hence B c= A ` by SPRECT_1:5; ::_thesis: verum
end;
theorem :: JORDAN2C:16
for n being Element of NAT
for A, B being Subset of (TOP-REAL n) st B is_outside_component_of A holds
B c= A `
proof
let n be Element of NAT ; ::_thesis: for A, B being Subset of (TOP-REAL n) st B is_outside_component_of A holds
B c= A `
let A, B be Subset of (TOP-REAL n); ::_thesis: ( B is_outside_component_of A implies B c= A ` )
assume B is_outside_component_of A ; ::_thesis: B c= A `
then B is_a_component_of A ` by Def3;
hence B c= A ` by SPRECT_1:5; ::_thesis: verum
end;
definition
let n be Nat;
let A be Subset of (TOP-REAL n);
func BDD A -> Subset of (TOP-REAL n) equals :: JORDAN2C:def 4
union { B where B is Subset of (TOP-REAL n) : B is_inside_component_of A } ;
correctness
coherence
union { B where B is Subset of (TOP-REAL n) : B is_inside_component_of A } is Subset of (TOP-REAL n);
proof
union { B where B is Subset of (TOP-REAL n) : B is_inside_component_of A } c= the carrier of (TOP-REAL n)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in union { B where B is Subset of (TOP-REAL n) : B is_inside_component_of A } or x in the carrier of (TOP-REAL n) )
assume x in union { B where B is Subset of (TOP-REAL n) : B is_inside_component_of A } ; ::_thesis: x in the carrier of (TOP-REAL n)
then consider y being set such that
A1: x in y and
A2: y in { B where B is Subset of (TOP-REAL n) : B is_inside_component_of A } by TARSKI:def_4;
ex B being Subset of (TOP-REAL n) st
( y = B & B is_inside_component_of A ) by A2;
hence x in the carrier of (TOP-REAL n) by A1; ::_thesis: verum
end;
hence union { B where B is Subset of (TOP-REAL n) : B is_inside_component_of A } is Subset of (TOP-REAL n) ; ::_thesis: verum
end;
end;
:: deftheorem defines BDD JORDAN2C:def_4_:_
for n being Nat
for A being Subset of (TOP-REAL n) holds BDD A = union { B where B is Subset of (TOP-REAL n) : B is_inside_component_of A } ;
definition
let n be Nat;
let A be Subset of (TOP-REAL n);
func UBD A -> Subset of (TOP-REAL n) equals :: JORDAN2C:def 5
union { B where B is Subset of (TOP-REAL n) : B is_outside_component_of A } ;
correctness
coherence
union { B where B is Subset of (TOP-REAL n) : B is_outside_component_of A } is Subset of (TOP-REAL n);
proof
union { B where B is Subset of (TOP-REAL n) : B is_outside_component_of A } c= the carrier of (TOP-REAL n)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in union { B where B is Subset of (TOP-REAL n) : B is_outside_component_of A } or x in the carrier of (TOP-REAL n) )
assume x in union { B where B is Subset of (TOP-REAL n) : B is_outside_component_of A } ; ::_thesis: x in the carrier of (TOP-REAL n)
then consider y being set such that
A1: x in y and
A2: y in { B where B is Subset of (TOP-REAL n) : B is_outside_component_of A } by TARSKI:def_4;
ex B being Subset of (TOP-REAL n) st
( y = B & B is_outside_component_of A ) by A2;
hence x in the carrier of (TOP-REAL n) by A1; ::_thesis: verum
end;
hence union { B where B is Subset of (TOP-REAL n) : B is_outside_component_of A } is Subset of (TOP-REAL n) ; ::_thesis: verum
end;
end;
:: deftheorem defines UBD JORDAN2C:def_5_:_
for n being Nat
for A being Subset of (TOP-REAL n) holds UBD A = union { B where B is Subset of (TOP-REAL n) : B is_outside_component_of A } ;
registration
let n be Nat;
cluster [#] (TOP-REAL n) -> convex ;
coherence
[#] (TOP-REAL n) is convex
proof
let w1, w2 be Point of (TOP-REAL n); :: according to CONVEX1:def_2 ::_thesis: for b1 being Element of REAL holds
( b1 <= 0 or 1 <= b1 or not w1 in [#] (TOP-REAL n) or not w2 in [#] (TOP-REAL n) or K258((TOP-REAL n),(b1 * w1),((1 - b1) * w2)) in [#] (TOP-REAL n) )
thus for b1 being Element of REAL holds
( b1 <= 0 or 1 <= b1 or not w1 in [#] (TOP-REAL n) or not w2 in [#] (TOP-REAL n) or K258((TOP-REAL n),(b1 * w1),((1 - b1) * w2)) in [#] (TOP-REAL n) ) ; ::_thesis: verum
end;
end;
registration
let n be Element of NAT ;
cluster [#] (TOP-REAL n) -> a_component ;
coherence
[#] (TOP-REAL n) is a_component
proof
set A = [#] (TOP-REAL n);
for B being Subset of (TOP-REAL n) st B is connected & [#] (TOP-REAL n) c= B holds
[#] (TOP-REAL n) = B by XBOOLE_0:def_10;
hence [#] (TOP-REAL n) is a_component by CONNSP_1:def_5; ::_thesis: verum
end;
end;
theorem :: JORDAN2C:17
canceled;
theorem :: JORDAN2C:18
canceled;
theorem :: JORDAN2C:19
canceled;
theorem Th20: :: JORDAN2C:20
for n being Element of NAT
for A being Subset of (TOP-REAL n) holds BDD A is a_union_of_components of (TOP-REAL n) | (A `)
proof
let n be Element of NAT ; ::_thesis: for A being Subset of (TOP-REAL n) holds BDD A is a_union_of_components of (TOP-REAL n) | (A `)
let A be Subset of (TOP-REAL n); ::_thesis: BDD A is a_union_of_components of (TOP-REAL n) | (A `)
{ B where B is Subset of (TOP-REAL n) : B is_inside_component_of A } c= bool the carrier of ((TOP-REAL n) | (A `))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { B where B is Subset of (TOP-REAL n) : B is_inside_component_of A } or x in bool the carrier of ((TOP-REAL n) | (A `)) )
assume x in { B where B is Subset of (TOP-REAL n) : B is_inside_component_of A } ; ::_thesis: x in bool the carrier of ((TOP-REAL n) | (A `))
then consider B being Subset of (TOP-REAL n) such that
A1: x = B and
A2: B is_inside_component_of A ;
ex C being Subset of ((TOP-REAL n) | (A `)) st
( C = B & C is a_component & C is bounded Subset of (Euclid n) ) by A2, Th13;
hence x in bool the carrier of ((TOP-REAL n) | (A `)) by A1; ::_thesis: verum
end;
then reconsider F0 = { B where B is Subset of (TOP-REAL n) : B is_inside_component_of A } as Subset-Family of the carrier of ((TOP-REAL n) | (A `)) ;
reconsider F0 = F0 as Subset-Family of ((TOP-REAL n) | (A `)) ;
A3: for B0 being Subset of ((TOP-REAL n) | (A `)) st B0 in F0 holds
B0 is a_component
proof
let B0 be Subset of ((TOP-REAL n) | (A `)); ::_thesis: ( B0 in F0 implies B0 is a_component )
assume B0 in F0 ; ::_thesis: B0 is a_component
then consider B being Subset of (TOP-REAL n) such that
A4: B = B0 and
A5: B is_inside_component_of A ;
ex C being Subset of ((TOP-REAL n) | (A `)) st
( C = B & C is a_component & C is bounded Subset of (Euclid n) ) by A5, Th13;
hence B0 is a_component by A4; ::_thesis: verum
end;
BDD A = union F0 ;
hence BDD A is a_union_of_components of (TOP-REAL n) | (A `) by A3, CONNSP_3:def_2; ::_thesis: verum
end;
theorem Th21: :: JORDAN2C:21
for n being Element of NAT
for A being Subset of (TOP-REAL n) holds UBD A is a_union_of_components of (TOP-REAL n) | (A `)
proof
let n be Element of NAT ; ::_thesis: for A being Subset of (TOP-REAL n) holds UBD A is a_union_of_components of (TOP-REAL n) | (A `)
let A be Subset of (TOP-REAL n); ::_thesis: UBD A is a_union_of_components of (TOP-REAL n) | (A `)
{ B where B is Subset of (TOP-REAL n) : B is_outside_component_of A } c= bool the carrier of ((TOP-REAL n) | (A `))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { B where B is Subset of (TOP-REAL n) : B is_outside_component_of A } or x in bool the carrier of ((TOP-REAL n) | (A `)) )
assume x in { B where B is Subset of (TOP-REAL n) : B is_outside_component_of A } ; ::_thesis: x in bool the carrier of ((TOP-REAL n) | (A `))
then consider B being Subset of (TOP-REAL n) such that
A1: x = B and
A2: B is_outside_component_of A ;
ex C being Subset of ((TOP-REAL n) | (A `)) st
( C = B & C is a_component & C is not bounded Subset of (Euclid n) ) by A2, Th14;
hence x in bool the carrier of ((TOP-REAL n) | (A `)) by A1; ::_thesis: verum
end;
then reconsider F0 = { B where B is Subset of (TOP-REAL n) : B is_outside_component_of A } as Subset-Family of the carrier of ((TOP-REAL n) | (A `)) ;
reconsider F0 = F0 as Subset-Family of ((TOP-REAL n) | (A `)) ;
A3: for B0 being Subset of ((TOP-REAL n) | (A `)) st B0 in F0 holds
B0 is a_component
proof
let B0 be Subset of ((TOP-REAL n) | (A `)); ::_thesis: ( B0 in F0 implies B0 is a_component )
assume B0 in F0 ; ::_thesis: B0 is a_component
then consider B being Subset of (TOP-REAL n) such that
A4: B = B0 and
A5: B is_outside_component_of A ;
ex C being Subset of ((TOP-REAL n) | (A `)) st
( C = B & C is a_component & C is not bounded Subset of (Euclid n) ) by A5, Th14;
hence B0 is a_component by A4; ::_thesis: verum
end;
UBD A = union F0 ;
hence UBD A is a_union_of_components of (TOP-REAL n) | (A `) by A3, CONNSP_3:def_2; ::_thesis: verum
end;
theorem Th22: :: JORDAN2C:22
for n being Element of NAT
for A, B being Subset of (TOP-REAL n) st B is_inside_component_of A holds
B c= BDD A
proof
let n be Element of NAT ; ::_thesis: for A, B being Subset of (TOP-REAL n) st B is_inside_component_of A holds
B c= BDD A
let A, B be Subset of (TOP-REAL n); ::_thesis: ( B is_inside_component_of A implies B c= BDD A )
assume B is_inside_component_of A ; ::_thesis: B c= BDD A
then A1: B in { B2 where B2 is Subset of (TOP-REAL n) : B2 is_inside_component_of A } ;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in B or x in BDD A )
assume x in B ; ::_thesis: x in BDD A
hence x in BDD A by A1, TARSKI:def_4; ::_thesis: verum
end;
theorem Th23: :: JORDAN2C:23
for n being Element of NAT
for A, B being Subset of (TOP-REAL n) st B is_outside_component_of A holds
B c= UBD A
proof
let n be Element of NAT ; ::_thesis: for A, B being Subset of (TOP-REAL n) st B is_outside_component_of A holds
B c= UBD A
let A, B be Subset of (TOP-REAL n); ::_thesis: ( B is_outside_component_of A implies B c= UBD A )
assume B is_outside_component_of A ; ::_thesis: B c= UBD A
then A1: B in { B2 where B2 is Subset of (TOP-REAL n) : B2 is_outside_component_of A } ;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in B or x in UBD A )
assume x in B ; ::_thesis: x in UBD A
hence x in UBD A by A1, TARSKI:def_4; ::_thesis: verum
end;
theorem Th24: :: JORDAN2C:24
for n being Element of NAT
for A being Subset of (TOP-REAL n) holds BDD A misses UBD A
proof
let n be Element of NAT ; ::_thesis: for A being Subset of (TOP-REAL n) holds BDD A misses UBD A
let A be Subset of (TOP-REAL n); ::_thesis: BDD A misses UBD A
set x = the Element of (BDD A) /\ (UBD A);
assume A1: (BDD A) /\ (UBD A) <> {} ; :: according to XBOOLE_0:def_7 ::_thesis: contradiction
then the Element of (BDD A) /\ (UBD A) in union { B where B is Subset of (TOP-REAL n) : B is_inside_component_of A } by XBOOLE_0:def_4;
then consider y being set such that
A2: the Element of (BDD A) /\ (UBD A) in y and
A3: y in { B where B is Subset of (TOP-REAL n) : B is_inside_component_of A } by TARSKI:def_4;
the Element of (BDD A) /\ (UBD A) in union { B2 where B2 is Subset of (TOP-REAL n) : B2 is_outside_component_of A } by A1, XBOOLE_0:def_4;
then consider y2 being set such that
A4: the Element of (BDD A) /\ (UBD A) in y2 and
A5: y2 in { B2 where B2 is Subset of (TOP-REAL n) : B2 is_outside_component_of A } by TARSKI:def_4;
consider B being Subset of (TOP-REAL n) such that
A6: y = B and
A7: B is_inside_component_of A by A3;
consider B2 being Subset of (TOP-REAL n) such that
A8: y2 = B2 and
A9: B2 is_outside_component_of A by A5;
consider C being Subset of ((TOP-REAL n) | (A `)) such that
A10: C = B and
A11: ( C is a_component & C is bounded Subset of (Euclid n) ) by A7, Th13;
consider C2 being Subset of ((TOP-REAL n) | (A `)) such that
A12: C2 = B2 and
A13: ( C2 is a_component & C2 is not bounded Subset of (Euclid n) ) by A9, Th14;
C /\ C2 <> {} ((TOP-REAL n) | (A `)) by A2, A6, A10, A4, A8, A12, XBOOLE_0:def_4;
then C meets C2 by XBOOLE_0:def_7;
hence contradiction by A11, A13, CONNSP_1:35; ::_thesis: verum
end;
theorem Th25: :: JORDAN2C:25
for n being Element of NAT
for A being Subset of (TOP-REAL n) holds BDD A c= A `
proof
let n be Element of NAT ; ::_thesis: for A being Subset of (TOP-REAL n) holds BDD A c= A `
let A be Subset of (TOP-REAL n); ::_thesis: BDD A c= A `
reconsider D = BDD A as Subset of ((TOP-REAL n) | (A `)) by Th20;
D c= the carrier of ((TOP-REAL n) | (A `)) ;
hence BDD A c= A ` by PRE_TOPC:8; ::_thesis: verum
end;
theorem Th26: :: JORDAN2C:26
for n being Element of NAT
for A being Subset of (TOP-REAL n) holds UBD A c= A `
proof
let n be Element of NAT ; ::_thesis: for A being Subset of (TOP-REAL n) holds UBD A c= A `
let A be Subset of (TOP-REAL n); ::_thesis: UBD A c= A `
reconsider D = UBD A as Subset of ((TOP-REAL n) | (A `)) by Th21;
D c= the carrier of ((TOP-REAL n) | (A `)) ;
hence UBD A c= A ` by PRE_TOPC:8; ::_thesis: verum
end;
theorem Th27: :: JORDAN2C:27
for n being Element of NAT
for A being Subset of (TOP-REAL n) holds (BDD A) \/ (UBD A) = A `
proof
let n be Element of NAT ; ::_thesis: for A being Subset of (TOP-REAL n) holds (BDD A) \/ (UBD A) = A `
let A be Subset of (TOP-REAL n); ::_thesis: (BDD A) \/ (UBD A) = A `
A1: A ` c= (BDD A) \/ (UBD A)
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in A ` or z in (BDD A) \/ (UBD A) )
assume A2: z in A ` ; ::_thesis: z in (BDD A) \/ (UBD A)
then reconsider p = z as Element of A ` ;
reconsider B = A ` as non empty Subset of (TOP-REAL n) by A2;
reconsider q = p as Point of ((TOP-REAL n) | (A `)) by PRE_TOPC:8;
Component_of q is Subset of ([#] ((TOP-REAL n) | (A `))) ;
then Component_of q is Subset of (A `) by PRE_TOPC:def_5;
then reconsider G = Component_of q as Subset of (TOP-REAL n) by XBOOLE_1:1;
A3: not (TOP-REAL n) | B is empty ;
then A4: q in G by CONNSP_1:38;
Component_of q is a_component by A3, CONNSP_1:40;
then A5: G is_a_component_of A ` by CONNSP_1:def_6;
percases ( G is bounded or not G is bounded ) ;
suppose G is bounded ; ::_thesis: z in (BDD A) \/ (UBD A)
then G is_inside_component_of A by A5, Def2;
then G c= BDD A by Th22;
hence z in (BDD A) \/ (UBD A) by A4, XBOOLE_0:def_3; ::_thesis: verum
end;
suppose not G is bounded ; ::_thesis: z in (BDD A) \/ (UBD A)
then G is_outside_component_of A by A5, Def3;
then G c= UBD A by Th23;
hence z in (BDD A) \/ (UBD A) by A4, XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
( BDD A c= A ` & UBD A c= A ` ) by Th25, Th26;
then (BDD A) \/ (UBD A) c= A ` by XBOOLE_1:8;
hence (BDD A) \/ (UBD A) = A ` by A1, XBOOLE_0:def_10; ::_thesis: verum
end;
theorem Th28: :: JORDAN2C:28
for n being Element of NAT
for P being Subset of (TOP-REAL n) st P = REAL n holds
P is connected
proof
let n be Element of NAT ; ::_thesis: for P being Subset of (TOP-REAL n) st P = REAL n holds
P is connected
let P be Subset of (TOP-REAL n); ::_thesis: ( P = REAL n implies P is connected )
assume A1: P = REAL n ; ::_thesis: P is connected
for p1, p2 being Point of (TOP-REAL n) st p1 in P & p2 in P holds
LSeg (p1,p2) c= P
proof
let p1, p2 be Point of (TOP-REAL n); ::_thesis: ( p1 in P & p2 in P implies LSeg (p1,p2) c= P )
assume that
p1 in P and
p2 in P ; ::_thesis: LSeg (p1,p2) c= P
the carrier of (TOP-REAL n) = REAL n by EUCLID:22;
hence LSeg (p1,p2) c= P by A1; ::_thesis: verum
end;
then P is convex by JORDAN1:def_1;
hence P is connected ; ::_thesis: verum
end;
theorem :: JORDAN2C:29
canceled;
theorem :: JORDAN2C:30
canceled;
theorem :: JORDAN2C:31
canceled;
theorem :: JORDAN2C:32
canceled;
theorem Th33: :: JORDAN2C:33
for n being Element of NAT
for W being Subset of (Euclid n) st n >= 1 & W = REAL n holds
not W is bounded
proof
let n be Element of NAT ; ::_thesis: for W being Subset of (Euclid n) st n >= 1 & W = REAL n holds
not W is bounded
let W be Subset of (Euclid n); ::_thesis: ( n >= 1 & W = REAL n implies not W is bounded )
assume that
A1: n >= 1 and
A2: W = REAL n ; ::_thesis: not W is bounded
reconsider y0 = 0. (TOP-REAL n) as Point of (Euclid n) by EUCLID:67;
assume W is bounded ; ::_thesis: contradiction
then consider r being Real such that
A3: 0 < r and
A4: for x, y being Point of (Euclid n) st x in W & y in W holds
dist (x,y) <= r by TBSP_1:def_7;
reconsider x0 = (r + 1) * (1.REAL n) as Point of (Euclid n) by TOPREAL3:8;
dist (x0,y0) <= r by A2, A4;
then |.(((r + 1) * (1.REAL n)) - (0. (TOP-REAL n))).| <= r by JGRAPH_1:28;
then |.((r + 1) * (1.REAL n)).| <= r by RLVECT_1:13;
then (abs (r + 1)) * |.(1.REAL n).| <= r by TOPRNS_1:7;
then (abs (r + 1)) * (sqrt n) <= r by EUCLID:73;
then A5: (r + 1) * (sqrt n) <= r by A3, ABSVALUE:def_1;
sqrt 1 <= sqrt n by A1, SQUARE_1:26;
then (r + 1) * 1 <= (r + 1) * (sqrt n) by A3, SQUARE_1:18, XREAL_1:64;
then (r + 1) * 1 <= r by A5, XXREAL_0:2;
then (r + 1) - r <= r - r by XREAL_1:9;
then 1 <= 0 ;
hence contradiction ; ::_thesis: verum
end;
theorem Th34: :: JORDAN2C:34
for n being Element of NAT
for A being Subset of (TOP-REAL n) holds
( A is bounded iff ex r being Real st
for q being Point of (TOP-REAL n) st q in A holds
|.q.| < r )
proof
let n be Element of NAT ; ::_thesis: for A being Subset of (TOP-REAL n) holds
( A is bounded iff ex r being Real st
for q being Point of (TOP-REAL n) st q in A holds
|.q.| < r )
let A be Subset of (TOP-REAL n); ::_thesis: ( A is bounded iff ex r being Real st
for q being Point of (TOP-REAL n) st q in A holds
|.q.| < r )
reconsider C = A as Subset of (Euclid n) by TOPREAL3:8;
hereby ::_thesis: ( ex r being Real st
for q being Point of (TOP-REAL n) st q in A holds
|.q.| < r implies A is bounded )
assume A is bounded ; ::_thesis: ex r2 being Real st
for q being Point of (TOP-REAL n) st q in A holds
|.q.| < r2
then reconsider C = A as bounded Subset of (Euclid n) by Th11;
percases ( C <> {} or C = {} ) ;
supposeA1: C <> {} ; ::_thesis: ex r2 being Real st
for q being Point of (TOP-REAL n) st q in A holds
|.q.| < r2
reconsider o = 0. (TOP-REAL n) as Point of (Euclid n) by EUCLID:67;
set x0 = the Element of C;
the Element of C in C by A1;
then reconsider x0 = the Element of C as Point of (Euclid n) ;
consider r being Real such that
0 < r and
A2: for x, y being Point of (Euclid n) st x in C & y in C holds
dist (x,y) <= r by TBSP_1:def_7;
set R0 = (r + (dist (o,x0))) + 1;
for q being Point of (TOP-REAL n) st q in A holds
|.q.| < (r + (dist (o,x0))) + 1
proof
let q1 be Point of (TOP-REAL n); ::_thesis: ( q1 in A implies |.q1.| < (r + (dist (o,x0))) + 1 )
reconsider z = q1 as Point of (Euclid n) by TOPREAL3:8;
|.(q1 - (0. (TOP-REAL n))).| = dist (o,z) by JGRAPH_1:28;
then A3: |.q1.| = dist (o,z) by RLVECT_1:13;
assume q1 in A ; ::_thesis: |.q1.| < (r + (dist (o,x0))) + 1
then dist (x0,z) <= r by A2;
then ( dist (o,z) <= (dist (o,x0)) + (dist (x0,z)) & (dist (o,x0)) + (dist (x0,z)) <= (dist (o,x0)) + r ) by METRIC_1:4, XREAL_1:6;
then A4: dist (o,z) <= (dist (o,x0)) + r by XXREAL_0:2;
r + (dist (o,x0)) < (r + (dist (o,x0))) + 1 by XREAL_1:29;
hence |.q1.| < (r + (dist (o,x0))) + 1 by A3, A4, XXREAL_0:2; ::_thesis: verum
end;
hence ex r2 being Real st
for q being Point of (TOP-REAL n) st q in A holds
|.q.| < r2 ; ::_thesis: verum
end;
suppose C = {} ; ::_thesis: ex r2 being Real st
for q being Point of (TOP-REAL n) st q in A holds
|.q.| < r2
then for q being Point of (TOP-REAL n) st q in A holds
|.q.| < 1 ;
hence ex r2 being Real st
for q being Point of (TOP-REAL n) st q in A holds
|.q.| < r2 ; ::_thesis: verum
end;
end;
end;
given r being Real such that A5: for q being Point of (TOP-REAL n) st q in A holds
|.q.| < r ; ::_thesis: A is bounded
now__::_thesis:_C_is_bounded
percases ( C <> {} or C = {} ) ;
supposeA6: C <> {} ; ::_thesis: C is bounded
set x0 = the Element of C;
the Element of C in C by A6;
then reconsider x0 = the Element of C as Point of (Euclid n) ;
reconsider q0 = x0 as Point of (TOP-REAL n) by TOPREAL3:8;
reconsider o = 0. (TOP-REAL n) as Point of (Euclid n) by EUCLID:67;
set R0 = r + r;
A7: for x, y being Point of (Euclid n) st x in C & y in C holds
dist (x,y) <= r + r
proof
let x, y be Point of (Euclid n); ::_thesis: ( x in C & y in C implies dist (x,y) <= r + r )
assume that
A8: x in C and
A9: y in C ; ::_thesis: dist (x,y) <= r + r
reconsider q2 = y as Point of (TOP-REAL n) by A9;
dist (o,y) = |.(q2 - (0. (TOP-REAL n))).| by JGRAPH_1:28
.= |.q2.| by RLVECT_1:13 ;
then A10: dist (o,y) < r by A5, A9;
reconsider q1 = x as Point of (TOP-REAL n) by A8;
dist (x,o) = |.(q1 - (0. (TOP-REAL n))).| by JGRAPH_1:28
.= |.q1.| by RLVECT_1:13 ;
then dist (x,o) < r by A5, A8;
then ( dist (x,y) <= (dist (x,o)) + (dist (o,y)) & (dist (x,o)) + (dist (o,y)) <= r + r ) by A10, METRIC_1:4, XREAL_1:7;
hence dist (x,y) <= r + r by XXREAL_0:2; ::_thesis: verum
end;
|.q0.| < r by A5, A6;
hence C is bounded by A7, TBSP_1:def_7; ::_thesis: verum
end;
suppose C = {} ; ::_thesis: C is bounded
hence C is bounded ; ::_thesis: verum
end;
end;
end;
hence A is bounded by Th11; ::_thesis: verum
end;
theorem Th35: :: JORDAN2C:35
for n being Element of NAT st n >= 1 holds
not [#] (TOP-REAL n) is bounded
proof
let n be Element of NAT ; ::_thesis: ( n >= 1 implies not [#] (TOP-REAL n) is bounded )
assume A1: n >= 1 ; ::_thesis: not [#] (TOP-REAL n) is bounded
assume [#] (TOP-REAL n) is bounded ; ::_thesis: contradiction
then reconsider C = [#] (TOP-REAL n) as bounded Subset of (Euclid n) by Th11;
C = REAL n by EUCLID:22;
hence contradiction by A1, Th33; ::_thesis: verum
end;
theorem Th36: :: JORDAN2C:36
for n being Element of NAT st n >= 1 holds
UBD ({} (TOP-REAL n)) = REAL n
proof
let n be Element of NAT ; ::_thesis: ( n >= 1 implies UBD ({} (TOP-REAL n)) = REAL n )
set A = {} (TOP-REAL n);
A1: (TOP-REAL n) | ([#] (TOP-REAL n)) = TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) by TSEP_1:93;
assume A2: n >= 1 ; ::_thesis: UBD ({} (TOP-REAL n)) = REAL n
A3: now__::_thesis:_ex_D_being_Subset_of_(Euclid_n)_st_
(_D_=_[#]_((TOP-REAL_n)_|_(({}_(TOP-REAL_n))_`))_&_not_D_is_bounded_)
reconsider D1 = [#] ((TOP-REAL n) | (({} (TOP-REAL n)) `)) as Subset of (Euclid n) by A1, TOPREAL3:8;
assume for D being Subset of (Euclid n) st D = [#] ((TOP-REAL n) | (({} (TOP-REAL n)) `)) holds
D is bounded ; ::_thesis: contradiction
then D1 is bounded ;
then [#] (TOP-REAL n) is bounded by A1, Th11;
hence contradiction by A2, Th35; ::_thesis: verum
end;
[#] ((TOP-REAL n) | (({} (TOP-REAL n)) `)) is a_component by A1, CONNSP_1:45;
then [#] (TOP-REAL n) is_outside_component_of {} (TOP-REAL n) by A1, A3, Th14;
then A4: [#] (TOP-REAL n) in { B2 where B2 is Subset of (TOP-REAL n) : B2 is_outside_component_of {} (TOP-REAL n) } ;
UBD ({} (TOP-REAL n)) c= the carrier of (TOP-REAL n) ;
hence UBD ({} (TOP-REAL n)) c= REAL n by EUCLID:22; :: according to XBOOLE_0:def_10 ::_thesis: REAL n c= UBD ({} (TOP-REAL n))
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in REAL n or x in UBD ({} (TOP-REAL n)) )
assume x in REAL n ; ::_thesis: x in UBD ({} (TOP-REAL n))
then x in [#] (TOP-REAL n) by EUCLID:22;
hence x in UBD ({} (TOP-REAL n)) by A4, TARSKI:def_4; ::_thesis: verum
end;
theorem Th37: :: JORDAN2C:37
for n being Element of NAT
for w1, w2, w3 being Point of (TOP-REAL n)
for P being non empty Subset of (TOP-REAL n)
for h1, h2 being Function of I[01],((TOP-REAL n) | P) st h1 is continuous & w1 = h1 . 0 & w2 = h1 . 1 & h2 is continuous & w2 = h2 . 0 & w3 = h2 . 1 holds
ex h3 being Function of I[01],((TOP-REAL n) | P) st
( h3 is continuous & w1 = h3 . 0 & w3 = h3 . 1 )
proof
let n be Element of NAT ; ::_thesis: for w1, w2, w3 being Point of (TOP-REAL n)
for P being non empty Subset of (TOP-REAL n)
for h1, h2 being Function of I[01],((TOP-REAL n) | P) st h1 is continuous & w1 = h1 . 0 & w2 = h1 . 1 & h2 is continuous & w2 = h2 . 0 & w3 = h2 . 1 holds
ex h3 being Function of I[01],((TOP-REAL n) | P) st
( h3 is continuous & w1 = h3 . 0 & w3 = h3 . 1 )
let w1, w2, w3 be Point of (TOP-REAL n); ::_thesis: for P being non empty Subset of (TOP-REAL n)
for h1, h2 being Function of I[01],((TOP-REAL n) | P) st h1 is continuous & w1 = h1 . 0 & w2 = h1 . 1 & h2 is continuous & w2 = h2 . 0 & w3 = h2 . 1 holds
ex h3 being Function of I[01],((TOP-REAL n) | P) st
( h3 is continuous & w1 = h3 . 0 & w3 = h3 . 1 )
let P be non empty Subset of (TOP-REAL n); ::_thesis: for h1, h2 being Function of I[01],((TOP-REAL n) | P) st h1 is continuous & w1 = h1 . 0 & w2 = h1 . 1 & h2 is continuous & w2 = h2 . 0 & w3 = h2 . 1 holds
ex h3 being Function of I[01],((TOP-REAL n) | P) st
( h3 is continuous & w1 = h3 . 0 & w3 = h3 . 1 )
let h1, h2 be Function of I[01],((TOP-REAL n) | P); ::_thesis: ( h1 is continuous & w1 = h1 . 0 & w2 = h1 . 1 & h2 is continuous & w2 = h2 . 0 & w3 = h2 . 1 implies ex h3 being Function of I[01],((TOP-REAL n) | P) st
( h3 is continuous & w1 = h3 . 0 & w3 = h3 . 1 ) )
assume that
A1: h1 is continuous and
A2: w1 = h1 . 0 and
A3: w2 = h1 . 1 and
A4: h2 is continuous and
A5: w2 = h2 . 0 and
A6: w3 = h2 . 1 ; ::_thesis: ex h3 being Function of I[01],((TOP-REAL n) | P) st
( h3 is continuous & w1 = h3 . 0 & w3 = h3 . 1 )
( 0 in [.0,1.] & 1 in [.0,1.] ) by XXREAL_1:1;
then reconsider p1 = w1, p2 = w2, p3 = w3 as Point of ((TOP-REAL n) | P) by A2, A3, A6, BORSUK_1:40, FUNCT_2:5;
p2,p3 are_connected by A4, A5, A6, BORSUK_2:def_1;
then reconsider P2 = h2 as Path of p2,p3 by A4, A5, A6, BORSUK_2:def_2;
p1,p2 are_connected by A1, A2, A3, BORSUK_2:def_1;
then reconsider P1 = h1 as Path of p1,p2 by A1, A2, A3, BORSUK_2:def_2;
ex P0 being Path of p1,p3 st
( P0 is continuous & P0 . 0 = p1 & P0 . 1 = p3 & ( for t being Point of I[01]
for t9 being Real st t = t9 holds
( ( 0 <= t9 & t9 <= 1 / 2 implies P0 . t = P1 . (2 * t9) ) & ( 1 / 2 <= t9 & t9 <= 1 implies P0 . t = P2 . ((2 * t9) - 1) ) ) ) )
proof
1 / 2 in { r where r is Real : ( 0 <= r & r <= 1 ) } ;
then reconsider pol = 1 / 2 as Point of I[01] by BORSUK_1:40, RCOMP_1:def_1;
reconsider T1 = Closed-Interval-TSpace (0,(1 / 2)), T2 = Closed-Interval-TSpace ((1 / 2),1) as SubSpace of I[01] by TOPMETR:20, TREAL_1:3;
set e2 = P[01] ((1 / 2),1,((#) (0,1)),((0,1) (#)));
set e1 = P[01] (0,(1 / 2),((#) (0,1)),((0,1) (#)));
set E1 = P1 * (P[01] (0,(1 / 2),((#) (0,1)),((0,1) (#))));
set E2 = P2 * (P[01] ((1 / 2),1,((#) (0,1)),((0,1) (#))));
set f = (P1 * (P[01] (0,(1 / 2),((#) (0,1)),((0,1) (#))))) +* (P2 * (P[01] ((1 / 2),1,((#) (0,1)),((0,1) (#)))));
A7: dom (P[01] (0,(1 / 2),((#) (0,1)),((0,1) (#)))) = the carrier of (Closed-Interval-TSpace (0,(1 / 2))) by FUNCT_2:def_1
.= [.0,(1 / 2).] by TOPMETR:18 ;
A8: dom (P[01] ((1 / 2),1,((#) (0,1)),((0,1) (#)))) = the carrier of (Closed-Interval-TSpace ((1 / 2),1)) by FUNCT_2:def_1
.= [.(1 / 2),1.] by TOPMETR:18 ;
reconsider gg = P2 * (P[01] ((1 / 2),1,((#) (0,1)),((0,1) (#)))) as Function of T2,((TOP-REAL n) | P) by TOPMETR:20;
reconsider ff = P1 * (P[01] (0,(1 / 2),((#) (0,1)),((0,1) (#)))) as Function of T1,((TOP-REAL n) | P) by TOPMETR:20;
A9: for t9 being Real st 1 / 2 <= t9 & t9 <= 1 holds
(P2 * (P[01] ((1 / 2),1,((#) (0,1)),((0,1) (#))))) . t9 = P2 . ((2 * t9) - 1)
proof
reconsider r1 = (#) (0,1), r2 = (0,1) (#) as Real by BORSUK_1:def_14, BORSUK_1:def_15, TREAL_1:5;
dom (P[01] ((1 / 2),1,((#) (0,1)),((0,1) (#)))) = the carrier of (Closed-Interval-TSpace ((1 / 2),1)) by FUNCT_2:def_1;
then A10: dom (P[01] ((1 / 2),1,((#) (0,1)),((0,1) (#)))) = [.(1 / 2),1.] by TOPMETR:18
.= { r where r is Real : ( 1 / 2 <= r & r <= 1 ) } by RCOMP_1:def_1 ;
let t9 be Real; ::_thesis: ( 1 / 2 <= t9 & t9 <= 1 implies (P2 * (P[01] ((1 / 2),1,((#) (0,1)),((0,1) (#))))) . t9 = P2 . ((2 * t9) - 1) )
assume ( 1 / 2 <= t9 & t9 <= 1 ) ; ::_thesis: (P2 * (P[01] ((1 / 2),1,((#) (0,1)),((0,1) (#))))) . t9 = P2 . ((2 * t9) - 1)
then A11: t9 in dom (P[01] ((1 / 2),1,((#) (0,1)),((0,1) (#)))) by A10;
then reconsider s = t9 as Point of (Closed-Interval-TSpace ((1 / 2),1)) ;
(P[01] ((1 / 2),1,((#) (0,1)),((0,1) (#)))) . s = (((r2 - r1) / (1 - (1 / 2))) * t9) + (((1 * r1) - ((1 / 2) * r2)) / (1 - (1 / 2))) by TREAL_1:11
.= (2 * t9) - 1 by BORSUK_1:def_14, BORSUK_1:def_15, TREAL_1:5 ;
hence (P2 * (P[01] ((1 / 2),1,((#) (0,1)),((0,1) (#))))) . t9 = P2 . ((2 * t9) - 1) by A11, FUNCT_1:13; ::_thesis: verum
end;
A12: for t9 being Real st 0 <= t9 & t9 <= 1 / 2 holds
(P1 * (P[01] (0,(1 / 2),((#) (0,1)),((0,1) (#))))) . t9 = P1 . (2 * t9)
proof
reconsider r1 = (#) (0,1), r2 = (0,1) (#) as Real by BORSUK_1:def_14, BORSUK_1:def_15, TREAL_1:5;
dom (P[01] (0,(1 / 2),((#) (0,1)),((0,1) (#)))) = the carrier of (Closed-Interval-TSpace (0,(1 / 2))) by FUNCT_2:def_1;
then A13: dom (P[01] (0,(1 / 2),((#) (0,1)),((0,1) (#)))) = [.0,(1 / 2).] by TOPMETR:18
.= { r where r is Real : ( 0 <= r & r <= 1 / 2 ) } by RCOMP_1:def_1 ;
let t9 be Real; ::_thesis: ( 0 <= t9 & t9 <= 1 / 2 implies (P1 * (P[01] (0,(1 / 2),((#) (0,1)),((0,1) (#))))) . t9 = P1 . (2 * t9) )
assume ( 0 <= t9 & t9 <= 1 / 2 ) ; ::_thesis: (P1 * (P[01] (0,(1 / 2),((#) (0,1)),((0,1) (#))))) . t9 = P1 . (2 * t9)
then A14: t9 in dom (P[01] (0,(1 / 2),((#) (0,1)),((0,1) (#)))) by A13;
then reconsider s = t9 as Point of (Closed-Interval-TSpace (0,(1 / 2))) ;
(P[01] (0,(1 / 2),((#) (0,1)),((0,1) (#)))) . s = (((r2 - r1) / ((1 / 2) - 0)) * t9) + ((((1 / 2) * r1) - (0 * r2)) / ((1 / 2) - 0)) by TREAL_1:11
.= 2 * t9 by BORSUK_1:def_14, BORSUK_1:def_15, TREAL_1:5 ;
hence (P1 * (P[01] (0,(1 / 2),((#) (0,1)),((0,1) (#))))) . t9 = P1 . (2 * t9) by A14, FUNCT_1:13; ::_thesis: verum
end;
then A15: ff . (1 / 2) = P2 . ((2 * (1 / 2)) - 1) by A3, A5
.= gg . pol by A9 ;
( [#] T1 = [.0,(1 / 2).] & [#] T2 = [.(1 / 2),1.] ) by TOPMETR:18;
then A16: ( ([#] T1) \/ ([#] T2) = [#] I[01] & ([#] T1) /\ ([#] T2) = {pol} ) by BORSUK_1:40, XXREAL_1:174, XXREAL_1:418;
rng ((P1 * (P[01] (0,(1 / 2),((#) (0,1)),((0,1) (#))))) +* (P2 * (P[01] ((1 / 2),1,((#) (0,1)),((0,1) (#)))))) c= (rng (P1 * (P[01] (0,(1 / 2),((#) (0,1)),((0,1) (#)))))) \/ (rng (P2 * (P[01] ((1 / 2),1,((#) (0,1)),((0,1) (#)))))) by FUNCT_4:17;
then A17: rng ((P1 * (P[01] (0,(1 / 2),((#) (0,1)),((0,1) (#))))) +* (P2 * (P[01] ((1 / 2),1,((#) (0,1)),((0,1) (#)))))) c= the carrier of ((TOP-REAL n) | P) by XBOOLE_1:1;
A18: ( T1 is compact & T2 is compact ) by HEINE:4;
dom P1 = the carrier of I[01] by FUNCT_2:def_1;
then A19: rng (P[01] (0,(1 / 2),((#) (0,1)),((0,1) (#)))) c= dom P1 by TOPMETR:20;
( dom P2 = the carrier of I[01] & rng (P[01] ((1 / 2),1,((#) (0,1)),((0,1) (#)))) c= the carrier of (Closed-Interval-TSpace (0,1)) ) by FUNCT_2:def_1;
then A20: dom (P2 * (P[01] ((1 / 2),1,((#) (0,1)),((0,1) (#))))) = dom (P[01] ((1 / 2),1,((#) (0,1)),((0,1) (#)))) by RELAT_1:27, TOPMETR:20;
not 0 in { r where r is Real : ( 1 / 2 <= r & r <= 1 ) }
proof
assume 0 in { r where r is Real : ( 1 / 2 <= r & r <= 1 ) } ; ::_thesis: contradiction
then ex rr being Real st
( rr = 0 & 1 / 2 <= rr & rr <= 1 ) ;
hence contradiction ; ::_thesis: verum
end;
then not 0 in dom (P2 * (P[01] ((1 / 2),1,((#) (0,1)),((0,1) (#))))) by A8, A20, RCOMP_1:def_1;
then A21: ((P1 * (P[01] (0,(1 / 2),((#) (0,1)),((0,1) (#))))) +* (P2 * (P[01] ((1 / 2),1,((#) (0,1)),((0,1) (#)))))) . 0 = (P1 * (P[01] (0,(1 / 2),((#) (0,1)),((0,1) (#))))) . 0 by FUNCT_4:11
.= P1 . (2 * 0) by A12
.= p1 by A2 ;
dom ((P1 * (P[01] (0,(1 / 2),((#) (0,1)),((0,1) (#))))) +* (P2 * (P[01] ((1 / 2),1,((#) (0,1)),((0,1) (#)))))) = (dom (P1 * (P[01] (0,(1 / 2),((#) (0,1)),((0,1) (#)))))) \/ (dom (P2 * (P[01] ((1 / 2),1,((#) (0,1)),((0,1) (#)))))) by FUNCT_4:def_1
.= [.0,(1 / 2).] \/ [.(1 / 2),1.] by A7, A8, A19, A20, RELAT_1:27
.= the carrier of I[01] by BORSUK_1:40, XXREAL_1:174 ;
then reconsider f = (P1 * (P[01] (0,(1 / 2),((#) (0,1)),((0,1) (#))))) +* (P2 * (P[01] ((1 / 2),1,((#) (0,1)),((0,1) (#))))) as Function of I[01],((TOP-REAL n) | P) by A17, FUNCT_2:def_1, RELSET_1:4;
( P[01] (0,(1 / 2),((#) (0,1)),((0,1) (#))) is continuous & P[01] ((1 / 2),1,((#) (0,1)),((0,1) (#))) is continuous ) by TREAL_1:12;
then reconsider f = f as continuous Function of I[01],((TOP-REAL n) | P) by A1, A4, A15, A16, A18, COMPTS_1:20, TOPMETR:20;
1 in { r where r is Real : ( 1 / 2 <= r & r <= 1 ) } ;
then 1 in dom (P2 * (P[01] ((1 / 2),1,((#) (0,1)),((0,1) (#))))) by A8, A20, RCOMP_1:def_1;
then A22: f . 1 = (P2 * (P[01] ((1 / 2),1,((#) (0,1)),((0,1) (#))))) . 1 by FUNCT_4:13
.= P2 . ((2 * 1) - 1) by A9
.= p3 by A6 ;
then p1,p3 are_connected by A21, BORSUK_2:def_1;
then reconsider f = f as Path of p1,p3 by A21, A22, BORSUK_2:def_2;
for t being Point of I[01]
for t9 being Real st t = t9 holds
( ( 0 <= t9 & t9 <= 1 / 2 implies f . t = P1 . (2 * t9) ) & ( 1 / 2 <= t9 & t9 <= 1 implies f . t = P2 . ((2 * t9) - 1) ) )
proof
let t be Point of I[01]; ::_thesis: for t9 being Real st t = t9 holds
( ( 0 <= t9 & t9 <= 1 / 2 implies f . t = P1 . (2 * t9) ) & ( 1 / 2 <= t9 & t9 <= 1 implies f . t = P2 . ((2 * t9) - 1) ) )
let t9 be Real; ::_thesis: ( t = t9 implies ( ( 0 <= t9 & t9 <= 1 / 2 implies f . t = P1 . (2 * t9) ) & ( 1 / 2 <= t9 & t9 <= 1 implies f . t = P2 . ((2 * t9) - 1) ) ) )
assume A23: t = t9 ; ::_thesis: ( ( 0 <= t9 & t9 <= 1 / 2 implies f . t = P1 . (2 * t9) ) & ( 1 / 2 <= t9 & t9 <= 1 implies f . t = P2 . ((2 * t9) - 1) ) )
thus ( 0 <= t9 & t9 <= 1 / 2 implies f . t = P1 . (2 * t9) ) ::_thesis: ( 1 / 2 <= t9 & t9 <= 1 implies f . t = P2 . ((2 * t9) - 1) )
proof
assume A24: ( 0 <= t9 & t9 <= 1 / 2 ) ; ::_thesis: f . t = P1 . (2 * t9)
then t9 in { r where r is Real : ( 0 <= r & r <= 1 / 2 ) } ;
then A25: t9 in [.0,(1 / 2).] by RCOMP_1:def_1;
percases ( t9 <> 1 / 2 or t9 = 1 / 2 ) ;
supposeA26: t9 <> 1 / 2 ; ::_thesis: f . t = P1 . (2 * t9)
not t9 in dom (P2 * (P[01] ((1 / 2),1,((#) (0,1)),((0,1) (#)))))
proof
assume t9 in dom (P2 * (P[01] ((1 / 2),1,((#) (0,1)),((0,1) (#))))) ; ::_thesis: contradiction
then t9 in [.0,(1 / 2).] /\ [.(1 / 2),1.] by A8, A20, A25, XBOOLE_0:def_4;
then t9 in {(1 / 2)} by XXREAL_1:418;
hence contradiction by A26, TARSKI:def_1; ::_thesis: verum
end;
then f . t = (P1 * (P[01] (0,(1 / 2),((#) (0,1)),((0,1) (#))))) . t by A23, FUNCT_4:11
.= P1 . (2 * t9) by A12, A23, A24 ;
hence f . t = P1 . (2 * t9) ; ::_thesis: verum
end;
supposeA27: t9 = 1 / 2 ; ::_thesis: f . t = P1 . (2 * t9)
1 / 2 in { r where r is Real : ( 1 / 2 <= r & r <= 1 ) } ;
then 1 / 2 in [.(1 / 2),1.] by RCOMP_1:def_1;
then 1 / 2 in the carrier of (Closed-Interval-TSpace ((1 / 2),1)) by TOPMETR:18;
then t in dom (P2 * (P[01] ((1 / 2),1,((#) (0,1)),((0,1) (#))))) by A23, A27, FUNCT_2:def_1, TOPMETR:20;
then f . t = (P2 * (P[01] ((1 / 2),1,((#) (0,1)),((0,1) (#))))) . (1 / 2) by A23, A27, FUNCT_4:13
.= P1 . (2 * t9) by A12, A15, A27 ;
hence f . t = P1 . (2 * t9) ; ::_thesis: verum
end;
end;
end;
thus ( 1 / 2 <= t9 & t9 <= 1 implies f . t = P2 . ((2 * t9) - 1) ) ::_thesis: verum
proof
assume A28: ( 1 / 2 <= t9 & t9 <= 1 ) ; ::_thesis: f . t = P2 . ((2 * t9) - 1)
then t9 in { r where r is Real : ( 1 / 2 <= r & r <= 1 ) } ;
then t9 in [.(1 / 2),1.] by RCOMP_1:def_1;
then f . t = (P2 * (P[01] ((1 / 2),1,((#) (0,1)),((0,1) (#))))) . t by A8, A20, A23, FUNCT_4:13
.= P2 . ((2 * t9) - 1) by A9, A23, A28 ;
hence f . t = P2 . ((2 * t9) - 1) ; ::_thesis: verum
end;
end;
hence ex P0 being Path of p1,p3 st
( P0 is continuous & P0 . 0 = p1 & P0 . 1 = p3 & ( for t being Point of I[01]
for t9 being Real st t = t9 holds
( ( 0 <= t9 & t9 <= 1 / 2 implies P0 . t = P1 . (2 * t9) ) & ( 1 / 2 <= t9 & t9 <= 1 implies P0 . t = P2 . ((2 * t9) - 1) ) ) ) ) by A21, A22; ::_thesis: verum
end;
hence ex h3 being Function of I[01],((TOP-REAL n) | P) st
( h3 is continuous & w1 = h3 . 0 & w3 = h3 . 1 ) ; ::_thesis: verum
end;
theorem Th38: :: JORDAN2C:38
for n being Element of NAT
for P being Subset of (TOP-REAL n)
for w1, w2, w3 being Point of (TOP-REAL n) st w1 in P & w2 in P & w3 in P & LSeg (w1,w2) c= P & LSeg (w2,w3) c= P holds
ex h being Function of I[01],((TOP-REAL n) | P) st
( h is continuous & w1 = h . 0 & w3 = h . 1 )
proof
let n be Element of NAT ; ::_thesis: for P being Subset of (TOP-REAL n)
for w1, w2, w3 being Point of (TOP-REAL n) st w1 in P & w2 in P & w3 in P & LSeg (w1,w2) c= P & LSeg (w2,w3) c= P holds
ex h being Function of I[01],((TOP-REAL n) | P) st
( h is continuous & w1 = h . 0 & w3 = h . 1 )
let P be Subset of (TOP-REAL n); ::_thesis: for w1, w2, w3 being Point of (TOP-REAL n) st w1 in P & w2 in P & w3 in P & LSeg (w1,w2) c= P & LSeg (w2,w3) c= P holds
ex h being Function of I[01],((TOP-REAL n) | P) st
( h is continuous & w1 = h . 0 & w3 = h . 1 )
let w1, w2, w3 be Point of (TOP-REAL n); ::_thesis: ( w1 in P & w2 in P & w3 in P & LSeg (w1,w2) c= P & LSeg (w2,w3) c= P implies ex h being Function of I[01],((TOP-REAL n) | P) st
( h is continuous & w1 = h . 0 & w3 = h . 1 ) )
assume that
A1: w1 in P and
A2: w2 in P and
A3: w3 in P and
A4: LSeg (w1,w2) c= P and
A5: LSeg (w2,w3) c= P ; ::_thesis: ex h being Function of I[01],((TOP-REAL n) | P) st
( h is continuous & w1 = h . 0 & w3 = h . 1 )
reconsider Y = P as non empty Subset of (TOP-REAL n) by A1;
percases ( w1 <> w2 or w1 = w2 ) ;
supposeA6: w1 <> w2 ; ::_thesis: ex h being Function of I[01],((TOP-REAL n) | P) st
( h is continuous & w1 = h . 0 & w3 = h . 1 )
then LSeg (w1,w2) is_an_arc_of w1,w2 by TOPREAL1:9;
then consider f being Function of I[01],((TOP-REAL n) | (LSeg (w1,w2))) such that
A7: f is being_homeomorphism and
A8: f . 0 = w1 and
A9: f . 1 = w2 by TOPREAL1:def_1;
A10: rng f = [#] ((TOP-REAL n) | (LSeg (w1,w2))) by A7, TOPS_2:def_5;
then A11: rng f c= P by A4, PRE_TOPC:def_5;
then [#] ((TOP-REAL n) | (LSeg (w1,w2))) c= [#] ((TOP-REAL n) | P) by A10, PRE_TOPC:def_5;
then A12: (TOP-REAL n) | (LSeg (w1,w2)) is SubSpace of (TOP-REAL n) | P by TOPMETR:3;
dom f = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1;
then reconsider g = f as Function of [.0,1.],P by A11, FUNCT_2:2;
reconsider gt = g as Function of I[01],((TOP-REAL n) | Y) by BORSUK_1:40, PRE_TOPC:8;
A13: f is continuous by A7, TOPS_2:def_5;
now__::_thesis:_ex_h_being_Function_of_I[01],((TOP-REAL_n)_|_P)_st_
(_h_is_continuous_&_w1_=_h_._0_&_w3_=_h_._1_)
percases ( w2 <> w3 or w2 = w3 ) ;
suppose w2 <> w3 ; ::_thesis: ex h being Function of I[01],((TOP-REAL n) | P) st
( h is continuous & w1 = h . 0 & w3 = h . 1 )
then LSeg (w2,w3) is_an_arc_of w2,w3 by TOPREAL1:9;
then consider f2 being Function of I[01],((TOP-REAL n) | (LSeg (w2,w3))) such that
A14: f2 is being_homeomorphism and
A15: ( f2 . 0 = w2 & f2 . 1 = w3 ) by TOPREAL1:def_1;
A16: rng f2 = [#] ((TOP-REAL n) | (LSeg (w2,w3))) by A14, TOPS_2:def_5;
then A17: rng f2 c= P by A5, PRE_TOPC:def_5;
then [#] ((TOP-REAL n) | (LSeg (w2,w3))) c= [#] ((TOP-REAL n) | P) by A16, PRE_TOPC:def_5;
then A18: (TOP-REAL n) | (LSeg (w2,w3)) is SubSpace of (TOP-REAL n) | P by TOPMETR:3;
[#] ((TOP-REAL n) | P) = P by PRE_TOPC:def_5;
then reconsider w19 = w1, w29 = w2, w39 = w3 as Point of ((TOP-REAL n) | P) by A1, A2, A3;
A19: ( gt is continuous & w29 = gt . 1 ) by A9, A13, A12, PRE_TOPC:26;
dom f2 = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1;
then reconsider g2 = f2 as Function of [.0,1.],P by A17, FUNCT_2:2;
reconsider gt2 = g2 as Function of I[01],((TOP-REAL n) | Y) by BORSUK_1:40, PRE_TOPC:8;
f2 is continuous by A14, TOPS_2:def_5;
then gt2 is continuous by A18, PRE_TOPC:26;
then ex h being Function of I[01],((TOP-REAL n) | Y) st
( h is continuous & w19 = h . 0 & w39 = h . 1 & rng h c= (rng gt) \/ (rng gt2) ) by A8, A15, A19, BORSUK_2:13;
hence ex h being Function of I[01],((TOP-REAL n) | P) st
( h is continuous & w1 = h . 0 & w3 = h . 1 ) ; ::_thesis: verum
end;
supposeA20: w2 = w3 ; ::_thesis: ex h being Function of I[01],((TOP-REAL n) | P) st
( h is continuous & w1 = h . 0 & w3 = h . 1 )
then LSeg (w1,w3) is_an_arc_of w1,w3 by A6, TOPREAL1:9;
then consider f being Function of I[01],((TOP-REAL n) | (LSeg (w1,w3))) such that
A21: f is being_homeomorphism and
A22: ( f . 0 = w1 & f . 1 = w3 ) by TOPREAL1:def_1;
A23: rng f = [#] ((TOP-REAL n) | (LSeg (w1,w3))) by A21, TOPS_2:def_5;
then A24: rng f c= P by A4, A20, PRE_TOPC:def_5;
then [#] ((TOP-REAL n) | (LSeg (w1,w3))) c= [#] ((TOP-REAL n) | P) by A23, PRE_TOPC:def_5;
then A25: (TOP-REAL n) | (LSeg (w1,w3)) is SubSpace of (TOP-REAL n) | P by TOPMETR:3;
dom f = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1;
then reconsider g = f as Function of [.0,1.],P by A24, FUNCT_2:2;
reconsider gt = g as Function of I[01],((TOP-REAL n) | Y) by BORSUK_1:40, PRE_TOPC:8;
f is continuous by A21, TOPS_2:def_5;
then gt is continuous by A25, PRE_TOPC:26;
hence ex h being Function of I[01],((TOP-REAL n) | P) st
( h is continuous & w1 = h . 0 & w3 = h . 1 ) by A22; ::_thesis: verum
end;
end;
end;
hence ex h being Function of I[01],((TOP-REAL n) | P) st
( h is continuous & w1 = h . 0 & w3 = h . 1 ) ; ::_thesis: verum
end;
supposeA26: w1 = w2 ; ::_thesis: ex h being Function of I[01],((TOP-REAL n) | P) st
( h is continuous & w1 = h . 0 & w3 = h . 1 )
now__::_thesis:_(_(_w2_<>_w3_&_ex_h_being_Function_of_I[01],((TOP-REAL_n)_|_P)_st_
(_h_is_continuous_&_w1_=_h_._0_&_w3_=_h_._1_)_)_or_(_w2_=_w3_&_ex_h_being_Function_of_I[01],((TOP-REAL_n)_|_P)_st_
(_h_is_continuous_&_w1_=_h_._0_&_w3_=_h_._1_)_)_)
percases ( w2 <> w3 or w2 = w3 ) ;
case w2 <> w3 ; ::_thesis: ex h being Function of I[01],((TOP-REAL n) | P) st
( h is continuous & w1 = h . 0 & w3 = h . 1 )
then LSeg (w1,w3) is_an_arc_of w1,w3 by A26, TOPREAL1:9;
then consider f being Function of I[01],((TOP-REAL n) | (LSeg (w1,w3))) such that
A27: f is being_homeomorphism and
A28: ( f . 0 = w1 & f . 1 = w3 ) by TOPREAL1:def_1;
A29: rng f = [#] ((TOP-REAL n) | (LSeg (w1,w3))) by A27, TOPS_2:def_5;
then A30: rng f c= P by A5, A26, PRE_TOPC:def_5;
then [#] ((TOP-REAL n) | (LSeg (w1,w3))) c= [#] ((TOP-REAL n) | P) by A29, PRE_TOPC:def_5;
then A31: (TOP-REAL n) | (LSeg (w1,w3)) is SubSpace of (TOP-REAL n) | P by TOPMETR:3;
dom f = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1;
then reconsider g = f as Function of [.0,1.],P by A30, FUNCT_2:2;
reconsider gt = g as Function of I[01],((TOP-REAL n) | Y) by BORSUK_1:40, PRE_TOPC:8;
f is continuous by A27, TOPS_2:def_5;
then gt is continuous by A31, PRE_TOPC:26;
hence ex h being Function of I[01],((TOP-REAL n) | P) st
( h is continuous & w1 = h . 0 & w3 = h . 1 ) by A28; ::_thesis: verum
end;
caseA32: w2 = w3 ; ::_thesis: ex h being Function of I[01],((TOP-REAL n) | P) st
( h is continuous & w1 = h . 0 & w3 = h . 1 )
[#] ((TOP-REAL n) | P) = P by PRE_TOPC:def_5;
then reconsider w19 = w1, w39 = w3 as Point of ((TOP-REAL n) | P) by A1, A3;
ex f being Function of I[01],((TOP-REAL n) | Y) st
( f is continuous & f . 0 = w19 & f . 1 = w39 ) by A26, A32, BORSUK_2:3;
hence ex h being Function of I[01],((TOP-REAL n) | P) st
( h is continuous & w1 = h . 0 & w3 = h . 1 ) ; ::_thesis: verum
end;
end;
end;
hence ex h being Function of I[01],((TOP-REAL n) | P) st
( h is continuous & w1 = h . 0 & w3 = h . 1 ) ; ::_thesis: verum
end;
end;
end;
theorem Th39: :: JORDAN2C:39
for n being Element of NAT
for P being Subset of (TOP-REAL n)
for w1, w2, w3, w4 being Point of (TOP-REAL n) st w1 in P & w2 in P & w3 in P & w4 in P & LSeg (w1,w2) c= P & LSeg (w2,w3) c= P & LSeg (w3,w4) c= P holds
ex h being Function of I[01],((TOP-REAL n) | P) st
( h is continuous & w1 = h . 0 & w4 = h . 1 )
proof
let n be Element of NAT ; ::_thesis: for P being Subset of (TOP-REAL n)
for w1, w2, w3, w4 being Point of (TOP-REAL n) st w1 in P & w2 in P & w3 in P & w4 in P & LSeg (w1,w2) c= P & LSeg (w2,w3) c= P & LSeg (w3,w4) c= P holds
ex h being Function of I[01],((TOP-REAL n) | P) st
( h is continuous & w1 = h . 0 & w4 = h . 1 )
let P be Subset of (TOP-REAL n); ::_thesis: for w1, w2, w3, w4 being Point of (TOP-REAL n) st w1 in P & w2 in P & w3 in P & w4 in P & LSeg (w1,w2) c= P & LSeg (w2,w3) c= P & LSeg (w3,w4) c= P holds
ex h being Function of I[01],((TOP-REAL n) | P) st
( h is continuous & w1 = h . 0 & w4 = h . 1 )
let w1, w2, w3, w4 be Point of (TOP-REAL n); ::_thesis: ( w1 in P & w2 in P & w3 in P & w4 in P & LSeg (w1,w2) c= P & LSeg (w2,w3) c= P & LSeg (w3,w4) c= P implies ex h being Function of I[01],((TOP-REAL n) | P) st
( h is continuous & w1 = h . 0 & w4 = h . 1 ) )
assume that
A1: w1 in P and
A2: w2 in P and
A3: w3 in P and
A4: w4 in P and
A5: ( LSeg (w1,w2) c= P & LSeg (w2,w3) c= P ) and
A6: LSeg (w3,w4) c= P ; ::_thesis: ex h being Function of I[01],((TOP-REAL n) | P) st
( h is continuous & w1 = h . 0 & w4 = h . 1 )
reconsider Y = P as non empty Subset of (TOP-REAL n) by A1;
consider h2 being Function of I[01],((TOP-REAL n) | P) such that
A7: ( h2 is continuous & w1 = h2 . 0 ) and
A8: w3 = h2 . 1 by A1, A2, A3, A5, Th38;
percases ( w3 <> w4 or w3 = w4 ) ;
suppose w3 <> w4 ; ::_thesis: ex h being Function of I[01],((TOP-REAL n) | P) st
( h is continuous & w1 = h . 0 & w4 = h . 1 )
then LSeg (w3,w4) is_an_arc_of w3,w4 by TOPREAL1:9;
then consider f being Function of I[01],((TOP-REAL n) | (LSeg (w3,w4))) such that
A9: f is being_homeomorphism and
A10: ( f . 0 = w3 & f . 1 = w4 ) by TOPREAL1:def_1;
A11: rng f = [#] ((TOP-REAL n) | (LSeg (w3,w4))) by A9, TOPS_2:def_5;
then A12: rng f c= P by A6, PRE_TOPC:def_5;
then [#] ((TOP-REAL n) | (LSeg (w3,w4))) c= [#] ((TOP-REAL n) | P) by A11, PRE_TOPC:def_5;
then A13: (TOP-REAL n) | (LSeg (w3,w4)) is SubSpace of (TOP-REAL n) | P by TOPMETR:3;
[#] ((TOP-REAL n) | P) = P by PRE_TOPC:def_5;
then reconsider w19 = w1, w39 = w3, w49 = w4 as Point of ((TOP-REAL n) | P) by A1, A3, A4;
A14: w39 = h2 . 1 by A8;
dom f = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1;
then reconsider g = f as Function of [.0,1.],P by A12, FUNCT_2:2;
reconsider gt = g as Function of I[01],((TOP-REAL n) | Y) by BORSUK_1:40, PRE_TOPC:8;
f is continuous by A9, TOPS_2:def_5;
then gt is continuous by A13, PRE_TOPC:26;
then ex h being Function of I[01],((TOP-REAL n) | Y) st
( h is continuous & w19 = h . 0 & w49 = h . 1 & rng h c= (rng h2) \/ (rng gt) ) by A7, A10, A14, BORSUK_2:13;
hence ex h being Function of I[01],((TOP-REAL n) | P) st
( h is continuous & w1 = h . 0 & w4 = h . 1 ) ; ::_thesis: verum
end;
suppose w3 = w4 ; ::_thesis: ex h being Function of I[01],((TOP-REAL n) | P) st
( h is continuous & w1 = h . 0 & w4 = h . 1 )
hence ex h being Function of I[01],((TOP-REAL n) | P) st
( h is continuous & w1 = h . 0 & w4 = h . 1 ) by A7, A8; ::_thesis: verum
end;
end;
end;
theorem Th40: :: JORDAN2C:40
for n being Element of NAT
for P being Subset of (TOP-REAL n)
for w1, w2, w3, w4, w5, w6, w7 being Point of (TOP-REAL n) st w1 in P & w2 in P & w3 in P & w4 in P & w5 in P & w6 in P & w7 in P & LSeg (w1,w2) c= P & LSeg (w2,w3) c= P & LSeg (w3,w4) c= P & LSeg (w4,w5) c= P & LSeg (w5,w6) c= P & LSeg (w6,w7) c= P holds
ex h being Function of I[01],((TOP-REAL n) | P) st
( h is continuous & w1 = h . 0 & w7 = h . 1 )
proof
let n be Element of NAT ; ::_thesis: for P being Subset of (TOP-REAL n)
for w1, w2, w3, w4, w5, w6, w7 being Point of (TOP-REAL n) st w1 in P & w2 in P & w3 in P & w4 in P & w5 in P & w6 in P & w7 in P & LSeg (w1,w2) c= P & LSeg (w2,w3) c= P & LSeg (w3,w4) c= P & LSeg (w4,w5) c= P & LSeg (w5,w6) c= P & LSeg (w6,w7) c= P holds
ex h being Function of I[01],((TOP-REAL n) | P) st
( h is continuous & w1 = h . 0 & w7 = h . 1 )
let P be Subset of (TOP-REAL n); ::_thesis: for w1, w2, w3, w4, w5, w6, w7 being Point of (TOP-REAL n) st w1 in P & w2 in P & w3 in P & w4 in P & w5 in P & w6 in P & w7 in P & LSeg (w1,w2) c= P & LSeg (w2,w3) c= P & LSeg (w3,w4) c= P & LSeg (w4,w5) c= P & LSeg (w5,w6) c= P & LSeg (w6,w7) c= P holds
ex h being Function of I[01],((TOP-REAL n) | P) st
( h is continuous & w1 = h . 0 & w7 = h . 1 )
let w1, w2, w3, w4, w5, w6, w7 be Point of (TOP-REAL n); ::_thesis: ( w1 in P & w2 in P & w3 in P & w4 in P & w5 in P & w6 in P & w7 in P & LSeg (w1,w2) c= P & LSeg (w2,w3) c= P & LSeg (w3,w4) c= P & LSeg (w4,w5) c= P & LSeg (w5,w6) c= P & LSeg (w6,w7) c= P implies ex h being Function of I[01],((TOP-REAL n) | P) st
( h is continuous & w1 = h . 0 & w7 = h . 1 ) )
assume that
A1: w1 in P and
A2: ( w2 in P & w3 in P & w4 in P & w5 in P & w6 in P & w7 in P & LSeg (w1,w2) c= P & LSeg (w2,w3) c= P & LSeg (w3,w4) c= P & LSeg (w4,w5) c= P & LSeg (w5,w6) c= P & LSeg (w6,w7) c= P ) ; ::_thesis: ex h being Function of I[01],((TOP-REAL n) | P) st
( h is continuous & w1 = h . 0 & w7 = h . 1 )
( ex h2 being Function of I[01],((TOP-REAL n) | P) st
( h2 is continuous & w1 = h2 . 0 & w4 = h2 . 1 ) & ex h4 being Function of I[01],((TOP-REAL n) | P) st
( h4 is continuous & w4 = h4 . 0 & w7 = h4 . 1 ) ) by A1, A2, Th39;
hence ex h being Function of I[01],((TOP-REAL n) | P) st
( h is continuous & w1 = h . 0 & w7 = h . 1 ) by A1, Th37; ::_thesis: verum
end;
theorem Th41: :: JORDAN2C:41
for n being Element of NAT
for w1, w2 being Point of (TOP-REAL n)
for P being Subset of (TopSpaceMetr (Euclid n)) st P = LSeg (w1,w2) & not 0. (TOP-REAL n) in LSeg (w1,w2) holds
ex w0 being Point of (TOP-REAL n) st
( w0 in LSeg (w1,w2) & |.w0.| > 0 & |.w0.| = (dist_min P) . (0. (TOP-REAL n)) )
proof
let n be Element of NAT ; ::_thesis: for w1, w2 being Point of (TOP-REAL n)
for P being Subset of (TopSpaceMetr (Euclid n)) st P = LSeg (w1,w2) & not 0. (TOP-REAL n) in LSeg (w1,w2) holds
ex w0 being Point of (TOP-REAL n) st
( w0 in LSeg (w1,w2) & |.w0.| > 0 & |.w0.| = (dist_min P) . (0. (TOP-REAL n)) )
let w1, w2 be Point of (TOP-REAL n); ::_thesis: for P being Subset of (TopSpaceMetr (Euclid n)) st P = LSeg (w1,w2) & not 0. (TOP-REAL n) in LSeg (w1,w2) holds
ex w0 being Point of (TOP-REAL n) st
( w0 in LSeg (w1,w2) & |.w0.| > 0 & |.w0.| = (dist_min P) . (0. (TOP-REAL n)) )
let P be Subset of (TopSpaceMetr (Euclid n)); ::_thesis: ( P = LSeg (w1,w2) & not 0. (TOP-REAL n) in LSeg (w1,w2) implies ex w0 being Point of (TOP-REAL n) st
( w0 in LSeg (w1,w2) & |.w0.| > 0 & |.w0.| = (dist_min P) . (0. (TOP-REAL n)) ) )
assume that
A1: P = LSeg (w1,w2) and
A2: not 0. (TOP-REAL n) in LSeg (w1,w2) ; ::_thesis: ex w0 being Point of (TOP-REAL n) st
( w0 in LSeg (w1,w2) & |.w0.| > 0 & |.w0.| = (dist_min P) . (0. (TOP-REAL n)) )
set M = Euclid n;
reconsider P0 = P as Subset of (TopSpaceMetr (Euclid n)) ;
A3: TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def_8;
then reconsider Q = {(0. (TOP-REAL n))} as Subset of (TopSpaceMetr (Euclid n)) ;
P0 is compact by A1, A3, COMPTS_1:23;
then consider x1, x2 being Point of (Euclid n) such that
A4: x1 in P0 and
A5: x2 in Q and
A6: dist (x1,x2) = min_dist_min (P0,Q) by A1, A3, WEIERSTR:30;
reconsider w01 = x1 as Point of (TOP-REAL n) by EUCLID:67;
A7: x2 = 0. (TOP-REAL n) by A5, TARSKI:def_1;
reconsider o = 0. (TOP-REAL n) as Point of (Euclid n) by EUCLID:67;
reconsider o2 = 0. (TOP-REAL n) as Point of (TopSpaceMetr (Euclid n)) by A3;
for x being set holds
( x in (dist_min P0) .: Q iff x = (dist_min P0) . o )
proof
let x be set ; ::_thesis: ( x in (dist_min P0) .: Q iff x = (dist_min P0) . o )
hereby ::_thesis: ( x = (dist_min P0) . o implies x in (dist_min P0) .: Q )
assume x in (dist_min P0) .: Q ; ::_thesis: x = (dist_min P0) . o
then ex y being set st
( y in dom (dist_min P0) & y in Q & x = (dist_min P0) . y ) by FUNCT_1:def_6;
hence x = (dist_min P0) . o by TARSKI:def_1; ::_thesis: verum
end;
o2 in the carrier of (TopSpaceMetr (Euclid n)) by A3;
then A8: ( o in Q & o in dom (dist_min P0) ) by FUNCT_2:def_1, TARSKI:def_1;
assume x = (dist_min P0) . o ; ::_thesis: x in (dist_min P0) .: Q
hence x in (dist_min P0) .: Q by A8, FUNCT_1:def_6; ::_thesis: verum
end;
then A9: (dist_min P0) .: Q = {((dist_min P0) . o)} by TARSKI:def_1;
( [#] ((dist_min P0) .: Q) = (dist_min P0) .: Q & lower_bound ([#] ((dist_min P0) .: Q)) = lower_bound ((dist_min P0) .: Q) ) by WEIERSTR:def_1, WEIERSTR:def_3;
then A10: lower_bound ((dist_min P0) .: Q) = (dist_min P0) . o by A9, SEQ_4:9;
A11: |.w01.| = |.(w01 - (0. (TOP-REAL n))).| by RLVECT_1:13
.= dist (x1,x2) by A7, JGRAPH_1:28 ;
|.w01.| <> 0 by A1, A2, A4, TOPRNS_1:24;
hence ex w0 being Point of (TOP-REAL n) st
( w0 in LSeg (w1,w2) & |.w0.| > 0 & |.w0.| = (dist_min P) . (0. (TOP-REAL n)) ) by A1, A4, A6, A10, A11, WEIERSTR:def_7; ::_thesis: verum
end;
theorem Th42: :: JORDAN2C:42
for n being Element of NAT
for a being Real
for Q being Subset of (TOP-REAL n)
for w1, w4 being Point of (TOP-REAL n) st Q = { q where q is Point of (TOP-REAL n) : |.q.| > a } & w1 in Q & w4 in Q & ( for r being Real holds
( not w1 = r * w4 & not w4 = r * w1 ) ) holds
ex w2, w3 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q )
proof
let n be Element of NAT ; ::_thesis: for a being Real
for Q being Subset of (TOP-REAL n)
for w1, w4 being Point of (TOP-REAL n) st Q = { q where q is Point of (TOP-REAL n) : |.q.| > a } & w1 in Q & w4 in Q & ( for r being Real holds
( not w1 = r * w4 & not w4 = r * w1 ) ) holds
ex w2, w3 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q )
let a be Real; ::_thesis: for Q being Subset of (TOP-REAL n)
for w1, w4 being Point of (TOP-REAL n) st Q = { q where q is Point of (TOP-REAL n) : |.q.| > a } & w1 in Q & w4 in Q & ( for r being Real holds
( not w1 = r * w4 & not w4 = r * w1 ) ) holds
ex w2, w3 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q )
let Q be Subset of (TOP-REAL n); ::_thesis: for w1, w4 being Point of (TOP-REAL n) st Q = { q where q is Point of (TOP-REAL n) : |.q.| > a } & w1 in Q & w4 in Q & ( for r being Real holds
( not w1 = r * w4 & not w4 = r * w1 ) ) holds
ex w2, w3 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q )
let w1, w4 be Point of (TOP-REAL n); ::_thesis: ( Q = { q where q is Point of (TOP-REAL n) : |.q.| > a } & w1 in Q & w4 in Q & ( for r being Real holds
( not w1 = r * w4 & not w4 = r * w1 ) ) implies ex w2, w3 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q ) )
TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def_8;
then reconsider P = LSeg (w1,w4) as Subset of (TopSpaceMetr (Euclid n)) ;
assume A1: ( Q = { q where q is Point of (TOP-REAL n) : |.q.| > a } & w1 in Q & w4 in Q & ( for r being Real holds
( not w1 = r * w4 & not w4 = r * w1 ) ) ) ; ::_thesis: ex w2, w3 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q )
then not 0. (TOP-REAL n) in LSeg (w1,w4) by RLTOPSP1:71;
then consider w0 being Point of (TOP-REAL n) such that
w0 in LSeg (w1,w4) and
A2: |.w0.| > 0 and
A3: |.w0.| = (dist_min P) . (0. (TOP-REAL n)) by Th41;
set l9 = (a + 1) / |.w0.|;
set w2 = ((a + 1) / |.w0.|) * w1;
set w3 = ((a + 1) / |.w0.|) * w4;
A4: LSeg ((((a + 1) / |.w0.|) * w1),(((a + 1) / |.w0.|) * w4)) c= Q
proof
TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def_8;
then reconsider P = LSeg (w1,w4) as Subset of (TopSpaceMetr (Euclid n)) ;
reconsider o = 0. (TOP-REAL n) as Point of (Euclid n) by EUCLID:67;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg ((((a + 1) / |.w0.|) * w1),(((a + 1) / |.w0.|) * w4)) or x in Q )
A5: abs ((a + 1) / |.w0.|) = (abs (a + 1)) / (abs |.w0.|) by COMPLEX1:67
.= (abs (a + 1)) / |.w0.| by ABSVALUE:def_1 ;
(dist o) .: P c= REAL
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (dist o) .: P or x in REAL )
assume x in (dist o) .: P ; ::_thesis: x in REAL
then consider z being set such that
z in dom (dist o) and
z in P and
A6: x = (dist o) . z by FUNCT_1:def_6;
thus x in REAL by A6; ::_thesis: verum
end;
then reconsider F = (dist o) .: P as Subset of REAL ;
assume x in LSeg ((((a + 1) / |.w0.|) * w1),(((a + 1) / |.w0.|) * w4)) ; ::_thesis: x in Q
then consider r being Real such that
A7: x = ((1 - r) * (((a + 1) / |.w0.|) * w1)) + (r * (((a + 1) / |.w0.|) * w4)) and
A8: ( 0 <= r & r <= 1 ) ;
reconsider w5 = ((1 - r) * w1) + (r * w4) as Point of (TOP-REAL n) ;
reconsider w59 = w5 as Point of (Euclid n) by TOPREAL3:8;
A9: dist (w59,o) = (dist o) . w59 by WEIERSTR:def_4;
0 is LowerBound of (dist o) .: P
proof
let r be ext-real number ; :: according to XXREAL_2:def_2 ::_thesis: ( not r in (dist o) .: P or 0 <= r )
assume r in (dist o) .: P ; ::_thesis: 0 <= r
then consider x being set such that
x in dom (dist o) and
A10: x in P and
A11: r = (dist o) . x by FUNCT_1:def_6;
reconsider w0 = x as Point of (Euclid n) by A10, TOPREAL3:8;
r = dist (w0,o) by A11, WEIERSTR:def_4;
hence 0 <= r by METRIC_1:5; ::_thesis: verum
end;
then A12: F is bounded_below by XXREAL_2:def_9;
TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def_8;
then w59 in the carrier of (TopSpaceMetr (Euclid n)) ;
then A13: w59 in dom (dist o) by FUNCT_2:def_1;
w5 in LSeg (w1,w4) by A8;
then dist (w59,o) in (dist o) .: P by A13, A9, FUNCT_1:def_6;
then lower_bound F <= dist (w59,o) by A12, SEQ_4:def_2;
then dist (w59,o) >= lower_bound ([#] ((dist o) .: P)) by WEIERSTR:def_1;
then dist (w59,o) >= lower_bound ((dist o) .: P) by WEIERSTR:def_3;
then dist (w59,o) >= |.w0.| by A3, WEIERSTR:def_6;
then |.(w5 - (0. (TOP-REAL n))).| >= |.w0.| by JGRAPH_1:28;
then |.w5.| >= |.w0.| by RLVECT_1:13;
then ( abs (a + 1) >= 0 & |.w5.| / |.w0.| >= 1 ) by A2, COMPLEX1:46, XREAL_1:181;
then (abs (a + 1)) * (|.w5.| / |.w0.|) >= (abs (a + 1)) * 1 by XREAL_1:66;
then (abs (a + 1)) * ((|.w0.| ") * |.w5.|) >= abs (a + 1) by XCMPLX_0:def_9;
then ((abs (a + 1)) * (|.w0.| ")) * |.w5.| >= abs (a + 1) ;
then A14: ((abs (a + 1)) / |.w0.|) * |.w5.| >= abs (a + 1) by XCMPLX_0:def_9;
( a + 1 > a & abs (a + 1) >= a + 1 ) by ABSVALUE:4, XREAL_1:29;
then abs (a + 1) > a by XXREAL_0:2;
then ((abs (a + 1)) / |.w0.|) * |.w5.| > a by A14, XXREAL_0:2;
then |.(((a + 1) / |.w0.|) * (((1 - r) * w1) + (r * w4))).| > a by A5, TOPRNS_1:7;
then |.((((a + 1) / |.w0.|) * ((1 - r) * w1)) + (((a + 1) / |.w0.|) * (r * w4))).| > a by EUCLID:32;
then |.((((a + 1) / |.w0.|) * ((1 - r) * w1)) + ((((a + 1) / |.w0.|) * r) * w4)).| > a by EUCLID:30;
then |.(((((a + 1) / |.w0.|) * (1 - r)) * w1) + ((((a + 1) / |.w0.|) * r) * w4)).| > a by EUCLID:30;
then |.((((1 - r) * ((a + 1) / |.w0.|)) * w1) + (r * (((a + 1) / |.w0.|) * w4))).| > a by EUCLID:30;
then |.(((1 - r) * (((a + 1) / |.w0.|) * w1)) + (r * (((a + 1) / |.w0.|) * w4))).| > a by EUCLID:30;
hence x in Q by A1, A7; ::_thesis: verum
end;
A15: ((a + 1) / |.w0.|) * w4 in LSeg ((((a + 1) / |.w0.|) * w1),(((a + 1) / |.w0.|) * w4)) by RLTOPSP1:68;
then A16: ((a + 1) / |.w0.|) * w4 in Q by A4;
A17: LSeg (w4,(((a + 1) / |.w0.|) * w4)) c= Q
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg (w4,(((a + 1) / |.w0.|) * w4)) or x in Q )
assume x in LSeg (w4,(((a + 1) / |.w0.|) * w4)) ; ::_thesis: x in Q
then consider r being Real such that
A18: x = ((1 - r) * w4) + (r * (((a + 1) / |.w0.|) * w4)) and
A19: 0 <= r and
A20: r <= 1 ;
now__::_thesis:_(_(_a_>=_0_&_|.(((1_-_r)_*_w4)_+_(r_*_(((a_+_1)_/_|.w0.|)_*_w4))).|_>_a_)_or_(_a_<_0_&_|.(((1_-_r)_*_w4)_+_(r_*_(((a_+_1)_/_|.w0.|)_*_w4))).|_>_a_)_)
percases ( a >= 0 or a < 0 ) ;
caseA21: a >= 0 ; ::_thesis: |.(((1 - r) * w4) + (r * (((a + 1) / |.w0.|) * w4))).| > a
TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def_8;
then reconsider P = LSeg (w4,w1) as Subset of (TopSpaceMetr (Euclid n)) ;
reconsider o = 0. (TOP-REAL n) as Point of (Euclid n) by EUCLID:67;
reconsider w5 = ((1 - 0) * w4) + (0 * w1) as Point of (TOP-REAL n) ;
A22: ((1 - 0) * w4) + (0 * w1) = ((1 - 0) * w4) + (0. (TOP-REAL n)) by EUCLID:29
.= (1 - 0) * w4 by EUCLID:27
.= w4 by EUCLID:29 ;
(dist o) .: P c= REAL
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (dist o) .: P or x in REAL )
assume x in (dist o) .: P ; ::_thesis: x in REAL
then consider z being set such that
z in dom (dist o) and
z in P and
A23: x = (dist o) . z by FUNCT_1:def_6;
thus x in REAL by A23; ::_thesis: verum
end;
then reconsider F = (dist o) .: P as Subset of REAL ;
reconsider w59 = w5 as Point of (Euclid n) by TOPREAL3:8;
A24: dist (w59,o) = (dist o) . w59 by WEIERSTR:def_4;
0 is LowerBound of (dist o) .: P
proof
let r be ext-real number ; :: according to XXREAL_2:def_2 ::_thesis: ( not r in (dist o) .: P or 0 <= r )
assume r in (dist o) .: P ; ::_thesis: 0 <= r
then consider x being set such that
x in dom (dist o) and
A25: x in P and
A26: r = (dist o) . x by FUNCT_1:def_6;
reconsider w0 = x as Point of (Euclid n) by A25, TOPREAL3:8;
r = dist (w0,o) by A26, WEIERSTR:def_4;
hence 0 <= r by METRIC_1:5; ::_thesis: verum
end;
then A27: F is bounded_below by XXREAL_2:def_9;
TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def_8;
then w59 in the carrier of (TopSpaceMetr (Euclid n)) ;
then A28: w59 in dom (dist o) by FUNCT_2:def_1;
w5 in { (((1 - r1) * w4) + (r1 * w1)) where r1 is Real : ( 0 <= r1 & r1 <= 1 ) } ;
then dist (w59,o) in (dist o) .: P by A28, A24, FUNCT_1:def_6;
then lower_bound F <= dist (w59,o) by A27, SEQ_4:def_2;
then dist (w59,o) >= lower_bound ([#] ((dist o) .: P)) by WEIERSTR:def_1;
then dist (w59,o) >= lower_bound ((dist o) .: P) by WEIERSTR:def_3;
then dist (w59,o) >= |.w0.| by A3, WEIERSTR:def_6;
then |.(w5 - (0. (TOP-REAL n))).| >= |.w0.| by JGRAPH_1:28;
then A29: |.w5.| >= |.w0.| by RLVECT_1:13;
(r * ((a + 1) / |.w0.|)) * |.w0.| = ((r * (a + 1)) / |.w0.|) * |.w0.| by XCMPLX_1:74
.= r * (a + 1) by A2, XCMPLX_1:87 ;
then A30: (r * ((a + 1) / |.w0.|)) * |.w4.| >= r * (a + 1) by A19, A21, A22, A29, XREAL_1:64;
A31: 1 - r >= 0 by A20, XREAL_1:48;
A32: a + r >= a + 0 by A19, XREAL_1:6;
A33: ex q1 being Point of (TOP-REAL n) st
( q1 = w4 & |.q1.| > a ) by A1;
now__::_thesis:_(_(_1_-_r_>_0_&_|.(((1_-_r)_*_w4)_+_(r_*_(((a_+_1)_/_|.w0.|)_*_w4))).|_>_a_)_or_(_1_-_r_<=_0_&_|.(((1_-_r)_*_w4)_+_(r_*_(((a_+_1)_/_|.w0.|)_*_w4))).|_>_a_)_)
percases ( 1 - r > 0 or 1 - r <= 0 ) ;
case 1 - r > 0 ; ::_thesis: |.(((1 - r) * w4) + (r * (((a + 1) / |.w0.|) * w4))).| > a
then A34: (1 - r) * |.w4.| > (1 - r) * a by A33, XREAL_1:68;
(abs ((1 - r) + (r * ((a + 1) / |.w0.|)))) * |.w4.| = ((1 - r) + (r * ((a + 1) / |.w0.|))) * |.w4.| by A19, A21, A31, ABSVALUE:def_1
.= ((1 - r) * |.w4.|) + ((r * ((a + 1) / |.w0.|)) * |.w4.|) ;
then (abs ((1 - r) + (r * ((a + 1) / |.w0.|)))) * |.w4.| > (r * (a + 1)) + ((1 - r) * a) by A30, A34, XREAL_1:8;
then (abs ((1 - r) + (r * ((a + 1) / |.w0.|)))) * |.w4.| > a by A32, XXREAL_0:2;
then |.(((1 - r) + (r * ((a + 1) / |.w0.|))) * w4).| > a by TOPRNS_1:7;
then |.(((1 - r) * w4) + ((r * ((a + 1) / |.w0.|)) * w4)).| > a by EUCLID:33;
hence |.(((1 - r) * w4) + (r * (((a + 1) / |.w0.|) * w4))).| > a by EUCLID:30; ::_thesis: verum
end;
case 1 - r <= 0 ; ::_thesis: |.(((1 - r) * w4) + (r * (((a + 1) / |.w0.|) * w4))).| > a
then (1 - r) + r <= 0 + r by XREAL_1:6;
then r = 1 by A20, XXREAL_0:1;
then A35: ((1 - r) * w4) + (r * (((a + 1) / |.w0.|) * w4)) = (0. (TOP-REAL n)) + (1 * (((a + 1) / |.w0.|) * w4)) by EUCLID:29
.= (0. (TOP-REAL n)) + (((a + 1) / |.w0.|) * w4) by EUCLID:29
.= ((a + 1) / |.w0.|) * w4 by EUCLID:27 ;
ex q3 being Point of (TOP-REAL n) st
( q3 = ((a + 1) / |.w0.|) * w4 & |.q3.| > a ) by A1, A16;
hence |.(((1 - r) * w4) + (r * (((a + 1) / |.w0.|) * w4))).| > a by A35; ::_thesis: verum
end;
end;
end;
hence |.(((1 - r) * w4) + (r * (((a + 1) / |.w0.|) * w4))).| > a ; ::_thesis: verum
end;
case a < 0 ; ::_thesis: |.(((1 - r) * w4) + (r * (((a + 1) / |.w0.|) * w4))).| > a
hence |.(((1 - r) * w4) + (r * (((a + 1) / |.w0.|) * w4))).| > a ; ::_thesis: verum
end;
end;
end;
hence x in Q by A1, A18; ::_thesis: verum
end;
A36: ((a + 1) / |.w0.|) * w1 in LSeg ((((a + 1) / |.w0.|) * w1),(((a + 1) / |.w0.|) * w4)) by RLTOPSP1:68;
then A37: ((a + 1) / |.w0.|) * w1 in Q by A4;
LSeg (w1,(((a + 1) / |.w0.|) * w1)) c= Q
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg (w1,(((a + 1) / |.w0.|) * w1)) or x in Q )
assume x in LSeg (w1,(((a + 1) / |.w0.|) * w1)) ; ::_thesis: x in Q
then consider r being Real such that
A38: x = ((1 - r) * w1) + (r * (((a + 1) / |.w0.|) * w1)) and
A39: 0 <= r and
A40: r <= 1 ;
now__::_thesis:_(_(_a_>=_0_&_|.(((1_-_r)_*_w1)_+_(r_*_(((a_+_1)_/_|.w0.|)_*_w1))).|_>_a_)_or_(_a_<_0_&_|.(((1_-_r)_*_w1)_+_(r_*_(((a_+_1)_/_|.w0.|)_*_w1))).|_>_a_)_)
percases ( a >= 0 or a < 0 ) ;
caseA41: a >= 0 ; ::_thesis: |.(((1 - r) * w1) + (r * (((a + 1) / |.w0.|) * w1))).| > a
TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def_8;
then reconsider P = LSeg (w1,w4) as Subset of (TopSpaceMetr (Euclid n)) ;
reconsider o = 0. (TOP-REAL n) as Point of (Euclid n) by EUCLID:67;
reconsider w5 = ((1 - 0) * w1) + (0 * w4) as Point of (TOP-REAL n) ;
A42: ((1 - 0) * w1) + (0 * w4) = ((1 - 0) * w1) + (0. (TOP-REAL n)) by EUCLID:29
.= (1 - 0) * w1 by EUCLID:27
.= w1 by EUCLID:29 ;
(dist o) .: P c= REAL
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (dist o) .: P or x in REAL )
assume x in (dist o) .: P ; ::_thesis: x in REAL
then consider z being set such that
z in dom (dist o) and
z in P and
A43: x = (dist o) . z by FUNCT_1:def_6;
thus x in REAL by A43; ::_thesis: verum
end;
then reconsider F = (dist o) .: P as Subset of REAL ;
reconsider w59 = w5 as Point of (Euclid n) by TOPREAL3:8;
0 is LowerBound of (dist o) .: P
proof
let r be ext-real number ; :: according to XXREAL_2:def_2 ::_thesis: ( not r in (dist o) .: P or 0 <= r )
assume r in (dist o) .: P ; ::_thesis: 0 <= r
then consider x being set such that
x in dom (dist o) and
A44: x in P and
A45: r = (dist o) . x by FUNCT_1:def_6;
reconsider w0 = x as Point of (Euclid n) by A44, TOPREAL3:8;
r = dist (w0,o) by A45, WEIERSTR:def_4;
hence 0 <= r by METRIC_1:5; ::_thesis: verum
end;
then A46: F is bounded_below by XXREAL_2:def_9;
TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def_8;
then w59 in the carrier of (TopSpaceMetr (Euclid n)) ;
then A47: w59 in dom (dist o) by FUNCT_2:def_1;
( w5 in LSeg (w1,w4) & dist (w59,o) = (dist o) . w59 ) by WEIERSTR:def_4;
then dist (w59,o) in (dist o) .: P by A47, FUNCT_1:def_6;
then lower_bound F <= dist (w59,o) by A46, SEQ_4:def_2;
then dist (w59,o) >= lower_bound ([#] ((dist o) .: P)) by WEIERSTR:def_1;
then dist (w59,o) >= lower_bound ((dist o) .: P) by WEIERSTR:def_3;
then dist (w59,o) >= |.w0.| by A3, WEIERSTR:def_6;
then |.(w5 - (0. (TOP-REAL n))).| >= |.w0.| by JGRAPH_1:28;
then A48: |.w5.| >= |.w0.| by RLVECT_1:13;
(r * ((a + 1) / |.w0.|)) * |.w0.| = ((r * (a + 1)) / |.w0.|) * |.w0.| by XCMPLX_1:74
.= r * (a + 1) by A2, XCMPLX_1:87 ;
then A49: (r * ((a + 1) / |.w0.|)) * |.w1.| >= r * (a + 1) by A39, A41, A42, A48, XREAL_1:64;
A50: ex q1 being Point of (TOP-REAL n) st
( q1 = w1 & |.q1.| > a ) by A1;
A51: a + r >= a + 0 by A39, XREAL_1:6;
A52: 1 - r >= 0 by A40, XREAL_1:48;
A53: ex q2 being Point of (TOP-REAL n) st
( q2 = ((a + 1) / |.w0.|) * w1 & |.q2.| > a ) by A1, A37;
now__::_thesis:_(_(_1_-_r_>_0_&_|.(((1_-_r)_*_w1)_+_(r_*_(((a_+_1)_/_|.w0.|)_*_w1))).|_>_a_)_or_(_1_-_r_<=_0_&_|.(((1_-_r)_*_w1)_+_(r_*_(((a_+_1)_/_|.w0.|)_*_w1))).|_>_a_)_)
percases ( 1 - r > 0 or 1 - r <= 0 ) ;
case 1 - r > 0 ; ::_thesis: |.(((1 - r) * w1) + (r * (((a + 1) / |.w0.|) * w1))).| > a
then A54: (1 - r) * |.w1.| > (1 - r) * a by A50, XREAL_1:68;
(abs ((1 - r) + (r * ((a + 1) / |.w0.|)))) * |.w1.| = ((1 - r) + (r * ((a + 1) / |.w0.|))) * |.w1.| by A39, A41, A52, ABSVALUE:def_1
.= ((1 - r) * |.w1.|) + ((r * ((a + 1) / |.w0.|)) * |.w1.|) ;
then (abs ((1 - r) + (r * ((a + 1) / |.w0.|)))) * |.w1.| > (r * (a + 1)) + ((1 - r) * a) by A49, A54, XREAL_1:8;
then (abs ((1 - r) + (r * ((a + 1) / |.w0.|)))) * |.w1.| > a by A51, XXREAL_0:2;
then |.(((1 - r) + (r * ((a + 1) / |.w0.|))) * w1).| > a by TOPRNS_1:7;
then |.(((1 - r) * w1) + ((r * ((a + 1) / |.w0.|)) * w1)).| > a by EUCLID:33;
hence |.(((1 - r) * w1) + (r * (((a + 1) / |.w0.|) * w1))).| > a by EUCLID:30; ::_thesis: verum
end;
case 1 - r <= 0 ; ::_thesis: |.(((1 - r) * w1) + (r * (((a + 1) / |.w0.|) * w1))).| > a
then (1 - r) + r <= 0 + r by XREAL_1:6;
then r = 1 by A40, XXREAL_0:1;
then ((1 - r) * w1) + (r * (((a + 1) / |.w0.|) * w1)) = (0. (TOP-REAL n)) + (1 * (((a + 1) / |.w0.|) * w1)) by EUCLID:29
.= (0. (TOP-REAL n)) + (((a + 1) / |.w0.|) * w1) by EUCLID:29
.= ((a + 1) / |.w0.|) * w1 by EUCLID:27 ;
hence |.(((1 - r) * w1) + (r * (((a + 1) / |.w0.|) * w1))).| > a by A53; ::_thesis: verum
end;
end;
end;
hence |.(((1 - r) * w1) + (r * (((a + 1) / |.w0.|) * w1))).| > a ; ::_thesis: verum
end;
case a < 0 ; ::_thesis: |.(((1 - r) * w1) + (r * (((a + 1) / |.w0.|) * w1))).| > a
hence |.(((1 - r) * w1) + (r * (((a + 1) / |.w0.|) * w1))).| > a ; ::_thesis: verum
end;
end;
end;
hence x in Q by A1, A38; ::_thesis: verum
end;
hence ex w2, w3 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q ) by A4, A36, A15, A17; ::_thesis: verum
end;
theorem Th43: :: JORDAN2C:43
for n being Element of NAT
for a being Real
for Q being Subset of (TOP-REAL n)
for w1, w4 being Point of (TOP-REAL n) st Q = (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } & w1 in Q & w4 in Q & ( for r being Real holds
( not w1 = r * w4 & not w4 = r * w1 ) ) holds
ex w2, w3 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q )
proof
let n be Element of NAT ; ::_thesis: for a being Real
for Q being Subset of (TOP-REAL n)
for w1, w4 being Point of (TOP-REAL n) st Q = (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } & w1 in Q & w4 in Q & ( for r being Real holds
( not w1 = r * w4 & not w4 = r * w1 ) ) holds
ex w2, w3 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q )
let a be Real; ::_thesis: for Q being Subset of (TOP-REAL n)
for w1, w4 being Point of (TOP-REAL n) st Q = (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } & w1 in Q & w4 in Q & ( for r being Real holds
( not w1 = r * w4 & not w4 = r * w1 ) ) holds
ex w2, w3 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q )
let Q be Subset of (TOP-REAL n); ::_thesis: for w1, w4 being Point of (TOP-REAL n) st Q = (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } & w1 in Q & w4 in Q & ( for r being Real holds
( not w1 = r * w4 & not w4 = r * w1 ) ) holds
ex w2, w3 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q )
let w1, w4 be Point of (TOP-REAL n); ::_thesis: ( Q = (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } & w1 in Q & w4 in Q & ( for r being Real holds
( not w1 = r * w4 & not w4 = r * w1 ) ) implies ex w2, w3 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q ) )
TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def_8;
then reconsider P = LSeg (w1,w4) as Subset of (TopSpaceMetr (Euclid n)) ;
assume A1: ( Q = (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } & w1 in Q & w4 in Q & ( for r being Real holds
( not w1 = r * w4 & not w4 = r * w1 ) ) ) ; ::_thesis: ex w2, w3 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q )
then not 0. (TOP-REAL n) in LSeg (w1,w4) by RLTOPSP1:71;
then consider w0 being Point of (TOP-REAL n) such that
w0 in LSeg (w1,w4) and
A2: |.w0.| > 0 and
A3: |.w0.| = (dist_min P) . (0. (TOP-REAL n)) by Th41;
set l9 = a / |.w0.|;
set w2 = (a / |.w0.|) * w1;
set w3 = (a / |.w0.|) * w4;
A4: (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } = { q1 where q1 is Point of (TOP-REAL n) : |.q1.| >= a }
proof
thus (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } c= { q1 where q1 is Point of (TOP-REAL n) : |.q1.| >= a } :: according to XBOOLE_0:def_10 ::_thesis: { q1 where q1 is Point of (TOP-REAL n) : |.q1.| >= a } c= (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a }
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } or z in { q1 where q1 is Point of (TOP-REAL n) : |.q1.| >= a } )
assume A5: z in (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } ; ::_thesis: z in { q1 where q1 is Point of (TOP-REAL n) : |.q1.| >= a }
then reconsider q2 = z as Point of (TOP-REAL n) by EUCLID:22;
not z in { q where q is Point of (TOP-REAL n) : |.q.| < a } by A5, XBOOLE_0:def_5;
then |.q2.| >= a ;
hence z in { q1 where q1 is Point of (TOP-REAL n) : |.q1.| >= a } ; ::_thesis: verum
end;
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in { q1 where q1 is Point of (TOP-REAL n) : |.q1.| >= a } or z in (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } )
assume z in { q1 where q1 is Point of (TOP-REAL n) : |.q1.| >= a } ; ::_thesis: z in (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a }
then consider q1 being Point of (TOP-REAL n) such that
A6: z = q1 and
A7: |.q1.| >= a ;
q1 in the carrier of (TOP-REAL n) ;
then A8: z in REAL n by A6, EUCLID:22;
for q being Point of (TOP-REAL n) st q = z holds
|.q.| >= a by A6, A7;
then not z in { q where q is Point of (TOP-REAL n) : |.q.| < a } ;
hence z in (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } by A8, XBOOLE_0:def_5; ::_thesis: verum
end;
A9: LSeg (w1,((a / |.w0.|) * w1)) c= Q
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg (w1,((a / |.w0.|) * w1)) or x in Q )
assume x in LSeg (w1,((a / |.w0.|) * w1)) ; ::_thesis: x in Q
then consider r being Real such that
A10: x = ((1 - r) * w1) + (r * ((a / |.w0.|) * w1)) and
A11: 0 <= r and
A12: r <= 1 ;
now__::_thesis:_(_(_a_>_0_&_|.(((1_-_r)_*_w1)_+_(r_*_((a_/_|.w0.|)_*_w1))).|_>=_a_)_or_(_a_<=_0_&_|.(((1_-_r)_*_w1)_+_(r_*_((a_/_|.w0.|)_*_w1))).|_>=_a_)_)
percases ( a > 0 or a <= 0 ) ;
caseA13: a > 0 ; ::_thesis: |.(((1 - r) * w1) + (r * ((a / |.w0.|) * w1))).| >= a
TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def_8;
then reconsider P = LSeg (w1,w4) as Subset of (TopSpaceMetr (Euclid n)) ;
reconsider o = 0. (TOP-REAL n) as Point of (Euclid n) by EUCLID:67;
reconsider w5 = ((1 - 0) * w1) + (0 * w4) as Point of (TOP-REAL n) ;
A14: ((1 - 0) * w1) + (0 * w4) = ((1 - 0) * w1) + (0. (TOP-REAL n)) by EUCLID:29
.= (1 - 0) * w1 by EUCLID:27
.= w1 by EUCLID:29 ;
(dist o) .: P c= REAL
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (dist o) .: P or x in REAL )
assume x in (dist o) .: P ; ::_thesis: x in REAL
then consider z being set such that
z in dom (dist o) and
z in P and
A15: x = (dist o) . z by FUNCT_1:def_6;
thus x in REAL by A15; ::_thesis: verum
end;
then reconsider F = (dist o) .: P as Subset of REAL ;
reconsider w59 = w5 as Point of (Euclid n) by TOPREAL3:8;
0 is LowerBound of (dist o) .: P
proof
let r be ext-real number ; :: according to XXREAL_2:def_2 ::_thesis: ( not r in (dist o) .: P or 0 <= r )
assume r in (dist o) .: P ; ::_thesis: 0 <= r
then consider x being set such that
x in dom (dist o) and
A16: x in P and
A17: r = (dist o) . x by FUNCT_1:def_6;
reconsider w0 = x as Point of (Euclid n) by A16, TOPREAL3:8;
r = dist (w0,o) by A17, WEIERSTR:def_4;
hence 0 <= r by METRIC_1:5; ::_thesis: verum
end;
then A18: F is bounded_below by XXREAL_2:def_9;
TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def_8;
then w59 in the carrier of (TopSpaceMetr (Euclid n)) ;
then A19: w59 in dom (dist o) by FUNCT_2:def_1;
( w5 in LSeg (w1,w4) & dist (w59,o) = (dist o) . w59 ) by WEIERSTR:def_4;
then dist (w59,o) in (dist o) .: P by A19, FUNCT_1:def_6;
then lower_bound F <= dist (w59,o) by A18, SEQ_4:def_2;
then dist (w59,o) >= lower_bound ([#] ((dist o) .: P)) by WEIERSTR:def_1;
then dist (w59,o) >= lower_bound ((dist o) .: P) by WEIERSTR:def_3;
then dist (w59,o) >= |.w0.| by A3, WEIERSTR:def_6;
then |.(w5 - (0. (TOP-REAL n))).| >= |.w0.| by JGRAPH_1:28;
then A20: |.w5.| >= |.w0.| by RLVECT_1:13;
A21: 1 - r >= 0 by A12, XREAL_1:48;
then A22: (abs ((1 - r) + (r * (a / |.w0.|)))) * |.w1.| = ((1 - r) + (r * (a / |.w0.|))) * |.w1.| by A11, A13, ABSVALUE:def_1
.= ((1 - r) * |.w1.|) + ((r * (a / |.w0.|)) * |.w1.|) ;
ex q1 being Point of (TOP-REAL n) st
( q1 = w1 & |.q1.| >= a ) by A1, A4;
then A23: (1 - r) * |.w1.| >= (1 - r) * a by A21, XREAL_1:64;
(r * (a / |.w0.|)) * |.w0.| = ((r * a) / |.w0.|) * |.w0.| by XCMPLX_1:74
.= r * a by A2, XCMPLX_1:87 ;
then (r * (a / |.w0.|)) * |.w1.| >= r * a by A11, A13, A14, A20, XREAL_1:64;
then (abs ((1 - r) + (r * (a / |.w0.|)))) * |.w1.| >= (r * a) + ((1 - r) * a) by A23, A22, XREAL_1:7;
then |.(((1 - r) + (r * (a / |.w0.|))) * w1).| >= a by TOPRNS_1:7;
then |.(((1 - r) * w1) + ((r * (a / |.w0.|)) * w1)).| >= a by EUCLID:33;
hence |.(((1 - r) * w1) + (r * ((a / |.w0.|) * w1))).| >= a by EUCLID:30; ::_thesis: verum
end;
case a <= 0 ; ::_thesis: |.(((1 - r) * w1) + (r * ((a / |.w0.|) * w1))).| >= a
hence |.(((1 - r) * w1) + (r * ((a / |.w0.|) * w1))).| >= a ; ::_thesis: verum
end;
end;
end;
hence x in Q by A1, A4, A10; ::_thesis: verum
end;
A24: LSeg (w4,((a / |.w0.|) * w4)) c= Q
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg (w4,((a / |.w0.|) * w4)) or x in Q )
assume x in LSeg (w4,((a / |.w0.|) * w4)) ; ::_thesis: x in Q
then consider r being Real such that
A25: x = ((1 - r) * w4) + (r * ((a / |.w0.|) * w4)) and
A26: 0 <= r and
A27: r <= 1 ;
now__::_thesis:_(_(_a_>_0_&_|.(((1_-_r)_*_w4)_+_(r_*_((a_/_|.w0.|)_*_w4))).|_>=_a_)_or_(_a_<=_0_&_|.(((1_-_r)_*_w4)_+_(r_*_((a_/_|.w0.|)_*_w4))).|_>=_a_)_)
percases ( a > 0 or a <= 0 ) ;
caseA28: a > 0 ; ::_thesis: |.(((1 - r) * w4) + (r * ((a / |.w0.|) * w4))).| >= a
TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def_8;
then reconsider P = LSeg (w4,w1) as Subset of (TopSpaceMetr (Euclid n)) ;
reconsider o = 0. (TOP-REAL n) as Point of (Euclid n) by EUCLID:67;
reconsider w5 = ((1 - 0) * w4) + (0 * w1) as Point of (TOP-REAL n) ;
A29: ((1 - 0) * w4) + (0 * w1) = ((1 - 0) * w4) + (0. (TOP-REAL n)) by EUCLID:29
.= (1 - 0) * w4 by EUCLID:27
.= w4 by EUCLID:29 ;
(dist o) .: P c= REAL
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (dist o) .: P or x in REAL )
assume x in (dist o) .: P ; ::_thesis: x in REAL
then consider z being set such that
z in dom (dist o) and
z in P and
A30: x = (dist o) . z by FUNCT_1:def_6;
thus x in REAL by A30; ::_thesis: verum
end;
then reconsider F = (dist o) .: P as Subset of REAL ;
reconsider w59 = w5 as Point of (Euclid n) by TOPREAL3:8;
A31: dist (w59,o) = (dist o) . w59 by WEIERSTR:def_4;
0 is LowerBound of (dist o) .: P
proof
let r be ext-real number ; :: according to XXREAL_2:def_2 ::_thesis: ( not r in (dist o) .: P or 0 <= r )
assume r in (dist o) .: P ; ::_thesis: 0 <= r
then consider x being set such that
x in dom (dist o) and
A32: x in P and
A33: r = (dist o) . x by FUNCT_1:def_6;
reconsider w0 = x as Point of (Euclid n) by A32, TOPREAL3:8;
r = dist (w0,o) by A33, WEIERSTR:def_4;
hence 0 <= r by METRIC_1:5; ::_thesis: verum
end;
then A34: F is bounded_below by XXREAL_2:def_9;
TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def_8;
then w59 in the carrier of (TopSpaceMetr (Euclid n)) ;
then A35: w59 in dom (dist o) by FUNCT_2:def_1;
w5 in { (((1 - r1) * w4) + (r1 * w1)) where r1 is Real : ( 0 <= r1 & r1 <= 1 ) } ;
then dist (w59,o) in (dist o) .: P by A35, A31, FUNCT_1:def_6;
then lower_bound F <= dist (w59,o) by A34, SEQ_4:def_2;
then dist (w59,o) >= lower_bound ([#] ((dist o) .: P)) by WEIERSTR:def_1;
then dist (w59,o) >= lower_bound ((dist o) .: P) by WEIERSTR:def_3;
then dist (w59,o) >= |.w0.| by A3, WEIERSTR:def_6;
then |.(w5 - (0. (TOP-REAL n))).| >= |.w0.| by JGRAPH_1:28;
then A36: |.w5.| >= |.w0.| by RLVECT_1:13;
A37: 1 - r >= 0 by A27, XREAL_1:48;
then A38: (abs ((1 - r) + (r * (a / |.w0.|)))) * |.w4.| = ((1 - r) + (r * (a / |.w0.|))) * |.w4.| by A26, A28, ABSVALUE:def_1
.= ((1 - r) * |.w4.|) + ((r * (a / |.w0.|)) * |.w4.|) ;
ex q1 being Point of (TOP-REAL n) st
( q1 = w4 & |.q1.| >= a ) by A1, A4;
then A39: (1 - r) * |.w4.| >= (1 - r) * a by A37, XREAL_1:64;
(r * (a / |.w0.|)) * |.w0.| = ((r * a) / |.w0.|) * |.w0.| by XCMPLX_1:74
.= r * a by A2, XCMPLX_1:87 ;
then (r * (a / |.w0.|)) * |.w4.| >= r * a by A26, A28, A29, A36, XREAL_1:64;
then (abs ((1 - r) + (r * (a / |.w0.|)))) * |.w4.| >= (r * a) + ((1 - r) * a) by A39, A38, XREAL_1:7;
then |.(((1 - r) + (r * (a / |.w0.|))) * w4).| >= a by TOPRNS_1:7;
then |.(((1 - r) * w4) + ((r * (a / |.w0.|)) * w4)).| >= a by EUCLID:33;
hence |.(((1 - r) * w4) + (r * ((a / |.w0.|) * w4))).| >= a by EUCLID:30; ::_thesis: verum
end;
case a <= 0 ; ::_thesis: |.(((1 - r) * w4) + (r * ((a / |.w0.|) * w4))).| >= a
hence |.(((1 - r) * w4) + (r * ((a / |.w0.|) * w4))).| >= a ; ::_thesis: verum
end;
end;
end;
hence x in Q by A1, A4, A25; ::_thesis: verum
end;
A40: LSeg (((a / |.w0.|) * w1),((a / |.w0.|) * w4)) c= Q
proof
TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def_8;
then reconsider P = LSeg (w1,w4) as Subset of (TopSpaceMetr (Euclid n)) ;
reconsider o = 0. (TOP-REAL n) as Point of (Euclid n) by EUCLID:67;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg (((a / |.w0.|) * w1),((a / |.w0.|) * w4)) or x in Q )
A41: abs (a / |.w0.|) = (abs a) / (abs |.w0.|) by COMPLEX1:67
.= (abs a) / |.w0.| by ABSVALUE:def_1 ;
(dist o) .: P c= REAL
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (dist o) .: P or x in REAL )
assume x in (dist o) .: P ; ::_thesis: x in REAL
then consider z being set such that
z in dom (dist o) and
z in P and
A42: x = (dist o) . z by FUNCT_1:def_6;
thus x in REAL by A42; ::_thesis: verum
end;
then reconsider F = (dist o) .: P as Subset of REAL ;
assume x in LSeg (((a / |.w0.|) * w1),((a / |.w0.|) * w4)) ; ::_thesis: x in Q
then consider r being Real such that
A43: x = ((1 - r) * ((a / |.w0.|) * w1)) + (r * ((a / |.w0.|) * w4)) and
A44: ( 0 <= r & r <= 1 ) ;
reconsider w5 = ((1 - r) * w1) + (r * w4) as Point of (TOP-REAL n) ;
reconsider w59 = w5 as Point of (Euclid n) by TOPREAL3:8;
A45: dist (w59,o) = (dist o) . w59 by WEIERSTR:def_4;
0 is LowerBound of (dist o) .: P
proof
let r be ext-real number ; :: according to XXREAL_2:def_2 ::_thesis: ( not r in (dist o) .: P or 0 <= r )
assume r in (dist o) .: P ; ::_thesis: 0 <= r
then consider x being set such that
x in dom (dist o) and
A46: x in P and
A47: r = (dist o) . x by FUNCT_1:def_6;
reconsider w0 = x as Point of (Euclid n) by A46, TOPREAL3:8;
r = dist (w0,o) by A47, WEIERSTR:def_4;
hence 0 <= r by METRIC_1:5; ::_thesis: verum
end;
then A48: F is bounded_below by XXREAL_2:def_9;
TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def_8;
then w59 in the carrier of (TopSpaceMetr (Euclid n)) ;
then A49: w59 in dom (dist o) by FUNCT_2:def_1;
w5 in LSeg (w1,w4) by A44;
then dist (w59,o) in (dist o) .: P by A49, A45, FUNCT_1:def_6;
then lower_bound F <= dist (w59,o) by A48, SEQ_4:def_2;
then dist (w59,o) >= lower_bound ([#] ((dist o) .: P)) by WEIERSTR:def_1;
then dist (w59,o) >= lower_bound ((dist o) .: P) by WEIERSTR:def_3;
then dist (w59,o) >= |.w0.| by A3, WEIERSTR:def_6;
then |.(w5 - (0. (TOP-REAL n))).| >= |.w0.| by JGRAPH_1:28;
then |.w5.| >= |.w0.| by RLVECT_1:13;
then ( abs a >= 0 & |.w5.| / |.w0.| >= 1 ) by A2, COMPLEX1:46, XREAL_1:181;
then (abs a) * (|.w5.| / |.w0.|) >= (abs a) * 1 by XREAL_1:66;
then (abs a) * (|.w5.| * (|.w0.| ")) >= abs a by XCMPLX_0:def_9;
then ((abs a) * (|.w0.| ")) * |.w5.| >= abs a ;
then A50: ((abs a) / |.w0.|) * |.w5.| >= abs a by XCMPLX_0:def_9;
abs a >= a by ABSVALUE:4;
then ((abs a) / |.w0.|) * |.w5.| >= a by A50, XXREAL_0:2;
then |.((a / |.w0.|) * (((1 - r) * w1) + (r * w4))).| >= a by A41, TOPRNS_1:7;
then |.(((a / |.w0.|) * ((1 - r) * w1)) + ((a / |.w0.|) * (r * w4))).| >= a by EUCLID:32;
then |.(((a / |.w0.|) * ((1 - r) * w1)) + (((a / |.w0.|) * r) * w4)).| >= a by EUCLID:30;
then |.((((a / |.w0.|) * (1 - r)) * w1) + (((a / |.w0.|) * r) * w4)).| >= a by EUCLID:30;
then |.((((1 - r) * (a / |.w0.|)) * w1) + (r * ((a / |.w0.|) * w4))).| >= a by EUCLID:30;
then |.(((1 - r) * ((a / |.w0.|) * w1)) + (r * ((a / |.w0.|) * w4))).| >= a by EUCLID:30;
hence x in Q by A1, A4, A43; ::_thesis: verum
end;
( (a / |.w0.|) * w1 in LSeg (((a / |.w0.|) * w1),((a / |.w0.|) * w4)) & (a / |.w0.|) * w4 in LSeg (((a / |.w0.|) * w1),((a / |.w0.|) * w4)) ) by RLTOPSP1:68;
hence ex w2, w3 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q ) by A40, A9, A24; ::_thesis: verum
end;
theorem :: JORDAN2C:44
for f being FinSequence of REAL holds
( f is Element of REAL (len f) & f is Point of (TOP-REAL (len f)) ) by EUCLID:76;
theorem Th45: :: JORDAN2C:45
for n being Element of NAT
for x being Element of REAL n
for f, g being FinSequence of REAL
for r being Real st f = x & g = r * x holds
( len f = len g & ( for i being Element of NAT st 1 <= i & i <= len f holds
g /. i = r * (f /. i) ) )
proof
let n be Element of NAT ; ::_thesis: for x being Element of REAL n
for f, g being FinSequence of REAL
for r being Real st f = x & g = r * x holds
( len f = len g & ( for i being Element of NAT st 1 <= i & i <= len f holds
g /. i = r * (f /. i) ) )
reconsider h2 = id REAL as Function ;
let x be Element of REAL n; ::_thesis: for f, g being FinSequence of REAL
for r being Real st f = x & g = r * x holds
( len f = len g & ( for i being Element of NAT st 1 <= i & i <= len f holds
g /. i = r * (f /. i) ) )
let f, g be FinSequence of REAL ; ::_thesis: for r being Real st f = x & g = r * x holds
( len f = len g & ( for i being Element of NAT st 1 <= i & i <= len f holds
g /. i = r * (f /. i) ) )
let r be Real; ::_thesis: ( f = x & g = r * x implies ( len f = len g & ( for i being Element of NAT st 1 <= i & i <= len f holds
g /. i = r * (f /. i) ) ) )
assume that
A1: f = x and
A2: g = r * x ; ::_thesis: ( len f = len g & ( for i being Element of NAT st 1 <= i & i <= len f holds
g /. i = r * (f /. i) ) )
A3: len f = n by A1, CARD_1:def_7;
set h1 = (dom (id REAL)) --> r;
A4: dom <:((dom (id REAL)) --> r),h2:> = (dom ((dom (id REAL)) --> r)) /\ (dom (id REAL)) by FUNCT_3:def_7;
A5: len g = n by A2, CARD_1:def_7;
A6: g = (multreal * <:((dom (id REAL)) --> r),h2:>) * x by A2, FUNCOP_1:def_5;
for i being Element of NAT st 1 <= i & i <= len f holds
g /. i = r * (f /. i)
proof
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= len f implies g /. i = r * (f /. i) )
A7: dom ((dom (id REAL)) --> r) = dom (id REAL) by FUNCOP_1:13
.= REAL by FUNCT_1:17 ;
dom h2 = REAL by FUNCT_1:17;
then A8: ((dom (id REAL)) --> r) . (x . i) = r by FUNCOP_1:7;
A9: h2 . (x . i) = x . i by FUNCT_1:17;
assume A10: ( 1 <= i & i <= len f ) ; ::_thesis: g /. i = r * (f /. i)
then A11: f . i = f /. i by FINSEQ_4:15;
i in Seg (len f) by A10, FINSEQ_1:1;
then i in dom g by A3, A5, FINSEQ_1:def_3;
then A12: g . i = (multreal * <:((dom (id REAL)) --> r),h2:>) . (x . i) by A6, FUNCT_1:12;
A13: dom <:((dom (id REAL)) --> r),h2:> = (dom ((dom (id REAL)) --> r)) /\ REAL by A4, FUNCT_1:17;
then <:((dom (id REAL)) --> r),h2:> . (x . i) = [(((dom (id REAL)) --> r) . (x . i)),(h2 . (x . i))] by A7, FUNCT_3:def_7;
then g . i = multreal . (r,(f . i)) by A1, A12, A13, A7, A8, A9, FUNCT_1:13;
then g . i = r * (f /. i) by A11, BINOP_2:def_11;
hence g /. i = r * (f /. i) by A3, A5, A10, FINSEQ_4:15; ::_thesis: verum
end;
hence ( len f = len g & ( for i being Element of NAT st 1 <= i & i <= len f holds
g /. i = r * (f /. i) ) ) by A2, A3, CARD_1:def_7; ::_thesis: verum
end;
theorem Th46: :: JORDAN2C:46
for n being Element of NAT
for x being Element of REAL n
for f being FinSequence st x <> 0* n & x = f holds
ex i being Element of NAT st
( 1 <= i & i <= n & f . i <> 0 )
proof
let n be Element of NAT ; ::_thesis: for x being Element of REAL n
for f being FinSequence st x <> 0* n & x = f holds
ex i being Element of NAT st
( 1 <= i & i <= n & f . i <> 0 )
let x be Element of REAL n; ::_thesis: for f being FinSequence st x <> 0* n & x = f holds
ex i being Element of NAT st
( 1 <= i & i <= n & f . i <> 0 )
let f be FinSequence; ::_thesis: ( x <> 0* n & x = f implies ex i being Element of NAT st
( 1 <= i & i <= n & f . i <> 0 ) )
assume that
A1: x <> 0* n and
A2: x = f ; ::_thesis: ex i being Element of NAT st
( 1 <= i & i <= n & f . i <> 0 )
A3: len f = n by A2, CARD_1:def_7;
assume A4: for i being Element of NAT holds
( not 1 <= i or not i <= n or not f . i <> 0 ) ; ::_thesis: contradiction
for z being set holds
( z in f iff ex x, y being set st
( x in Seg n & y in {0} & z = [x,y] ) )
proof
let z be set ; ::_thesis: ( z in f iff ex x, y being set st
( x in Seg n & y in {0} & z = [x,y] ) )
hereby ::_thesis: ( ex x, y being set st
( x in Seg n & y in {0} & z = [x,y] ) implies z in f )
assume A5: z in f ; ::_thesis: ex x, y being set st
( x in Seg n & y in {0} & z = [x,y] )
then consider x0, y0 being set such that
A6: z = [x0,y0] by RELAT_1:def_1;
A7: y0 = f . x0 by A5, A6, FUNCT_1:1;
A8: x0 in dom f by A5, A6, XTUPLE_0:def_12;
then reconsider n1 = x0 as Element of NAT ;
A9: x0 in Seg (len f) by A8, FINSEQ_1:def_3;
then ( 1 <= n1 & n1 <= len f ) by FINSEQ_1:1;
then f . n1 = 0 by A3, A4;
then y0 in {0} by A7, TARSKI:def_1;
hence ex x, y being set st
( x in Seg n & y in {0} & z = [x,y] ) by A3, A6, A9; ::_thesis: verum
end;
given x, y being set such that A10: x in Seg n and
A11: y in {0} and
A12: z = [x,y] ; ::_thesis: z in f
reconsider n1 = x as Element of NAT by A10;
A13: n1 <= n by A10, FINSEQ_1:1;
A14: x in dom f by A3, A10, FINSEQ_1:def_3;
( y = 0 & 1 <= n1 ) by A10, A11, FINSEQ_1:1, TARSKI:def_1;
then y = f . x by A4, A13;
hence z in f by A12, A14, FUNCT_1:1; ::_thesis: verum
end;
then f = [:(Seg n),{0}:] by ZFMISC_1:def_2;
hence contradiction by A1, A2, FUNCOP_1:def_2; ::_thesis: verum
end;
theorem Th47: :: JORDAN2C:47
for n being Element of NAT
for x being Element of REAL n st n >= 2 & x <> 0* n holds
ex y being Element of REAL n st
for r being Real holds
( not y = r * x & not x = r * y )
proof
let n be Element of NAT ; ::_thesis: for x being Element of REAL n st n >= 2 & x <> 0* n holds
ex y being Element of REAL n st
for r being Real holds
( not y = r * x & not x = r * y )
let x be Element of REAL n; ::_thesis: ( n >= 2 & x <> 0* n implies ex y being Element of REAL n st
for r being Real holds
( not y = r * x & not x = r * y ) )
assume that
A1: n >= 2 and
A2: x <> 0* n ; ::_thesis: ex y being Element of REAL n st
for r being Real holds
( not y = r * x & not x = r * y )
reconsider f = x as FinSequence of REAL ;
consider i2 being Element of NAT such that
A3: 1 <= i2 and
A4: i2 <= n and
A5: f . i2 <> 0 by A2, Th46;
A6: len f = n by CARD_1:def_7;
then A7: 1 <= len f by A1, XXREAL_0:2;
percases ( i2 > 1 or i2 <= 1 ) ;
supposeA8: i2 > 1 ; ::_thesis: ex y being Element of REAL n st
for r being Real holds
( not y = r * x & not x = r * y )
reconsider g = <*((f /. 1) + 1)*> ^ (mid (f,2,(len f))) as FinSequence of REAL ;
A9: len (mid (f,2,(len f))) = ((len f) -' 2) + 1 by A1, A6, A7, FINSEQ_6:118
.= ((len f) - 2) + 1 by A1, A6, XREAL_1:233 ;
len g = (len <*((f /. 1) + 1)*>) + (len (mid (f,2,(len f)))) by FINSEQ_1:22;
then A10: len g = 1 + (((len f) - 2) + 1) by A9, FINSEQ_1:39
.= len f ;
then reconsider y2 = g as Element of REAL n by A6, EUCLID:76;
A11: len <*((f /. 1) + 1)*> = 1 by FINSEQ_1:39;
now__::_thesis:_for_r_being_Real_holds_
(_not_y2_=_r_*_x_&_not_x_=_r_*_y2_)
given r being Real such that A12: ( y2 = r * x or x = r * y2 ) ; ::_thesis: contradiction
percases ( y2 = r * x or x = r * y2 ) by A12;
supposeA13: y2 = r * x ; ::_thesis: contradiction
i2 <= ((len f) - (1 + 1)) + (1 + 1) by A4, CARD_1:def_7;
then A14: i2 - 1 <= ((((len f) - (1 + 1)) + 1) + 1) - 1 by XREAL_1:9;
A15: ( i2 -' 1 = i2 - 1 & 1 <= i2 -' 1 ) by A8, NAT_D:49, XREAL_1:233;
A16: 1 <= len f by A1, A6, XXREAL_0:2;
then A17: g /. 1 = g . 1 by A10, FINSEQ_4:15;
A18: g /. i2 = g . i2 by A3, A4, A6, A10, FINSEQ_4:15;
A19: ((i2 -' 1) + 2) -' 1 = (((i2 -' 1) + 1) + 1) -' 1
.= (i2 -' 1) + 1 by NAT_D:34
.= (i2 - 1) + 1 by A3, XREAL_1:233
.= i2 ;
A20: f /. i2 = f . i2 by A3, A4, A6, FINSEQ_4:15;
( 1 + 1 <= i2 & i2 <= 1 + (len (mid (f,2,(len f)))) ) by A4, A8, A9, CARD_1:def_7, NAT_1:13;
then g . i2 = (mid (f,2,(len f))) . (i2 - 1) by A11, FINSEQ_1:23
.= f . i2 by A1, A6, A9, A16, A15, A14, A19, FINSEQ_6:118 ;
then 1 * (f /. i2) = r * (f /. i2) by A3, A4, A6, A13, A18, A20, Th45;
then A21: 1 = r by A5, A20, XCMPLX_1:5;
g /. 1 = r * (f /. 1) by A13, A16, Th45;
then (f /. 1) + 1 = 1 * (f /. 1) by A21, A17, FINSEQ_1:41;
hence contradiction ; ::_thesis: verum
end;
supposeA22: x = r * y2 ; ::_thesis: contradiction
i2 <= ((len f) - (1 + 1)) + (1 + 1) by A4, CARD_1:def_7;
then A23: i2 - 1 <= ((((len f) - (1 + 1)) + 1) + 1) - 1 by XREAL_1:9;
A24: ( i2 -' 1 = i2 - 1 & 1 <= i2 -' 1 ) by A8, NAT_D:49, XREAL_1:233;
A25: 1 <= len f by A1, A6, XXREAL_0:2;
then A26: g /. 1 = g . 1 by A10, FINSEQ_4:15;
A27: g /. i2 = g . i2 by A3, A4, A6, A10, FINSEQ_4:15;
A28: ((i2 -' 1) + 2) -' 1 = (((i2 -' 1) + 1) + 1) -' 1
.= (i2 -' 1) + 1 by NAT_D:34
.= (i2 - 1) + 1 by A3, XREAL_1:233
.= i2 ;
A29: f /. i2 = f . i2 by A3, A4, A6, FINSEQ_4:15;
( 1 + 1 <= i2 & i2 <= 1 + (len (mid (f,2,(len f)))) ) by A4, A8, A9, CARD_1:def_7, NAT_1:13;
then g . i2 = (mid (f,2,(len f))) . (i2 - 1) by A11, FINSEQ_1:23
.= f . i2 by A1, A6, A9, A25, A24, A23, A28, FINSEQ_6:118 ;
then 1 * (f /. i2) = r * (f /. i2) by A3, A4, A6, A10, A22, A27, A29, Th45;
then A30: 1 = r by A5, A29, XCMPLX_1:5;
f /. 1 = r * (g /. 1) by A10, A22, A25, Th45;
then (f /. 1) + 1 = 1 * (f /. 1) by A30, A26, FINSEQ_1:41;
hence contradiction ; ::_thesis: verum
end;
end;
end;
hence ex y being Element of REAL n st
for r being Real holds
( not y = r * x & not x = r * y ) ; ::_thesis: verum
end;
supposeA31: i2 <= 1 ; ::_thesis: ex y being Element of REAL n st
for r being Real holds
( not y = r * x & not x = r * y )
reconsider g = (mid (f,1,((len f) -' 1))) ^ <*((f /. (len f)) + 1)*> as FinSequence of REAL ;
A32: (len f) -' 1 <= len f by NAT_D:44;
A33: (1 + 1) - 1 <= (len f) - 1 by A1, A6, XREAL_1:9;
A34: (len f) -' 1 = (len f) - 1 by A1, A6, XREAL_1:233, XXREAL_0:2;
then A35: (((len f) -' 1) -' 1) + 1 = (((len f) - 1) - 1) + 1 by A33, XREAL_1:233
.= ((len f) - (1 + 1)) + 1 ;
then A36: len (mid (f,1,((len f) -' 1))) = (len f) - 1 by A7, A34, A32, A33, FINSEQ_6:118;
( len <*((f /. (len f)) + 1)*> = 1 & len (mid (f,1,((len f) -' 1))) = ((len f) - 2) + 1 ) by A7, A34, A32, A33, A35, FINSEQ_1:39, FINSEQ_6:118;
then A37: len g = (((len f) - 2) + 1) + 1 by FINSEQ_1:22
.= len f ;
then reconsider y2 = g as Element of REAL n by A6, EUCLID:76;
A38: i2 = 1 by A3, A31, XXREAL_0:1;
now__::_thesis:_for_r_being_Real_holds_
(_not_y2_=_r_*_x_&_not_x_=_r_*_y2_)
given r being Real such that A39: ( y2 = r * x or x = r * y2 ) ; ::_thesis: contradiction
percases ( y2 = r * x or x = r * y2 ) by A39;
supposeA40: y2 = r * x ; ::_thesis: contradiction
A41: g /. i2 = g . i2 by A3, A4, A6, A37, FINSEQ_4:15;
A42: f /. i2 = f . i2 by A3, A4, A6, FINSEQ_4:15;
g . i2 = (mid (f,1,((len f) -' 1))) . i2 by A38, A33, A36, FINSEQ_6:109
.= f . i2 by A38, A34, A32, A33, FINSEQ_6:123 ;
then 1 * (f /. i2) = r * (f /. i2) by A3, A4, A6, A40, A41, A42, Th45;
then A43: 1 = r by A5, A42, XCMPLX_1:5;
A44: g . (len f) = g . (((len f) - 1) + 1)
.= (f /. (len f)) + 1 by A36, FINSEQ_1:42 ;
A45: 1 <= len f by A1, A6, XXREAL_0:2;
then A46: g /. (len f) = g . (len f) by A37, FINSEQ_4:15;
g /. (len f) = r * (f /. (len f)) by A40, A45, Th45;
hence contradiction by A43, A46, A44; ::_thesis: verum
end;
supposeA47: x = r * y2 ; ::_thesis: contradiction
A48: g /. i2 = g . i2 by A3, A4, A6, A37, FINSEQ_4:15;
A49: f /. i2 = f . i2 by A3, A4, A6, FINSEQ_4:15;
g . i2 = (mid (f,1,((len f) -' 1))) . i2 by A38, A33, A36, FINSEQ_6:109
.= f . i2 by A38, A34, A32, A33, FINSEQ_6:123 ;
then 1 * (f /. i2) = r * (f /. i2) by A3, A4, A6, A37, A47, A48, A49, Th45;
then A50: 1 = r by A5, A49, XCMPLX_1:5;
A51: g . (len f) = g . (((len f) - 1) + 1)
.= (f /. (len f)) + 1 by A36, FINSEQ_1:42 ;
A52: 1 <= len f by A1, A6, XXREAL_0:2;
then A53: g /. (len f) = g . (len f) by A37, FINSEQ_4:15;
f /. (len f) = r * (g /. (len f)) by A37, A47, A52, Th45;
hence contradiction by A50, A53, A51; ::_thesis: verum
end;
end;
end;
hence ex y being Element of REAL n st
for r being Real holds
( not y = r * x & not x = r * y ) ; ::_thesis: verum
end;
end;
end;
theorem Th48: :: JORDAN2C:48
for n being Element of NAT
for a being Real
for Q being Subset of (TOP-REAL n)
for w1, w7 being Point of (TOP-REAL n) st n >= 2 & Q = { q where q is Point of (TOP-REAL n) : |.q.| > a } & w1 in Q & w7 in Q & ex r being Real st
( w1 = r * w7 or w7 = r * w1 ) holds
ex w2, w3, w4, w5, w6 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & w4 in Q & w5 in Q & w6 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q & LSeg (w4,w5) c= Q & LSeg (w5,w6) c= Q & LSeg (w6,w7) c= Q )
proof
let n be Element of NAT ; ::_thesis: for a being Real
for Q being Subset of (TOP-REAL n)
for w1, w7 being Point of (TOP-REAL n) st n >= 2 & Q = { q where q is Point of (TOP-REAL n) : |.q.| > a } & w1 in Q & w7 in Q & ex r being Real st
( w1 = r * w7 or w7 = r * w1 ) holds
ex w2, w3, w4, w5, w6 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & w4 in Q & w5 in Q & w6 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q & LSeg (w4,w5) c= Q & LSeg (w5,w6) c= Q & LSeg (w6,w7) c= Q )
let a be Real; ::_thesis: for Q being Subset of (TOP-REAL n)
for w1, w7 being Point of (TOP-REAL n) st n >= 2 & Q = { q where q is Point of (TOP-REAL n) : |.q.| > a } & w1 in Q & w7 in Q & ex r being Real st
( w1 = r * w7 or w7 = r * w1 ) holds
ex w2, w3, w4, w5, w6 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & w4 in Q & w5 in Q & w6 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q & LSeg (w4,w5) c= Q & LSeg (w5,w6) c= Q & LSeg (w6,w7) c= Q )
let Q be Subset of (TOP-REAL n); ::_thesis: for w1, w7 being Point of (TOP-REAL n) st n >= 2 & Q = { q where q is Point of (TOP-REAL n) : |.q.| > a } & w1 in Q & w7 in Q & ex r being Real st
( w1 = r * w7 or w7 = r * w1 ) holds
ex w2, w3, w4, w5, w6 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & w4 in Q & w5 in Q & w6 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q & LSeg (w4,w5) c= Q & LSeg (w5,w6) c= Q & LSeg (w6,w7) c= Q )
let w1, w7 be Point of (TOP-REAL n); ::_thesis: ( n >= 2 & Q = { q where q is Point of (TOP-REAL n) : |.q.| > a } & w1 in Q & w7 in Q & ex r being Real st
( w1 = r * w7 or w7 = r * w1 ) implies ex w2, w3, w4, w5, w6 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & w4 in Q & w5 in Q & w6 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q & LSeg (w4,w5) c= Q & LSeg (w5,w6) c= Q & LSeg (w6,w7) c= Q ) )
assume A1: ( n >= 2 & Q = { q where q is Point of (TOP-REAL n) : |.q.| > a } & w1 in Q & w7 in Q ) ; ::_thesis: ( for r being Real holds
( not w1 = r * w7 & not w7 = r * w1 ) or ex w2, w3, w4, w5, w6 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & w4 in Q & w5 in Q & w6 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q & LSeg (w4,w5) c= Q & LSeg (w5,w6) c= Q & LSeg (w6,w7) c= Q ) )
reconsider y1 = w1 as Element of REAL n by EUCLID:22;
given r8 being Real such that A2: ( w1 = r8 * w7 or w7 = r8 * w1 ) ; ::_thesis: ex w2, w3, w4, w5, w6 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & w4 in Q & w5 in Q & w6 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q & LSeg (w4,w5) c= Q & LSeg (w5,w6) c= Q & LSeg (w6,w7) c= Q )
percases ( a >= 0 or a < 0 ) ;
supposeA3: a >= 0 ; ::_thesis: ex w2, w3, w4, w5, w6 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & w4 in Q & w5 in Q & w6 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q & LSeg (w4,w5) c= Q & LSeg (w5,w6) c= Q & LSeg (w6,w7) c= Q )
now__::_thesis:_not_w1_=_0._(TOP-REAL_n)
assume A4: w1 = 0. (TOP-REAL n) ; ::_thesis: contradiction
ex q being Point of (TOP-REAL n) st
( q = w1 & |.q.| > a ) by A1;
hence contradiction by A3, A4, TOPRNS_1:23; ::_thesis: verum
end;
then w1 <> 0* n by EUCLID:70;
then consider y being Element of REAL n such that
A5: for r being Real holds
( not y = r * y1 & not y1 = r * y ) by A1, Th47;
set y4 = ((a + 1) / |.y.|) * y;
reconsider w4 = ((a + 1) / |.y.|) * y as Point of (TOP-REAL n) by EUCLID:22;
A6: now__::_thesis:_not_|.y.|_=_0
A7: 0 * y1 = 0 * w1
.= 0. (TOP-REAL n) by EUCLID:29
.= 0* n by EUCLID:70 ;
assume |.y.| = 0 ; ::_thesis: contradiction
hence contradiction by A5, A7, EUCLID:8; ::_thesis: verum
end;
then A8: (a + 1) / |.y.| > 0 by A3, XREAL_1:139;
A9: now__::_thesis:_for_r_being_Real_holds_
(_not_w1_=_r_*_w4_&_not_w4_=_r_*_w1_)
reconsider y9 = y, y19 = y1 as Element of n -tuples_on REAL ;
given r being Real such that A10: ( w1 = r * w4 or w4 = r * w1 ) ; ::_thesis: contradiction
percases ( w1 = r * w4 or w4 = r * w1 ) by A10;
suppose w1 = r * w4 ; ::_thesis: contradiction
then y1 = (r * ((a + 1) / |.y.|)) * y by RVSUM_1:49;
hence contradiction by A5; ::_thesis: verum
end;
suppose w4 = r * w1 ; ::_thesis: contradiction
then ((((a + 1) / |.y.|) ") * ((a + 1) / |.y.|)) * y9 = (((a + 1) / |.y.|) ") * (r * y1) by RVSUM_1:49;
then ((((a + 1) / |.y.|) ") * ((a + 1) / |.y.|)) * y = ((((a + 1) / |.y.|) ") * r) * y19 by RVSUM_1:49;
then 1 * y = ((((a + 1) / |.y.|) ") * r) * y1 by A8, XCMPLX_0:def_7;
then y = ((((a + 1) / |.y.|) ") * r) * y1 by RVSUM_1:52;
hence contradiction by A5; ::_thesis: verum
end;
end;
end;
A11: |.w4.| = (abs ((a + 1) / |.y.|)) * |.y.| by EUCLID:11
.= ((a + 1) / |.y.|) * |.y.| by A3, ABSVALUE:def_1
.= a + 1 by A6, XCMPLX_1:87 ;
then |.w4.| > a by XREAL_1:29;
then A12: w4 in Q by A1;
now__::_thesis:_for_r1_being_Real_holds_
(_not_w4_=_r1_*_w7_&_not_w7_=_r1_*_w4_)
given r1 being Real such that A13: ( w4 = r1 * w7 or w7 = r1 * w4 ) ; ::_thesis: contradiction
A14: now__::_thesis:_not_r1_=_0
assume r1 = 0 ; ::_thesis: contradiction
then A15: ( w4 = 0. (TOP-REAL n) or w7 = 0. (TOP-REAL n) ) by A13, EUCLID:29;
ex q7 being Point of (TOP-REAL n) st
( q7 = w7 & |.q7.| > a ) by A1;
hence contradiction by A3, A11, A15, TOPRNS_1:23; ::_thesis: verum
end;
percases ( w1 = r8 * w7 or w7 = r8 * w1 ) by A2;
supposeA16: w1 = r8 * w7 ; ::_thesis: contradiction
now__::_thesis:_(_(_w4_=_r1_*_w7_&_contradiction_)_or_(_w7_=_r1_*_w4_&_contradiction_)_)
percases ( w4 = r1 * w7 or w7 = r1 * w4 ) by A13;
case w4 = r1 * w7 ; ::_thesis: contradiction
then (r1 ") * w4 = ((r1 ") * r1) * w7 by EUCLID:30;
then (r1 ") * w4 = 1 * w7 by A14, XCMPLX_0:def_7;
then (r1 ") * w4 = w7 by EUCLID:29;
then w1 = (r8 * (r1 ")) * w4 by A16, EUCLID:30;
hence contradiction by A9; ::_thesis: verum
end;
case w7 = r1 * w4 ; ::_thesis: contradiction
then (r1 ") * w7 = ((r1 ") * r1) * w4 by EUCLID:30;
then (r1 ") * w7 = 1 * w4 by A14, XCMPLX_0:def_7;
then (r1 ") * w7 = w4 by EUCLID:29;
then ((r1 ") ") * w4 = (((r1 ") ") * (r1 ")) * w7 by EUCLID:30;
then ((r1 ") ") * w4 = 1 * w7 by A14, XCMPLX_0:def_7;
then ((r1 ") ") * w4 = w7 by EUCLID:29;
then w1 = (r8 * ((r1 ") ")) * w4 by A16, EUCLID:30;
hence contradiction by A9; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
supposeA17: w7 = r8 * w1 ; ::_thesis: contradiction
A18: now__::_thesis:_not_r8_=_0
assume r8 = 0 ; ::_thesis: contradiction
then A19: w7 = 0. (TOP-REAL n) by A17, EUCLID:29;
ex q7 being Point of (TOP-REAL n) st
( q7 = w7 & |.q7.| > a ) by A1;
hence contradiction by A3, A19, TOPRNS_1:23; ::_thesis: verum
end;
(r8 ") * w7 = ((r8 ") * r8) * w1 by A17, EUCLID:30;
then (r8 ") * w7 = 1 * w1 by A18, XCMPLX_0:def_7;
then A20: (r8 ") * w7 = w1 by EUCLID:29;
now__::_thesis:_(_(_w4_=_r1_*_w7_&_contradiction_)_or_(_w7_=_r1_*_w4_&_contradiction_)_)
percases ( w4 = r1 * w7 or w7 = r1 * w4 ) by A13;
case w4 = r1 * w7 ; ::_thesis: contradiction
then (r1 ") * w4 = ((r1 ") * r1) * w7 by EUCLID:30;
then (r1 ") * w4 = 1 * w7 by A14, XCMPLX_0:def_7;
then (r1 ") * w4 = w7 by EUCLID:29;
then w1 = ((r8 ") * (r1 ")) * w4 by A20, EUCLID:30;
hence contradiction by A9; ::_thesis: verum
end;
case w7 = r1 * w4 ; ::_thesis: contradiction
then (r1 ") * w7 = ((r1 ") * r1) * w4 by EUCLID:30;
then (r1 ") * w7 = 1 * w4 by A14, XCMPLX_0:def_7;
then (r1 ") * w7 = w4 by EUCLID:29;
then ((r1 ") ") * w4 = (((r1 ") ") * (r1 ")) * w7 by EUCLID:30;
then ((r1 ") ") * w4 = 1 * w7 by A14, XCMPLX_0:def_7;
then ((r1 ") ") * w4 = w7 by EUCLID:29;
then w1 = ((r8 ") * ((r1 ") ")) * w4 by A20, EUCLID:30;
hence contradiction by A9; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
end;
end;
then A21: ex w29, w39 being Point of (TOP-REAL n) st
( w29 in Q & w39 in Q & LSeg (w4,w29) c= Q & LSeg (w29,w39) c= Q & LSeg (w39,w7) c= Q ) by A1, A12, Th42;
ex w2, w3 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q ) by A1, A12, A9, Th42;
hence ex w2, w3, w4, w5, w6 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & w4 in Q & w5 in Q & w6 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q & LSeg (w4,w5) c= Q & LSeg (w5,w6) c= Q & LSeg (w6,w7) c= Q ) by A12, A21; ::_thesis: verum
end;
supposeA22: a < 0 ; ::_thesis: ex w2, w3, w4, w5, w6 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & w4 in Q & w5 in Q & w6 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q & LSeg (w4,w5) c= Q & LSeg (w5,w6) c= Q & LSeg (w6,w7) c= Q )
set w2 = 0. (TOP-REAL n);
A23: REAL n c= Q
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in REAL n or x in Q )
assume x in REAL n ; ::_thesis: x in Q
then reconsider w = x as Point of (TOP-REAL n) by EUCLID:22;
|.w.| >= 0 ;
hence x in Q by A1, A22; ::_thesis: verum
end;
the carrier of (TOP-REAL n) = REAL n by EUCLID:22;
then A24: Q = the carrier of (TOP-REAL n) by A23, XBOOLE_0:def_10;
then ( LSeg (w1,(0. (TOP-REAL n))) c= Q & LSeg ((0. (TOP-REAL n)),w7) c= Q ) ;
hence ex w2, w3, w4, w5, w6 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & w4 in Q & w5 in Q & w6 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q & LSeg (w4,w5) c= Q & LSeg (w5,w6) c= Q & LSeg (w6,w7) c= Q ) by A24; ::_thesis: verum
end;
end;
end;
theorem Th49: :: JORDAN2C:49
for n being Element of NAT
for a being Real
for Q being Subset of (TOP-REAL n)
for w1, w7 being Point of (TOP-REAL n) st n >= 2 & Q = (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } & w1 in Q & w7 in Q & ex r being Real st
( w1 = r * w7 or w7 = r * w1 ) holds
ex w2, w3, w4, w5, w6 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & w4 in Q & w5 in Q & w6 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q & LSeg (w4,w5) c= Q & LSeg (w5,w6) c= Q & LSeg (w6,w7) c= Q )
proof
let n be Element of NAT ; ::_thesis: for a being Real
for Q being Subset of (TOP-REAL n)
for w1, w7 being Point of (TOP-REAL n) st n >= 2 & Q = (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } & w1 in Q & w7 in Q & ex r being Real st
( w1 = r * w7 or w7 = r * w1 ) holds
ex w2, w3, w4, w5, w6 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & w4 in Q & w5 in Q & w6 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q & LSeg (w4,w5) c= Q & LSeg (w5,w6) c= Q & LSeg (w6,w7) c= Q )
let a be Real; ::_thesis: for Q being Subset of (TOP-REAL n)
for w1, w7 being Point of (TOP-REAL n) st n >= 2 & Q = (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } & w1 in Q & w7 in Q & ex r being Real st
( w1 = r * w7 or w7 = r * w1 ) holds
ex w2, w3, w4, w5, w6 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & w4 in Q & w5 in Q & w6 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q & LSeg (w4,w5) c= Q & LSeg (w5,w6) c= Q & LSeg (w6,w7) c= Q )
let Q be Subset of (TOP-REAL n); ::_thesis: for w1, w7 being Point of (TOP-REAL n) st n >= 2 & Q = (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } & w1 in Q & w7 in Q & ex r being Real st
( w1 = r * w7 or w7 = r * w1 ) holds
ex w2, w3, w4, w5, w6 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & w4 in Q & w5 in Q & w6 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q & LSeg (w4,w5) c= Q & LSeg (w5,w6) c= Q & LSeg (w6,w7) c= Q )
let w1, w7 be Point of (TOP-REAL n); ::_thesis: ( n >= 2 & Q = (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } & w1 in Q & w7 in Q & ex r being Real st
( w1 = r * w7 or w7 = r * w1 ) implies ex w2, w3, w4, w5, w6 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & w4 in Q & w5 in Q & w6 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q & LSeg (w4,w5) c= Q & LSeg (w5,w6) c= Q & LSeg (w6,w7) c= Q ) )
reconsider y1 = w1 as Element of REAL n by EUCLID:22;
assume A1: ( n >= 2 & Q = (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } & w1 in Q & w7 in Q & ex r being Real st
( w1 = r * w7 or w7 = r * w1 ) ) ; ::_thesis: ex w2, w3, w4, w5, w6 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & w4 in Q & w5 in Q & w6 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q & LSeg (w4,w5) c= Q & LSeg (w5,w6) c= Q & LSeg (w6,w7) c= Q )
then consider r8 being Real such that
A2: ( w1 = r8 * w7 or w7 = r8 * w1 ) ;
percases ( a > 0 or a <= 0 ) ;
supposeA3: a > 0 ; ::_thesis: ex w2, w3, w4, w5, w6 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & w4 in Q & w5 in Q & w6 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q & LSeg (w4,w5) c= Q & LSeg (w5,w6) c= Q & LSeg (w6,w7) c= Q )
now__::_thesis:_not_w1_=_0._(TOP-REAL_n)
assume w1 = 0. (TOP-REAL n) ; ::_thesis: contradiction
then |.w1.| = 0 by TOPRNS_1:23;
then w1 in { q where q is Point of (TOP-REAL n) : |.q.| < a } by A3;
hence contradiction by A1, XBOOLE_0:def_5; ::_thesis: verum
end;
then w1 <> 0* n by EUCLID:70;
then consider y being Element of REAL n such that
A4: for r being Real holds
( not y = r * y1 & not y1 = r * y ) by A1, Th47;
set y4 = (a / |.y.|) * y;
reconsider w4 = (a / |.y.|) * y as Point of (TOP-REAL n) by EUCLID:22;
A5: now__::_thesis:_not_|.y.|_=_0
A6: 0 * y1 = 0 * w1
.= 0. (TOP-REAL n) by EUCLID:29
.= 0* n by EUCLID:70 ;
assume |.y.| = 0 ; ::_thesis: contradiction
hence contradiction by A4, A6, EUCLID:8; ::_thesis: verum
end;
then A7: a / |.y.| > 0 by A3, XREAL_1:139;
A8: now__::_thesis:_for_r_being_Real_holds_
(_not_w1_=_r_*_w4_&_not_w4_=_r_*_w1_)
reconsider y9 = y, y19 = y1 as Element of n -tuples_on REAL ;
given r being Real such that A9: ( w1 = r * w4 or w4 = r * w1 ) ; ::_thesis: contradiction
( y1 = (r * (a / |.y.|)) * y or (((a / |.y.|) ") * (a / |.y.|)) * y9 = ((a / |.y.|) ") * (r * y1) ) by A9, RVSUM_1:49;
then ( y1 = (r * (a / |.y.|)) * y or (((a / |.y.|) ") * (a / |.y.|)) * y = (((a / |.y.|) ") * r) * y19 ) by RVSUM_1:49;
then A10: ( y1 = (r * (a / |.y.|)) * y9 or 1 * y = (((a / |.y.|) ") * r) * y1 ) by A7, XCMPLX_0:def_7;
percases ( y1 = (r * (a / |.y.|)) * y or y = (((a / |.y.|) ") * r) * y1 ) by A10, RVSUM_1:52;
suppose y1 = (r * (a / |.y.|)) * y ; ::_thesis: contradiction
hence contradiction by A4; ::_thesis: verum
end;
suppose y = (((a / |.y.|) ") * r) * y1 ; ::_thesis: contradiction
hence contradiction by A4; ::_thesis: verum
end;
end;
end;
A11: |.w4.| = (abs (a / |.y.|)) * |.y.| by EUCLID:11
.= (a / |.y.|) * |.y.| by A3, ABSVALUE:def_1
.= a by A5, XCMPLX_1:87 ;
A12: now__::_thesis:_not_w4_in__{__q_where_q_is_Point_of_(TOP-REAL_n)_:_|.q.|_<_a__}_
assume w4 in { q where q is Point of (TOP-REAL n) : |.q.| < a } ; ::_thesis: contradiction
then ex q being Point of (TOP-REAL n) st
( q = w4 & |.q.| < a ) ;
hence contradiction by A11; ::_thesis: verum
end;
then A13: w4 in Q by A1, XBOOLE_0:def_5;
now__::_thesis:_for_r1_being_Real_holds_
(_not_w4_=_r1_*_w7_&_not_w7_=_r1_*_w4_)
given r1 being Real such that A14: ( w4 = r1 * w7 or w7 = r1 * w4 ) ; ::_thesis: contradiction
A15: now__::_thesis:_not_r1_=_0
assume r1 = 0 ; ::_thesis: contradiction
then ( w4 = 0. (TOP-REAL n) or w7 = 0. (TOP-REAL n) ) by A14, EUCLID:29;
then ( |.w4.| = 0 or |.w7.| = 0 ) by TOPRNS_1:23;
then ( w4 in { q where q is Point of (TOP-REAL n) : |.q.| < a } or w7 in { q2 where q2 is Point of (TOP-REAL n) : |.q2.| < a } ) by A3;
hence contradiction by A1, A12, XBOOLE_0:def_5; ::_thesis: verum
end;
now__::_thesis:_(_(_w1_=_r8_*_w7_&_contradiction_)_or_(_w7_=_r8_*_w1_&_contradiction_)_)
percases ( w1 = r8 * w7 or w7 = r8 * w1 ) by A2;
caseA16: w1 = r8 * w7 ; ::_thesis: contradiction
now__::_thesis:_(_(_w4_=_r1_*_w7_&_contradiction_)_or_(_w7_=_r1_*_w4_&_contradiction_)_)
percases ( w4 = r1 * w7 or w7 = r1 * w4 ) by A14;
case w4 = r1 * w7 ; ::_thesis: contradiction
then (r1 ") * w4 = ((r1 ") * r1) * w7 by EUCLID:30;
then (r1 ") * w4 = 1 * w7 by A15, XCMPLX_0:def_7;
then (r1 ") * w4 = w7 by EUCLID:29;
then w1 = (r8 * (r1 ")) * w4 by A16, EUCLID:30;
hence contradiction by A8; ::_thesis: verum
end;
case w7 = r1 * w4 ; ::_thesis: contradiction
then (r1 ") * w7 = ((r1 ") * r1) * w4 by EUCLID:30;
then (r1 ") * w7 = 1 * w4 by A15, XCMPLX_0:def_7;
then (r1 ") * w7 = w4 by EUCLID:29;
then ((r1 ") ") * w4 = (((r1 ") ") * (r1 ")) * w7 by EUCLID:30;
then ((r1 ") ") * w4 = 1 * w7 by A15, XCMPLX_0:def_7;
then ((r1 ") ") * w4 = w7 by EUCLID:29;
then w1 = (r8 * ((r1 ") ")) * w4 by A16, EUCLID:30;
hence contradiction by A8; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
caseA17: w7 = r8 * w1 ; ::_thesis: contradiction
A18: now__::_thesis:_not_r8_=_0
assume r8 = 0 ; ::_thesis: contradiction
then w7 = 0. (TOP-REAL n) by A17, EUCLID:29;
then |.w7.| = 0 by TOPRNS_1:23;
then w7 in { q where q is Point of (TOP-REAL n) : |.q.| < a } by A3;
hence contradiction by A1, XBOOLE_0:def_5; ::_thesis: verum
end;
(r8 ") * w7 = ((r8 ") * r8) * w1 by A17, EUCLID:30;
then (r8 ") * w7 = 1 * w1 by A18, XCMPLX_0:def_7;
then A19: (r8 ") * w7 = w1 by EUCLID:29;
now__::_thesis:_(_(_w4_=_r1_*_w7_&_contradiction_)_or_(_w7_=_r1_*_w4_&_contradiction_)_)
percases ( w4 = r1 * w7 or w7 = r1 * w4 ) by A14;
case w4 = r1 * w7 ; ::_thesis: contradiction
then (r1 ") * w4 = ((r1 ") * r1) * w7 by EUCLID:30;
then (r1 ") * w4 = 1 * w7 by A15, XCMPLX_0:def_7;
then (r1 ") * w4 = w7 by EUCLID:29;
then w1 = ((r8 ") * (r1 ")) * w4 by A19, EUCLID:30;
hence contradiction by A8; ::_thesis: verum
end;
case w7 = r1 * w4 ; ::_thesis: contradiction
then (r1 ") * w7 = ((r1 ") * r1) * w4 by EUCLID:30;
then (r1 ") * w7 = 1 * w4 by A15, XCMPLX_0:def_7;
then (r1 ") * w7 = w4 by EUCLID:29;
then ((r1 ") ") * w4 = (((r1 ") ") * (r1 ")) * w7 by EUCLID:30;
then ((r1 ") ") * w4 = 1 * w7 by A15, XCMPLX_0:def_7;
then ((r1 ") ") * w4 = w7 by EUCLID:29;
then w1 = ((r8 ") * ((r1 ") ")) * w4 by A19, EUCLID:30;
hence contradiction by A8; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
then A20: ex w29, w39 being Point of (TOP-REAL n) st
( w29 in Q & w39 in Q & LSeg (w4,w29) c= Q & LSeg (w29,w39) c= Q & LSeg (w39,w7) c= Q ) by A1, A13, Th43;
ex w2, w3 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q ) by A1, A13, A8, Th43;
hence ex w2, w3, w4, w5, w6 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & w4 in Q & w5 in Q & w6 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q & LSeg (w4,w5) c= Q & LSeg (w5,w6) c= Q & LSeg (w6,w7) c= Q ) by A13, A20; ::_thesis: verum
end;
supposeA21: a <= 0 ; ::_thesis: ex w2, w3, w4, w5, w6 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & w4 in Q & w5 in Q & w6 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q & LSeg (w4,w5) c= Q & LSeg (w5,w6) c= Q & LSeg (w6,w7) c= Q )
set w2 = 0. (TOP-REAL n);
A22: REAL n c= Q
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in REAL n or x in Q )
A23: now__::_thesis:_not_x_in__{__q_where_q_is_Point_of_(TOP-REAL_n)_:_|.q.|_<_a__}_
assume x in { q where q is Point of (TOP-REAL n) : |.q.| < a } ; ::_thesis: contradiction
then ex q being Point of (TOP-REAL n) st
( q = x & |.q.| < a ) ;
hence contradiction by A21; ::_thesis: verum
end;
assume x in REAL n ; ::_thesis: x in Q
hence x in Q by A1, A23, XBOOLE_0:def_5; ::_thesis: verum
end;
the carrier of (TOP-REAL n) = REAL n by EUCLID:22;
then A24: Q = the carrier of (TOP-REAL n) by A22, XBOOLE_0:def_10;
then ( LSeg (w1,(0. (TOP-REAL n))) c= Q & LSeg ((0. (TOP-REAL n)),w7) c= Q ) ;
hence ex w2, w3, w4, w5, w6 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & w4 in Q & w5 in Q & w6 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q & LSeg (w4,w5) c= Q & LSeg (w5,w6) c= Q & LSeg (w6,w7) c= Q ) by A24; ::_thesis: verum
end;
end;
end;
theorem Th50: :: JORDAN2C:50
for n being Element of NAT
for a being Real st n >= 1 holds
{ q where q is Point of (TOP-REAL n) : |.q.| > a } <> {}
proof
let n be Element of NAT ; ::_thesis: for a being Real st n >= 1 holds
{ q where q is Point of (TOP-REAL n) : |.q.| > a } <> {}
let a be Real; ::_thesis: ( n >= 1 implies { q where q is Point of (TOP-REAL n) : |.q.| > a } <> {} )
assume A1: n >= 1 ; ::_thesis: { q where q is Point of (TOP-REAL n) : |.q.| > a } <> {}
now__::_thesis:_(a_+_1)_*_(1.REAL_n)_in__{__q_where_q_is_Point_of_(TOP-REAL_n)_:_|.q.|_>_a__}_
( abs (a + 1) >= 0 & sqrt 1 <= sqrt n ) by A1, COMPLEX1:46, SQUARE_1:26;
then A2: (abs (a + 1)) * 1 <= (abs (a + 1)) * (sqrt n) by SQUARE_1:18, XREAL_1:64;
A3: a + 1 <= abs (a + 1) by ABSVALUE:4;
assume not (a + 1) * (1.REAL n) in { q where q is Point of (TOP-REAL n) : |.q.| > a } ; ::_thesis: contradiction
then A4: |.((a + 1) * (1.REAL n)).| <= a ;
A5: a < a + 1 by XREAL_1:29;
|.((a + 1) * (1.REAL n)).| = (abs (a + 1)) * |.(1.REAL n).| by TOPRNS_1:7
.= (abs (a + 1)) * (sqrt n) by EUCLID:73 ;
then a + 1 <= |.((a + 1) * (1.REAL n)).| by A2, A3, XXREAL_0:2;
hence contradiction by A4, A5, XXREAL_0:2; ::_thesis: verum
end;
hence { q where q is Point of (TOP-REAL n) : |.q.| > a } <> {} ; ::_thesis: verum
end;
theorem Th51: :: JORDAN2C:51
for n being Element of NAT
for a being Real
for P being Subset of (TOP-REAL n) st n >= 2 & P = { q where q is Point of (TOP-REAL n) : |.q.| > a } holds
P is connected
proof
let n be Element of NAT ; ::_thesis: for a being Real
for P being Subset of (TOP-REAL n) st n >= 2 & P = { q where q is Point of (TOP-REAL n) : |.q.| > a } holds
P is connected
let a be Real; ::_thesis: for P being Subset of (TOP-REAL n) st n >= 2 & P = { q where q is Point of (TOP-REAL n) : |.q.| > a } holds
P is connected
let P be Subset of (TOP-REAL n); ::_thesis: ( n >= 2 & P = { q where q is Point of (TOP-REAL n) : |.q.| > a } implies P is connected )
assume A1: ( n >= 2 & P = { q where q is Point of (TOP-REAL n) : |.q.| > a } ) ; ::_thesis: P is connected
then reconsider Q = P as non empty Subset of (TOP-REAL n) by Th50, XXREAL_0:2;
for w1, w7 being Point of (TOP-REAL n) st w1 in Q & w7 in Q & w1 <> w7 holds
ex f being Function of I[01],((TOP-REAL n) | Q) st
( f is continuous & w1 = f . 0 & w7 = f . 1 )
proof
let w1, w7 be Point of (TOP-REAL n); ::_thesis: ( w1 in Q & w7 in Q & w1 <> w7 implies ex f being Function of I[01],((TOP-REAL n) | Q) st
( f is continuous & w1 = f . 0 & w7 = f . 1 ) )
assume that
A2: ( w1 in Q & w7 in Q ) and
w1 <> w7 ; ::_thesis: ex f being Function of I[01],((TOP-REAL n) | Q) st
( f is continuous & w1 = f . 0 & w7 = f . 1 )
percases ( for r being Real holds
( not w1 = r * w7 & not w7 = r * w1 ) or ex r being Real st
( w1 = r * w7 or w7 = r * w1 ) ) ;
suppose for r being Real holds
( not w1 = r * w7 & not w7 = r * w1 ) ; ::_thesis: ex f being Function of I[01],((TOP-REAL n) | Q) st
( f is continuous & w1 = f . 0 & w7 = f . 1 )
then ex w2, w3 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w7) c= Q ) by A1, A2, Th42;
hence ex f being Function of I[01],((TOP-REAL n) | Q) st
( f is continuous & w1 = f . 0 & w7 = f . 1 ) by A2, Th39; ::_thesis: verum
end;
suppose ex r being Real st
( w1 = r * w7 or w7 = r * w1 ) ; ::_thesis: ex f being Function of I[01],((TOP-REAL n) | Q) st
( f is continuous & w1 = f . 0 & w7 = f . 1 )
then ex w2, w3, w4, w5, w6 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & w4 in Q & w5 in Q & w6 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q & LSeg (w4,w5) c= Q & LSeg (w5,w6) c= Q & LSeg (w6,w7) c= Q ) by A1, A2, Th48;
hence ex f being Function of I[01],((TOP-REAL n) | Q) st
( f is continuous & w1 = f . 0 & w7 = f . 1 ) by A2, Th40; ::_thesis: verum
end;
end;
end;
hence P is connected by JORDAN1:2; ::_thesis: verum
end;
theorem Th52: :: JORDAN2C:52
for n being Element of NAT
for a being Real st n >= 1 holds
(REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } <> {}
proof
let n be Element of NAT ; ::_thesis: for a being Real st n >= 1 holds
(REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } <> {}
let a be Real; ::_thesis: ( n >= 1 implies (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } <> {} )
A1: { q where q is Point of (TOP-REAL n) : |.q.| > a } c= (REAL n) \ { q2 where q2 is Point of (TOP-REAL n) : |.q2.| < a }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { q where q is Point of (TOP-REAL n) : |.q.| > a } or x in (REAL n) \ { q2 where q2 is Point of (TOP-REAL n) : |.q2.| < a } )
assume x in { q where q is Point of (TOP-REAL n) : |.q.| > a } ; ::_thesis: x in (REAL n) \ { q2 where q2 is Point of (TOP-REAL n) : |.q2.| < a }
then consider q being Point of (TOP-REAL n) such that
A2: q = x and
A3: |.q.| > a ;
A4: now__::_thesis:_not_x_in__{__q2_where_q2_is_Point_of_(TOP-REAL_n)_:_|.q2.|_<_a__}_
assume x in { q2 where q2 is Point of (TOP-REAL n) : |.q2.| < a } ; ::_thesis: contradiction
then ex q2 being Point of (TOP-REAL n) st
( q2 = x & |.q2.| < a ) ;
hence contradiction by A2, A3; ::_thesis: verum
end;
q in the carrier of (TOP-REAL n) ;
then q in REAL n by EUCLID:22;
hence x in (REAL n) \ { q2 where q2 is Point of (TOP-REAL n) : |.q2.| < a } by A2, A4, XBOOLE_0:def_5; ::_thesis: verum
end;
assume n >= 1 ; ::_thesis: (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } <> {}
hence (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } <> {} by A1, Th50, XBOOLE_1:3; ::_thesis: verum
end;
theorem Th53: :: JORDAN2C:53
for n being Element of NAT
for a being Real
for P being Subset of (TOP-REAL n) st n >= 2 & P = (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } holds
P is connected
proof
let n be Element of NAT ; ::_thesis: for a being Real
for P being Subset of (TOP-REAL n) st n >= 2 & P = (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } holds
P is connected
let a be Real; ::_thesis: for P being Subset of (TOP-REAL n) st n >= 2 & P = (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } holds
P is connected
let P be Subset of (TOP-REAL n); ::_thesis: ( n >= 2 & P = (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } implies P is connected )
assume A1: ( n >= 2 & P = (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } ) ; ::_thesis: P is connected
then reconsider Q = P as non empty Subset of (TOP-REAL n) by Th52, XXREAL_0:2;
for w1, w7 being Point of (TOP-REAL n) st w1 in Q & w7 in Q & w1 <> w7 holds
ex f being Function of I[01],((TOP-REAL n) | Q) st
( f is continuous & w1 = f . 0 & w7 = f . 1 )
proof
let w1, w7 be Point of (TOP-REAL n); ::_thesis: ( w1 in Q & w7 in Q & w1 <> w7 implies ex f being Function of I[01],((TOP-REAL n) | Q) st
( f is continuous & w1 = f . 0 & w7 = f . 1 ) )
assume that
A2: ( w1 in Q & w7 in Q ) and
w1 <> w7 ; ::_thesis: ex f being Function of I[01],((TOP-REAL n) | Q) st
( f is continuous & w1 = f . 0 & w7 = f . 1 )
percases ( for r being Real holds
( not w1 = r * w7 & not w7 = r * w1 ) or ex r being Real st
( w1 = r * w7 or w7 = r * w1 ) ) ;
suppose for r being Real holds
( not w1 = r * w7 & not w7 = r * w1 ) ; ::_thesis: ex f being Function of I[01],((TOP-REAL n) | Q) st
( f is continuous & w1 = f . 0 & w7 = f . 1 )
then ex w2, w3 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w7) c= Q ) by A1, A2, Th43;
hence ex f being Function of I[01],((TOP-REAL n) | Q) st
( f is continuous & w1 = f . 0 & w7 = f . 1 ) by A2, Th39; ::_thesis: verum
end;
suppose ex r being Real st
( w1 = r * w7 or w7 = r * w1 ) ; ::_thesis: ex f being Function of I[01],((TOP-REAL n) | Q) st
( f is continuous & w1 = f . 0 & w7 = f . 1 )
then ex w2, w3, w4, w5, w6 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & w4 in Q & w5 in Q & w6 in Q & LSeg (w1,w2) c= Q & LSeg (w2,w3) c= Q & LSeg (w3,w4) c= Q & LSeg (w4,w5) c= Q & LSeg (w5,w6) c= Q & LSeg (w6,w7) c= Q ) by A1, A2, Th49;
hence ex f being Function of I[01],((TOP-REAL n) | Q) st
( f is continuous & w1 = f . 0 & w7 = f . 1 ) by A2, Th40; ::_thesis: verum
end;
end;
end;
hence P is connected by JORDAN1:2; ::_thesis: verum
end;
theorem Th54: :: JORDAN2C:54
for a being Real
for n being Element of NAT
for P being Subset of (TOP-REAL n) st n >= 1 & P = (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } holds
not P is bounded
proof
let a be Real; ::_thesis: for n being Element of NAT
for P being Subset of (TOP-REAL n) st n >= 1 & P = (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } holds
not P is bounded
let n be Element of NAT ; ::_thesis: for P being Subset of (TOP-REAL n) st n >= 1 & P = (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } holds
not P is bounded
let P be Subset of (TOP-REAL n); ::_thesis: ( n >= 1 & P = (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } implies not P is bounded )
assume that
A1: n >= 1 and
A2: P = (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } ; ::_thesis: not P is bounded
percases ( a > 0 or a <= 0 ) ;
supposeA3: a > 0 ; ::_thesis: not P is bounded
now__::_thesis:_not_P_is_bounded
set p = the Element of P;
assume P is bounded ; ::_thesis: contradiction
then consider r being Real such that
A4: for q being Point of (TOP-REAL n) st q in P holds
|.q.| < r by Th34;
A5: P <> {} by A1, A2, Th52;
then the Element of P in P ;
then reconsider p = the Element of P as Point of (TOP-REAL n) ;
A6: |.p.| < r by A4, A5;
A7: now__::_thesis:_((a_+_r)_+_1)_*_(1.REAL_n)_in_(REAL_n)_\__{__q_where_q_is_Point_of_(TOP-REAL_n)_:_|.q.|_<_a__}_
assume not ((a + r) + 1) * (1.REAL n) in (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } ; ::_thesis: contradiction
then A8: ( not ((a + r) + 1) * (1.REAL n) in REAL n or ((a + r) + 1) * (1.REAL n) in { q where q is Point of (TOP-REAL n) : |.q.| < a } ) by XBOOLE_0:def_5;
((a + r) + 1) * (1.REAL n) in the carrier of (TOP-REAL n) ;
then A9: ex q being Point of (TOP-REAL n) st
( q = ((a + r) + 1) * (1.REAL n) & |.q.| < a ) by A8, EUCLID:22;
A10: (a + r) + 1 <= abs ((a + r) + 1) by ABSVALUE:4;
( a + r < (a + r) + 1 & a < a + r ) by A6, XREAL_1:29;
then A11: a < (a + r) + 1 by XXREAL_0:2;
( abs ((a + r) + 1) >= 0 & sqrt 1 <= sqrt n ) by A1, COMPLEX1:46, SQUARE_1:26;
then A12: (abs ((a + r) + 1)) * 1 <= (abs ((a + r) + 1)) * (sqrt n) by SQUARE_1:18, XREAL_1:64;
|.(((a + r) + 1) * (1.REAL n)).| = (abs ((a + r) + 1)) * |.(1.REAL n).| by TOPRNS_1:7
.= (abs ((a + r) + 1)) * (sqrt n) by EUCLID:73 ;
then (a + r) + 1 <= |.(((a + r) + 1) * (1.REAL n)).| by A12, A10, XXREAL_0:2;
hence contradiction by A9, A11, XXREAL_0:2; ::_thesis: verum
end;
A13: (a + r) + 1 <= abs ((a + r) + 1) by ABSVALUE:4;
( abs ((a + r) + 1) >= 0 & sqrt 1 <= sqrt n ) by A1, COMPLEX1:46, SQUARE_1:26;
then A14: (abs ((a + r) + 1)) * 1 <= (abs ((a + r) + 1)) * (sqrt n) by SQUARE_1:18, XREAL_1:64;
A15: a + r < (a + r) + 1 by XREAL_1:29;
|.(((a + r) + 1) * (1.REAL n)).| = (abs ((a + r) + 1)) * |.(1.REAL n).| by TOPRNS_1:7
.= (abs ((a + r) + 1)) * (sqrt n) by EUCLID:73 ;
then (a + r) + 1 <= |.(((a + r) + 1) * (1.REAL n)).| by A14, A13, XXREAL_0:2;
then A16: a + r < |.(((a + r) + 1) * (1.REAL n)).| by A15, XXREAL_0:2;
r < r + a by A3, XREAL_1:29;
hence contradiction by A2, A4, A7, A16, XXREAL_0:2; ::_thesis: verum
end;
hence not P is bounded ; ::_thesis: verum
end;
supposeA17: a <= 0 ; ::_thesis: not P is bounded
now__::_thesis:_not__{__q_where_q_is_Point_of_(TOP-REAL_n)_:_|.q.|_<_a__}__<>_{}
{ q where q is Point of (TOP-REAL n) : |.q.| < a } c= the carrier of (TOP-REAL n)
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in { q where q is Point of (TOP-REAL n) : |.q.| < a } or z in the carrier of (TOP-REAL n) )
assume z in { q where q is Point of (TOP-REAL n) : |.q.| < a } ; ::_thesis: z in the carrier of (TOP-REAL n)
then ex q being Point of (TOP-REAL n) st
( q = z & |.q.| < a ) ;
hence z in the carrier of (TOP-REAL n) ; ::_thesis: verum
end;
then reconsider Q = { q where q is Point of (TOP-REAL n) : |.q.| < a } as Subset of (TOP-REAL n) ;
set d = the Element of Q;
assume { q where q is Point of (TOP-REAL n) : |.q.| < a } <> {} ; ::_thesis: contradiction
then the Element of Q in { q where q is Point of (TOP-REAL n) : |.q.| < a } ;
then ex q being Point of (TOP-REAL n) st
( q = the Element of Q & |.q.| < a ) ;
hence contradiction by A17; ::_thesis: verum
end;
then P = [#] (TOP-REAL n) by A2, EUCLID:22;
hence not P is bounded by A1, Th35; ::_thesis: verum
end;
end;
end;
theorem Th55: :: JORDAN2C:55
for a being Real
for P being Subset of (TOP-REAL 1) st P = { q where q is Point of (TOP-REAL 1) : ex r being Real st
( q = <*r*> & r > a ) } holds
P is convex
proof
let a be Real; ::_thesis: for P being Subset of (TOP-REAL 1) st P = { q where q is Point of (TOP-REAL 1) : ex r being Real st
( q = <*r*> & r > a ) } holds
P is convex
let P be Subset of (TOP-REAL 1); ::_thesis: ( P = { q where q is Point of (TOP-REAL 1) : ex r being Real st
( q = <*r*> & r > a ) } implies P is convex )
assume A1: P = { q where q is Point of (TOP-REAL 1) : ex r being Real st
( q = <*r*> & r > a ) } ; ::_thesis: P is convex
for w1, w2 being Point of (TOP-REAL 1) st w1 in P & w2 in P holds
LSeg (w1,w2) c= P
proof
let w1, w2 be Point of (TOP-REAL 1); ::_thesis: ( w1 in P & w2 in P implies LSeg (w1,w2) c= P )
assume that
A2: w1 in P and
A3: w2 in P ; ::_thesis: LSeg (w1,w2) c= P
consider q2 being Point of (TOP-REAL 1) such that
A4: q2 = w2 and
A5: ex r being Real st
( q2 = <*r*> & r > a ) by A1, A3;
consider q1 being Point of (TOP-REAL 1) such that
A6: q1 = w1 and
A7: ex r being Real st
( q1 = <*r*> & r > a ) by A1, A2;
consider r2 being Real such that
A8: q2 = <*r2*> and
A9: r2 > a by A5;
consider r1 being Real such that
A10: q1 = <*r1*> and
A11: r1 > a by A7;
thus LSeg (w1,w2) c= P ::_thesis: verum
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg (w1,w2) or x in P )
assume x in LSeg (w1,w2) ; ::_thesis: x in P
then consider r3 being Real such that
A12: x = ((1 - r3) * w1) + (r3 * w2) and
A13: 0 <= r3 and
A14: r3 <= 1 ;
A15: 1 - r3 >= 0 by A14, XREAL_1:48;
percases ( r3 > 0 or r3 <= 0 ) ;
supposeA16: r3 > 0 ; ::_thesis: x in P
A17: ( (1 - r3) * r1 >= (1 - r3) * a & ((1 - r3) * a) + (r3 * a) = a ) by A11, A15, XREAL_1:64;
r3 * r2 > r3 * a by A9, A16, XREAL_1:68;
then A18: ((1 - r3) * r1) + (r3 * r2) > a by A17, XREAL_1:8;
<*(((1 - r3) * r1) + (r3 * r2))*> = |[((1 - r3) * r1)]| + |[(r3 * r2)]| by JORDAN2B:22
.= ((1 - r3) * |[r1]|) + |[(r3 * r2)]| by JORDAN2B:21
.= ((1 - r3) * |[r1]|) + (r3 * |[r2]|) by JORDAN2B:21 ;
hence x in P by A1, A6, A10, A4, A8, A12, A18; ::_thesis: verum
end;
suppose r3 <= 0 ; ::_thesis: x in P
then r3 = 0 by A13;
then x = w1 + (0 * w2) by A12, EUCLID:29
.= w1 + (0. (TOP-REAL 1)) by EUCLID:29
.= w1 by EUCLID:27 ;
hence x in P by A2; ::_thesis: verum
end;
end;
end;
end;
hence P is convex by JORDAN1:def_1; ::_thesis: verum
end;
theorem Th56: :: JORDAN2C:56
for a being Real
for P being Subset of (TOP-REAL 1) st P = { q where q is Point of (TOP-REAL 1) : ex r being Real st
( q = <*r*> & r < - a ) } holds
P is convex
proof
let a be Real; ::_thesis: for P being Subset of (TOP-REAL 1) st P = { q where q is Point of (TOP-REAL 1) : ex r being Real st
( q = <*r*> & r < - a ) } holds
P is convex
let P be Subset of (TOP-REAL 1); ::_thesis: ( P = { q where q is Point of (TOP-REAL 1) : ex r being Real st
( q = <*r*> & r < - a ) } implies P is convex )
assume A1: P = { q where q is Point of (TOP-REAL 1) : ex r being Real st
( q = <*r*> & r < - a ) } ; ::_thesis: P is convex
for w1, w2 being Point of (TOP-REAL 1) st w1 in P & w2 in P holds
LSeg (w1,w2) c= P
proof
let w1, w2 be Point of (TOP-REAL 1); ::_thesis: ( w1 in P & w2 in P implies LSeg (w1,w2) c= P )
assume that
A2: w1 in P and
A3: w2 in P ; ::_thesis: LSeg (w1,w2) c= P
consider q2 being Point of (TOP-REAL 1) such that
A4: q2 = w2 and
A5: ex r being Real st
( q2 = <*r*> & r < - a ) by A1, A3;
consider q1 being Point of (TOP-REAL 1) such that
A6: q1 = w1 and
A7: ex r being Real st
( q1 = <*r*> & r < - a ) by A1, A2;
consider r2 being Real such that
A8: q2 = <*r2*> and
A9: r2 < - a by A5;
consider r1 being Real such that
A10: q1 = <*r1*> and
A11: r1 < - a by A7;
thus LSeg (w1,w2) c= P ::_thesis: verum
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg (w1,w2) or x in P )
assume x in LSeg (w1,w2) ; ::_thesis: x in P
then consider r3 being Real such that
A12: x = ((1 - r3) * w1) + (r3 * w2) and
A13: 0 <= r3 and
A14: r3 <= 1 ;
A15: 1 - r3 >= 0 by A14, XREAL_1:48;
percases ( r3 > 0 or r3 <= 0 ) ;
supposeA16: r3 > 0 ; ::_thesis: x in P
A17: ( (1 - r3) * r1 <= (1 - r3) * (- a) & ((1 - r3) * (- a)) + (r3 * (- a)) = - a ) by A11, A15, XREAL_1:64;
r3 * r2 < r3 * (- a) by A9, A16, XREAL_1:68;
then A18: ((1 - r3) * r1) + (r3 * r2) < - a by A17, XREAL_1:8;
<*(((1 - r3) * r1) + (r3 * r2))*> = |[((1 - r3) * r1)]| + |[(r3 * r2)]| by JORDAN2B:22
.= ((1 - r3) * |[r1]|) + |[(r3 * r2)]| by JORDAN2B:21
.= ((1 - r3) * |[r1]|) + (r3 * |[r2]|) by JORDAN2B:21 ;
hence x in P by A1, A6, A10, A4, A8, A12, A18; ::_thesis: verum
end;
suppose r3 <= 0 ; ::_thesis: x in P
then r3 = 0 by A13;
then x = w1 + (0 * w2) by A12, EUCLID:29
.= w1 + (0. (TOP-REAL 1)) by EUCLID:29
.= w1 by EUCLID:27 ;
hence x in P by A2; ::_thesis: verum
end;
end;
end;
end;
hence P is convex by JORDAN1:def_1; ::_thesis: verum
end;
theorem :: JORDAN2C:57
canceled;
theorem :: JORDAN2C:58
canceled;
theorem Th59: :: JORDAN2C:59
for W being Subset of (Euclid 1)
for a being Real st W = { q where q is Point of (TOP-REAL 1) : ex r being Real st
( q = <*r*> & r > a ) } holds
not W is bounded
proof
let W be Subset of (Euclid 1); ::_thesis: for a being Real st W = { q where q is Point of (TOP-REAL 1) : ex r being Real st
( q = <*r*> & r > a ) } holds
not W is bounded
let a be Real; ::_thesis: ( W = { q where q is Point of (TOP-REAL 1) : ex r being Real st
( q = <*r*> & r > a ) } implies not W is bounded )
assume A1: W = { q where q is Point of (TOP-REAL 1) : ex r being Real st
( q = <*r*> & r > a ) } ; ::_thesis: not W is bounded
abs a >= 0 by COMPLEX1:46;
then A2: ((abs a) + (abs a)) + (abs a) >= 0 + (abs a) by XREAL_1:6;
assume W is bounded ; ::_thesis: contradiction
then consider r being Real such that
A3: 0 < r and
A4: for x, y being Point of (Euclid 1) st x in W & y in W holds
dist (x,y) <= r by TBSP_1:def_7;
A5: (r + (abs a)) * (1.REAL 1) = (r + (abs a)) * <*1*> by FINSEQ_2:59
.= <*((r + (abs a)) * 1)*> by RVSUM_1:47 ;
reconsider z2 = (r + (abs a)) * (1.REAL 1) as Point of (Euclid 1) by FINSEQ_2:131;
( a <= abs a & 0 + (abs a) < r + (abs a) ) by A3, ABSVALUE:4, XREAL_1:6;
then a < r + (abs a) by XXREAL_0:2;
then A6: (r + (abs a)) * (1.REAL 1) in W by A1, A5;
A7: (3 * (r + (abs a))) * (1.REAL 1) = (3 * (r + (abs a))) * <*1*> by FINSEQ_2:59
.= <*((3 * (r + (abs a))) * 1)*> by RVSUM_1:47 ;
reconsider z1 = (3 * (r + (abs a))) * (1.REAL 1) as Point of (Euclid 1) by FINSEQ_2:131;
dist (z1,z2) = |.(((3 * (r + (abs a))) * (1.REAL 1)) - ((r + (abs a)) * (1.REAL 1))).| by JGRAPH_1:28
.= |.(((((r + (abs a)) + (r + (abs a))) + (r + (abs a))) - (r + (abs a))) * (1.REAL 1)).| by EUCLID:50
.= (abs ((r + (abs a)) + (r + (abs a)))) * |.(1.REAL 1).| by TOPRNS_1:7
.= (abs ((r + (abs a)) + (r + (abs a)))) * (sqrt 1) by EUCLID:73 ;
then A8: (r + (abs a)) + (r + (abs a)) <= dist (z1,z2) by ABSVALUE:4, SQUARE_1:18;
A9: 0 <= abs a by COMPLEX1:46;
then (r + (abs a)) + 0 < (r + (abs a)) + (r + (abs a)) by A3, XREAL_1:6;
then A10: r + (abs a) < dist (z1,z2) by A8, XXREAL_0:2;
r + 0 <= r + (abs a) by A9, XREAL_1:6;
then A11: r < dist (z1,z2) by A10, XXREAL_0:2;
3 * r > 0 by A3, XREAL_1:129;
then ( a <= abs a & 0 + (abs a) < (3 * r) + (3 * (abs a)) ) by A2, ABSVALUE:4, XREAL_1:8;
then a < 3 * (r + (abs a)) by XXREAL_0:2;
then (3 * (r + (abs a))) * (1.REAL 1) in W by A1, A7;
hence contradiction by A4, A6, A11; ::_thesis: verum
end;
theorem Th60: :: JORDAN2C:60
for W being Subset of (Euclid 1)
for a being Real st W = { q where q is Point of (TOP-REAL 1) : ex r being Real st
( q = <*r*> & r < - a ) } holds
not W is bounded
proof
let W be Subset of (Euclid 1); ::_thesis: for a being Real st W = { q where q is Point of (TOP-REAL 1) : ex r being Real st
( q = <*r*> & r < - a ) } holds
not W is bounded
let a be Real; ::_thesis: ( W = { q where q is Point of (TOP-REAL 1) : ex r being Real st
( q = <*r*> & r < - a ) } implies not W is bounded )
abs a >= 0 by COMPLEX1:46;
then A1: ((abs a) + (abs a)) + (abs a) >= 0 + (abs a) by XREAL_1:6;
assume A2: W = { q where q is Point of (TOP-REAL 1) : ex r being Real st
( q = <*r*> & r < - a ) } ; ::_thesis: not W is bounded
assume W is bounded ; ::_thesis: contradiction
then consider r being Real such that
A3: 0 < r and
A4: for x, y being Point of (Euclid 1) st x in W & y in W holds
dist (x,y) <= r by TBSP_1:def_7;
A5: (- (3 * (r + (abs a)))) * (1.REAL 1) = (- (3 * (r + (abs a)))) * <*1*> by FINSEQ_2:59
.= <*((- (3 * (r + (abs a)))) * 1)*> by RVSUM_1:47 ;
reconsider z1 = (- (3 * (r + (abs a)))) * (1.REAL 1) as Point of (Euclid 1) by FINSEQ_2:131;
3 * r > 0 by A3, XREAL_1:129;
then ( a <= abs a & 0 + (abs a) < (3 * r) + (3 * (abs a)) ) by A1, ABSVALUE:4, XREAL_1:8;
then a < 3 * (r + (abs a)) by XXREAL_0:2;
then - a > - (3 * (r + (abs a))) by XREAL_1:24;
then A6: (- (3 * (r + (abs a)))) * (1.REAL 1) in { q where q is Point of (TOP-REAL 1) : ex r being Real st
( q = <*r*> & r < - a ) } by A5;
A7: (- (r + (abs a))) * (1.REAL 1) = (- (r + (abs a))) * <*1*> by FINSEQ_2:59
.= <*((- (r + (abs a))) * 1)*> by RVSUM_1:47 ;
reconsider z2 = (- (r + (abs a))) * (1.REAL 1) as Point of (Euclid 1) by FINSEQ_2:131;
dist (z1,z2) = |.(((- (3 * (r + (abs a)))) * (1.REAL 1)) - ((- (r + (abs a))) * (1.REAL 1))).| by JGRAPH_1:28
.= |.(((- (3 * (r + (abs a)))) - (- (r + (abs a)))) * (1.REAL 1)).| by EUCLID:50
.= |.(- (((- (3 * (r + (abs a)))) - (- (r + (abs a)))) * (1.REAL 1))).| by TOPRNS_1:26
.= |.((- ((- (3 * (r + (abs a)))) + (- (- (r + (abs a)))))) * (1.REAL 1)).| by EUCLID:40
.= (abs ((r + (abs a)) + (r + (abs a)))) * |.(1.REAL 1).| by TOPRNS_1:7
.= (abs ((r + (abs a)) + (r + (abs a)))) * (sqrt 1) by EUCLID:73 ;
then A8: (r + (abs a)) + (r + (abs a)) <= dist (z1,z2) by ABSVALUE:4, SQUARE_1:18;
A9: 0 <= abs a by COMPLEX1:46;
then (r + (abs a)) + 0 < (r + (abs a)) + (r + (abs a)) by A3, XREAL_1:6;
then A10: r + (abs a) < dist (z1,z2) by A8, XXREAL_0:2;
r + 0 <= r + (abs a) by A9, XREAL_1:6;
then A11: r < dist (z1,z2) by A10, XXREAL_0:2;
( a <= abs a & 0 + (abs a) < r + (abs a) ) by A3, ABSVALUE:4, XREAL_1:6;
then a < r + (abs a) by XXREAL_0:2;
then - a > - (r + (abs a)) by XREAL_1:24;
then (- (r + (abs a))) * (1.REAL 1) in W by A2, A7;
hence contradiction by A2, A4, A6, A11; ::_thesis: verum
end;
theorem Th61: :: JORDAN2C:61
for n being Element of NAT
for W being Subset of (Euclid n)
for a being Real st n >= 2 & W = { q where q is Point of (TOP-REAL n) : |.q.| > a } holds
not W is bounded
proof
let n be Element of NAT ; ::_thesis: for W being Subset of (Euclid n)
for a being Real st n >= 2 & W = { q where q is Point of (TOP-REAL n) : |.q.| > a } holds
not W is bounded
let W be Subset of (Euclid n); ::_thesis: for a being Real st n >= 2 & W = { q where q is Point of (TOP-REAL n) : |.q.| > a } holds
not W is bounded
let a be Real; ::_thesis: ( n >= 2 & W = { q where q is Point of (TOP-REAL n) : |.q.| > a } implies not W is bounded )
assume A1: ( n >= 2 & W = { q where q is Point of (TOP-REAL n) : |.q.| > a } ) ; ::_thesis: not W is bounded
A2: 1 <= n by A1, XXREAL_0:2;
then A3: 1 <= sqrt n by SQUARE_1:18, SQUARE_1:26;
assume W is bounded ; ::_thesis: contradiction
then consider r being Real such that
A4: 0 < r and
A5: for x, y being Point of (Euclid n) st x in W & y in W holds
dist (x,y) <= r by TBSP_1:def_7;
A6: r + (abs a) <= abs (r + (abs a)) by ABSVALUE:4;
( abs (r + (abs a)) >= 0 & 1 <= sqrt n ) by A2, COMPLEX1:46, SQUARE_1:18, SQUARE_1:26;
then A7: (abs (r + (abs a))) * 1 <= (abs (r + (abs a))) * (sqrt n) by XREAL_1:64;
( a <= abs a & abs a < r + (abs a) ) by A4, ABSVALUE:4, XREAL_1:29;
then A8: a < r + (abs a) by XXREAL_0:2;
|.(- ((r + (abs a)) * (1.REAL n))).| = |.((r + (abs a)) * (1.REAL n)).| by TOPRNS_1:26
.= (abs (r + (abs a))) * |.(1.REAL n).| by TOPRNS_1:7
.= (abs (r + (abs a))) * (sqrt n) by EUCLID:73 ;
then r + (abs a) <= |.(- ((r + (abs a)) * (1.REAL n))).| by A7, A6, XXREAL_0:2;
then a < |.(- ((r + (abs a)) * (1.REAL n))).| by A8, XXREAL_0:2;
then A9: - ((r + (abs a)) * (1.REAL n)) in W by A1;
then reconsider z2 = - ((r + (abs a)) * (1.REAL n)) as Point of (Euclid n) ;
A10: r + (abs a) <= abs (r + (abs a)) by ABSVALUE:4;
abs (r + (abs a)) >= 0 by COMPLEX1:46;
then A11: (abs (r + (abs a))) * 1 <= (abs (r + (abs a))) * (sqrt n) by A3, XREAL_1:64;
|.((r + (abs a)) * (1.REAL n)).| = (abs (r + (abs a))) * |.(1.REAL n).| by TOPRNS_1:7
.= (abs (r + (abs a))) * (sqrt n) by EUCLID:73 ;
then r + (abs a) <= |.((r + (abs a)) * (1.REAL n)).| by A11, A10, XXREAL_0:2;
then a < |.((r + (abs a)) * (1.REAL n)).| by A8, XXREAL_0:2;
then A12: (r + (abs a)) * (1.REAL n) in W by A1;
then reconsider z1 = (r + (abs a)) * (1.REAL n) as Point of (Euclid n) ;
A13: (r + (abs a)) + (r + (abs a)) <= abs ((r + (abs a)) + (r + (abs a))) by ABSVALUE:4;
abs ((r + (abs a)) + (r + (abs a))) >= 0 by COMPLEX1:46;
then A14: (abs ((r + (abs a)) + (r + (abs a)))) * 1 <= (abs ((r + (abs a)) + (r + (abs a)))) * (sqrt n) by A3, XREAL_1:64;
A15: 0 <= abs a by COMPLEX1:46;
then A16: r + 0 <= r + (abs a) by XREAL_1:6;
A17: (r + (abs a)) + 0 < (r + (abs a)) + (r + (abs a)) by A4, A15, XREAL_1:6;
dist (z1,z2) = |.(((r + (abs a)) * (1.REAL n)) - (- ((r + (abs a)) * (1.REAL n)))).| by JGRAPH_1:28
.= |.(((r + (abs a)) * (1.REAL n)) + ((r + (abs a)) * (1.REAL n))).|
.= |.(((r + (abs a)) + (r + (abs a))) * (1.REAL n)).| by EUCLID:33
.= (abs ((r + (abs a)) + (r + (abs a)))) * |.(1.REAL n).| by TOPRNS_1:7
.= (abs ((r + (abs a)) + (r + (abs a)))) * (sqrt n) by EUCLID:73 ;
then (r + (abs a)) + (r + (abs a)) <= dist (z1,z2) by A14, A13, XXREAL_0:2;
then r + (abs a) < dist (z1,z2) by A17, XXREAL_0:2;
then r < dist (z1,z2) by A16, XXREAL_0:2;
hence contradiction by A5, A12, A9; ::_thesis: verum
end;
theorem Th62: :: JORDAN2C:62
for n being Element of NAT
for W being Subset of (Euclid n)
for a being Real st n >= 2 & W = (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } holds
not W is bounded
proof
let n be Element of NAT ; ::_thesis: for W being Subset of (Euclid n)
for a being Real st n >= 2 & W = (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } holds
not W is bounded
let W be Subset of (Euclid n); ::_thesis: for a being Real st n >= 2 & W = (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } holds
not W is bounded
let a be Real; ::_thesis: ( n >= 2 & W = (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } implies not W is bounded )
reconsider 1R = 1.REAL n as Point of (TOP-REAL n) ;
assume A1: ( n >= 2 & W = (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } ) ; ::_thesis: not W is bounded
assume W is bounded ; ::_thesis: contradiction
then consider r being Real such that
A2: 0 < r and
A3: for x, y being Point of (Euclid n) st x in W & y in W holds
dist (x,y) <= r by TBSP_1:def_7;
A4: 0 <= abs a by COMPLEX1:46;
then A5: (r + (abs a)) + 0 < (r + (abs a)) + (r + (abs a)) by A2, XREAL_1:6;
n >= 1 by A1, XXREAL_0:2;
then A6: 1 <= sqrt n by SQUARE_1:18, SQUARE_1:26;
A7: now__::_thesis:_not_-_((r_+_(abs_a))_*_(1.REAL_n))_in__{__q_where_q_is_Point_of_(TOP-REAL_n)_:_|.q.|_<_a__}_
( a <= abs a & abs a < r + (abs a) ) by A2, ABSVALUE:4, XREAL_1:29;
then A8: a < r + (abs a) by XXREAL_0:2;
assume - ((r + (abs a)) * (1.REAL n)) in { q where q is Point of (TOP-REAL n) : |.q.| < a } ; ::_thesis: contradiction
then A9: ex q being Point of (TOP-REAL n) st
( q = - ((r + (abs a)) * (1.REAL n)) & |.q.| < a ) ;
abs (r + (abs a)) >= 0 by COMPLEX1:46;
then A10: (abs (r + (abs a))) * 1 <= (abs (r + (abs a))) * (sqrt n) by A6, XREAL_1:64;
A11: r + (abs a) <= abs (r + (abs a)) by ABSVALUE:4;
|.(- ((r + (abs a)) * (1.REAL n))).| = |.((r + (abs a)) * (1.REAL n)).| by TOPRNS_1:26
.= (abs (r + (abs a))) * |.(1.REAL n).| by TOPRNS_1:7
.= (abs (r + (abs a))) * (sqrt n) by EUCLID:73 ;
then r + (abs a) <= |.(- ((r + (abs a)) * (1.REAL n))).| by A10, A11, XXREAL_0:2;
hence contradiction by A9, A8, XXREAL_0:2; ::_thesis: verum
end;
A12: now__::_thesis:_not_(r_+_(abs_a))_*_(1.REAL_n)_in__{__q_where_q_is_Point_of_(TOP-REAL_n)_:_|.q.|_<_a__}_
( a <= abs a & abs a < r + (abs a) ) by A2, ABSVALUE:4, XREAL_1:29;
then A13: a < r + (abs a) by XXREAL_0:2;
assume (r + (abs a)) * (1.REAL n) in { q where q is Point of (TOP-REAL n) : |.q.| < a } ; ::_thesis: contradiction
then A14: ex q being Point of (TOP-REAL n) st
( q = (r + (abs a)) * (1.REAL n) & |.q.| < a ) ;
abs (r + (abs a)) >= 0 by COMPLEX1:46;
then A15: (abs (r + (abs a))) * 1 <= (abs (r + (abs a))) * (sqrt n) by A6, XREAL_1:64;
A16: r + (abs a) <= abs (r + (abs a)) by ABSVALUE:4;
|.((r + (abs a)) * (1.REAL n)).| = (abs (r + (abs a))) * |.(1.REAL n).| by TOPRNS_1:7
.= (abs (r + (abs a))) * (sqrt n) by EUCLID:73 ;
then r + (abs a) <= |.((r + (abs a)) * (1.REAL n)).| by A15, A16, XXREAL_0:2;
hence contradiction by A14, A13, XXREAL_0:2; ::_thesis: verum
end;
reconsider z2 = - ((r + (abs a)) * (1.REAL n)) as Point of (Euclid n) by EUCLID:22;
reconsider z1 = (r + (abs a)) * (1.REAL n) as Point of (Euclid n) by EUCLID:22;
A17: (r + (abs a)) + (r + (abs a)) <= abs ((r + (abs a)) + (r + (abs a))) by ABSVALUE:4;
abs ((r + (abs a)) + (r + (abs a))) >= 0 by COMPLEX1:46;
then A18: (abs ((r + (abs a)) + (r + (abs a)))) * 1 <= (abs ((r + (abs a)) + (r + (abs a)))) * (sqrt n) by A6, XREAL_1:64;
dist (z1,z2) = |.(((r + (abs a)) * (1.REAL n)) - (- ((r + (abs a)) * (1.REAL n)))).| by JGRAPH_1:28
.= |.(((r + (abs a)) * 1R) + (- (- ((r + (abs a)) * 1R)))).|
.= |.(((r + (abs a)) * 1R) + ((r + (abs a)) * 1R)).|
.= |.(((r + (abs a)) + (r + (abs a))) * 1R).| by EUCLID:33
.= |.(((r + (abs a)) + (r + (abs a))) * (1.REAL n)).|
.= (abs ((r + (abs a)) + (r + (abs a)))) * |.(1.REAL n).| by TOPRNS_1:7
.= (abs ((r + (abs a)) + (r + (abs a)))) * (sqrt n) by EUCLID:73 ;
then (r + (abs a)) + (r + (abs a)) <= dist (z1,z2) by A18, A17, XXREAL_0:2;
then A19: r + (abs a) < dist (z1,z2) by A5, XXREAL_0:2;
r + 0 <= r + (abs a) by A4, XREAL_1:6;
then A20: r < dist (z1,z2) by A19, XXREAL_0:2;
- ((r + (abs a)) * (1.REAL n)) in the carrier of (TOP-REAL n) ;
then - ((r + (abs a)) * (1.REAL n)) in REAL n by EUCLID:22;
then A21: - ((r + (abs a)) * (1.REAL n)) in W by A1, A7, XBOOLE_0:def_5;
(r + (abs a)) * (1.REAL n) in the carrier of (TOP-REAL n) ;
then (r + (abs a)) * (1.REAL n) in REAL n by EUCLID:22;
then (r + (abs a)) * (1.REAL n) in W by A1, A12, XBOOLE_0:def_5;
hence contradiction by A3, A21, A20; ::_thesis: verum
end;
theorem Th63: :: JORDAN2C:63
for n being Element of NAT
for P, P1, Q being Subset of (TOP-REAL n)
for W being Subset of (Euclid n) st P = W & P is connected & not W is bounded & P1 = Component_of (Down (P,(Q `))) & W misses Q holds
P1 is_outside_component_of Q
proof
let n be Element of NAT ; ::_thesis: for P, P1, Q being Subset of (TOP-REAL n)
for W being Subset of (Euclid n) st P = W & P is connected & not W is bounded & P1 = Component_of (Down (P,(Q `))) & W misses Q holds
P1 is_outside_component_of Q
let P, P1 be Subset of (TOP-REAL n); ::_thesis: for Q being Subset of (TOP-REAL n)
for W being Subset of (Euclid n) st P = W & P is connected & not W is bounded & P1 = Component_of (Down (P,(Q `))) & W misses Q holds
P1 is_outside_component_of Q
let Q be Subset of (TOP-REAL n); ::_thesis: for W being Subset of (Euclid n) st P = W & P is connected & not W is bounded & P1 = Component_of (Down (P,(Q `))) & W misses Q holds
P1 is_outside_component_of Q
let W be Subset of (Euclid n); ::_thesis: ( P = W & P is connected & not W is bounded & P1 = Component_of (Down (P,(Q `))) & W misses Q implies P1 is_outside_component_of Q )
assume that
A1: P = W and
A2: P is connected and
A3: not W is bounded and
A4: P1 = Component_of (Down (P,(Q `))) and
A5: W misses Q ; ::_thesis: P1 is_outside_component_of Q
A6: (TOP-REAL n) | P is connected by A2, CONNSP_1:def_3;
A7: Down (P,(Q `)) = P \ Q by SUBSET_1:13
.= P by A1, A5, XBOOLE_1:83 ;
then reconsider P0 = P as Subset of ((TOP-REAL n) | (Q `)) ;
reconsider W0 = Component_of P0 as Subset of (Euclid n) by A4, A7, TOPREAL3:8;
P0 c= Q ` by A1, A5, SUBSET_1:23;
then ((TOP-REAL n) | (Q `)) | P0 = (TOP-REAL n) | P by PRE_TOPC:7;
then A8: P0 is connected by A6, CONNSP_1:def_3;
A9: now__::_thesis:_ex_D_being_Subset_of_(Euclid_n)_st_
(_D_=_P1_&_not_D_is_bounded_)
assume for D being Subset of (Euclid n) st D = P1 holds
D is bounded ; ::_thesis: contradiction
then W0 is bounded by A4, A7;
hence contradiction by A1, A3, A8, CONNSP_3:1, TBSP_1:14; ::_thesis: verum
end;
A10: W <> {} (Euclid n) by A3;
A11: W /\ Q = {} by A5, XBOOLE_0:def_7;
now__::_thesis:_not_Q_`_=_{}
assume Q ` = {} ; ::_thesis: contradiction
then Q = ({} the carrier of (TOP-REAL n)) ` ;
hence contradiction by A1, A10, A11, XBOOLE_1:28; ::_thesis: verum
end;
then reconsider Q1 = Q ` as non empty Subset of (TOP-REAL n) ;
not (TOP-REAL n) | Q1 is empty ;
then Component_of P0 is a_component by A1, A10, A8, CONNSP_3:9;
hence P1 is_outside_component_of Q by A4, A7, A9, Th14; ::_thesis: verum
end;
theorem Th64: :: JORDAN2C:64
for n being Element of NAT
for A being Subset of (Euclid n)
for B being non empty Subset of (Euclid n)
for C being Subset of ((Euclid n) | B) st A = C & C is bounded holds
A is bounded
proof
let n be Element of NAT ; ::_thesis: for A being Subset of (Euclid n)
for B being non empty Subset of (Euclid n)
for C being Subset of ((Euclid n) | B) st A = C & C is bounded holds
A is bounded
let A be Subset of (Euclid n); ::_thesis: for B being non empty Subset of (Euclid n)
for C being Subset of ((Euclid n) | B) st A = C & C is bounded holds
A is bounded
let B be non empty Subset of (Euclid n); ::_thesis: for C being Subset of ((Euclid n) | B) st A = C & C is bounded holds
A is bounded
let C be Subset of ((Euclid n) | B); ::_thesis: ( A = C & C is bounded implies A is bounded )
assume that
A1: A = C and
A2: C is bounded ; ::_thesis: A is bounded
consider r0 being Real such that
A3: 0 < r0 and
A4: for x, y being Point of ((Euclid n) | B) st x in C & y in C holds
dist (x,y) <= r0 by A2, TBSP_1:def_7;
for x, y being Point of (Euclid n) st x in A & y in A holds
dist (x,y) <= r0
proof
let x, y be Point of (Euclid n); ::_thesis: ( x in A & y in A implies dist (x,y) <= r0 )
assume A5: ( x in A & y in A ) ; ::_thesis: dist (x,y) <= r0
then reconsider x0 = x, y0 = y as Point of ((Euclid n) | B) by A1;
( the distance of ((Euclid n) | B) . (x0,y0) = the distance of (Euclid n) . (x,y) & the distance of ((Euclid n) | B) . (x0,y0) = dist (x0,y0) ) by TOPMETR:def_1;
hence dist (x,y) <= r0 by A1, A4, A5; ::_thesis: verum
end;
hence A is bounded by A3, TBSP_1:def_7; ::_thesis: verum
end;
theorem Th65: :: JORDAN2C:65
for n being Element of NAT
for A being Subset of (TOP-REAL n) st A is compact holds
A is bounded
proof
let n be Element of NAT ; ::_thesis: for A being Subset of (TOP-REAL n) st A is compact holds
A is bounded
let A be Subset of (TOP-REAL n); ::_thesis: ( A is compact implies A is bounded )
assume A1: A is compact ; ::_thesis: A is bounded
A c= the carrier of ((TOP-REAL n) | A) by PRE_TOPC:8;
then reconsider A2 = A as Subset of ((TOP-REAL n) | A) ;
percases ( A <> {} or A = {} ) ;
suppose A <> {} ; ::_thesis: A is bounded
then reconsider A1 = A as non empty Subset of (Euclid n) by TOPREAL3:8;
[#] ((TOP-REAL n) | A) = A2 by PRE_TOPC:def_5;
then [#] ((TOP-REAL n) | A) is compact by A1, COMPTS_1:2;
then A2: (TOP-REAL n) | A is compact by COMPTS_1:1;
TopSpaceMetr ((Euclid n) | A1) = (TOP-REAL n) | A by EUCLID:63;
then (Euclid n) | A1 is totally_bounded by A2, TBSP_1:9;
then A3: (Euclid n) | A1 is bounded by TBSP_1:19;
[#] ((Euclid n) | A1) = A1 by TOPMETR:def_2;
then A1 is bounded by A3, Th64;
hence A is bounded by Th11; ::_thesis: verum
end;
suppose A = {} ; ::_thesis: A is bounded
hence A is bounded ; ::_thesis: verum
end;
end;
end;
registration
let n be Element of NAT ;
cluster compact -> bounded for Element of bool the carrier of (TOP-REAL n);
coherence
for b1 being Subset of (TOP-REAL n) st b1 is compact holds
b1 is bounded by Th65;
end;
theorem Th66: :: JORDAN2C:66
for n being Element of NAT
for A being Subset of (TOP-REAL n) st 1 <= n & A is bounded holds
A ` <> {}
proof
let n be Element of NAT ; ::_thesis: for A being Subset of (TOP-REAL n) st 1 <= n & A is bounded holds
A ` <> {}
let A be Subset of (TOP-REAL n); ::_thesis: ( 1 <= n & A is bounded implies A ` <> {} )
assume that
A1: 1 <= n and
A2: A is bounded ; ::_thesis: A ` <> {}
consider r being Real such that
A3: for q being Point of (TOP-REAL n) st q in A holds
|.q.| < r by A2, Th34;
abs r >= 0 by COMPLEX1:46;
then A4: (abs r) * |.(1.REAL n).| >= (abs r) * 1 by A1, EUCLID:75, XREAL_1:64;
( |.(r * (1.REAL n)).| = (abs r) * |.(1.REAL n).| & r <= abs r ) by ABSVALUE:4, TOPRNS_1:7;
then not r * (1.REAL n) in A by A3, A4, XXREAL_0:2;
hence A ` <> {} by XBOOLE_0:def_5; ::_thesis: verum
end;
theorem Th67: :: JORDAN2C:67
for n being Element of NAT
for r being Real holds
( ex B being Subset of (Euclid n) st B = { q where q is Point of (TOP-REAL n) : |.q.| < r } & ( for A being Subset of (Euclid n) st A = { q1 where q1 is Point of (TOP-REAL n) : |.q1.| < r } holds
A is bounded ) )
proof
let n be Element of NAT ; ::_thesis: for r being Real holds
( ex B being Subset of (Euclid n) st B = { q where q is Point of (TOP-REAL n) : |.q.| < r } & ( for A being Subset of (Euclid n) st A = { q1 where q1 is Point of (TOP-REAL n) : |.q1.| < r } holds
A is bounded ) )
let r be Real; ::_thesis: ( ex B being Subset of (Euclid n) st B = { q where q is Point of (TOP-REAL n) : |.q.| < r } & ( for A being Subset of (Euclid n) st A = { q1 where q1 is Point of (TOP-REAL n) : |.q1.| < r } holds
A is bounded ) )
A1: { q where q is Point of (TOP-REAL n) : |.q.| < r } c= the carrier of (Euclid n)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { q where q is Point of (TOP-REAL n) : |.q.| < r } or x in the carrier of (Euclid n) )
assume x in { q where q is Point of (TOP-REAL n) : |.q.| < r } ; ::_thesis: x in the carrier of (Euclid n)
then ex q being Point of (TOP-REAL n) st
( q = x & |.q.| < r ) ;
then x in the carrier of (TOP-REAL n) ;
hence x in the carrier of (Euclid n) by TOPREAL3:8; ::_thesis: verum
end;
hence ex B being Subset of (Euclid n) st B = { q where q is Point of (TOP-REAL n) : |.q.| < r } ; ::_thesis: for A being Subset of (Euclid n) st A = { q1 where q1 is Point of (TOP-REAL n) : |.q1.| < r } holds
A is bounded
reconsider C = { q1 where q1 is Point of (TOP-REAL n) : |.q1.| < r } as Subset of (TOP-REAL n) by A1, TOPREAL3:8;
let A be Subset of (Euclid n); ::_thesis: ( A = { q1 where q1 is Point of (TOP-REAL n) : |.q1.| < r } implies A is bounded )
for q being Point of (TOP-REAL n) st q in C holds
|.q.| < r
proof
let q be Point of (TOP-REAL n); ::_thesis: ( q in C implies |.q.| < r )
assume q in C ; ::_thesis: |.q.| < r
then ex q1 being Point of (TOP-REAL n) st
( q1 = q & |.q1.| < r ) ;
hence |.q.| < r ; ::_thesis: verum
end;
then A2: C is bounded by Th34;
assume A = { q1 where q1 is Point of (TOP-REAL n) : |.q1.| < r } ; ::_thesis: A is bounded
hence A is bounded by A2, Th11; ::_thesis: verum
end;
theorem Th68: :: JORDAN2C:68
for n being Element of NAT
for A being Subset of (TOP-REAL n) st n >= 2 & A is bounded holds
UBD A is_outside_component_of A
proof
let n be Element of NAT ; ::_thesis: for A being Subset of (TOP-REAL n) st n >= 2 & A is bounded holds
UBD A is_outside_component_of A
let A be Subset of (TOP-REAL n); ::_thesis: ( n >= 2 & A is bounded implies UBD A is_outside_component_of A )
assume that
A1: n >= 2 and
A2: A is bounded ; ::_thesis: UBD A is_outside_component_of A
reconsider C = A as bounded Subset of (Euclid n) by A2, Th11;
percases ( C <> {} or C = {} ) ;
supposeA3: C <> {} ; ::_thesis: UBD A is_outside_component_of A
set x0 = the Element of C;
A4: the Element of C in C by A3;
then reconsider q1 = the Element of C as Point of (TOP-REAL n) ;
reconsider o = 0. (TOP-REAL n) as Point of (Euclid n) by EUCLID:67;
reconsider x0 = the Element of C as Point of (Euclid n) by A4;
consider r being Real such that
0 < r and
A5: for x, y being Point of (Euclid n) st x in C & y in C holds
dist (x,y) <= r by TBSP_1:def_7;
set R0 = (r + (dist (o,x0))) + 1;
reconsider W = (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < (r + (dist (o,x0))) + 1 } as Subset of (Euclid n) ;
A6: now__::_thesis:_not_W_meets_A
assume W meets A ; ::_thesis: contradiction
then consider z being set such that
A7: z in W and
A8: z in A by XBOOLE_0:3;
A9: not z in { q where q is Point of (TOP-REAL n) : |.q.| < (r + (dist (o,x0))) + 1 } by A7, XBOOLE_0:def_5;
reconsider z = z as Point of (Euclid n) by A7;
dist (x0,z) <= r by A5, A8;
then ( dist (o,z) <= (dist (o,x0)) + (dist (x0,z)) & (dist (o,x0)) + (dist (x0,z)) <= (dist (o,x0)) + r ) by METRIC_1:4, XREAL_1:6;
then A10: dist (o,z) <= (dist (o,x0)) + r by XXREAL_0:2;
reconsider q1 = z as Point of (TOP-REAL n) by TOPREAL3:8;
A11: |.(q1 - (0. (TOP-REAL n))).| = dist (o,z) by JGRAPH_1:28;
|.q1.| >= (r + (dist (o,x0))) + 1 by A9;
then dist (o,z) >= (r + (dist (o,x0))) + 1 by A11, RLVECT_1:13;
then r + ((dist (o,x0)) + 1) <= r + (dist (o,x0)) by A10, XXREAL_0:2;
then (dist (o,x0)) + 1 <= (dist (o,x0)) + 0 by XREAL_1:6;
hence contradiction by XREAL_1:6; ::_thesis: verum
end;
reconsider P = W as Subset of (TOP-REAL n) by TOPREAL3:8;
reconsider P = P as Subset of (TOP-REAL n) ;
the carrier of ((TOP-REAL n) | (A `)) = A ` by PRE_TOPC:8;
then reconsider P1 = Component_of (Down (P,(A `))) as Subset of (TOP-REAL n) by XBOOLE_1:1;
A12: P is connected by A1, Th53;
A13: UBD A c= P1
proof
A14: (TOP-REAL n) | P is connected by A12, CONNSP_1:def_3;
A15: P c= A ` by A6, SUBSET_1:23;
then Down (P,(A `)) = P by XBOOLE_1:28;
then ((TOP-REAL n) | (A `)) | (Down (P,(A `))) is connected by A15, A14, PRE_TOPC:7;
then A16: Down (P,(A `)) is connected by CONNSP_1:def_3;
reconsider G = A ` as non empty Subset of (TOP-REAL n) by A1, A2, Th66, XXREAL_0:2;
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in UBD A or z in P1 )
A17: (TOP-REAL n) | G is non empty TopSpace ;
assume z in UBD A ; ::_thesis: z in P1
then consider y being set such that
A18: z in y and
A19: y in { B where B is Subset of (TOP-REAL n) : B is_outside_component_of A } by TARSKI:def_4;
consider B being Subset of (TOP-REAL n) such that
A20: y = B and
A21: B is_outside_component_of A by A19;
consider C2 being Subset of ((TOP-REAL n) | (A `)) such that
A22: C2 = B and
A23: C2 is a_component and
A24: C2 is not bounded Subset of (Euclid n) by A21, Th14;
consider D2 being Subset of (Euclid n) such that
A25: D2 = { q where q is Point of (TOP-REAL n) : |.q.| < (r + (dist (o,x0))) + 1 } by Th67;
reconsider D2 = D2 as Subset of (Euclid n) ;
A26: A c= D2
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in A or z in D2 )
A27: |.q1.| = |.(q1 - (0. (TOP-REAL n))).| by RLVECT_1:13
.= dist (x0,o) by JGRAPH_1:28 ;
assume A28: z in A ; ::_thesis: z in D2
then reconsider q2 = z as Point of (TOP-REAL n) ;
reconsider x1 = q2 as Point of (Euclid n) by TOPREAL3:8;
( |.(q2 - q1).| = dist (x1,x0) & dist (x1,x0) <= r ) by A5, A28, JGRAPH_1:28;
then A29: |.(q2 - q1).| + |.q1.| <= r + (dist (o,x0)) by A27, XREAL_1:6;
A30: r + (dist (o,x0)) < (r + (dist (o,x0))) + 1 by XREAL_1:29;
( |.q2.| = |.((q2 - q1) + q1).| & |.((q2 - q1) + q1).| <= |.(q2 - q1).| + |.q1.| ) by EUCLID:48, TOPRNS_1:29;
then |.q2.| <= r + (dist (o,x0)) by A29, XXREAL_0:2;
then |.q2.| < (r + (dist (o,x0))) + 1 by A30, XXREAL_0:2;
hence z in D2 by A25; ::_thesis: verum
end;
the carrier of (Euclid n) = the carrier of (TOP-REAL n) by TOPREAL3:8;
then D2 ` c= the carrier of (TOP-REAL n) \ A by A26, XBOOLE_1:34;
then A31: P /\ (D2 `) c= P /\ (A `) by XBOOLE_1:26;
now__::_thesis:_not_W_/\_C2_=_{}
reconsider D = C2 as Subset of (Euclid n) by A22, TOPREAL3:8;
assume A32: W /\ C2 = {} ; ::_thesis: contradiction
A33: C2 c= { q where q is Point of (TOP-REAL n) : |.q.| < (r + (dist (o,x0))) + 1 }
proof
let x8 be set ; :: according to TARSKI:def_3 ::_thesis: ( not x8 in C2 or x8 in { q where q is Point of (TOP-REAL n) : |.q.| < (r + (dist (o,x0))) + 1 } )
assume A34: x8 in C2 ; ::_thesis: x8 in { q where q is Point of (TOP-REAL n) : |.q.| < (r + (dist (o,x0))) + 1 }
assume not x8 in { q where q is Point of (TOP-REAL n) : |.q.| < (r + (dist (o,x0))) + 1 } ; ::_thesis: contradiction
then x8 in W by A22, A24, A34, EUCLID:22, XBOOLE_0:def_5;
hence contradiction by A32, A34, XBOOLE_0:def_4; ::_thesis: verum
end;
not D is bounded by A24;
hence contradiction by A25, A33, Th67, TBSP_1:14; ::_thesis: verum
end;
then (Down (P,(A `))) /\ C2 <> {} by A25, A31, XBOOLE_1:3, XBOOLE_1:26;
then A35: Down (P,(A `)) meets C2 by XBOOLE_0:def_7;
C2 is connected by A23, CONNSP_1:def_5;
then C2 c= Component_of (Down (P,(A `))) by A16, A35, A17, CONNSP_3:16;
hence z in P1 by A18, A20, A22; ::_thesis: verum
end;
not W is bounded by A1, Th62;
then ( P1 is_outside_component_of A & P1 c= UBD A ) by A12, A6, Th23, Th63;
hence UBD A is_outside_component_of A by A13, XBOOLE_0:def_10; ::_thesis: verum
end;
supposeA36: C = {} ; ::_thesis: UBD A is_outside_component_of A
REAL n c= the carrier of (Euclid n) ;
then reconsider W = REAL n as Subset of (Euclid n) ;
W /\ A = {} by A36;
then A37: W misses A by XBOOLE_0:def_7;
reconsider P = W as Subset of (TOP-REAL n) by TOPREAL3:8;
reconsider P = P as Subset of (TOP-REAL n) ;
the carrier of ((TOP-REAL n) | (A `)) = A ` by PRE_TOPC:8;
then reconsider P1 = Component_of (Down (P,(A `))) as Subset of (TOP-REAL n) by XBOOLE_1:1;
[#] (TOP-REAL n) is a_component ;
then A38: [#] TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) is a_component by CONNSP_1:45;
not W is bounded by A1, Th33, XXREAL_0:2;
then A39: P1 is_outside_component_of A by A37, Th28, Th63;
A = {} (TOP-REAL n) by A36;
then A40: UBD A = REAL n by A1, Th36, XXREAL_0:2;
( [#] (TOP-REAL n) = REAL n & (TOP-REAL n) | ([#] (TOP-REAL n)) = TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) ) by EUCLID:22, TSEP_1:93;
hence UBD A is_outside_component_of A by A36, A39, A40, A38, CONNSP_3:7; ::_thesis: verum
end;
end;
end;
theorem Th69: :: JORDAN2C:69
for n being Element of NAT
for a being Real
for P being Subset of (TOP-REAL n) st P = { q where q is Point of (TOP-REAL n) : |.q.| < a } holds
P is convex
proof
let n be Element of NAT ; ::_thesis: for a being Real
for P being Subset of (TOP-REAL n) st P = { q where q is Point of (TOP-REAL n) : |.q.| < a } holds
P is convex
let a be Real; ::_thesis: for P being Subset of (TOP-REAL n) st P = { q where q is Point of (TOP-REAL n) : |.q.| < a } holds
P is convex
let P be Subset of (TOP-REAL n); ::_thesis: ( P = { q where q is Point of (TOP-REAL n) : |.q.| < a } implies P is convex )
assume A1: P = { q where q is Point of (TOP-REAL n) : |.q.| < a } ; ::_thesis: P is convex
for p1, p2 being Point of (TOP-REAL n) st p1 in P & p2 in P holds
LSeg (p1,p2) c= P
proof
let p1, p2 be Point of (TOP-REAL n); ::_thesis: ( p1 in P & p2 in P implies LSeg (p1,p2) c= P )
assume that
A2: p1 in P and
A3: p2 in P ; ::_thesis: LSeg (p1,p2) c= P
A4: ex q2 being Point of (TOP-REAL n) st
( q2 = p2 & |.q2.| < a ) by A1, A3;
A5: ex q1 being Point of (TOP-REAL n) st
( q1 = p1 & |.q1.| < a ) by A1, A2;
LSeg (p1,p2) c= P
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg (p1,p2) or x in P )
assume A6: x in LSeg (p1,p2) ; ::_thesis: x in P
then consider r being Real such that
A7: x = ((1 - r) * p1) + (r * p2) and
A8: 0 <= r and
A9: r <= 1 ;
A10: |.((1 - r) * p1).| = (abs (1 - r)) * |.p1.| by TOPRNS_1:7;
reconsider q = x as Point of (TOP-REAL n) by A6;
A11: |.(((1 - r) * p1) + (r * p2)).| <= |.((1 - r) * p1).| + |.(r * p2).| by TOPRNS_1:29;
A12: 1 - r >= 0 by A9, XREAL_1:48;
then A13: abs (1 - r) = 1 - r by ABSVALUE:def_1;
percases ( 1 - r > 0 or 1 - r <= 0 ) ;
supposeA14: 1 - r > 0 ; ::_thesis: x in P
A15: ( |.(r * p2).| = (abs r) * |.p2.| & r = abs r ) by A8, ABSVALUE:def_1, TOPRNS_1:7;
0 <= abs r by COMPLEX1:46;
then A16: (abs r) * |.p2.| <= (abs r) * a by A4, XREAL_1:64;
(abs (1 - r)) * |.p1.| < (abs (1 - r)) * a by A5, A13, A14, XREAL_1:68;
then |.((1 - r) * p1).| + |.(r * p2).| < ((1 - r) * a) + (r * a) by A10, A13, A16, A15, XREAL_1:8;
then |.q.| < a by A7, A11, XXREAL_0:2;
hence x in P by A1; ::_thesis: verum
end;
suppose 1 - r <= 0 ; ::_thesis: x in P
then (1 - r) + r = 0 + r by A12;
then 0 < abs r by ABSVALUE:def_1;
then A17: (abs r) * |.p2.| < (abs r) * a by A4, XREAL_1:68;
A18: r = abs r by A8, ABSVALUE:def_1;
( (abs (1 - r)) * |.p1.| <= (abs (1 - r)) * a & |.(r * p2).| = (abs r) * |.p2.| ) by A5, A12, A13, TOPRNS_1:7, XREAL_1:64;
then |.((1 - r) * p1).| + |.(r * p2).| < ((1 - r) * a) + (r * a) by A10, A13, A17, A18, XREAL_1:8;
then |.q.| < a by A7, A11, XXREAL_0:2;
hence x in P by A1; ::_thesis: verum
end;
end;
end;
hence LSeg (p1,p2) c= P ; ::_thesis: verum
end;
hence P is convex by JORDAN1:def_1; ::_thesis: verum
end;
theorem Th70: :: JORDAN2C:70
for n being Element of NAT
for u being Point of (Euclid n)
for a being Real
for P being Subset of (TOP-REAL n) st P = Ball (u,a) holds
P is convex
proof
let n be Element of NAT ; ::_thesis: for u being Point of (Euclid n)
for a being Real
for P being Subset of (TOP-REAL n) st P = Ball (u,a) holds
P is convex
let u be Point of (Euclid n); ::_thesis: for a being Real
for P being Subset of (TOP-REAL n) st P = Ball (u,a) holds
P is convex
let a be Real; ::_thesis: for P being Subset of (TOP-REAL n) st P = Ball (u,a) holds
P is convex
let P be Subset of (TOP-REAL n); ::_thesis: ( P = Ball (u,a) implies P is convex )
assume A1: P = Ball (u,a) ; ::_thesis: P is convex
for p1, p2 being Point of (TOP-REAL n) st p1 in P & p2 in P holds
LSeg (p1,p2) c= P
proof
reconsider p = u as Point of (TOP-REAL n) by TOPREAL3:8;
let p1, p2 be Point of (TOP-REAL n); ::_thesis: ( p1 in P & p2 in P implies LSeg (p1,p2) c= P )
assume that
A2: p1 in P and
A3: p2 in P ; ::_thesis: LSeg (p1,p2) c= P
A4: P = { q where q is Element of (Euclid n) : dist (u,q) < a } by A1, METRIC_1:17;
then ex q2 being Point of (Euclid n) st
( q2 = p2 & dist (u,q2) < a ) by A3;
then A5: |.(p - p2).| < a by JGRAPH_1:28;
A6: for p3 being Point of (TOP-REAL n) st |.(p - p3).| < a holds
p3 in P
proof
let p3 be Point of (TOP-REAL n); ::_thesis: ( |.(p - p3).| < a implies p3 in P )
reconsider u3 = p3 as Point of (Euclid n) by TOPREAL3:8;
assume |.(p - p3).| < a ; ::_thesis: p3 in P
then dist (u,u3) < a by JGRAPH_1:28;
hence p3 in P by A4; ::_thesis: verum
end;
ex q1 being Point of (Euclid n) st
( q1 = p1 & dist (u,q1) < a ) by A2, A4;
then A7: |.(p - p1).| < a by JGRAPH_1:28;
LSeg (p1,p2) c= P
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg (p1,p2) or x in P )
assume A8: x in LSeg (p1,p2) ; ::_thesis: x in P
then consider r being Real such that
A9: x = ((1 - r) * p1) + (r * p2) and
A10: 0 <= r and
A11: r <= 1 ;
reconsider q = x as Point of (TOP-REAL n) by A8;
A12: |.((1 - r) * (p - p1)).| = (abs (1 - r)) * |.(p - p1).| by TOPRNS_1:7;
((1 - r) * p) + (r * p) = ((1 - r) + r) * p by EUCLID:33
.= p by EUCLID:29 ;
then |.(p - (((1 - r) * p1) + (r * p2))).| = |.(((((1 - r) * p) + (r * p)) - ((1 - r) * p1)) - (r * p2)).| by EUCLID:46
.= |.(((((1 - r) * p) + (- ((1 - r) * p1))) + (r * p)) + (- (r * p2))).| by EUCLID:26
.= |.((((1 - r) * p) + (- ((1 - r) * p1))) + ((r * p) + (- (r * p2)))).| by EUCLID:26
.= |.((((1 - r) * p) + ((1 - r) * (- p1))) + ((r * p) + (- (r * p2)))).| by EUCLID:40
.= |.(((1 - r) * (p - p1)) + ((r * p) + (- (r * p2)))).| by EUCLID:32
.= |.(((1 - r) * (p - p1)) + ((r * p) + (r * (- p2)))).| by EUCLID:40
.= |.(((1 - r) * (p - p1)) + (r * (p - p2))).| by EUCLID:32 ;
then A13: |.(p - (((1 - r) * p1) + (r * p2))).| <= |.((1 - r) * (p - p1)).| + |.(r * (p - p2)).| by TOPRNS_1:29;
A14: 1 - r >= 0 by A11, XREAL_1:48;
then A15: abs (1 - r) = 1 - r by ABSVALUE:def_1;
percases ( 1 - r > 0 or 1 - r <= 0 ) ;
supposeA16: 1 - r > 0 ; ::_thesis: x in P
A17: ( |.(r * (p - p2)).| = (abs r) * |.(p - p2).| & r = abs r ) by A10, ABSVALUE:def_1, TOPRNS_1:7;
0 <= abs r by COMPLEX1:46;
then A18: (abs r) * |.(p - p2).| <= (abs r) * a by A5, XREAL_1:64;
(abs (1 - r)) * |.(p - p1).| < (abs (1 - r)) * a by A7, A15, A16, XREAL_1:68;
then |.((1 - r) * (p - p1)).| + |.(r * (p - p2)).| < ((1 - r) * a) + (r * a) by A12, A15, A18, A17, XREAL_1:8;
then |.(p - q).| < a by A9, A13, XXREAL_0:2;
hence x in P by A6; ::_thesis: verum
end;
suppose 1 - r <= 0 ; ::_thesis: x in P
then (1 - r) + r = 0 + r by A14;
then 0 < abs r by ABSVALUE:def_1;
then A19: (abs r) * |.(p - p2).| < (abs r) * a by A5, XREAL_1:68;
A20: r = abs r by A10, ABSVALUE:def_1;
( (abs (1 - r)) * |.(p - p1).| <= (abs (1 - r)) * a & |.(r * (p - p2)).| = (abs r) * |.(p - p2).| ) by A7, A14, A15, TOPRNS_1:7, XREAL_1:64;
then |.((1 - r) * (p - p1)).| + |.(r * (p - p2)).| < ((1 - r) * a) + (r * a) by A12, A15, A19, A20, XREAL_1:8;
then |.(p - q).| < a by A9, A13, XXREAL_0:2;
hence x in P by A6; ::_thesis: verum
end;
end;
end;
hence LSeg (p1,p2) c= P ; ::_thesis: verum
end;
hence P is convex by JORDAN1:def_1; ::_thesis: verum
end;
theorem :: JORDAN2C:71
canceled;
theorem Th72: :: JORDAN2C:72
for n being Element of NAT
for r being Real
for p, q being Point of (TOP-REAL n)
for u being Point of (Euclid n) st p <> q & p in Ball (u,r) & q in Ball (u,r) holds
ex h being Function of I[01],(TOP-REAL n) st
( h is continuous & h . 0 = p & h . 1 = q & rng h c= Ball (u,r) )
proof
let n be Element of NAT ; ::_thesis: for r being Real
for p, q being Point of (TOP-REAL n)
for u being Point of (Euclid n) st p <> q & p in Ball (u,r) & q in Ball (u,r) holds
ex h being Function of I[01],(TOP-REAL n) st
( h is continuous & h . 0 = p & h . 1 = q & rng h c= Ball (u,r) )
let r be Real; ::_thesis: for p, q being Point of (TOP-REAL n)
for u being Point of (Euclid n) st p <> q & p in Ball (u,r) & q in Ball (u,r) holds
ex h being Function of I[01],(TOP-REAL n) st
( h is continuous & h . 0 = p & h . 1 = q & rng h c= Ball (u,r) )
let p, q be Point of (TOP-REAL n); ::_thesis: for u being Point of (Euclid n) st p <> q & p in Ball (u,r) & q in Ball (u,r) holds
ex h being Function of I[01],(TOP-REAL n) st
( h is continuous & h . 0 = p & h . 1 = q & rng h c= Ball (u,r) )
let u be Point of (Euclid n); ::_thesis: ( p <> q & p in Ball (u,r) & q in Ball (u,r) implies ex h being Function of I[01],(TOP-REAL n) st
( h is continuous & h . 0 = p & h . 1 = q & rng h c= Ball (u,r) ) )
assume that
A1: p <> q and
A2: ( p in Ball (u,r) & q in Ball (u,r) ) ; ::_thesis: ex h being Function of I[01],(TOP-REAL n) st
( h is continuous & h . 0 = p & h . 1 = q & rng h c= Ball (u,r) )
reconsider Q = Ball (u,r) as Subset of (TOP-REAL n) by TOPREAL3:8;
Q is convex by Th70;
then A3: LSeg (p,q) c= Ball (u,r) by A2, JORDAN1:def_1;
reconsider P = LSeg (p,q) as Subset of (TOP-REAL n) ;
LSeg (p,q) is_an_arc_of p,q by A1, TOPREAL1:9;
then consider f being Function of I[01],((TOP-REAL n) | P) such that
A4: f is being_homeomorphism and
A5: ( f . 0 = p & f . 1 = q ) by TOPREAL1:def_1;
reconsider h = f as Function of I[01],(TOP-REAL n) by JORDAN6:3;
take h ; ::_thesis: ( h is continuous & h . 0 = p & h . 1 = q & rng h c= Ball (u,r) )
( rng f = [#] ((TOP-REAL n) | P) & f is continuous ) by A4, TOPS_2:def_5;
hence ( h is continuous & h . 0 = p & h . 1 = q & rng h c= Ball (u,r) ) by A3, A5, JORDAN6:3, PRE_TOPC:def_5; ::_thesis: verum
end;
theorem Th73: :: JORDAN2C:73
for n being Element of NAT
for r being Real
for p1, p2, p being Point of (TOP-REAL n)
for u being Point of (Euclid n)
for f being Function of I[01],(TOP-REAL n) st f is continuous & f . 0 = p1 & f . 1 = p2 & p in Ball (u,r) & p2 in Ball (u,r) holds
ex h being Function of I[01],(TOP-REAL n) st
( h is continuous & h . 0 = p1 & h . 1 = p & rng h c= (rng f) \/ (Ball (u,r)) )
proof
let n be Element of NAT ; ::_thesis: for r being Real
for p1, p2, p being Point of (TOP-REAL n)
for u being Point of (Euclid n)
for f being Function of I[01],(TOP-REAL n) st f is continuous & f . 0 = p1 & f . 1 = p2 & p in Ball (u,r) & p2 in Ball (u,r) holds
ex h being Function of I[01],(TOP-REAL n) st
( h is continuous & h . 0 = p1 & h . 1 = p & rng h c= (rng f) \/ (Ball (u,r)) )
let r be Real; ::_thesis: for p1, p2, p being Point of (TOP-REAL n)
for u being Point of (Euclid n)
for f being Function of I[01],(TOP-REAL n) st f is continuous & f . 0 = p1 & f . 1 = p2 & p in Ball (u,r) & p2 in Ball (u,r) holds
ex h being Function of I[01],(TOP-REAL n) st
( h is continuous & h . 0 = p1 & h . 1 = p & rng h c= (rng f) \/ (Ball (u,r)) )
let p1, p2, p be Point of (TOP-REAL n); ::_thesis: for u being Point of (Euclid n)
for f being Function of I[01],(TOP-REAL n) st f is continuous & f . 0 = p1 & f . 1 = p2 & p in Ball (u,r) & p2 in Ball (u,r) holds
ex h being Function of I[01],(TOP-REAL n) st
( h is continuous & h . 0 = p1 & h . 1 = p & rng h c= (rng f) \/ (Ball (u,r)) )
let u be Point of (Euclid n); ::_thesis: for f being Function of I[01],(TOP-REAL n) st f is continuous & f . 0 = p1 & f . 1 = p2 & p in Ball (u,r) & p2 in Ball (u,r) holds
ex h being Function of I[01],(TOP-REAL n) st
( h is continuous & h . 0 = p1 & h . 1 = p & rng h c= (rng f) \/ (Ball (u,r)) )
let f be Function of I[01],(TOP-REAL n); ::_thesis: ( f is continuous & f . 0 = p1 & f . 1 = p2 & p in Ball (u,r) & p2 in Ball (u,r) implies ex h being Function of I[01],(TOP-REAL n) st
( h is continuous & h . 0 = p1 & h . 1 = p & rng h c= (rng f) \/ (Ball (u,r)) ) )
assume that
A1: ( f is continuous & f . 0 = p1 & f . 1 = p2 ) and
A2: ( p in Ball (u,r) & p2 in Ball (u,r) ) ; ::_thesis: ex h being Function of I[01],(TOP-REAL n) st
( h is continuous & h . 0 = p1 & h . 1 = p & rng h c= (rng f) \/ (Ball (u,r)) )
percases ( p2 <> p or p2 = p ) ;
suppose p2 <> p ; ::_thesis: ex h being Function of I[01],(TOP-REAL n) st
( h is continuous & h . 0 = p1 & h . 1 = p & rng h c= (rng f) \/ (Ball (u,r)) )
then LSeg (p2,p) is_an_arc_of p2,p by TOPREAL1:9;
then consider f1 being Function of I[01],((TOP-REAL n) | (LSeg (p2,p))) such that
A3: f1 is being_homeomorphism and
A4: ( f1 . 0 = p2 & f1 . 1 = p ) by TOPREAL1:def_1;
reconsider f2 = f1 as Function of I[01],(TOP-REAL n) by JORDAN6:3;
rng f1 = [#] ((TOP-REAL n) | (LSeg (p2,p))) by A3, TOPS_2:def_5;
then rng f2 = LSeg (p2,p) by PRE_TOPC:def_5;
then A5: (rng f) \/ (rng f2) c= (rng f) \/ (Ball (u,r)) by A2, TOPREAL3:21, XBOOLE_1:9;
f1 is continuous by A3, TOPS_2:def_5;
then f2 is continuous by JORDAN6:3;
then ex h3 being Function of I[01],(TOP-REAL n) st
( h3 is continuous & p1 = h3 . 0 & p = h3 . 1 & rng h3 c= (rng f) \/ (rng f2) ) by A1, A4, BORSUK_2:13;
hence ex h being Function of I[01],(TOP-REAL n) st
( h is continuous & h . 0 = p1 & h . 1 = p & rng h c= (rng f) \/ (Ball (u,r)) ) by A5, XBOOLE_1:1; ::_thesis: verum
end;
suppose p2 = p ; ::_thesis: ex h being Function of I[01],(TOP-REAL n) st
( h is continuous & h . 0 = p1 & h . 1 = p & rng h c= (rng f) \/ (Ball (u,r)) )
hence ex h being Function of I[01],(TOP-REAL n) st
( h is continuous & h . 0 = p1 & h . 1 = p & rng h c= (rng f) \/ (Ball (u,r)) ) by A1, XBOOLE_1:7; ::_thesis: verum
end;
end;
end;
theorem Th74: :: JORDAN2C:74
for n being Element of NAT
for r being Real
for p1, p2, p being Point of (TOP-REAL n)
for u being Point of (Euclid n)
for P being Subset of (TOP-REAL n)
for f being Function of I[01],(TOP-REAL n) st f is continuous & rng f c= P & f . 0 = p1 & f . 1 = p2 & p in Ball (u,r) & p2 in Ball (u,r) & Ball (u,r) c= P holds
ex f1 being Function of I[01],(TOP-REAL n) st
( f1 is continuous & rng f1 c= P & f1 . 0 = p1 & f1 . 1 = p )
proof
let n be Element of NAT ; ::_thesis: for r being Real
for p1, p2, p being Point of (TOP-REAL n)
for u being Point of (Euclid n)
for P being Subset of (TOP-REAL n)
for f being Function of I[01],(TOP-REAL n) st f is continuous & rng f c= P & f . 0 = p1 & f . 1 = p2 & p in Ball (u,r) & p2 in Ball (u,r) & Ball (u,r) c= P holds
ex f1 being Function of I[01],(TOP-REAL n) st
( f1 is continuous & rng f1 c= P & f1 . 0 = p1 & f1 . 1 = p )
let r be Real; ::_thesis: for p1, p2, p being Point of (TOP-REAL n)
for u being Point of (Euclid n)
for P being Subset of (TOP-REAL n)
for f being Function of I[01],(TOP-REAL n) st f is continuous & rng f c= P & f . 0 = p1 & f . 1 = p2 & p in Ball (u,r) & p2 in Ball (u,r) & Ball (u,r) c= P holds
ex f1 being Function of I[01],(TOP-REAL n) st
( f1 is continuous & rng f1 c= P & f1 . 0 = p1 & f1 . 1 = p )
let p1, p2, p be Point of (TOP-REAL n); ::_thesis: for u being Point of (Euclid n)
for P being Subset of (TOP-REAL n)
for f being Function of I[01],(TOP-REAL n) st f is continuous & rng f c= P & f . 0 = p1 & f . 1 = p2 & p in Ball (u,r) & p2 in Ball (u,r) & Ball (u,r) c= P holds
ex f1 being Function of I[01],(TOP-REAL n) st
( f1 is continuous & rng f1 c= P & f1 . 0 = p1 & f1 . 1 = p )
let u be Point of (Euclid n); ::_thesis: for P being Subset of (TOP-REAL n)
for f being Function of I[01],(TOP-REAL n) st f is continuous & rng f c= P & f . 0 = p1 & f . 1 = p2 & p in Ball (u,r) & p2 in Ball (u,r) & Ball (u,r) c= P holds
ex f1 being Function of I[01],(TOP-REAL n) st
( f1 is continuous & rng f1 c= P & f1 . 0 = p1 & f1 . 1 = p )
let P be Subset of (TOP-REAL n); ::_thesis: for f being Function of I[01],(TOP-REAL n) st f is continuous & rng f c= P & f . 0 = p1 & f . 1 = p2 & p in Ball (u,r) & p2 in Ball (u,r) & Ball (u,r) c= P holds
ex f1 being Function of I[01],(TOP-REAL n) st
( f1 is continuous & rng f1 c= P & f1 . 0 = p1 & f1 . 1 = p )
let f be Function of I[01],(TOP-REAL n); ::_thesis: ( f is continuous & rng f c= P & f . 0 = p1 & f . 1 = p2 & p in Ball (u,r) & p2 in Ball (u,r) & Ball (u,r) c= P implies ex f1 being Function of I[01],(TOP-REAL n) st
( f1 is continuous & rng f1 c= P & f1 . 0 = p1 & f1 . 1 = p ) )
assume ( f is continuous & rng f c= P & f . 0 = p1 & f . 1 = p2 & p in Ball (u,r) & p2 in Ball (u,r) & Ball (u,r) c= P ) ; ::_thesis: ex f1 being Function of I[01],(TOP-REAL n) st
( f1 is continuous & rng f1 c= P & f1 . 0 = p1 & f1 . 1 = p )
then ( ex f3 being Function of I[01],(TOP-REAL n) st
( f3 is continuous & f3 . 0 = p1 & f3 . 1 = p & rng f3 c= (rng f) \/ (Ball (u,r)) ) & (rng f) \/ (Ball (u,r)) c= P ) by Th73, XBOOLE_1:8;
hence ex f1 being Function of I[01],(TOP-REAL n) st
( f1 is continuous & rng f1 c= P & f1 . 0 = p1 & f1 . 1 = p ) by XBOOLE_1:1; ::_thesis: verum
end;
theorem Th75: :: JORDAN2C:75
for n being Element of NAT
for R being Subset of (TOP-REAL n)
for p being Point of (TOP-REAL n)
for P being Subset of (TOP-REAL n) st R is connected & R is open & P = { q where q is Point of (TOP-REAL n) : ( q <> p & q in R & ( for f being Function of I[01],(TOP-REAL n) holds
( not f is continuous or not rng f c= R or not f . 0 = p or not f . 1 = q ) ) ) } holds
P is open
proof
let n be Element of NAT ; ::_thesis: for R being Subset of (TOP-REAL n)
for p being Point of (TOP-REAL n)
for P being Subset of (TOP-REAL n) st R is connected & R is open & P = { q where q is Point of (TOP-REAL n) : ( q <> p & q in R & ( for f being Function of I[01],(TOP-REAL n) holds
( not f is continuous or not rng f c= R or not f . 0 = p or not f . 1 = q ) ) ) } holds
P is open
let R be Subset of (TOP-REAL n); ::_thesis: for p being Point of (TOP-REAL n)
for P being Subset of (TOP-REAL n) st R is connected & R is open & P = { q where q is Point of (TOP-REAL n) : ( q <> p & q in R & ( for f being Function of I[01],(TOP-REAL n) holds
( not f is continuous or not rng f c= R or not f . 0 = p or not f . 1 = q ) ) ) } holds
P is open
let p be Point of (TOP-REAL n); ::_thesis: for P being Subset of (TOP-REAL n) st R is connected & R is open & P = { q where q is Point of (TOP-REAL n) : ( q <> p & q in R & ( for f being Function of I[01],(TOP-REAL n) holds
( not f is continuous or not rng f c= R or not f . 0 = p or not f . 1 = q ) ) ) } holds
P is open
let P be Subset of (TOP-REAL n); ::_thesis: ( R is connected & R is open & P = { q where q is Point of (TOP-REAL n) : ( q <> p & q in R & ( for f being Function of I[01],(TOP-REAL n) holds
( not f is continuous or not rng f c= R or not f . 0 = p or not f . 1 = q ) ) ) } implies P is open )
assume that
A1: ( R is connected & R is open ) and
A2: P = { q where q is Point of (TOP-REAL n) : ( q <> p & q in R & ( for f being Function of I[01],(TOP-REAL n) holds
( not f is continuous or not rng f c= R or not f . 0 = p or not f . 1 = q ) ) ) } ; ::_thesis: P is open
A3: TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def_8;
then reconsider P9 = P as Subset of (TopSpaceMetr (Euclid n)) ;
A4: P c= R
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in P or x in R )
assume x in P ; ::_thesis: x in R
then ex q being Point of (TOP-REAL n) st
( q = x & q <> p & q in R & ( for f being Function of I[01],(TOP-REAL n) holds
( not f is continuous or not rng f c= R or not f . 0 = p or not f . 1 = q ) ) ) by A2;
hence x in R ; ::_thesis: verum
end;
now__::_thesis:_for_u_being_Point_of_(Euclid_n)_st_u_in_P_holds_
ex_r_being_real_number_st_
(_r_>_0_&_Ball_(u,r)_c=_P9_)
A5: TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def_8;
then reconsider R9 = R as Subset of (TopSpaceMetr (Euclid n)) ;
let u be Point of (Euclid n); ::_thesis: ( u in P implies ex r being real number st
( r > 0 & Ball (u,r) c= P9 ) )
reconsider p2 = u as Point of (TOP-REAL n) by TOPREAL3:8;
assume A6: u in P ; ::_thesis: ex r being real number st
( r > 0 & Ball (u,r) c= P9 )
R9 is open by A1, A5, PRE_TOPC:30;
then consider r being real number such that
A7: r > 0 and
A8: Ball (u,r) c= R9 by A4, A6, TOPMETR:15;
take r = r; ::_thesis: ( r > 0 & Ball (u,r) c= P9 )
thus r > 0 by A7; ::_thesis: Ball (u,r) c= P9
reconsider r9 = r as Real by XREAL_0:def_1;
A9: p2 in Ball (u,r9) by A7, TBSP_1:11;
Ball (u,r) c= P9
proof
assume not Ball (u,r) c= P9 ; ::_thesis: contradiction
then consider x being set such that
A10: x in Ball (u,r) and
A11: not x in P by TARSKI:def_3;
x in R by A8, A10;
then reconsider q = x as Point of (TOP-REAL n) ;
percases ( q = p or ex f1 being Function of I[01],(TOP-REAL n) st
( f1 is continuous & rng f1 c= R & f1 . 0 = p & f1 . 1 = q ) ) by A2, A8, A10, A11;
supposeA12: q = p ; ::_thesis: contradiction
A13: now__::_thesis:_not_q_=_p2
assume A14: q = p2 ; ::_thesis: contradiction
ex p3 being Point of (TOP-REAL n) st
( p3 = p2 & p3 <> p & p3 in R & ( for f1 being Function of I[01],(TOP-REAL n) holds
( not f1 is continuous or not rng f1 c= R or not f1 . 0 = p or not f1 . 1 = p3 ) ) ) by A2, A6;
hence contradiction by A12, A14; ::_thesis: verum
end;
u in Ball (u,r9) by A7, TBSP_1:11;
then A15: ex f2 being Function of I[01],(TOP-REAL n) st
( f2 is continuous & f2 . 0 = q & f2 . 1 = p2 & rng f2 c= Ball (u,r9) ) by A10, A13, Th72;
not p2 in P
proof
assume p2 in P ; ::_thesis: contradiction
then ex q4 being Point of (TOP-REAL n) st
( q4 = p2 & q4 <> p & q4 in R & ( for f1 being Function of I[01],(TOP-REAL n) holds
( not f1 is continuous or not rng f1 c= R or not f1 . 0 = p or not f1 . 1 = q4 ) ) ) by A2;
hence contradiction by A8, A12, A15, XBOOLE_1:1; ::_thesis: verum
end;
hence contradiction by A6; ::_thesis: verum
end;
supposeA16: ex f1 being Function of I[01],(TOP-REAL n) st
( f1 is continuous & rng f1 c= R & f1 . 0 = p & f1 . 1 = q ) ; ::_thesis: contradiction
not p2 in P
proof
assume p2 in P ; ::_thesis: contradiction
then ex q4 being Point of (TOP-REAL n) st
( q4 = p2 & q4 <> p & q4 in R & ( for f1 being Function of I[01],(TOP-REAL n) holds
( not f1 is continuous or not rng f1 c= R or not f1 . 0 = p or not f1 . 1 = q4 ) ) ) by A2;
hence contradiction by A8, A9, A10, A16, Th74; ::_thesis: verum
end;
hence contradiction by A6; ::_thesis: verum
end;
end;
end;
hence Ball (u,r) c= P9 ; ::_thesis: verum
end;
then P9 is open by TOPMETR:15;
hence P is open by A3, PRE_TOPC:30; ::_thesis: verum
end;
theorem Th76: :: JORDAN2C:76
for n being Element of NAT
for p being Point of (TOP-REAL n)
for R, P being Subset of (TOP-REAL n) st R is connected & R is open & p in R & P = { q where q is Point of (TOP-REAL n) : ( q = p or ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & rng f c= R & f . 0 = p & f . 1 = q ) ) } holds
P is open
proof
let n be Element of NAT ; ::_thesis: for p being Point of (TOP-REAL n)
for R, P being Subset of (TOP-REAL n) st R is connected & R is open & p in R & P = { q where q is Point of (TOP-REAL n) : ( q = p or ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & rng f c= R & f . 0 = p & f . 1 = q ) ) } holds
P is open
let p be Point of (TOP-REAL n); ::_thesis: for R, P being Subset of (TOP-REAL n) st R is connected & R is open & p in R & P = { q where q is Point of (TOP-REAL n) : ( q = p or ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & rng f c= R & f . 0 = p & f . 1 = q ) ) } holds
P is open
let R, P be Subset of (TOP-REAL n); ::_thesis: ( R is connected & R is open & p in R & P = { q where q is Point of (TOP-REAL n) : ( q = p or ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & rng f c= R & f . 0 = p & f . 1 = q ) ) } implies P is open )
assume that
A1: ( R is connected & R is open ) and
A2: p in R and
A3: P = { q where q is Point of (TOP-REAL n) : ( q = p or ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & rng f c= R & f . 0 = p & f . 1 = q ) ) } ; ::_thesis: P is open
A4: TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def_8;
then reconsider P9 = P as Subset of (TopSpaceMetr (Euclid n)) ;
reconsider RR = R as Subset of (TopSpaceMetr (Euclid n)) by A4;
now__::_thesis:_for_u_being_Point_of_(Euclid_n)_st_u_in_P9_holds_
ex_r_being_real_number_st_
(_r_>_0_&_Ball_(u,r)_c=_P_)
let u be Point of (Euclid n); ::_thesis: ( u in P9 implies ex r being real number st
( r > 0 & Ball (u,r) c= P ) )
reconsider p2 = u as Point of (TOP-REAL n) by TOPREAL3:8;
assume u in P9 ; ::_thesis: ex r being real number st
( r > 0 & Ball (u,r) c= P )
then consider q1 being Point of (TOP-REAL n) such that
A5: q1 = u and
A6: ( q1 = p or ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & rng f c= R & f . 0 = p & f . 1 = q1 ) ) by A3;
A7: now__::_thesis:_p2_in_R
percases ( q1 = p or ( q1 <> p & ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & rng f c= R & f . 0 = p & f . 1 = q1 ) ) ) by A6;
suppose q1 = p ; ::_thesis: p2 in R
hence p2 in R by A2, A5; ::_thesis: verum
end;
suppose ( q1 <> p & ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & rng f c= R & f . 0 = p & f . 1 = q1 ) ) ; ::_thesis: p2 in R
then consider f2 being Function of I[01],(TOP-REAL n) such that
f2 is continuous and
A8: rng f2 c= R and
f2 . 0 = p and
A9: f2 . 1 = q1 ;
dom f2 = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1;
then 1 in dom f2 by XXREAL_1:1;
then u in rng f2 by A5, A9, FUNCT_1:def_3;
hence p2 in R by A8; ::_thesis: verum
end;
end;
end;
RR is open by A1, A4, PRE_TOPC:30;
then consider r being real number such that
A10: r > 0 and
A11: Ball (u,r) c= R by A7, TOPMETR:15;
take r = r; ::_thesis: ( r > 0 & Ball (u,r) c= P )
thus r > 0 by A10; ::_thesis: Ball (u,r) c= P
reconsider r9 = r as Real by XREAL_0:def_1;
A12: p2 in Ball (u,r9) by A10, TBSP_1:11;
thus Ball (u,r) c= P ::_thesis: verum
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Ball (u,r) or x in P )
assume A13: x in Ball (u,r) ; ::_thesis: x in P
then reconsider q = x as Point of (TOP-REAL n) by A11, TARSKI:def_3;
percases ( q = p or q <> p ) ;
suppose q = p ; ::_thesis: x in P
hence x in P by A3; ::_thesis: verum
end;
supposeA14: q <> p ; ::_thesis: x in P
A15: now__::_thesis:_(_q1_=_p_implies_x_in_P_)
assume q1 = p ; ::_thesis: x in P
then p in Ball (u,r9) by A5, A10, TBSP_1:11;
then consider f2 being Function of I[01],(TOP-REAL n) such that
A16: ( f2 is continuous & f2 . 0 = p & f2 . 1 = q ) and
A17: rng f2 c= Ball (u,r9) by A13, A14, Th72;
rng f2 c= R by A11, A17, XBOOLE_1:1;
hence x in P by A3, A16; ::_thesis: verum
end;
now__::_thesis:_(_q1_<>_p_implies_x_in_P_)
assume q1 <> p ; ::_thesis: x in P
then ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & rng f c= R & f . 0 = p & f . 1 = q ) by A5, A6, A11, A12, A13, Th74;
hence x in P by A3; ::_thesis: verum
end;
hence x in P by A15; ::_thesis: verum
end;
end;
end;
end;
then P9 is open by TOPMETR:15;
hence P is open by A4, PRE_TOPC:30; ::_thesis: verum
end;
theorem Th77: :: JORDAN2C:77
for n being Element of NAT
for p being Point of (TOP-REAL n)
for P, R being Subset of (TOP-REAL n) st p in R & P = { q where q is Point of (TOP-REAL n) : ( q = p or ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & rng f c= R & f . 0 = p & f . 1 = q ) ) } holds
P c= R
proof
let n be Element of NAT ; ::_thesis: for p being Point of (TOP-REAL n)
for P, R being Subset of (TOP-REAL n) st p in R & P = { q where q is Point of (TOP-REAL n) : ( q = p or ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & rng f c= R & f . 0 = p & f . 1 = q ) ) } holds
P c= R
let p be Point of (TOP-REAL n); ::_thesis: for P, R being Subset of (TOP-REAL n) st p in R & P = { q where q is Point of (TOP-REAL n) : ( q = p or ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & rng f c= R & f . 0 = p & f . 1 = q ) ) } holds
P c= R
let P, R be Subset of (TOP-REAL n); ::_thesis: ( p in R & P = { q where q is Point of (TOP-REAL n) : ( q = p or ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & rng f c= R & f . 0 = p & f . 1 = q ) ) } implies P c= R )
assume that
A1: p in R and
A2: P = { q where q is Point of (TOP-REAL n) : ( q = p or ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & rng f c= R & f . 0 = p & f . 1 = q ) ) } ; ::_thesis: P c= R
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in P or x in R )
assume x in P ; ::_thesis: x in R
then consider q being Point of (TOP-REAL n) such that
A3: q = x and
A4: ( q = p or ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & rng f c= R & f . 0 = p & f . 1 = q ) ) by A2;
percases ( q = p or ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & rng f c= R & f . 0 = p & f . 1 = q ) ) by A4;
suppose q = p ; ::_thesis: x in R
hence x in R by A1, A3; ::_thesis: verum
end;
suppose ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & rng f c= R & f . 0 = p & f . 1 = q ) ; ::_thesis: x in R
then consider f1 being Function of I[01],(TOP-REAL n) such that
f1 is continuous and
A5: rng f1 c= R and
f1 . 0 = p and
A6: f1 . 1 = q ;
reconsider P1 = rng f1 as Subset of (TOP-REAL n) ;
dom f1 = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1;
then 1 in dom f1 by XXREAL_1:1;
then q in P1 by A6, FUNCT_1:def_3;
hence x in R by A3, A5; ::_thesis: verum
end;
end;
end;
theorem Th78: :: JORDAN2C:78
for n being Element of NAT
for P, R being Subset of (TOP-REAL n)
for p being Point of (TOP-REAL n) st R is connected & R is open & p in R & P = { q where q is Point of (TOP-REAL n) : ( q = p or ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & rng f c= R & f . 0 = p & f . 1 = q ) ) } holds
R c= P
proof
let n be Element of NAT ; ::_thesis: for P, R being Subset of (TOP-REAL n)
for p being Point of (TOP-REAL n) st R is connected & R is open & p in R & P = { q where q is Point of (TOP-REAL n) : ( q = p or ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & rng f c= R & f . 0 = p & f . 1 = q ) ) } holds
R c= P
let P, R be Subset of (TOP-REAL n); ::_thesis: for p being Point of (TOP-REAL n) st R is connected & R is open & p in R & P = { q where q is Point of (TOP-REAL n) : ( q = p or ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & rng f c= R & f . 0 = p & f . 1 = q ) ) } holds
R c= P
let p be Point of (TOP-REAL n); ::_thesis: ( R is connected & R is open & p in R & P = { q where q is Point of (TOP-REAL n) : ( q = p or ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & rng f c= R & f . 0 = p & f . 1 = q ) ) } implies R c= P )
assume that
A1: ( R is connected & R is open ) and
A2: p in R and
A3: P = { q where q is Point of (TOP-REAL n) : ( q = p or ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & rng f c= R & f . 0 = p & f . 1 = q ) ) } ; ::_thesis: R c= P
reconsider R9 = R as non empty Subset of (TOP-REAL n) by A2;
A4: p in P by A3;
set P2 = R \ P;
reconsider P22 = R \ P as Subset of (TOP-REAL n) ;
A5: [#] ((TOP-REAL n) | R) = R by PRE_TOPC:def_5;
then reconsider P11 = P, P12 = P22 as Subset of ((TOP-REAL n) | R) by A2, A3, Th77, XBOOLE_1:36;
reconsider P11 = P11, P12 = P12 as Subset of ((TOP-REAL n) | R) ;
P \/ (R \ P) = P \/ R by XBOOLE_1:39;
then A6: ( P11 misses P12 & [#] ((TOP-REAL n) | R) = P11 \/ P12 ) by A5, XBOOLE_1:12, XBOOLE_1:79;
now__::_thesis:_for_x_being_set_holds_
(_x_in_R_\_P_iff_x_in__{__q_where_q_is_Point_of_(TOP-REAL_n)_:_(_q_<>_p_&_q_in_R_&_(_for_f_being_Function_of_I[01],(TOP-REAL_n)_holds_
(_not_f_is_continuous_or_not_rng_f_c=_R_or_not_f_._0_=_p_or_not_f_._1_=_q_)_)_)__}__)
let x be set ; ::_thesis: ( x in R \ P iff x in { q where q is Point of (TOP-REAL n) : ( q <> p & q in R & ( for f being Function of I[01],(TOP-REAL n) holds
( not f is continuous or not rng f c= R or not f . 0 = p or not f . 1 = q ) ) ) } )
A7: now__::_thesis:_(_x_in_R_\_P_implies_x_in__{__q_where_q_is_Point_of_(TOP-REAL_n)_:_(_q_<>_p_&_q_in_R_&_(_for_f_being_Function_of_I[01],(TOP-REAL_n)_holds_
(_not_f_is_continuous_or_not_rng_f_c=_R_or_not_f_._0_=_p_or_not_f_._1_=_q_)_)_)__}__)
assume A8: x in R \ P ; ::_thesis: x in { q where q is Point of (TOP-REAL n) : ( q <> p & q in R & ( for f being Function of I[01],(TOP-REAL n) holds
( not f is continuous or not rng f c= R or not f . 0 = p or not f . 1 = q ) ) ) }
then reconsider q2 = x as Point of (TOP-REAL n) ;
not x in P by A8, XBOOLE_0:def_5;
then A9: ( q2 <> p & ( for f being Function of I[01],(TOP-REAL n) holds
( not f is continuous or not rng f c= R or not f . 0 = p or not f . 1 = q2 ) ) ) by A3;
q2 in R by A8, XBOOLE_0:def_5;
hence x in { q where q is Point of (TOP-REAL n) : ( q <> p & q in R & ( for f being Function of I[01],(TOP-REAL n) holds
( not f is continuous or not rng f c= R or not f . 0 = p or not f . 1 = q ) ) ) } by A9; ::_thesis: verum
end;
now__::_thesis:_(_x_in__{__q_where_q_is_Point_of_(TOP-REAL_n)_:_(_q_<>_p_&_q_in_R_&_(_for_f_being_Function_of_I[01],(TOP-REAL_n)_holds_
(_not_f_is_continuous_or_not_rng_f_c=_R_or_not_f_._0_=_p_or_not_f_._1_=_q_)_)_)__}__implies_x_in_R_\_P_)
assume x in { q where q is Point of (TOP-REAL n) : ( q <> p & q in R & ( for f being Function of I[01],(TOP-REAL n) holds
( not f is continuous or not rng f c= R or not f . 0 = p or not f . 1 = q ) ) ) } ; ::_thesis: x in R \ P
then A10: ex q3 being Point of (TOP-REAL n) st
( q3 = x & q3 <> p & q3 in R & ( for f being Function of I[01],(TOP-REAL n) holds
( not f is continuous or not rng f c= R or not f . 0 = p or not f . 1 = q3 ) ) ) ;
then for q being Point of (TOP-REAL n) holds
( not q = x or ( not q = p & ( for f being Function of I[01],(TOP-REAL n) holds
( not f is continuous or not rng f c= R or not f . 0 = p or not f . 1 = q ) ) ) ) ;
then not x in P by A3;
hence x in R \ P by A10, XBOOLE_0:def_5; ::_thesis: verum
end;
hence ( x in R \ P iff x in { q where q is Point of (TOP-REAL n) : ( q <> p & q in R & ( for f being Function of I[01],(TOP-REAL n) holds
( not f is continuous or not rng f c= R or not f . 0 = p or not f . 1 = q ) ) ) } ) by A7; ::_thesis: verum
end;
then R \ P = { q where q is Point of (TOP-REAL n) : ( q <> p & q in R & ( for f being Function of I[01],(TOP-REAL n) holds
( not f is continuous or not rng f c= R or not f . 0 = p or not f . 1 = q ) ) ) } by TARSKI:1;
then P22 is open by A1, Th75;
then A11: P22 in the topology of (TOP-REAL n) by PRE_TOPC:def_2;
reconsider PPP = P as Subset of (TOP-REAL n) ;
PPP is open by A1, A2, A3, Th76;
then A12: P in the topology of (TOP-REAL n) by PRE_TOPC:def_2;
P11 = P /\ ([#] ((TOP-REAL n) | R)) by XBOOLE_1:28;
then P11 in the topology of ((TOP-REAL n) | R) by A12, PRE_TOPC:def_4;
then A13: P11 is open by PRE_TOPC:def_2;
P12 = P22 /\ ([#] ((TOP-REAL n) | R)) by XBOOLE_1:28;
then P12 in the topology of ((TOP-REAL n) | R) by A11, PRE_TOPC:def_4;
then A14: P12 is open by PRE_TOPC:def_2;
(TOP-REAL n) | R9 is connected by A1, CONNSP_1:def_3;
then ( P11 = {} ((TOP-REAL n) | R) or P12 = {} ((TOP-REAL n) | R) ) by A6, A13, A14, CONNSP_1:11;
hence R c= P by A4, XBOOLE_1:37; ::_thesis: verum
end;
theorem Th79: :: JORDAN2C:79
for n being Element of NAT
for R being Subset of (TOP-REAL n)
for p, q being Point of (TOP-REAL n) st R is connected & R is open & p in R & q in R & p <> q holds
ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & rng f c= R & f . 0 = p & f . 1 = q )
proof
let n be Element of NAT ; ::_thesis: for R being Subset of (TOP-REAL n)
for p, q being Point of (TOP-REAL n) st R is connected & R is open & p in R & q in R & p <> q holds
ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & rng f c= R & f . 0 = p & f . 1 = q )
let R be Subset of (TOP-REAL n); ::_thesis: for p, q being Point of (TOP-REAL n) st R is connected & R is open & p in R & q in R & p <> q holds
ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & rng f c= R & f . 0 = p & f . 1 = q )
let p, q be Point of (TOP-REAL n); ::_thesis: ( R is connected & R is open & p in R & q in R & p <> q implies ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & rng f c= R & f . 0 = p & f . 1 = q ) )
assume that
A1: ( R is connected & R is open & p in R ) and
A2: q in R and
A3: p <> q ; ::_thesis: ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & rng f c= R & f . 0 = p & f . 1 = q )
set RR = { q2 where q2 is Point of (TOP-REAL n) : ( q2 = p or ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & rng f c= R & f . 0 = p & f . 1 = q2 ) ) } ;
{ q2 where q2 is Point of (TOP-REAL n) : ( q2 = p or ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & rng f c= R & f . 0 = p & f . 1 = q2 ) ) } c= the carrier of (TOP-REAL n)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { q2 where q2 is Point of (TOP-REAL n) : ( q2 = p or ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & rng f c= R & f . 0 = p & f . 1 = q2 ) ) } or x in the carrier of (TOP-REAL n) )
assume x in { q2 where q2 is Point of (TOP-REAL n) : ( q2 = p or ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & rng f c= R & f . 0 = p & f . 1 = q2 ) ) } ; ::_thesis: x in the carrier of (TOP-REAL n)
then ex q2 being Point of (TOP-REAL n) st
( q2 = x & ( q2 = p or ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & rng f c= R & f . 0 = p & f . 1 = q2 ) ) ) ;
hence x in the carrier of (TOP-REAL n) ; ::_thesis: verum
end;
then R c= { q2 where q2 is Point of (TOP-REAL n) : ( q2 = p or ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & rng f c= R & f . 0 = p & f . 1 = q2 ) ) } by A1, Th78;
then q in { q2 where q2 is Point of (TOP-REAL n) : ( q2 = p or ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & rng f c= R & f . 0 = p & f . 1 = q2 ) ) } by A2;
then ex q2 being Point of (TOP-REAL n) st
( q = q2 & ( q2 = p or ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & rng f c= R & f . 0 = p & f . 1 = q2 ) ) ) ;
hence ex f being Function of I[01],(TOP-REAL n) st
( f is continuous & rng f c= R & f . 0 = p & f . 1 = q ) by A3; ::_thesis: verum
end;
theorem Th80: :: JORDAN2C:80
for n being Element of NAT
for A being Subset of (TOP-REAL n)
for a being real number st A = { q where q is Point of (TOP-REAL n) : |.q.| = a } holds
( A ` is open & A is closed )
proof
let n be Element of NAT ; ::_thesis: for A being Subset of (TOP-REAL n)
for a being real number st A = { q where q is Point of (TOP-REAL n) : |.q.| = a } holds
( A ` is open & A is closed )
let A be Subset of (TOP-REAL n); ::_thesis: for a being real number st A = { q where q is Point of (TOP-REAL n) : |.q.| = a } holds
( A ` is open & A is closed )
let a be real number ; ::_thesis: ( A = { q where q is Point of (TOP-REAL n) : |.q.| = a } implies ( A ` is open & A is closed ) )
assume A1: A = { q where q is Point of (TOP-REAL n) : |.q.| = a } ; ::_thesis: ( A ` is open & A is closed )
reconsider a = a as Real by XREAL_0:def_1;
A2: TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def_8;
then reconsider P1 = A ` as Subset of (TopSpaceMetr (Euclid n)) ;
for p being Point of (Euclid n) st p in P1 holds
ex r being real number st
( r > 0 & Ball (p,r) c= P1 )
proof
let p be Point of (Euclid n); ::_thesis: ( p in P1 implies ex r being real number st
( r > 0 & Ball (p,r) c= P1 ) )
reconsider q1 = p as Point of (TOP-REAL n) by TOPREAL3:8;
assume p in P1 ; ::_thesis: ex r being real number st
( r > 0 & Ball (p,r) c= P1 )
then not p in A by XBOOLE_0:def_5;
then A3: |.q1.| <> a by A1;
now__::_thesis:_(_(_|.q1.|_<=_a_&_ex_r_being_real_number_st_
(_r_>_0_&_Ball_(p,r)_c=_P1_)_)_or_(_|.q1.|_>_a_&_ex_r_being_real_number_st_
(_r_>_0_&_Ball_(p,r)_c=_P1_)_)_)
percases ( |.q1.| <= a or |.q1.| > a ) ;
caseA4: |.q1.| <= a ; ::_thesis: ex r being real number st
( r > 0 & Ball (p,r) c= P1 )
set r1 = (a - |.q1.|) / 2;
|.q1.| < a by A3, A4, XXREAL_0:1;
then A5: a - |.q1.| > 0 by XREAL_1:50;
Ball (p,((a - |.q1.|) / 2)) c= P1
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Ball (p,((a - |.q1.|) / 2)) or x in P1 )
assume A6: x in Ball (p,((a - |.q1.|) / 2)) ; ::_thesis: x in P1
then reconsider p2 = x as Point of (Euclid n) ;
reconsider q2 = p2 as Point of (TOP-REAL n) by TOPREAL3:8;
dist (p,p2) < (a - |.q1.|) / 2 by A6, METRIC_1:11;
then A7: |.(q2 - q1).| < (a - |.q1.|) / 2 by JGRAPH_1:28;
now__::_thesis:_not_q2_in_A
assume q2 in A ; ::_thesis: contradiction
then A8: ex q being Point of (TOP-REAL n) st
( q = q2 & |.q.| = a ) by A1;
|.(q2 - q1).| >= |.q2.| - |.q1.| by TOPRNS_1:32;
then (a - |.q1.|) / 2 > ((a - |.q1.|) / 2) + ((a - |.q1.|) / 2) by A7, A8, XXREAL_0:2;
then ((a - |.q1.|) / 2) - ((a - |.q1.|) / 2) > (a - |.q1.|) / 2 by XREAL_1:20;
hence contradiction by A5; ::_thesis: verum
end;
hence x in P1 by XBOOLE_0:def_5; ::_thesis: verum
end;
hence ex r being real number st
( r > 0 & Ball (p,r) c= P1 ) by A5, XREAL_1:139; ::_thesis: verum
end;
caseA9: |.q1.| > a ; ::_thesis: ex r being real number st
( r > 0 & Ball (p,r) c= P1 )
set r1 = (|.q1.| - a) / 2;
A10: |.q1.| - a > 0 by A9, XREAL_1:50;
Ball (p,((|.q1.| - a) / 2)) c= P1
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Ball (p,((|.q1.| - a) / 2)) or x in P1 )
assume A11: x in Ball (p,((|.q1.| - a) / 2)) ; ::_thesis: x in P1
then reconsider p2 = x as Point of (Euclid n) ;
reconsider q2 = p2 as Point of (TOP-REAL n) by TOPREAL3:8;
dist (p,p2) < (|.q1.| - a) / 2 by A11, METRIC_1:11;
then A12: |.(q1 - q2).| < (|.q1.| - a) / 2 by JGRAPH_1:28;
now__::_thesis:_not_q2_in_A
assume q2 in A ; ::_thesis: contradiction
then A13: ex q being Point of (TOP-REAL n) st
( q = q2 & |.q.| = a ) by A1;
|.(q1 - q2).| >= |.q1.| - |.q2.| by TOPRNS_1:32;
then (|.q1.| - a) / 2 > ((|.q1.| - a) / 2) + ((|.q1.| - a) / 2) by A12, A13, XXREAL_0:2;
then ((|.q1.| - a) / 2) - ((|.q1.| - a) / 2) > (|.q1.| - a) / 2 by XREAL_1:20;
hence contradiction by A10; ::_thesis: verum
end;
hence x in P1 by XBOOLE_0:def_5; ::_thesis: verum
end;
hence ex r being real number st
( r > 0 & Ball (p,r) c= P1 ) by A10, XREAL_1:139; ::_thesis: verum
end;
end;
end;
hence ex r being real number st
( r > 0 & Ball (p,r) c= P1 ) ; ::_thesis: verum
end;
then P1 is open by TOPMETR:15;
hence A ` is open by A2, PRE_TOPC:30; ::_thesis: A is closed
hence A is closed by TOPS_1:3; ::_thesis: verum
end;
theorem Th81: :: JORDAN2C:81
for n being Element of NAT
for B being non empty Subset of (TOP-REAL n) st B is open holds
(TOP-REAL n) | B is locally_connected
proof
let n be Element of NAT ; ::_thesis: for B being non empty Subset of (TOP-REAL n) st B is open holds
(TOP-REAL n) | B is locally_connected
let B be non empty Subset of (TOP-REAL n); ::_thesis: ( B is open implies (TOP-REAL n) | B is locally_connected )
A1: TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def_8;
assume A2: B is open ; ::_thesis: (TOP-REAL n) | B is locally_connected
for A being non empty Subset of ((TOP-REAL n) | B)
for C being Subset of ((TOP-REAL n) | B) st A is open & C is_a_component_of A holds
C is open
proof
let A be non empty Subset of ((TOP-REAL n) | B); ::_thesis: for C being Subset of ((TOP-REAL n) | B) st A is open & C is_a_component_of A holds
C is open
let C be Subset of ((TOP-REAL n) | B); ::_thesis: ( A is open & C is_a_component_of A implies C is open )
assume that
A3: A is open and
A4: C is_a_component_of A ; ::_thesis: C is open
consider C1 being Subset of (((TOP-REAL n) | B) | A) such that
A5: C1 = C and
A6: C1 is a_component by A4, CONNSP_1:def_6;
C1 c= [#] (((TOP-REAL n) | B) | A) ;
then A7: C1 c= A by PRE_TOPC:def_5;
A c= the carrier of ((TOP-REAL n) | B) ;
then A c= B by PRE_TOPC:8;
then C c= B by A5, A7, XBOOLE_1:1;
then reconsider C0 = C as Subset of (TOP-REAL n) by XBOOLE_1:1;
reconsider CC = C0 as Subset of (TopSpaceMetr (Euclid n)) by A1;
for p being Point of (Euclid n) st p in C0 holds
ex r being real number st
( r > 0 & Ball (p,r) c= C0 )
proof
consider A0 being Subset of (TOP-REAL n) such that
A8: A0 is open and
A9: A0 /\ ([#] ((TOP-REAL n) | B)) = A by A3, TOPS_2:24;
A10: A0 /\ B = A by A9, PRE_TOPC:def_5;
A11: TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def_8;
then reconsider AA = A0 /\ B as Subset of (TopSpaceMetr (Euclid n)) ;
let p be Point of (Euclid n); ::_thesis: ( p in C0 implies ex r being real number st
( r > 0 & Ball (p,r) c= C0 ) )
assume A12: p in C0 ; ::_thesis: ex r being real number st
( r > 0 & Ball (p,r) c= C0 )
AA is open by A2, A8, A11, PRE_TOPC:30;
then consider r1 being real number such that
A13: r1 > 0 and
A14: Ball (p,r1) c= AA by A5, A7, A12, A10, TOPMETR:15;
reconsider r1 = r1 as Real by XREAL_0:def_1;
A15: Ball (p,r1) c= A by A9, A14, PRE_TOPC:def_5;
then reconsider BL2 = Ball (p,r1) as Subset of ((TOP-REAL n) | B) by XBOOLE_1:1;
Ball (p,r1) c= [#] (((TOP-REAL n) | B) | A) by A15, PRE_TOPC:def_5;
then reconsider BL = Ball (p,r1) as Subset of (((TOP-REAL n) | B) | A) ;
reconsider BL = BL as Subset of (((TOP-REAL n) | B) | A) ;
reconsider BL2 = BL2 as Subset of ((TOP-REAL n) | B) ;
reconsider BL1 = Ball (p,r1) as Subset of (TOP-REAL n) by TOPREAL3:8;
reconsider BL1 = BL1 as Subset of (TOP-REAL n) ;
now__::_thesis:_Ball_(p,r1)_c=_C0
p in BL by A13, GOBOARD6:1;
then BL /\ C <> {} (((TOP-REAL n) | B) | A) by A12, XBOOLE_0:def_4;
then A16: BL meets C by XBOOLE_0:def_7;
BL1 is convex by Th70;
then A17: BL2 is connected by CONNSP_1:46;
assume not Ball (p,r1) c= C0 ; ::_thesis: contradiction
hence contradiction by A5, A6, A17, A16, CONNSP_1:36, CONNSP_1:46; ::_thesis: verum
end;
hence ex r being real number st
( r > 0 & Ball (p,r) c= C0 ) by A13; ::_thesis: verum
end;
then CC is open by TOPMETR:15;
then A18: ( [#] ((TOP-REAL n) | B) = B & C0 is open ) by A1, PRE_TOPC:30, PRE_TOPC:def_5;
C c= the carrier of ((TOP-REAL n) | B) ;
then C c= B by PRE_TOPC:8;
then C0 /\ B = C by XBOOLE_1:28;
hence C is open by A18, TOPS_2:24; ::_thesis: verum
end;
hence (TOP-REAL n) | B is locally_connected by CONNSP_2:18; ::_thesis: verum
end;
theorem Th82: :: JORDAN2C:82
for n being Element of NAT
for B being non empty Subset of (TOP-REAL n)
for A being Subset of (TOP-REAL n)
for a being real number st A = { q where q is Point of (TOP-REAL n) : |.q.| = a } & A ` = B holds
(TOP-REAL n) | B is locally_connected
proof
let n be Element of NAT ; ::_thesis: for B being non empty Subset of (TOP-REAL n)
for A being Subset of (TOP-REAL n)
for a being real number st A = { q where q is Point of (TOP-REAL n) : |.q.| = a } & A ` = B holds
(TOP-REAL n) | B is locally_connected
let B be non empty Subset of (TOP-REAL n); ::_thesis: for A being Subset of (TOP-REAL n)
for a being real number st A = { q where q is Point of (TOP-REAL n) : |.q.| = a } & A ` = B holds
(TOP-REAL n) | B is locally_connected
let A be Subset of (TOP-REAL n); ::_thesis: for a being real number st A = { q where q is Point of (TOP-REAL n) : |.q.| = a } & A ` = B holds
(TOP-REAL n) | B is locally_connected
let a be real number ; ::_thesis: ( A = { q where q is Point of (TOP-REAL n) : |.q.| = a } & A ` = B implies (TOP-REAL n) | B is locally_connected )
assume A1: ( A = { q where q is Point of (TOP-REAL n) : |.q.| = a } & A ` = B ) ; ::_thesis: (TOP-REAL n) | B is locally_connected
then A ` is open by Th80;
hence (TOP-REAL n) | B is locally_connected by A1, Th81; ::_thesis: verum
end;
theorem Th83: :: JORDAN2C:83
for n being Element of NAT
for f being Function of (TOP-REAL n),R^1 st ( for q being Point of (TOP-REAL n) holds f . q = |.q.| ) holds
f is continuous
proof
let n be Element of NAT ; ::_thesis: for f being Function of (TOP-REAL n),R^1 st ( for q being Point of (TOP-REAL n) holds f . q = |.q.| ) holds
f is continuous
let f be Function of (TOP-REAL n),R^1; ::_thesis: ( ( for q being Point of (TOP-REAL n) holds f . q = |.q.| ) implies f is continuous )
A1: TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def_8;
then reconsider f1 = f as Function of (TopSpaceMetr (Euclid n)),(TopSpaceMetr RealSpace) by TOPMETR:def_6;
assume A2: for q being Point of (TOP-REAL n) holds f . q = |.q.| ; ::_thesis: f is continuous
now__::_thesis:_for_r_being_real_number_
for_u_being_Element_of_(Euclid_n)
for_u1_being_Element_of_RealSpace_st_r_>_0_&_u1_=_f1_._u_holds_
ex_s_being_real_number_st_
(_s_>_0_&_(_for_w_being_Element_of_(Euclid_n)
for_w1_being_Element_of_RealSpace_st_w1_=_f1_._w_&_dist_(u,w)_<_s_holds_
dist_(u1,w1)_<_r_)_)
let r be real number ; ::_thesis: for u being Element of (Euclid n)
for u1 being Element of RealSpace st r > 0 & u1 = f1 . u holds
ex s being real number st
( s > 0 & ( for w being Element of (Euclid n)
for w1 being Element of RealSpace st w1 = f1 . w & dist (u,w) < s holds
dist (u1,w1) < r ) )
let u be Element of (Euclid n); ::_thesis: for u1 being Element of RealSpace st r > 0 & u1 = f1 . u holds
ex s being real number st
( s > 0 & ( for w being Element of (Euclid n)
for w1 being Element of RealSpace st w1 = f1 . w & dist (u,w) < s holds
dist (u1,w1) < r ) )
let u1 be Element of RealSpace; ::_thesis: ( r > 0 & u1 = f1 . u implies ex s being real number st
( s > 0 & ( for w being Element of (Euclid n)
for w1 being Element of RealSpace st w1 = f1 . w & dist (u,w) < s holds
dist (u1,w1) < r ) ) )
assume that
A3: r > 0 and
A4: u1 = f1 . u ; ::_thesis: ex s being real number st
( s > 0 & ( for w being Element of (Euclid n)
for w1 being Element of RealSpace st w1 = f1 . w & dist (u,w) < s holds
dist (u1,w1) < r ) )
set s1 = r;
for w being Element of (Euclid n)
for w1 being Element of RealSpace st w1 = f1 . w & dist (u,w) < r holds
dist (u1,w1) < r
proof
let w be Element of (Euclid n); ::_thesis: for w1 being Element of RealSpace st w1 = f1 . w & dist (u,w) < r holds
dist (u1,w1) < r
let w1 be Element of RealSpace; ::_thesis: ( w1 = f1 . w & dist (u,w) < r implies dist (u1,w1) < r )
assume that
A5: w1 = f1 . w and
A6: dist (u,w) < r ; ::_thesis: dist (u1,w1) < r
reconsider tu = u1, tw = w1 as Real ;
reconsider qw = w, qu = u as Point of (TOP-REAL n) by TOPREAL3:8;
A7: dist (u1,w1) = the distance of RealSpace . (u1,w1)
.= abs (tu - tw) by METRIC_1:def_12 ;
A8: tu = |.qu.| by A2, A4;
w1 = |.qw.| by A2, A5;
then ( dist (u,w) = |.(qu - qw).| & dist (u1,w1) <= |.(qu - qw).| ) by A7, A8, Th9, JGRAPH_1:28;
hence dist (u1,w1) < r by A6, XXREAL_0:2; ::_thesis: verum
end;
hence ex s being real number st
( s > 0 & ( for w being Element of (Euclid n)
for w1 being Element of RealSpace st w1 = f1 . w & dist (u,w) < s holds
dist (u1,w1) < r ) ) by A3; ::_thesis: verum
end;
then f1 is continuous by UNIFORM1:3;
hence f is continuous by A1, PRE_TOPC:32, TOPMETR:def_6; ::_thesis: verum
end;
theorem Th84: :: JORDAN2C:84
for n being Element of NAT ex f being Function of (TOP-REAL n),R^1 st
( ( for q being Point of (TOP-REAL n) holds f . q = |.q.| ) & f is continuous )
proof
let n be Element of NAT ; ::_thesis: ex f being Function of (TOP-REAL n),R^1 st
( ( for q being Point of (TOP-REAL n) holds f . q = |.q.| ) & f is continuous )
defpred S1[ set , set ] means ex q being Point of (TOP-REAL n) st
( q = $1 & $2 = |.q.| );
A1: for x being set st x in the carrier of (TOP-REAL n) holds
ex y being set st S1[x,y]
proof
let x be set ; ::_thesis: ( x in the carrier of (TOP-REAL n) implies ex y being set st S1[x,y] )
assume x in the carrier of (TOP-REAL n) ; ::_thesis: ex y being set st S1[x,y]
then reconsider q3 = x as Point of (TOP-REAL n) ;
take |.q3.| ; ::_thesis: S1[x,|.q3.|]
thus S1[x,|.q3.|] ; ::_thesis: verum
end;
consider f1 being Function such that
A2: ( dom f1 = the carrier of (TOP-REAL n) & ( for x being set st x in the carrier of (TOP-REAL n) holds
S1[x,f1 . x] ) ) from CLASSES1:sch_1(A1);
rng f1 c= the carrier of R^1
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in rng f1 or z in the carrier of R^1 )
assume z in rng f1 ; ::_thesis: z in the carrier of R^1
then consider xz being set such that
A3: xz in dom f1 and
A4: z = f1 . xz by FUNCT_1:def_3;
ex q4 being Point of (TOP-REAL n) st
( q4 = xz & f1 . xz = |.q4.| ) by A2, A3;
hence z in the carrier of R^1 by A4, TOPMETR:17; ::_thesis: verum
end;
then reconsider f2 = f1 as Function of (TOP-REAL n),R^1 by A2, FUNCT_2:def_1, RELSET_1:4;
A5: for q being Point of (TOP-REAL n) holds f1 . q = |.q.|
proof
let q be Point of (TOP-REAL n); ::_thesis: f1 . q = |.q.|
ex q2 being Point of (TOP-REAL n) st
( q2 = q & f1 . q = |.q2.| ) by A2;
hence f1 . q = |.q.| ; ::_thesis: verum
end;
then f2 is continuous by Th83;
hence ex f being Function of (TOP-REAL n),R^1 st
( ( for q being Point of (TOP-REAL n) holds f . q = |.q.| ) & f is continuous ) by A5; ::_thesis: verum
end;
theorem Th85: :: JORDAN2C:85
for n being Element of NAT
for g being Function of I[01],(TOP-REAL n) st g is continuous holds
ex f being Function of I[01],R^1 st
( ( for t being Point of I[01] holds f . t = |.(g . t).| ) & f is continuous )
proof
let n be Element of NAT ; ::_thesis: for g being Function of I[01],(TOP-REAL n) st g is continuous holds
ex f being Function of I[01],R^1 st
( ( for t being Point of I[01] holds f . t = |.(g . t).| ) & f is continuous )
let g be Function of I[01],(TOP-REAL n); ::_thesis: ( g is continuous implies ex f being Function of I[01],R^1 st
( ( for t being Point of I[01] holds f . t = |.(g . t).| ) & f is continuous ) )
consider h being Function of (TOP-REAL n),R^1 such that
A1: for q being Point of (TOP-REAL n) holds h . q = |.q.| and
A2: h is continuous by Th84;
set f1 = h * g;
A3: for t being Point of I[01] holds (h * g) . t = |.(g . t).|
proof
let t be Point of I[01]; ::_thesis: (h * g) . t = |.(g . t).|
reconsider q = g . t as Point of (TOP-REAL n) ;
dom g = the carrier of I[01] by FUNCT_2:def_1;
then (h * g) . t = h . (g . t) by FUNCT_1:13
.= |.q.| by A1 ;
hence (h * g) . t = |.(g . t).| ; ::_thesis: verum
end;
assume g is continuous ; ::_thesis: ex f being Function of I[01],R^1 st
( ( for t being Point of I[01] holds f . t = |.(g . t).| ) & f is continuous )
hence ex f being Function of I[01],R^1 st
( ( for t being Point of I[01] holds f . t = |.(g . t).| ) & f is continuous ) by A2, A3; ::_thesis: verum
end;
theorem Th86: :: JORDAN2C:86
for n being Element of NAT
for g being Function of I[01],(TOP-REAL n)
for a being Real st g is continuous & |.(g /. 0).| <= a & a <= |.(g /. 1).| holds
ex s being Point of I[01] st |.(g /. s).| = a
proof
let n be Element of NAT ; ::_thesis: for g being Function of I[01],(TOP-REAL n)
for a being Real st g is continuous & |.(g /. 0).| <= a & a <= |.(g /. 1).| holds
ex s being Point of I[01] st |.(g /. s).| = a
reconsider I = 1 as Point of I[01] by BORSUK_1:40, XXREAL_1:1;
A1: 0 in [.0,1.] by XXREAL_1:1;
reconsider o = 0 as Point of I[01] by BORSUK_1:40, XXREAL_1:1;
let g be Function of I[01],(TOP-REAL n); ::_thesis: for a being Real st g is continuous & |.(g /. 0).| <= a & a <= |.(g /. 1).| holds
ex s being Point of I[01] st |.(g /. s).| = a
let a be Real; ::_thesis: ( g is continuous & |.(g /. 0).| <= a & a <= |.(g /. 1).| implies ex s being Point of I[01] st |.(g /. s).| = a )
assume that
A2: g is continuous and
A3: ( |.(g /. 0).| <= a & a <= |.(g /. 1).| ) ; ::_thesis: ex s being Point of I[01] st |.(g /. s).| = a
consider f being Function of I[01],R^1 such that
A4: for t being Point of I[01] holds f . t = |.(g . t).| and
A5: f is continuous by A2, Th85;
A6: f . 0 = |.(g . o).| by A4
.= |.(g /. 0).| by FUNCT_2:def_13 ;
set c = |.(g /. 0).|;
set b = |.(g /. 1).|;
A7: 1 in the carrier of I[01] by BORSUK_1:40, XXREAL_1:1;
A8: f . 1 = |.(g . I).| by A4
.= |.(g /. 1).| by FUNCT_2:def_13 ;
percases ( ( |.(g /. 0).| < a & a < |.(g /. 1).| ) or |.(g /. 0).| = a or a = |.(g /. 1).| ) by A3, XXREAL_0:1;
suppose ( |.(g /. 0).| < a & a < |.(g /. 1).| ) ; ::_thesis: ex s being Point of I[01] st |.(g /. s).| = a
then consider rc being Real such that
A9: f . rc = a and
A10: ( 0 < rc & rc < 1 ) by A5, A6, A8, TOPMETR:20, TOPREAL5:6;
reconsider rc1 = rc as Point of I[01] by A10, BORSUK_1:40, XXREAL_1:1;
A11: rc in the carrier of I[01] by A10, BORSUK_1:40, XXREAL_1:1;
|.(g /. rc).| = |.(g . rc1).| by FUNCT_2:def_13
.= a by A4, A9 ;
hence ex s being Point of I[01] st |.(g /. s).| = a by A11; ::_thesis: verum
end;
suppose |.(g /. 0).| = a ; ::_thesis: ex s being Point of I[01] st |.(g /. s).| = a
hence ex s being Point of I[01] st |.(g /. s).| = a by A1, BORSUK_1:40; ::_thesis: verum
end;
suppose a = |.(g /. 1).| ; ::_thesis: ex s being Point of I[01] st |.(g /. s).| = a
hence ex s being Point of I[01] st |.(g /. s).| = a by A7; ::_thesis: verum
end;
end;
end;
theorem Th87: :: JORDAN2C:87
for n being Element of NAT
for r being Real
for q being Point of (TOP-REAL n) st q = <*r*> holds
|.q.| = abs r
proof
let n be Element of NAT ; ::_thesis: for r being Real
for q being Point of (TOP-REAL n) st q = <*r*> holds
|.q.| = abs r
let r be Real; ::_thesis: for q being Point of (TOP-REAL n) st q = <*r*> holds
|.q.| = abs r
let q be Point of (TOP-REAL n); ::_thesis: ( q = <*r*> implies |.q.| = abs r )
assume A1: q = <*r*> ; ::_thesis: |.q.| = abs r
reconsider xr = <*r*> as Element of REAL 1 by FINSEQ_2:131;
len xr = 1 by FINSEQ_1:39;
then A2: q /. 1 = xr . 1 by A1, FINSEQ_4:15;
then ( len (sqr xr) = 1 & (sqr xr) . 1 = (q /. 1) ^2 ) by CARD_1:def_7, VALUED_1:11;
then A3: sqr xr = <*((q /. 1) ^2)*> by FINSEQ_1:40;
sqrt ((q /. 1) ^2) = abs (q /. 1) by COMPLEX1:72
.= abs r by A2, FINSEQ_1:40 ;
hence |.q.| = abs r by A1, A3, FINSOP_1:11; ::_thesis: verum
end;
theorem :: JORDAN2C:88
for n being Element of NAT
for A being Subset of (TOP-REAL n)
for a being Real st n >= 1 & a > 0 & A = { q where q is Point of (TOP-REAL n) : |.q.| = a } holds
BDD A is_inside_component_of A
proof
let n be Element of NAT ; ::_thesis: for A being Subset of (TOP-REAL n)
for a being Real st n >= 1 & a > 0 & A = { q where q is Point of (TOP-REAL n) : |.q.| = a } holds
BDD A is_inside_component_of A
let A be Subset of (TOP-REAL n); ::_thesis: for a being Real st n >= 1 & a > 0 & A = { q where q is Point of (TOP-REAL n) : |.q.| = a } holds
BDD A is_inside_component_of A
let a be Real; ::_thesis: ( n >= 1 & a > 0 & A = { q where q is Point of (TOP-REAL n) : |.q.| = a } implies BDD A is_inside_component_of A )
{ q where q is Point of (TOP-REAL n) : |.q.| < a } c= the carrier of (TOP-REAL n)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { q where q is Point of (TOP-REAL n) : |.q.| < a } or x in the carrier of (TOP-REAL n) )
assume x in { q where q is Point of (TOP-REAL n) : |.q.| < a } ; ::_thesis: x in the carrier of (TOP-REAL n)
then ex q being Point of (TOP-REAL n) st
( q = x & |.q.| < a ) ;
hence x in the carrier of (TOP-REAL n) ; ::_thesis: verum
end;
then reconsider W = { q where q is Point of (TOP-REAL n) : |.q.| < a } as Subset of (Euclid n) by TOPREAL3:8;
reconsider P = W as Subset of (TOP-REAL n) by TOPREAL3:8;
reconsider P = P as Subset of (TOP-REAL n) ;
A1: the carrier of ((TOP-REAL n) | (A `)) = A ` by PRE_TOPC:8;
then reconsider P1 = Component_of (Down (P,(A `))) as Subset of (TOP-REAL n) by XBOOLE_1:1;
assume A2: ( n >= 1 & a > 0 & A = { q where q is Point of (TOP-REAL n) : |.q.| = a } ) ; ::_thesis: BDD A is_inside_component_of A
A3: P c= A `
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in P or x in A ` )
assume A4: x in P ; ::_thesis: x in A `
then reconsider q = x as Point of (TOP-REAL n) ;
A5: ex q1 being Point of (TOP-REAL n) st
( q1 = q & |.q1.| < a ) by A4;
now__::_thesis:_not_q_in_A
assume q in A ; ::_thesis: contradiction
then ex q2 being Point of (TOP-REAL n) st
( q2 = q & |.q2.| = a ) by A2;
hence contradiction by A5; ::_thesis: verum
end;
hence x in A ` by XBOOLE_0:def_5; ::_thesis: verum
end;
then A6: Down (P,(A `)) = P by XBOOLE_1:28;
P is convex by Th69;
then (TOP-REAL n) | P is connected by CONNSP_1:def_3;
then ((TOP-REAL n) | (A `)) | (Down (P,(A `))) is connected by A3, A6, PRE_TOPC:7;
then A7: Down (P,(A `)) is connected by CONNSP_1:def_3;
|.(0. (TOP-REAL n)).| = 0 by TOPRNS_1:23;
then A8: 0. (TOP-REAL n) in P by A2;
then reconsider G = A ` as non empty Subset of (TOP-REAL n) by A3;
A9: not (TOP-REAL n) | G is empty ;
A10: P c= Component_of (Down (P,(A `))) by A8, A6, A7, CONNSP_3:13;
A11: Down (P,(A `)) <> {} by A3, A8, XBOOLE_0:def_4;
then A12: Component_of (Down (P,(A `))) is a_component by A9, A7, CONNSP_3:9;
then A13: Component_of (Down (P,(A `))) is connected by CONNSP_1:def_5;
Component_of (Down (P,(A `))) is bounded Subset of (Euclid n)
proof
reconsider D2 = Component_of (Down (P,(A `))) as Subset of (TOP-REAL n) by A1, XBOOLE_1:1;
reconsider D = D2 as Subset of (Euclid n) by TOPREAL3:8;
reconsider D = D as Subset of (Euclid n) ;
now__::_thesis:_D2_is_bounded
reconsider B = A ` as non empty Subset of (TOP-REAL n) by A3, A8;
set p = 0. (TOP-REAL n);
reconsider RR = (TOP-REAL n) | B as non empty TopSpace ;
assume not D2 is bounded ; ::_thesis: contradiction
then consider q being Point of (TOP-REAL n) such that
A14: q in D2 and
A15: |.q.| >= a by Th34;
A16: ( A ` is open & D2 is connected ) by A2, A13, Th80, CONNSP_1:23;
D c= the carrier of ((TOP-REAL n) | (A `)) ;
then A17: D2 c= A ` by PRE_TOPC:8;
then A18: D2 = Down (D2,(A `)) by XBOOLE_1:28;
RR is locally_connected by A2, Th82;
then Component_of (Down (P,(A `))) is open by A11, A7, CONNSP_2:15, CONNSP_3:9;
then consider G being Subset of (TOP-REAL n) such that
A19: G is open and
A20: Down (D2,(A `)) = G /\ ([#] ((TOP-REAL n) | (A `))) by A18, TOPS_2:24;
A21: G /\ (A `) = D2 by A18, A20, PRE_TOPC:def_5;
0. (TOP-REAL n) <> q by A2, A15, TOPRNS_1:23;
then consider f1 being Function of I[01],(TOP-REAL n) such that
A22: f1 is continuous and
A23: rng f1 c= D2 and
A24: f1 . 0 = 0. (TOP-REAL n) and
A25: f1 . 1 = q by A8, A10, A14, A19, A21, A16, Th79;
A26: |.(f1 /. 1).| >= a by A15, A25, BORSUK_1:def_15, FUNCT_2:def_13;
|.(0. (TOP-REAL n)).| < a by A2, TOPRNS_1:23;
then |.(f1 /. 0).| < a by A24, BORSUK_1:def_14, FUNCT_2:def_13;
then consider t0 being Point of I[01] such that
A27: |.(f1 /. t0).| = a by A22, A26, Th86;
reconsider q2 = f1 . t0 as Point of (TOP-REAL n) ;
t0 in [#] I[01] ;
then t0 in dom f1 by FUNCT_2:def_1;
then q2 in rng f1 by FUNCT_1:def_3;
then A28: q2 in D2 by A23;
q2 in A by A2, A27;
then A /\ (A `) <> {} the carrier of (TOP-REAL n) by A17, A28, XBOOLE_0:def_4;
then A meets A ` by XBOOLE_0:def_7;
hence contradiction by XBOOLE_1:79; ::_thesis: verum
end;
hence Component_of (Down (P,(A `))) is bounded Subset of (Euclid n) by Th11; ::_thesis: verum
end;
then A29: P1 is_inside_component_of A by A12, Th13;
A30: P1 c= union { B where B is Subset of (TOP-REAL n) : B is_inside_component_of A }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in P1 or x in union { B where B is Subset of (TOP-REAL n) : B is_inside_component_of A } )
assume A31: x in P1 ; ::_thesis: x in union { B where B is Subset of (TOP-REAL n) : B is_inside_component_of A }
P1 in { B where B is Subset of (TOP-REAL n) : B is_inside_component_of A } by A29;
hence x in union { B where B is Subset of (TOP-REAL n) : B is_inside_component_of A } by A31, TARSKI:def_4; ::_thesis: verum
end;
now__::_thesis:_(_(_n_>=_2_&_ex_B_being_Subset_of_(TOP-REAL_n)_st_
(_B_is_inside_component_of_A_&_B_=_BDD_A_)_)_or_(_n_<_2_&_ex_B_being_Subset_of_(TOP-REAL_n)_st_
(_B_is_inside_component_of_A_&_B_=_BDD_A_)_)_)
percases ( n >= 2 or n < 2 ) ;
caseA32: n >= 2 ; ::_thesis: ex B being Subset of (TOP-REAL n) st
( B is_inside_component_of A & B = BDD A )
union { B where B is Subset of (TOP-REAL n) : B is_inside_component_of A } c= P1
proof
reconsider E = A ` as non empty Subset of (TOP-REAL n) by A3, A8;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in union { B where B is Subset of (TOP-REAL n) : B is_inside_component_of A } or x in P1 )
assume x in union { B where B is Subset of (TOP-REAL n) : B is_inside_component_of A } ; ::_thesis: x in P1
then consider y being set such that
A33: x in y and
A34: y in { B where B is Subset of (TOP-REAL n) : B is_inside_component_of A } by TARSKI:def_4;
consider B being Subset of (TOP-REAL n) such that
A35: B = y and
A36: B is_inside_component_of A by A34;
ex C being Subset of ((TOP-REAL n) | (A `)) st
( C = B & C is a_component & C is bounded Subset of (Euclid n) ) by A36, Th13;
then reconsider p = x as Point of ((TOP-REAL n) | (A `)) by A33, A35;
A37: ( the carrier of ((TOP-REAL n) | (A `)) = A ` & p in the carrier of ((TOP-REAL n) | E) ) by PRE_TOPC:8;
then reconsider q2 = p as Point of (TOP-REAL n) ;
not p in A by A37, XBOOLE_0:def_5;
then |.q2.| <> a by A2;
then A38: ( |.q2.| < a or |.q2.| > a ) by XXREAL_0:1;
now__::_thesis:_(_(_p_in__{__q_where_q_is_Point_of_(TOP-REAL_n)_:_|.q.|_>_a__}__&_x_in_P1_)_or_(_p_in__{__q1_where_q1_is_Point_of_(TOP-REAL_n)_:_|.q1.|_<_a__}__&_x_in_P1_)_)
percases ( p in { q where q is Point of (TOP-REAL n) : |.q.| > a } or p in { q1 where q1 is Point of (TOP-REAL n) : |.q1.| < a } ) by A38;
caseA39: p in { q where q is Point of (TOP-REAL n) : |.q.| > a } ; ::_thesis: x in P1
{ q where q is Point of (TOP-REAL n) : |.q.| > a } c= A `
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in { q where q is Point of (TOP-REAL n) : |.q.| > a } or z in A ` )
assume z in { q where q is Point of (TOP-REAL n) : |.q.| > a } ; ::_thesis: z in A `
then consider q being Point of (TOP-REAL n) such that
A40: q = z and
A41: |.q.| > a ;
now__::_thesis:_not_q_in_A
assume q in A ; ::_thesis: contradiction
then ex q2 being Point of (TOP-REAL n) st
( q2 = q & |.q2.| = a ) by A2;
hence contradiction by A41; ::_thesis: verum
end;
hence z in A ` by A40, XBOOLE_0:def_5; ::_thesis: verum
end;
then reconsider Q = { q where q is Point of (TOP-REAL n) : |.q.| > a } as Subset of ((TOP-REAL n) | (A `)) by PRE_TOPC:8;
reconsider Q = Q as Subset of ((TOP-REAL n) | (A `)) ;
{ q where q is Point of (TOP-REAL n) : |.q.| > a } c= the carrier of (TOP-REAL n)
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in { q where q is Point of (TOP-REAL n) : |.q.| > a } or z in the carrier of (TOP-REAL n) )
assume z in { q where q is Point of (TOP-REAL n) : |.q.| > a } ; ::_thesis: z in the carrier of (TOP-REAL n)
then ex q being Point of (TOP-REAL n) st
( q = z & |.q.| > a ) ;
hence z in the carrier of (TOP-REAL n) ; ::_thesis: verum
end;
then reconsider P2 = { q where q is Point of (TOP-REAL n) : |.q.| > a } as Subset of (TOP-REAL n) ;
P2 is Subset of (Euclid n) by TOPREAL3:8;
then reconsider W2 = { q where q is Point of (TOP-REAL n) : |.q.| > a } as Subset of (Euclid n) ;
P2 is connected by A32, Th51;
then A42: (TOP-REAL n) | P2 is connected by CONNSP_1:def_3;
A43: not W2 is bounded by A32, Th61;
A44: now__::_thesis:_not_W2_meets_A
assume W2 meets A ; ::_thesis: contradiction
then consider z being set such that
A45: ( z in W2 & z in A ) by XBOOLE_0:3;
( ex q being Point of (TOP-REAL n) st
( q = z & |.q.| > a ) & ex q2 being Point of (TOP-REAL n) st
( q2 = z & |.q2.| = a ) ) by A2, A45;
hence contradiction ; ::_thesis: verum
end;
then W2 /\ ((A `) `) = {} by XBOOLE_0:def_7;
then P2 \ (A `) = {} by SUBSET_1:13;
then A46: W2 c= A ` by XBOOLE_1:37;
then Q = Down (P2,(A `)) by XBOOLE_1:28;
then Up (Component_of Q) is_outside_component_of A by A32, A43, A44, Th51, Th63;
then A47: Component_of Q c= UBD A by Th23;
(TOP-REAL n) | P2 = ((TOP-REAL n) | (A `)) | Q by A46, PRE_TOPC:7;
then Q is connected by A42, CONNSP_1:def_3;
then Q c= Component_of Q by CONNSP_3:1;
then A48: p in Component_of Q by A39;
B c= BDD A by A36, Th22;
then p in (BDD A) /\ (UBD A) by A33, A35, A47, A48, XBOOLE_0:def_4;
then BDD A meets UBD A by XBOOLE_0:4;
hence x in P1 by Th24; ::_thesis: verum
end;
caseA49: p in { q1 where q1 is Point of (TOP-REAL n) : |.q1.| < a } ; ::_thesis: x in P1
Down (P,(A `)) c= Component_of (Down (P,(A `))) by A7, CONNSP_3:1;
hence x in P1 by A6, A49; ::_thesis: verum
end;
end;
end;
hence x in P1 ; ::_thesis: verum
end;
then P1 = union { B where B is Subset of (TOP-REAL n) : B is_inside_component_of A } by A30, XBOOLE_0:def_10;
hence ex B being Subset of (TOP-REAL n) st
( B is_inside_component_of A & B = BDD A ) by A29; ::_thesis: verum
end;
case n < 2 ; ::_thesis: ex B being Subset of (TOP-REAL n) st
( B is_inside_component_of A & B = BDD A )
then n < 1 + 1 ;
then A50: n <= 1 by NAT_1:13;
then A51: n = 1 by A2, XXREAL_0:1;
union { B where B is Subset of (TOP-REAL n) : B is_inside_component_of A } c= P1
proof
reconsider E = A ` as non empty Subset of (TOP-REAL n) by A3, A8;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in union { B where B is Subset of (TOP-REAL n) : B is_inside_component_of A } or x in P1 )
assume x in union { B where B is Subset of (TOP-REAL n) : B is_inside_component_of A } ; ::_thesis: x in P1
then consider y being set such that
A52: x in y and
A53: y in { B where B is Subset of (TOP-REAL n) : B is_inside_component_of A } by TARSKI:def_4;
consider B being Subset of (TOP-REAL n) such that
A54: B = y and
A55: B is_inside_component_of A by A53;
ex C being Subset of ((TOP-REAL n) | (A `)) st
( C = B & C is a_component & C is bounded Subset of (Euclid n) ) by A55, Th13;
then reconsider p = x as Point of ((TOP-REAL n) | (A `)) by A52, A54;
A56: ( the carrier of ((TOP-REAL n) | (A `)) = A ` & p in the carrier of ((TOP-REAL n) | E) ) by PRE_TOPC:8;
then reconsider q2 = p as Point of (TOP-REAL n) ;
not p in A by A56, XBOOLE_0:def_5;
then |.q2.| <> a by A2;
then A57: ( |.q2.| < a or |.q2.| > a ) by XXREAL_0:1;
now__::_thesis:_(_(_p_in__{__q_where_q_is_Point_of_(TOP-REAL_n)_:_|.q.|_>_a__}__&_x_in_P1_)_or_(_p_in__{__q1_where_q1_is_Point_of_(TOP-REAL_n)_:_|.q1.|_<_a__}__&_x_in_P1_)_)
percases ( p in { q where q is Point of (TOP-REAL n) : |.q.| > a } or p in { q1 where q1 is Point of (TOP-REAL n) : |.q1.| < a } ) by A57;
case p in { q where q is Point of (TOP-REAL n) : |.q.| > a } ; ::_thesis: x in P1
then consider q0 being Point of (TOP-REAL n) such that
A58: q0 = p and
A59: |.q0.| > a ;
q0 is Element of REAL n by EUCLID:22;
then consider r0 being Real such that
A60: q0 = <*r0*> by A51, FINSEQ_2:97;
A61: |.q0.| = abs r0 by A60, Th87;
A62: now__::_thesis:_(_p_in__{__q_where_q_is_Point_of_(TOP-REAL_n)_:_ex_r_being_Real_st_
(_q_=_<*r*>_&_r_>_a_)__}__or_p_in__{__q1_where_q1_is_Point_of_(TOP-REAL_n)_:_ex_r1_being_Real_st_
(_q1_=_<*r1*>_&_r1_<_-_a_)__}__)
percases ( r0 >= 0 or r0 < 0 ) ;
suppose r0 >= 0 ; ::_thesis: ( p in { q where q is Point of (TOP-REAL n) : ex r being Real st
( q = <*r*> & r > a ) } or p in { q1 where q1 is Point of (TOP-REAL n) : ex r1 being Real st
( q1 = <*r1*> & r1 < - a ) } )
then r0 = abs r0 by ABSVALUE:def_1;
hence ( p in { q where q is Point of (TOP-REAL n) : ex r being Real st
( q = <*r*> & r > a ) } or p in { q1 where q1 is Point of (TOP-REAL n) : ex r1 being Real st
( q1 = <*r1*> & r1 < - a ) } ) by A58, A59, A60, A61; ::_thesis: verum
end;
suppose r0 < 0 ; ::_thesis: ( p in { q where q is Point of (TOP-REAL n) : ex r being Real st
( q = <*r*> & r > a ) } or p in { q1 where q1 is Point of (TOP-REAL n) : ex r1 being Real st
( q1 = <*r1*> & r1 < - a ) } )
then - r0 > a by A59, A61, ABSVALUE:def_1;
then - (- r0) < - a by XREAL_1:24;
hence ( p in { q where q is Point of (TOP-REAL n) : ex r being Real st
( q = <*r*> & r > a ) } or p in { q1 where q1 is Point of (TOP-REAL n) : ex r1 being Real st
( q1 = <*r1*> & r1 < - a ) } ) by A58, A60; ::_thesis: verum
end;
end;
end;
now__::_thesis:_x_in_P1
percases ( p in { q where q is Point of (TOP-REAL n) : ex r being Real st
( q = <*r*> & r > a ) } or p in { q1 where q1 is Point of (TOP-REAL n) : ex r1 being Real st
( q1 = <*r1*> & r1 < - a ) } ) by A62;
supposeA63: p in { q where q is Point of (TOP-REAL n) : ex r being Real st
( q = <*r*> & r > a ) } ; ::_thesis: x in P1
{ q where q is Point of (TOP-REAL n) : ex r being Real st
( q = <*r*> & r > a ) } c= A `
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in { q where q is Point of (TOP-REAL n) : ex r being Real st
( q = <*r*> & r > a ) } or z in A ` )
assume z in { q where q is Point of (TOP-REAL n) : ex r being Real st
( q = <*r*> & r > a ) } ; ::_thesis: z in A `
then consider q being Point of (TOP-REAL n) such that
A64: q = z and
A65: ex r being Real st
( q = <*r*> & r > a ) ;
consider r being Real such that
A66: q = <*r*> and
A67: r > a by A65;
n = 1 by A2, A50, XXREAL_0:1;
then reconsider xr = <*r*> as Element of REAL n by FINSEQ_2:131;
len xr = 1 by FINSEQ_1:39;
then A68: q /. 1 = xr . 1 by A66, FINSEQ_4:15;
then A69: (sqr xr) . 1 = (q /. 1) ^2 by VALUED_1:11;
A70: sqrt ((q /. 1) ^2) = abs (q /. 1) by COMPLEX1:72
.= abs r by A68, FINSEQ_1:40 ;
len (sqr xr) = 1 by A51, CARD_1:def_7;
then sqr xr = <*((q /. 1) ^2)*> by A69, FINSEQ_1:40;
then A71: |.q.| = abs r by A66, A70, FINSOP_1:11
.= r by A2, A67, ABSVALUE:def_1 ;
now__::_thesis:_not_q_in_A
assume q in A ; ::_thesis: contradiction
then ex q2 being Point of (TOP-REAL n) st
( q2 = q & |.q2.| = a ) by A2;
hence contradiction by A67, A71; ::_thesis: verum
end;
hence z in A ` by A64, XBOOLE_0:def_5; ::_thesis: verum
end;
then reconsider Q = { q where q is Point of (TOP-REAL n) : ex r being Real st
( q = <*r*> & r > a ) } as Subset of ((TOP-REAL n) | (A `)) by PRE_TOPC:8;
{ q where q is Point of (TOP-REAL n) : ex r being Real st
( q = <*r*> & r > a ) } c= the carrier of (TOP-REAL n)
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in { q where q is Point of (TOP-REAL n) : ex r being Real st
( q = <*r*> & r > a ) } or z in the carrier of (TOP-REAL n) )
assume z in { q where q is Point of (TOP-REAL n) : ex r being Real st
( q = <*r*> & r > a ) } ; ::_thesis: z in the carrier of (TOP-REAL n)
then ex q being Point of (TOP-REAL n) st
( q = z & ex r being Real st
( q = <*r*> & r > a ) ) ;
hence z in the carrier of (TOP-REAL n) ; ::_thesis: verum
end;
then reconsider P3 = { q where q is Point of (TOP-REAL n) : ex r being Real st
( q = <*r*> & r > a ) } as Subset of (TOP-REAL n) ;
reconsider W3 = P3 as Subset of (Euclid n) by TOPREAL3:8;
reconsider Q = Q as Subset of ((TOP-REAL n) | (A `)) ;
{ q where q is Point of (TOP-REAL n) : |.q.| > a } c= the carrier of (TOP-REAL n)
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in { q where q is Point of (TOP-REAL n) : |.q.| > a } or z in the carrier of (TOP-REAL n) )
assume z in { q where q is Point of (TOP-REAL n) : |.q.| > a } ; ::_thesis: z in the carrier of (TOP-REAL n)
then ex q being Point of (TOP-REAL n) st
( q = z & |.q.| > a ) ;
hence z in the carrier of (TOP-REAL n) ; ::_thesis: verum
end;
then reconsider P2 = { q where q is Point of (TOP-REAL n) : |.q.| > a } as Subset of (TOP-REAL n) ;
P2 is Subset of (Euclid n) by TOPREAL3:8;
then reconsider W2 = { q where q is Point of (TOP-REAL n) : |.q.| > a } as Subset of (Euclid n) ;
A72: W3 c= W2
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in W3 or z in W2 )
assume z in W3 ; ::_thesis: z in W2
then consider q being Point of (TOP-REAL n) such that
A73: q = z and
A74: ex r being Real st
( q = <*r*> & r > a ) ;
consider r being Real such that
A75: q = <*r*> and
A76: r > a by A74;
A77: r = abs r by A2, A76, ABSVALUE:def_1;
n = 1 by A2, A50, XXREAL_0:1;
then reconsider xr = <*r*> as Element of REAL n by FINSEQ_2:131;
len xr = 1 by FINSEQ_1:39;
then A78: q /. 1 = xr . 1 by A75, FINSEQ_4:15;
then A79: (sqr xr) . 1 = (q /. 1) ^2 by VALUED_1:11;
len (sqr xr) = 1 by A51, CARD_1:def_7;
then A80: sqr xr = <*((q /. 1) ^2)*> by A79, FINSEQ_1:40;
sqrt ((q /. 1) ^2) = abs (q /. 1) by COMPLEX1:72
.= abs r by A78, FINSEQ_1:40 ;
then |.q.| = abs r by A75, A80, FINSOP_1:11;
hence z in W2 by A73, A76, A77; ::_thesis: verum
end;
A81: now__::_thesis:_W2_/\_A_=_{}
set z = the Element of W2 /\ A;
assume A82: not W2 /\ A = {} ; ::_thesis: contradiction
then the Element of W2 /\ A in W2 by XBOOLE_0:def_4;
then A83: ex q being Point of (TOP-REAL n) st
( q = the Element of W2 /\ A & |.q.| > a ) ;
the Element of W2 /\ A in A by A82, XBOOLE_0:def_4;
then ex q2 being Point of (TOP-REAL n) st
( q2 = the Element of W2 /\ A & |.q2.| = a ) by A2;
hence contradiction by A83; ::_thesis: verum
end;
then W3 /\ A = {} by A72, XBOOLE_1:3, XBOOLE_1:26;
then A84: W3 misses A by XBOOLE_0:def_7;
W3 /\ ((A `) `) = {} by A81, A72, XBOOLE_1:3, XBOOLE_1:26;
then W3 \ (A `) = {} by SUBSET_1:13;
then A85: W3 c= A ` by XBOOLE_1:37;
then A86: (TOP-REAL n) | P3 = ((TOP-REAL n) | (A `)) | Q by PRE_TOPC:7;
A87: P3 is convex by A51, Th55;
then (TOP-REAL n) | P3 is connected by CONNSP_1:def_3;
then Q is connected by A86, CONNSP_1:def_3;
then Q c= Component_of Q by CONNSP_3:1;
then A88: p in Component_of Q by A63;
A89: Q = Down (P3,(A `)) by A85, XBOOLE_1:28;
not W3 is bounded by A51, Th59;
then Up (Component_of Q) is_outside_component_of A by A87, A84, A89, Th63;
then A90: Component_of Q c= UBD A by Th23;
B c= BDD A by A55, Th22;
then (BDD A) /\ (UBD A) <> {} by A52, A54, A90, A88, XBOOLE_0:def_4;
then BDD A meets UBD A by XBOOLE_0:def_7;
hence x in P1 by Th24; ::_thesis: verum
end;
supposeA91: p in { q1 where q1 is Point of (TOP-REAL n) : ex r1 being Real st
( q1 = <*r1*> & r1 < - a ) } ; ::_thesis: x in P1
{ q where q is Point of (TOP-REAL n) : ex r being Real st
( q = <*r*> & r < - a ) } c= A `
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in { q where q is Point of (TOP-REAL n) : ex r being Real st
( q = <*r*> & r < - a ) } or z in A ` )
assume z in { q where q is Point of (TOP-REAL n) : ex r being Real st
( q = <*r*> & r < - a ) } ; ::_thesis: z in A `
then consider q being Point of (TOP-REAL n) such that
A92: q = z and
A93: ex r being Real st
( q = <*r*> & r < - a ) ;
consider r being Real such that
A94: q = <*r*> and
A95: r < - a by A93;
A96: r < - 0 by A2, A95;
n = 1 by A2, A50, XXREAL_0:1;
then reconsider xr = <*r*> as Element of REAL n by FINSEQ_2:131;
len xr = 1 by FINSEQ_1:39;
then A97: q /. 1 = xr . 1 by A94, FINSEQ_4:15;
then A98: (sqr xr) . 1 = (q /. 1) ^2 by VALUED_1:11;
len (sqr xr) = 1 by A51, CARD_1:def_7;
then A99: sqr xr = <*((q /. 1) ^2)*> by A98, FINSEQ_1:40;
sqrt ((q /. 1) ^2) = abs (q /. 1) by COMPLEX1:72
.= abs r by A97, FINSEQ_1:40 ;
then A100: |.q.| = abs r by A94, A99, FINSOP_1:11
.= - r by A96, ABSVALUE:def_1 ;
now__::_thesis:_not_q_in_A
assume q in A ; ::_thesis: contradiction
then ex q2 being Point of (TOP-REAL n) st
( q2 = q & |.q2.| = a ) by A2;
hence contradiction by A95, A100; ::_thesis: verum
end;
hence z in A ` by A92, XBOOLE_0:def_5; ::_thesis: verum
end;
then reconsider Q = { q where q is Point of (TOP-REAL n) : ex r being Real st
( q = <*r*> & r < - a ) } as Subset of ((TOP-REAL n) | (A `)) by PRE_TOPC:8;
{ q where q is Point of (TOP-REAL n) : ex r being Real st
( q = <*r*> & r < - a ) } c= the carrier of (TOP-REAL n)
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in { q where q is Point of (TOP-REAL n) : ex r being Real st
( q = <*r*> & r < - a ) } or z in the carrier of (TOP-REAL n) )
assume z in { q where q is Point of (TOP-REAL n) : ex r being Real st
( q = <*r*> & r < - a ) } ; ::_thesis: z in the carrier of (TOP-REAL n)
then ex q being Point of (TOP-REAL n) st
( q = z & ex r being Real st
( q = <*r*> & r < - a ) ) ;
hence z in the carrier of (TOP-REAL n) ; ::_thesis: verum
end;
then reconsider P3 = { q where q is Point of (TOP-REAL n) : ex r being Real st
( q = <*r*> & r < - a ) } as Subset of (TOP-REAL n) ;
reconsider W3 = P3 as Subset of (Euclid n) by TOPREAL3:8;
reconsider Q = Q as Subset of ((TOP-REAL n) | (A `)) ;
{ q where q is Point of (TOP-REAL n) : |.q.| > a } c= the carrier of (TOP-REAL n)
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in { q where q is Point of (TOP-REAL n) : |.q.| > a } or z in the carrier of (TOP-REAL n) )
assume z in { q where q is Point of (TOP-REAL n) : |.q.| > a } ; ::_thesis: z in the carrier of (TOP-REAL n)
then ex q being Point of (TOP-REAL n) st
( q = z & |.q.| > a ) ;
hence z in the carrier of (TOP-REAL n) ; ::_thesis: verum
end;
then reconsider P2 = { q where q is Point of (TOP-REAL n) : |.q.| > a } as Subset of (TOP-REAL n) ;
P2 is Subset of (Euclid n) by TOPREAL3:8;
then reconsider W2 = { q where q is Point of (TOP-REAL n) : |.q.| > a } as Subset of (Euclid n) ;
A101: W3 c= W2
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in W3 or z in W2 )
assume z in W3 ; ::_thesis: z in W2
then consider q being Point of (TOP-REAL n) such that
A102: q = z and
A103: ex r being Real st
( q = <*r*> & r < - a ) ;
consider r being Real such that
A104: q = <*r*> and
A105: r < - a by A103;
A106: ( r < - 0 & - r > - (- a) ) by A2, A105, XREAL_1:24;
n = 1 by A2, A50, XXREAL_0:1;
then reconsider xr = <*r*> as Element of REAL n by FINSEQ_2:131;
len xr = 1 by FINSEQ_1:39;
then A107: q /. 1 = xr . 1 by A104, FINSEQ_4:15;
then A108: (sqr xr) . 1 = (q /. 1) ^2 by VALUED_1:11;
len (sqr xr) = 1 by A51, CARD_1:def_7;
then A109: sqr xr = <*((q /. 1) ^2)*> by A108, FINSEQ_1:40;
sqrt ((q /. 1) ^2) = abs (q /. 1) by COMPLEX1:72
.= abs r by A107, FINSEQ_1:40 ;
then |.q.| = abs r by A104, A109, FINSOP_1:11;
then |.q.| > a by A106, ABSVALUE:def_1;
hence z in W2 by A102; ::_thesis: verum
end;
A110: now__::_thesis:_W2_/\_A_=_{}
set z = the Element of W2 /\ A;
assume A111: not W2 /\ A = {} ; ::_thesis: contradiction
then the Element of W2 /\ A in W2 by XBOOLE_0:def_4;
then A112: ex q being Point of (TOP-REAL n) st
( q = the Element of W2 /\ A & |.q.| > a ) ;
the Element of W2 /\ A in A by A111, XBOOLE_0:def_4;
then ex q2 being Point of (TOP-REAL n) st
( q2 = the Element of W2 /\ A & |.q2.| = a ) by A2;
hence contradiction by A112; ::_thesis: verum
end;
then W3 /\ A = {} by A101, XBOOLE_1:3, XBOOLE_1:26;
then A113: W3 misses A by XBOOLE_0:def_7;
W3 /\ ((A `) `) = {} by A110, A101, XBOOLE_1:3, XBOOLE_1:26;
then W3 \ (A `) = {} by SUBSET_1:13;
then A114: W3 c= A ` by XBOOLE_1:37;
then A115: (TOP-REAL n) | P3 = ((TOP-REAL n) | (A `)) | Q by PRE_TOPC:7;
A116: P3 is convex by A51, Th56;
then (TOP-REAL n) | P3 is connected by CONNSP_1:def_3;
then Q is connected by A115, CONNSP_1:def_3;
then Q c= Component_of Q by CONNSP_3:1;
then A117: p in Component_of Q by A91;
A118: Q = Down (P3,(A `)) by A114, XBOOLE_1:28;
Up (Component_of Q) is_outside_component_of A by A116, A113, A118, Th63, A51, Th60;
then A119: Component_of Q c= UBD A by Th23;
B c= BDD A by A55, Th22;
then p in (BDD A) /\ (UBD A) by A52, A54, A119, A117, XBOOLE_0:def_4;
then BDD A meets UBD A by XBOOLE_0:def_7;
hence x in P1 by Th24; ::_thesis: verum
end;
end;
end;
hence x in P1 ; ::_thesis: verum
end;
caseA120: p in { q1 where q1 is Point of (TOP-REAL n) : |.q1.| < a } ; ::_thesis: x in P1
Down (P,(A `)) c= Component_of (Down (P,(A `))) by A7, CONNSP_3:1;
hence x in P1 by A6, A120; ::_thesis: verum
end;
end;
end;
hence x in P1 ; ::_thesis: verum
end;
then P1 = union { B where B is Subset of (TOP-REAL n) : B is_inside_component_of A } by A30, XBOOLE_0:def_10;
hence ex B being Subset of (TOP-REAL n) st
( B is_inside_component_of A & B = BDD A ) by A29; ::_thesis: verum
end;
end;
end;
hence BDD A is_inside_component_of A ; ::_thesis: verum
end;
begin
theorem Th89: :: JORDAN2C:89
for D being non empty compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( len (GoB (SpStSeq D)) = 2 & width (GoB (SpStSeq D)) = 2 & (SpStSeq D) /. 1 = (GoB (SpStSeq D)) * (1,2) & (SpStSeq D) /. 2 = (GoB (SpStSeq D)) * (2,2) & (SpStSeq D) /. 3 = (GoB (SpStSeq D)) * (2,1) & (SpStSeq D) /. 4 = (GoB (SpStSeq D)) * (1,1) & (SpStSeq D) /. 5 = (GoB (SpStSeq D)) * (1,2) )
proof
let D be non empty compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( len (GoB (SpStSeq D)) = 2 & width (GoB (SpStSeq D)) = 2 & (SpStSeq D) /. 1 = (GoB (SpStSeq D)) * (1,2) & (SpStSeq D) /. 2 = (GoB (SpStSeq D)) * (2,2) & (SpStSeq D) /. 3 = (GoB (SpStSeq D)) * (2,1) & (SpStSeq D) /. 4 = (GoB (SpStSeq D)) * (1,1) & (SpStSeq D) /. 5 = (GoB (SpStSeq D)) * (1,2) )
set f = SpStSeq D;
A1: S-bound (L~ (SpStSeq D)) < N-bound (L~ (SpStSeq D)) by SPRECT_1:32;
A2: len (SpStSeq D) = 5 by SPRECT_1:82;
then A3: (SpStSeq D) /. 5 = (SpStSeq D) /. 1 by FINSEQ_6:def_1;
4 in Seg (len (SpStSeq D)) by A2, FINSEQ_1:1;
then A4: 4 in dom (SpStSeq D) by FINSEQ_1:def_3;
then 4 in dom (X_axis (SpStSeq D)) by SPRECT_2:15;
then ( (SpStSeq D) /. 4 = W-min (L~ (SpStSeq D)) & (X_axis (SpStSeq D)) . 4 = ((SpStSeq D) /. 4) `1 ) by GOBOARD1:def_1, SPRECT_1:86;
then A5: (X_axis (SpStSeq D)) . 4 = W-bound (L~ (SpStSeq D)) by EUCLID:52;
A6: (SpStSeq D) /. 3 = S-max (L~ (SpStSeq D)) by SPRECT_1:85;
3 in Seg (len (SpStSeq D)) by A2, FINSEQ_1:1;
then A7: 3 in dom (SpStSeq D) by FINSEQ_1:def_3;
then 3 in dom (X_axis (SpStSeq D)) by SPRECT_2:15;
then ( (SpStSeq D) /. 3 = E-min (L~ (SpStSeq D)) & (X_axis (SpStSeq D)) . 3 = ((SpStSeq D) /. 3) `1 ) by GOBOARD1:def_1, SPRECT_1:85;
then A8: (X_axis (SpStSeq D)) . 3 = E-bound (L~ (SpStSeq D)) by EUCLID:52;
A9: (SpStSeq D) /. (1 + 1) = N-max (L~ (SpStSeq D)) by SPRECT_1:84;
3 in dom (Y_axis (SpStSeq D)) by A7, SPRECT_2:16;
then (Y_axis (SpStSeq D)) . 3 = ((SpStSeq D) /. 3) `2 by GOBOARD1:def_2;
then A10: (Y_axis (SpStSeq D)) . 3 = S-bound (L~ (SpStSeq D)) by A6, EUCLID:52;
A11: (SpStSeq D) /. 1 = N-min (L~ (SpStSeq D)) by SPRECT_1:83;
1 in Seg (len (SpStSeq D)) by A2, FINSEQ_1:1;
then A12: 1 in dom (SpStSeq D) by FINSEQ_1:def_3;
then 1 in dom (Y_axis (SpStSeq D)) by SPRECT_2:16;
then (Y_axis (SpStSeq D)) . 1 = ((SpStSeq D) /. 1) `2 by GOBOARD1:def_2;
then A13: (Y_axis (SpStSeq D)) . 1 = N-bound (L~ (SpStSeq D)) by A11, EUCLID:52;
A14: (SpStSeq D) /. 4 = S-min (L~ (SpStSeq D)) by SPRECT_1:86;
2 in Seg (len (SpStSeq D)) by A2, FINSEQ_1:1;
then A15: 2 in dom (SpStSeq D) by FINSEQ_1:def_3;
then 2 in dom (X_axis (SpStSeq D)) by SPRECT_2:15;
then ( (SpStSeq D) /. (1 + 1) = E-max (L~ (SpStSeq D)) & (X_axis (SpStSeq D)) . 2 = ((SpStSeq D) /. 2) `1 ) by GOBOARD1:def_1, SPRECT_1:84;
then A16: (X_axis (SpStSeq D)) . 2 = E-bound (L~ (SpStSeq D)) by EUCLID:52;
4 in dom (Y_axis (SpStSeq D)) by A4, SPRECT_2:16;
then (Y_axis (SpStSeq D)) . 4 = ((SpStSeq D) /. 4) `2 by GOBOARD1:def_2;
then A17: (Y_axis (SpStSeq D)) . 4 = S-bound (L~ (SpStSeq D)) by A14, EUCLID:52;
2 in dom (Y_axis (SpStSeq D)) by A15, SPRECT_2:16;
then (Y_axis (SpStSeq D)) . 2 = ((SpStSeq D) /. 2) `2 by GOBOARD1:def_2;
then A18: (Y_axis (SpStSeq D)) . 2 = N-bound (L~ (SpStSeq D)) by A9, EUCLID:52;
A19: {(S-bound (L~ (SpStSeq D))),(N-bound (L~ (SpStSeq D)))} c= rng (Y_axis (SpStSeq D))
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in {(S-bound (L~ (SpStSeq D))),(N-bound (L~ (SpStSeq D)))} or z in rng (Y_axis (SpStSeq D)) )
assume A20: z in {(S-bound (L~ (SpStSeq D))),(N-bound (L~ (SpStSeq D)))} ; ::_thesis: z in rng (Y_axis (SpStSeq D))
now__::_thesis:_(_(_z_=_S-bound_(L~_(SpStSeq_D))_&_z_in_rng_(Y_axis_(SpStSeq_D))_)_or_(_z_=_N-bound_(L~_(SpStSeq_D))_&_z_in_rng_(Y_axis_(SpStSeq_D))_)_)
percases ( z = S-bound (L~ (SpStSeq D)) or z = N-bound (L~ (SpStSeq D)) ) by A20, TARSKI:def_2;
caseA21: z = S-bound (L~ (SpStSeq D)) ; ::_thesis: z in rng (Y_axis (SpStSeq D))
4 in dom (Y_axis (SpStSeq D)) by A4, SPRECT_2:16;
hence z in rng (Y_axis (SpStSeq D)) by A17, A21, FUNCT_1:def_3; ::_thesis: verum
end;
caseA22: z = N-bound (L~ (SpStSeq D)) ; ::_thesis: z in rng (Y_axis (SpStSeq D))
2 in dom (Y_axis (SpStSeq D)) by A15, SPRECT_2:16;
hence z in rng (Y_axis (SpStSeq D)) by A18, A22, FUNCT_1:def_3; ::_thesis: verum
end;
end;
end;
hence z in rng (Y_axis (SpStSeq D)) ; ::_thesis: verum
end;
A23: (SpStSeq D) /. 1 = W-max (L~ (SpStSeq D)) by SPRECT_1:83;
1 in dom (X_axis (SpStSeq D)) by A12, SPRECT_2:15;
then (X_axis (SpStSeq D)) . 1 = ((SpStSeq D) /. 1) `1 by GOBOARD1:def_1;
then A24: (X_axis (SpStSeq D)) . 1 = W-bound (L~ (SpStSeq D)) by A23, EUCLID:52;
A25: {(W-bound (L~ (SpStSeq D))),(E-bound (L~ (SpStSeq D)))} c= rng (X_axis (SpStSeq D))
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in {(W-bound (L~ (SpStSeq D))),(E-bound (L~ (SpStSeq D)))} or z in rng (X_axis (SpStSeq D)) )
assume A26: z in {(W-bound (L~ (SpStSeq D))),(E-bound (L~ (SpStSeq D)))} ; ::_thesis: z in rng (X_axis (SpStSeq D))
now__::_thesis:_(_(_z_=_W-bound_(L~_(SpStSeq_D))_&_z_in_rng_(X_axis_(SpStSeq_D))_)_or_(_z_=_E-bound_(L~_(SpStSeq_D))_&_z_in_rng_(X_axis_(SpStSeq_D))_)_)
percases ( z = W-bound (L~ (SpStSeq D)) or z = E-bound (L~ (SpStSeq D)) ) by A26, TARSKI:def_2;
caseA27: z = W-bound (L~ (SpStSeq D)) ; ::_thesis: z in rng (X_axis (SpStSeq D))
1 in dom (X_axis (SpStSeq D)) by A12, SPRECT_2:15;
hence z in rng (X_axis (SpStSeq D)) by A24, A27, FUNCT_1:def_3; ::_thesis: verum
end;
caseA28: z = E-bound (L~ (SpStSeq D)) ; ::_thesis: z in rng (X_axis (SpStSeq D))
2 in dom (X_axis (SpStSeq D)) by A15, SPRECT_2:15;
hence z in rng (X_axis (SpStSeq D)) by A16, A28, FUNCT_1:def_3; ::_thesis: verum
end;
end;
end;
hence z in rng (X_axis (SpStSeq D)) ; ::_thesis: verum
end;
A29: GoB (SpStSeq D) = GoB ((Incr (X_axis (SpStSeq D))),(Incr (Y_axis (SpStSeq D)))) by GOBOARD2:def_2;
5 in Seg (len (SpStSeq D)) by A2, FINSEQ_1:1;
then A30: 5 in dom (SpStSeq D) by FINSEQ_1:def_3;
then 5 in dom (X_axis (SpStSeq D)) by SPRECT_2:15;
then (X_axis (SpStSeq D)) . 5 = ((SpStSeq D) /. 5) `1 by GOBOARD1:def_1;
then A31: (X_axis (SpStSeq D)) . 5 = W-bound (L~ (SpStSeq D)) by A23, A3, EUCLID:52;
rng (X_axis (SpStSeq D)) c= {(W-bound (L~ (SpStSeq D))),(E-bound (L~ (SpStSeq D)))}
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in rng (X_axis (SpStSeq D)) or z in {(W-bound (L~ (SpStSeq D))),(E-bound (L~ (SpStSeq D)))} )
assume z in rng (X_axis (SpStSeq D)) ; ::_thesis: z in {(W-bound (L~ (SpStSeq D))),(E-bound (L~ (SpStSeq D)))}
then consider u being set such that
A32: u in dom (X_axis (SpStSeq D)) and
A33: z = (X_axis (SpStSeq D)) . u by FUNCT_1:def_3;
reconsider mu = u as Element of NAT by A32;
u in dom (SpStSeq D) by A32, SPRECT_2:15;
then u in Seg (len (SpStSeq D)) by FINSEQ_1:def_3;
then ( 1 <= mu & mu <= 5 ) by A2, FINSEQ_1:1;
then A34: ( mu = 1 or mu = 1 + 1 or mu = 1 + 2 or mu = 1 + 3 or mu = 1 + 4 ) by NAT_1:58;
percases ( mu = 1 or mu = 2 or mu = 3 or mu = 4 or mu = 5 ) by A34;
suppose mu = 1 ; ::_thesis: z in {(W-bound (L~ (SpStSeq D))),(E-bound (L~ (SpStSeq D)))}
hence z in {(W-bound (L~ (SpStSeq D))),(E-bound (L~ (SpStSeq D)))} by A24, A33, TARSKI:def_2; ::_thesis: verum
end;
suppose mu = 2 ; ::_thesis: z in {(W-bound (L~ (SpStSeq D))),(E-bound (L~ (SpStSeq D)))}
hence z in {(W-bound (L~ (SpStSeq D))),(E-bound (L~ (SpStSeq D)))} by A16, A33, TARSKI:def_2; ::_thesis: verum
end;
suppose mu = 3 ; ::_thesis: z in {(W-bound (L~ (SpStSeq D))),(E-bound (L~ (SpStSeq D)))}
hence z in {(W-bound (L~ (SpStSeq D))),(E-bound (L~ (SpStSeq D)))} by A8, A33, TARSKI:def_2; ::_thesis: verum
end;
suppose mu = 4 ; ::_thesis: z in {(W-bound (L~ (SpStSeq D))),(E-bound (L~ (SpStSeq D)))}
hence z in {(W-bound (L~ (SpStSeq D))),(E-bound (L~ (SpStSeq D)))} by A5, A33, TARSKI:def_2; ::_thesis: verum
end;
suppose mu = 5 ; ::_thesis: z in {(W-bound (L~ (SpStSeq D))),(E-bound (L~ (SpStSeq D)))}
hence z in {(W-bound (L~ (SpStSeq D))),(E-bound (L~ (SpStSeq D)))} by A31, A33, TARSKI:def_2; ::_thesis: verum
end;
end;
end;
then A35: rng (X_axis (SpStSeq D)) = {(W-bound (L~ (SpStSeq D))),(E-bound (L~ (SpStSeq D)))} by A25, XBOOLE_0:def_10;
then A36: rng (Incr (X_axis (SpStSeq D))) = {(W-bound (L~ (SpStSeq D))),(E-bound (L~ (SpStSeq D)))} by SEQ_4:def_21;
5 in dom (Y_axis (SpStSeq D)) by A30, SPRECT_2:16;
then (Y_axis (SpStSeq D)) . 5 = ((SpStSeq D) /. 5) `2 by GOBOARD1:def_2;
then A37: (Y_axis (SpStSeq D)) . 5 = N-bound (L~ (SpStSeq D)) by A11, A3, EUCLID:52;
rng (Y_axis (SpStSeq D)) c= {(S-bound (L~ (SpStSeq D))),(N-bound (L~ (SpStSeq D)))}
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in rng (Y_axis (SpStSeq D)) or z in {(S-bound (L~ (SpStSeq D))),(N-bound (L~ (SpStSeq D)))} )
assume z in rng (Y_axis (SpStSeq D)) ; ::_thesis: z in {(S-bound (L~ (SpStSeq D))),(N-bound (L~ (SpStSeq D)))}
then consider u being set such that
A38: u in dom (Y_axis (SpStSeq D)) and
A39: z = (Y_axis (SpStSeq D)) . u by FUNCT_1:def_3;
reconsider mu = u as Element of NAT by A38;
u in dom (SpStSeq D) by A38, SPRECT_2:16;
then u in Seg (len (SpStSeq D)) by FINSEQ_1:def_3;
then ( 1 <= mu & mu <= 5 ) by A2, FINSEQ_1:1;
then A40: ( mu = 1 or mu = 1 + 1 or mu = 1 + 2 or mu = 1 + 3 or mu = 1 + 4 ) by NAT_1:58;
percases ( mu = 1 or mu = 2 or mu = 3 or mu = 4 or mu = 5 ) by A40;
suppose mu = 1 ; ::_thesis: z in {(S-bound (L~ (SpStSeq D))),(N-bound (L~ (SpStSeq D)))}
hence z in {(S-bound (L~ (SpStSeq D))),(N-bound (L~ (SpStSeq D)))} by A13, A39, TARSKI:def_2; ::_thesis: verum
end;
suppose mu = 2 ; ::_thesis: z in {(S-bound (L~ (SpStSeq D))),(N-bound (L~ (SpStSeq D)))}
hence z in {(S-bound (L~ (SpStSeq D))),(N-bound (L~ (SpStSeq D)))} by A18, A39, TARSKI:def_2; ::_thesis: verum
end;
suppose mu = 3 ; ::_thesis: z in {(S-bound (L~ (SpStSeq D))),(N-bound (L~ (SpStSeq D)))}
hence z in {(S-bound (L~ (SpStSeq D))),(N-bound (L~ (SpStSeq D)))} by A10, A39, TARSKI:def_2; ::_thesis: verum
end;
suppose mu = 4 ; ::_thesis: z in {(S-bound (L~ (SpStSeq D))),(N-bound (L~ (SpStSeq D)))}
hence z in {(S-bound (L~ (SpStSeq D))),(N-bound (L~ (SpStSeq D)))} by A17, A39, TARSKI:def_2; ::_thesis: verum
end;
suppose mu = 5 ; ::_thesis: z in {(S-bound (L~ (SpStSeq D))),(N-bound (L~ (SpStSeq D)))}
hence z in {(S-bound (L~ (SpStSeq D))),(N-bound (L~ (SpStSeq D)))} by A37, A39, TARSKI:def_2; ::_thesis: verum
end;
end;
end;
then A41: rng (Y_axis (SpStSeq D)) = {(S-bound (L~ (SpStSeq D))),(N-bound (L~ (SpStSeq D)))} by A19, XBOOLE_0:def_10;
then card (rng (Y_axis (SpStSeq D))) = 2 by A1, CARD_2:57;
then A42: len (Incr (Y_axis (SpStSeq D))) = 2 by SEQ_4:def_21;
A43: W-bound (L~ (SpStSeq D)) < E-bound (L~ (SpStSeq D)) by SPRECT_1:31;
then A44: card (rng (X_axis (SpStSeq D))) = 2 by A35, CARD_2:57;
then A45: len (Incr (X_axis (SpStSeq D))) = 2 by SEQ_4:def_21;
A46: len (GoB (SpStSeq D)) = card (rng (X_axis (SpStSeq D))) by GOBOARD2:13
.= 1 + 1 by A43, A35, CARD_2:57 ;
then A47: 1 in Seg (len (GoB (SpStSeq D))) by FINSEQ_1:1;
A48: width (GoB (SpStSeq D)) = card (rng (Y_axis (SpStSeq D))) by GOBOARD2:13
.= 1 + 1 by A1, A41, CARD_2:57 ;
for p being FinSequence of the carrier of (TOP-REAL 2) st p in rng (GoB (SpStSeq D)) holds
len p = 2
proof
len (GoB ((Incr (X_axis (SpStSeq D))),(Incr (Y_axis (SpStSeq D))))) = len (Incr (X_axis (SpStSeq D))) by GOBOARD2:def_1
.= 2 by A44, SEQ_4:def_21 ;
then consider s1 being FinSequence such that
A49: s1 in rng (GoB ((Incr (X_axis (SpStSeq D))),(Incr (Y_axis (SpStSeq D))))) and
A50: len s1 = width (GoB ((Incr (X_axis (SpStSeq D))),(Incr (Y_axis (SpStSeq D))))) by MATRIX_1:def_3;
let p be FinSequence of the carrier of (TOP-REAL 2); ::_thesis: ( p in rng (GoB (SpStSeq D)) implies len p = 2 )
consider n being Nat such that
A51: for x being set st x in rng (GoB (SpStSeq D)) holds
ex s being FinSequence st
( s = x & len s = n ) by MATRIX_1:def_1;
assume p in rng (GoB (SpStSeq D)) ; ::_thesis: len p = 2
then A52: ex s2 being FinSequence st
( s2 = p & len s2 = n ) by A51;
s1 in rng (GoB (SpStSeq D)) by A49, GOBOARD2:def_2;
then ex s being FinSequence st
( s = s1 & len s = n ) by A51;
hence len p = 2 by A48, A50, A52, GOBOARD2:def_2; ::_thesis: verum
end;
then A53: GoB (SpStSeq D) is Matrix of 2,2, the carrier of (TOP-REAL 2) by A46, MATRIX_1:def_2;
A54: 1 in Seg (width (GoB (SpStSeq D))) by A48, FINSEQ_1:1;
then [1,1] in [:(Seg (len (GoB (SpStSeq D)))),(Seg (width (GoB (SpStSeq D)))):] by A47, ZFMISC_1:87;
then A55: [1,1] in Indices (GoB (SpStSeq D)) by A46, A48, A53, MATRIX_1:24;
A56: width (GoB (SpStSeq D)) in Seg (width (GoB (SpStSeq D))) by A48, FINSEQ_1:1;
then [1,(width (GoB (SpStSeq D)))] in [:(Seg (len (GoB (SpStSeq D)))),(Seg (width (GoB (SpStSeq D)))):] by A47, ZFMISC_1:87;
then A57: [1,(width (GoB (SpStSeq D)))] in Indices (GoB (SpStSeq D)) by A46, A48, A53, MATRIX_1:24;
A58: len (GoB (SpStSeq D)) in Seg (len (GoB (SpStSeq D))) by A46, FINSEQ_1:1;
then [(len (GoB (SpStSeq D))),1] in [:(Seg (len (GoB (SpStSeq D)))),(Seg (width (GoB (SpStSeq D)))):] by A54, ZFMISC_1:87;
then A59: [(len (GoB (SpStSeq D))),1] in Indices (GoB (SpStSeq D)) by A46, A48, A53, MATRIX_1:24;
(S-max (L~ (SpStSeq D))) `1 = (SE-corner D) `1 by SPRECT_1:81
.= E-bound D by EUCLID:52
.= E-bound (L~ (SpStSeq D)) by SPRECT_1:61
.= (Incr (X_axis (SpStSeq D))) . 2 by A43, A36, A45, Th6 ;
then (S-max (L~ (SpStSeq D))) `1 = |[((Incr (X_axis (SpStSeq D))) . 2),((Incr (Y_axis (SpStSeq D))) . 1)]| `1 by EUCLID:52;
then A60: (S-max (L~ (SpStSeq D))) `1 = ((GoB (SpStSeq D)) * ((len (GoB (SpStSeq D))),1)) `1 by A29, A46, A59, GOBOARD2:def_1;
(S-min (L~ (SpStSeq D))) `1 = (SW-corner D) `1 by SPRECT_1:80
.= W-bound D by EUCLID:52
.= W-bound (L~ (SpStSeq D)) by SPRECT_1:58
.= (Incr (X_axis (SpStSeq D))) . 1 by A43, A36, A45, Th6 ;
then (S-min (L~ (SpStSeq D))) `1 = |[((Incr (X_axis (SpStSeq D))) . 1),((Incr (Y_axis (SpStSeq D))) . 1)]| `1 by EUCLID:52;
then A61: (S-min (L~ (SpStSeq D))) `1 = ((GoB (SpStSeq D)) * (1,1)) `1 by A29, A55, GOBOARD2:def_1;
[(len (GoB (SpStSeq D))),(width (GoB (SpStSeq D)))] in [:(Seg (len (GoB (SpStSeq D)))),(Seg (width (GoB (SpStSeq D)))):] by A58, A56, ZFMISC_1:87;
then A62: [(len (GoB (SpStSeq D))),(width (GoB (SpStSeq D)))] in Indices (GoB (SpStSeq D)) by A46, A48, A53, MATRIX_1:24;
W-bound (L~ (SpStSeq D)) = (Incr (X_axis (SpStSeq D))) . 1 by A43, A36, A45, Th6;
then (W-max (L~ (SpStSeq D))) `1 = (Incr (X_axis (SpStSeq D))) . 1 by EUCLID:52;
then (W-max (L~ (SpStSeq D))) `1 = |[((Incr (X_axis (SpStSeq D))) . 1),((Incr (Y_axis (SpStSeq D))) . (1 + 1))]| `1 by EUCLID:52;
then A63: (W-max (L~ (SpStSeq D))) `1 = ((GoB (SpStSeq D)) * (1,(width (GoB (SpStSeq D))))) `1 by A29, A48, A57, GOBOARD2:def_1;
A64: ( (SpStSeq D) /. 3 = |[(((SpStSeq D) /. 3) `1),(((SpStSeq D) /. 3) `2)]| & (SpStSeq D) /. 4 = |[(((SpStSeq D) /. 4) `1),(((SpStSeq D) /. 4) `2)]| ) by EUCLID:53;
A65: ( (SpStSeq D) /. 1 = |[(((SpStSeq D) /. 1) `1),(((SpStSeq D) /. 1) `2)]| & (SpStSeq D) /. (1 + 1) = |[(((SpStSeq D) /. (1 + 1)) `1),(((SpStSeq D) /. (1 + 1)) `2)]| ) by EUCLID:53;
A66: rng (Incr (Y_axis (SpStSeq D))) = {(S-bound (L~ (SpStSeq D))),(N-bound (L~ (SpStSeq D)))} by A41, SEQ_4:def_21;
then A67: N-bound (L~ (SpStSeq D)) = (Incr (Y_axis (SpStSeq D))) . 2 by A1, A42, Th6;
then (N-min (L~ (SpStSeq D))) `2 = (Incr (Y_axis (SpStSeq D))) . 2 by EUCLID:52;
then (N-min (L~ (SpStSeq D))) `2 = |[((Incr (X_axis (SpStSeq D))) . 1),((Incr (Y_axis (SpStSeq D))) . 2)]| `2 by EUCLID:52;
then A68: (N-min (L~ (SpStSeq D))) `2 = ((GoB (SpStSeq D)) * (1,(width (GoB (SpStSeq D))))) `2 by A29, A48, A57, GOBOARD2:def_1;
A69: S-bound (L~ (SpStSeq D)) = (Incr (Y_axis (SpStSeq D))) . 1 by A1, A66, A42, Th6;
then (S-min (L~ (SpStSeq D))) `2 = (Incr (Y_axis (SpStSeq D))) . 1 by EUCLID:52;
then (S-min (L~ (SpStSeq D))) `2 = |[((Incr (X_axis (SpStSeq D))) . 1),((Incr (Y_axis (SpStSeq D))) . 1)]| `2 by EUCLID:52;
then A70: (S-min (L~ (SpStSeq D))) `2 = ((GoB (SpStSeq D)) * (1,1)) `2 by A29, A55, GOBOARD2:def_1;
(N-max (L~ (SpStSeq D))) `2 = N-bound (L~ (SpStSeq D)) by EUCLID:52;
then (N-max (L~ (SpStSeq D))) `2 = |[((Incr (X_axis (SpStSeq D))) . 2),((Incr (Y_axis (SpStSeq D))) . 2)]| `2 by A67, EUCLID:52;
then A71: (N-max (L~ (SpStSeq D))) `2 = ((GoB (SpStSeq D)) * ((len (GoB (SpStSeq D))),(width (GoB (SpStSeq D))))) `2 by A29, A46, A48, A62, GOBOARD2:def_1;
(S-max (L~ (SpStSeq D))) `2 = (Incr (Y_axis (SpStSeq D))) . 1 by A69, EUCLID:52;
then (S-max (L~ (SpStSeq D))) `2 = |[((Incr (X_axis (SpStSeq D))) . 2),((Incr (Y_axis (SpStSeq D))) . 1)]| `2 by EUCLID:52;
then A72: (S-max (L~ (SpStSeq D))) `2 = ((GoB (SpStSeq D)) * ((len (GoB (SpStSeq D))),1)) `2 by A29, A46, A59, GOBOARD2:def_1;
(N-max (L~ (SpStSeq D))) `1 = (NE-corner D) `1 by SPRECT_1:77
.= E-bound D by EUCLID:52
.= E-bound (L~ (SpStSeq D)) by SPRECT_1:61
.= (Incr (X_axis (SpStSeq D))) . 2 by A43, A36, A45, Th6 ;
then (N-max (L~ (SpStSeq D))) `1 = |[((Incr (X_axis (SpStSeq D))) . (1 + 1)),((Incr (Y_axis (SpStSeq D))) . (1 + 1))]| `1 by EUCLID:52;
then (N-max (L~ (SpStSeq D))) `1 = ((GoB (SpStSeq D)) * ((len (GoB (SpStSeq D))),(width (GoB (SpStSeq D))))) `1 by A29, A46, A48, A62, GOBOARD2:def_1;
hence ( len (GoB (SpStSeq D)) = 2 & width (GoB (SpStSeq D)) = 2 & (SpStSeq D) /. 1 = (GoB (SpStSeq D)) * (1,2) & (SpStSeq D) /. 2 = (GoB (SpStSeq D)) * (2,2) & (SpStSeq D) /. 3 = (GoB (SpStSeq D)) * (2,1) & (SpStSeq D) /. 4 = (GoB (SpStSeq D)) * (1,1) & (SpStSeq D) /. 5 = (GoB (SpStSeq D)) * (1,2) ) by A11, A23, A9, A6, A14, A3, A46, A48, A63, A60, A61, A68, A71, A72, A70, A65, A64, EUCLID:53; ::_thesis: verum
end;
theorem Th90: :: JORDAN2C:90
for D being non empty compact non horizontal non vertical Subset of (TOP-REAL 2) holds not LeftComp (SpStSeq D) is bounded
proof
let D be non empty compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: not LeftComp (SpStSeq D) is bounded
set f = SpStSeq D;
set q3 = the Element of LeftComp (SpStSeq D);
reconsider q4 = the Element of LeftComp (SpStSeq D) as Point of (TOP-REAL 2) ;
set r1 = |.((1 / 2) * (((SpStSeq D) /. 1) + ((SpStSeq D) /. 2))).|;
reconsider f1 = SpStSeq D as non constant standard special_circular_sequence ;
A1: W-bound (L~ f1) < E-bound (L~ f1) by SPRECT_1:31;
A2: (N-min (L~ f1)) `2 = N-bound (L~ f1) by EUCLID:52;
then ((N-min (L~ f1)) `2) + ((N-max (L~ f1)) `2) = (N-bound (L~ f1)) + (N-bound (L~ f1)) by EUCLID:52
.= 2 * (N-bound (L~ (SpStSeq D))) ;
then A3: (1 / 2) * (((N-min (L~ (SpStSeq D))) `2) + ((N-max (L~ (SpStSeq D))) `2)) = N-bound (L~ (SpStSeq D)) ;
A4: len f1 = 5 by SPRECT_1:82;
then 5 in Seg (len f1) by FINSEQ_1:1;
then A5: 5 in dom f1 by FINSEQ_1:def_3;
then 5 in dom (Y_axis f1) by SPRECT_2:16;
then A6: (Y_axis f1) . 5 = (f1 /. 5) `2 by GOBOARD1:def_2;
4 in Seg (len f1) by A4, FINSEQ_1:1;
then A7: 4 in dom f1 by FINSEQ_1:def_3;
then 4 in dom (Y_axis f1) by SPRECT_2:16;
then ( f1 /. 4 = S-min (L~ f1) & (Y_axis f1) . 4 = (f1 /. 4) `2 ) by GOBOARD1:def_2, SPRECT_1:86;
then A8: (Y_axis f1) . 4 = S-bound (L~ f1) by EUCLID:52;
3 in Seg (len f1) by A4, FINSEQ_1:1;
then A9: 3 in dom f1 by FINSEQ_1:def_3;
then 3 in dom (Y_axis f1) by SPRECT_2:16;
then ( f1 /. 3 = S-max (L~ f1) & (Y_axis f1) . 3 = (f1 /. 3) `2 ) by GOBOARD1:def_2, SPRECT_1:85;
then A10: (Y_axis f1) . 3 = S-bound (L~ f1) by EUCLID:52;
3 in dom (X_axis f1) by A9, SPRECT_2:15;
then ( f1 /. 3 = E-min (L~ f1) & (X_axis f1) . 3 = (f1 /. 3) `1 ) by GOBOARD1:def_1, SPRECT_1:85;
then A11: (X_axis f1) . 3 = E-bound (L~ f1) by EUCLID:52;
5 in dom (X_axis f1) by A5, SPRECT_2:15;
then A12: (X_axis f1) . 5 = (f1 /. 5) `1 by GOBOARD1:def_1;
assume LeftComp (SpStSeq D) is bounded ; ::_thesis: contradiction
then consider r being Real such that
A13: for q being Point of (TOP-REAL 2) st q in LeftComp (SpStSeq D) holds
|.q.| < r by Th34;
set q1 = |[0,((|.((1 / 2) * (((SpStSeq D) /. 1) + ((SpStSeq D) /. 2))).| + r) + 1)]| + ((1 / 2) * (((SpStSeq D) /. 1) + ((SpStSeq D) /. 2)));
A14: f1 /. 1 = N-min (L~ f1) by SPRECT_1:83;
4 in dom (X_axis f1) by A7, SPRECT_2:15;
then ( f1 /. 4 = W-min (L~ f1) & (X_axis f1) . 4 = (f1 /. 4) `1 ) by GOBOARD1:def_1, SPRECT_1:86;
then A15: (X_axis f1) . 4 = W-bound (L~ f1) by EUCLID:52;
A16: GoB f1 = GoB ((Incr (X_axis f1)),(Incr (Y_axis f1))) by GOBOARD2:def_2;
A17: f1 /. 2 = E-max (L~ f1) by SPRECT_1:84;
2 in Seg (len f1) by A4, FINSEQ_1:1;
then A18: 2 in dom f1 by FINSEQ_1:def_3;
then A19: 2 in dom (X_axis f1) by SPRECT_2:15;
then (X_axis f1) . 2 = (f1 /. 2) `1 by GOBOARD1:def_1;
then A20: (X_axis f1) . 2 = E-bound (L~ f1) by A17, EUCLID:52;
A21: 1 in Seg (len f1) by A4, FINSEQ_1:1;
then A22: 1 in dom f1 by FINSEQ_1:def_3;
then 1 in dom (Y_axis f1) by SPRECT_2:16;
then (Y_axis f1) . 1 = (f1 /. 1) `2 by GOBOARD1:def_2;
then A23: (Y_axis f1) . 1 = N-bound (L~ f1) by A14, EUCLID:52;
(X_axis f1) . 2 = (f1 /. 2) `1 by A19, GOBOARD1:def_1;
then A24: (f1 /. 2) `1 in rng (X_axis f1) by A19, FUNCT_1:def_3;
len (X_axis f1) = len f1 by GOBOARD1:def_1;
then A25: dom (X_axis f1) = Seg (len f1) by FINSEQ_1:def_3;
then (X_axis f1) . 1 = (f1 /. 1) `1 by A21, GOBOARD1:def_1;
then A26: (f1 /. 1) `1 in rng (X_axis f1) by A21, A25, FUNCT_1:def_3;
{((f1 /. 1) `1),((f1 /. 2) `1)} c= rng (X_axis f1)
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in {((f1 /. 1) `1),((f1 /. 2) `1)} or z in rng (X_axis f1) )
assume z in {((f1 /. 1) `1),((f1 /. 2) `1)} ; ::_thesis: z in rng (X_axis f1)
hence z in rng (X_axis f1) by A26, A24, TARSKI:def_2; ::_thesis: verum
end;
then {((f1 /. 1) `1)} \/ {((f1 /. 2) `1)} c= rng (X_axis f1) by ENUMSET1:1;
then A27: card ({((f1 /. 1) `1)} \/ {((f1 /. 2) `1)}) <= card (rng (X_axis f1)) by NAT_1:43;
A28: f1 /. (1 + 1) = N-max (L~ f1) by SPRECT_1:84;
then (f1 /. 1) `1 < (f1 /. 2) `1 by A14, SPRECT_2:51;
then not (f1 /. 2) `1 in {((f1 /. 1) `1)} by TARSKI:def_1;
then A29: card ({((f1 /. 1) `1)} \/ {((f1 /. 2) `1)}) = (card {((f1 /. 1) `1)}) + 1 by CARD_2:41
.= 1 + 1 by CARD_1:30
.= 2 ;
A30: 1 <> (len (GoB f1)) + 1 by A27, GOBOARD2:13, XREAL_1:29;
2 in dom (Y_axis f1) by A18, SPRECT_2:16;
then (Y_axis f1) . 2 = (f1 /. 2) `2 by GOBOARD1:def_2;
then A31: (Y_axis f1) . 2 = N-bound (L~ f1) by A28, EUCLID:52;
f1 /. 5 = f1 /. 1 by A4, FINSEQ_6:def_1;
then A32: (Y_axis f1) . 5 = N-bound (L~ f1) by A14, A6, EUCLID:52;
A33: rng (Y_axis f1) c= {(S-bound (L~ f1)),(N-bound (L~ f1))}
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in rng (Y_axis f1) or z in {(S-bound (L~ f1)),(N-bound (L~ f1))} )
assume z in rng (Y_axis f1) ; ::_thesis: z in {(S-bound (L~ f1)),(N-bound (L~ f1))}
then consider u being set such that
A34: u in dom (Y_axis f1) and
A35: z = (Y_axis f1) . u by FUNCT_1:def_3;
reconsider mu = u as Element of NAT by A34;
u in dom f1 by A34, SPRECT_2:16;
then u in Seg (len f1) by FINSEQ_1:def_3;
then ( 1 <= mu & mu <= 5 ) by A4, FINSEQ_1:1;
then A36: ( mu = 1 or mu = 1 + 1 or mu = 1 + 2 or mu = 1 + 3 or mu = 1 + 4 ) by NAT_1:58;
percases ( mu = 1 or mu = 2 or mu = 3 or mu = 4 or mu = 5 ) by A36;
suppose mu = 1 ; ::_thesis: z in {(S-bound (L~ f1)),(N-bound (L~ f1))}
hence z in {(S-bound (L~ f1)),(N-bound (L~ f1))} by A23, A35, TARSKI:def_2; ::_thesis: verum
end;
suppose mu = 2 ; ::_thesis: z in {(S-bound (L~ f1)),(N-bound (L~ f1))}
hence z in {(S-bound (L~ f1)),(N-bound (L~ f1))} by A31, A35, TARSKI:def_2; ::_thesis: verum
end;
suppose mu = 3 ; ::_thesis: z in {(S-bound (L~ f1)),(N-bound (L~ f1))}
hence z in {(S-bound (L~ f1)),(N-bound (L~ f1))} by A10, A35, TARSKI:def_2; ::_thesis: verum
end;
suppose mu = 4 ; ::_thesis: z in {(S-bound (L~ f1)),(N-bound (L~ f1))}
hence z in {(S-bound (L~ f1)),(N-bound (L~ f1))} by A8, A35, TARSKI:def_2; ::_thesis: verum
end;
suppose mu = 5 ; ::_thesis: z in {(S-bound (L~ f1)),(N-bound (L~ f1))}
hence z in {(S-bound (L~ f1)),(N-bound (L~ f1))} by A32, A35, TARSKI:def_2; ::_thesis: verum
end;
end;
end;
{(S-bound (L~ f1)),(N-bound (L~ f1))} c= rng (Y_axis f1)
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in {(S-bound (L~ f1)),(N-bound (L~ f1))} or z in rng (Y_axis f1) )
assume A37: z in {(S-bound (L~ f1)),(N-bound (L~ f1))} ; ::_thesis: z in rng (Y_axis f1)
percases ( z = S-bound (L~ f1) or z = N-bound (L~ f1) ) by A37, TARSKI:def_2;
supposeA38: z = S-bound (L~ f1) ; ::_thesis: z in rng (Y_axis f1)
4 in dom (Y_axis f1) by A7, SPRECT_2:16;
hence z in rng (Y_axis f1) by A8, A38, FUNCT_1:def_3; ::_thesis: verum
end;
supposeA39: z = N-bound (L~ f1) ; ::_thesis: z in rng (Y_axis f1)
2 in dom (Y_axis f1) by A18, SPRECT_2:16;
hence z in rng (Y_axis f1) by A31, A39, FUNCT_1:def_3; ::_thesis: verum
end;
end;
end;
then A40: ( S-bound (L~ f1) < N-bound (L~ f1) & rng (Y_axis f1) = {(S-bound (L~ f1)),(N-bound (L~ f1))} ) by A33, SPRECT_1:32, XBOOLE_0:def_10;
A41: width (GoB f1) = card (rng (Y_axis f1)) by GOBOARD2:13
.= 1 + 1 by A40, CARD_2:57 ;
then A42: width (GoB f1) in Seg (width (GoB f1)) by FINSEQ_1:1;
f1 /. (1 + 1) = E-max (L~ f1) by SPRECT_1:84;
then A43: (SpStSeq D) /. 2 = |[((E-max (L~ (SpStSeq D))) `1),((N-max (L~ (SpStSeq D))) `2)]| by A28, EUCLID:53;
A44: f1 /. 1 = W-max (L~ f1) by SPRECT_1:83;
then (SpStSeq D) /. 1 = |[((W-max (L~ (SpStSeq D))) `1),((N-min (L~ (SpStSeq D))) `2)]| by A14, EUCLID:53;
then ((SpStSeq D) /. 1) + ((SpStSeq D) /. 2) = |[(((W-max (L~ (SpStSeq D))) `1) + ((E-max (L~ (SpStSeq D))) `1)),(((N-min (L~ (SpStSeq D))) `2) + ((N-max (L~ (SpStSeq D))) `2))]| by A43, EUCLID:56;
then (1 / 2) * (((SpStSeq D) /. 1) + ((SpStSeq D) /. 2)) = |[((1 / 2) * (((W-max (L~ (SpStSeq D))) `1) + ((E-max (L~ (SpStSeq D))) `1))),(N-bound (L~ (SpStSeq D)))]| by A3, EUCLID:58;
then A45: |[0,((|.((1 / 2) * (((SpStSeq D) /. 1) + ((SpStSeq D) /. 2))).| + r) + 1)]| + ((1 / 2) * (((SpStSeq D) /. 1) + ((SpStSeq D) /. 2))) = |[(0 + ((1 / 2) * (((W-max (L~ (SpStSeq D))) `1) + ((E-max (L~ (SpStSeq D))) `1)))),(((|.((1 / 2) * (((SpStSeq D) /. 1) + ((SpStSeq D) /. 2))).| + r) + 1) + (N-bound (L~ (SpStSeq D))))]| by EUCLID:56
.= |[((1 / 2) * (((W-max (L~ (SpStSeq D))) `1) + ((E-max (L~ (SpStSeq D))) `1))),(((|.((1 / 2) * (((SpStSeq D) /. 1) + ((SpStSeq D) /. 2))).| + r) + 1) + (N-bound (L~ (SpStSeq D))))]| ;
(W-max (L~ (SpStSeq D))) `1 = W-bound (L~ (SpStSeq D)) by EUCLID:52;
then A46: (W-max (L~ (SpStSeq D))) `1 < (E-max (L~ (SpStSeq D))) `1 by A1, EUCLID:52;
A47: f1 /. 1 = W-max (L~ f1) by SPRECT_1:83;
then A48: ((GoB f1) * (1,1)) `1 <= (W-max (L~ (SpStSeq D))) `1 by A4, A41, JORDAN5D:5;
then ((GoB f1) * (1,1)) `1 < (E-max (L~ (SpStSeq D))) `1 by A46, XXREAL_0:2;
then (((GoB f1) * (1,1)) `1) + (((GoB f1) * (1,1)) `1) < ((W-max (L~ (SpStSeq D))) `1) + ((E-max (L~ (SpStSeq D))) `1) by A48, XREAL_1:8;
then A49: (1 / 2) * (2 * (((GoB f1) * (1,1)) `1)) < (1 / 2) * (((W-max (L~ (SpStSeq D))) `1) + ((E-max (L~ (SpStSeq D))) `1)) by XREAL_1:68;
1 in dom (X_axis f1) by A22, SPRECT_2:15;
then (X_axis f1) . 1 = (f1 /. 1) `1 by GOBOARD1:def_1;
then A50: (X_axis f1) . 1 = W-bound (L~ f1) by A47, EUCLID:52;
f1 /. 5 = W-max (L~ f1) by A4, A44, FINSEQ_6:def_1;
then A51: (X_axis f1) . 5 = W-bound (L~ f1) by A12, EUCLID:52;
A52: rng (X_axis f1) c= {(W-bound (L~ f1)),(E-bound (L~ f1))}
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in rng (X_axis f1) or z in {(W-bound (L~ f1)),(E-bound (L~ f1))} )
assume z in rng (X_axis f1) ; ::_thesis: z in {(W-bound (L~ f1)),(E-bound (L~ f1))}
then consider u being set such that
A53: u in dom (X_axis f1) and
A54: z = (X_axis f1) . u by FUNCT_1:def_3;
reconsider mu = u as Element of NAT by A53;
u in dom f1 by A53, SPRECT_2:15;
then u in Seg (len f1) by FINSEQ_1:def_3;
then ( 1 <= mu & mu <= 5 ) by A4, FINSEQ_1:1;
then A55: ( mu = 1 or mu = 1 + 1 or mu = 1 + 2 or mu = 1 + 3 or mu = 1 + 4 ) by NAT_1:58;
percases ( mu = 1 or mu = 2 or mu = 3 or mu = 4 or mu = 5 ) by A55;
suppose mu = 1 ; ::_thesis: z in {(W-bound (L~ f1)),(E-bound (L~ f1))}
hence z in {(W-bound (L~ f1)),(E-bound (L~ f1))} by A50, A54, TARSKI:def_2; ::_thesis: verum
end;
suppose mu = 2 ; ::_thesis: z in {(W-bound (L~ f1)),(E-bound (L~ f1))}
hence z in {(W-bound (L~ f1)),(E-bound (L~ f1))} by A20, A54, TARSKI:def_2; ::_thesis: verum
end;
suppose mu = 3 ; ::_thesis: z in {(W-bound (L~ f1)),(E-bound (L~ f1))}
hence z in {(W-bound (L~ f1)),(E-bound (L~ f1))} by A11, A54, TARSKI:def_2; ::_thesis: verum
end;
suppose mu = 4 ; ::_thesis: z in {(W-bound (L~ f1)),(E-bound (L~ f1))}
hence z in {(W-bound (L~ f1)),(E-bound (L~ f1))} by A15, A54, TARSKI:def_2; ::_thesis: verum
end;
suppose mu = 5 ; ::_thesis: z in {(W-bound (L~ f1)),(E-bound (L~ f1))}
hence z in {(W-bound (L~ f1)),(E-bound (L~ f1))} by A51, A54, TARSKI:def_2; ::_thesis: verum
end;
end;
end;
{(W-bound (L~ f1)),(E-bound (L~ f1))} c= rng (X_axis f1)
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in {(W-bound (L~ f1)),(E-bound (L~ f1))} or z in rng (X_axis f1) )
assume A56: z in {(W-bound (L~ f1)),(E-bound (L~ f1))} ; ::_thesis: z in rng (X_axis f1)
percases ( z = W-bound (L~ f1) or z = E-bound (L~ f1) ) by A56, TARSKI:def_2;
supposeA57: z = W-bound (L~ f1) ; ::_thesis: z in rng (X_axis f1)
1 in dom (X_axis f1) by A22, SPRECT_2:15;
hence z in rng (X_axis f1) by A50, A57, FUNCT_1:def_3; ::_thesis: verum
end;
supposeA58: z = E-bound (L~ f1) ; ::_thesis: z in rng (X_axis f1)
2 in dom (X_axis f1) by A18, SPRECT_2:15;
hence z in rng (X_axis f1) by A20, A58, FUNCT_1:def_3; ::_thesis: verum
end;
end;
end;
then A59: rng (X_axis f1) = {(W-bound (L~ f1)),(E-bound (L~ f1))} by A52, XBOOLE_0:def_10;
A60: len (GoB f1) = card (rng (X_axis f1)) by GOBOARD2:13
.= 1 + 1 by A1, A59, CARD_2:57 ;
then A61: ((GoB f1) * ((1 + 1),1)) `1 >= (E-max (L~ (SpStSeq D))) `1 by A4, A17, A41, JORDAN5D:5;
then (W-max (L~ (SpStSeq D))) `1 < ((GoB f1) * ((1 + 1),1)) `1 by A46, XXREAL_0:2;
then ((W-max (L~ (SpStSeq D))) `1) + ((E-max (L~ (SpStSeq D))) `1) < (((GoB f1) * ((1 + 1),1)) `1) + (((GoB f1) * ((1 + 1),1)) `1) by A61, XREAL_1:8;
then A62: (1 / 2) * (((W-max (L~ (SpStSeq D))) `1) + ((E-max (L~ (SpStSeq D))) `1)) < (1 / 2) * (2 * (((GoB f1) * ((1 + 1),1)) `1)) by XREAL_1:68;
A63: card (rng (X_axis f1)) = 2 by A1, A59, CARD_2:57;
for p being FinSequence of the carrier of (TOP-REAL 2) st p in rng (GoB f1) holds
len p = 2
proof
len (GoB ((Incr (X_axis f1)),(Incr (Y_axis f1)))) = len (Incr (X_axis f1)) by GOBOARD2:def_1
.= 2 by A63, SEQ_4:def_21 ;
then consider s1 being FinSequence such that
A64: s1 in rng (GoB ((Incr (X_axis f1)),(Incr (Y_axis f1)))) and
A65: len s1 = width (GoB ((Incr (X_axis f1)),(Incr (Y_axis f1)))) by MATRIX_1:def_3;
let p be FinSequence of the carrier of (TOP-REAL 2); ::_thesis: ( p in rng (GoB f1) implies len p = 2 )
consider n being Nat such that
A66: for x being set st x in rng (GoB f1) holds
ex s being FinSequence st
( s = x & len s = n ) by MATRIX_1:def_1;
assume p in rng (GoB f1) ; ::_thesis: len p = 2
then A67: ex s2 being FinSequence st
( s2 = p & len s2 = n ) by A66;
ex s being FinSequence st
( s = s1 & len s = n ) by A16, A64, A66;
hence len p = 2 by A41, A65, A67, GOBOARD2:def_2; ::_thesis: verum
end;
then A68: GoB f1 is Matrix of 2,2, the carrier of (TOP-REAL 2) by A60, MATRIX_1:def_2;
len (GoB f1) in Seg (len (GoB f1)) by A60, FINSEQ_1:1;
then [(len (GoB f1)),(width (GoB f1))] in [:(Seg (len (GoB f1))),(Seg (width (GoB f1))):] by A42, ZFMISC_1:87;
then A69: [(len (GoB f1)),(width (GoB f1))] in Indices (GoB f1) by A60, A41, A68, MATRIX_1:24;
1 in Seg (len (GoB f1)) by A60, FINSEQ_1:1;
then [1,(width (GoB f1))] in [:(Seg (len (GoB f1))),(Seg (width (GoB f1))):] by A42, ZFMISC_1:87;
then A70: [1,(width (GoB f1))] in Indices (GoB f1) by A60, A41, A68, MATRIX_1:24;
card (rng (X_axis f1)) > 1 by A27, A29, XXREAL_0:2;
then A71: 1 < len (GoB f1) by GOBOARD2:13;
A72: f1 /. 1 = (GoB f1) * (1,(width (GoB f1))) by A41, Th89;
set p = |[0,((|.((1 / 2) * (((SpStSeq D) /. 1) + ((SpStSeq D) /. 2))).| + r) + 1)]|;
A73: ( |[0,((|.((1 / 2) * (((SpStSeq D) /. 1) + ((SpStSeq D) /. 2))).| + r) + 1)]| `1 = 0 & |[0,((|.((1 / 2) * (((SpStSeq D) /. 1) + ((SpStSeq D) /. 2))).| + r) + 1)]| `2 = (|.((1 / 2) * (((SpStSeq D) /. 1) + ((SpStSeq D) /. 2))).| + r) + 1 ) by EUCLID:52;
A74: ( |.(|[0,((|.((1 / 2) * (((SpStSeq D) /. 1) + ((SpStSeq D) /. 2))).| + r) + 1)]| + ((1 / 2) * (((SpStSeq D) /. 1) + ((SpStSeq D) /. 2)))).| >= |.|[0,((|.((1 / 2) * (((SpStSeq D) /. 1) + ((SpStSeq D) /. 2))).| + r) + 1)]|.| - |.((1 / 2) * (((SpStSeq D) /. 1) + ((SpStSeq D) /. 2))).| & r < r + 1 ) by TOPRNS_1:31, XREAL_1:29;
A75: Int (left_cell (f1,1)) c= LeftComp (SpStSeq D) by GOBOARD9:def_1;
A76: width (GoB f1) <> (width (GoB f1)) + 1 ;
f1 /. (1 + 1) = (GoB f1) * ((len (GoB f1)),(width (GoB f1))) by A60, A41, Th89;
then left_cell (f1,1) = cell ((GoB f1),1,(width (GoB f1))) by A4, A70, A69, A72, A30, A76, GOBOARD5:def_7;
then A77: Int (left_cell (f1,1)) = { |[r2,s]| where r2, s is Real : ( ((GoB f1) * (1,1)) `1 < r2 & r2 < ((GoB f1) * ((1 + 1),1)) `1 & ((GoB f1) * (1,(width (GoB f1)))) `2 < s ) } by A71, GOBOARD6:25;
A78: |.q4.| < r by A13;
A79: |.|[0,((|.((1 / 2) * (((SpStSeq D) /. 1) + ((SpStSeq D) /. 2))).| + r) + 1)]|.| = sqrt (((|[0,((|.((1 / 2) * (((SpStSeq D) /. 1) + ((SpStSeq D) /. 2))).| + r) + 1)]| `1) ^2) + ((|[0,((|.((1 / 2) * (((SpStSeq D) /. 1) + ((SpStSeq D) /. 2))).| + r) + 1)]| `2) ^2)) by JGRAPH_1:30
.= (|.((1 / 2) * (((SpStSeq D) /. 1) + ((SpStSeq D) /. 2))).| + r) + 1 by A78, A73, SQUARE_1:22 ;
((GoB f1) * (1,(width (GoB f1)))) `2 < (N-bound (L~ (SpStSeq D))) + ((|.((1 / 2) * (((SpStSeq D) /. 1) + ((SpStSeq D) /. 2))).| + r) + 1) by A78, A14, A72, A2, XREAL_1:29;
then |[0,((|.((1 / 2) * (((SpStSeq D) /. 1) + ((SpStSeq D) /. 2))).| + r) + 1)]| + ((1 / 2) * (((SpStSeq D) /. 1) + ((SpStSeq D) /. 2))) in Int (left_cell (f1,1)) by A77, A45, A49, A62;
hence contradiction by A13, A79, A74, A75, XXREAL_0:2; ::_thesis: verum
end;
theorem Th91: :: JORDAN2C:91
for D being non empty compact non horizontal non vertical Subset of (TOP-REAL 2) holds LeftComp (SpStSeq D) c= UBD (L~ (SpStSeq D))
proof
let D be non empty compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: LeftComp (SpStSeq D) c= UBD (L~ (SpStSeq D))
set f = SpStSeq D;
set A = L~ (SpStSeq D);
( LeftComp (SpStSeq D) is_a_component_of (L~ (SpStSeq D)) ` & not LeftComp (SpStSeq D) is bounded ) by Th90, GOBOARD9:def_1;
then A1: LeftComp (SpStSeq D) is_outside_component_of L~ (SpStSeq D) by Def3;
LeftComp (SpStSeq D) c= union { B where B is Subset of (TOP-REAL 2) : B is_outside_component_of L~ (SpStSeq D) }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LeftComp (SpStSeq D) or x in union { B where B is Subset of (TOP-REAL 2) : B is_outside_component_of L~ (SpStSeq D) } )
assume A2: x in LeftComp (SpStSeq D) ; ::_thesis: x in union { B where B is Subset of (TOP-REAL 2) : B is_outside_component_of L~ (SpStSeq D) }
LeftComp (SpStSeq D) in { B where B is Subset of (TOP-REAL 2) : B is_outside_component_of L~ (SpStSeq D) } by A1;
hence x in union { B where B is Subset of (TOP-REAL 2) : B is_outside_component_of L~ (SpStSeq D) } by A2, TARSKI:def_4; ::_thesis: verum
end;
hence LeftComp (SpStSeq D) c= UBD (L~ (SpStSeq D)) ; ::_thesis: verum
end;
theorem Th92: :: JORDAN2C:92
for G being TopSpace
for A, B, C being Subset of G st A is a_component & B is a_component & C is connected & A meets C & B meets C holds
A = B
proof
let G be TopSpace; ::_thesis: for A, B, C being Subset of G st A is a_component & B is a_component & C is connected & A meets C & B meets C holds
A = B
let A, B, C be Subset of G; ::_thesis: ( A is a_component & B is a_component & C is connected & A meets C & B meets C implies A = B )
assume that
A1: A is a_component and
A2: B is a_component and
A3: C is connected and
A4: A meets C and
A5: B meets C ; ::_thesis: A = B
A6: ( C /\ A = {} G or C c= A ) by A1, A3, A4, CONNSP_1:36;
A7: ( C misses B or C c= B ) by A2, A3, CONNSP_1:36;
percases ( A = B or A misses B ) by A1, A2, CONNSP_1:1, CONNSP_1:34;
suppose A = B ; ::_thesis: A = B
hence A = B ; ::_thesis: verum
end;
suppose A misses B ; ::_thesis: A = B
then A8: A /\ B = {} by XBOOLE_0:def_7;
C c= A /\ B by A4, A5, A6, A7, XBOOLE_0:def_7, XBOOLE_1:19;
then C = {} by A8;
then C /\ A = {} ;
hence A = B by A4, XBOOLE_0:def_7; ::_thesis: verum
end;
end;
end;
theorem Th93: :: JORDAN2C:93
for D being non empty compact non horizontal non vertical Subset of (TOP-REAL 2)
for B being Subset of (TOP-REAL 2) st B is_a_component_of (L~ (SpStSeq D)) ` & not B is bounded holds
B = LeftComp (SpStSeq D)
proof
let D be non empty compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for B being Subset of (TOP-REAL 2) st B is_a_component_of (L~ (SpStSeq D)) ` & not B is bounded holds
B = LeftComp (SpStSeq D)
let B be Subset of (TOP-REAL 2); ::_thesis: ( B is_a_component_of (L~ (SpStSeq D)) ` & not B is bounded implies B = LeftComp (SpStSeq D) )
set f = SpStSeq D;
assume that
A1: B is_a_component_of (L~ (SpStSeq D)) ` and
A2: not B is bounded ; ::_thesis: B = LeftComp (SpStSeq D)
A3: ex B1 being Subset of ((TOP-REAL 2) | ((L~ (SpStSeq D)) `)) st
( B1 = B & B1 is a_component ) by A1, CONNSP_1:def_6;
consider r1 being Real such that
A4: for q being Point of (TOP-REAL 2) st q in L~ (SpStSeq D) holds
|.q.| < r1 by Th34;
consider q4 being Point of (TOP-REAL 2) such that
A5: q4 in B and
A6: |.q4.| >= r1 by A2, Th34;
A7: now__::_thesis:_not_q4_in__{__q_where_q_is_Point_of_(TOP-REAL_2)_:_|.q.|_<_r1__}_
assume q4 in { q where q is Point of (TOP-REAL 2) : |.q.| < r1 } ; ::_thesis: contradiction
then ex q being Point of (TOP-REAL 2) st
( q = q4 & |.q.| < r1 ) ;
hence contradiction by A6; ::_thesis: verum
end;
reconsider P = (REAL 2) \ { q where q is Point of (TOP-REAL 2) : |.q.| < r1 } as Subset of (TOP-REAL 2) by EUCLID:22;
P c= the carrier of (TOP-REAL 2) \ (L~ (SpStSeq D))
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in P or z in the carrier of (TOP-REAL 2) \ (L~ (SpStSeq D)) )
assume A8: z in P ; ::_thesis: z in the carrier of (TOP-REAL 2) \ (L~ (SpStSeq D))
now__::_thesis:_not_z_in_L~_(SpStSeq_D)
assume A9: z in L~ (SpStSeq D) ; ::_thesis: contradiction
then reconsider q3 = z as Point of (TOP-REAL 2) ;
A10: not q3 in { q where q is Point of (TOP-REAL 2) : |.q.| < r1 } by A8, XBOOLE_0:def_5;
|.q3.| < r1 by A4, A9;
hence contradiction by A10; ::_thesis: verum
end;
hence z in the carrier of (TOP-REAL 2) \ (L~ (SpStSeq D)) by A8, XBOOLE_0:def_5; ::_thesis: verum
end;
then A11: P /\ ( the carrier of (TOP-REAL 2) \ (L~ (SpStSeq D))) = P by XBOOLE_1:28;
then A12: Down (P,((L~ (SpStSeq D)) `)) is connected by Th53, CONNSP_1:46;
not LeftComp (SpStSeq D) is bounded by Th90;
then consider q3 being Point of (TOP-REAL 2) such that
A13: q3 in LeftComp (SpStSeq D) and
A14: |.q3.| >= r1 by Th34;
A15: now__::_thesis:_not_q3_in__{__q_where_q_is_Point_of_(TOP-REAL_2)_:_|.q.|_<_r1__}_
assume q3 in { q where q is Point of (TOP-REAL 2) : |.q.| < r1 } ; ::_thesis: contradiction
then ex q being Point of (TOP-REAL 2) st
( q = q3 & |.q.| < r1 ) ;
hence contradiction by A14; ::_thesis: verum
end;
q4 in the carrier of (TOP-REAL 2) ;
then q4 in REAL 2 by EUCLID:22;
then q4 in P by A7, XBOOLE_0:def_5;
then A16: B meets P by A5, XBOOLE_0:3;
LeftComp (SpStSeq D) is_a_component_of (L~ (SpStSeq D)) ` by GOBOARD9:def_1;
then consider L1 being Subset of ((TOP-REAL 2) | ((L~ (SpStSeq D)) `)) such that
A17: L1 = LeftComp (SpStSeq D) and
A18: L1 is a_component by CONNSP_1:def_6;
q3 in the carrier of (TOP-REAL 2) ;
then q3 in REAL 2 by EUCLID:22;
then q3 in P by A15, XBOOLE_0:def_5;
then L1 meets P by A17, A13, XBOOLE_0:3;
hence B = LeftComp (SpStSeq D) by A3, A17, A18, A11, A12, A16, Th92; ::_thesis: verum
end;
theorem Th94: :: JORDAN2C:94
for D being non empty compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( RightComp (SpStSeq D) c= BDD (L~ (SpStSeq D)) & RightComp (SpStSeq D) is bounded )
proof
let D be non empty compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( RightComp (SpStSeq D) c= BDD (L~ (SpStSeq D)) & RightComp (SpStSeq D) is bounded )
set f = SpStSeq D;
set A = L~ (SpStSeq D);
A1: RightComp (SpStSeq D) is_a_component_of (L~ (SpStSeq D)) ` by GOBOARD9:def_2;
A2: now__::_thesis:_RightComp_(SpStSeq_D)_is_bounded
A3: LeftComp (SpStSeq D) misses RightComp (SpStSeq D) by SPRECT_1:88;
assume not RightComp (SpStSeq D) is bounded ; ::_thesis: contradiction
hence contradiction by A1, A3, Th93; ::_thesis: verum
end;
then A4: RightComp (SpStSeq D) is_inside_component_of L~ (SpStSeq D) by A1, Def2;
RightComp (SpStSeq D) c= union { B where B is Subset of (TOP-REAL 2) : B is_inside_component_of L~ (SpStSeq D) }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in RightComp (SpStSeq D) or x in union { B where B is Subset of (TOP-REAL 2) : B is_inside_component_of L~ (SpStSeq D) } )
assume A5: x in RightComp (SpStSeq D) ; ::_thesis: x in union { B where B is Subset of (TOP-REAL 2) : B is_inside_component_of L~ (SpStSeq D) }
RightComp (SpStSeq D) in { B where B is Subset of (TOP-REAL 2) : B is_inside_component_of L~ (SpStSeq D) } by A4;
hence x in union { B where B is Subset of (TOP-REAL 2) : B is_inside_component_of L~ (SpStSeq D) } by A5, TARSKI:def_4; ::_thesis: verum
end;
hence RightComp (SpStSeq D) c= BDD (L~ (SpStSeq D)) ; ::_thesis: RightComp (SpStSeq D) is bounded
thus RightComp (SpStSeq D) is bounded by A2; ::_thesis: verum
end;
theorem Th95: :: JORDAN2C:95
for D being non empty compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( LeftComp (SpStSeq D) = UBD (L~ (SpStSeq D)) & RightComp (SpStSeq D) = BDD (L~ (SpStSeq D)) )
proof
let D be non empty compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( LeftComp (SpStSeq D) = UBD (L~ (SpStSeq D)) & RightComp (SpStSeq D) = BDD (L~ (SpStSeq D)) )
set f = SpStSeq D;
A1: (L~ (SpStSeq D)) ` = (LeftComp (SpStSeq D)) \/ (RightComp (SpStSeq D)) by GOBRD12:10;
A2: LeftComp (SpStSeq D) c= UBD (L~ (SpStSeq D)) by Th91;
A3: RightComp (SpStSeq D) c= BDD (L~ (SpStSeq D)) by Th94;
A4: now__::_thesis:_LeftComp_(SpStSeq_D)_=_UBD_(L~_(SpStSeq_D))
assume not LeftComp (SpStSeq D) = UBD (L~ (SpStSeq D)) ; ::_thesis: contradiction
then not UBD (L~ (SpStSeq D)) c= LeftComp (SpStSeq D) by A2, XBOOLE_0:def_10;
then consider z being set such that
A5: z in UBD (L~ (SpStSeq D)) and
A6: not z in LeftComp (SpStSeq D) by TARSKI:def_3;
UBD (L~ (SpStSeq D)) c= (L~ (SpStSeq D)) ` by Th26;
then ( z in LeftComp (SpStSeq D) or z in RightComp (SpStSeq D) ) by A1, A5, XBOOLE_0:def_3;
then BDD (L~ (SpStSeq D)) meets UBD (L~ (SpStSeq D)) by A3, A5, A6, XBOOLE_0:3;
hence contradiction by Th24; ::_thesis: verum
end;
now__::_thesis:_RightComp_(SpStSeq_D)_=_BDD_(L~_(SpStSeq_D))
assume not RightComp (SpStSeq D) = BDD (L~ (SpStSeq D)) ; ::_thesis: contradiction
then not BDD (L~ (SpStSeq D)) c= RightComp (SpStSeq D) by A3, XBOOLE_0:def_10;
then consider z being set such that
A7: z in BDD (L~ (SpStSeq D)) and
A8: not z in RightComp (SpStSeq D) by TARSKI:def_3;
BDD (L~ (SpStSeq D)) c= (L~ (SpStSeq D)) ` by Th25;
then ( z in LeftComp (SpStSeq D) or z in RightComp (SpStSeq D) ) by A1, A7, XBOOLE_0:def_3;
then BDD (L~ (SpStSeq D)) meets UBD (L~ (SpStSeq D)) by A2, A7, A8, XBOOLE_0:3;
hence contradiction by Th24; ::_thesis: verum
end;
hence ( LeftComp (SpStSeq D) = UBD (L~ (SpStSeq D)) & RightComp (SpStSeq D) = BDD (L~ (SpStSeq D)) ) by A4; ::_thesis: verum
end;
theorem Th96: :: JORDAN2C:96
for D being non empty compact non horizontal non vertical Subset of (TOP-REAL 2) holds
( UBD (L~ (SpStSeq D)) <> {} & UBD (L~ (SpStSeq D)) is_outside_component_of L~ (SpStSeq D) & BDD (L~ (SpStSeq D)) <> {} & BDD (L~ (SpStSeq D)) is_inside_component_of L~ (SpStSeq D) )
proof
let D be non empty compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( UBD (L~ (SpStSeq D)) <> {} & UBD (L~ (SpStSeq D)) is_outside_component_of L~ (SpStSeq D) & BDD (L~ (SpStSeq D)) <> {} & BDD (L~ (SpStSeq D)) is_inside_component_of L~ (SpStSeq D) )
set f = SpStSeq D;
A1: UBD (L~ (SpStSeq D)) = LeftComp (SpStSeq D) by Th95;
hence UBD (L~ (SpStSeq D)) <> {} ; ::_thesis: ( UBD (L~ (SpStSeq D)) is_outside_component_of L~ (SpStSeq D) & BDD (L~ (SpStSeq D)) <> {} & BDD (L~ (SpStSeq D)) is_inside_component_of L~ (SpStSeq D) )
( LeftComp (SpStSeq D) is_a_component_of (L~ (SpStSeq D)) ` & not LeftComp (SpStSeq D) is bounded ) by Th90, GOBOARD9:def_1;
hence UBD (L~ (SpStSeq D)) is_outside_component_of L~ (SpStSeq D) by A1, Def3; ::_thesis: ( BDD (L~ (SpStSeq D)) <> {} & BDD (L~ (SpStSeq D)) is_inside_component_of L~ (SpStSeq D) )
A2: BDD (L~ (SpStSeq D)) = RightComp (SpStSeq D) by Th95;
hence BDD (L~ (SpStSeq D)) <> {} ; ::_thesis: BDD (L~ (SpStSeq D)) is_inside_component_of L~ (SpStSeq D)
( RightComp (SpStSeq D) is_a_component_of (L~ (SpStSeq D)) ` & RightComp (SpStSeq D) is bounded ) by Th94, GOBOARD9:def_2;
hence BDD (L~ (SpStSeq D)) is_inside_component_of L~ (SpStSeq D) by A2, Def2; ::_thesis: verum
end;
begin
theorem Th97: :: JORDAN2C:97
for G being non empty TopSpace
for A being Subset of G st A ` <> {} holds
( A is boundary iff for x being set
for V being Subset of G st x in A & x in V & V is open holds
ex B being Subset of G st
( B is_a_component_of A ` & V meets B ) )
proof
let G be non empty TopSpace; ::_thesis: for A being Subset of G st A ` <> {} holds
( A is boundary iff for x being set
for V being Subset of G st x in A & x in V & V is open holds
ex B being Subset of G st
( B is_a_component_of A ` & V meets B ) )
let A be Subset of G; ::_thesis: ( A ` <> {} implies ( A is boundary iff for x being set
for V being Subset of G st x in A & x in V & V is open holds
ex B being Subset of G st
( B is_a_component_of A ` & V meets B ) ) )
assume A1: A ` <> {} ; ::_thesis: ( A is boundary iff for x being set
for V being Subset of G st x in A & x in V & V is open holds
ex B being Subset of G st
( B is_a_component_of A ` & V meets B ) )
hereby ::_thesis: ( ( for x being set
for V being Subset of G st x in A & x in V & V is open holds
ex B being Subset of G st
( B is_a_component_of A ` & V meets B ) ) implies A is boundary )
reconsider A1 = A ` as non empty Subset of G by A1;
reconsider A2 = A ` as Subset of G ;
assume A is boundary ; ::_thesis: for x being set
for V being Subset of G st x in A & x in V & V is open holds
ex B being Subset of G st
( B is_a_component_of A ` & V meets B )
then A ` is dense by TOPS_1:def_4;
then A2: Cl (A `) = [#] G by TOPS_1:def_3;
let x be set ; ::_thesis: for V being Subset of G st x in A & x in V & V is open holds
ex B being Subset of G st
( B is_a_component_of A ` & V meets B )
let V be Subset of G; ::_thesis: ( x in A & x in V & V is open implies ex B being Subset of G st
( B is_a_component_of A ` & V meets B ) )
assume that
x in A and
A3: ( x in V & V is open ) ; ::_thesis: ex B being Subset of G st
( B is_a_component_of A ` & V meets B )
A2 meets V by A3, A2, PRE_TOPC:def_7;
then consider z being set such that
A4: z in A ` and
A5: z in V by XBOOLE_0:3;
reconsider p = z as Point of (G | (A `)) by A4, PRE_TOPC:8;
Component_of p c= the carrier of (G | (A `)) ;
then Component_of p c= A ` by PRE_TOPC:8;
then reconsider B0 = Component_of p as Subset of G by XBOOLE_1:1;
A6: not G | A1 is empty ;
then p in Component_of p by CONNSP_1:38;
then p in V /\ B0 by A5, XBOOLE_0:def_4;
then A7: V meets B0 by XBOOLE_0:4;
Component_of p is a_component by A6, CONNSP_1:40;
then B0 is_a_component_of A ` by CONNSP_1:def_6;
hence ex B being Subset of G st
( B is_a_component_of A ` & V meets B ) by A7; ::_thesis: verum
end;
assume A8: for x being set
for V being Subset of G st x in A & x in V & V is open holds
ex B being Subset of G st
( B is_a_component_of A ` & V meets B ) ; ::_thesis: A is boundary
the carrier of G c= Cl (A `)
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in the carrier of G or z in Cl (A `) )
assume A9: z in the carrier of G ; ::_thesis: z in Cl (A `)
percases ( z in A or not z in A ) ;
supposeA10: z in A ; ::_thesis: z in Cl (A `)
for G1 being Subset of G st G1 is open & z in G1 holds
A ` meets G1
proof
let G1 be Subset of G; ::_thesis: ( G1 is open & z in G1 implies A ` meets G1 )
assume A11: G1 is open ; ::_thesis: ( not z in G1 or A ` meets G1 )
assume z in G1 ; ::_thesis: A ` meets G1
then consider B being Subset of G such that
A12: B is_a_component_of A ` and
A13: G1 meets B by A8, A10, A11;
A14: G1 /\ B <> {} by A13, XBOOLE_0:def_7;
consider B1 being Subset of (G | (A `)) such that
A15: B1 = B and
B1 is a_component by A12, CONNSP_1:def_6;
B1 c= the carrier of (G | (A `)) ;
then B1 c= A ` by PRE_TOPC:8;
then (A `) /\ G1 <> {} G by A15, A14, XBOOLE_1:3, XBOOLE_1:26;
hence A ` meets G1 by XBOOLE_0:def_7; ::_thesis: verum
end;
hence z in Cl (A `) by A9, PRE_TOPC:def_7; ::_thesis: verum
end;
supposeA16: not z in A ; ::_thesis: z in Cl (A `)
A17: A ` c= Cl (A `) by PRE_TOPC:18;
z in the carrier of G \ A by A9, A16, XBOOLE_0:def_5;
hence z in Cl (A `) by A17; ::_thesis: verum
end;
end;
end;
then Cl (A `) = [#] G by XBOOLE_0:def_10;
then A ` is dense by TOPS_1:def_3;
hence A is boundary by TOPS_1:def_4; ::_thesis: verum
end;
theorem Th98: :: JORDAN2C:98
for A being Subset of (TOP-REAL 2) st A ` <> {} holds
( ( A is boundary & A is Jordan ) iff ex A1, A2 being Subset of (TOP-REAL 2) st
( A ` = A1 \/ A2 & A1 misses A2 & (Cl A1) \ A1 = (Cl A2) \ A2 & A = (Cl A1) \ A1 & ( for C1, C2 being Subset of ((TOP-REAL 2) | (A `)) st C1 = A1 & C2 = A2 holds
( C1 is a_component & C2 is a_component ) ) ) )
proof
let A be Subset of (TOP-REAL 2); ::_thesis: ( A ` <> {} implies ( ( A is boundary & A is Jordan ) iff ex A1, A2 being Subset of (TOP-REAL 2) st
( A ` = A1 \/ A2 & A1 misses A2 & (Cl A1) \ A1 = (Cl A2) \ A2 & A = (Cl A1) \ A1 & ( for C1, C2 being Subset of ((TOP-REAL 2) | (A `)) st C1 = A1 & C2 = A2 holds
( C1 is a_component & C2 is a_component ) ) ) ) )
assume A1: A ` <> {} ; ::_thesis: ( ( A is boundary & A is Jordan ) iff ex A1, A2 being Subset of (TOP-REAL 2) st
( A ` = A1 \/ A2 & A1 misses A2 & (Cl A1) \ A1 = (Cl A2) \ A2 & A = (Cl A1) \ A1 & ( for C1, C2 being Subset of ((TOP-REAL 2) | (A `)) st C1 = A1 & C2 = A2 holds
( C1 is a_component & C2 is a_component ) ) ) )
hereby ::_thesis: ( ex A1, A2 being Subset of (TOP-REAL 2) st
( A ` = A1 \/ A2 & A1 misses A2 & (Cl A1) \ A1 = (Cl A2) \ A2 & A = (Cl A1) \ A1 & ( for C1, C2 being Subset of ((TOP-REAL 2) | (A `)) st C1 = A1 & C2 = A2 holds
( C1 is a_component & C2 is a_component ) ) ) implies ( A is boundary & A is Jordan ) )
assume that
A2: A is boundary and
A3: A is Jordan ; ::_thesis: ex A1, A2 being Subset of (TOP-REAL 2) st
( A ` = A1 \/ A2 & A1 misses A2 & (Cl A1) \ A1 = (Cl A2) \ A2 & A = (Cl A1) \ A1 & ( for C1, C2 being Subset of ((TOP-REAL 2) | (A `)) st C1 = A1 & C2 = A2 holds
( C1 is a_component & C2 is a_component ) ) )
consider A1, A2 being Subset of (TOP-REAL 2) such that
A4: A ` = A1 \/ A2 and
A5: A1 misses A2 and
A6: (Cl A1) \ A1 = (Cl A2) \ A2 and
A7: for C1, C2 being Subset of ((TOP-REAL 2) | (A `)) st C1 = A1 & C2 = A2 holds
( C1 is a_component & C2 is a_component ) by A3, JORDAN1:def_2;
A = (A1 \/ A2) ` by A4;
then A8: A = (A1 `) /\ (A2 `) by XBOOLE_1:53;
A2 c= A ` by A4, XBOOLE_1:7;
then reconsider D2 = A2 as Subset of ((TOP-REAL 2) | (A `)) by PRE_TOPC:8;
A1 c= A ` by A4, XBOOLE_1:7;
then reconsider D1 = A1 as Subset of ((TOP-REAL 2) | (A `)) by PRE_TOPC:8;
D2 = A2 ;
then A9: D1 is a_component by A7;
A10: A c= (Cl A1) \ A1
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in A or z in (Cl A1) \ A1 )
assume A11: z in A ; ::_thesis: z in (Cl A1) \ A1
for G being Subset of (TOP-REAL 2) st G is open & z in G holds
A1 \/ A2 meets G
proof
let G be Subset of (TOP-REAL 2); ::_thesis: ( G is open & z in G implies A1 \/ A2 meets G )
assume A12: G is open ; ::_thesis: ( not z in G or A1 \/ A2 meets G )
hereby ::_thesis: verum
assume z in G ; ::_thesis: A1 \/ A2 meets G
then consider B being Subset of (TOP-REAL 2) such that
A13: B is_a_component_of A ` and
A14: G meets B by A1, A2, A11, A12, Th97;
consider B1 being Subset of ((TOP-REAL 2) | (A `)) such that
A15: B1 = B and
A16: B1 is a_component by A13, CONNSP_1:def_6;
A17: now__::_thesis:_(_(_B1_=_D1_&_B1_c=_A1_\/_A2_)_or_(_B1,D1_are_separated_&_B1_c=_A1_\/_A2_)_)
percases ( B1 = D1 or B1,D1 are_separated ) by A9, A16, CONNSP_1:34;
case B1 = D1 ; ::_thesis: B1 c= A1 \/ A2
hence B1 c= A1 \/ A2 by XBOOLE_1:7; ::_thesis: verum
end;
case B1,D1 are_separated ; ::_thesis: B1 c= A1 \/ A2
then A18: ( Cl B1 misses D1 or B1 misses Cl D1 ) by CONNSP_1:def_1;
( B1 is closed & D1 is closed ) by A9, A16, CONNSP_1:33;
then B1 misses D1 by A18, PRE_TOPC:22;
then A19: B1 /\ D1 = {} by XBOOLE_0:def_7;
B1 c= the carrier of ((TOP-REAL 2) | (A `)) ;
then B1 c= A ` by PRE_TOPC:8;
then B1 = B1 /\ (A `) by XBOOLE_1:28
.= (B1 /\ A1) \/ (B1 /\ A2) by A4, XBOOLE_1:23
.= B1 /\ A2 by A19 ;
then A20: B1 c= A2 by XBOOLE_1:17;
A2 c= A1 \/ A2 by XBOOLE_1:7;
hence B1 c= A1 \/ A2 by A20, XBOOLE_1:1; ::_thesis: verum
end;
end;
end;
G /\ B <> {} by A14, XBOOLE_0:def_7;
then (A1 \/ A2) /\ G <> {} by A15, A17, XBOOLE_1:3, XBOOLE_1:26;
hence A1 \/ A2 meets G by XBOOLE_0:def_7; ::_thesis: verum
end;
end;
then z in Cl (A1 \/ A2) by A11, PRE_TOPC:def_7;
then z in (Cl A1) \/ (Cl A2) by PRE_TOPC:20;
then A21: ( z in Cl A1 or z in Cl A2 ) by XBOOLE_0:def_3;
not z in A ` by A11, XBOOLE_0:def_5;
then ( not z in A1 & not z in A2 ) by A4, XBOOLE_0:def_3;
hence z in (Cl A1) \ A1 by A6, A21, XBOOLE_0:def_5; ::_thesis: verum
end;
( (Cl A1) \ A1 c= A1 ` & (Cl A2) \ A2 c= A2 ` ) by XBOOLE_1:33;
then (Cl A1) \ A1 c= A by A6, A8, XBOOLE_1:19;
then A = (Cl A1) \ A1 by A10, XBOOLE_0:def_10;
hence ex A1, A2 being Subset of (TOP-REAL 2) st
( A ` = A1 \/ A2 & A1 misses A2 & (Cl A1) \ A1 = (Cl A2) \ A2 & A = (Cl A1) \ A1 & ( for C1, C2 being Subset of ((TOP-REAL 2) | (A `)) st C1 = A1 & C2 = A2 holds
( C1 is a_component & C2 is a_component ) ) ) by A4, A5, A6, A7; ::_thesis: verum
end;
hereby ::_thesis: verum
assume ex A1, A2 being Subset of (TOP-REAL 2) st
( A ` = A1 \/ A2 & A1 misses A2 & (Cl A1) \ A1 = (Cl A2) \ A2 & A = (Cl A1) \ A1 & ( for C1, C2 being Subset of ((TOP-REAL 2) | (A `)) st C1 = A1 & C2 = A2 holds
( C1 is a_component & C2 is a_component ) ) ) ; ::_thesis: ( A is boundary & A is Jordan )
then consider A1, A2 being Subset of (TOP-REAL 2) such that
A22: A ` = A1 \/ A2 and
A23: ( A1 misses A2 & (Cl A1) \ A1 = (Cl A2) \ A2 ) and
A24: A = (Cl A1) \ A1 and
A25: for C1, C2 being Subset of ((TOP-REAL 2) | (A `)) st C1 = A1 & C2 = A2 holds
( C1 is a_component & C2 is a_component ) ;
for x being set
for V being Subset of (TOP-REAL 2) st x in A & x in V & V is open holds
ex B being Subset of (TOP-REAL 2) st
( B is_a_component_of A ` & V meets B )
proof
A2 c= A ` by A22, XBOOLE_1:7;
then reconsider D2 = A2 as Subset of ((TOP-REAL 2) | (A `)) by PRE_TOPC:8;
A1 c= A ` by A22, XBOOLE_1:7;
then reconsider D1 = A1 as Subset of ((TOP-REAL 2) | (A `)) by PRE_TOPC:8;
let x be set ; ::_thesis: for V being Subset of (TOP-REAL 2) st x in A & x in V & V is open holds
ex B being Subset of (TOP-REAL 2) st
( B is_a_component_of A ` & V meets B )
let V be Subset of (TOP-REAL 2); ::_thesis: ( x in A & x in V & V is open implies ex B being Subset of (TOP-REAL 2) st
( B is_a_component_of A ` & V meets B ) )
assume that
A26: x in A and
A27: ( x in V & V is open ) ; ::_thesis: ex B being Subset of (TOP-REAL 2) st
( B is_a_component_of A ` & V meets B )
D2 = A2 ;
then D1 is a_component by A25;
then A28: A1 is_a_component_of A ` by CONNSP_1:def_6;
x in Cl A1 by A24, A26, XBOOLE_0:def_5;
then A1 meets V by A27, PRE_TOPC:def_7;
hence ex B being Subset of (TOP-REAL 2) st
( B is_a_component_of A ` & V meets B ) by A28; ::_thesis: verum
end;
hence ( A is boundary & A is Jordan ) by A1, A22, A23, A25, Th97, JORDAN1:def_2; ::_thesis: verum
end;
end;
theorem Th99: :: JORDAN2C:99
for n being Element of NAT
for p being Point of (TOP-REAL n)
for P being Subset of (TOP-REAL n) st n >= 1 & P = {p} holds
P is boundary
proof
let n be Element of NAT ; ::_thesis: for p being Point of (TOP-REAL n)
for P being Subset of (TOP-REAL n) st n >= 1 & P = {p} holds
P is boundary
let p be Point of (TOP-REAL n); ::_thesis: for P being Subset of (TOP-REAL n) st n >= 1 & P = {p} holds
P is boundary
let P be Subset of (TOP-REAL n); ::_thesis: ( n >= 1 & P = {p} implies P is boundary )
assume that
A1: n >= 1 and
A2: P = {p} ; ::_thesis: P is boundary
the carrier of (TOP-REAL n) c= Cl (P `)
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in the carrier of (TOP-REAL n) or z in Cl (P `) )
assume A3: z in the carrier of (TOP-REAL n) ; ::_thesis: z in Cl (P `)
percases ( z = p or z <> p ) ;
supposeA4: z = p ; ::_thesis: z in Cl (P `)
reconsider ez = z as Point of (Euclid n) by A3, TOPREAL3:8;
for G1 being Subset of (TOP-REAL n) st G1 is open & z in G1 holds
P ` meets G1
proof
let G1 be Subset of (TOP-REAL n); ::_thesis: ( G1 is open & z in G1 implies P ` meets G1 )
assume A5: G1 is open ; ::_thesis: ( not z in G1 or P ` meets G1 )
thus ( z in G1 implies P ` meets G1 ) ::_thesis: verum
proof
A6: TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def_8;
then reconsider GG = G1 as Subset of (TopSpaceMetr (Euclid n)) ;
assume A7: z in G1 ; ::_thesis: P ` meets G1
GG is open by A5, A6, PRE_TOPC:30;
then consider r being real number such that
A8: r > 0 and
A9: Ball (ez,r) c= GG by A7, TOPMETR:15;
reconsider r = r as Real by XREAL_0:def_1;
set p2 = p - (((r / 2) / (sqrt n)) * (1.REAL n));
reconsider ep2 = p - (((r / 2) / (sqrt n)) * (1.REAL n)) as Point of (Euclid n) by TOPREAL3:8;
A10: 0 < sqrt n by A1, SQUARE_1:25;
A11: |.(p - (p - (((r / 2) / (sqrt n)) * (1.REAL n)))).| = |.((p - p) + (((r / 2) / (sqrt n)) * (1.REAL n))).| by EUCLID:47
.= |.(((((r / 2) / (sqrt n)) * (1.REAL n)) + p) - p).| by EUCLID:45
.= |.(((r / 2) / (sqrt n)) * (1.REAL n)).| by EUCLID:48
.= (abs ((r / 2) / (sqrt n))) * |.(1.REAL n).| by TOPRNS_1:7
.= (abs ((r / 2) / (sqrt n))) * (sqrt n) by EUCLID:73
.= ((abs (r / 2)) / (abs (sqrt n))) * (sqrt n) by COMPLEX1:67
.= ((abs (r / 2)) / (sqrt n)) * (sqrt n) by A10, ABSVALUE:def_1
.= abs (r / 2) by A10, XCMPLX_1:87
.= r / 2 by A8, ABSVALUE:def_1 ;
r / 2 > 0 by A8, XREAL_1:139;
then p <> p - (((r / 2) / (sqrt n)) * (1.REAL n)) by A11, TOPRNS_1:28;
then not p - (((r / 2) / (sqrt n)) * (1.REAL n)) in P by A2, TARSKI:def_1;
then A12: p - (((r / 2) / (sqrt n)) * (1.REAL n)) in P ` by XBOOLE_0:def_5;
r / 2 < r by A8, XREAL_1:216;
then dist (ez,ep2) < r by A4, A11, JGRAPH_1:28;
then p - (((r / 2) / (sqrt n)) * (1.REAL n)) in Ball (ez,r) by METRIC_1:11;
hence P ` meets G1 by A9, A12, XBOOLE_0:3; ::_thesis: verum
end;
end;
hence z in Cl (P `) by A3, PRE_TOPC:def_7; ::_thesis: verum
end;
suppose z <> p ; ::_thesis: z in Cl (P `)
then not z in P by A2, TARSKI:def_1;
then A13: z in P ` by A3, XBOOLE_0:def_5;
P ` c= Cl (P `) by PRE_TOPC:18;
hence z in Cl (P `) by A13; ::_thesis: verum
end;
end;
end;
then Cl (P `) = [#] (TOP-REAL n) by XBOOLE_0:def_10;
then P ` is dense by TOPS_1:def_3;
hence P is boundary by TOPS_1:def_4; ::_thesis: verum
end;
theorem Th100: :: JORDAN2C:100
for p, q being Point of (TOP-REAL 2)
for r being Real st p `1 = q `2 & - (p `2) = q `1 & p = r * q holds
( p `1 = 0 & p `2 = 0 & p = 0. (TOP-REAL 2) )
proof
let p, q be Point of (TOP-REAL 2); ::_thesis: for r being Real st p `1 = q `2 & - (p `2) = q `1 & p = r * q holds
( p `1 = 0 & p `2 = 0 & p = 0. (TOP-REAL 2) )
let r be Real; ::_thesis: ( p `1 = q `2 & - (p `2) = q `1 & p = r * q implies ( p `1 = 0 & p `2 = 0 & p = 0. (TOP-REAL 2) ) )
A1: 1 + (r * r) > 0 + 0 by XREAL_1:8, XREAL_1:63;
assume ( p `1 = q `2 & - (p `2) = q `1 & p = r * q ) ; ::_thesis: ( p `1 = 0 & p `2 = 0 & p = 0. (TOP-REAL 2) )
then A2: p = |[(r * (- (p `2))),(r * (p `1))]| by EUCLID:57;
then p `2 = r * (p `1) by EUCLID:52;
then p `1 = - (r * (r * (p `1))) by A2, EUCLID:52
.= - ((r * r) * (p `1)) ;
then (1 + (r * r)) * (p `1) = 0 ;
hence A3: p `1 = 0 by A1, XCMPLX_1:6; ::_thesis: ( p `2 = 0 & p = 0. (TOP-REAL 2) )
p `1 = r * (- (p `2)) by A2, EUCLID:52;
then p `2 = - ((r * r) * (p `2)) by A2, EUCLID:52;
then (1 + (r * r)) * (p `2) = 0 ;
hence p `2 = 0 by A1, XCMPLX_1:6; ::_thesis: p = 0. (TOP-REAL 2)
hence p = 0. (TOP-REAL 2) by A3, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
theorem Th101: :: JORDAN2C:101
for q1, q2 being Point of (TOP-REAL 2) holds LSeg (q1,q2) is boundary
proof
let q1, q2 be Point of (TOP-REAL 2); ::_thesis: LSeg (q1,q2) is boundary
percases ( q1 = q2 or q1 <> q2 ) ;
suppose q1 = q2 ; ::_thesis: LSeg (q1,q2) is boundary
then LSeg (q1,q2) = {q1} by RLTOPSP1:70;
hence LSeg (q1,q2) is boundary by Th99; ::_thesis: verum
end;
supposeA1: q1 <> q2 ; ::_thesis: LSeg (q1,q2) is boundary
set P = LSeg (q1,q2);
the carrier of (TOP-REAL 2) c= Cl ((LSeg (q1,q2)) `)
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in the carrier of (TOP-REAL 2) or z in Cl ((LSeg (q1,q2)) `) )
assume A2: z in the carrier of (TOP-REAL 2) ; ::_thesis: z in Cl ((LSeg (q1,q2)) `)
percases ( z in LSeg (q1,q2) or not z in LSeg (q1,q2) ) ;
supposeA3: z in LSeg (q1,q2) ; ::_thesis: z in Cl ((LSeg (q1,q2)) `)
reconsider ez = z as Point of (Euclid 2) by A2, TOPREAL3:8;
set p1 = q1 - q2;
consider s being Real such that
A4: z = ((1 - s) * q1) + (s * q2) and
0 <= s and
s <= 1 by A3;
set p = ((1 - s) * q1) + (s * q2);
A5: now__::_thesis:_not_|.(q1_-_q2).|_=_0
assume |.(q1 - q2).| = 0 ; ::_thesis: contradiction
then q1 - q2 = 0. (TOP-REAL 2) by TOPRNS_1:24;
hence contradiction by A1, EUCLID:43; ::_thesis: verum
end;
for G1 being Subset of (TOP-REAL 2) st G1 is open & z in G1 holds
(LSeg (q1,q2)) ` meets G1
proof
let G1 be Subset of (TOP-REAL 2); ::_thesis: ( G1 is open & z in G1 implies (LSeg (q1,q2)) ` meets G1 )
assume A6: G1 is open ; ::_thesis: ( not z in G1 or (LSeg (q1,q2)) ` meets G1 )
thus ( z in G1 implies (LSeg (q1,q2)) ` meets G1 ) ::_thesis: verum
proof
A7: TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) = TopSpaceMetr (Euclid 2) by EUCLID:def_8;
then reconsider GG = G1 as Subset of (TopSpaceMetr (Euclid 2)) ;
assume A8: z in G1 ; ::_thesis: (LSeg (q1,q2)) ` meets G1
GG is open by A6, A7, PRE_TOPC:30;
then consider r being real number such that
A9: r > 0 and
A10: Ball (ez,r) c= G1 by A8, TOPMETR:15;
reconsider r = r as Real by XREAL_0:def_1;
A11: r / 2 < r by A9, XREAL_1:216;
set p2 = (((r / 2) / |.(q1 - q2).|) * |[(- ((q1 - q2) `2)),((q1 - q2) `1)]|) + (((1 - s) * q1) + (s * q2));
now__::_thesis:_not_(((r_/_2)_/_|.(q1_-_q2).|)_*_|[(-_((q1_-_q2)_`2)),((q1_-_q2)_`1)]|)_+_(((1_-_s)_*_q1)_+_(s_*_q2))_in_LSeg_(q1,q2)
assume (((r / 2) / |.(q1 - q2).|) * |[(- ((q1 - q2) `2)),((q1 - q2) `1)]|) + (((1 - s) * q1) + (s * q2)) in LSeg (q1,q2) ; ::_thesis: contradiction
then consider s2 being Real such that
A12: (((r / 2) / |.(q1 - q2).|) * |[(- ((q1 - q2) `2)),((q1 - q2) `1)]|) + (((1 - s) * q1) + (s * q2)) = ((1 - s2) * q1) + (s2 * q2) and
0 <= s2 and
s2 <= 1 ;
A13: now__::_thesis:_not_s_-_s2_=_0
assume s - s2 = 0 ; ::_thesis: contradiction
then ((r / 2) / |.(q1 - q2).|) * |[(- ((q1 - q2) `2)),((q1 - q2) `1)]| = (((1 - s) * q1) + (s * q2)) - (((1 - s) * q1) + (s * q2)) by A12, EUCLID:48;
then A14: ((r / 2) / |.(q1 - q2).|) * |[(- ((q1 - q2) `2)),((q1 - q2) `1)]| = 0. (TOP-REAL 2) by EUCLID:42;
A15: (r / 2) / |.(q1 - q2).| = (r * (2 ")) * (|.(q1 - q2).| ") by XCMPLX_0:def_9
.= r * ((2 ") * (|.(q1 - q2).| ")) ;
(2 ") * (|.(q1 - q2).| ") <> 0 by A5;
then |[(- ((q1 - q2) `2)),((q1 - q2) `1)]| = 0. (TOP-REAL 2) by A9, A14, A15, EUCLID:31, XCMPLX_1:6;
then A16: ( (0. (TOP-REAL 2)) `1 = - ((q1 - q2) `2) & (0. (TOP-REAL 2)) `2 = (q1 - q2) `1 ) by EUCLID:52;
( (0. (TOP-REAL 2)) `1 = 0 & (0. (TOP-REAL 2)) `2 = 0 ) by EUCLID:52, EUCLID:54;
hence contradiction by A1, A16, EUCLID:43, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
A17: ((((r / 2) / |.(q1 - q2).|) * |[(- ((q1 - q2) `2)),((q1 - q2) `1)]|) + (((1 - s) * q1) + (s * q2))) - (((1 - s) * q1) + (s * q2)) = ((r / 2) / |.(q1 - q2).|) * |[(- ((q1 - q2) `2)),((q1 - q2) `1)]| by EUCLID:48;
((((r / 2) / |.(q1 - q2).|) * |[(- ((q1 - q2) `2)),((q1 - q2) `1)]|) + (((1 - s) * q1) + (s * q2))) - (((1 - s) * q1) + (s * q2)) = ((((1 - s2) * q1) + (s2 * q2)) - ((1 - s) * q1)) - (s * q2) by A12, EUCLID:46
.= ((((1 - s2) * q1) - ((1 - s) * q1)) + (s2 * q2)) - (s * q2) by EUCLID:26
.= ((((1 - s2) - (1 - s)) * q1) + (s2 * q2)) - (s * q2) by EUCLID:50
.= ((s - s2) * q1) + ((s2 * q2) - (s * q2)) by EUCLID:45
.= ((s - s2) * q1) + ((s2 - s) * q2) by EUCLID:50
.= ((s - s2) * q1) + ((- (s - s2)) * q2)
.= ((s - s2) * q1) - ((s - s2) * q2) by EUCLID:40
.= (s - s2) * (q1 - q2) by EUCLID:49 ;
then ((1 / (s - s2)) * (s - s2)) * (q1 - q2) = (1 / (s - s2)) * (((r / 2) / |.(q1 - q2).|) * |[(- ((q1 - q2) `2)),((q1 - q2) `1)]|) by A17, EUCLID:30;
then 1 * (q1 - q2) = (1 / (s - s2)) * (((r / 2) / |.(q1 - q2).|) * |[(- ((q1 - q2) `2)),((q1 - q2) `1)]|) by A13, XCMPLX_1:106;
then q1 - q2 = (1 / (s - s2)) * (((r / 2) / |.(q1 - q2).|) * |[(- ((q1 - q2) `2)),((q1 - q2) `1)]|) by EUCLID:29;
then A18: q1 - q2 = ((1 / (s - s2)) * ((r / 2) / |.(q1 - q2).|)) * |[(- ((q1 - q2) `2)),((q1 - q2) `1)]| by EUCLID:30;
( (q1 - q2) `1 = |[(- ((q1 - q2) `2)),((q1 - q2) `1)]| `2 & - ((q1 - q2) `2) = |[(- ((q1 - q2) `2)),((q1 - q2) `1)]| `1 ) by EUCLID:52;
then q1 - q2 = 0. (TOP-REAL 2) by A18, Th100;
hence contradiction by A1, EUCLID:43; ::_thesis: verum
end;
then A19: (((r / 2) / |.(q1 - q2).|) * |[(- ((q1 - q2) `2)),((q1 - q2) `1)]|) + (((1 - s) * q1) + (s * q2)) in the carrier of (TOP-REAL 2) \ (LSeg (q1,q2)) by XBOOLE_0:def_5;
reconsider ep2 = (((r / 2) / |.(q1 - q2).|) * |[(- ((q1 - q2) `2)),((q1 - q2) `1)]|) + (((1 - s) * q1) + (s * q2)) as Point of (Euclid 2) by TOPREAL3:8;
A20: ((((1 - s) * q1) + (s * q2)) + (- (((r / 2) / |.(q1 - q2).|) * |[(- ((q1 - q2) `2)),((q1 - q2) `1)]|))) - (((1 - s) * q1) + (s * q2)) = - (((r / 2) / |.(q1 - q2).|) * |[(- ((q1 - q2) `2)),((q1 - q2) `1)]|) by EUCLID:48;
A21: ( |[(- ((q1 - q2) `2)),((q1 - q2) `1)]| `1 = - ((q1 - q2) `2) & |[(- ((q1 - q2) `2)),((q1 - q2) `1)]| `2 = (q1 - q2) `1 ) by EUCLID:52;
|.((((1 - s) * q1) + (s * q2)) - ((((r / 2) / |.(q1 - q2).|) * |[(- ((q1 - q2) `2)),((q1 - q2) `1)]|) + (((1 - s) * q1) + (s * q2)))).| = |.(((((1 - s) * q1) + (s * q2)) - (((r / 2) / |.(q1 - q2).|) * |[(- ((q1 - q2) `2)),((q1 - q2) `1)]|)) - (((1 - s) * q1) + (s * q2))).| by EUCLID:46
.= |.(- (((r / 2) / |.(q1 - q2).|) * |[(- ((q1 - q2) `2)),((q1 - q2) `1)]|)).| by A20
.= |.(((r / 2) / |.(q1 - q2).|) * |[(- ((q1 - q2) `2)),((q1 - q2) `1)]|).| by TOPRNS_1:26
.= (abs ((r / 2) / |.(q1 - q2).|)) * |.|[(- ((q1 - q2) `2)),((q1 - q2) `1)]|.| by TOPRNS_1:7
.= (abs ((r / 2) / |.(q1 - q2).|)) * (sqrt (((- ((q1 - q2) `2)) ^2) + (((q1 - q2) `1) ^2))) by A21, JGRAPH_1:30
.= (abs ((r / 2) / |.(q1 - q2).|)) * (sqrt ((((q1 - q2) `1) ^2) + (((q1 - q2) `2) ^2)))
.= (abs ((r / 2) / |.(q1 - q2).|)) * |.(q1 - q2).| by JGRAPH_1:30
.= ((abs (r / 2)) / (abs |.(q1 - q2).|)) * |.(q1 - q2).| by COMPLEX1:67
.= ((abs (r / 2)) / |.(q1 - q2).|) * |.(q1 - q2).| by ABSVALUE:def_1
.= abs (r / 2) by A5, XCMPLX_1:87
.= r / 2 by A9, ABSVALUE:def_1 ;
then dist (ez,ep2) < r by A4, A11, JGRAPH_1:28;
then (((r / 2) / |.(q1 - q2).|) * |[(- ((q1 - q2) `2)),((q1 - q2) `1)]|) + (((1 - s) * q1) + (s * q2)) in Ball (ez,r) by METRIC_1:11;
hence (LSeg (q1,q2)) ` meets G1 by A10, A19, XBOOLE_0:3; ::_thesis: verum
end;
end;
hence z in Cl ((LSeg (q1,q2)) `) by A2, PRE_TOPC:def_7; ::_thesis: verum
end;
supposeA22: not z in LSeg (q1,q2) ; ::_thesis: z in Cl ((LSeg (q1,q2)) `)
A23: (LSeg (q1,q2)) ` c= Cl ((LSeg (q1,q2)) `) by PRE_TOPC:18;
z in the carrier of (TOP-REAL 2) \ (LSeg (q1,q2)) by A2, A22, XBOOLE_0:def_5;
hence z in Cl ((LSeg (q1,q2)) `) by A23; ::_thesis: verum
end;
end;
end;
then Cl ((LSeg (q1,q2)) `) = [#] (TOP-REAL 2) by XBOOLE_0:def_10;
then (LSeg (q1,q2)) ` is dense by TOPS_1:def_3;
hence LSeg (q1,q2) is boundary by TOPS_1:def_4; ::_thesis: verum
end;
end;
end;
registration
let q1, q2 be Point of (TOP-REAL 2);
cluster LSeg (q1,q2) -> boundary ;
coherence
LSeg (q1,q2) is boundary by Th101;
end;
theorem Th102: :: JORDAN2C:102
for f being FinSequence of (TOP-REAL 2) holds L~ f is boundary
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: L~ f is boundary
A1: L~ f = union { (LSeg (f,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len f ) } by TOPREAL1:def_4;
defpred S1[ Element of NAT ] means for R1 being Subset of (TOP-REAL 2) st R1 = union { (LSeg (f,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= $1 ) } holds
R1 is boundary ;
A2: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
A3: now__::_thesis:_(_(_1_<=_k_&_k_+_1_<=_len_f_&_LSeg_(f,k)_is_boundary_)_or_(_(_not_1_<=_k_or_not_k_+_1_<=_len_f_)_&_LSeg_(f,k)_is_boundary_)_)
percases ( ( 1 <= k & k + 1 <= len f ) or not 1 <= k or not k + 1 <= len f ) ;
case ( 1 <= k & k + 1 <= len f ) ; ::_thesis: LSeg (f,k) is boundary
then LSeg (f,k) = LSeg ((f /. k),(f /. (k + 1))) by TOPREAL1:def_3;
hence LSeg (f,k) is boundary ; ::_thesis: verum
end;
case ( not 1 <= k or not k + 1 <= len f ) ; ::_thesis: LSeg (f,k) is boundary
then LSeg (f,k) = {} by TOPREAL1:def_3;
hence LSeg (f,k) is boundary ; ::_thesis: verum
end;
end;
end;
union { (LSeg (f,i2)) where i2 is Element of NAT : ( 1 <= i2 & i2 + 1 <= k ) } c= the carrier of (TOP-REAL 2)
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in union { (LSeg (f,i2)) where i2 is Element of NAT : ( 1 <= i2 & i2 + 1 <= k ) } or z in the carrier of (TOP-REAL 2) )
assume z in union { (LSeg (f,i2)) where i2 is Element of NAT : ( 1 <= i2 & i2 + 1 <= k ) } ; ::_thesis: z in the carrier of (TOP-REAL 2)
then consider x being set such that
A4: ( z in x & x in { (LSeg (f,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= k ) } ) by TARSKI:def_4;
ex i being Element of NAT st
( x = LSeg (f,i) & 1 <= i & i + 1 <= k ) by A4;
hence z in the carrier of (TOP-REAL 2) by A4; ::_thesis: verum
end;
then reconsider R3 = union { (LSeg (f,i2)) where i2 is Element of NAT : ( 1 <= i2 & i2 + 1 <= k ) } as Subset of (TOP-REAL 2) ;
assume for R1 being Subset of (TOP-REAL 2) st R1 = union { (LSeg (f,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= k ) } holds
R1 is boundary ; ::_thesis: S1[k + 1]
then A5: R3 is boundary ;
thus for R2 being Subset of (TOP-REAL 2) st R2 = union { (LSeg (f,i2)) where i2 is Element of NAT : ( 1 <= i2 & i2 + 1 <= k + 1 ) } holds
R2 is boundary ::_thesis: verum
proof
let R2 be Subset of (TOP-REAL 2); ::_thesis: ( R2 = union { (LSeg (f,i2)) where i2 is Element of NAT : ( 1 <= i2 & i2 + 1 <= k + 1 ) } implies R2 is boundary )
assume A6: R2 = union { (LSeg (f,i2)) where i2 is Element of NAT : ( 1 <= i2 & i2 + 1 <= k + 1 ) } ; ::_thesis: R2 is boundary
A7: R3 \/ (LSeg (f,k)) c= R2
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in R3 \/ (LSeg (f,k)) or z in R2 )
assume A8: z in R3 \/ (LSeg (f,k)) ; ::_thesis: z in R2
percases ( z in R3 or z in LSeg (f,k) ) by A8, XBOOLE_0:def_3;
suppose z in R3 ; ::_thesis: z in R2
then consider x being set such that
A9: ( z in x & x in { (LSeg (f,i2)) where i2 is Element of NAT : ( 1 <= i2 & i2 + 1 <= k ) } ) by TARSKI:def_4;
consider i2 being Element of NAT such that
A10: ( x = LSeg (f,i2) & 1 <= i2 ) and
A11: i2 + 1 <= k by A9;
i2 + 1 < k + 1 by A11, NAT_1:13;
then x in { (LSeg (f,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= k + 1 ) } by A10;
hence z in R2 by A6, A9, TARSKI:def_4; ::_thesis: verum
end;
supposeA12: z in LSeg (f,k) ; ::_thesis: z in R2
now__::_thesis:_z_in_R2
percases ( 1 <= k or k < 1 ) ;
suppose 1 <= k ; ::_thesis: z in R2
then LSeg (f,k) in { (LSeg (f,i2)) where i2 is Element of NAT : ( 1 <= i2 & i2 + 1 <= k + 1 ) } ;
hence z in R2 by A6, A12, TARSKI:def_4; ::_thesis: verum
end;
suppose k < 1 ; ::_thesis: z in R2
hence z in R2 by A12, TOPREAL1:def_3; ::_thesis: verum
end;
end;
end;
hence z in R2 ; ::_thesis: verum
end;
end;
end;
R2 c= R3 \/ (LSeg (f,k))
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in R2 or z in R3 \/ (LSeg (f,k)) )
assume z in R2 ; ::_thesis: z in R3 \/ (LSeg (f,k))
then consider x being set such that
A13: ( z in x & x in { (LSeg (f,i2)) where i2 is Element of NAT : ( 1 <= i2 & i2 + 1 <= k + 1 ) } ) by A6, TARSKI:def_4;
consider i2 being Element of NAT such that
A14: x = LSeg (f,i2) and
A15: 1 <= i2 and
A16: i2 + 1 <= k + 1 by A13;
now__::_thesis:_(_(_i2_+_1_<=_k_&_(_z_in_R3_or_z_in_LSeg_(f,k)_)_)_or_(_i2_+_1_>_k_&_(_z_in_R3_or_z_in_LSeg_(f,k)_)_)_)
percases ( i2 + 1 <= k or i2 + 1 > k ) ;
case i2 + 1 <= k ; ::_thesis: ( z in R3 or z in LSeg (f,k) )
then x in { (LSeg (f,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= k ) } by A14, A15;
hence ( z in R3 or z in LSeg (f,k) ) by A13, TARSKI:def_4; ::_thesis: verum
end;
case i2 + 1 > k ; ::_thesis: ( z in R3 or z in LSeg (f,k) )
then k + 1 <= i2 + 1 by NAT_1:13;
then i2 + 1 = k + 1 by A16, XXREAL_0:1;
hence ( z in R3 or z in LSeg (f,k) ) by A13, A14; ::_thesis: verum
end;
end;
end;
hence z in R3 \/ (LSeg (f,k)) by XBOOLE_0:def_3; ::_thesis: verum
end;
then R2 = R3 \/ (LSeg (f,k)) by A7, XBOOLE_0:def_10;
hence R2 is boundary by A5, A3, TOPS_1:49; ::_thesis: verum
end;
end;
union { (LSeg (f,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= 0 ) } c= {}
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in union { (LSeg (f,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= 0 ) } or z in {} )
assume z in union { (LSeg (f,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= 0 ) } ; ::_thesis: z in {}
then consider x being set such that
A17: ( z in x & x in { (LSeg (f,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= 0 ) } ) by TARSKI:def_4;
ex i being Element of NAT st
( x = LSeg (f,i) & 1 <= i & i + 1 <= 0 ) by A17;
hence z in {} ; ::_thesis: verum
end;
then A18: S1[ 0 ] ;
for j being Element of NAT holds S1[j] from NAT_1:sch_1(A18, A2);
hence L~ f is boundary by A1; ::_thesis: verum
end;
registration
let f be FinSequence of (TOP-REAL 2);
cluster L~ f -> boundary ;
coherence
L~ f is boundary by Th102;
end;
theorem Th103: :: JORDAN2C:103
for n being Element of NAT
for r being Real
for ep being Point of (Euclid n)
for p, q being Point of (TOP-REAL n) st p = ep & q in Ball (ep,r) holds
( |.(p - q).| < r & |.(q - p).| < r )
proof
let n be Element of NAT ; ::_thesis: for r being Real
for ep being Point of (Euclid n)
for p, q being Point of (TOP-REAL n) st p = ep & q in Ball (ep,r) holds
( |.(p - q).| < r & |.(q - p).| < r )
let r be Real; ::_thesis: for ep being Point of (Euclid n)
for p, q being Point of (TOP-REAL n) st p = ep & q in Ball (ep,r) holds
( |.(p - q).| < r & |.(q - p).| < r )
let ep be Point of (Euclid n); ::_thesis: for p, q being Point of (TOP-REAL n) st p = ep & q in Ball (ep,r) holds
( |.(p - q).| < r & |.(q - p).| < r )
let p, q be Point of (TOP-REAL n); ::_thesis: ( p = ep & q in Ball (ep,r) implies ( |.(p - q).| < r & |.(q - p).| < r ) )
assume that
A1: p = ep and
A2: q in Ball (ep,r) ; ::_thesis: ( |.(p - q).| < r & |.(q - p).| < r )
reconsider eq = q as Point of (Euclid n) by TOPREAL3:8;
dist (ep,eq) < r by A2, METRIC_1:11;
hence ( |.(p - q).| < r & |.(q - p).| < r ) by A1, JGRAPH_1:28; ::_thesis: verum
end;
theorem :: JORDAN2C:104
for D being non empty compact non horizontal non vertical Subset of (TOP-REAL 2)
for a being Real
for p being Point of (TOP-REAL 2) st a > 0 & p in L~ (SpStSeq D) holds
ex q being Point of (TOP-REAL 2) st
( q in UBD (L~ (SpStSeq D)) & |.(p - q).| < a )
proof
let D be non empty compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for a being Real
for p being Point of (TOP-REAL 2) st a > 0 & p in L~ (SpStSeq D) holds
ex q being Point of (TOP-REAL 2) st
( q in UBD (L~ (SpStSeq D)) & |.(p - q).| < a )
let a be Real; ::_thesis: for p being Point of (TOP-REAL 2) st a > 0 & p in L~ (SpStSeq D) holds
ex q being Point of (TOP-REAL 2) st
( q in UBD (L~ (SpStSeq D)) & |.(p - q).| < a )
let p be Point of (TOP-REAL 2); ::_thesis: ( a > 0 & p in L~ (SpStSeq D) implies ex q being Point of (TOP-REAL 2) st
( q in UBD (L~ (SpStSeq D)) & |.(p - q).| < a ) )
assume that
A1: a > 0 and
A2: p in L~ (SpStSeq D) ; ::_thesis: ex q being Point of (TOP-REAL 2) st
( q in UBD (L~ (SpStSeq D)) & |.(p - q).| < a )
set q1 = the Element of UBD (L~ (SpStSeq D));
set A = L~ (SpStSeq D);
(L~ (SpStSeq D)) ` <> {} by SPRECT_1:def_3;
then consider A1, A2 being Subset of (TOP-REAL 2) such that
A3: (L~ (SpStSeq D)) ` = A1 \/ A2 and
A1 misses A2 and
A4: (Cl A1) \ A1 = (Cl A2) \ A2 and
A5: L~ (SpStSeq D) = (Cl A1) \ A1 and
A6: for C1, C2 being Subset of ((TOP-REAL 2) | ((L~ (SpStSeq D)) `)) st C1 = A1 & C2 = A2 holds
( C1 is a_component & C2 is a_component ) by Th98;
A7: Down (A2,((L~ (SpStSeq D)) `)) = A2 by A3, XBOOLE_1:21;
UBD (L~ (SpStSeq D)) is_outside_component_of L~ (SpStSeq D) by Th68;
then UBD (L~ (SpStSeq D)) is_a_component_of (L~ (SpStSeq D)) ` by Def3;
then consider B1 being Subset of ((TOP-REAL 2) | ((L~ (SpStSeq D)) `)) such that
A8: B1 = UBD (L~ (SpStSeq D)) and
A9: B1 is a_component by CONNSP_1:def_6;
B1 c= [#] ((TOP-REAL 2) | ((L~ (SpStSeq D)) `)) ;
then A10: UBD (L~ (SpStSeq D)) c= A1 \/ A2 by A3, A8, PRE_TOPC:def_5;
A11: Down (A1,((L~ (SpStSeq D)) `)) = A1 by A3, XBOOLE_1:21;
then A12: Down (A1,((L~ (SpStSeq D)) `)) is a_component by A6, A7;
A13: Down (A2,((L~ (SpStSeq D)) `)) is a_component by A6, A11, A7;
A14: UBD (L~ (SpStSeq D)) <> {} by Th96;
then A15: the Element of UBD (L~ (SpStSeq D)) in UBD (L~ (SpStSeq D)) ;
percases ( the Element of UBD (L~ (SpStSeq D)) in A1 or the Element of UBD (L~ (SpStSeq D)) in A2 ) by A10, A15, XBOOLE_0:def_3;
suppose the Element of UBD (L~ (SpStSeq D)) in A1 ; ::_thesis: ex q being Point of (TOP-REAL 2) st
( q in UBD (L~ (SpStSeq D)) & |.(p - q).| < a )
then B1 /\ (Down (A1,((L~ (SpStSeq D)) `))) <> {} ((TOP-REAL 2) | ((L~ (SpStSeq D)) `)) by A11, A8, A14, XBOOLE_0:def_4;
then B1 meets Down (A1,((L~ (SpStSeq D)) `)) by XBOOLE_0:def_7;
then B1 = Down (A1,((L~ (SpStSeq D)) `)) by A12, A9, CONNSP_1:35;
then A16: p in Cl (UBD (L~ (SpStSeq D))) by A2, A5, A11, A8, XBOOLE_0:def_5;
reconsider ep = p as Point of (Euclid 2) by TOPREAL3:8;
reconsider G2 = Ball (ep,a) as Subset of (TOP-REAL 2) by TOPREAL3:8;
the distance of (Euclid 2) is Reflexive by METRIC_1:def_6;
then dist (ep,ep) = 0 by METRIC_1:def_2;
then A17: p in Ball (ep,a) by A1, METRIC_1:11;
G2 is open by GOBOARD6:3;
then UBD (L~ (SpStSeq D)) meets G2 by A16, A17, PRE_TOPC:def_7;
then consider t2 being set such that
A18: t2 in UBD (L~ (SpStSeq D)) and
A19: t2 in G2 by XBOOLE_0:3;
reconsider qt2 = t2 as Point of (TOP-REAL 2) by A18;
|.(p - qt2).| < a by A19, Th103;
hence ex q being Point of (TOP-REAL 2) st
( q in UBD (L~ (SpStSeq D)) & |.(p - q).| < a ) by A18; ::_thesis: verum
end;
suppose the Element of UBD (L~ (SpStSeq D)) in A2 ; ::_thesis: ex q being Point of (TOP-REAL 2) st
( q in UBD (L~ (SpStSeq D)) & |.(p - q).| < a )
then B1 /\ (Down (A2,((L~ (SpStSeq D)) `))) <> {} ((TOP-REAL 2) | ((L~ (SpStSeq D)) `)) by A7, A8, A14, XBOOLE_0:def_4;
then B1 meets Down (A2,((L~ (SpStSeq D)) `)) by XBOOLE_0:def_7;
then B1 = Down (A2,((L~ (SpStSeq D)) `)) by A13, A9, CONNSP_1:35;
then A20: p in Cl (UBD (L~ (SpStSeq D))) by A2, A4, A5, A7, A8, XBOOLE_0:def_5;
reconsider ep = p as Point of (Euclid 2) by TOPREAL3:8;
reconsider G2 = Ball (ep,a) as Subset of (TOP-REAL 2) by TOPREAL3:8;
the distance of (Euclid 2) is Reflexive by METRIC_1:def_6;
then dist (ep,ep) = 0 by METRIC_1:def_2;
then A21: p in Ball (ep,a) by A1, METRIC_1:11;
G2 is open by GOBOARD6:3;
then UBD (L~ (SpStSeq D)) meets G2 by A20, A21, PRE_TOPC:def_7;
then consider t2 being set such that
A22: t2 in UBD (L~ (SpStSeq D)) and
A23: t2 in G2 by XBOOLE_0:3;
reconsider qt2 = t2 as Point of (TOP-REAL 2) by A22;
|.(p - qt2).| < a by A23, Th103;
hence ex q being Point of (TOP-REAL 2) st
( q in UBD (L~ (SpStSeq D)) & |.(p - q).| < a ) by A22; ::_thesis: verum
end;
end;
end;
theorem :: JORDAN2C:105
REAL 0 = {(0. (TOP-REAL 0))} by EUCLID:77;
theorem Th106: :: JORDAN2C:106
for n being Element of NAT
for A being Subset of (TOP-REAL n) st A is bounded holds
BDD A is bounded
proof
let n be Element of NAT ; ::_thesis: for A being Subset of (TOP-REAL n) st A is bounded holds
BDD A is bounded
let A be Subset of (TOP-REAL n); ::_thesis: ( A is bounded implies BDD A is bounded )
assume A is bounded ; ::_thesis: BDD A is bounded
then consider r being Real such that
A1: for q being Point of (TOP-REAL n) st q in A holds
|.q.| < r by Th34;
percases ( n >= 1 or n < 1 ) ;
supposeA2: n >= 1 ; ::_thesis: BDD A is bounded
set a = r;
reconsider P = (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < r } as Subset of (TOP-REAL n) by EUCLID:22;
A3: P c= A `
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in P or z in A ` )
assume A4: z in P ; ::_thesis: z in A `
then reconsider q0 = z as Point of (TOP-REAL n) ;
not z in { q where q is Point of (TOP-REAL n) : |.q.| < r } by A4, XBOOLE_0:def_5;
then |.q0.| >= r ;
then not q0 in A by A1;
hence z in A ` by XBOOLE_0:def_5; ::_thesis: verum
end;
then A5: Down (P,(A `)) = P by XBOOLE_1:28;
now__::_thesis:_BDD_A_is_bounded
percases ( n >= 2 or n < 2 ) ;
suppose n >= 2 ; ::_thesis: BDD A is bounded
then A6: P is connected by Th53;
now__::_thesis:_BDD_A_is_bounded
assume not BDD A is bounded ; ::_thesis: contradiction
then consider q being Point of (TOP-REAL n) such that
A7: q in BDD A and
A8: not |.q.| < r by Th34;
consider y being set such that
A9: q in y and
A10: y in { B3 where B3 is Subset of (TOP-REAL n) : B3 is_inside_component_of A } by A7, TARSKI:def_4;
consider B3 being Subset of (TOP-REAL n) such that
A11: y = B3 and
A12: B3 is_inside_component_of A by A10;
q in the carrier of (TOP-REAL n) ;
then A13: q in REAL n by EUCLID:22;
B3 is_a_component_of A ` by A12, Def2;
then consider B4 being Subset of ((TOP-REAL n) | (A `)) such that
A14: B4 = B3 and
A15: B4 is a_component by CONNSP_1:def_6;
for q2 being Point of (TOP-REAL n) st q2 = q holds
|.q2.| >= r by A8;
then not q in { q2 where q2 is Point of (TOP-REAL n) : |.q2.| < r } ;
then q in P by A13, XBOOLE_0:def_5;
then P /\ B4 <> {} ((TOP-REAL n) | (A `)) by A9, A11, A14, XBOOLE_0:def_4;
then P meets B4 by XBOOLE_0:def_7;
then A16: P c= B4 by A5, A6, A15, CONNSP_1:36, CONNSP_1:46;
B3 is bounded by A12, Def2;
hence contradiction by A2, A14, A16, Th54, RLTOPSP1:42; ::_thesis: verum
end;
hence BDD A is bounded ; ::_thesis: verum
end;
supposeA17: n < 2 ; ::_thesis: BDD A is bounded
{ q where q is Point of (TOP-REAL n) : for r2 being Real st q = |[r2]| holds
r2 >= r } c= the carrier of (TOP-REAL n)
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in { q where q is Point of (TOP-REAL n) : for r2 being Real st q = |[r2]| holds
r2 >= r } or z in the carrier of (TOP-REAL n) )
assume z in { q where q is Point of (TOP-REAL n) : for r2 being Real st q = |[r2]| holds
r2 >= r } ; ::_thesis: z in the carrier of (TOP-REAL n)
then ex q being Point of (TOP-REAL n) st
( q = z & ( for r2 being Real st q = |[r2]| holds
r2 >= r ) ) ;
hence z in the carrier of (TOP-REAL n) ; ::_thesis: verum
end;
then reconsider P2 = { q where q is Point of (TOP-REAL n) : for r2 being Real st q = |[r2]| holds
r2 >= r } as Subset of (TOP-REAL n) ;
{ q where q is Point of (TOP-REAL n) : for r2 being Real st q = |[r2]| holds
r2 <= - r } c= the carrier of (TOP-REAL n)
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in { q where q is Point of (TOP-REAL n) : for r2 being Real st q = |[r2]| holds
r2 <= - r } or z in the carrier of (TOP-REAL n) )
assume z in { q where q is Point of (TOP-REAL n) : for r2 being Real st q = |[r2]| holds
r2 <= - r } ; ::_thesis: z in the carrier of (TOP-REAL n)
then ex q being Point of (TOP-REAL n) st
( q = z & ( for r2 being Real st q = |[r2]| holds
r2 <= - r ) ) ;
hence z in the carrier of (TOP-REAL n) ; ::_thesis: verum
end;
then reconsider P1 = { q where q is Point of (TOP-REAL n) : for r2 being Real st q = |[r2]| holds
r2 <= - r } as Subset of (TOP-REAL n) ;
n < 1 + 1 by A17;
then n <= 1 by NAT_1:13;
then A18: n = 1 by A2, XXREAL_0:1;
A19: P c= P1 \/ P2
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in P or z in P1 \/ P2 )
assume A20: z in P ; ::_thesis: z in P1 \/ P2
then reconsider q0 = z as Point of (TOP-REAL n) ;
consider r3 being Real such that
A21: q0 = <*r3*> by A18, JORDAN2B:20;
not z in { q where q is Point of (TOP-REAL n) : |.q.| < r } by A20, XBOOLE_0:def_5;
then |.q0.| >= r ;
then A22: abs r3 >= r by A21, Th10;
percases ( - r >= r3 or r3 >= r ) by A22, SEQ_2:1;
suppose - r >= r3 ; ::_thesis: z in P1 \/ P2
then for r2 being Real st q0 = |[r2]| holds
r2 <= - r by A21, JORDAN2B:23;
then q0 in P1 ;
hence z in P1 \/ P2 by XBOOLE_0:def_3; ::_thesis: verum
end;
suppose r3 >= r ; ::_thesis: z in P1 \/ P2
then for r2 being Real st q0 = |[r2]| holds
r2 >= r by A21, JORDAN2B:23;
then q0 in P2 ;
hence z in P1 \/ P2 by XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
P1 \/ P2 c= P
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in P1 \/ P2 or z in P )
assume A23: z in P1 \/ P2 ; ::_thesis: z in P
percases ( z in P1 or z in P2 ) by A23, XBOOLE_0:def_3;
supposeA24: z in P1 ; ::_thesis: z in P
then A25: ex q being Point of (TOP-REAL n) st
( q = z & ( for r2 being Real st q = |[r2]| holds
r2 <= - r ) ) ;
for q2 being Point of (TOP-REAL n) st q2 = z holds
|.q2.| >= r
proof
let q2 be Point of (TOP-REAL n); ::_thesis: ( q2 = z implies |.q2.| >= r )
consider r3 being Real such that
A26: q2 = <*r3*> by A18, JORDAN2B:20;
assume A27: q2 = z ; ::_thesis: |.q2.| >= r
then A28: r3 <= - r by A25, A26;
now__::_thesis:_(_(_r_<_0_&_abs_r3_>=_r_)_or_(_r_>=_0_&_abs_r3_>=_r_)_)
percases ( r < 0 or r >= 0 ) ;
case r < 0 ; ::_thesis: abs r3 >= r
hence abs r3 >= r by COMPLEX1:46; ::_thesis: verum
end;
case r >= 0 ; ::_thesis: abs r3 >= r
then - r <= - 0 ;
then abs r3 = - r3 by A28, ABSVALUE:30;
hence abs r3 >= r by A25, A27, A26, XREAL_1:25; ::_thesis: verum
end;
end;
end;
hence |.q2.| >= r by A26, Th10; ::_thesis: verum
end;
then A29: not z in { q2 where q2 is Point of (TOP-REAL n) : |.q2.| < r } ;
z in the carrier of (TOP-REAL n) by A24;
then z in REAL n by EUCLID:22;
hence z in P by A29, XBOOLE_0:def_5; ::_thesis: verum
end;
supposeA30: z in P2 ; ::_thesis: z in P
then A31: ex q being Point of (TOP-REAL n) st
( q = z & ( for r2 being Real st q = |[r2]| holds
r2 >= r ) ) ;
for q2 being Point of (TOP-REAL n) st q2 = z holds
|.q2.| >= r
proof
let q2 be Point of (TOP-REAL n); ::_thesis: ( q2 = z implies |.q2.| >= r )
consider r3 being Real such that
A32: q2 = <*r3*> by A18, JORDAN2B:20;
assume q2 = z ; ::_thesis: |.q2.| >= r
then A33: r3 >= r by A31, A32;
now__::_thesis:_abs_r3_>=_r
percases ( r < 0 or r >= 0 ) ;
suppose r < 0 ; ::_thesis: abs r3 >= r
hence abs r3 >= r by COMPLEX1:46; ::_thesis: verum
end;
suppose r >= 0 ; ::_thesis: abs r3 >= r
hence abs r3 >= r by A33, ABSVALUE:def_1; ::_thesis: verum
end;
end;
end;
hence |.q2.| >= r by A32, Th10; ::_thesis: verum
end;
then A34: not z in { q2 where q2 is Point of (TOP-REAL n) : |.q2.| < r } ;
z in the carrier of (TOP-REAL n) by A30;
then z in REAL n by EUCLID:22;
hence z in P by A34, XBOOLE_0:def_5; ::_thesis: verum
end;
end;
end;
then A35: P = P1 \/ P2 by A19, XBOOLE_0:def_10;
then P2 c= P by XBOOLE_1:7;
then A36: Down (P2,(A `)) = P2 by A3, XBOOLE_1:1, XBOOLE_1:28;
for w1, w2 being Point of (TOP-REAL n) st w1 in P2 & w2 in P2 holds
LSeg (w1,w2) c= P2
proof
let w1, w2 be Point of (TOP-REAL n); ::_thesis: ( w1 in P2 & w2 in P2 implies LSeg (w1,w2) c= P2 )
assume that
A37: w1 in P2 and
A38: w2 in P2 ; ::_thesis: LSeg (w1,w2) c= P2
A39: ex q2 being Point of (TOP-REAL n) st
( q2 = w2 & ( for r2 being Real st q2 = |[r2]| holds
r2 >= r ) ) by A38;
consider r3 being Real such that
A40: w1 = <*r3*> by A18, JORDAN2B:20;
consider r4 being Real such that
A41: w2 = <*r4*> by A18, JORDAN2B:20;
A42: ex q1 being Point of (TOP-REAL n) st
( q1 = w1 & ( for r2 being Real st q1 = |[r2]| holds
r2 >= r ) ) by A37;
thus LSeg (w1,w2) c= P2 ::_thesis: verum
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in LSeg (w1,w2) or z in P2 )
assume z in LSeg (w1,w2) ; ::_thesis: z in P2
then consider r2 being Real such that
A43: z = ((1 - r2) * w1) + (r2 * w2) and
A44: 0 <= r2 and
A45: r2 <= 1 ;
reconsider q4 = z as Point of (TOP-REAL n) by A43;
( (1 - r2) * w1 = |[((1 - r2) * r3)]| & r2 * w2 = |[(r2 * r4)]| ) by A18, A40, A41, JORDAN2B:21;
then A46: z = |[(((1 - r2) * r3) + (r2 * r4))]| by A18, A43, JORDAN2B:22;
for r5 being Real st q4 = |[r5]| holds
r5 >= r
proof
let r5 be Real; ::_thesis: ( q4 = |[r5]| implies r5 >= r )
assume q4 = |[r5]| ; ::_thesis: r5 >= r
then A47: r5 = ((1 - r2) * r3) + (r2 * r4) by A46, JORDAN2B:23;
1 - r2 >= 0 by A45, XREAL_1:48;
then A48: (1 - r2) * r3 >= (1 - r2) * r by A42, A40, XREAL_1:64;
( r2 * r4 >= r2 * r & ((1 - r2) * r) + (r2 * r) = r ) by A39, A41, A44, XREAL_1:64;
hence r5 >= r by A47, A48, XREAL_1:7; ::_thesis: verum
end;
hence z in P2 ; ::_thesis: verum
end;
end;
then P2 is convex by JORDAN1:def_1;
then A49: Down (P2,(A `)) is connected by A36, CONNSP_1:46;
P1 c= P by A35, XBOOLE_1:7;
then A50: Down (P1,(A `)) = P1 by A3, XBOOLE_1:1, XBOOLE_1:28;
A51: now__::_thesis:_not_P2_is_bounded
assume P2 is bounded ; ::_thesis: contradiction
then consider r being Real such that
A52: for q being Point of (TOP-REAL n) st q in P2 holds
|.q.| < r by Th34;
( 0 <= abs r & 0 <= abs r ) by COMPLEX1:46;
then A53: abs ((abs r) + (abs r)) = (abs r) + (abs r) by ABSVALUE:def_1;
set p3 = |[((abs r) + (abs r))]|;
A54: abs r <= (abs r) + (abs r) by COMPLEX1:46, XREAL_1:31;
for r5 being Real st |[((abs r) + (abs r))]| = |[r5]| holds
r5 >= r
proof
let r5 be Real; ::_thesis: ( |[((abs r) + (abs r))]| = |[r5]| implies r5 >= r )
assume |[((abs r) + (abs r))]| = |[r5]| ; ::_thesis: r5 >= r
then A55: r5 = (abs r) + (abs r) by JORDAN2B:23;
( r <= abs r & abs r <= (abs r) + (abs r) ) by ABSVALUE:4, COMPLEX1:46, XREAL_1:31;
hence r5 >= r by A55, XXREAL_0:2; ::_thesis: verum
end;
then A56: |[((abs r) + (abs r))]| in P2 by A18;
( |.|[((abs r) + (abs r))]|.| = abs ((abs r) + (abs r)) & r <= abs r ) by Th10, ABSVALUE:4;
hence contradiction by A52, A56, A53, A54, XXREAL_0:2; ::_thesis: verum
end;
A57: now__::_thesis:_not_P1_is_bounded
assume P1 is bounded ; ::_thesis: contradiction
then consider r being Real such that
A58: for q being Point of (TOP-REAL n) st q in P1 holds
|.q.| < r by Th34;
( 0 <= abs r & 0 <= abs r ) by COMPLEX1:46;
then A59: abs ((abs r) + (abs r)) = (abs r) + (abs r) by ABSVALUE:def_1;
set p3 = |[(- ((abs r) + (abs r)))]|;
A60: ( r <= abs r & abs r <= (abs r) + (abs r) ) by ABSVALUE:4, COMPLEX1:46, XREAL_1:31;
for r5 being Real st |[(- ((abs r) + (abs r)))]| = |[r5]| holds
r5 <= - r
proof
let r5 be Real; ::_thesis: ( |[(- ((abs r) + (abs r)))]| = |[r5]| implies r5 <= - r )
r <= abs r by ABSVALUE:4;
then A61: - (abs r) <= - r by XREAL_1:24;
abs r <= (abs r) + (abs r) by COMPLEX1:46, XREAL_1:31;
then A62: - (abs r) >= - ((abs r) + (abs r)) by XREAL_1:24;
assume |[(- ((abs r) + (abs r)))]| = |[r5]| ; ::_thesis: r5 <= - r
then r5 = - ((abs r) + (abs r)) by JORDAN2B:23;
hence r5 <= - r by A61, A62, XXREAL_0:2; ::_thesis: verum
end;
then A63: |[(- ((abs r) + (abs r)))]| in P1 by A18;
|.|[(- ((abs r) + (abs r)))]|.| = abs (- ((abs r) + (abs r))) by Th10
.= abs ((abs r) + (abs r)) by COMPLEX1:52 ;
hence contradiction by A58, A63, A59, A60, XXREAL_0:2; ::_thesis: verum
end;
for w1, w2 being Point of (TOP-REAL n) st w1 in P1 & w2 in P1 holds
LSeg (w1,w2) c= P1
proof
let w1, w2 be Point of (TOP-REAL n); ::_thesis: ( w1 in P1 & w2 in P1 implies LSeg (w1,w2) c= P1 )
assume that
A64: w1 in P1 and
A65: w2 in P1 ; ::_thesis: LSeg (w1,w2) c= P1
consider r4 being Real such that
A66: w2 = <*r4*> by A18, JORDAN2B:20;
ex q2 being Point of (TOP-REAL n) st
( q2 = w2 & ( for r2 being Real st q2 = |[r2]| holds
r2 <= - r ) ) by A65;
then A67: r4 <= - r by A66;
consider r3 being Real such that
A68: w1 = <*r3*> by A18, JORDAN2B:20;
ex q1 being Point of (TOP-REAL n) st
( q1 = w1 & ( for r2 being Real st q1 = |[r2]| holds
r2 <= - r ) ) by A64;
then A69: r3 <= - r by A68;
thus LSeg (w1,w2) c= P1 ::_thesis: verum
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in LSeg (w1,w2) or z in P1 )
assume z in LSeg (w1,w2) ; ::_thesis: z in P1
then consider r2 being Real such that
A70: z = ((1 - r2) * w1) + (r2 * w2) and
A71: 0 <= r2 and
A72: r2 <= 1 ;
reconsider q4 = z as Point of (TOP-REAL n) by A70;
A73: r2 * w2 = |[(r2 * r4)]| by A18, A66, JORDAN2B:21;
(1 - r2) * w1 = (1 - r2) * |[r3]| by A68
.= |[((1 - r2) * r3)]| by JORDAN2B:21 ;
then A74: z = |[(((1 - r2) * r3) + (r2 * r4))]| by A18, A70, A73, JORDAN2B:22;
for r5 being Real st q4 = |[r5]| holds
r5 <= - r
proof
let r5 be Real; ::_thesis: ( q4 = |[r5]| implies r5 <= - r )
assume q4 = |[r5]| ; ::_thesis: r5 <= - r
then A75: r5 = ((1 - r2) * r3) + (r2 * r4) by A74, JORDAN2B:23;
1 - r2 >= 0 by A72, XREAL_1:48;
then A76: (1 - r2) * r3 <= (1 - r2) * (- r) by A69, XREAL_1:64;
( r2 * r4 <= r2 * (- r) & ((1 - r2) * (- r)) + (r2 * (- r)) = - r ) by A67, A71, XREAL_1:64;
hence r5 <= - r by A75, A76, XREAL_1:7; ::_thesis: verum
end;
hence z in P1 ; ::_thesis: verum
end;
end;
then P1 is convex by JORDAN1:def_1;
then A77: Down (P1,(A `)) is connected by A50, CONNSP_1:46;
now__::_thesis:_BDD_A_is_bounded
assume not BDD A is bounded ; ::_thesis: contradiction
then consider q being Point of (TOP-REAL n) such that
A78: q in BDD A and
A79: not |.q.| < r by Th34;
consider y being set such that
A80: q in y and
A81: y in { B3 where B3 is Subset of (TOP-REAL n) : B3 is_inside_component_of A } by A78, TARSKI:def_4;
consider B3 being Subset of (TOP-REAL n) such that
A82: y = B3 and
A83: B3 is_inside_component_of A by A81;
q in the carrier of (TOP-REAL n) ;
then A84: q in REAL n by EUCLID:22;
for q2 being Point of (TOP-REAL n) st q2 = q holds
|.q2.| >= r by A79;
then not q in { q2 where q2 is Point of (TOP-REAL n) : |.q2.| < r } ;
then A85: q in P by A84, XBOOLE_0:def_5;
B3 is_a_component_of A ` by A83, Def2;
then consider B4 being Subset of ((TOP-REAL n) | (A `)) such that
A86: B4 = B3 and
A87: B4 is a_component by CONNSP_1:def_6;
percases ( q in P1 or q in P2 ) by A19, A85, XBOOLE_0:def_3;
suppose q in P1 ; ::_thesis: contradiction
then P1 /\ B4 <> {} ((TOP-REAL n) | (A `)) by A80, A82, A86, XBOOLE_0:def_4;
then A88: P1 meets B4 by XBOOLE_0:def_7;
B3 is bounded by A83, Def2;
hence contradiction by A50, A57, A77, A86, A87, A88, CONNSP_1:36, RLTOPSP1:42; ::_thesis: verum
end;
suppose q in P2 ; ::_thesis: contradiction
then P2 /\ B4 <> {} ((TOP-REAL n) | (A `)) by A80, A82, A86, XBOOLE_0:def_4;
then A89: P2 meets B4 by XBOOLE_0:def_7;
B3 is bounded by A83, Def2;
hence contradiction by A36, A51, A49, A86, A87, A89, CONNSP_1:36, RLTOPSP1:42; ::_thesis: verum
end;
end;
end;
hence BDD A is bounded ; ::_thesis: verum
end;
end;
end;
hence BDD A is bounded ; ::_thesis: verum
end;
suppose n < 1 ; ::_thesis: BDD A is bounded
then n < 0 + 1 ;
then A90: n = 0 by NAT_1:13;
for q2 being Point of (TOP-REAL n) holds |.q2.| < 1
proof
let q2 be Point of (TOP-REAL n); ::_thesis: |.q2.| < 1
q2 = 0. (TOP-REAL n) by A90, EUCLID:77;
hence |.q2.| < 1 by TOPRNS_1:23; ::_thesis: verum
end;
then for q2 being Point of (TOP-REAL n) st q2 in [#] (TOP-REAL n) holds
|.q2.| < 1 ;
then [#] (TOP-REAL n) is bounded by Th34;
hence BDD A is bounded by RLTOPSP1:42; ::_thesis: verum
end;
end;
end;
theorem Th107: :: JORDAN2C:107
for G being non empty TopSpace
for A, B, C, D being Subset of G st B is a_component & C is a_component & A \/ B = the carrier of G & C misses A holds
C = B
proof
let G be non empty TopSpace; ::_thesis: for A, B, C, D being Subset of G st B is a_component & C is a_component & A \/ B = the carrier of G & C misses A holds
C = B
let A, B, C, D be Subset of G; ::_thesis: ( B is a_component & C is a_component & A \/ B = the carrier of G & C misses A implies C = B )
assume that
A1: B is a_component and
A2: C is a_component and
A3: A \/ B = the carrier of G and
A4: C misses A ; ::_thesis: C = B
now__::_thesis:_not_C_misses_B
C /\ the carrier of G = C by XBOOLE_1:28;
then A5: (C /\ A) \/ (C /\ B) = C by A3, XBOOLE_1:23;
assume C misses B ; ::_thesis: contradiction
then A6: C /\ B = {} by XBOOLE_0:def_7;
C <> {} G by A2, CONNSP_1:32;
hence contradiction by A4, A6, A5, XBOOLE_0:def_7; ::_thesis: verum
end;
hence C = B by A1, A2, CONNSP_1:35; ::_thesis: verum
end;
theorem Th108: :: JORDAN2C:108
for A being Subset of (TOP-REAL 2) st A is bounded & A is Jordan holds
BDD A is_inside_component_of A
proof
let A be Subset of (TOP-REAL 2); ::_thesis: ( A is bounded & A is Jordan implies BDD A is_inside_component_of A )
assume that
A1: A is bounded and
A2: A is Jordan ; ::_thesis: BDD A is_inside_component_of A
reconsider D = A ` as non empty Subset of (TOP-REAL 2) by A2, JORDAN1:def_2;
consider A1, A2 being Subset of (TOP-REAL 2) such that
A3: A ` = A1 \/ A2 and
A4: A1 misses A2 and
(Cl A1) \ A1 = (Cl A2) \ A2 and
A5: for C1, C2 being Subset of ((TOP-REAL 2) | (A `)) st C1 = A1 & C2 = A2 holds
( C1 is a_component & C2 is a_component ) by A2, JORDAN1:def_2;
A6: UBD A is_outside_component_of A by A1, Th68;
then UBD A is_a_component_of A ` by Def3;
then consider B1 being Subset of ((TOP-REAL 2) | (A `)) such that
A7: B1 = UBD A and
A8: B1 is a_component by CONNSP_1:def_6;
A9: Down (A1,(A `)) = A1 by A3, XBOOLE_1:21;
A10: Down (A2,(A `)) = A2 by A3, XBOOLE_1:21;
then A11: Down (A2,(A `)) is a_component by A5, A9;
then A12: A2 is_a_component_of A ` by A10, CONNSP_1:def_6;
A13: not (TOP-REAL 2) | D is empty ;
then A14: Down (A2,(A `)) <> {} ((TOP-REAL 2) | (A `)) by A11, CONNSP_1:32;
A15: Down (A1,(A `)) is a_component by A5, A9, A10;
then A16: A1 is_a_component_of A ` by A9, CONNSP_1:def_6;
percases ( B1 = A1 or B1 misses Down (A1,(A `)) ) by A9, A15, A8, CONNSP_1:35;
supposeA17: B1 = A1 ; ::_thesis: BDD A is_inside_component_of A
A18: now__::_thesis:_BDD_A_c=_A2
assume not BDD A c= A2 ; ::_thesis: contradiction
then consider x being set such that
A19: x in BDD A and
A20: not x in A2 by TARSKI:def_3;
consider y being set such that
A21: x in y and
A22: y in { B3 where B3 is Subset of (TOP-REAL 2) : B3 is_inside_component_of A } by A19, TARSKI:def_4;
consider B3 being Subset of (TOP-REAL 2) such that
A23: y = B3 and
A24: B3 is_inside_component_of A by A22;
A25: B3 is_a_component_of A ` by A24, Def2;
then consider B4 being Subset of ((TOP-REAL 2) | (A `)) such that
A26: B4 = B3 and
A27: B4 is a_component by CONNSP_1:def_6;
A28: B3 <> {} ((TOP-REAL 2) | (A `)) by A13, A26, A27, CONNSP_1:32;
now__::_thesis:_not_B4_=_Down_(A1,(A_`))
assume B4 = Down (A1,(A `)) ; ::_thesis: contradiction
then UBD A is bounded by A9, A7, A17, A24, A26, Def2;
hence contradiction by A6, Def3; ::_thesis: verum
end;
then A29: B3 misses A1 by A9, A15, A26, A27, CONNSP_1:35;
( B4 = Down (A2,(A `)) or B4 misses Down (A2,(A `)) ) by A11, A27, CONNSP_1:35;
then A30: ( B4 = Down (A2,(A `)) or B4 /\ (Down (A2,(A `))) = {} ((TOP-REAL 2) | (A `)) ) by XBOOLE_0:def_7;
B3 = B3 /\ (A1 \/ A2) by A3, A25, SPRECT_1:5, XBOOLE_1:28
.= (B3 /\ A1) \/ (B3 /\ A2) by XBOOLE_1:23
.= {} by A10, A20, A21, A23, A26, A30, A29, XBOOLE_0:def_7 ;
hence contradiction by A28; ::_thesis: verum
end;
now__::_thesis:_A2_is_bounded
assume not A2 is bounded ; ::_thesis: contradiction
then A2 is_outside_component_of A by A12, Def3;
then A2 /\ (UBD A) <> {} by A10, A14, Th23, XBOOLE_1:28;
hence contradiction by A4, A7, A17, XBOOLE_0:def_7; ::_thesis: verum
end;
then A31: A2 is_inside_component_of A by A12, Def2;
then A2 c= BDD A by Th22;
hence BDD A is_inside_component_of A by A31, A18, XBOOLE_0:def_10; ::_thesis: verum
end;
supposeA32: B1 misses Down (A1,(A `)) ; ::_thesis: BDD A is_inside_component_of A
set E1 = Down (A1,(A `));
set E2 = Down (A2,(A `));
(Down (A1,(A `))) \/ (Down (A2,(A `))) = [#] ((TOP-REAL 2) | (A `)) by A3, A9, A10, PRE_TOPC:def_5;
then A33: UBD A = A2 by A10, A11, A13, A7, A8, A32, Th107;
A34: (BDD A) \/ (UBD A) = A ` by Th27;
A35: BDD A misses UBD A by Th24;
A36: BDD A c= A1
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in BDD A or z in A1 )
assume z in BDD A ; ::_thesis: z in A1
then ( z in A ` & not z in UBD A ) by A35, A34, XBOOLE_0:3, XBOOLE_0:def_3;
hence z in A1 by A3, A33, XBOOLE_0:def_3; ::_thesis: verum
end;
A37: BDD A is bounded by A1, Th106;
A1 c= BDD A
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in A1 or z in BDD A )
assume z in A1 ; ::_thesis: z in BDD A
then ( z in A ` & not z in UBD A ) by A3, A4, A33, XBOOLE_0:3, XBOOLE_0:def_3;
hence z in BDD A by A34, XBOOLE_0:def_3; ::_thesis: verum
end;
then BDD A = A1 by A36, XBOOLE_0:def_10;
hence BDD A is_inside_component_of A by A16, A37, Def2; ::_thesis: verum
end;
end;
end;
theorem :: JORDAN2C:109
for D being non empty compact non horizontal non vertical Subset of (TOP-REAL 2)
for a being Real
for p being Point of (TOP-REAL 2) st a > 0 & p in L~ (SpStSeq D) holds
ex q being Point of (TOP-REAL 2) st
( q in BDD (L~ (SpStSeq D)) & |.(p - q).| < a )
proof
let D be non empty compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for a being Real
for p being Point of (TOP-REAL 2) st a > 0 & p in L~ (SpStSeq D) holds
ex q being Point of (TOP-REAL 2) st
( q in BDD (L~ (SpStSeq D)) & |.(p - q).| < a )
let a be Real; ::_thesis: for p being Point of (TOP-REAL 2) st a > 0 & p in L~ (SpStSeq D) holds
ex q being Point of (TOP-REAL 2) st
( q in BDD (L~ (SpStSeq D)) & |.(p - q).| < a )
let p be Point of (TOP-REAL 2); ::_thesis: ( a > 0 & p in L~ (SpStSeq D) implies ex q being Point of (TOP-REAL 2) st
( q in BDD (L~ (SpStSeq D)) & |.(p - q).| < a ) )
assume that
A1: a > 0 and
A2: p in L~ (SpStSeq D) ; ::_thesis: ex q being Point of (TOP-REAL 2) st
( q in BDD (L~ (SpStSeq D)) & |.(p - q).| < a )
set q1 = the Element of BDD (L~ (SpStSeq D));
set A = L~ (SpStSeq D);
(L~ (SpStSeq D)) ` <> {} by SPRECT_1:def_3;
then consider A1, A2 being Subset of (TOP-REAL 2) such that
A3: (L~ (SpStSeq D)) ` = A1 \/ A2 and
A1 misses A2 and
A4: (Cl A1) \ A1 = (Cl A2) \ A2 and
A5: L~ (SpStSeq D) = (Cl A1) \ A1 and
A6: for C1, C2 being Subset of ((TOP-REAL 2) | ((L~ (SpStSeq D)) `)) st C1 = A1 & C2 = A2 holds
( C1 is a_component & C2 is a_component ) by Th98;
A7: Down (A2,((L~ (SpStSeq D)) `)) = A2 by A3, XBOOLE_1:21;
BDD (L~ (SpStSeq D)) is_inside_component_of L~ (SpStSeq D) by Th108;
then BDD (L~ (SpStSeq D)) is_a_component_of (L~ (SpStSeq D)) ` by Def2;
then consider B1 being Subset of ((TOP-REAL 2) | ((L~ (SpStSeq D)) `)) such that
A8: B1 = BDD (L~ (SpStSeq D)) and
A9: B1 is a_component by CONNSP_1:def_6;
B1 c= the carrier of ((TOP-REAL 2) | ((L~ (SpStSeq D)) `)) ;
then A10: BDD (L~ (SpStSeq D)) c= A1 \/ A2 by A3, A8, PRE_TOPC:8;
A11: Down (A1,((L~ (SpStSeq D)) `)) = A1 by A3, XBOOLE_1:21;
then A12: Down (A1,((L~ (SpStSeq D)) `)) is a_component by A6, A7;
A13: Down (A2,((L~ (SpStSeq D)) `)) is a_component by A6, A11, A7;
A14: BDD (L~ (SpStSeq D)) <> {} by Th96;
then A15: the Element of BDD (L~ (SpStSeq D)) in BDD (L~ (SpStSeq D)) ;
percases ( the Element of BDD (L~ (SpStSeq D)) in A1 or the Element of BDD (L~ (SpStSeq D)) in A2 ) by A10, A15, XBOOLE_0:def_3;
suppose the Element of BDD (L~ (SpStSeq D)) in A1 ; ::_thesis: ex q being Point of (TOP-REAL 2) st
( q in BDD (L~ (SpStSeq D)) & |.(p - q).| < a )
then B1 /\ (Down (A1,((L~ (SpStSeq D)) `))) <> {} ((TOP-REAL 2) | ((L~ (SpStSeq D)) `)) by A11, A8, A14, XBOOLE_0:def_4;
then B1 meets Down (A1,((L~ (SpStSeq D)) `)) by XBOOLE_0:def_7;
then B1 = Down (A1,((L~ (SpStSeq D)) `)) by A12, A9, CONNSP_1:35;
then A16: p in Cl (BDD (L~ (SpStSeq D))) by A2, A5, A11, A8, XBOOLE_0:def_5;
reconsider ep = p as Point of (Euclid 2) by TOPREAL3:8;
reconsider G2 = Ball (ep,a) as Subset of (TOP-REAL 2) by TOPREAL3:8;
the distance of (Euclid 2) is Reflexive by METRIC_1:def_6;
then dist (ep,ep) = 0 by METRIC_1:def_2;
then A17: p in Ball (ep,a) by A1, METRIC_1:11;
G2 is open by GOBOARD6:3;
then BDD (L~ (SpStSeq D)) meets G2 by A16, A17, PRE_TOPC:def_7;
then consider t2 being set such that
A18: t2 in BDD (L~ (SpStSeq D)) and
A19: t2 in G2 by XBOOLE_0:3;
reconsider qt2 = t2 as Point of (TOP-REAL 2) by A18;
|.(p - qt2).| < a by A19, Th103;
hence ex q being Point of (TOP-REAL 2) st
( q in BDD (L~ (SpStSeq D)) & |.(p - q).| < a ) by A18; ::_thesis: verum
end;
suppose the Element of BDD (L~ (SpStSeq D)) in A2 ; ::_thesis: ex q being Point of (TOP-REAL 2) st
( q in BDD (L~ (SpStSeq D)) & |.(p - q).| < a )
then B1 /\ (Down (A2,((L~ (SpStSeq D)) `))) <> {} ((TOP-REAL 2) | ((L~ (SpStSeq D)) `)) by A7, A8, A14, XBOOLE_0:def_4;
then B1 meets Down (A2,((L~ (SpStSeq D)) `)) by XBOOLE_0:def_7;
then B1 = Down (A2,((L~ (SpStSeq D)) `)) by A13, A9, CONNSP_1:35;
then A20: p in Cl (BDD (L~ (SpStSeq D))) by A2, A4, A5, A7, A8, XBOOLE_0:def_5;
reconsider ep = p as Point of (Euclid 2) by TOPREAL3:8;
reconsider G2 = Ball (ep,a) as Subset of (TOP-REAL 2) by TOPREAL3:8;
the distance of (Euclid 2) is Reflexive by METRIC_1:def_6;
then dist (ep,ep) = 0 by METRIC_1:def_2;
then A21: p in Ball (ep,a) by A1, METRIC_1:11;
G2 is open by GOBOARD6:3;
then BDD (L~ (SpStSeq D)) meets G2 by A20, A21, PRE_TOPC:def_7;
then consider t2 being set such that
A22: t2 in BDD (L~ (SpStSeq D)) and
A23: t2 in G2 by XBOOLE_0:3;
reconsider qt2 = t2 as Point of (TOP-REAL 2) by A22;
|.(p - qt2).| < a by A23, Th103;
hence ex q being Point of (TOP-REAL 2) st
( q in BDD (L~ (SpStSeq D)) & |.(p - q).| < a ) by A22; ::_thesis: verum
end;
end;
end;
begin
theorem :: JORDAN2C:110
for f being non constant standard clockwise_oriented special_circular_sequence
for p being Point of (TOP-REAL 2) st f /. 1 = N-min (L~ f) & p `1 < W-bound (L~ f) holds
p in LeftComp f
proof
let f be non constant standard clockwise_oriented special_circular_sequence; ::_thesis: for p being Point of (TOP-REAL 2) st f /. 1 = N-min (L~ f) & p `1 < W-bound (L~ f) holds
p in LeftComp f
let p be Point of (TOP-REAL 2); ::_thesis: ( f /. 1 = N-min (L~ f) & p `1 < W-bound (L~ f) implies p in LeftComp f )
assume that
A1: f /. 1 = N-min (L~ f) and
A2: p `1 < W-bound (L~ f) ; ::_thesis: p in LeftComp f
set g = SpStSeq (L~ f);
A3: LeftComp (SpStSeq (L~ f)) c= LeftComp f by A1, SPRECT_3:41;
W-bound (L~ (SpStSeq (L~ f))) = W-bound (L~ f) by SPRECT_1:58;
then p in LeftComp (SpStSeq (L~ f)) by A2, SPRECT_3:40;
hence p in LeftComp f by A3; ::_thesis: verum
end;
theorem :: JORDAN2C:111
for f being non constant standard clockwise_oriented special_circular_sequence
for p being Point of (TOP-REAL 2) st f /. 1 = N-min (L~ f) & p `1 > E-bound (L~ f) holds
p in LeftComp f
proof
let f be non constant standard clockwise_oriented special_circular_sequence; ::_thesis: for p being Point of (TOP-REAL 2) st f /. 1 = N-min (L~ f) & p `1 > E-bound (L~ f) holds
p in LeftComp f
let p be Point of (TOP-REAL 2); ::_thesis: ( f /. 1 = N-min (L~ f) & p `1 > E-bound (L~ f) implies p in LeftComp f )
assume that
A1: f /. 1 = N-min (L~ f) and
A2: p `1 > E-bound (L~ f) ; ::_thesis: p in LeftComp f
set g = SpStSeq (L~ f);
A3: LeftComp (SpStSeq (L~ f)) c= LeftComp f by A1, SPRECT_3:41;
E-bound (L~ (SpStSeq (L~ f))) = E-bound (L~ f) by SPRECT_1:61;
then p in LeftComp (SpStSeq (L~ f)) by A2, SPRECT_3:40;
hence p in LeftComp f by A3; ::_thesis: verum
end;
theorem :: JORDAN2C:112
for f being non constant standard clockwise_oriented special_circular_sequence
for p being Point of (TOP-REAL 2) st f /. 1 = N-min (L~ f) & p `2 < S-bound (L~ f) holds
p in LeftComp f
proof
let f be non constant standard clockwise_oriented special_circular_sequence; ::_thesis: for p being Point of (TOP-REAL 2) st f /. 1 = N-min (L~ f) & p `2 < S-bound (L~ f) holds
p in LeftComp f
let p be Point of (TOP-REAL 2); ::_thesis: ( f /. 1 = N-min (L~ f) & p `2 < S-bound (L~ f) implies p in LeftComp f )
assume that
A1: f /. 1 = N-min (L~ f) and
A2: p `2 < S-bound (L~ f) ; ::_thesis: p in LeftComp f
set g = SpStSeq (L~ f);
A3: LeftComp (SpStSeq (L~ f)) c= LeftComp f by A1, SPRECT_3:41;
S-bound (L~ (SpStSeq (L~ f))) = S-bound (L~ f) by SPRECT_1:59;
then p in LeftComp (SpStSeq (L~ f)) by A2, SPRECT_3:40;
hence p in LeftComp f by A3; ::_thesis: verum
end;
theorem :: JORDAN2C:113
for f being non constant standard clockwise_oriented special_circular_sequence
for p being Point of (TOP-REAL 2) st f /. 1 = N-min (L~ f) & p `2 > N-bound (L~ f) holds
p in LeftComp f
proof
let f be non constant standard clockwise_oriented special_circular_sequence; ::_thesis: for p being Point of (TOP-REAL 2) st f /. 1 = N-min (L~ f) & p `2 > N-bound (L~ f) holds
p in LeftComp f
let p be Point of (TOP-REAL 2); ::_thesis: ( f /. 1 = N-min (L~ f) & p `2 > N-bound (L~ f) implies p in LeftComp f )
assume that
A1: f /. 1 = N-min (L~ f) and
A2: p `2 > N-bound (L~ f) ; ::_thesis: p in LeftComp f
set g = SpStSeq (L~ f);
A3: LeftComp (SpStSeq (L~ f)) c= LeftComp f by A1, SPRECT_3:41;
N-bound (L~ (SpStSeq (L~ f))) = N-bound (L~ f) by SPRECT_1:60;
then p in LeftComp (SpStSeq (L~ f)) by A2, SPRECT_3:40;
hence p in LeftComp f by A3; ::_thesis: verum
end;
theorem :: JORDAN2C:114
for T being TopSpace
for A, B being Subset of T st B is_a_component_of A holds
B is connected
proof
let T be TopSpace; ::_thesis: for A, B being Subset of T st B is_a_component_of A holds
B is connected
let A, B be Subset of T; ::_thesis: ( B is_a_component_of A implies B is connected )
assume B is_a_component_of A ; ::_thesis: B is connected
then consider C being Subset of (T | A) such that
A1: C = B and
A2: C is a_component by CONNSP_1:def_6;
C is connected by A2, CONNSP_1:def_5;
hence B is connected by A1, CONNSP_1:23; ::_thesis: verum
end;
theorem :: JORDAN2C:115
for n being Element of NAT
for A, B being Subset of (TOP-REAL n) st B is_inside_component_of A holds
B is connected
proof
let n be Element of NAT ; ::_thesis: for A, B being Subset of (TOP-REAL n) st B is_inside_component_of A holds
B is connected
let A, B be Subset of (TOP-REAL n); ::_thesis: ( B is_inside_component_of A implies B is connected )
assume B is_inside_component_of A ; ::_thesis: B is connected
then consider C being Subset of ((TOP-REAL n) | (A `)) such that
A1: C = B and
A2: C is a_component and
C is bounded Subset of (Euclid n) by Th13;
C is connected by A2, CONNSP_1:def_5;
hence B is connected by A1, CONNSP_1:23; ::_thesis: verum
end;
theorem Th116: :: JORDAN2C:116
for n being Element of NAT
for A, B being Subset of (TOP-REAL n) st B is_outside_component_of A holds
B is connected
proof
let n be Element of NAT ; ::_thesis: for A, B being Subset of (TOP-REAL n) st B is_outside_component_of A holds
B is connected
let A, B be Subset of (TOP-REAL n); ::_thesis: ( B is_outside_component_of A implies B is connected )
assume B is_outside_component_of A ; ::_thesis: B is connected
then consider C being Subset of ((TOP-REAL n) | (A `)) such that
A1: C = B and
A2: C is a_component and
C is not bounded Subset of (Euclid n) by Th14;
C is connected by A2, CONNSP_1:def_5;
hence B is connected by A1, CONNSP_1:23; ::_thesis: verum
end;
theorem :: JORDAN2C:117
for n being Element of NAT
for A, B being Subset of (TOP-REAL n) st B is_a_component_of A ` holds
A misses B
proof
let n be Element of NAT ; ::_thesis: for A, B being Subset of (TOP-REAL n) st B is_a_component_of A ` holds
A misses B
let A, B be Subset of (TOP-REAL n); ::_thesis: ( B is_a_component_of A ` implies A misses B )
assume B is_a_component_of A ` ; ::_thesis: A misses B
then B c= A ` by SPRECT_1:5;
then A misses B by SUBSET_1:23;
hence A /\ B = {} by XBOOLE_0:def_7; :: according to XBOOLE_0:def_7 ::_thesis: verum
end;
theorem :: JORDAN2C:118
for n being Element of NAT
for R, P, Q being Subset of (TOP-REAL n) st P is_outside_component_of Q & R is_inside_component_of Q holds
P misses R
proof
let n be Element of NAT ; ::_thesis: for R, P, Q being Subset of (TOP-REAL n) st P is_outside_component_of Q & R is_inside_component_of Q holds
P misses R
let R, P, Q be Subset of (TOP-REAL n); ::_thesis: ( P is_outside_component_of Q & R is_inside_component_of Q implies P misses R )
assume A1: P is_outside_component_of Q ; ::_thesis: ( not R is_inside_component_of Q or P misses R )
assume A2: R is_inside_component_of Q ; ::_thesis: P misses R
BDD Q misses UBD Q by Th24;
then P misses BDD Q by A1, Th23, XBOOLE_1:63;
hence P misses R by A2, Th22, XBOOLE_1:63; ::_thesis: verum
end;
theorem :: JORDAN2C:119
for n being Element of NAT st 2 <= n holds
for A, B, P being Subset of (TOP-REAL n) st P is bounded & A is_outside_component_of P & B is_outside_component_of P holds
A = B
proof
let n be Element of NAT ; ::_thesis: ( 2 <= n implies for A, B, P being Subset of (TOP-REAL n) st P is bounded & A is_outside_component_of P & B is_outside_component_of P holds
A = B )
assume A1: 2 <= n ; ::_thesis: for A, B, P being Subset of (TOP-REAL n) st P is bounded & A is_outside_component_of P & B is_outside_component_of P holds
A = B
let A, B, P be Subset of (TOP-REAL n); ::_thesis: ( P is bounded & A is_outside_component_of P & B is_outside_component_of P implies A = B )
assume that
A2: P is bounded and
A3: A is_outside_component_of P and
A4: B is_outside_component_of P ; ::_thesis: A = B
A5: B is_a_component_of P ` by A4, Def3;
UBD P is_outside_component_of P by A1, A2, Th68;
then A6: UBD P is_a_component_of P ` by Def3;
A7: not P ` is empty by A1, A2, Th66, XXREAL_0:2;
A8: B <> {} by A4, Def3;
A9: B c= UBD P by A4, Th23;
A10: A c= UBD P by A3, Th23;
A11: A is_a_component_of P ` by A3, Def3;
then A <> {} by A7, SPRECT_1:4;
then A = UBD P by A11, A6, A10, GOBOARD9:1, XBOOLE_1:69;
hence A = B by A5, A8, A6, A9, GOBOARD9:1, XBOOLE_1:69; ::_thesis: verum
end;
registration
let C be closed Subset of (TOP-REAL 2);
cluster BDD C -> open ;
coherence
BDD C is open
proof
set F = { B where B is Subset of (TOP-REAL 2) : B is_inside_component_of C } ;
{ B where B is Subset of (TOP-REAL 2) : B is_inside_component_of C } c= bool the carrier of (TOP-REAL 2)
proof
let f be set ; :: according to TARSKI:def_3 ::_thesis: ( not f in { B where B is Subset of (TOP-REAL 2) : B is_inside_component_of C } or f in bool the carrier of (TOP-REAL 2) )
assume f in { B where B is Subset of (TOP-REAL 2) : B is_inside_component_of C } ; ::_thesis: f in bool the carrier of (TOP-REAL 2)
then ex B being Subset of (TOP-REAL 2) st
( f = B & B is_inside_component_of C ) ;
hence f in bool the carrier of (TOP-REAL 2) ; ::_thesis: verum
end;
then reconsider F = { B where B is Subset of (TOP-REAL 2) : B is_inside_component_of C } as Subset-Family of (TOP-REAL 2) ;
F is open
proof
let P be Subset of (TOP-REAL 2); :: according to TOPS_2:def_1 ::_thesis: ( not P in F or P is open )
assume P in F ; ::_thesis: P is open
then consider B being Subset of (TOP-REAL 2) such that
A1: P = B and
A2: B is_inside_component_of C ;
B is_a_component_of C ` by A2, Def2;
hence P is open by A1, SPRECT_3:8; ::_thesis: verum
end;
hence BDD C is open by TOPS_2:19; ::_thesis: verum
end;
cluster UBD C -> open ;
coherence
UBD C is open
proof
set F = { B where B is Subset of (TOP-REAL 2) : B is_outside_component_of C } ;
{ B where B is Subset of (TOP-REAL 2) : B is_outside_component_of C } c= bool the carrier of (TOP-REAL 2)
proof
let f be set ; :: according to TARSKI:def_3 ::_thesis: ( not f in { B where B is Subset of (TOP-REAL 2) : B is_outside_component_of C } or f in bool the carrier of (TOP-REAL 2) )
assume f in { B where B is Subset of (TOP-REAL 2) : B is_outside_component_of C } ; ::_thesis: f in bool the carrier of (TOP-REAL 2)
then ex B being Subset of (TOP-REAL 2) st
( f = B & B is_outside_component_of C ) ;
hence f in bool the carrier of (TOP-REAL 2) ; ::_thesis: verum
end;
then reconsider F = { B where B is Subset of (TOP-REAL 2) : B is_outside_component_of C } as Subset-Family of (TOP-REAL 2) ;
F is open
proof
let P be Subset of (TOP-REAL 2); :: according to TOPS_2:def_1 ::_thesis: ( not P in F or P is open )
assume P in F ; ::_thesis: P is open
then consider B being Subset of (TOP-REAL 2) such that
A3: P = B and
A4: B is_outside_component_of C ;
B is_a_component_of C ` by A4, Def3;
hence P is open by A3, SPRECT_3:8; ::_thesis: verum
end;
hence UBD C is open by TOPS_2:19; ::_thesis: verum
end;
end;
registration
let C be compact Subset of (TOP-REAL 2);
cluster UBD C -> connected ;
coherence
UBD C is connected by Th68, Th116;
end;
theorem Th120: :: JORDAN2C:120
for p being Point of (TOP-REAL 2) holds not west_halfline p is bounded
proof
let p be Point of (TOP-REAL 2); ::_thesis: not west_halfline p is bounded
set Wp = west_halfline p;
set p11 = p `1 ;
set p12 = p `2 ;
assume west_halfline p is bounded ; ::_thesis: contradiction
then reconsider C = west_halfline p as bounded Subset of (Euclid 2) by Th11;
consider r being Real such that
A1: 0 < r and
A2: for x, y being Point of (Euclid 2) st x in C & y in C holds
dist (x,y) <= r by TBSP_1:def_7;
set EX1 = (p `1) - (2 * r);
reconsider p1 = p, EX = |[((p `1) - (2 * r)),(p `2)]| as Point of (Euclid 2) by EUCLID:67;
0 + (p `1) <= (2 * r) + (p `1) by A1, XREAL_1:6;
then (p `1) - (2 * r) <= p `1 by XREAL_1:20;
then A3: |[((p `1) - (2 * r)),(p `2)]| `1 <= p `1 by EUCLID:52;
then A4: p1 in west_halfline p by TOPREAL1:def_13;
|[((p `1) - (2 * r)),(p `2)]| `2 = p `2 by EUCLID:52;
then A5: EX in west_halfline p by A3, TOPREAL1:def_13;
p = |[(p `1),(p `2)]| by EUCLID:53;
then dist (p1,EX) = sqrt ((((p `1) - ((p `1) - (2 * r))) ^2) + (((p `2) - (p `2)) ^2)) by GOBOARD6:6
.= 2 * r by A1, SQUARE_1:22 ;
hence contradiction by A1, A2, A5, A4, XREAL_1:155; ::_thesis: verum
end;
theorem Th121: :: JORDAN2C:121
for p being Point of (TOP-REAL 2) holds not east_halfline p is bounded
proof
let p be Point of (TOP-REAL 2); ::_thesis: not east_halfline p is bounded
set Wp = east_halfline p;
set p11 = p `1 ;
set p12 = p `2 ;
assume east_halfline p is bounded ; ::_thesis: contradiction
then reconsider C = east_halfline p as bounded Subset of (Euclid 2) by Th11;
consider r being Real such that
A1: 0 < r and
A2: for x, y being Point of (Euclid 2) st x in C & y in C holds
dist (x,y) <= r by TBSP_1:def_7;
set EX1 = (p `1) + (2 * r);
set EX2 = p `2 ;
reconsider p1 = p, EX = |[((p `1) + (2 * r)),(p `2)]| as Point of (Euclid 2) by EUCLID:67;
0 + (p `1) <= (2 * r) + (p `1) by A1, XREAL_1:6;
then A3: |[((p `1) + (2 * r)),(p `2)]| `1 >= p `1 by EUCLID:52;
then A4: p1 in east_halfline p by TOPREAL1:def_11;
|[((p `1) + (2 * r)),(p `2)]| `2 = p `2 by EUCLID:52;
then A5: EX in east_halfline p by A3, TOPREAL1:def_11;
p = |[(p `1),(p `2)]| by EUCLID:53;
then dist (p1,EX) = sqrt ((((p `1) - ((p `1) + (2 * r))) ^2) + (((p `2) - (p `2)) ^2)) by GOBOARD6:6
.= sqrt (((((p `1) + (2 * r)) - (p `1)) ^2) + 0)
.= 2 * r by A1, SQUARE_1:22 ;
hence contradiction by A1, A2, A5, A4, XREAL_1:155; ::_thesis: verum
end;
theorem Th122: :: JORDAN2C:122
for p being Point of (TOP-REAL 2) holds not north_halfline p is bounded
proof
let p be Point of (TOP-REAL 2); ::_thesis: not north_halfline p is bounded
set Wp = north_halfline p;
set p11 = p `1 ;
set p12 = p `2 ;
assume north_halfline p is bounded ; ::_thesis: contradiction
then reconsider C = north_halfline p as bounded Subset of (Euclid 2) by Th11;
consider r being Real such that
A1: 0 < r and
A2: for x, y being Point of (Euclid 2) st x in C & y in C holds
dist (x,y) <= r by TBSP_1:def_7;
set EX2 = (p `2) + (2 * r);
set EX1 = p `1 ;
reconsider p1 = p, EX = |[(p `1),((p `2) + (2 * r))]| as Point of (Euclid 2) by EUCLID:67;
A3: |[(p `1),((p `2) + (2 * r))]| `1 = p `1 by EUCLID:52;
then A4: p1 in north_halfline p by TOPREAL1:def_10;
0 + (p `2) <= (2 * r) + (p `2) by A1, XREAL_1:6;
then |[(p `1),((p `2) + (2 * r))]| `2 >= p `2 by EUCLID:52;
then A5: EX in north_halfline p by A3, TOPREAL1:def_10;
p = |[(p `1),(p `2)]| by EUCLID:53;
then dist (p1,EX) = sqrt ((((p `1) - (p `1)) ^2) + (((p `2) - ((p `2) + (2 * r))) ^2)) by GOBOARD6:6
.= sqrt (((((p `2) + (2 * r)) - (p `2)) ^2) + 0)
.= 2 * r by A1, SQUARE_1:22 ;
hence contradiction by A1, A2, A5, A4, XREAL_1:155; ::_thesis: verum
end;
theorem Th123: :: JORDAN2C:123
for p being Point of (TOP-REAL 2) holds not south_halfline p is bounded
proof
let p be Point of (TOP-REAL 2); ::_thesis: not south_halfline p is bounded
set Wp = south_halfline p;
set p11 = p `1 ;
set p12 = p `2 ;
assume south_halfline p is bounded ; ::_thesis: contradiction
then reconsider C = south_halfline p as bounded Subset of (Euclid 2) by Th11;
consider r being Real such that
A1: 0 < r and
A2: for x, y being Point of (Euclid 2) st x in C & y in C holds
dist (x,y) <= r by TBSP_1:def_7;
set EX2 = (p `2) - (2 * r);
set EX1 = p `1 ;
reconsider p1 = p, EX = |[(p `1),((p `2) - (2 * r))]| as Point of (Euclid 2) by EUCLID:67;
p = |[(p `1),(p `2)]| by EUCLID:53;
then A3: dist (p1,EX) = sqrt ((((p `1) - (p `1)) ^2) + (((p `2) - ((p `2) - (2 * r))) ^2)) by GOBOARD6:6
.= 2 * r by A1, SQUARE_1:22 ;
A4: |[(p `1),((p `2) - (2 * r))]| `1 = p `1 by EUCLID:52;
then A5: p1 in south_halfline p by TOPREAL1:def_12;
0 + (p `2) <= (2 * r) + (p `2) by A1, XREAL_1:6;
then (p `2) - (2 * r) <= p `2 by XREAL_1:20;
then |[(p `1),((p `2) - (2 * r))]| `2 <= p `2 by EUCLID:52;
then EX in south_halfline p by A4, TOPREAL1:def_12;
hence contradiction by A1, A2, A5, A3, XREAL_1:155; ::_thesis: verum
end;
registration
let C be compact Subset of (TOP-REAL 2);
cluster UBD C -> non empty ;
coherence
not UBD C is empty
proof
A1: UBD C is_outside_component_of C by Th68;
thus not UBD C is empty by A1, Def3; ::_thesis: verum
end;
end;
theorem Th124: :: JORDAN2C:124
for C being compact Subset of (TOP-REAL 2) holds UBD C is_a_component_of C `
proof
let C be compact Subset of (TOP-REAL 2); ::_thesis: UBD C is_a_component_of C `
UBD C is_outside_component_of C by Th68;
hence UBD C is_a_component_of C ` by Def3; ::_thesis: verum
end;
theorem Th125: :: JORDAN2C:125
for C being compact Subset of (TOP-REAL 2)
for WH being connected Subset of (TOP-REAL 2) st not WH is bounded & WH misses C holds
WH c= UBD C
proof
let C be compact Subset of (TOP-REAL 2); ::_thesis: for WH being connected Subset of (TOP-REAL 2) st not WH is bounded & WH misses C holds
WH c= UBD C
let WH be connected Subset of (TOP-REAL 2); ::_thesis: ( not WH is bounded & WH misses C implies WH c= UBD C )
assume that
A1: not WH is bounded and
A2: WH misses C ; ::_thesis: WH c= UBD C
A3: WH meets UBD C
proof
( (BDD C) \/ (UBD C) = C ` & [#] the carrier of (TOP-REAL 2) = C \/ (C `) ) by Th27, SUBSET_1:10;
then A4: WH c= (UBD C) \/ (BDD C) by A2, XBOOLE_1:73;
assume A5: WH misses UBD C ; ::_thesis: contradiction
BDD C is bounded by Th106;
hence contradiction by A1, A5, A4, RLTOPSP1:42, XBOOLE_1:73; ::_thesis: verum
end;
WH c= C ` by A2, SUBSET_1:23;
hence WH c= UBD C by A3, Th124, GOBOARD9:4; ::_thesis: verum
end;
theorem :: JORDAN2C:126
for C being compact Subset of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st west_halfline p misses C holds
west_halfline p c= UBD C
proof
let C be compact Subset of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st west_halfline p misses C holds
west_halfline p c= UBD C
let p be Point of (TOP-REAL 2); ::_thesis: ( west_halfline p misses C implies west_halfline p c= UBD C )
set WH = west_halfline p;
assume A1: west_halfline p misses C ; ::_thesis: west_halfline p c= UBD C
( not west_halfline p is bounded & not west_halfline p is empty & west_halfline p is connected ) by Th120;
hence west_halfline p c= UBD C by A1, Th125; ::_thesis: verum
end;
theorem :: JORDAN2C:127
for C being compact Subset of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st east_halfline p misses C holds
east_halfline p c= UBD C
proof
let C be compact Subset of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st east_halfline p misses C holds
east_halfline p c= UBD C
let p be Point of (TOP-REAL 2); ::_thesis: ( east_halfline p misses C implies east_halfline p c= UBD C )
set WH = east_halfline p;
assume A1: east_halfline p misses C ; ::_thesis: east_halfline p c= UBD C
( not east_halfline p is bounded & not east_halfline p is empty & east_halfline p is connected ) by Th121;
hence east_halfline p c= UBD C by A1, Th125; ::_thesis: verum
end;
theorem :: JORDAN2C:128
for C being compact Subset of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st south_halfline p misses C holds
south_halfline p c= UBD C
proof
let C be compact Subset of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st south_halfline p misses C holds
south_halfline p c= UBD C
let p be Point of (TOP-REAL 2); ::_thesis: ( south_halfline p misses C implies south_halfline p c= UBD C )
set WH = south_halfline p;
assume A1: south_halfline p misses C ; ::_thesis: south_halfline p c= UBD C
( not south_halfline p is bounded & not south_halfline p is empty & south_halfline p is connected ) by Th123;
hence south_halfline p c= UBD C by A1, Th125; ::_thesis: verum
end;
theorem :: JORDAN2C:129
for C being compact Subset of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st north_halfline p misses C holds
north_halfline p c= UBD C
proof
let C be compact Subset of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st north_halfline p misses C holds
north_halfline p c= UBD C
let p be Point of (TOP-REAL 2); ::_thesis: ( north_halfline p misses C implies north_halfline p c= UBD C )
set WH = north_halfline p;
assume A1: north_halfline p misses C ; ::_thesis: north_halfline p c= UBD C
( not north_halfline p is bounded & not north_halfline p is empty & north_halfline p is connected ) by Th122;
hence north_halfline p c= UBD C by A1, Th125; ::_thesis: verum
end;
theorem :: JORDAN2C:130
for n being Nat
for r being Real st r > 0 holds
for x, y, z being Element of (Euclid n) st x = 0* n holds
for p being Element of (TOP-REAL n) st p = y & r * p = z holds
r * (dist (x,y)) = dist (x,z) by Lm1;
theorem :: JORDAN2C:131
for n being Nat
for r, s being Real st r > 0 holds
for x being Element of (Euclid n) st x = 0* n holds
for A being Subset of (TOP-REAL n) st A = Ball (x,s) holds
r * A = Ball (x,(r * s)) by Lm2;
theorem :: JORDAN2C:132
for n being Nat
for r, s, t being Real st 0 < s & s <= t holds
for x being Element of (Euclid n) st x = 0* n holds
for BA being Subset of (TOP-REAL n) st BA = Ball (x,r) holds
s * BA c= t * BA by Lm3;