:: JORDAN1A semantic presentation

begin

                    3 = (2 * 1) + 1
;

then Lm1:  3 div 2 = 1
by NAT_D:def_1;

 1 = (2 * 0) + 1
;

then Lm2:  1 div 2 = 0
by NAT_D:def_1;

definition
let f be FinSequence;
func Center f ->  Element of NAT  equals :: JORDAN1A:def 1
((len f) div 2) + 1;
coherence
 ((len f) div 2) + 1 is Element of NAT ;
end;

:: deftheorem  defines Center JORDAN1A:def_1_:_
 for f being FinSequence holds Center f = ((len f) div 2) + 1;

theorem :: JORDAN1A:1
for f being FinSequence st len f is odd holds
len f = (2 * (Center f)) - 1
proof
let f be FinSequence; ::_thesis: ( len f is odd implies len f = (2 * (Center f)) - 1 )
assume  len f is odd ; ::_thesis: len f = (2 * (Center f)) - 1
then consider k being Element of NAT  such that
A1: len f = (2 * k) + 1 by ABIAN:9;
A2: (2 * k) mod 2 = 0 by NAT_D:13;
thus  len f = (2 * (((2 * k) div 2) + (1 div 2))) + 1 by A1, Lm2, NAT_D:18
.= (2 * ((len f) div 2)) + ((2 * 1) - 1) by A1, A2, NAT_D:19
.= (2 * (Center f)) - 1 ; ::_thesis: verum
end;

theorem :: JORDAN1A:2
for f being FinSequence st len f is even holds
len f = (2 * (Center f)) - 2
proof
let f be FinSequence; ::_thesis: ( len f is even implies len f = (2 * (Center f)) - 2 )
assume  ex k being Element of NAT st len f = 2 * k ; :: according to ABIAN:def_2 ::_thesis: len f = (2 * (Center f)) - 2
hence  len f = ((2 * ((len f) div 2)) + (2 * 1)) - 2 by NAT_D:18
.= (2 * (Center f)) - 2 ;
::_thesis: verum
end;

begin

registration
cluster non empty being_simple_closed_curve compact non horizontal non vertical for Element of K6( the U1 of (TOP-REAL 2));
existence
 ex b1 being Subset of (TOP-REAL 2) st
( b1 is compact & not b1 is vertical & not b1 is horizontal & b1 is being_simple_closed_curve & not b1 is empty )
proof
set f = the non constant standard special_circular_sequence;
take  L~ the non constant standard special_circular_sequence ; ::_thesis: ( L~ the non constant standard special_circular_sequence is compact & not L~ the non constant standard special_circular_sequence is vertical & not L~ the non constant standard special_circular_sequence is horizontal & L~ the non constant standard special_circular_sequence is being_simple_closed_curve & not L~ the non constant standard special_circular_sequence is empty )
thus  ( L~ the non constant standard special_circular_sequence is compact & not L~ the non constant standard special_circular_sequence is vertical & not L~ the non constant standard special_circular_sequence is horizontal & L~ the non constant standard special_circular_sequence is being_simple_closed_curve & not L~ the non constant standard special_circular_sequence is empty ) ; ::_thesis: verum
end;
end;

theorem Th3: :: JORDAN1A:3
for D being non empty Subset of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in N-most D holds
p `2 = N-bound D
proof
let D be non empty Subset of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st p in N-most D holds
p `2 = N-bound D
let p be Point of (TOP-REAL 2); ::_thesis: ( p in N-most D implies p `2 = N-bound D )
assume  p in N-most D ; ::_thesis: p `2 = N-bound D
then A1: p in LSeg ((NW-corner D),(NE-corner D)) by XBOOLE_0:def_4;
(NE-corner D) `2 = N-bound D by EUCLID:52
.= (NW-corner D) `2 by EUCLID:52 ;
hence p `2 = (NW-corner D) `2 by A1, GOBOARD7:6
.= N-bound D by EUCLID:52 ;
::_thesis: verum
end;

theorem Th4: :: JORDAN1A:4
for D being non empty Subset of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in E-most D holds
p `1 = E-bound D
proof
let D be non empty Subset of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st p in E-most D holds
p `1 = E-bound D
let p be Point of (TOP-REAL 2); ::_thesis: ( p in E-most D implies p `1 = E-bound D )
assume  p in E-most D ; ::_thesis: p `1 = E-bound D
then A1: p in LSeg ((SE-corner D),(NE-corner D)) by XBOOLE_0:def_4;
(SE-corner D) `1 = E-bound D by EUCLID:52
.= (NE-corner D) `1 by EUCLID:52 ;
hence p `1 = (SE-corner D) `1 by A1, GOBOARD7:5
.= E-bound D by EUCLID:52 ;
::_thesis: verum
end;

theorem Th5: :: JORDAN1A:5
for D being non empty Subset of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in S-most D holds
p `2 = S-bound D
proof
let D be non empty Subset of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st p in S-most D holds
p `2 = S-bound D
let p be Point of (TOP-REAL 2); ::_thesis: ( p in S-most D implies p `2 = S-bound D )
assume  p in S-most D ; ::_thesis: p `2 = S-bound D
then A1: p in LSeg ((SW-corner D),(SE-corner D)) by XBOOLE_0:def_4;
(SE-corner D) `2 = S-bound D by EUCLID:52
.= (SW-corner D) `2 by EUCLID:52 ;
hence p `2 = (SW-corner D) `2 by A1, GOBOARD7:6
.= S-bound D by EUCLID:52 ;
::_thesis: verum
end;

theorem Th6: :: JORDAN1A:6
for D being non empty Subset of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in W-most D holds
p `1 = W-bound D
proof
let D be non empty Subset of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st p in W-most D holds
p `1 = W-bound D
let p be Point of (TOP-REAL 2); ::_thesis: ( p in W-most D implies p `1 = W-bound D )
assume  p in W-most D ; ::_thesis: p `1 = W-bound D
then A1: p in LSeg ((SW-corner D),(NW-corner D)) by XBOOLE_0:def_4;
(SW-corner D) `1 = W-bound D by EUCLID:52
.= (NW-corner D) `1 by EUCLID:52 ;
hence p `1 = (SW-corner D) `1 by A1, GOBOARD7:5
.= W-bound D by EUCLID:52 ;
::_thesis: verum
end;

theorem :: JORDAN1A:7
for D being Subset of (TOP-REAL 2) holds BDD D misses D
proof
let D be Subset of (TOP-REAL 2); ::_thesis: BDD D misses D
 D misses D ` by SUBSET_1:24;
hence  BDD D misses D by JORDAN2C:25, XBOOLE_1:63; ::_thesis: verum
end;

theorem Th8: :: JORDAN1A:8
for p being Point of (TOP-REAL 2) holds p in Vertical_Line (p `1)
proof
let p be Point of (TOP-REAL 2); ::_thesis: p in Vertical_Line (p `1)
 p in  {  q where q is Point of (TOP-REAL 2) : p `1 = q `1  }  ;
hence  p in Vertical_Line (p `1) by JORDAN6:def_6; ::_thesis: verum
end;

theorem :: JORDAN1A:9
for r, s being real number holds |[r,s]| in Vertical_Line r
proof
let r, s be real number ; ::_thesis: |[r,s]| in Vertical_Line r
 |[r,s]| `1 = r by EUCLID:52;
hence  |[r,s]| in Vertical_Line r by Th8; ::_thesis: verum
end;

theorem :: JORDAN1A:10
for s being real number
for A being Subset of (TOP-REAL 2) st A c= Vertical_Line s holds
A is vertical
proof
let s be real number ; ::_thesis: for A being Subset of (TOP-REAL 2) st A c= Vertical_Line s holds
A is vertical
A1: Vertical_Line s =  {  p where p is Point of (TOP-REAL 2) : p `1 = s  }  by JORDAN6:def_6;
let A be Subset of (TOP-REAL 2); ::_thesis: ( A c= Vertical_Line s implies A is vertical )
assume A2: A c= Vertical_Line s ; ::_thesis: A is vertical
 for p, q being Point of (TOP-REAL 2) st p in A & q in A holds
p `1 = q `1
proof
let p, q be Point of (TOP-REAL 2); ::_thesis: ( p in A & q in A implies p `1 = q `1 )
assume  p in A ; ::_thesis: ( not q in A or p `1 = q `1 )
then  p in Vertical_Line s by A2;
then A3: ex p1 being Point of (TOP-REAL 2) st
( p1 = p & p1 `1 = s ) by A1;
assume  q in A ; ::_thesis: p `1 = q `1
then  q in Vertical_Line s by A2;
then  ex p1 being Point of (TOP-REAL 2) st
( p1 = q & p1 `1 = s ) by A1;
hence  p `1 = q `1 by A3; ::_thesis: verum
end;
hence  A is vertical by SPPOL_1:def_3; ::_thesis: verum
end;

theorem :: JORDAN1A:11
for p, q being Point of (TOP-REAL 2)
for r being real number st p `1 = q `1 & r in [.(proj2 . p),(proj2 . q).] holds
|[(p `1),r]| in LSeg (p,q)
proof
let p, q be Point of (TOP-REAL 2); ::_thesis: for r being real number st p `1 = q `1 & r in [.(proj2 . p),(proj2 . q).] holds
|[(p `1),r]| in LSeg (p,q)
let r be real number ; ::_thesis: ( p `1 = q `1 & r in [.(proj2 . p),(proj2 . q).] implies |[(p `1),r]| in LSeg (p,q) )
assume A1: p `1 = q `1 ; ::_thesis: ( not r in [.(proj2 . p),(proj2 . q).] or |[(p `1),r]| in LSeg (p,q) )
assume A2: r in [.(proj2 . p),(proj2 . q).] ; ::_thesis: |[(p `1),r]| in LSeg (p,q)
A3: |[(p `1),r]| `2 = r by EUCLID:52;
 proj2 . q = q `2 by PSCOMP_1:def_6;
then A4: |[(p `1),r]| `2 <= q `2 by A2, A3, XXREAL_1:1;
 proj2 . p = p `2 by PSCOMP_1:def_6;
then  ( p `1 = |[(p `1),r]| `1 & p `2 <= |[(p `1),r]| `2 ) by A2, A3, EUCLID:52, XXREAL_1:1;
hence  |[(p `1),r]| in LSeg (p,q) by A1, A4, GOBOARD7:7; ::_thesis: verum
end;

theorem :: JORDAN1A:12
for p, q being Point of (TOP-REAL 2)
for r being real number st p `2 = q `2 & r in [.(proj1 . p),(proj1 . q).] holds
|[r,(p `2)]| in LSeg (p,q)
proof
let p, q be Point of (TOP-REAL 2); ::_thesis: for r being real number st p `2 = q `2 & r in [.(proj1 . p),(proj1 . q).] holds
|[r,(p `2)]| in LSeg (p,q)
let r be real number ; ::_thesis: ( p `2 = q `2 & r in [.(proj1 . p),(proj1 . q).] implies |[r,(p `2)]| in LSeg (p,q) )
assume A1: p `2 = q `2 ; ::_thesis: ( not r in [.(proj1 . p),(proj1 . q).] or |[r,(p `2)]| in LSeg (p,q) )
assume A2: r in [.(proj1 . p),(proj1 . q).] ; ::_thesis: |[r,(p `2)]| in LSeg (p,q)
A3: |[r,(p `2)]| `1 = r by EUCLID:52;
 proj1 . q = q `1 by PSCOMP_1:def_5;
then A4: |[r,(p `2)]| `1 <= q `1 by A2, A3, XXREAL_1:1;
 proj1 . p = p `1 by PSCOMP_1:def_5;
then  ( p `2 = |[r,(p `2)]| `2 & p `1 <= |[r,(p `2)]| `1 ) by A2, A3, EUCLID:52, XXREAL_1:1;
hence  |[r,(p `2)]| in LSeg (p,q) by A1, A4, GOBOARD7:8; ::_thesis: verum
end;

theorem :: JORDAN1A:13
for p, q being Point of (TOP-REAL 2)
for s being real number st p in Vertical_Line s & q in Vertical_Line s holds
LSeg (p,q) c= Vertical_Line s
proof
let p, q be Point of (TOP-REAL 2); ::_thesis: for s being real number st p in Vertical_Line s & q in Vertical_Line s holds
LSeg (p,q) c= Vertical_Line s
let s be real number ; ::_thesis: ( p in Vertical_Line s & q in Vertical_Line s implies LSeg (p,q) c= Vertical_Line s )
A1: Vertical_Line s =  {  p1 where p1 is Point of (TOP-REAL 2) : p1 `1 = s  }  by JORDAN6:def_6;
assume  ( p in Vertical_Line s & q in Vertical_Line s ) ; ::_thesis: LSeg (p,q) c= Vertical_Line s
then A2: ( p `1 = s & q `1 = s ) by JORDAN6:31;
let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in LSeg (p,q) or u in Vertical_Line s )
assume A3: u in LSeg (p,q) ; ::_thesis: u in Vertical_Line s
then reconsider p1 = u as Point of (TOP-REAL 2) ;
 p1 `1 = s by A2, A3, GOBOARD7:5;
hence  u in Vertical_Line s by A1; ::_thesis: verum
end;

theorem :: JORDAN1A:14
for A, B being Subset of (TOP-REAL 2) st A meets B holds
proj2 .: A meets proj2 .: B
proof
let A, B be Subset of (TOP-REAL 2); ::_thesis: ( A meets B implies proj2 .: A meets proj2 .: B )
assume  A meets B ; ::_thesis: proj2 .: A meets proj2 .: B
then consider e being set  such that
A1: e in A and
A2: e in B by XBOOLE_0:3;
reconsider e = e as Point of (TOP-REAL 2) by A1;
 e `2 = proj2 . e by PSCOMP_1:def_6;
then  ( e `2 in proj2 .: A & e `2 in proj2 .: B ) by A1, A2, FUNCT_2:35;
hence  proj2 .: A meets proj2 .: B by XBOOLE_0:3; ::_thesis: verum
end;

theorem :: JORDAN1A:15
for s being real number
for A, B being Subset of (TOP-REAL 2) st A misses B & A c= Vertical_Line s & B c= Vertical_Line s holds
proj2 .: A misses proj2 .: B
proof
let s be real number ; ::_thesis: for A, B being Subset of (TOP-REAL 2) st A misses B & A c= Vertical_Line s & B c= Vertical_Line s holds
proj2 .: A misses proj2 .: B
let A, B be Subset of (TOP-REAL 2); ::_thesis: ( A misses B & A c= Vertical_Line s & B c= Vertical_Line s implies proj2 .: A misses proj2 .: B )
assume that
A1: A misses B and
A2: A c= Vertical_Line s and
A3: B c= Vertical_Line s ; ::_thesis: proj2 .: A misses proj2 .: B
assume  proj2 .: A meets proj2 .: B ; ::_thesis: contradiction
then consider e being set  such that
A4: e in proj2 .: A and
A5: e in proj2 .: B by XBOOLE_0:3;
reconsider e = e as Real by A4;
consider d being Point of (TOP-REAL 2) such that
A6: d in B and
A7: e = proj2 . d by A5, FUNCT_2:65;
A8: d `1 = s by A3, A6, JORDAN6:31;
consider c being Point of (TOP-REAL 2) such that
A9: c in A and
A10: e = proj2 . c by A4, FUNCT_2:65;
 c `1 = s by A2, A9, JORDAN6:31;
then c = |[(d `1),(c `2)]| by A8, EUCLID:53
.= |[(d `1),e]| by A10, PSCOMP_1:def_6
.= |[(d `1),(d `2)]| by A7, PSCOMP_1:def_6
.= d by EUCLID:53 ;
hence  contradiction by A1, A9, A6, XBOOLE_0:3; ::_thesis: verum
end;

begin

theorem :: JORDAN1A:16
for i, j being Element of NAT
for G being Go-board st 1 <= i & i <= len G & 1 <= j & j <= width G holds
G * (i,j) in LSeg ((G * (i,1)),(G * (i,(width G))))
proof
let i, j be Element of NAT ; ::_thesis: for G being Go-board st 1 <= i & i <= len G & 1 <= j & j <= width G holds
G * (i,j) in LSeg ((G * (i,1)),(G * (i,(width G))))
let G be Go-board; ::_thesis: ( 1 <= i & i <= len G & 1 <= j & j <= width G implies G * (i,j) in LSeg ((G * (i,1)),(G * (i,(width G)))) )
assume that
A1: ( 1 <= i & i <= len G ) and
A2: ( 1 <= j & j <= width G ) ; ::_thesis: G * (i,j) in LSeg ((G * (i,1)),(G * (i,(width G))))
A3: (G * (i,j)) `2 <= (G * (i,(width G))) `2 by A1, A2, SPRECT_3:12;
 1 <= width G by A2, XXREAL_0:2;
then A4: (G * (i,1)) `1 = (G * (i,(width G))) `1 by A1, GOBOARD5:2;
 ( (G * (i,1)) `1 = (G * (i,j)) `1 & (G * (i,1)) `2 <= (G * (i,j)) `2 ) by A1, A2, GOBOARD5:2, SPRECT_3:12;
hence  G * (i,j) in LSeg ((G * (i,1)),(G * (i,(width G)))) by A4, A3, GOBOARD7:7; ::_thesis: verum
end;

theorem :: JORDAN1A:17
for i, j being Element of NAT
for G being Go-board st 1 <= i & i <= len G & 1 <= j & j <= width G holds
G * (i,j) in LSeg ((G * (1,j)),(G * ((len G),j)))
proof
let i, j be Element of NAT ; ::_thesis: for G being Go-board st 1 <= i & i <= len G & 1 <= j & j <= width G holds
G * (i,j) in LSeg ((G * (1,j)),(G * ((len G),j)))
let G be Go-board; ::_thesis: ( 1 <= i & i <= len G & 1 <= j & j <= width G implies G * (i,j) in LSeg ((G * (1,j)),(G * ((len G),j))) )
assume that
A1: ( 1 <= i & i <= len G ) and
A2: ( 1 <= j & j <= width G ) ; ::_thesis: G * (i,j) in LSeg ((G * (1,j)),(G * ((len G),j)))
A3: (G * (i,j)) `1 <= (G * ((len G),j)) `1 by A1, A2, SPRECT_3:13;
 1 <= len G by A1, XXREAL_0:2;
then A4: (G * (1,j)) `2 = (G * ((len G),j)) `2 by A2, GOBOARD5:1;
 ( (G * (1,j)) `2 = (G * (i,j)) `2 & (G * (1,j)) `1 <= (G * (i,j)) `1 ) by A1, A2, GOBOARD5:1, SPRECT_3:13;
hence  G * (i,j) in LSeg ((G * (1,j)),(G * ((len G),j))) by A4, A3, GOBOARD7:8; ::_thesis: verum
end;

theorem Th18: :: JORDAN1A:18
for j1, j2, i1, i2 being Element of NAT
for G being Go-board st 1 <= j1 & j1 <= width G & 1 <= j2 & j2 <= width G & 1 <= i1 & i1 <= i2 & i2 <= len G holds
(G * (i1,j1)) `1 <= (G * (i2,j2)) `1
proof
let j1, j2, i1, i2 be Element of NAT ; ::_thesis: for G being Go-board st 1 <= j1 & j1 <= width G & 1 <= j2 & j2 <= width G & 1 <= i1 & i1 <= i2 & i2 <= len G holds
(G * (i1,j1)) `1 <= (G * (i2,j2)) `1
let G be Go-board; ::_thesis: ( 1 <= j1 & j1 <= width G & 1 <= j2 & j2 <= width G & 1 <= i1 & i1 <= i2 & i2 <= len G implies (G * (i1,j1)) `1 <= (G * (i2,j2)) `1 )
assume that
A1: ( 1 <= j1 & j1 <= width G ) and
A2: ( 1 <= j2 & j2 <= width G ) and
A3: ( 1 <= i1 & i1 <= i2 ) and
A4: i2 <= len G ; ::_thesis: (G * (i1,j1)) `1 <= (G * (i2,j2)) `1
A5: 1 <= i2 by A3, XXREAL_0:2;
then (G * (i2,j1)) `1 = (G * (i2,1)) `1 by A1, A4, GOBOARD5:2
.= (G * (i2,j2)) `1 by A2, A4, A5, GOBOARD5:2 ;
hence  (G * (i1,j1)) `1 <= (G * (i2,j2)) `1 by A1, A3, A4, SPRECT_3:13; ::_thesis: verum
end;

theorem Th19: :: JORDAN1A:19
for i1, i2, j1, j2 being Element of NAT
for G being Go-board st 1 <= i1 & i1 <= len G & 1 <= i2 & i2 <= len G & 1 <= j1 & j1 <= j2 & j2 <= width G holds
(G * (i1,j1)) `2 <= (G * (i2,j2)) `2
proof
let i1, i2, j1, j2 be Element of NAT ; ::_thesis: for G being Go-board st 1 <= i1 & i1 <= len G & 1 <= i2 & i2 <= len G & 1 <= j1 & j1 <= j2 & j2 <= width G holds
(G * (i1,j1)) `2 <= (G * (i2,j2)) `2
let G be Go-board; ::_thesis: ( 1 <= i1 & i1 <= len G & 1 <= i2 & i2 <= len G & 1 <= j1 & j1 <= j2 & j2 <= width G implies (G * (i1,j1)) `2 <= (G * (i2,j2)) `2 )
assume that
A1: ( 1 <= i1 & i1 <= len G ) and
A2: ( 1 <= i2 & i2 <= len G ) and
A3: ( 1 <= j1 & j1 <= j2 ) and
A4: j2 <= width G ; ::_thesis: (G * (i1,j1)) `2 <= (G * (i2,j2)) `2
A5: 1 <= j2 by A3, XXREAL_0:2;
then (G * (i1,j2)) `2 = (G * (1,j2)) `2 by A1, A4, GOBOARD5:1
.= (G * (i2,j2)) `2 by A2, A4, A5, GOBOARD5:1 ;
hence  (G * (i1,j1)) `2 <= (G * (i2,j2)) `2 by A1, A3, A4, SPRECT_3:12; ::_thesis: verum
end;

theorem Th20: :: JORDAN1A:20
for t being Element of NAT
for G being Go-board
for f being non constant standard special_circular_sequence st f is_sequence_on G & 1 <= t & t <= len G holds
(G * (t,(width G))) `2 >= N-bound (L~ f)
proof
let t be Element of NAT ; ::_thesis: for G being Go-board
for f being non constant standard special_circular_sequence st f is_sequence_on G & 1 <= t & t <= len G holds
(G * (t,(width G))) `2 >= N-bound (L~ f)
let G be Go-board; ::_thesis: for f being non constant standard special_circular_sequence st f is_sequence_on G & 1 <= t & t <= len G holds
(G * (t,(width G))) `2 >= N-bound (L~ f)
let f be non constant standard special_circular_sequence; ::_thesis: ( f is_sequence_on G & 1 <= t & t <= len G implies (G * (t,(width G))) `2 >= N-bound (L~ f) )
 N-min (L~ f) in rng f by SPRECT_2:39;
then consider x being set  such that
A1: x in dom f and
A2: f . x = N-min (L~ f) by FUNCT_1:def_3;
reconsider x = x as Element of NAT by A1;
assume  f is_sequence_on G ; ::_thesis: ( not 1 <= t or not t <= len G or (G * (t,(width G))) `2 >= N-bound (L~ f) )
then consider i, j being Element of NAT  such that
A3: [i,j] in Indices G and
A4: f /. x = G * (i,j) by A1, GOBOARD1:def_9;
A5: ( 1 <= i & i <= len G ) by A3, MATRIX_1:38;
assume A6: ( 1 <= t & t <= len G ) ; ::_thesis: (G * (t,(width G))) `2 >= N-bound (L~ f)
 ( 1 <= j & j <= width G ) by A3, MATRIX_1:38;
then  ( N-bound (L~ f) = (N-min (L~ f)) `2 & (G * (t,(width G))) `2 >= (G * (i,j)) `2 ) by A6, A5, Th19, EUCLID:52;
hence  (G * (t,(width G))) `2 >= N-bound (L~ f) by A1, A2, A4, PARTFUN1:def_6; ::_thesis: verum
end;

theorem Th21: :: JORDAN1A:21
for t being Element of NAT
for G being Go-board
for f being non constant standard special_circular_sequence st f is_sequence_on G & 1 <= t & t <= width G holds
(G * (1,t)) `1 <= W-bound (L~ f)
proof
let t be Element of NAT ; ::_thesis: for G being Go-board
for f being non constant standard special_circular_sequence st f is_sequence_on G & 1 <= t & t <= width G holds
(G * (1,t)) `1 <= W-bound (L~ f)
let G be Go-board; ::_thesis: for f being non constant standard special_circular_sequence st f is_sequence_on G & 1 <= t & t <= width G holds
(G * (1,t)) `1 <= W-bound (L~ f)
let f be non constant standard special_circular_sequence; ::_thesis: ( f is_sequence_on G & 1 <= t & t <= width G implies (G * (1,t)) `1 <= W-bound (L~ f) )
 W-min (L~ f) in rng f by SPRECT_2:43;
then consider x being set  such that
A1: x in dom f and
A2: f . x = W-min (L~ f) by FUNCT_1:def_3;
reconsider x = x as Element of NAT by A1;
assume  f is_sequence_on G ; ::_thesis: ( not 1 <= t or not t <= width G or (G * (1,t)) `1 <= W-bound (L~ f) )
then consider i, j being Element of NAT  such that
A3: [i,j] in Indices G and
A4: f /. x = G * (i,j) by A1, GOBOARD1:def_9;
A5: ( 1 <= i & i <= len G ) by A3, MATRIX_1:38;
assume A6: ( 1 <= t & t <= width G ) ; ::_thesis: (G * (1,t)) `1 <= W-bound (L~ f)
 ( 1 <= j & j <= width G ) by A3, MATRIX_1:38;
then  ( W-bound (L~ f) = (W-min (L~ f)) `1 & (G * (1,t)) `1 <= (G * (i,j)) `1 ) by A6, A5, Th18, EUCLID:52;
hence  (G * (1,t)) `1 <= W-bound (L~ f) by A1, A2, A4, PARTFUN1:def_6; ::_thesis: verum
end;

theorem Th22: :: JORDAN1A:22
for t being Element of NAT
for G being Go-board
for f being non constant standard special_circular_sequence st f is_sequence_on G & 1 <= t & t <= len G holds
(G * (t,1)) `2 <= S-bound (L~ f)
proof
let t be Element of NAT ; ::_thesis: for G being Go-board
for f being non constant standard special_circular_sequence st f is_sequence_on G & 1 <= t & t <= len G holds
(G * (t,1)) `2 <= S-bound (L~ f)
let G be Go-board; ::_thesis: for f being non constant standard special_circular_sequence st f is_sequence_on G & 1 <= t & t <= len G holds
(G * (t,1)) `2 <= S-bound (L~ f)
let f be non constant standard special_circular_sequence; ::_thesis: ( f is_sequence_on G & 1 <= t & t <= len G implies (G * (t,1)) `2 <= S-bound (L~ f) )
 S-min (L~ f) in rng f by SPRECT_2:41;
then consider x being set  such that
A1: x in dom f and
A2: f . x = S-min (L~ f) by FUNCT_1:def_3;
reconsider x = x as Element of NAT by A1;
assume  f is_sequence_on G ; ::_thesis: ( not 1 <= t or not t <= len G or (G * (t,1)) `2 <= S-bound (L~ f) )
then consider i, j being Element of NAT  such that
A3: [i,j] in Indices G and
A4: f /. x = G * (i,j) by A1, GOBOARD1:def_9;
A5: ( 1 <= i & i <= len G ) by A3, MATRIX_1:38;
assume A6: ( 1 <= t & t <= len G ) ; ::_thesis: (G * (t,1)) `2 <= S-bound (L~ f)
 ( 1 <= j & j <= width G ) by A3, MATRIX_1:38;
then  ( S-bound (L~ f) = (S-min (L~ f)) `2 & (G * (t,1)) `2 <= (G * (i,j)) `2 ) by A6, A5, Th19, EUCLID:52;
hence  (G * (t,1)) `2 <= S-bound (L~ f) by A1, A2, A4, PARTFUN1:def_6; ::_thesis: verum
end;

theorem Th23: :: JORDAN1A:23
for t being Element of NAT
for G being Go-board
for f being non constant standard special_circular_sequence st f is_sequence_on G & 1 <= t & t <= width G holds
(G * ((len G),t)) `1 >= E-bound (L~ f)
proof
let t be Element of NAT ; ::_thesis: for G being Go-board
for f being non constant standard special_circular_sequence st f is_sequence_on G & 1 <= t & t <= width G holds
(G * ((len G),t)) `1 >= E-bound (L~ f)
let G be Go-board; ::_thesis: for f being non constant standard special_circular_sequence st f is_sequence_on G & 1 <= t & t <= width G holds
(G * ((len G),t)) `1 >= E-bound (L~ f)
let f be non constant standard special_circular_sequence; ::_thesis: ( f is_sequence_on G & 1 <= t & t <= width G implies (G * ((len G),t)) `1 >= E-bound (L~ f) )
 E-min (L~ f) in rng f by SPRECT_2:45;
then consider x being set  such that
A1: x in dom f and
A2: f . x = E-min (L~ f) by FUNCT_1:def_3;
reconsider x = x as Element of NAT by A1;
assume  f is_sequence_on G ; ::_thesis: ( not 1 <= t or not t <= width G or (G * ((len G),t)) `1 >= E-bound (L~ f) )
then consider i, j being Element of NAT  such that
A3: [i,j] in Indices G and
A4: f /. x = G * (i,j) by A1, GOBOARD1:def_9;
A5: ( 1 <= i & i <= len G ) by A3, MATRIX_1:38;
assume A6: ( 1 <= t & t <= width G ) ; ::_thesis: (G * ((len G),t)) `1 >= E-bound (L~ f)
 ( 1 <= j & j <= width G ) by A3, MATRIX_1:38;
then  ( E-bound (L~ f) = (E-min (L~ f)) `1 & (G * ((len G),t)) `1 >= (G * (i,j)) `1 ) by A6, A5, Th18, EUCLID:52;
hence  (G * ((len G),t)) `1 >= E-bound (L~ f) by A1, A2, A4, PARTFUN1:def_6; ::_thesis: verum
end;

theorem Th24: :: JORDAN1A:24
for i, j being Element of NAT
for G being Go-board st i <= len G & j <= width G holds
not cell (G,i,j) is empty
proof
let i, j be Element of NAT ; ::_thesis: for G being Go-board st i <= len G & j <= width G holds
not cell (G,i,j) is empty
let G be Go-board; ::_thesis: ( i <= len G & j <= width G implies not cell (G,i,j) is empty )
assume  ( i <= len G & j <= width G ) ; ::_thesis: not cell (G,i,j) is empty
then  not Int (cell (G,i,j)) is empty by GOBOARD9:14;
hence  not cell (G,i,j) is empty by TOPS_1:16, XBOOLE_1:3; ::_thesis: verum
end;

theorem Th25: :: JORDAN1A:25
for i, j being Element of NAT
for G being Go-board st i <= len G & j <= width G holds
cell (G,i,j) is connected
proof
let i, j be Element of NAT ; ::_thesis: for G being Go-board st i <= len G & j <= width G holds
cell (G,i,j) is connected
let G be Go-board; ::_thesis: ( i <= len G & j <= width G implies cell (G,i,j) is connected )
assume A1: ( i <= len G & j <= width G ) ; ::_thesis: cell (G,i,j) is connected
then  Int (cell (G,i,j)) is convex by GOBOARD9:17;
then  Cl (Int (cell (G,i,j))) is connected by CONNSP_1:19;
hence  cell (G,i,j) is connected by A1, GOBRD11:35; ::_thesis: verum
end;

theorem Th26: :: JORDAN1A:26
for i being Element of NAT
for G being Go-board st i <= len G holds
not cell (G,i,0) is bounded
proof
let i be Element of NAT ; ::_thesis: for G being Go-board st i <= len G holds
not cell (G,i,0) is bounded
let G be Go-board; ::_thesis: ( i <= len G implies not cell (G,i,0) is bounded )
assume A1: i <= len G ; ::_thesis: not cell (G,i,0) is bounded
percases ( i = 0 or ( i >= 1 & i < len G ) or i = len G ) by A1, NAT_1:14, XXREAL_0:1;
suppose i = 0 ; ::_thesis: not cell (G,i,0) is bounded
then A2: cell (G,i,0) =  {  |[r,s]| where r, s is Real : ( r <= (G * (1,1)) `1 & s <= (G * (1,1)) `2 )  }  by GOBRD11:24;
 for r being Real ex q being Point of (TOP-REAL 2) st
( q in cell (G,i,0) & not |.q.| < r )
proof
let r be Real; ::_thesis: ex q being Point of (TOP-REAL 2) st
( q in cell (G,i,0) & not |.q.| < r )
take q = |[(min ((- r),((G * (1,1)) `1))),(min ((- r),((G * (1,1)) `2)))]|; ::_thesis: ( q in cell (G,i,0) & not |.q.| < r )
A3: abs (q `1) <= |.q.| by JGRAPH_1:33;
 ( min ((- r),((G * (1,1)) `1)) <= (G * (1,1)) `1 & min ((- r),((G * (1,1)) `2)) <= (G * (1,1)) `2 ) by XXREAL_0:17;
hence  q in cell (G,i,0) by A2; ::_thesis: not |.q.| < r
percases ( r <= 0 or r > 0 ) ;
suppose r <= 0 ; ::_thesis: not |.q.| < r
hence  not |.q.| < r ; ::_thesis: verum
end;
supposeA4: r > 0 ; ::_thesis: not |.q.| < r
 q `1 = min ((- r),((G * (1,1)) `1)) by EUCLID:52;
then A5: abs (- r) <= abs (q `1) by A4, TOPREAL6:3, XXREAL_0:17;
 - (- r) > 0 by A4;
then  - r < 0 ;
then  - (- r) <= abs (q `1) by A5, ABSVALUE:def_1;
hence  not |.q.| < r by A3, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
hence  not cell (G,i,0) is bounded by JORDAN2C:34; ::_thesis: verum
end;
supposeA6: ( i >= 1 & i < len G ) ; ::_thesis: not cell (G,i,0) is bounded
then A7: cell (G,i,0) =  {  |[r,s]| where r, s is Element of REAL : ( (G * (i,1)) `1 <= r & r <= (G * ((i + 1),1)) `1 & s <= (G * (1,1)) `2 )  }  by GOBRD11:30;
 for r being Real ex q being Point of (TOP-REAL 2) st
( q in cell (G,i,0) & not |.q.| < r )
proof
let r be Real; ::_thesis: ex q being Point of (TOP-REAL 2) st
( q in cell (G,i,0) & not |.q.| < r )
take q = |[((G * (i,1)) `1),(min ((- r),((G * (1,1)) `2)))]|; ::_thesis: ( q in cell (G,i,0) & not |.q.| < r )
A8: min ((- r),((G * (1,1)) `2)) <= (G * (1,1)) `2 by XXREAL_0:17;
 width G <> 0 by GOBOARD1:def_3;
then A9: 1 <= width G by NAT_1:14;
 ( i < i + 1 & i + 1 <= len G ) by A6, NAT_1:13;
then  (G * (i,1)) `1 <= (G * ((i + 1),1)) `1 by A6, A9, GOBOARD5:3;
hence  q in cell (G,i,0) by A7, A8; ::_thesis: not |.q.| < r
A10: abs (q `2) <= |.q.| by JGRAPH_1:33;
percases ( r <= 0 or r > 0 ) ;
suppose r <= 0 ; ::_thesis: not |.q.| < r
hence  not |.q.| < r ; ::_thesis: verum
end;
supposeA11: r > 0 ; ::_thesis: not |.q.| < r
 q `2 = min ((- r),((G * (1,1)) `2)) by EUCLID:52;
then A12: abs (- r) <= abs (q `2) by A11, TOPREAL6:3, XXREAL_0:17;
 - (- r) > 0 by A11;
then  - r < 0 ;
then  - (- r) <= abs (q `2) by A12, ABSVALUE:def_1;
hence  not |.q.| < r by A10, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
hence  not cell (G,i,0) is bounded by JORDAN2C:34; ::_thesis: verum
end;
suppose i = len G ; ::_thesis: not cell (G,i,0) is bounded
then A13: cell (G,i,0) =  {  |[r,s]| where r, s is Element of REAL : ( (G * ((len G),1)) `1 <= r & s <= (G * (1,1)) `2 )  }  by GOBRD11:27;
 for r being Real ex q being Point of (TOP-REAL 2) st
( q in cell (G,i,0) & not |.q.| < r )
proof
let r be Real; ::_thesis: ex q being Point of (TOP-REAL 2) st
( q in cell (G,i,0) & not |.q.| < r )
take q = |[(max (r,((G * ((len G),1)) `1))),((G * (1,1)) `2)]|; ::_thesis: ( q in cell (G,i,0) & not |.q.| < r )
A14: abs (q `1) <= |.q.| by JGRAPH_1:33;
 (G * ((len G),1)) `1 <= max (r,((G * ((len G),1)) `1)) by XXREAL_0:25;
hence  q in cell (G,i,0) by A13; ::_thesis: not |.q.| < r
percases ( r <= 0 or r > 0 ) ;
suppose r <= 0 ; ::_thesis: not |.q.| < r
hence  not |.q.| < r ; ::_thesis: verum
end;
supposeA15: r > 0 ; ::_thesis: not |.q.| < r
 q `1 = max (r,((G * ((len G),1)) `1)) by EUCLID:52;
then  r <= q `1 by XXREAL_0:25;
then  r <= abs (q `1) by A15, ABSVALUE:def_1;
hence  not |.q.| < r by A14, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
hence  not cell (G,i,0) is bounded by JORDAN2C:34; ::_thesis: verum
end;
end;
end;

theorem Th27: :: JORDAN1A:27
for i being Element of NAT
for G being Go-board st i <= len G holds
not cell (G,i,(width G)) is bounded
proof
let i be Element of NAT ; ::_thesis: for G being Go-board st i <= len G holds
not cell (G,i,(width G)) is bounded
let G be Go-board; ::_thesis: ( i <= len G implies not cell (G,i,(width G)) is bounded )
assume A1: i <= len G ; ::_thesis: not cell (G,i,(width G)) is bounded
percases ( i = 0 or ( i >= 1 & i < len G ) or i = len G ) by A1, NAT_1:14, XXREAL_0:1;
supposeA2: i = 0 ; ::_thesis: not cell (G,i,(width G)) is bounded
A3: cell (G,0,(width G)) =  {  |[r,s]| where r, s is Element of REAL : ( r <= (G * (1,1)) `1 & (G * (1,(width G))) `2 <= s )  }  by GOBRD11:25;
 for r being Real ex q being Point of (TOP-REAL 2) st
( q in cell (G,0,(width G)) & not |.q.| < r )
proof
let r be Real; ::_thesis: ex q being Point of (TOP-REAL 2) st
( q in cell (G,0,(width G)) & not |.q.| < r )
take q = |[(min ((- r),((G * (1,1)) `1))),((G * (1,(width G))) `2)]|; ::_thesis: ( q in cell (G,0,(width G)) & not |.q.| < r )
A4: abs (q `1) <= |.q.| by JGRAPH_1:33;
 min ((- r),((G * (1,1)) `1)) <= (G * (1,1)) `1 by XXREAL_0:17;
hence  q in cell (G,0,(width G)) by A3; ::_thesis: not |.q.| < r
percases ( r <= 0 or r > 0 ) ;
suppose r <= 0 ; ::_thesis: not |.q.| < r
hence  not |.q.| < r ; ::_thesis: verum
end;
supposeA5: r > 0 ; ::_thesis: not |.q.| < r
 q `1 = min ((- r),((G * (1,1)) `1)) by EUCLID:52;
then A6: abs (- r) <= abs (q `1) by A5, TOPREAL6:3, XXREAL_0:17;
 - (- r) > 0 by A5;
then  - r < 0 ;
then  - (- r) <= abs (q `1) by A6, ABSVALUE:def_1;
hence  not |.q.| < r by A4, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
hence  not cell (G,i,(width G)) is bounded by A2, JORDAN2C:34; ::_thesis: verum
end;
supposeA7: ( i >= 1 & i < len G ) ; ::_thesis: not cell (G,i,(width G)) is bounded
then A8: cell (G,i,(width G)) =  {  |[r,s]| where r, s is Element of REAL : ( (G * (i,1)) `1 <= r & r <= (G * ((i + 1),1)) `1 & (G * (1,(width G))) `2 <= s )  }  by GOBRD11:31;
 for r being Real ex q being Point of (TOP-REAL 2) st
( q in cell (G,i,(width G)) & not |.q.| < r )
proof
let r be Real; ::_thesis: ex q being Point of (TOP-REAL 2) st
( q in cell (G,i,(width G)) & not |.q.| < r )
take q = |[((G * (i,1)) `1),(max (r,((G * (1,(width G))) `2)))]|; ::_thesis: ( q in cell (G,i,(width G)) & not |.q.| < r )
A9: max (r,((G * (1,(width G))) `2)) >= (G * (1,(width G))) `2 by XXREAL_0:25;
 width G <> 0 by GOBOARD1:def_3;
then A10: 1 <= width G by NAT_1:14;
 ( i < i + 1 & i + 1 <= len G ) by A7, NAT_1:13;
then  (G * (i,1)) `1 <= (G * ((i + 1),1)) `1 by A7, A10, GOBOARD5:3;
hence  q in cell (G,i,(width G)) by A8, A9; ::_thesis: not |.q.| < r
A11: abs (q `2) <= |.q.| by JGRAPH_1:33;
percases ( r <= 0 or r > 0 ) ;
suppose r <= 0 ; ::_thesis: not |.q.| < r
hence  not |.q.| < r ; ::_thesis: verum
end;
supposeA12: r > 0 ; ::_thesis: not |.q.| < r
 q `2 = max (r,((G * (1,(width G))) `2)) by EUCLID:52;
then  q `2 >= r by XXREAL_0:25;
then  r <= abs (q `2) by A12, ABSVALUE:def_1;
hence  not |.q.| < r by A11, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
hence  not cell (G,i,(width G)) is bounded by JORDAN2C:34; ::_thesis: verum
end;
supposeA13: i = len G ; ::_thesis: not cell (G,i,(width G)) is bounded
A14: cell (G,(len G),(width G)) =  {  |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 <= r & (G * (1,(width G))) `2 <= s )  }  by GOBRD11:28;
 for r being Real ex q being Point of (TOP-REAL 2) st
( q in cell (G,i,(width G)) & not |.q.| < r )
proof
let r be Real; ::_thesis: ex q being Point of (TOP-REAL 2) st
( q in cell (G,i,(width G)) & not |.q.| < r )
take q = |[(max (r,((G * ((len G),1)) `1))),((G * (1,(width G))) `2)]|; ::_thesis: ( q in cell (G,i,(width G)) & not |.q.| < r )
A15: abs (q `1) <= |.q.| by JGRAPH_1:33;
 (G * ((len G),1)) `1 <= max (r,((G * ((len G),1)) `1)) by XXREAL_0:25;
hence  q in cell (G,i,(width G)) by A13, A14; ::_thesis: not |.q.| < r
percases ( r <= 0 or r > 0 ) ;
suppose r <= 0 ; ::_thesis: not |.q.| < r
hence  not |.q.| < r ; ::_thesis: verum
end;
supposeA16: r > 0 ; ::_thesis: not |.q.| < r
 q `1 = max (r,((G * ((len G),1)) `1)) by EUCLID:52;
then  r <= q `1 by XXREAL_0:25;
then  r <= abs (q `1) by A16, ABSVALUE:def_1;
hence  not |.q.| < r by A15, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
hence  not cell (G,i,(width G)) is bounded by JORDAN2C:34; ::_thesis: verum
end;
end;
end;

begin

theorem :: JORDAN1A:28
for n being Element of NAT
for D being non empty Subset of (TOP-REAL 2) holds width (Gauge (D,n)) = (2 |^ n) + 3
proof
let n be Element of NAT ; ::_thesis: for D being non empty Subset of (TOP-REAL 2) holds width (Gauge (D,n)) = (2 |^ n) + 3
let D be non empty Subset of (TOP-REAL 2); ::_thesis: width (Gauge (D,n)) = (2 |^ n) + 3
thus  width (Gauge (D,n)) = len (Gauge (D,n)) by JORDAN8:def_1
.= (2 |^ n) + 3 by JORDAN8:def_1 ; ::_thesis: verum
end;

theorem :: JORDAN1A:29
for i, j being Element of NAT
for D being non empty Subset of (TOP-REAL 2) st i < j holds
len (Gauge (D,i)) < len (Gauge (D,j))
proof
let i, j be Element of NAT ; ::_thesis: for D being non empty Subset of (TOP-REAL 2) st i < j holds
len (Gauge (D,i)) < len (Gauge (D,j))
let D be non empty Subset of (TOP-REAL 2); ::_thesis: ( i < j implies len (Gauge (D,i)) < len (Gauge (D,j)) )
assume  i < j ; ::_thesis: len (Gauge (D,i)) < len (Gauge (D,j))
then A1: 2 |^ i < 2 |^ j by PEPIN:66;
 ( len (Gauge (D,i)) = (2 |^ i) + 3 & len (Gauge (D,j)) = (2 |^ j) + 3 ) by JORDAN8:def_1;
hence  len (Gauge (D,i)) < len (Gauge (D,j)) by A1, XREAL_1:6; ::_thesis: verum
end;

theorem :: JORDAN1A:30
for i, j being Element of NAT
for D being non empty Subset of (TOP-REAL 2) st i <= j holds
len (Gauge (D,i)) <= len (Gauge (D,j))
proof
let i, j be Element of NAT ; ::_thesis: for D being non empty Subset of (TOP-REAL 2) st i <= j holds
len (Gauge (D,i)) <= len (Gauge (D,j))
let D be non empty Subset of (TOP-REAL 2); ::_thesis: ( i <= j implies len (Gauge (D,i)) <= len (Gauge (D,j)) )
assume  i <= j ; ::_thesis: len (Gauge (D,i)) <= len (Gauge (D,j))
then A1: 2 |^ i <= 2 |^ j by PREPOWER:93;
 ( len (Gauge (D,i)) = (2 |^ i) + 3 & len (Gauge (D,j)) = (2 |^ j) + 3 ) by JORDAN8:def_1;
hence  len (Gauge (D,i)) <= len (Gauge (D,j)) by A1, XREAL_1:6; ::_thesis: verum
end;

theorem Th31: :: JORDAN1A:31
for m, n, i being Element of NAT
for D being non empty Subset of (TOP-REAL 2) st m <= n & 1 < i & i < len (Gauge (D,m)) holds
( 1 < ((2 |^ (n -' m)) * (i - 2)) + 2 & ((2 |^ (n -' m)) * (i - 2)) + 2 < len (Gauge (D,n)) )
proof
let m, n, i be Element of NAT ; ::_thesis: for D being non empty Subset of (TOP-REAL 2) st m <= n & 1 < i & i < len (Gauge (D,m)) holds
( 1 < ((2 |^ (n -' m)) * (i - 2)) + 2 & ((2 |^ (n -' m)) * (i - 2)) + 2 < len (Gauge (D,n)) )
let D be non empty Subset of (TOP-REAL 2); ::_thesis: ( m <= n & 1 < i & i < len (Gauge (D,m)) implies ( 1 < ((2 |^ (n -' m)) * (i - 2)) + 2 & ((2 |^ (n -' m)) * (i - 2)) + 2 < len (Gauge (D,n)) ) )
assume that
A1: m <= n and
A2: 1 < i and
A3: i < len (Gauge (D,m)) ; ::_thesis: ( 1 < ((2 |^ (n -' m)) * (i - 2)) + 2 & ((2 |^ (n -' m)) * (i - 2)) + 2 < len (Gauge (D,n)) )
 1 + 1 <= i by A2, NAT_1:13;
then reconsider i2 = i - 2 as Element of NAT by INT_1:5;
 0 < ((2 |^ (n -' m)) * i2) + 1 ;
then  0 + 1 < (((2 |^ (n -' m)) * (i - 2)) + 1) + 1 by XREAL_1:6;
hence  1 < ((2 |^ (n -' m)) * (i - 2)) + 2 ; ::_thesis: ((2 |^ (n -' m)) * (i - 2)) + 2 < len (Gauge (D,n))
len (Gauge (D,m)) = (2 |^ m) + (2 + 1) by JORDAN8:def_1
.= ((2 |^ m) + 2) + 1 ;
then  i <= (2 |^ m) + 2 by A3, NAT_1:13;
then  i2 <= 2 |^ m by XREAL_1:20;
then  (2 |^ (n -' m)) * i2 <= (2 |^ (n -' m)) * (2 |^ m) by XREAL_1:64;
then  (2 |^ (n -' m)) * i2 <= 2 |^ ((n -' m) + m) by NEWTON:8;
then  (2 |^ (n -' m)) * i2 <= 2 |^ n by A1, XREAL_1:235;
then  (2 |^ (n -' m)) * i2 < (2 |^ n) + 1 by NAT_1:13;
then  ((2 |^ (n -' m)) * (i - 2)) + 2 < ((2 |^ n) + 1) + 2 by XREAL_1:6;
then  ((2 |^ (n -' m)) * (i - 2)) + 2 < (2 |^ n) + (1 + 2) ;
hence  ((2 |^ (n -' m)) * (i - 2)) + 2 < len (Gauge (D,n)) by JORDAN8:def_1; ::_thesis: verum
end;

theorem Th32: :: JORDAN1A:32
for m, n, i being Element of NAT
for D being non empty Subset of (TOP-REAL 2) st m <= n & 1 < i & i < width (Gauge (D,m)) holds
( 1 < ((2 |^ (n -' m)) * (i - 2)) + 2 & ((2 |^ (n -' m)) * (i - 2)) + 2 < width (Gauge (D,n)) )
proof
let m, n, i be Element of NAT ; ::_thesis: for D being non empty Subset of (TOP-REAL 2) st m <= n & 1 < i & i < width (Gauge (D,m)) holds
( 1 < ((2 |^ (n -' m)) * (i - 2)) + 2 & ((2 |^ (n -' m)) * (i - 2)) + 2 < width (Gauge (D,n)) )
let D be non empty Subset of (TOP-REAL 2); ::_thesis: ( m <= n & 1 < i & i < width (Gauge (D,m)) implies ( 1 < ((2 |^ (n -' m)) * (i - 2)) + 2 & ((2 |^ (n -' m)) * (i - 2)) + 2 < width (Gauge (D,n)) ) )
 ( len (Gauge (D,n)) = width (Gauge (D,n)) & len (Gauge (D,m)) = width (Gauge (D,m)) ) by JORDAN8:def_1;
hence  ( m <= n & 1 < i & i < width (Gauge (D,m)) implies ( 1 < ((2 |^ (n -' m)) * (i - 2)) + 2 & ((2 |^ (n -' m)) * (i - 2)) + 2 < width (Gauge (D,n)) ) ) by Th31; ::_thesis: verum
end;

theorem Th33: :: JORDAN1A:33
for m, n, i, j being Element of NAT
for D being non empty Subset of (TOP-REAL 2) st m <= n & 1 < i & i < len (Gauge (D,m)) & 1 < j & j < width (Gauge (D,m)) holds
for i1, j1 being Element of NAT st i1 = ((2 |^ (n -' m)) * (i - 2)) + 2 & j1 = ((2 |^ (n -' m)) * (j - 2)) + 2 holds
(Gauge (D,m)) * (i,j) = (Gauge (D,n)) * (i1,j1)
proof
let m, n, i, j be Element of NAT ; ::_thesis: for D being non empty Subset of (TOP-REAL 2) st m <= n & 1 < i & i < len (Gauge (D,m)) & 1 < j & j < width (Gauge (D,m)) holds
for i1, j1 being Element of NAT st i1 = ((2 |^ (n -' m)) * (i - 2)) + 2 & j1 = ((2 |^ (n -' m)) * (j - 2)) + 2 holds
(Gauge (D,m)) * (i,j) = (Gauge (D,n)) * (i1,j1)
let D be non empty Subset of (TOP-REAL 2); ::_thesis: ( m <= n & 1 < i & i < len (Gauge (D,m)) & 1 < j & j < width (Gauge (D,m)) implies for i1, j1 being Element of NAT st i1 = ((2 |^ (n -' m)) * (i - 2)) + 2 & j1 = ((2 |^ (n -' m)) * (j - 2)) + 2 holds
(Gauge (D,m)) * (i,j) = (Gauge (D,n)) * (i1,j1) )
assume that
A1: m <= n and
A2: ( 1 < i & i < len (Gauge (D,m)) ) and
A3: ( 1 < j & j < width (Gauge (D,m)) ) ; ::_thesis: for i1, j1 being Element of NAT st i1 = ((2 |^ (n -' m)) * (i - 2)) + 2 & j1 = ((2 |^ (n -' m)) * (j - 2)) + 2 holds
(Gauge (D,m)) * (i,j) = (Gauge (D,n)) * (i1,j1)
let i1, j1 be Element of NAT ; ::_thesis: ( i1 = ((2 |^ (n -' m)) * (i - 2)) + 2 & j1 = ((2 |^ (n -' m)) * (j - 2)) + 2 implies (Gauge (D,m)) * (i,j) = (Gauge (D,n)) * (i1,j1) )
assume that
A4: i1 = ((2 |^ (n -' m)) * (i - 2)) + 2 and
A5: j1 = ((2 |^ (n -' m)) * (j - 2)) + 2 ; ::_thesis: (Gauge (D,m)) * (i,j) = (Gauge (D,n)) * (i1,j1)
A6: ( 1 < i1 & i1 < len (Gauge (D,n)) ) by A1, A2, A4, Th31;
(j - 2) / (2 |^ m) = (j - 2) / (2 |^ (n -' (n -' m))) by A1, NAT_D:58
.= (j - 2) / ((2 |^ n) / (2 |^ (n -' m))) by NAT_D:50, TOPREAL6:10
.= (j1 - 2) / (2 |^ n) by A5, XCMPLX_1:77 ;
then A7: (((N-bound D) - (S-bound D)) / (2 |^ m)) * (j - 2) = ((N-bound D) - (S-bound D)) * ((j1 - 2) / (2 |^ n)) by XCMPLX_1:75
.= (((N-bound D) - (S-bound D)) / (2 |^ n)) * (j1 - 2) by XCMPLX_1:75 ;
(i - 2) / (2 |^ m) = (i - 2) / (2 |^ (n -' (n -' m))) by A1, NAT_D:58
.= (i - 2) / ((2 |^ n) / (2 |^ (n -' m))) by NAT_D:50, TOPREAL6:10
.= (i1 - 2) / (2 |^ n) by A4, XCMPLX_1:77 ;
then A8: (((E-bound D) - (W-bound D)) / (2 |^ m)) * (i - 2) = ((E-bound D) - (W-bound D)) * ((i1 - 2) / (2 |^ n)) by XCMPLX_1:75
.= (((E-bound D) - (W-bound D)) / (2 |^ n)) * (i1 - 2) by XCMPLX_1:75 ;
 ( 1 < j1 & j1 < width (Gauge (D,n)) ) by A1, A3, A5, Th32;
then A9: [i1,j1] in Indices (Gauge (D,n)) by A6, MATRIX_1:36;
 [i,j] in Indices (Gauge (D,m)) by A2, A3, MATRIX_1:36;
hence (Gauge (D,m)) * (i,j) = |[((W-bound D) + ((((E-bound D) - (W-bound D)) / (2 |^ m)) * (i - 2))),((S-bound D) + ((((N-bound D) - (S-bound D)) / (2 |^ m)) * (j - 2)))]| by JORDAN8:def_1
.= (Gauge (D,n)) * (i1,j1) by A9, A8, A7, JORDAN8:def_1 ;
::_thesis: verum
end;

theorem Th34: :: JORDAN1A:34
for m, n, i being Element of NAT
for D being non empty Subset of (TOP-REAL 2) st m <= n & 1 < i & i + 1 < len (Gauge (D,m)) holds
( 1 < ((2 |^ (n -' m)) * (i - 1)) + 2 & ((2 |^ (n -' m)) * (i - 1)) + 2 <= len (Gauge (D,n)) )
proof
let m, n, i be Element of NAT ; ::_thesis: for D being non empty Subset of (TOP-REAL 2) st m <= n & 1 < i & i + 1 < len (Gauge (D,m)) holds
( 1 < ((2 |^ (n -' m)) * (i - 1)) + 2 & ((2 |^ (n -' m)) * (i - 1)) + 2 <= len (Gauge (D,n)) )
let D be non empty Subset of (TOP-REAL 2); ::_thesis: ( m <= n & 1 < i & i + 1 < len (Gauge (D,m)) implies ( 1 < ((2 |^ (n -' m)) * (i - 1)) + 2 & ((2 |^ (n -' m)) * (i - 1)) + 2 <= len (Gauge (D,n)) ) )
assume that
A1: m <= n and
A2: 1 < i and
A3: i + 1 < len (Gauge (D,m)) ; ::_thesis: ( 1 < ((2 |^ (n -' m)) * (i - 1)) + 2 & ((2 |^ (n -' m)) * (i - 1)) + 2 <= len (Gauge (D,n)) )
reconsider i2 = i - 1 as Element of NAT by A2, INT_1:5;
 0 < ((2 |^ (n -' m)) * i2) + 1 ;
then  0 + 1 < (((2 |^ (n -' m)) * (i - 1)) + 1) + 1 by XREAL_1:6;
hence  1 < ((2 |^ (n -' m)) * (i - 1)) + 2 ; ::_thesis: ((2 |^ (n -' m)) * (i - 1)) + 2 <= len (Gauge (D,n))
len (Gauge (D,m)) = (2 |^ m) + (2 + 1) by JORDAN8:def_1
.= ((2 |^ m) + 2) + 1 ;
then  i + 1 <= ((2 |^ m) + 1) + 1 by A3, NAT_1:13;
then  i <= (2 |^ m) + 1 by XREAL_1:6;
then  i2 <= 2 |^ m by XREAL_1:20;
then  (2 |^ (n -' m)) * i2 <= (2 |^ (n -' m)) * (2 |^ m) by XREAL_1:64;
then  (2 |^ (n -' m)) * i2 <= 2 |^ ((n -' m) + m) by NEWTON:8;
then  (2 |^ (n -' m)) * i2 <= 2 |^ n by A1, XREAL_1:235;
then  (2 |^ (n -' m)) * i2 <= (2 |^ n) + 1 by NAT_1:13;
then  ((2 |^ (n -' m)) * (i - 1)) + 2 <= ((2 |^ n) + 1) + 2 by XREAL_1:6;
then  ((2 |^ (n -' m)) * (i - 1)) + 2 <= (2 |^ n) + (1 + 2) ;
hence  ((2 |^ (n -' m)) * (i - 1)) + 2 <= len (Gauge (D,n)) by JORDAN8:def_1; ::_thesis: verum
end;

theorem Th35: :: JORDAN1A:35
for m, n, i being Element of NAT
for D being non empty Subset of (TOP-REAL 2) st m <= n & 1 < i & i + 1 < width (Gauge (D,m)) holds
( 1 < ((2 |^ (n -' m)) * (i - 1)) + 2 & ((2 |^ (n -' m)) * (i - 1)) + 2 <= width (Gauge (D,n)) )
proof
let m, n, i be Element of NAT ; ::_thesis: for D being non empty Subset of (TOP-REAL 2) st m <= n & 1 < i & i + 1 < width (Gauge (D,m)) holds
( 1 < ((2 |^ (n -' m)) * (i - 1)) + 2 & ((2 |^ (n -' m)) * (i - 1)) + 2 <= width (Gauge (D,n)) )
let D be non empty Subset of (TOP-REAL 2); ::_thesis: ( m <= n & 1 < i & i + 1 < width (Gauge (D,m)) implies ( 1 < ((2 |^ (n -' m)) * (i - 1)) + 2 & ((2 |^ (n -' m)) * (i - 1)) + 2 <= width (Gauge (D,n)) ) )
 ( len (Gauge (D,n)) = width (Gauge (D,n)) & len (Gauge (D,m)) = width (Gauge (D,m)) ) by JORDAN8:def_1;
hence  ( m <= n & 1 < i & i + 1 < width (Gauge (D,m)) implies ( 1 < ((2 |^ (n -' m)) * (i - 1)) + 2 & ((2 |^ (n -' m)) * (i - 1)) + 2 <= width (Gauge (D,n)) ) ) by Th34; ::_thesis: verum
end;

Lm3: now__::_thesis:_for_D_being_non_empty_Subset_of_(TOP-REAL_2)
for_n_being_Element_of_NAT_holds_
(_1_<=_Center_(Gauge_(D,n))_&_Center_(Gauge_(D,n))_<=_len_(Gauge_(D,n))_)
let D be non empty Subset of (TOP-REAL 2); ::_thesis: for n being Element of NAT holds
( 1 <= Center (Gauge (D,n)) & Center (Gauge (D,n)) <= len (Gauge (D,n)) )
let n be Element of NAT ; ::_thesis: ( 1 <= Center (Gauge (D,n)) & Center (Gauge (D,n)) <= len (Gauge (D,n)) )
set G = Gauge (D,n);
 0 + 1 <= ((len (Gauge (D,n))) div 2) + 1 by XREAL_1:6;
hence  1 <= Center (Gauge (D,n)) ; ::_thesis: Center (Gauge (D,n)) <= len (Gauge (D,n))
 0 < len (Gauge (D,n)) by JORDAN8:10;
then  (len (Gauge (D,n))) div 2 < len (Gauge (D,n)) by INT_1:56;
hence  Center (Gauge (D,n)) <= len (Gauge (D,n)) by NAT_1:13; ::_thesis: verum
end;

Lm4: now__::_thesis:_for_D_being_non_empty_Subset_of_(TOP-REAL_2)
for_n,_j_being_Element_of_NAT_st_1_<=_j_&_j_<=_len_(Gauge_(D,n))_holds_
[(Center_(Gauge_(D,n))),j]_in_Indices_(Gauge_(D,n))
let D be non empty Subset of (TOP-REAL 2); ::_thesis: for n, j being Element of NAT st 1 <= j & j <= len (Gauge (D,n)) holds
[(Center (Gauge (D,n))),j] in Indices (Gauge (D,n))
let n, j be Element of NAT ; ::_thesis: ( 1 <= j & j <= len (Gauge (D,n)) implies [(Center (Gauge (D,n))),j] in Indices (Gauge (D,n)) )
set G = Gauge (D,n);
assume A1: ( 1 <= j & j <= len (Gauge (D,n)) ) ; ::_thesis: [(Center (Gauge (D,n))),j] in Indices (Gauge (D,n))
A2: ( len (Gauge (D,n)) = width (Gauge (D,n)) & 0 + 1 <= ((len (Gauge (D,n))) div 2) + 1 ) by JORDAN8:def_1, XREAL_1:6;
 Center (Gauge (D,n)) <= len (Gauge (D,n)) by Lm3;
hence  [(Center (Gauge (D,n))),j] in Indices (Gauge (D,n)) by A1, A2, MATRIX_1:36; ::_thesis: verum
end;

Lm5: now__::_thesis:_for_D_being_non_empty_Subset_of_(TOP-REAL_2)
for_n,_j_being_Element_of_NAT_st_1_<=_j_&_j_<=_len_(Gauge_(D,n))_holds_
[j,(Center_(Gauge_(D,n)))]_in_Indices_(Gauge_(D,n))
let D be non empty Subset of (TOP-REAL 2); ::_thesis: for n, j being Element of NAT st 1 <= j & j <= len (Gauge (D,n)) holds
[j,(Center (Gauge (D,n)))] in Indices (Gauge (D,n))
let n, j be Element of NAT ; ::_thesis: ( 1 <= j & j <= len (Gauge (D,n)) implies [j,(Center (Gauge (D,n)))] in Indices (Gauge (D,n)) )
set G = Gauge (D,n);
assume A1: ( 1 <= j & j <= len (Gauge (D,n)) ) ; ::_thesis: [j,(Center (Gauge (D,n)))] in Indices (Gauge (D,n))
A2: ( len (Gauge (D,n)) = width (Gauge (D,n)) & 0 + 1 <= ((len (Gauge (D,n))) div 2) + 1 ) by JORDAN8:def_1, XREAL_1:6;
 Center (Gauge (D,n)) <= len (Gauge (D,n)) by Lm3;
hence  [j,(Center (Gauge (D,n)))] in Indices (Gauge (D,n)) by A1, A2, MATRIX_1:36; ::_thesis: verum
end;

Lm6:  for n being Element of NAT
for D being non empty Subset of (TOP-REAL 2)
for w being real number st n > 0 holds
(w / (2 |^ n)) * ((Center (Gauge (D,n))) - 2) = w / 2
proof
let n be Element of NAT ; ::_thesis: for D being non empty Subset of (TOP-REAL 2)
for w being real number st n > 0 holds
(w / (2 |^ n)) * ((Center (Gauge (D,n))) - 2) = w / 2
let D be non empty Subset of (TOP-REAL 2); ::_thesis: for w being real number st n > 0 holds
(w / (2 |^ n)) * ((Center (Gauge (D,n))) - 2) = w / 2
let w be real number ; ::_thesis: ( n > 0 implies (w / (2 |^ n)) * ((Center (Gauge (D,n))) - 2) = w / 2 )
set G = Gauge (D,n);
A1: 2 |^ n <> 0 by NEWTON:83;
assume A2: n > 0 ; ::_thesis: (w / (2 |^ n)) * ((Center (Gauge (D,n))) - 2) = w / 2
then A3: (2 |^ n) mod 2 = 0 by PEPIN:36;
thus (w / (2 |^ n)) * ((Center (Gauge (D,n))) - 2) = (w / (2 |^ n)) * (((((2 |^ n) + 3) div 2) + 1) - 2) by JORDAN8:def_1
.= (w / (2 |^ n)) * ((((2 |^ n) + 3) div 2) + (1 - 2))
.= (w / (2 |^ n)) * ((((2 |^ n) div 2) + 1) + (- 1)) by A3, Lm1, NAT_D:19
.= (w / (2 |^ n)) * ((2 |^ n) / 2) by A2, PEPIN:64
.= w / 2 by A1, XCMPLX_1:98 ; ::_thesis: verum
end;

Lm7: now__::_thesis:_for_m,_n_being_Element_of_NAT_
for_c,_d_being_real_number_st_0_<=_c_&_m_<=_n_holds_
c_/_(2_|^_n)_<=_c_/_(2_|^_m)
let m, n be Element of NAT ; ::_thesis: for c, d being real number st 0 <= c & m <= n holds
c / (2 |^ n) <= c / (2 |^ m)
let c, d be real number ; ::_thesis: ( 0 <= c & m <= n implies c / (2 |^ n) <= c / (2 |^ m) )
assume A1: 0 <= c ; ::_thesis: ( m <= n implies c / (2 |^ n) <= c / (2 |^ m) )
assume  m <= n ; ::_thesis: c / (2 |^ n) <= c / (2 |^ m)
then  ( 0 < 2 |^ m & 2 |^ m <= 2 |^ n ) by NEWTON:83, PREPOWER:93;
hence  c / (2 |^ n) <= c / (2 |^ m) by A1, XREAL_1:118; ::_thesis: verum
end;

Lm8: now__::_thesis:_for_m,_n_being_Element_of_NAT_
for_c,_d_being_real_number_st_0_<=_c_&_m_<=_n_holds_
d_+_(c_/_(2_|^_n))_<=_d_+_(c_/_(2_|^_m))
let m, n be Element of NAT ; ::_thesis: for c, d being real number st 0 <= c & m <= n holds
d + (c / (2 |^ n)) <= d + (c / (2 |^ m))
let c, d be real number ; ::_thesis: ( 0 <= c & m <= n implies d + (c / (2 |^ n)) <= d + (c / (2 |^ m)) )
assume  ( 0 <= c & m <= n ) ; ::_thesis: d + (c / (2 |^ n)) <= d + (c / (2 |^ m))
then  c / (2 |^ n) <= c / (2 |^ m) by Lm7;
hence  d + (c / (2 |^ n)) <= d + (c / (2 |^ m)) by XREAL_1:6; ::_thesis: verum
end;

Lm9: now__::_thesis:_for_m,_n_being_Element_of_NAT_
for_c,_d_being_real_number_st_0_<=_c_&_m_<=_n_holds_
d_-_(c_/_(2_|^_m))_<=_d_-_(c_/_(2_|^_n))
let m, n be Element of NAT ; ::_thesis: for c, d being real number st 0 <= c & m <= n holds
d - (c / (2 |^ m)) <= d - (c / (2 |^ n))
let c, d be real number ; ::_thesis: ( 0 <= c & m <= n implies d - (c / (2 |^ m)) <= d - (c / (2 |^ n)) )
assume  ( 0 <= c & m <= n ) ; ::_thesis: d - (c / (2 |^ m)) <= d - (c / (2 |^ n))
then  c / (2 |^ n) <= c / (2 |^ m) by Lm7;
hence  d - (c / (2 |^ m)) <= d - (c / (2 |^ n)) by XREAL_1:13; ::_thesis: verum
end;

theorem Th36: :: JORDAN1A:36
for i, n, j, m being Element of NAT
for D being non empty Subset of (TOP-REAL 2) st 1 <= i & i <= len (Gauge (D,n)) & 1 <= j & j <= len (Gauge (D,m)) & ( ( n > 0 & m > 0 ) or ( n = 0 & m = 0 ) ) holds
((Gauge (D,n)) * ((Center (Gauge (D,n))),i)) `1 = ((Gauge (D,m)) * ((Center (Gauge (D,m))),j)) `1
proof
let i, n, j, m be Element of NAT ; ::_thesis: for D being non empty Subset of (TOP-REAL 2) st 1 <= i & i <= len (Gauge (D,n)) & 1 <= j & j <= len (Gauge (D,m)) & ( ( n > 0 & m > 0 ) or ( n = 0 & m = 0 ) ) holds
((Gauge (D,n)) * ((Center (Gauge (D,n))),i)) `1 = ((Gauge (D,m)) * ((Center (Gauge (D,m))),j)) `1
let D be non empty Subset of (TOP-REAL 2); ::_thesis: ( 1 <= i & i <= len (Gauge (D,n)) & 1 <= j & j <= len (Gauge (D,m)) & ( ( n > 0 & m > 0 ) or ( n = 0 & m = 0 ) ) implies ((Gauge (D,n)) * ((Center (Gauge (D,n))),i)) `1 = ((Gauge (D,m)) * ((Center (Gauge (D,m))),j)) `1 )
set a = N-bound D;
set s = S-bound D;
set w = W-bound D;
set e = E-bound D;
set G = Gauge (D,n);
set M = Gauge (D,m);
assume  ( 1 <= i & i <= len (Gauge (D,n)) ) ; ::_thesis: ( not 1 <= j or not j <= len (Gauge (D,m)) or ( not ( n > 0 & m > 0 ) & not ( n = 0 & m = 0 ) ) or ((Gauge (D,n)) * ((Center (Gauge (D,n))),i)) `1 = ((Gauge (D,m)) * ((Center (Gauge (D,m))),j)) `1 )
then A1: [(Center (Gauge (D,n))),i] in Indices (Gauge (D,n)) by Lm4;
assume  ( 1 <= j & j <= len (Gauge (D,m)) ) ; ::_thesis: ( ( not ( n > 0 & m > 0 ) & not ( n = 0 & m = 0 ) ) or ((Gauge (D,n)) * ((Center (Gauge (D,n))),i)) `1 = ((Gauge (D,m)) * ((Center (Gauge (D,m))),j)) `1 )
then A2: [(Center (Gauge (D,m))),j] in Indices (Gauge (D,m)) by Lm4;
assume A3: ( ( n > 0 & m > 0 ) or ( n = 0 & m = 0 ) ) ; ::_thesis: ((Gauge (D,n)) * ((Center (Gauge (D,n))),i)) `1 = ((Gauge (D,m)) * ((Center (Gauge (D,m))),j)) `1
percases ( ( n > 0 & m > 0 ) or ( n = 0 & m = 0 ) ) by A3;
supposethat A4: n > 0 and
A5: m > 0 ; ::_thesis: ((Gauge (D,n)) * ((Center (Gauge (D,n))),i)) `1 = ((Gauge (D,m)) * ((Center (Gauge (D,m))),j)) `1
thus ((Gauge (D,n)) * ((Center (Gauge (D,n))),i)) `1 = |[((W-bound D) + ((((E-bound D) - (W-bound D)) / (2 |^ n)) * ((Center (Gauge (D,n))) - 2))),((S-bound D) + ((((N-bound D) - (S-bound D)) / (2 |^ n)) * (i - 2)))]| `1 by A1, JORDAN8:def_1
.= (W-bound D) + ((((E-bound D) - (W-bound D)) / (2 |^ n)) * ((Center (Gauge (D,n))) - 2)) by EUCLID:52
.= (W-bound D) + (((E-bound D) - (W-bound D)) / 2) by A4, Lm6
.= (W-bound D) + ((((E-bound D) - (W-bound D)) / (2 |^ m)) * ((Center (Gauge (D,m))) - 2)) by A5, Lm6
.= |[((W-bound D) + ((((E-bound D) - (W-bound D)) / (2 |^ m)) * ((Center (Gauge (D,m))) - 2))),((S-bound D) + ((((N-bound D) - (S-bound D)) / (2 |^ m)) * (j - 2)))]| `1 by EUCLID:52
.= ((Gauge (D,m)) * ((Center (Gauge (D,m))),j)) `1 by A2, JORDAN8:def_1 ; ::_thesis: verum
end;
supposeA6: ( n = 0 & m = 0 ) ; ::_thesis: ((Gauge (D,n)) * ((Center (Gauge (D,n))),i)) `1 = ((Gauge (D,m)) * ((Center (Gauge (D,m))),j)) `1
thus ((Gauge (D,n)) * ((Center (Gauge (D,n))),i)) `1 = |[((W-bound D) + ((((E-bound D) - (W-bound D)) / (2 |^ n)) * ((Center (Gauge (D,n))) - 2))),((S-bound D) + ((((N-bound D) - (S-bound D)) / (2 |^ n)) * (i - 2)))]| `1 by A1, JORDAN8:def_1
.= (W-bound D) + ((((E-bound D) - (W-bound D)) / (2 |^ m)) * ((Center (Gauge (D,m))) - 2)) by A6, EUCLID:52
.= |[((W-bound D) + ((((E-bound D) - (W-bound D)) / (2 |^ m)) * ((Center (Gauge (D,m))) - 2))),((S-bound D) + ((((N-bound D) - (S-bound D)) / (2 |^ m)) * (j - 2)))]| `1 by EUCLID:52
.= ((Gauge (D,m)) * ((Center (Gauge (D,m))),j)) `1 by A2, JORDAN8:def_1 ; ::_thesis: verum
end;
end;
end;

theorem :: JORDAN1A:37
for i, n, j, m being Element of NAT
for D being non empty Subset of (TOP-REAL 2) st 1 <= i & i <= len (Gauge (D,n)) & 1 <= j & j <= len (Gauge (D,m)) & ( ( n > 0 & m > 0 ) or ( n = 0 & m = 0 ) ) holds
((Gauge (D,n)) * (i,(Center (Gauge (D,n))))) `2 = ((Gauge (D,m)) * (j,(Center (Gauge (D,m))))) `2
proof
let i, n, j, m be Element of NAT ; ::_thesis: for D being non empty Subset of (TOP-REAL 2) st 1 <= i & i <= len (Gauge (D,n)) & 1 <= j & j <= len (Gauge (D,m)) & ( ( n > 0 & m > 0 ) or ( n = 0 & m = 0 ) ) holds
((Gauge (D,n)) * (i,(Center (Gauge (D,n))))) `2 = ((Gauge (D,m)) * (j,(Center (Gauge (D,m))))) `2
let D be non empty Subset of (TOP-REAL 2); ::_thesis: ( 1 <= i & i <= len (Gauge (D,n)) & 1 <= j & j <= len (Gauge (D,m)) & ( ( n > 0 & m > 0 ) or ( n = 0 & m = 0 ) ) implies ((Gauge (D,n)) * (i,(Center (Gauge (D,n))))) `2 = ((Gauge (D,m)) * (j,(Center (Gauge (D,m))))) `2 )
set a = N-bound D;
set s = S-bound D;
set w = W-bound D;
set e = E-bound D;
set G = Gauge (D,n);
set M = Gauge (D,m);
assume  ( 1 <= i & i <= len (Gauge (D,n)) ) ; ::_thesis: ( not 1 <= j or not j <= len (Gauge (D,m)) or ( not ( n > 0 & m > 0 ) & not ( n = 0 & m = 0 ) ) or ((Gauge (D,n)) * (i,(Center (Gauge (D,n))))) `2 = ((Gauge (D,m)) * (j,(Center (Gauge (D,m))))) `2 )
then A1: [i,(Center (Gauge (D,n)))] in Indices (Gauge (D,n)) by Lm5;
assume  ( 1 <= j & j <= len (Gauge (D,m)) ) ; ::_thesis: ( ( not ( n > 0 & m > 0 ) & not ( n = 0 & m = 0 ) ) or ((Gauge (D,n)) * (i,(Center (Gauge (D,n))))) `2 = ((Gauge (D,m)) * (j,(Center (Gauge (D,m))))) `2 )
then A2: [j,(Center (Gauge (D,m)))] in Indices (Gauge (D,m)) by Lm5;
assume A3: ( ( n > 0 & m > 0 ) or ( n = 0 & m = 0 ) ) ; ::_thesis: ((Gauge (D,n)) * (i,(Center (Gauge (D,n))))) `2 = ((Gauge (D,m)) * (j,(Center (Gauge (D,m))))) `2
percases ( ( n > 0 & m > 0 ) or ( n = 0 & m = 0 ) ) by A3;
supposethat A4: n > 0 and
A5: m > 0 ; ::_thesis: ((Gauge (D,n)) * (i,(Center (Gauge (D,n))))) `2 = ((Gauge (D,m)) * (j,(Center (Gauge (D,m))))) `2
thus ((Gauge (D,n)) * (i,(Center (Gauge (D,n))))) `2 = |[((W-bound D) + ((((E-bound D) - (W-bound D)) / (2 |^ n)) * (i - 2))),((S-bound D) + ((((N-bound D) - (S-bound D)) / (2 |^ n)) * ((Center (Gauge (D,n))) - 2)))]| `2 by A1, JORDAN8:def_1
.= (S-bound D) + ((((N-bound D) - (S-bound D)) / (2 |^ n)) * ((Center (Gauge (D,n))) - 2)) by EUCLID:52
.= (S-bound D) + (((N-bound D) - (S-bound D)) / 2) by A4, Lm6
.= (S-bound D) + ((((N-bound D) - (S-bound D)) / (2 |^ m)) * ((Center (Gauge (D,m))) - 2)) by A5, Lm6
.= |[((W-bound D) + ((((E-bound D) - (W-bound D)) / (2 |^ m)) * (j - 2))),((S-bound D) + ((((N-bound D) - (S-bound D)) / (2 |^ m)) * ((Center (Gauge (D,m))) - 2)))]| `2 by EUCLID:52
.= ((Gauge (D,m)) * (j,(Center (Gauge (D,m))))) `2 by A2, JORDAN8:def_1 ; ::_thesis: verum
end;
supposeA6: ( n = 0 & m = 0 ) ; ::_thesis: ((Gauge (D,n)) * (i,(Center (Gauge (D,n))))) `2 = ((Gauge (D,m)) * (j,(Center (Gauge (D,m))))) `2
thus ((Gauge (D,n)) * (i,(Center (Gauge (D,n))))) `2 = |[((W-bound D) + ((((E-bound D) - (W-bound D)) / (2 |^ n)) * (i - 2))),((S-bound D) + ((((N-bound D) - (S-bound D)) / (2 |^ n)) * ((Center (Gauge (D,n))) - 2)))]| `2 by A1, JORDAN8:def_1
.= (S-bound D) + ((((N-bound D) - (S-bound D)) / (2 |^ m)) * ((Center (Gauge (D,m))) - 2)) by A6, EUCLID:52
.= |[((W-bound D) + ((((E-bound D) - (W-bound D)) / (2 |^ m)) * (j - 2))),((S-bound D) + ((((N-bound D) - (S-bound D)) / (2 |^ m)) * ((Center (Gauge (D,m))) - 2)))]| `2 by EUCLID:52
.= ((Gauge (D,m)) * (j,(Center (Gauge (D,m))))) `2 by A2, JORDAN8:def_1 ; ::_thesis: verum
end;
end;
end;

Lm10: now__::_thesis:_for_D_being_non_empty_Subset_of_(TOP-REAL_2)
for_n_being_Element_of_NAT_
for_e,_w_being_real_number_holds_w_+_(((e_-_w)_/_(2_|^_n))_*_((len_(Gauge_(D,n)))_-_2))_=_e_+_((e_-_w)_/_(2_|^_n))
let D be non empty Subset of (TOP-REAL 2); ::_thesis: for n being Element of NAT
for e, w being real number holds w + (((e - w) / (2 |^ n)) * ((len (Gauge (D,n))) - 2)) = e + ((e - w) / (2 |^ n))
let n be Element of NAT ; ::_thesis: for e, w being real number holds w + (((e - w) / (2 |^ n)) * ((len (Gauge (D,n))) - 2)) = e + ((e - w) / (2 |^ n))
let e, w be real number ; ::_thesis: w + (((e - w) / (2 |^ n)) * ((len (Gauge (D,n))) - 2)) = e + ((e - w) / (2 |^ n))
A1: 2 |^ n <> 0 by NEWTON:83;
thus w + (((e - w) / (2 |^ n)) * ((len (Gauge (D,n))) - 2)) = w + (((e - w) / (2 |^ n)) * (((2 |^ n) + 3) - 2)) by JORDAN8:def_1
.= (w + (((e - w) / (2 |^ n)) * (2 |^ n))) + ((e - w) / (2 |^ n))
.= (w + (e - w)) + ((e - w) / (2 |^ n)) by A1, XCMPLX_1:87
.= e + ((e - w) / (2 |^ n)) ; ::_thesis: verum
end;

Lm11: now__::_thesis:_for_D_being_non_empty_Subset_of_(TOP-REAL_2)
for_n,_i_being_Element_of_NAT_st_[i,1]_in_Indices_(Gauge_(D,n))_holds_
((Gauge_(D,n))_*_(i,1))_`2_=_(S-bound_D)_-_(((N-bound_D)_-_(S-bound_D))_/_(2_|^_n))
let D be non empty Subset of (TOP-REAL 2); ::_thesis: for n, i being Element of NAT st [i,1] in Indices (Gauge (D,n)) holds
((Gauge (D,n)) * (i,1)) `2 = (S-bound D) - (((N-bound D) - (S-bound D)) / (2 |^ n))
let n, i be Element of NAT ; ::_thesis: ( [i,1] in Indices (Gauge (D,n)) implies ((Gauge (D,n)) * (i,1)) `2 = (S-bound D) - (((N-bound D) - (S-bound D)) / (2 |^ n)) )
set a = N-bound D;
set s = S-bound D;
set w = W-bound D;
set e = E-bound D;
set G = Gauge (D,n);
assume  [i,1] in Indices (Gauge (D,n)) ; ::_thesis: ((Gauge (D,n)) * (i,1)) `2 = (S-bound D) - (((N-bound D) - (S-bound D)) / (2 |^ n))
hence ((Gauge (D,n)) * (i,1)) `2 = |[((W-bound D) + ((((E-bound D) - (W-bound D)) / (2 |^ n)) * (i - 2))),((S-bound D) + ((((N-bound D) - (S-bound D)) / (2 |^ n)) * (1 - 2)))]| `2 by JORDAN8:def_1
.= (S-bound D) - (((N-bound D) - (S-bound D)) / (2 |^ n)) by EUCLID:52 ;
::_thesis: verum
end;

Lm12: now__::_thesis:_for_D_being_non_empty_Subset_of_(TOP-REAL_2)
for_n,_i_being_Element_of_NAT_st_[1,i]_in_Indices_(Gauge_(D,n))_holds_
((Gauge_(D,n))_*_(1,i))_`1_=_(W-bound_D)_-_(((E-bound_D)_-_(W-bound_D))_/_(2_|^_n))
let D be non empty Subset of (TOP-REAL 2); ::_thesis: for n, i being Element of NAT st [1,i] in Indices (Gauge (D,n)) holds
((Gauge (D,n)) * (1,i)) `1 = (W-bound D) - (((E-bound D) - (W-bound D)) / (2 |^ n))
let n, i be Element of NAT ; ::_thesis: ( [1,i] in Indices (Gauge (D,n)) implies ((Gauge (D,n)) * (1,i)) `1 = (W-bound D) - (((E-bound D) - (W-bound D)) / (2 |^ n)) )
set a = N-bound D;
set s = S-bound D;
set w = W-bound D;
set e = E-bound D;
set G = Gauge (D,n);
assume  [1,i] in Indices (Gauge (D,n)) ; ::_thesis: ((Gauge (D,n)) * (1,i)) `1 = (W-bound D) - (((E-bound D) - (W-bound D)) / (2 |^ n))
hence ((Gauge (D,n)) * (1,i)) `1 = |[((W-bound D) + ((((E-bound D) - (W-bound D)) / (2 |^ n)) * (1 - 2))),((S-bound D) + ((((N-bound D) - (S-bound D)) / (2 |^ n)) * (i - 2)))]| `1 by JORDAN8:def_1
.= (W-bound D) - (((E-bound D) - (W-bound D)) / (2 |^ n)) by EUCLID:52 ;
::_thesis: verum
end;

Lm13: now__::_thesis:_for_D_being_non_empty_Subset_of_(TOP-REAL_2)
for_n,_i_being_Element_of_NAT_st_[i,(len_(Gauge_(D,n)))]_in_Indices_(Gauge_(D,n))_holds_
((Gauge_(D,n))_*_(i,(len_(Gauge_(D,n)))))_`2_=_(N-bound_D)_+_(((N-bound_D)_-_(S-bound_D))_/_(2_|^_n))
let D be non empty Subset of (TOP-REAL 2); ::_thesis: for n, i being Element of NAT st [i,(len (Gauge (D,n)))] in Indices (Gauge (D,n)) holds
((Gauge (D,n)) * (i,(len (Gauge (D,n))))) `2 = (N-bound D) + (((N-bound D) - (S-bound D)) / (2 |^ n))
let n, i be Element of NAT ; ::_thesis: ( [i,(len (Gauge (D,n)))] in Indices (Gauge (D,n)) implies ((Gauge (D,n)) * (i,(len (Gauge (D,n))))) `2 = (N-bound D) + (((N-bound D) - (S-bound D)) / (2 |^ n)) )
set a = N-bound D;
set s = S-bound D;
set w = W-bound D;
set e = E-bound D;
set G = Gauge (D,n);
assume  [i,(len (Gauge (D,n)))] in Indices (Gauge (D,n)) ; ::_thesis: ((Gauge (D,n)) * (i,(len (Gauge (D,n))))) `2 = (N-bound D) + (((N-bound D) - (S-bound D)) / (2 |^ n))
hence ((Gauge (D,n)) * (i,(len (Gauge (D,n))))) `2 = |[((W-bound D) + ((((E-bound D) - (W-bound D)) / (2 |^ n)) * (i - 2))),((S-bound D) + ((((N-bound D) - (S-bound D)) / (2 |^ n)) * ((len (Gauge (D,n))) - 2)))]| `2 by JORDAN8:def_1
.= (S-bound D) + ((((N-bound D) - (S-bound D)) / (2 |^ n)) * ((len (Gauge (D,n))) - 2)) by EUCLID:52
.= (N-bound D) + (((N-bound D) - (S-bound D)) / (2 |^ n)) by Lm10 ;
::_thesis: verum
end;

Lm14: now__::_thesis:_for_D_being_non_empty_Subset_of_(TOP-REAL_2)
for_n,_i_being_Element_of_NAT_st_[(len_(Gauge_(D,n))),i]_in_Indices_(Gauge_(D,n))_holds_
((Gauge_(D,n))_*_((len_(Gauge_(D,n))),i))_`1_=_(E-bound_D)_+_(((E-bound_D)_-_(W-bound_D))_/_(2_|^_n))
let D be non empty Subset of (TOP-REAL 2); ::_thesis: for n, i being Element of NAT st [(len (Gauge (D,n))),i] in Indices (Gauge (D,n)) holds
((Gauge (D,n)) * ((len (Gauge (D,n))),i)) `1 = (E-bound D) + (((E-bound D) - (W-bound D)) / (2 |^ n))
let n, i be Element of NAT ; ::_thesis: ( [(len (Gauge (D,n))),i] in Indices (Gauge (D,n)) implies ((Gauge (D,n)) * ((len (Gauge (D,n))),i)) `1 = (E-bound D) + (((E-bound D) - (W-bound D)) / (2 |^ n)) )
set a = N-bound D;
set s = S-bound D;
set w = W-bound D;
set e = E-bound D;
set G = Gauge (D,n);
assume  [(len (Gauge (D,n))),i] in Indices (Gauge (D,n)) ; ::_thesis: ((Gauge (D,n)) * ((len (Gauge (D,n))),i)) `1 = (E-bound D) + (((E-bound D) - (W-bound D)) / (2 |^ n))
hence ((Gauge (D,n)) * ((len (Gauge (D,n))),i)) `1 = |[((W-bound D) + ((((E-bound D) - (W-bound D)) / (2 |^ n)) * ((len (Gauge (D,n))) - 2))),((S-bound D) + ((((N-bound D) - (S-bound D)) / (2 |^ n)) * (i - 2)))]| `1 by JORDAN8:def_1
.= (W-bound D) + ((((E-bound D) - (W-bound D)) / (2 |^ n)) * ((len (Gauge (D,n))) - 2)) by EUCLID:52
.= (E-bound D) + (((E-bound D) - (W-bound D)) / (2 |^ n)) by Lm10 ;
::_thesis: verum
end;

theorem :: JORDAN1A:38
for i being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= i & i <= len (Gauge (C,1)) holds
((Gauge (C,1)) * ((Center (Gauge (C,1))),i)) `1 = ((W-bound C) + (E-bound C)) / 2
proof
let i be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= i & i <= len (Gauge (C,1)) holds
((Gauge (C,1)) * ((Center (Gauge (C,1))),i)) `1 = ((W-bound C) + (E-bound C)) / 2
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( 1 <= i & i <= len (Gauge (C,1)) implies ((Gauge (C,1)) * ((Center (Gauge (C,1))),i)) `1 = ((W-bound C) + (E-bound C)) / 2 )
set a = N-bound C;
set s = S-bound C;
set w = W-bound C;
set e = E-bound C;
set G = Gauge (C,1);
assume  ( 1 <= i & i <= len (Gauge (C,1)) ) ; ::_thesis: ((Gauge (C,1)) * ((Center (Gauge (C,1))),i)) `1 = ((W-bound C) + (E-bound C)) / 2
then  [(Center (Gauge (C,1))),i] in Indices (Gauge (C,1)) by Lm4;
hence ((Gauge (C,1)) * ((Center (Gauge (C,1))),i)) `1 = |[((W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ 1)) * ((Center (Gauge (C,1))) - 2))),((S-bound C) + ((((N-bound C) - (S-bound C)) / (2 |^ 1)) * (i - 2)))]| `1 by JORDAN8:def_1
.= (W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ 1)) * ((Center (Gauge (C,1))) - 2)) by EUCLID:52
.= (W-bound C) + (((E-bound C) - (W-bound C)) / 2) by Lm6
.= ((W-bound C) + (E-bound C)) / 2 ;
::_thesis: verum
end;

theorem :: JORDAN1A:39
for i being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= i & i <= len (Gauge (C,1)) holds
((Gauge (C,1)) * (i,(Center (Gauge (C,1))))) `2 = ((S-bound C) + (N-bound C)) / 2
proof
let i be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= i & i <= len (Gauge (C,1)) holds
((Gauge (C,1)) * (i,(Center (Gauge (C,1))))) `2 = ((S-bound C) + (N-bound C)) / 2
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( 1 <= i & i <= len (Gauge (C,1)) implies ((Gauge (C,1)) * (i,(Center (Gauge (C,1))))) `2 = ((S-bound C) + (N-bound C)) / 2 )
set a = N-bound C;
set s = S-bound C;
set w = W-bound C;
set e = E-bound C;
set G = Gauge (C,1);
assume  ( 1 <= i & i <= len (Gauge (C,1)) ) ; ::_thesis: ((Gauge (C,1)) * (i,(Center (Gauge (C,1))))) `2 = ((S-bound C) + (N-bound C)) / 2
then  [i,(Center (Gauge (C,1)))] in Indices (Gauge (C,1)) by Lm5;
hence ((Gauge (C,1)) * (i,(Center (Gauge (C,1))))) `2 = |[((W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ 1)) * (i - 2))),((S-bound C) + ((((N-bound C) - (S-bound C)) / (2 |^ 1)) * ((Center (Gauge (C,1))) - 2)))]| `2 by JORDAN8:def_1
.= (S-bound C) + ((((N-bound C) - (S-bound C)) / (2 |^ 1)) * ((Center (Gauge (C,1))) - 2)) by EUCLID:52
.= (S-bound C) + (((N-bound C) - (S-bound C)) / 2) by Lm6
.= ((S-bound C) + (N-bound C)) / 2 ;
::_thesis: verum
end;

theorem Th40: :: JORDAN1A:40
for i, n, j, m being Element of NAT
for E being compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= i & i <= len (Gauge (E,n)) & 1 <= j & j <= len (Gauge (E,m)) & m <= n holds
((Gauge (E,n)) * (i,(len (Gauge (E,n))))) `2 <= ((Gauge (E,m)) * (j,(len (Gauge (E,m))))) `2
proof
let i, n, j, m be Element of NAT ; ::_thesis: for E being compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= i & i <= len (Gauge (E,n)) & 1 <= j & j <= len (Gauge (E,m)) & m <= n holds
((Gauge (E,n)) * (i,(len (Gauge (E,n))))) `2 <= ((Gauge (E,m)) * (j,(len (Gauge (E,m))))) `2
let E be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( 1 <= i & i <= len (Gauge (E,n)) & 1 <= j & j <= len (Gauge (E,m)) & m <= n implies ((Gauge (E,n)) * (i,(len (Gauge (E,n))))) `2 <= ((Gauge (E,m)) * (j,(len (Gauge (E,m))))) `2 )
set a = N-bound E;
set s = S-bound E;
set G = Gauge (E,n);
set M = Gauge (E,m);
assume that
A1: ( 1 <= i & i <= len (Gauge (E,n)) ) and
A2: ( 1 <= j & j <= len (Gauge (E,m)) ) and
A3: m <= n ; ::_thesis: ((Gauge (E,n)) * (i,(len (Gauge (E,n))))) `2 <= ((Gauge (E,m)) * (j,(len (Gauge (E,m))))) `2
A4: len (Gauge (E,m)) = width (Gauge (E,m)) by JORDAN8:def_1;
 1 <= len (Gauge (E,m)) by A2, XXREAL_0:2;
then  [j,(len (Gauge (E,m)))] in Indices (Gauge (E,m)) by A2, A4, MATRIX_1:36;
then A5: ((Gauge (E,m)) * (j,(len (Gauge (E,m))))) `2 = (N-bound E) + (((N-bound E) - (S-bound E)) / (2 |^ m)) by Lm13;
A6: len (Gauge (E,n)) = width (Gauge (E,n)) by JORDAN8:def_1;
 1 <= len (Gauge (E,n)) by A1, XXREAL_0:2;
then  [i,(len (Gauge (E,n)))] in Indices (Gauge (E,n)) by A1, A6, MATRIX_1:36;
then  ( 0 < (N-bound E) - (S-bound E) & ((Gauge (E,n)) * (i,(len (Gauge (E,n))))) `2 = (N-bound E) + (((N-bound E) - (S-bound E)) / (2 |^ n)) ) by Lm13, SPRECT_1:32, XREAL_1:50;
hence  ((Gauge (E,n)) * (i,(len (Gauge (E,n))))) `2 <= ((Gauge (E,m)) * (j,(len (Gauge (E,m))))) `2 by A3, A5, Lm8; ::_thesis: verum
end;

theorem :: JORDAN1A:41
for i, n, j, m being Element of NAT
for E being compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= i & i <= len (Gauge (E,n)) & 1 <= j & j <= len (Gauge (E,m)) & m <= n holds
((Gauge (E,n)) * ((len (Gauge (E,n))),i)) `1 <= ((Gauge (E,m)) * ((len (Gauge (E,m))),j)) `1
proof
let i, n, j, m be Element of NAT ; ::_thesis: for E being compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= i & i <= len (Gauge (E,n)) & 1 <= j & j <= len (Gauge (E,m)) & m <= n holds
((Gauge (E,n)) * ((len (Gauge (E,n))),i)) `1 <= ((Gauge (E,m)) * ((len (Gauge (E,m))),j)) `1
let E be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( 1 <= i & i <= len (Gauge (E,n)) & 1 <= j & j <= len (Gauge (E,m)) & m <= n implies ((Gauge (E,n)) * ((len (Gauge (E,n))),i)) `1 <= ((Gauge (E,m)) * ((len (Gauge (E,m))),j)) `1 )
set w = W-bound E;
set e = E-bound E;
set G = Gauge (E,n);
set M = Gauge (E,m);
assume that
A1: ( 1 <= i & i <= len (Gauge (E,n)) ) and
A2: ( 1 <= j & j <= len (Gauge (E,m)) ) and
A3: m <= n ; ::_thesis: ((Gauge (E,n)) * ((len (Gauge (E,n))),i)) `1 <= ((Gauge (E,m)) * ((len (Gauge (E,m))),j)) `1
A4: len (Gauge (E,m)) = width (Gauge (E,m)) by JORDAN8:def_1;
 1 <= len (Gauge (E,m)) by A2, XXREAL_0:2;
then  [(len (Gauge (E,m))),j] in Indices (Gauge (E,m)) by A2, A4, MATRIX_1:36;
then A5: ((Gauge (E,m)) * ((len (Gauge (E,m))),j)) `1 = (E-bound E) + (((E-bound E) - (W-bound E)) / (2 |^ m)) by Lm14;
A6: len (Gauge (E,n)) = width (Gauge (E,n)) by JORDAN8:def_1;
 1 <= len (Gauge (E,n)) by A1, XXREAL_0:2;
then  [(len (Gauge (E,n))),i] in Indices (Gauge (E,n)) by A1, A6, MATRIX_1:36;
then  ( 0 < (E-bound E) - (W-bound E) & ((Gauge (E,n)) * ((len (Gauge (E,n))),i)) `1 = (E-bound E) + (((E-bound E) - (W-bound E)) / (2 |^ n)) ) by Lm14, SPRECT_1:31, XREAL_1:50;
hence  ((Gauge (E,n)) * ((len (Gauge (E,n))),i)) `1 <= ((Gauge (E,m)) * ((len (Gauge (E,m))),j)) `1 by A3, A5, Lm8; ::_thesis: verum
end;

theorem :: JORDAN1A:42
for i, n, j, m being Element of NAT
for E being compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= i & i <= len (Gauge (E,n)) & 1 <= j & j <= len (Gauge (E,m)) & m <= n holds
((Gauge (E,m)) * (1,j)) `1 <= ((Gauge (E,n)) * (1,i)) `1
proof
let i, n, j, m be Element of NAT ; ::_thesis: for E being compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= i & i <= len (Gauge (E,n)) & 1 <= j & j <= len (Gauge (E,m)) & m <= n holds
((Gauge (E,m)) * (1,j)) `1 <= ((Gauge (E,n)) * (1,i)) `1
let E be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( 1 <= i & i <= len (Gauge (E,n)) & 1 <= j & j <= len (Gauge (E,m)) & m <= n implies ((Gauge (E,m)) * (1,j)) `1 <= ((Gauge (E,n)) * (1,i)) `1 )
set w = W-bound E;
set e = E-bound E;
set G = Gauge (E,n);
set M = Gauge (E,m);
assume that
A1: ( 1 <= i & i <= len (Gauge (E,n)) ) and
A2: ( 1 <= j & j <= len (Gauge (E,m)) ) and
A3: m <= n ; ::_thesis: ((Gauge (E,m)) * (1,j)) `1 <= ((Gauge (E,n)) * (1,i)) `1
A4: len (Gauge (E,m)) = width (Gauge (E,m)) by JORDAN8:def_1;
 1 <= len (Gauge (E,m)) by A2, XXREAL_0:2;
then  [1,j] in Indices (Gauge (E,m)) by A2, A4, MATRIX_1:36;
then A5: ((Gauge (E,m)) * (1,j)) `1 = (W-bound E) - (((E-bound E) - (W-bound E)) / (2 |^ m)) by Lm12;
A6: len (Gauge (E,n)) = width (Gauge (E,n)) by JORDAN8:def_1;
 1 <= len (Gauge (E,n)) by A1, XXREAL_0:2;
then  [1,i] in Indices (Gauge (E,n)) by A1, A6, MATRIX_1:36;
then  ( 0 < (E-bound E) - (W-bound E) & ((Gauge (E,n)) * (1,i)) `1 = (W-bound E) - (((E-bound E) - (W-bound E)) / (2 |^ n)) ) by Lm12, SPRECT_1:31, XREAL_1:50;
hence  ((Gauge (E,m)) * (1,j)) `1 <= ((Gauge (E,n)) * (1,i)) `1 by A3, A5, Lm9; ::_thesis: verum
end;

theorem :: JORDAN1A:43
for i, n, j, m being Element of NAT
for E being compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= i & i <= len (Gauge (E,n)) & 1 <= j & j <= len (Gauge (E,m)) & m <= n holds
((Gauge (E,m)) * (j,1)) `2 <= ((Gauge (E,n)) * (i,1)) `2
proof
let i, n, j, m be Element of NAT ; ::_thesis: for E being compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= i & i <= len (Gauge (E,n)) & 1 <= j & j <= len (Gauge (E,m)) & m <= n holds
((Gauge (E,m)) * (j,1)) `2 <= ((Gauge (E,n)) * (i,1)) `2
let E be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( 1 <= i & i <= len (Gauge (E,n)) & 1 <= j & j <= len (Gauge (E,m)) & m <= n implies ((Gauge (E,m)) * (j,1)) `2 <= ((Gauge (E,n)) * (i,1)) `2 )
set a = N-bound E;
set s = S-bound E;
set G = Gauge (E,n);
set M = Gauge (E,m);
assume that
A1: ( 1 <= i & i <= len (Gauge (E,n)) ) and
A2: ( 1 <= j & j <= len (Gauge (E,m)) ) and
A3: m <= n ; ::_thesis: ((Gauge (E,m)) * (j,1)) `2 <= ((Gauge (E,n)) * (i,1)) `2
A4: len (Gauge (E,m)) = width (Gauge (E,m)) by JORDAN8:def_1;
 1 <= len (Gauge (E,m)) by A2, XXREAL_0:2;
then  [j,1] in Indices (Gauge (E,m)) by A2, A4, MATRIX_1:36;
then A5: ((Gauge (E,m)) * (j,1)) `2 = (S-bound E) - (((N-bound E) - (S-bound E)) / (2 |^ m)) by Lm11;
A6: len (Gauge (E,n)) = width (Gauge (E,n)) by JORDAN8:def_1;
 1 <= len (Gauge (E,n)) by A1, XXREAL_0:2;
then  [i,1] in Indices (Gauge (E,n)) by A1, A6, MATRIX_1:36;
then  ( 0 < (N-bound E) - (S-bound E) & ((Gauge (E,n)) * (i,1)) `2 = (S-bound E) - (((N-bound E) - (S-bound E)) / (2 |^ n)) ) by Lm11, SPRECT_1:32, XREAL_1:50;
hence  ((Gauge (E,m)) * (j,1)) `2 <= ((Gauge (E,n)) * (i,1)) `2 by A3, A5, Lm9; ::_thesis: verum
end;

theorem :: JORDAN1A:44
for m, n being Element of NAT
for E being compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= m & m <= n holds
LSeg (((Gauge (E,n)) * ((Center (Gauge (E,n))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),(len (Gauge (E,n)))))) c= LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,m)) * ((Center (Gauge (E,m))),(len (Gauge (E,m))))))
proof
let m, n be Element of NAT ; ::_thesis: for E being compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= m & m <= n holds
LSeg (((Gauge (E,n)) * ((Center (Gauge (E,n))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),(len (Gauge (E,n)))))) c= LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,m)) * ((Center (Gauge (E,m))),(len (Gauge (E,m))))))
let E be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( 1 <= m & m <= n implies LSeg (((Gauge (E,n)) * ((Center (Gauge (E,n))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),(len (Gauge (E,n)))))) c= LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,m)) * ((Center (Gauge (E,m))),(len (Gauge (E,m)))))) )
set a = N-bound E;
set s = S-bound E;
set G = Gauge (E,n);
set M = Gauge (E,m);
set sn = Center (Gauge (E,n));
set sm = Center (Gauge (E,m));
assume A1: 1 <= m ; ::_thesis: ( not m <= n or LSeg (((Gauge (E,n)) * ((Center (Gauge (E,n))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),(len (Gauge (E,n)))))) c= LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,m)) * ((Center (Gauge (E,m))),(len (Gauge (E,m)))))) )
A2: 1 <= len (Gauge (E,m)) by GOBRD11:34;
then  [(Center (Gauge (E,m))),1] in Indices (Gauge (E,m)) by Lm4;
then A3: ((Gauge (E,m)) * ((Center (Gauge (E,m))),1)) `2 = (S-bound E) - (((N-bound E) - (S-bound E)) / (2 |^ m)) by Lm11;
 [(Center (Gauge (E,m))),(len (Gauge (E,m)))] in Indices (Gauge (E,m)) by A2, Lm4;
then A4: ((Gauge (E,m)) * ((Center (Gauge (E,m))),(len (Gauge (E,m))))) `2 = (N-bound E) + (((N-bound E) - (S-bound E)) / (2 |^ m)) by Lm13;
A5: Center (Gauge (E,n)) <= len (Gauge (E,n)) by Lm3;
A6: 1 <= len (Gauge (E,n)) by GOBRD11:34;
assume A7: m <= n ; ::_thesis: LSeg (((Gauge (E,n)) * ((Center (Gauge (E,n))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),(len (Gauge (E,n)))))) c= LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,m)) * ((Center (Gauge (E,m))),(len (Gauge (E,m))))))
then A8: ( ((Gauge (E,m)) * ((Center (Gauge (E,m))),1)) `1 = ((Gauge (E,n)) * ((Center (Gauge (E,n))),(len (Gauge (E,n))))) `1 & ((Gauge (E,n)) * ((Center (Gauge (E,n))),(len (Gauge (E,n))))) `1 = ((Gauge (E,m)) * ((Center (Gauge (E,m))),(len (Gauge (E,m))))) `1 ) by A1, A6, A2, Th36;
 0 < (N-bound E) - (S-bound E) by SPRECT_1:32, XREAL_1:50;
then A9: ((N-bound E) - (S-bound E)) / (2 |^ n) <= ((N-bound E) - (S-bound E)) / (2 |^ m) by A7, Lm7;
 ( len (Gauge (E,n)) = width (Gauge (E,n)) & 1 <= Center (Gauge (E,n)) ) by Lm3, JORDAN8:def_1;
then A10: ((Gauge (E,n)) * ((Center (Gauge (E,n))),1)) `2 <= ((Gauge (E,n)) * ((Center (Gauge (E,n))),(len (Gauge (E,n))))) `2 by A6, A5, SPRECT_3:12;
 [(Center (Gauge (E,n))),(len (Gauge (E,n)))] in Indices (Gauge (E,n)) by A6, Lm4;
then  ((Gauge (E,n)) * ((Center (Gauge (E,n))),(len (Gauge (E,n))))) `2 = (N-bound E) + (((N-bound E) - (S-bound E)) / (2 |^ n)) by Lm13;
then A11: ((Gauge (E,n)) * ((Center (Gauge (E,n))),(len (Gauge (E,n))))) `2 <= ((Gauge (E,m)) * ((Center (Gauge (E,m))),(len (Gauge (E,m))))) `2 by A9, A4, XREAL_1:7;
then A12: ((Gauge (E,n)) * ((Center (Gauge (E,n))),1)) `2 <= ((Gauge (E,m)) * ((Center (Gauge (E,m))),(len (Gauge (E,m))))) `2 by A10, XXREAL_0:2;
 [(Center (Gauge (E,n))),1] in Indices (Gauge (E,n)) by A6, Lm4;
then  ((Gauge (E,n)) * ((Center (Gauge (E,n))),1)) `2 = (S-bound E) - (((N-bound E) - (S-bound E)) / (2 |^ n)) by Lm11;
then A13: ((Gauge (E,m)) * ((Center (Gauge (E,m))),1)) `2 <= ((Gauge (E,n)) * ((Center (Gauge (E,n))),1)) `2 by A9, A3, XREAL_1:13;
then  ((Gauge (E,m)) * ((Center (Gauge (E,m))),1)) `2 <= ((Gauge (E,n)) * ((Center (Gauge (E,n))),(len (Gauge (E,n))))) `2 by A10, XXREAL_0:2;
then A14: (Gauge (E,n)) * ((Center (Gauge (E,n))),(len (Gauge (E,n)))) in LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,m)) * ((Center (Gauge (E,m))),(len (Gauge (E,m)))))) by A11, A8, GOBOARD7:7;
 ( ((Gauge (E,m)) * ((Center (Gauge (E,m))),1)) `1 = ((Gauge (E,n)) * ((Center (Gauge (E,n))),1)) `1 & ((Gauge (E,n)) * ((Center (Gauge (E,n))),1)) `1 = ((Gauge (E,m)) * ((Center (Gauge (E,m))),(len (Gauge (E,m))))) `1 ) by A1, A7, A6, A2, Th36;
then  (Gauge (E,n)) * ((Center (Gauge (E,n))),1) in LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,m)) * ((Center (Gauge (E,m))),(len (Gauge (E,m)))))) by A13, A12, GOBOARD7:7;
hence  LSeg (((Gauge (E,n)) * ((Center (Gauge (E,n))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),(len (Gauge (E,n)))))) c= LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,m)) * ((Center (Gauge (E,m))),(len (Gauge (E,m)))))) by A14, TOPREAL1:6; ::_thesis: verum
end;

theorem :: JORDAN1A:45
for m, n, j being Element of NAT
for E being compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= m & m <= n & 1 <= j & j <= width (Gauge (E,n)) holds
LSeg (((Gauge (E,n)) * ((Center (Gauge (E,n))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) c= LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),j)))
proof
let m, n, j be Element of NAT ; ::_thesis: for E being compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= m & m <= n & 1 <= j & j <= width (Gauge (E,n)) holds
LSeg (((Gauge (E,n)) * ((Center (Gauge (E,n))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) c= LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),j)))
let E be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( 1 <= m & m <= n & 1 <= j & j <= width (Gauge (E,n)) implies LSeg (((Gauge (E,n)) * ((Center (Gauge (E,n))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) c= LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) )
set a = N-bound E;
set s = S-bound E;
set w = W-bound E;
set e = E-bound E;
set G = Gauge (E,n);
set M = Gauge (E,m);
set sn = Center (Gauge (E,n));
set sm = Center (Gauge (E,m));
assume that
A1: 1 <= m and
A2: m <= n and
A3: 1 <= j and
A4: j <= width (Gauge (E,n)) ; ::_thesis: LSeg (((Gauge (E,n)) * ((Center (Gauge (E,n))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) c= LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),j)))
now__::_thesis:_for_t_being_Element_of_NAT_st_1_<=_t_&_t_<=_j_holds_
(Gauge_(E,n))_*_((Center_(Gauge_(E,n))),t)_in_LSeg_(((Gauge_(E,m))_*_((Center_(Gauge_(E,m))),1)),((Gauge_(E,n))_*_((Center_(Gauge_(E,n))),j)))
A5: 0 < (N-bound E) - (S-bound E) by SPRECT_1:32, XREAL_1:50;
then A6: (S-bound E) - (((N-bound E) - (S-bound E)) / (2 |^ m)) <= (S-bound E) - 0 by XREAL_1:13;
A7: ((N-bound E) - (S-bound E)) / (2 |^ n) <= ((N-bound E) - (S-bound E)) / (2 |^ m) by A2, A5, Lm7;
A8: 1 <= len (Gauge (E,m)) by GOBRD11:34;
then  [(Center (Gauge (E,m))),1] in Indices (Gauge (E,m)) by Lm4;
then A9: ((Gauge (E,m)) * ((Center (Gauge (E,m))),1)) `2 = (S-bound E) - (((N-bound E) - (S-bound E)) / (2 |^ m)) by Lm11;
let t be Element of NAT ; ::_thesis: ( 1 <= t & t <= j implies (Gauge (E,n)) * ((Center (Gauge (E,n))),t) in LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) )
assume that
A10: 1 <= t and
A11: t <= j ; ::_thesis: (Gauge (E,n)) * ((Center (Gauge (E,n))),t) in LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),j)))
 ( 1 <= Center (Gauge (E,n)) & Center (Gauge (E,n)) <= len (Gauge (E,n)) ) by Lm3;
then A12: ((Gauge (E,n)) * ((Center (Gauge (E,n))),t)) `2 <= ((Gauge (E,n)) * ((Center (Gauge (E,n))),j)) `2 by A4, A10, A11, SPRECT_3:12;
A13: len (Gauge (E,n)) = width (Gauge (E,n)) by JORDAN8:def_1;
then A14: t <= len (Gauge (E,n)) by A4, A11, XXREAL_0:2;
then A15: ((Gauge (E,m)) * ((Center (Gauge (E,m))),1)) `1 = ((Gauge (E,n)) * ((Center (Gauge (E,n))),t)) `1 by A1, A2, A10, A8, Th36;
A16: [(Center (Gauge (E,n))),t] in Indices (Gauge (E,n)) by A10, A14, Lm4;
then A17: ((Gauge (E,n)) * ((Center (Gauge (E,n))),t)) `2 = |[((W-bound E) + ((((E-bound E) - (W-bound E)) / (2 |^ n)) * ((Center (Gauge (E,n))) - 2))),((S-bound E) + ((((N-bound E) - (S-bound E)) / (2 |^ n)) * (t - 2)))]| `2 by JORDAN8:def_1
.= (S-bound E) + ((((N-bound E) - (S-bound E)) / (2 |^ n)) * (t - 2)) by EUCLID:52 ;
A18: now__::_thesis:_((Gauge_(E,m))_*_((Center_(Gauge_(E,m))),1))_`2_<=_((Gauge_(E,n))_*_((Center_(Gauge_(E,n))),t))_`2
percases ( t = 1 or t > 1 ) by A10, XXREAL_0:1;
suppose t = 1 ; ::_thesis: ((Gauge (E,m)) * ((Center (Gauge (E,m))),1)) `2 <= ((Gauge (E,n)) * ((Center (Gauge (E,n))),t)) `2
then  ((Gauge (E,n)) * ((Center (Gauge (E,n))),t)) `2 = (S-bound E) - (((N-bound E) - (S-bound E)) / (2 |^ n)) by A16, Lm11;
hence  ((Gauge (E,m)) * ((Center (Gauge (E,m))),1)) `2 <= ((Gauge (E,n)) * ((Center (Gauge (E,n))),t)) `2 by A7, A9, XREAL_1:13; ::_thesis: verum
end;
suppose t > 1 ; ::_thesis: ((Gauge (E,m)) * ((Center (Gauge (E,m))),1)) `2 <= ((Gauge (E,n)) * ((Center (Gauge (E,n))),t)) `2
then  t >= 1 + 1 by NAT_1:13;
then  t - 2 >= 2 - 2 by XREAL_1:9;
then  (S-bound E) + 0 <= (S-bound E) + ((((N-bound E) - (S-bound E)) / (2 |^ n)) * (t - 2)) by A5, XREAL_1:6;
hence  ((Gauge (E,m)) * ((Center (Gauge (E,m))),1)) `2 <= ((Gauge (E,n)) * ((Center (Gauge (E,n))),t)) `2 by A17, A6, A9, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
 ((Gauge (E,n)) * ((Center (Gauge (E,n))),t)) `1 = ((Gauge (E,n)) * ((Center (Gauge (E,n))),j)) `1 by A1, A2, A3, A4, A10, A13, A14, Th36;
hence  (Gauge (E,n)) * ((Center (Gauge (E,n))),t) in LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) by A15, A18, A12, GOBOARD7:7; ::_thesis: verum
end;
then  ( (Gauge (E,n)) * ((Center (Gauge (E,n))),1) in LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) & (Gauge (E,n)) * ((Center (Gauge (E,n))),j) in LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) ) by A3;
hence  LSeg (((Gauge (E,n)) * ((Center (Gauge (E,n))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) c= LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) by TOPREAL1:6; ::_thesis: verum
end;

theorem :: JORDAN1A:46
for m, n, j being Element of NAT
for E being compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= m & m <= n & 1 <= j & j <= width (Gauge (E,n)) holds
LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) c= LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,m)) * ((Center (Gauge (E,m))),(len (Gauge (E,m))))))
proof
let m, n, j be Element of NAT ; ::_thesis: for E being compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= m & m <= n & 1 <= j & j <= width (Gauge (E,n)) holds
LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) c= LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,m)) * ((Center (Gauge (E,m))),(len (Gauge (E,m))))))
let E be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( 1 <= m & m <= n & 1 <= j & j <= width (Gauge (E,n)) implies LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) c= LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,m)) * ((Center (Gauge (E,m))),(len (Gauge (E,m)))))) )
set a = N-bound E;
set s = S-bound E;
set w = W-bound E;
set e = E-bound E;
set G = Gauge (E,n);
set M = Gauge (E,m);
set sn = Center (Gauge (E,n));
set sm = Center (Gauge (E,m));
assume that
A1: 1 <= m and
A2: m <= n and
A3: 1 <= j and
A4: j <= width (Gauge (E,n)) ; ::_thesis: LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) c= LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,m)) * ((Center (Gauge (E,m))),(len (Gauge (E,m))))))
A5: ( 1 <= Center (Gauge (E,m)) & Center (Gauge (E,m)) <= len (Gauge (E,m)) ) by Lm3;
A6: ( 1 <= Center (Gauge (E,n)) & Center (Gauge (E,n)) <= len (Gauge (E,n)) ) by Lm3;
then A7: ((Gauge (E,n)) * ((Center (Gauge (E,n))),(len (Gauge (E,n))))) `2 <= ((Gauge (E,m)) * ((Center (Gauge (E,m))),(len (Gauge (E,m))))) `2 by A2, A5, Th40;
 len (Gauge (E,n)) = width (Gauge (E,n)) by JORDAN8:def_1;
then  ((Gauge (E,n)) * ((Center (Gauge (E,n))),j)) `2 <= ((Gauge (E,n)) * ((Center (Gauge (E,n))),(len (Gauge (E,n))))) `2 by A3, A4, A6, SPRECT_3:12;
then A8: ((Gauge (E,n)) * ((Center (Gauge (E,n))),j)) `2 <= ((Gauge (E,m)) * ((Center (Gauge (E,m))),(len (Gauge (E,m))))) `2 by A7, XXREAL_0:2;
A9: 0 < (N-bound E) - (S-bound E) by SPRECT_1:32, XREAL_1:50;
then A10: (S-bound E) - (((N-bound E) - (S-bound E)) / (2 |^ m)) <= (S-bound E) - 0 by XREAL_1:13;
A11: 1 <= len (Gauge (E,m)) by GOBRD11:34;
then  [(Center (Gauge (E,m))),1] in Indices (Gauge (E,m)) by Lm4;
then A12: ((Gauge (E,m)) * ((Center (Gauge (E,m))),1)) `2 = (S-bound E) - (((N-bound E) - (S-bound E)) / (2 |^ m)) by Lm11;
A13: ((N-bound E) - (S-bound E)) / (2 |^ n) <= ((N-bound E) - (S-bound E)) / (2 |^ m) by A2, A9, Lm7;
A14: len (Gauge (E,n)) = width (Gauge (E,n)) by JORDAN8:def_1;
then A15: [(Center (Gauge (E,n))),j] in Indices (Gauge (E,n)) by A3, A4, Lm4;
then A16: ((Gauge (E,n)) * ((Center (Gauge (E,n))),j)) `2 = |[((W-bound E) + ((((E-bound E) - (W-bound E)) / (2 |^ n)) * ((Center (Gauge (E,n))) - 2))),((S-bound E) + ((((N-bound E) - (S-bound E)) / (2 |^ n)) * (j - 2)))]| `2 by JORDAN8:def_1
.= (S-bound E) + ((((N-bound E) - (S-bound E)) / (2 |^ n)) * (j - 2)) by EUCLID:52 ;
A17: now__::_thesis:_((Gauge_(E,m))_*_((Center_(Gauge_(E,m))),1))_`2_<=_((Gauge_(E,n))_*_((Center_(Gauge_(E,n))),j))_`2
percases ( j = 1 or j > 1 ) by A3, XXREAL_0:1;
suppose j = 1 ; ::_thesis: ((Gauge (E,m)) * ((Center (Gauge (E,m))),1)) `2 <= ((Gauge (E,n)) * ((Center (Gauge (E,n))),j)) `2
then  ((Gauge (E,n)) * ((Center (Gauge (E,n))),j)) `2 = (S-bound E) - (((N-bound E) - (S-bound E)) / (2 |^ n)) by A15, Lm11;
hence  ((Gauge (E,m)) * ((Center (Gauge (E,m))),1)) `2 <= ((Gauge (E,n)) * ((Center (Gauge (E,n))),j)) `2 by A13, A12, XREAL_1:13; ::_thesis: verum
end;
suppose j > 1 ; ::_thesis: ((Gauge (E,m)) * ((Center (Gauge (E,m))),1)) `2 <= ((Gauge (E,n)) * ((Center (Gauge (E,n))),j)) `2
then  j >= 1 + 1 by NAT_1:13;
then  j - 2 >= 2 - 2 by XREAL_1:9;
then  (S-bound E) + 0 <= (S-bound E) + ((((N-bound E) - (S-bound E)) / (2 |^ n)) * (j - 2)) by A9, XREAL_1:6;
hence  ((Gauge (E,m)) * ((Center (Gauge (E,m))),1)) `2 <= ((Gauge (E,n)) * ((Center (Gauge (E,n))),j)) `2 by A12, A16, A10, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
 len (Gauge (E,m)) = width (Gauge (E,m)) by JORDAN8:def_1;
then A18: ((Gauge (E,m)) * ((Center (Gauge (E,m))),1)) `2 <= ((Gauge (E,m)) * ((Center (Gauge (E,m))),(len (Gauge (E,m))))) `2 by A11, A5, SPRECT_3:12;
 ((Gauge (E,m)) * ((Center (Gauge (E,m))),1)) `1 = ((Gauge (E,m)) * ((Center (Gauge (E,m))),(len (Gauge (E,m))))) `1 by A1, A11, Th36;
then A19: (Gauge (E,m)) * ((Center (Gauge (E,m))),1) in LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,m)) * ((Center (Gauge (E,m))),(len (Gauge (E,m)))))) by A18, GOBOARD7:7;
 ( ((Gauge (E,m)) * ((Center (Gauge (E,m))),1)) `1 = ((Gauge (E,n)) * ((Center (Gauge (E,n))),j)) `1 & ((Gauge (E,n)) * ((Center (Gauge (E,n))),j)) `1 = ((Gauge (E,m)) * ((Center (Gauge (E,m))),(len (Gauge (E,m))))) `1 ) by A1, A2, A3, A4, A11, A14, Th36;
then  (Gauge (E,n)) * ((Center (Gauge (E,n))),j) in LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,m)) * ((Center (Gauge (E,m))),(len (Gauge (E,m)))))) by A17, A8, GOBOARD7:7;
hence  LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) c= LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),1)),((Gauge (E,m)) * ((Center (Gauge (E,m))),(len (Gauge (E,m)))))) by A19, TOPREAL1:6; ::_thesis: verum
end;

theorem Th47: :: JORDAN1A:47
for m, n, i, j being Element of NAT
for E being compact non horizontal non vertical Subset of (TOP-REAL 2) st m <= n & 1 < i & i + 1 < len (Gauge (E,m)) & 1 < j & j + 1 < width (Gauge (E,m)) holds
for i1, j1 being Element of NAT st ((2 |^ (n -' m)) * (i - 2)) + 2 <= i1 & i1 < ((2 |^ (n -' m)) * (i - 1)) + 2 & ((2 |^ (n -' m)) * (j - 2)) + 2 <= j1 & j1 < ((2 |^ (n -' m)) * (j - 1)) + 2 holds
cell ((Gauge (E,n)),i1,j1) c= cell ((Gauge (E,m)),i,j)
proof
let m, n, i, j be Element of NAT ; ::_thesis: for E being compact non horizontal non vertical Subset of (TOP-REAL 2) st m <= n & 1 < i & i + 1 < len (Gauge (E,m)) & 1 < j & j + 1 < width (Gauge (E,m)) holds
for i1, j1 being Element of NAT st ((2 |^ (n -' m)) * (i - 2)) + 2 <= i1 & i1 < ((2 |^ (n -' m)) * (i - 1)) + 2 & ((2 |^ (n -' m)) * (j - 2)) + 2 <= j1 & j1 < ((2 |^ (n -' m)) * (j - 1)) + 2 holds
cell ((Gauge (E,n)),i1,j1) c= cell ((Gauge (E,m)),i,j)
let E be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( m <= n & 1 < i & i + 1 < len (Gauge (E,m)) & 1 < j & j + 1 < width (Gauge (E,m)) implies for i1, j1 being Element of NAT st ((2 |^ (n -' m)) * (i - 2)) + 2 <= i1 & i1 < ((2 |^ (n -' m)) * (i - 1)) + 2 & ((2 |^ (n -' m)) * (j - 2)) + 2 <= j1 & j1 < ((2 |^ (n -' m)) * (j - 1)) + 2 holds
cell ((Gauge (E,n)),i1,j1) c= cell ((Gauge (E,m)),i,j) )
set G = Gauge (E,m);
set G1 = Gauge (E,n);
assume that
A1: m <= n and
A2: 1 < i and
A3: i + 1 < len (Gauge (E,m)) and
A4: 1 < j and
A5: j + 1 < width (Gauge (E,m)) ; ::_thesis: for i1, j1 being Element of NAT st ((2 |^ (n -' m)) * (i - 2)) + 2 <= i1 & i1 < ((2 |^ (n -' m)) * (i - 1)) + 2 & ((2 |^ (n -' m)) * (j - 2)) + 2 <= j1 & j1 < ((2 |^ (n -' m)) * (j - 1)) + 2 holds
cell ((Gauge (E,n)),i1,j1) c= cell ((Gauge (E,m)),i,j)
set i2 = ((2 |^ (n -' m)) * (i -' 2)) + 2;
set j2 = ((2 |^ (n -' m)) * (j -' 2)) + 2;
set i3 = ((2 |^ (n -' m)) * (i -' 1)) + 2;
set j3 = ((2 |^ (n -' m)) * (j -' 1)) + 2;
let i1, j1 be Element of NAT ; ::_thesis: ( ((2 |^ (n -' m)) * (i - 2)) + 2 <= i1 & i1 < ((2 |^ (n -' m)) * (i - 1)) + 2 & ((2 |^ (n -' m)) * (j - 2)) + 2 <= j1 & j1 < ((2 |^ (n -' m)) * (j - 1)) + 2 implies cell ((Gauge (E,n)),i1,j1) c= cell ((Gauge (E,m)),i,j) )
assume that
A6: ((2 |^ (n -' m)) * (i - 2)) + 2 <= i1 and
A7: i1 < ((2 |^ (n -' m)) * (i - 1)) + 2 and
A8: ((2 |^ (n -' m)) * (j - 2)) + 2 <= j1 and
A9: j1 < ((2 |^ (n -' m)) * (j - 1)) + 2 ; ::_thesis: cell ((Gauge (E,n)),i1,j1) c= cell ((Gauge (E,m)),i,j)
A10: j - 1 = j -' 1 by A4, XREAL_1:233;
then A11: ((2 |^ (n -' m)) * (j -' 1)) + 2 <= width (Gauge (E,n)) by A1, A4, A5, Th35;
A12: 1 + 1 <= i by A2, NAT_1:13;
then A13: ((2 |^ (n -' m)) * (i -' 2)) + 2 = ((2 |^ (n -' m)) * (i - 2)) + 2 by XREAL_1:233;
 i < i + 1 by XREAL_1:29;
then A14: i < len (Gauge (E,m)) by A3, XXREAL_0:2;
then A15: 1 <= ((2 |^ (n -' m)) * (i - 2)) + 2 by A1, A2, Th31;
then A16: 1 <= i1 by A6, XXREAL_0:2;
 j1 + 1 <= ((2 |^ (n -' m)) * (j -' 1)) + 2 by A9, A10, NAT_1:13;
then A17: ( j1 + 1 < ((2 |^ (n -' m)) * (j -' 1)) + 2 or j1 + 1 = ((2 |^ (n -' m)) * (j -' 1)) + 2 ) by XXREAL_0:1;
let e be set ; :: according to TARSKI:def_3 ::_thesis: ( not e in cell ((Gauge (E,n)),i1,j1) or e in cell ((Gauge (E,m)),i,j) )
assume A18: e in cell ((Gauge (E,n)),i1,j1) ; ::_thesis: e in cell ((Gauge (E,m)),i,j)
then reconsider p = e as Point of (TOP-REAL 2) ;
 ((2 |^ (n -' m)) * (i - 1)) + 2 <= len (Gauge (E,n)) by A1, A2, A3, Th34;
then A19: i1 < len (Gauge (E,n)) by A7, XXREAL_0:2;
then A20: i1 + 1 <= len (Gauge (E,n)) by NAT_1:13;
A21: (j + 1) - (1 + 1) = j - 1
.= j -' 1 by A4, XREAL_1:233 ;
 1 < j + 1 by A4, XREAL_1:29;
then A22: (Gauge (E,m)) * (i,(j + 1)) = (Gauge (E,n)) * ((((2 |^ (n -' m)) * (i -' 2)) + 2),(((2 |^ (n -' m)) * (j -' 1)) + 2)) by A1, A2, A5, A14, A21, A13, Th33;
A23: i - 1 = i -' 1 by A2, XREAL_1:233;
then A24: ((2 |^ (n -' m)) * (i -' 1)) + 2 <= len (Gauge (E,n)) by A1, A2, A3, Th34;
 i1 + 1 <= ((2 |^ (n -' m)) * (i -' 1)) + 2 by A7, A23, NAT_1:13;
then A25: ( i1 + 1 < ((2 |^ (n -' m)) * (i -' 1)) + 2 or i1 + 1 = ((2 |^ (n -' m)) * (i -' 1)) + 2 ) by XXREAL_0:1;
A26: ((2 |^ (n -' m)) * (i -' 2)) + 2 = ((2 |^ (n -' m)) * (i - 2)) + 2 by A12, XREAL_1:233;
A27: (i + 1) - (1 + 1) = i - 1
.= i -' 1 by A2, XREAL_1:233 ;
A28: ((2 |^ (n -' m)) * (i -' 2)) + 2 = ((2 |^ (n -' m)) * (i - 2)) + 2 by A12, XREAL_1:233;
then A29: ((2 |^ (n -' m)) * (i -' 2)) + 2 <= len (Gauge (E,n)) by A6, A19, XXREAL_0:2;
 j < j + 1 by XREAL_1:29;
then A30: j < width (Gauge (E,m)) by A5, XXREAL_0:2;
then A31: 1 <= ((2 |^ (n -' m)) * (j - 2)) + 2 by A1, A4, Th32;
then A32: 1 <= j1 by A8, XXREAL_0:2;
then  1 < j1 + 1 by NAT_1:13;
then A33: ((Gauge (E,n)) * (i1,(j1 + 1))) `2 <= ((Gauge (E,n)) * (i1,(((2 |^ (n -' m)) * (j -' 1)) + 2))) `2 by A19, A16, A11, A17, GOBOARD5:4;
 ((2 |^ (n -' m)) * (j - 1)) + 2 <= width (Gauge (E,n)) by A1, A4, A5, Th35;
then A34: j1 < width (Gauge (E,n)) by A9, XXREAL_0:2;
then A35: j1 + 1 <= width (Gauge (E,n)) by NAT_1:13;
then A36: ((Gauge (E,n)) * (i1,j1)) `1 <= p `1 by A18, A20, A16, A32, JORDAN9:17;
A37: 1 + 1 <= j by A4, NAT_1:13;
then A38: ((2 |^ (n -' m)) * (j -' 2)) + 2 = ((2 |^ (n -' m)) * (j - 2)) + 2 by XREAL_1:233;
then  ( ((2 |^ (n -' m)) * (j -' 2)) + 2 < j1 or ((2 |^ (n -' m)) * (j -' 2)) + 2 = j1 ) by A8, XXREAL_0:1;
then A39: ((Gauge (E,n)) * (i1,(((2 |^ (n -' m)) * (j -' 2)) + 2))) `2 <= ((Gauge (E,n)) * (i1,j1)) `2 by A19, A34, A16, A31, A38, GOBOARD5:4;
A40: ((2 |^ (n -' m)) * (j -' 2)) + 2 = ((2 |^ (n -' m)) * (j - 2)) + 2 by A37, XREAL_1:233;
then A41: (Gauge (E,m)) * (i,j) = (Gauge (E,n)) * ((((2 |^ (n -' m)) * (i -' 2)) + 2),(((2 |^ (n -' m)) * (j -' 2)) + 2)) by A1, A2, A4, A14, A30, A28, Th33;
 1 < i + 1 by A2, XREAL_1:29;
then A42: (Gauge (E,m)) * ((i + 1),j) = (Gauge (E,n)) * ((((2 |^ (n -' m)) * (i -' 1)) + 2),(((2 |^ (n -' m)) * (j -' 2)) + 2)) by A1, A3, A4, A30, A27, A38, Th33;
A43: p `1 <= ((Gauge (E,n)) * ((i1 + 1),j1)) `1 by A18, A20, A35, A16, A32, JORDAN9:17;
 1 < i1 + 1 by A16, NAT_1:13;
then A44: ((Gauge (E,n)) * ((i1 + 1),j1)) `1 <= ((Gauge (E,n)) * ((((2 |^ (n -' m)) * (i -' 1)) + 2),j1)) `1 by A34, A32, A24, A25, GOBOARD5:3;
A45: ((Gauge (E,n)) * (i1,j1)) `2 <= p `2 by A18, A20, A35, A16, A32, JORDAN9:17;
A46: ((2 |^ (n -' m)) * (j -' 2)) + 2 <= width (Gauge (E,n)) by A8, A34, A40, XXREAL_0:2;
then ((Gauge (E,n)) * (i1,(((2 |^ (n -' m)) * (j -' 2)) + 2))) `2 = ((Gauge (E,n)) * (1,(((2 |^ (n -' m)) * (j -' 2)) + 2))) `2 by A19, A16, A31, A38, GOBOARD5:1
.= ((Gauge (E,n)) * ((((2 |^ (n -' m)) * (i -' 2)) + 2),(((2 |^ (n -' m)) * (j -' 2)) + 2))) `2 by A15, A31, A29, A46, A26, A38, GOBOARD5:1 ;
then A47: ((Gauge (E,m)) * (i,j)) `2 <= p `2 by A45, A41, A39, XXREAL_0:2;
A48: p `2 <= ((Gauge (E,n)) * (i1,(j1 + 1))) `2 by A18, A20, A35, A16, A32, JORDAN9:17;
A49: 1 < ((2 |^ (n -' m)) * (j -' 1)) + 2 by A9, A32, A10, XXREAL_0:2;
then ((Gauge (E,n)) * (i1,(((2 |^ (n -' m)) * (j -' 1)) + 2))) `2 = ((Gauge (E,n)) * (1,(((2 |^ (n -' m)) * (j -' 1)) + 2))) `2 by A19, A16, A11, GOBOARD5:1
.= ((Gauge (E,n)) * ((((2 |^ (n -' m)) * (i -' 2)) + 2),(((2 |^ (n -' m)) * (j -' 1)) + 2))) `2 by A15, A29, A13, A11, A49, GOBOARD5:1 ;
then A50: p `2 <= ((Gauge (E,m)) * (i,(j + 1))) `2 by A48, A22, A33, XXREAL_0:2;
 ( ((2 |^ (n -' m)) * (i -' 2)) + 2 < i1 or ((2 |^ (n -' m)) * (i -' 2)) + 2 = i1 ) by A6, A28, XXREAL_0:1;
then A51: ((Gauge (E,n)) * ((((2 |^ (n -' m)) * (i -' 2)) + 2),j1)) `1 <= ((Gauge (E,n)) * (i1,j1)) `1 by A19, A34, A15, A32, A28, GOBOARD5:3;
A52: 1 < ((2 |^ (n -' m)) * (i -' 1)) + 2 by A7, A16, A23, XXREAL_0:2;
then ((Gauge (E,n)) * ((((2 |^ (n -' m)) * (i -' 1)) + 2),j1)) `1 = ((Gauge (E,n)) * ((((2 |^ (n -' m)) * (i -' 1)) + 2),1)) `1 by A34, A32, A24, GOBOARD5:2
.= ((Gauge (E,n)) * ((((2 |^ (n -' m)) * (i -' 1)) + 2),(((2 |^ (n -' m)) * (j -' 2)) + 2))) `1 by A31, A46, A38, A24, A52, GOBOARD5:2 ;
then A53: p `1 <= ((Gauge (E,m)) * ((i + 1),j)) `1 by A43, A42, A44, XXREAL_0:2;
((Gauge (E,n)) * ((((2 |^ (n -' m)) * (i -' 2)) + 2),j1)) `1 = ((Gauge (E,n)) * ((((2 |^ (n -' m)) * (i -' 2)) + 2),1)) `1 by A34, A15, A32, A28, A29, GOBOARD5:2
.= ((Gauge (E,n)) * ((((2 |^ (n -' m)) * (i -' 2)) + 2),(((2 |^ (n -' m)) * (j -' 2)) + 2))) `1 by A15, A31, A28, A40, A29, A46, GOBOARD5:2 ;
then  ((Gauge (E,m)) * (i,j)) `1 <= p `1 by A36, A41, A51, XXREAL_0:2;
hence  e in cell ((Gauge (E,m)),i,j) by A2, A3, A4, A5, A53, A47, A50, JORDAN9:17; ::_thesis: verum
end;

theorem :: JORDAN1A:48
for m, n, i, j being Element of NAT
for E being compact non horizontal non vertical Subset of (TOP-REAL 2) st m <= n & 3 <= i & i < len (Gauge (E,m)) & 1 < j & j + 1 < width (Gauge (E,m)) holds
for i1, j1 being Element of NAT st i1 = ((2 |^ (n -' m)) * (i - 2)) + 2 & j1 = ((2 |^ (n -' m)) * (j - 2)) + 2 holds
cell ((Gauge (E,n)),(i1 -' 1),j1) c= cell ((Gauge (E,m)),(i -' 1),j)
proof
let m, n, i, j be Element of NAT ; ::_thesis: for E being compact non horizontal non vertical Subset of (TOP-REAL 2) st m <= n & 3 <= i & i < len (Gauge (E,m)) & 1 < j & j + 1 < width (Gauge (E,m)) holds
for i1, j1 being Element of NAT st i1 = ((2 |^ (n -' m)) * (i - 2)) + 2 & j1 = ((2 |^ (n -' m)) * (j - 2)) + 2 holds
cell ((Gauge (E,n)),(i1 -' 1),j1) c= cell ((Gauge (E,m)),(i -' 1),j)
let E be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( m <= n & 3 <= i & i < len (Gauge (E,m)) & 1 < j & j + 1 < width (Gauge (E,m)) implies for i1, j1 being Element of NAT st i1 = ((2 |^ (n -' m)) * (i - 2)) + 2 & j1 = ((2 |^ (n -' m)) * (j - 2)) + 2 holds
cell ((Gauge (E,n)),(i1 -' 1),j1) c= cell ((Gauge (E,m)),(i -' 1),j) )
assume that
A1: m <= n and
A2: 3 <= i and
A3: i < len (Gauge (E,m)) and
A4: ( 1 < j & j + 1 < width (Gauge (E,m)) ) ; ::_thesis: for i1, j1 being Element of NAT st i1 = ((2 |^ (n -' m)) * (i - 2)) + 2 & j1 = ((2 |^ (n -' m)) * (j - 2)) + 2 holds
cell ((Gauge (E,n)),(i1 -' 1),j1) c= cell ((Gauge (E,m)),(i -' 1),j)
A5: i - 2 = i -' 2 by A2, XREAL_1:233, XXREAL_0:2;
A6: 2 + 1 <= i by A2;
then  1 + 1 < i by NAT_1:13;
then A7: 1 < i - 1 by XREAL_1:20;
A8: 2 |^ (n -' m) > 0 by NEWTON:83;
A9: i - 3 = i -' 3 by A2, XREAL_1:233;
then  i -' 3 < i -' 2 by A5, XREAL_1:10;
then  (2 |^ (n -' m)) * (i -' 3) < (2 |^ (n -' m)) * (i -' 2) by A8, XREAL_1:68;
then  ((2 |^ (n -' m)) * (i -' 3)) + 1 <= (2 |^ (n -' m)) * (i -' 2) by NAT_1:13;
then  (2 |^ (n -' m)) * (i -' 3) <= ((2 |^ (n -' m)) * (i -' 2)) -' 1 by NAT_D:55;
then A10: ((2 |^ (n -' m)) * (i -' 3)) + 2 <= (((2 |^ (n -' m)) * (i -' 2)) -' 1) + 2 by XREAL_1:6;
A11: i -' 1 = i - 1 by A2, XREAL_1:233, XXREAL_0:2;
then A12: (i -' 1) - 1 = i - (1 + 1) ;
 i > 2 + 0 by A6, NAT_1:13;
then  i - 2 > 0 by XREAL_1:20;
then A13: (2 |^ (n -' m)) * (i -' 2) > 0 by A8, A5, XREAL_1:129;
then  (2 |^ (n -' m)) * (i -' 2) >= 0 + 1 by NAT_1:13;
then A14: ((2 |^ (n -' m)) * ((i -' 1) - 2)) + 2 <= (((2 |^ (n -' m)) * (i -' 2)) + 2) -' 1 by A9, A11, A10, NAT_D:38;
A15: (i -' 1) + 1 < len (Gauge (E,m)) by A2, A3, XREAL_1:235, XXREAL_0:2;
let i1, j1 be Element of NAT ; ::_thesis: ( i1 = ((2 |^ (n -' m)) * (i - 2)) + 2 & j1 = ((2 |^ (n -' m)) * (j - 2)) + 2 implies cell ((Gauge (E,n)),(i1 -' 1),j1) c= cell ((Gauge (E,m)),(i -' 1),j) )
assume that
A16: i1 = ((2 |^ (n -' m)) * (i - 2)) + 2 and
A17: j1 = ((2 |^ (n -' m)) * (j - 2)) + 2 ; ::_thesis: cell ((Gauge (E,n)),(i1 -' 1),j1) c= cell ((Gauge (E,m)),(i -' 1),j)
 i1 < i1 + 1 by XREAL_1:29;
then A18: i1 - 1 < i1 by XREAL_1:19;
 i1 > 0 + 2 by A16, A5, A13, XREAL_1:6;
then A19: i1 -' 1 < i1 by A18, XREAL_1:233, XXREAL_0:2;
 j - 2 < j - 1 by XREAL_1:10;
then  (2 |^ (n -' m)) * (j - 2) < (2 |^ (n -' m)) * (j - 1) by A8, XREAL_1:68;
then  j1 < ((2 |^ (n -' m)) * (j - 1)) + 2 by A17, XREAL_1:6;
hence  cell ((Gauge (E,n)),(i1 -' 1),j1) c= cell ((Gauge (E,m)),(i -' 1),j) by A1, A4, A16, A17, A7, A15, A5, A14, A12, A19, Th47; ::_thesis: verum
end;

theorem :: JORDAN1A:49
for i, n being Element of NAT
for C being compact non horizontal non vertical Subset of (TOP-REAL 2) st i <= len (Gauge (C,n)) holds
cell ((Gauge (C,n)),i,0) c= UBD C
proof
let i, n be Element of NAT ; ::_thesis: for C being compact non horizontal non vertical Subset of (TOP-REAL 2) st i <= len (Gauge (C,n)) holds
cell ((Gauge (C,n)),i,0) c= UBD C
let C be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( i <= len (Gauge (C,n)) implies cell ((Gauge (C,n)),i,0) c= UBD C )
A1: not C ` is empty by JORDAN2C:66;
assume A2: i <= len (Gauge (C,n)) ; ::_thesis: cell ((Gauge (C,n)),i,0) c= UBD C
then  cell ((Gauge (C,n)),i,0) misses C by JORDAN8:17;
then A3: cell ((Gauge (C,n)),i,0) c= C ` by SUBSET_1:23;
 0 <= width (Gauge (C,n)) ;
then  ( cell ((Gauge (C,n)),i,0) is connected & not cell ((Gauge (C,n)),i,0) is empty ) by A2, Th24, Th25;
then consider W being Subset of (TOP-REAL 2) such that
A4: W is_a_component_of C ` and
A5: cell ((Gauge (C,n)),i,0) c= W by A3, A1, GOBOARD9:3;
 not W is bounded by A2, A5, Th26, RLTOPSP1:42;
then  W is_outside_component_of C by A4, JORDAN2C:def_3;
then  W c= UBD C by JORDAN2C:23;
hence  cell ((Gauge (C,n)),i,0) c= UBD C by A5, XBOOLE_1:1; ::_thesis: verum
end;

theorem :: JORDAN1A:50
for i, n being Element of NAT
for E, C being compact non horizontal non vertical Subset of (TOP-REAL 2) st i <= len (Gauge (E,n)) holds
cell ((Gauge (E,n)),i,(width (Gauge (E,n)))) c= UBD E
proof
let i, n be Element of NAT ; ::_thesis: for E, C being compact non horizontal non vertical Subset of (TOP-REAL 2) st i <= len (Gauge (E,n)) holds
cell ((Gauge (E,n)),i,(width (Gauge (E,n)))) c= UBD E
let E, C be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( i <= len (Gauge (E,n)) implies cell ((Gauge (E,n)),i,(width (Gauge (E,n)))) c= UBD E )
assume A1: i <= len (Gauge (E,n)) ; ::_thesis: cell ((Gauge (E,n)),i,(width (Gauge (E,n)))) c= UBD E
 width (Gauge (E,n)) = len (Gauge (E,n)) by JORDAN8:def_1;
then  cell ((Gauge (E,n)),i,(width (Gauge (E,n)))) misses E by A1, JORDAN8:15;
then A2: cell ((Gauge (E,n)),i,(width (Gauge (E,n)))) c= E ` by SUBSET_1:23;
A3: not E ` is empty by JORDAN2C:66;
 ( cell ((Gauge (E,n)),i,(width (Gauge (E,n)))) is connected & not cell ((Gauge (E,n)),i,(width (Gauge (E,n)))) is empty ) by A1, Th24, Th25;
then consider W being Subset of (TOP-REAL 2) such that
A4: W is_a_component_of E ` and
A5: cell ((Gauge (E,n)),i,(width (Gauge (E,n)))) c= W by A2, A3, GOBOARD9:3;
 not W is bounded by A1, A5, Th27, RLTOPSP1:42;
then  W is_outside_component_of E by A4, JORDAN2C:def_3;
then  W c= UBD E by JORDAN2C:23;
hence  cell ((Gauge (E,n)),i,(width (Gauge (E,n)))) c= UBD E by A5, XBOOLE_1:1; ::_thesis: verum
end;

begin

theorem Th51: :: JORDAN1A:51
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in C holds
north_halfline p meets L~ (Cage (C,n))
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in C holds
north_halfline p meets L~ (Cage (C,n))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st p in C holds
north_halfline p meets L~ (Cage (C,n))
let p be Point of (TOP-REAL 2); ::_thesis: ( p in C implies north_halfline p meets L~ (Cage (C,n)) )
set f = Cage (C,n);
assume A1: p in C ; ::_thesis: north_halfline p meets L~ (Cage (C,n))
set X =  {  q where q is Point of (TOP-REAL 2) : ( q `1 = p `1 & q `2 >= p `2 )  }  ;
A2:  {  q where q is Point of (TOP-REAL 2) : ( q `1 = p `1 & q `2 >= p `2 )  }  = north_halfline p by TOPREAL1:30;
 (max ((N-bound (L~ (Cage (C,n)))),(p `2))) + 1 > (N-bound (L~ (Cage (C,n)))) + 0 by XREAL_1:8, XXREAL_0:25;
then  ( (Cage (C,n)) /. 1 = N-min (L~ (Cage (C,n))) & |[(p `1),((max ((N-bound (L~ (Cage (C,n)))),(p `2))) + 1)]| `2 > N-bound (L~ (Cage (C,n))) ) by EUCLID:52, JORDAN9:32;
then  |[(p `1),((max ((N-bound (L~ (Cage (C,n)))),(p `2))) + 1)]| in LeftComp (Cage (C,n)) by JORDAN2C:113;
then A3: |[(p `1),((max ((N-bound (L~ (Cage (C,n)))),(p `2))) + 1)]| in UBD (L~ (Cage (C,n))) by GOBRD14:36;
 LeftComp (Cage (C,n)) is_outside_component_of L~ (Cage (C,n)) by GOBRD14:34;
then  LeftComp (Cage (C,n)) is_a_component_of (L~ (Cage (C,n))) ` by JORDAN2C:def_3;
then A4: UBD (L~ (Cage (C,n))) is_a_component_of (L~ (Cage (C,n))) ` by GOBRD14:36;
reconsider X =  {  q where q is Point of (TOP-REAL 2) : ( q `1 = p `1 & q `2 >= p `2 )  }  as connected Subset of (TOP-REAL 2) by A2;
A5: ( C c= BDD (L~ (Cage (C,n))) & p in X ) by JORDAN10:12;
 max ((N-bound (L~ (Cage (C,n)))),(p `2)) >= p `2 by XXREAL_0:25;
then  (max ((N-bound (L~ (Cage (C,n)))),(p `2))) + 1 >= (p `2) + 0 by XREAL_1:7;
then A6: |[(p `1),((max ((N-bound (L~ (Cage (C,n)))),(p `2))) + 1)]| `2 >= p `2 by EUCLID:52;
 |[(p `1),((max ((N-bound (L~ (Cage (C,n)))),(p `2))) + 1)]| `1 = p `1 by EUCLID:52;
then  |[(p `1),((max ((N-bound (L~ (Cage (C,n)))),(p `2))) + 1)]| in X by A6;
then A7: X meets UBD (L~ (Cage (C,n))) by A3, XBOOLE_0:3;
assume  not north_halfline p meets L~ (Cage (C,n)) ; ::_thesis: contradiction
then  X c= (L~ (Cage (C,n))) ` by A2, SUBSET_1:23;
then  X c= UBD (L~ (Cage (C,n))) by A7, A4, GOBOARD9:4;
then  p in (BDD (L~ (Cage (C,n)))) /\ (UBD (L~ (Cage (C,n)))) by A1, A5, XBOOLE_0:def_4;
then  BDD (L~ (Cage (C,n))) meets UBD (L~ (Cage (C,n))) by XBOOLE_0:4;
hence  contradiction by JORDAN2C:24; ::_thesis: verum
end;

theorem Th52: :: JORDAN1A:52
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in C holds
east_halfline p meets L~ (Cage (C,n))
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in C holds
east_halfline p meets L~ (Cage (C,n))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st p in C holds
east_halfline p meets L~ (Cage (C,n))
let p be Point of (TOP-REAL 2); ::_thesis: ( p in C implies east_halfline p meets L~ (Cage (C,n)) )
set f = Cage (C,n);
assume A1: p in C ; ::_thesis: east_halfline p meets L~ (Cage (C,n))
set X =  {  q where q is Point of (TOP-REAL 2) : ( q `1 >= p `1 & q `2 = p `2 )  }  ;
A2:  {  q where q is Point of (TOP-REAL 2) : ( q `1 >= p `1 & q `2 = p `2 )  }  = east_halfline p by TOPREAL1:32;
 (max ((E-bound (L~ (Cage (C,n)))),(p `1))) + 1 > (E-bound (L~ (Cage (C,n)))) + 0 by XREAL_1:8, XXREAL_0:25;
then  ( (Cage (C,n)) /. 1 = N-min (L~ (Cage (C,n))) & |[((max ((E-bound (L~ (Cage (C,n)))),(p `1))) + 1),(p `2)]| `1 > E-bound (L~ (Cage (C,n))) ) by EUCLID:52, JORDAN9:32;
then  |[((max ((E-bound (L~ (Cage (C,n)))),(p `1))) + 1),(p `2)]| in LeftComp (Cage (C,n)) by JORDAN2C:111;
then A3: |[((max ((E-bound (L~ (Cage (C,n)))),(p `1))) + 1),(p `2)]| in UBD (L~ (Cage (C,n))) by GOBRD14:36;
 LeftComp (Cage (C,n)) is_outside_component_of L~ (Cage (C,n)) by GOBRD14:34;
then  LeftComp (Cage (C,n)) is_a_component_of (L~ (Cage (C,n))) ` by JORDAN2C:def_3;
then A4: UBD (L~ (Cage (C,n))) is_a_component_of (L~ (Cage (C,n))) ` by GOBRD14:36;
reconsider X =  {  q where q is Point of (TOP-REAL 2) : ( q `1 >= p `1 & q `2 = p `2 )  }  as connected Subset of (TOP-REAL 2) by A2;
A5: ( C c= BDD (L~ (Cage (C,n))) & p in X ) by JORDAN10:12;
 max ((E-bound (L~ (Cage (C,n)))),(p `1)) >= p `1 by XXREAL_0:25;
then  (max ((E-bound (L~ (Cage (C,n)))),(p `1))) + 1 >= (p `1) + 0 by XREAL_1:7;
then A6: |[((max ((E-bound (L~ (Cage (C,n)))),(p `1))) + 1),(p `2)]| `1 >= p `1 by EUCLID:52;
 |[((max ((E-bound (L~ (Cage (C,n)))),(p `1))) + 1),(p `2)]| `2 = p `2 by EUCLID:52;
then  |[((max ((E-bound (L~ (Cage (C,n)))),(p `1))) + 1),(p `2)]| in X by A6;
then A7: X meets UBD (L~ (Cage (C,n))) by A3, XBOOLE_0:3;
assume  not east_halfline p meets L~ (Cage (C,n)) ; ::_thesis: contradiction
then  X c= (L~ (Cage (C,n))) ` by A2, SUBSET_1:23;
then  X c= UBD (L~ (Cage (C,n))) by A7, A4, GOBOARD9:4;
then  p in (BDD (L~ (Cage (C,n)))) /\ (UBD (L~ (Cage (C,n)))) by A1, A5, XBOOLE_0:def_4;
then  BDD (L~ (Cage (C,n))) meets UBD (L~ (Cage (C,n))) by XBOOLE_0:4;
hence  contradiction by JORDAN2C:24; ::_thesis: verum
end;

theorem Th53: :: JORDAN1A:53
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in C holds
south_halfline p meets L~ (Cage (C,n))
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in C holds
south_halfline p meets L~ (Cage (C,n))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st p in C holds
south_halfline p meets L~ (Cage (C,n))
let p be Point of (TOP-REAL 2); ::_thesis: ( p in C implies south_halfline p meets L~ (Cage (C,n)) )
set f = Cage (C,n);
assume A1: p in C ; ::_thesis: south_halfline p meets L~ (Cage (C,n))
set X =  {  q where q is Point of (TOP-REAL 2) : ( q `1 = p `1 & q `2 <= p `2 )  }  ;
A2:  {  q where q is Point of (TOP-REAL 2) : ( q `1 = p `1 & q `2 <= p `2 )  }  = south_halfline p by TOPREAL1:34;
 (min ((S-bound (L~ (Cage (C,n)))),(p `2))) - 1 < (S-bound (L~ (Cage (C,n)))) - 0 by XREAL_1:15, XXREAL_0:17;
then  ( (Cage (C,n)) /. 1 = N-min (L~ (Cage (C,n))) & |[(p `1),((min ((S-bound (L~ (Cage (C,n)))),(p `2))) - 1)]| `2 < S-bound (L~ (Cage (C,n))) ) by EUCLID:52, JORDAN9:32;
then  |[(p `1),((min ((S-bound (L~ (Cage (C,n)))),(p `2))) - 1)]| in LeftComp (Cage (C,n)) by JORDAN2C:112;
then A3: |[(p `1),((min ((S-bound (L~ (Cage (C,n)))),(p `2))) - 1)]| in UBD (L~ (Cage (C,n))) by GOBRD14:36;
 LeftComp (Cage (C,n)) is_outside_component_of L~ (Cage (C,n)) by GOBRD14:34;
then  LeftComp (Cage (C,n)) is_a_component_of (L~ (Cage (C,n))) ` by JORDAN2C:def_3;
then A4: UBD (L~ (Cage (C,n))) is_a_component_of (L~ (Cage (C,n))) ` by GOBRD14:36;
reconsider X =  {  q where q is Point of (TOP-REAL 2) : ( q `1 = p `1 & q `2 <= p `2 )  }  as connected Subset of (TOP-REAL 2) by A2;
A5: ( C c= BDD (L~ (Cage (C,n))) & p in X ) by JORDAN10:12;
 min ((S-bound (L~ (Cage (C,n)))),(p `2)) <= p `2 by XXREAL_0:17;
then  (min ((S-bound (L~ (Cage (C,n)))),(p `2))) - 1 <= (p `2) - 0 by XREAL_1:13;
then A6: |[(p `1),((min ((S-bound (L~ (Cage (C,n)))),(p `2))) - 1)]| `2 <= p `2 by EUCLID:52;
 |[(p `1),((min ((S-bound (L~ (Cage (C,n)))),(p `2))) - 1)]| `1 = p `1 by EUCLID:52;
then  |[(p `1),((min ((S-bound (L~ (Cage (C,n)))),(p `2))) - 1)]| in X by A6;
then A7: X meets UBD (L~ (Cage (C,n))) by A3, XBOOLE_0:3;
assume  not south_halfline p meets L~ (Cage (C,n)) ; ::_thesis: contradiction
then  X c= (L~ (Cage (C,n))) ` by A2, SUBSET_1:23;
then  X c= UBD (L~ (Cage (C,n))) by A7, A4, GOBOARD9:4;
then  p in (BDD (L~ (Cage (C,n)))) /\ (UBD (L~ (Cage (C,n)))) by A1, A5, XBOOLE_0:def_4;
then  BDD (L~ (Cage (C,n))) meets UBD (L~ (Cage (C,n))) by XBOOLE_0:4;
hence  contradiction by JORDAN2C:24; ::_thesis: verum
end;

theorem Th54: :: JORDAN1A:54
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in C holds
west_halfline p meets L~ (Cage (C,n))
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in C holds
west_halfline p meets L~ (Cage (C,n))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st p in C holds
west_halfline p meets L~ (Cage (C,n))
let p be Point of (TOP-REAL 2); ::_thesis: ( p in C implies west_halfline p meets L~ (Cage (C,n)) )
set f = Cage (C,n);
assume A1: p in C ; ::_thesis: west_halfline p meets L~ (Cage (C,n))
set X =  {  q where q is Point of (TOP-REAL 2) : ( q `1 <= p `1 & q `2 = p `2 )  }  ;
A2:  {  q where q is Point of (TOP-REAL 2) : ( q `1 <= p `1 & q `2 = p `2 )  }  = west_halfline p by TOPREAL1:36;
 (min ((W-bound (L~ (Cage (C,n)))),(p `1))) - 1 < (W-bound (L~ (Cage (C,n)))) - 0 by XREAL_1:15, XXREAL_0:17;
then  ( (Cage (C,n)) /. 1 = N-min (L~ (Cage (C,n))) & |[((min ((W-bound (L~ (Cage (C,n)))),(p `1))) - 1),(p `2)]| `1 < W-bound (L~ (Cage (C,n))) ) by EUCLID:52, JORDAN9:32;
then  |[((min ((W-bound (L~ (Cage (C,n)))),(p `1))) - 1),(p `2)]| in LeftComp (Cage (C,n)) by JORDAN2C:110;
then A3: |[((min ((W-bound (L~ (Cage (C,n)))),(p `1))) - 1),(p `2)]| in UBD (L~ (Cage (C,n))) by GOBRD14:36;
 LeftComp (Cage (C,n)) is_outside_component_of L~ (Cage (C,n)) by GOBRD14:34;
then  LeftComp (Cage (C,n)) is_a_component_of (L~ (Cage (C,n))) ` by JORDAN2C:def_3;
then A4: UBD (L~ (Cage (C,n))) is_a_component_of (L~ (Cage (C,n))) ` by GOBRD14:36;
reconsider X =  {  q where q is Point of (TOP-REAL 2) : ( q `1 <= p `1 & q `2 = p `2 )  }  as connected Subset of (TOP-REAL 2) by A2;
A5: ( C c= BDD (L~ (Cage (C,n))) & p in X ) by JORDAN10:12;
 min ((W-bound (L~ (Cage (C,n)))),(p `1)) <= p `1 by XXREAL_0:17;
then  (min ((W-bound (L~ (Cage (C,n)))),(p `1))) - 1 <= (p `1) - 0 by XREAL_1:13;
then A6: |[((min ((W-bound (L~ (Cage (C,n)))),(p `1))) - 1),(p `2)]| `1 <= p `1 by EUCLID:52;
 |[((min ((W-bound (L~ (Cage (C,n)))),(p `1))) - 1),(p `2)]| `2 = p `2 by EUCLID:52;
then  |[((min ((W-bound (L~ (Cage (C,n)))),(p `1))) - 1),(p `2)]| in X by A6;
then A7: X meets UBD (L~ (Cage (C,n))) by A3, XBOOLE_0:3;
assume  not west_halfline p meets L~ (Cage (C,n)) ; ::_thesis: contradiction
then  X c= (L~ (Cage (C,n))) ` by A2, SUBSET_1:23;
then  X c= UBD (L~ (Cage (C,n))) by A7, A4, GOBOARD9:4;
then  p in (BDD (L~ (Cage (C,n)))) /\ (UBD (L~ (Cage (C,n)))) by A1, A5, XBOOLE_0:def_4;
then  BDD (L~ (Cage (C,n))) meets UBD (L~ (Cage (C,n))) by XBOOLE_0:4;
hence  contradiction by JORDAN2C:24; ::_thesis: verum
end;

Lm15:  for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) ex k, t being Element of NAT st
( 1 <= k & k <= len (Cage (C,n)) & 1 <= t & t <= width (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * (1,t) )
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) ex k, t being Element of NAT st
( 1 <= k & k <= len (Cage (C,n)) & 1 <= t & t <= width (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * (1,t) )
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ex k, t being Element of NAT st
( 1 <= k & k <= len (Cage (C,n)) & 1 <= t & t <= width (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * (1,t) )
consider x being set  such that
A1: x in W-most C by XBOOLE_0:def_1;
reconsider x = x as Point of (TOP-REAL 2) by A1;
A2: x in C by A1, XBOOLE_0:def_4;
set X =  {  q where q is Point of (TOP-REAL 2) : ( q `1 <= x `1 & q `2 = x `2 )  }  ;
A3:  {  q where q is Point of (TOP-REAL 2) : ( q `1 <= x `1 & q `2 = x `2 )  }  = west_halfline x by TOPREAL1:36;
then reconsider X =  {  q where q is Point of (TOP-REAL 2) : ( q `1 <= x `1 & q `2 = x `2 )  }  as connected Subset of (TOP-REAL 2) ;
assume A4: for k, t being Element of NAT st 1 <= k & k <= len (Cage (C,n)) & 1 <= t & t <= width (Gauge (C,n)) holds
(Cage (C,n)) /. k <> (Gauge (C,n)) * (1,t) ; ::_thesis: contradiction
A5: now__::_thesis:_x_in_UBD_(L~_(Cage_(C,n)))
 west_halfline x meets L~ (Cage (C,n)) by A2, Th54;
then consider y being set  such that
A6: y in X and
A7: y in L~ (Cage (C,n)) by A3, XBOOLE_0:3;
reconsider y = y as Point of (TOP-REAL 2) by A6;
consider q being Point of (TOP-REAL 2) such that
A8: y = q and
A9: q `1 <= x `1 and
A10: q `2 = x `2 by A6;
consider i being Element of NAT  such that
A11: 1 <= i and
A12: i + 1 <= len (Cage (C,n)) and
A13: y in LSeg ((Cage (C,n)),i) by A7, SPPOL_2:13;
A14: q `1 < x `1
proof
assume  q `1 >= x `1 ; ::_thesis: contradiction
then  q `1 = x `1 by A9, XXREAL_0:1;
then  q = x by A10, TOPREAL3:6;
then  x in C /\ (L~ (Cage (C,n))) by A2, A7, A8, XBOOLE_0:def_4;
then  C meets L~ (Cage (C,n)) by XBOOLE_0:4;
hence  contradiction by JORDAN10:5; ::_thesis: verum
end;
A15: y in LSeg (((Cage (C,n)) /. i),((Cage (C,n)) /. (i + 1))) by A11, A12, A13, TOPREAL1:def_3;
now__::_thesis:_x_in_UBD_(L~_(Cage_(C,n)))
percases ( ((Cage (C,n)) /. i) `1 <= ((Cage (C,n)) /. (i + 1)) `1 or ((Cage (C,n)) /. i) `1 >= ((Cage (C,n)) /. (i + 1)) `1 ) ;
suppose ((Cage (C,n)) /. i) `1 <= ((Cage (C,n)) /. (i + 1)) `1 ; ::_thesis: x in UBD (L~ (Cage (C,n)))
then  ((Cage (C,n)) /. i) `1 <= q `1 by A8, A15, TOPREAL1:3;
then A16: ((Cage (C,n)) /. i) `1 < x `1 by A14, XXREAL_0:2;
A17: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def_1;
A18: i < len (Cage (C,n)) by A12, NAT_1:13;
then  i in Seg (len (Cage (C,n))) by A11, FINSEQ_1:1;
then  i in dom (Cage (C,n)) by FINSEQ_1:def_3;
then consider i1, i2 being Element of NAT  such that
A19: [i1,i2] in Indices (Gauge (C,n)) and
A20: (Cage (C,n)) /. i = (Gauge (C,n)) * (i1,i2) by A17, GOBOARD1:def_9;
A21: 1 <= i2 by A19, MATRIX_1:38;
A22: i1 <= len (Gauge (C,n)) by A19, MATRIX_1:38;
A23: 1 <= i1 by A19, MATRIX_1:38;
A24: i2 <= width (Gauge (C,n)) by A19, MATRIX_1:38;
then A25: i2 <= len (Gauge (C,n)) by JORDAN8:def_1;
x `1 = (W-min C) `1 by A1, PSCOMP_1:31
.= W-bound C by EUCLID:52
.= ((Gauge (C,n)) * (2,i2)) `1 by A21, A25, JORDAN8:11 ;
then  i1 < 1 + 1 by A16, A20, A21, A24, A22, SPRECT_3:13;
then  i1 <= 1 by NAT_1:13;
then  (Cage (C,n)) /. i = (Gauge (C,n)) * (1,i2) by A20, A23, XXREAL_0:1;
hence  x in UBD (L~ (Cage (C,n))) by A4, A11, A18, A21, A24; ::_thesis: verum
end;
suppose ((Cage (C,n)) /. i) `1 >= ((Cage (C,n)) /. (i + 1)) `1 ; ::_thesis: x in UBD (L~ (Cage (C,n)))
then  q `1 >= ((Cage (C,n)) /. (i + 1)) `1 by A8, A15, TOPREAL1:3;
then A26: ((Cage (C,n)) /. (i + 1)) `1 < x `1 by A14, XXREAL_0:2;
A27: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def_1;
A28: i + 1 >= 1 by NAT_1:11;
then  i + 1 in Seg (len (Cage (C,n))) by A12, FINSEQ_1:1;
then  i + 1 in dom (Cage (C,n)) by FINSEQ_1:def_3;
then consider i1, i2 being Element of NAT  such that
A29: [i1,i2] in Indices (Gauge (C,n)) and
A30: (Cage (C,n)) /. (i + 1) = (Gauge (C,n)) * (i1,i2) by A27, GOBOARD1:def_9;
A31: 1 <= i2 by A29, MATRIX_1:38;
A32: i1 <= len (Gauge (C,n)) by A29, MATRIX_1:38;
A33: 1 <= i1 by A29, MATRIX_1:38;
A34: i2 <= width (Gauge (C,n)) by A29, MATRIX_1:38;
then A35: i2 <= len (Gauge (C,n)) by JORDAN8:def_1;
x `1 = (W-min C) `1 by A1, PSCOMP_1:31
.= W-bound C by EUCLID:52
.= ((Gauge (C,n)) * (2,i2)) `1 by A31, A35, JORDAN8:11 ;
then  i1 < 1 + 1 by A26, A30, A31, A34, A32, SPRECT_3:13;
then  i1 <= 1 by NAT_1:13;
then  (Cage (C,n)) /. (i + 1) = (Gauge (C,n)) * (1,i2) by A30, A33, XXREAL_0:1;
hence  x in UBD (L~ (Cage (C,n))) by A4, A12, A28, A31, A34; ::_thesis: verum
end;
end;
end;
hence  x in UBD (L~ (Cage (C,n))) ; ::_thesis: verum
end;
 C c= BDD (L~ (Cage (C,n))) by JORDAN10:12;
then  x in (BDD (L~ (Cage (C,n)))) /\ (UBD (L~ (Cage (C,n)))) by A2, A5, XBOOLE_0:def_4;
then  BDD (L~ (Cage (C,n))) meets UBD (L~ (Cage (C,n))) by XBOOLE_0:4;
hence  contradiction by JORDAN2C:24; ::_thesis: verum
end;

Lm16:  for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) ex k, t being Element of NAT st
( 1 <= k & k <= len (Cage (C,n)) & 1 <= t & t <= len (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * (t,1) )
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) ex k, t being Element of NAT st
( 1 <= k & k <= len (Cage (C,n)) & 1 <= t & t <= len (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * (t,1) )
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ex k, t being Element of NAT st
( 1 <= k & k <= len (Cage (C,n)) & 1 <= t & t <= len (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * (t,1) )
consider x being set  such that
A1: x in S-most C by XBOOLE_0:def_1;
reconsider x = x as Point of (TOP-REAL 2) by A1;
A2: x in C by A1, XBOOLE_0:def_4;
set X =  {  q where q is Point of (TOP-REAL 2) : ( q `1 = x `1 & q `2 <= x `2 )  }  ;
A3:  {  q where q is Point of (TOP-REAL 2) : ( q `1 = x `1 & q `2 <= x `2 )  }  = south_halfline x by TOPREAL1:34;
then reconsider X =  {  q where q is Point of (TOP-REAL 2) : ( q `1 = x `1 & q `2 <= x `2 )  }  as connected Subset of (TOP-REAL 2) ;
assume A4: for k, t being Element of NAT st 1 <= k & k <= len (Cage (C,n)) & 1 <= t & t <= len (Gauge (C,n)) holds
(Cage (C,n)) /. k <> (Gauge (C,n)) * (t,1) ; ::_thesis: contradiction
A5: now__::_thesis:_x_in_UBD_(L~_(Cage_(C,n)))
 south_halfline x meets L~ (Cage (C,n)) by A2, Th53;
then consider y being set  such that
A6: y in X and
A7: y in L~ (Cage (C,n)) by A3, XBOOLE_0:3;
reconsider y = y as Point of (TOP-REAL 2) by A6;
consider q being Point of (TOP-REAL 2) such that
A8: y = q and
A9: q `1 = x `1 and
A10: q `2 <= x `2 by A6;
consider i being Element of NAT  such that
A11: 1 <= i and
A12: i + 1 <= len (Cage (C,n)) and
A13: y in LSeg ((Cage (C,n)),i) by A7, SPPOL_2:13;
A14: q `2 < x `2
proof
assume  q `2 >= x `2 ; ::_thesis: contradiction
then  q `2 = x `2 by A10, XXREAL_0:1;
then  q = x by A9, TOPREAL3:6;
then  x in C /\ (L~ (Cage (C,n))) by A2, A7, A8, XBOOLE_0:def_4;
then  C meets L~ (Cage (C,n)) by XBOOLE_0:4;
hence  contradiction by JORDAN10:5; ::_thesis: verum
end;
A15: y in LSeg (((Cage (C,n)) /. i),((Cage (C,n)) /. (i + 1))) by A11, A12, A13, TOPREAL1:def_3;
now__::_thesis:_x_in_UBD_(L~_(Cage_(C,n)))
percases ( ((Cage (C,n)) /. i) `2 <= ((Cage (C,n)) /. (i + 1)) `2 or ((Cage (C,n)) /. i) `2 >= ((Cage (C,n)) /. (i + 1)) `2 ) ;
suppose ((Cage (C,n)) /. i) `2 <= ((Cage (C,n)) /. (i + 1)) `2 ; ::_thesis: x in UBD (L~ (Cage (C,n)))
then  ((Cage (C,n)) /. i) `2 <= q `2 by A8, A15, TOPREAL1:4;
then A16: ((Cage (C,n)) /. i) `2 < x `2 by A14, XXREAL_0:2;
A17: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def_1;
A18: i < len (Cage (C,n)) by A12, NAT_1:13;
then  i in Seg (len (Cage (C,n))) by A11, FINSEQ_1:1;
then  i in dom (Cage (C,n)) by FINSEQ_1:def_3;
then consider i1, i2 being Element of NAT  such that
A19: [i1,i2] in Indices (Gauge (C,n)) and
A20: (Cage (C,n)) /. i = (Gauge (C,n)) * (i1,i2) by A17, GOBOARD1:def_9;
A21: 1 <= i2 by A19, MATRIX_1:38;
A22: i2 <= width (Gauge (C,n)) by A19, MATRIX_1:38;
A23: ( 1 <= i1 & i1 <= len (Gauge (C,n)) ) by A19, MATRIX_1:38;
x `2 = (S-min C) `2 by A1, PSCOMP_1:55
.= S-bound C by EUCLID:52
.= ((Gauge (C,n)) * (i1,2)) `2 by A23, JORDAN8:13 ;
then  i2 < 1 + 1 by A16, A20, A22, A23, SPRECT_3:12;
then  i2 <= 1 by NAT_1:13;
then  (Cage (C,n)) /. i = (Gauge (C,n)) * (i1,1) by A20, A21, XXREAL_0:1;
hence  x in UBD (L~ (Cage (C,n))) by A4, A11, A18, A23; ::_thesis: verum
end;
suppose ((Cage (C,n)) /. i) `2 >= ((Cage (C,n)) /. (i + 1)) `2 ; ::_thesis: x in UBD (L~ (Cage (C,n)))
then  q `2 >= ((Cage (C,n)) /. (i + 1)) `2 by A8, A15, TOPREAL1:4;
then A24: ((Cage (C,n)) /. (i + 1)) `2 < x `2 by A14, XXREAL_0:2;
A25: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def_1;
A26: i + 1 >= 1 by NAT_1:11;
then  i + 1 in Seg (len (Cage (C,n))) by A12, FINSEQ_1:1;
then  i + 1 in dom (Cage (C,n)) by FINSEQ_1:def_3;
then consider i1, i2 being Element of NAT  such that
A27: [i1,i2] in Indices (Gauge (C,n)) and
A28: (Cage (C,n)) /. (i + 1) = (Gauge (C,n)) * (i1,i2) by A25, GOBOARD1:def_9;
A29: 1 <= i2 by A27, MATRIX_1:38;
A30: i2 <= width (Gauge (C,n)) by A27, MATRIX_1:38;
A31: ( 1 <= i1 & i1 <= len (Gauge (C,n)) ) by A27, MATRIX_1:38;
x `2 = (S-min C) `2 by A1, PSCOMP_1:55
.= S-bound C by EUCLID:52
.= ((Gauge (C,n)) * (i1,2)) `2 by A31, JORDAN8:13 ;
then  i2 < 1 + 1 by A24, A28, A30, A31, SPRECT_3:12;
then  i2 <= 1 by NAT_1:13;
then  (Cage (C,n)) /. (i + 1) = (Gauge (C,n)) * (i1,1) by A28, A29, XXREAL_0:1;
hence  x in UBD (L~ (Cage (C,n))) by A4, A12, A26, A31; ::_thesis: verum
end;
end;
end;
hence  x in UBD (L~ (Cage (C,n))) ; ::_thesis: verum
end;
 C c= BDD (L~ (Cage (C,n))) by JORDAN10:12;
then  x in (BDD (L~ (Cage (C,n)))) /\ (UBD (L~ (Cage (C,n)))) by A2, A5, XBOOLE_0:def_4;
then  BDD (L~ (Cage (C,n))) meets UBD (L~ (Cage (C,n))) by XBOOLE_0:4;
hence  contradiction by JORDAN2C:24; ::_thesis: verum
end;

Lm17:  for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) ex k, t being Element of NAT st
( 1 <= k & k <= len (Cage (C,n)) & 1 <= t & t <= width (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * ((len (Gauge (C,n))),t) )
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) ex k, t being Element of NAT st
( 1 <= k & k <= len (Cage (C,n)) & 1 <= t & t <= width (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * ((len (Gauge (C,n))),t) )
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ex k, t being Element of NAT st
( 1 <= k & k <= len (Cage (C,n)) & 1 <= t & t <= width (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * ((len (Gauge (C,n))),t) )
consider x being set  such that
A1: x in E-most C by XBOOLE_0:def_1;
reconsider x = x as Point of (TOP-REAL 2) by A1;
A2: x in C by A1, XBOOLE_0:def_4;
set X =  {  q where q is Point of (TOP-REAL 2) : ( q `1 >= x `1 & q `2 = x `2 )  }  ;
A3:  {  q where q is Point of (TOP-REAL 2) : ( q `1 >= x `1 & q `2 = x `2 )  }  = east_halfline x by TOPREAL1:32;
then reconsider X =  {  q where q is Point of (TOP-REAL 2) : ( q `1 >= x `1 & q `2 = x `2 )  }  as connected Subset of (TOP-REAL 2) ;
assume A4: for k, t being Element of NAT st 1 <= k & k <= len (Cage (C,n)) & 1 <= t & t <= width (Gauge (C,n)) holds
(Cage (C,n)) /. k <> (Gauge (C,n)) * ((len (Gauge (C,n))),t) ; ::_thesis: contradiction
A5: now__::_thesis:_x_in_UBD_(L~_(Cage_(C,n)))
 east_halfline x meets L~ (Cage (C,n)) by A2, Th52;
then consider y being set  such that
A6: y in X and
A7: y in L~ (Cage (C,n)) by A3, XBOOLE_0:3;
reconsider y = y as Point of (TOP-REAL 2) by A6;
consider q being Point of (TOP-REAL 2) such that
A8: y = q and
A9: q `1 >= x `1 and
A10: q `2 = x `2 by A6;
consider i being Element of NAT  such that
A11: 1 <= i and
A12: i + 1 <= len (Cage (C,n)) and
A13: y in LSeg ((Cage (C,n)),i) by A7, SPPOL_2:13;
A14: y in LSeg (((Cage (C,n)) /. i),((Cage (C,n)) /. (i + 1))) by A11, A12, A13, TOPREAL1:def_3;
A15: q `1 > x `1
proof
assume  q `1 <= x `1 ; ::_thesis: contradiction
then  q `1 = x `1 by A9, XXREAL_0:1;
then  q = x by A10, TOPREAL3:6;
then  x in C /\ (L~ (Cage (C,n))) by A2, A7, A8, XBOOLE_0:def_4;
then  C meets L~ (Cage (C,n)) by XBOOLE_0:4;
hence  contradiction by JORDAN10:5; ::_thesis: verum
end;
A16: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def_1;
A17: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def_1;
A18: 4 <= len (Gauge (C,n)) by JORDAN8:10;
A19: 1 <= i + 1 by A11, NAT_1:13;
then  i + 1 in Seg (len (Cage (C,n))) by A12, FINSEQ_1:1;
then  i + 1 in dom (Cage (C,n)) by FINSEQ_1:def_3;
then consider l1, l2 being Element of NAT  such that
A20: [l1,l2] in Indices (Gauge (C,n)) and
A21: (Cage (C,n)) /. (i + 1) = (Gauge (C,n)) * (l1,l2) by A16, GOBOARD1:def_9;
A22: 1 <= l2 by A20, MATRIX_1:38;
A23: l2 <= width (Gauge (C,n)) by A20, MATRIX_1:38;
then A24: l2 <= len (Gauge (C,n)) by JORDAN8:def_1;
A25: x `1 = (E-min C) `1 by A1, PSCOMP_1:47
.= E-bound C by EUCLID:52
.= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),l2)) `1 by A22, A24, JORDAN8:12 ;
A26: l1 <= len (Gauge (C,n)) by A20, MATRIX_1:38;
A27: 1 <= l1 by A20, MATRIX_1:38;
A28: (len (Gauge (C,n))) -' 1 <= len (Gauge (C,n)) by NAT_D:35;
A29: i < len (Cage (C,n)) by A12, NAT_1:13;
then  i in Seg (len (Cage (C,n))) by A11, FINSEQ_1:1;
then  i in dom (Cage (C,n)) by FINSEQ_1:def_3;
then consider i1, i2 being Element of NAT  such that
A30: [i1,i2] in Indices (Gauge (C,n)) and
A31: (Cage (C,n)) /. i = (Gauge (C,n)) * (i1,i2) by A17, GOBOARD1:def_9;
A32: 1 <= i2 by A30, MATRIX_1:38;
A33: i1 <= len (Gauge (C,n)) by A30, MATRIX_1:38;
A34: 1 <= i1 by A30, MATRIX_1:38;
A35: i2 <= width (Gauge (C,n)) by A30, MATRIX_1:38;
then A36: i2 <= len (Gauge (C,n)) by JORDAN8:def_1;
A37: x `1 = (E-min C) `1 by A1, PSCOMP_1:47
.= E-bound C by EUCLID:52
.= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),i2)) `1 by A32, A36, JORDAN8:12 ;
now__::_thesis:_contradiction
percases ( ((Cage (C,n)) /. i) `1 >= ((Cage (C,n)) /. (i + 1)) `1 or ((Cage (C,n)) /. i) `1 <= ((Cage (C,n)) /. (i + 1)) `1 ) ;
suppose ((Cage (C,n)) /. i) `1 >= ((Cage (C,n)) /. (i + 1)) `1 ; ::_thesis: contradiction
then  ((Cage (C,n)) /. i) `1 >= q `1 by A8, A14, TOPREAL1:3;
then  ((Cage (C,n)) /. i) `1 > x `1 by A15, XXREAL_0:2;
then  i1 > (len (Gauge (C,n))) -' 1 by A31, A32, A35, A34, A37, A28, SPRECT_3:13;
then  i1 >= ((len (Gauge (C,n))) -' 1) + 1 by NAT_1:13;
then  i1 >= len (Gauge (C,n)) by A18, XREAL_1:235, XXREAL_0:2;
then  (Cage (C,n)) /. i = (Gauge (C,n)) * ((len (Gauge (C,n))),i2) by A31, A33, XXREAL_0:1;
hence  contradiction by A4, A11, A29, A32, A35; ::_thesis: verum
end;
suppose ((Cage (C,n)) /. i) `1 <= ((Cage (C,n)) /. (i + 1)) `1 ; ::_thesis: contradiction
then  q `1 <= ((Cage (C,n)) /. (i + 1)) `1 by A8, A14, TOPREAL1:3;
then  ((Cage (C,n)) /. (i + 1)) `1 > x `1 by A15, XXREAL_0:2;
then  l1 > (len (Gauge (C,n))) -' 1 by A21, A22, A23, A27, A25, A28, SPRECT_3:13;
then  l1 >= ((len (Gauge (C,n))) -' 1) + 1 by NAT_1:13;
then  l1 >= len (Gauge (C,n)) by A18, XREAL_1:235, XXREAL_0:2;
then  (Cage (C,n)) /. (i + 1) = (Gauge (C,n)) * ((len (Gauge (C,n))),l2) by A21, A26, XXREAL_0:1;
hence  contradiction by A4, A12, A19, A22, A23; ::_thesis: verum
end;
end;
end;
hence  x in UBD (L~ (Cage (C,n))) ; ::_thesis: verum
end;
 C c= BDD (L~ (Cage (C,n))) by JORDAN10:12;
then  x in (BDD (L~ (Cage (C,n)))) /\ (UBD (L~ (Cage (C,n)))) by A2, A5, XBOOLE_0:def_4;
then  BDD (L~ (Cage (C,n))) meets UBD (L~ (Cage (C,n))) by XBOOLE_0:4;
hence  contradiction by JORDAN2C:24; ::_thesis: verum
end;

theorem Th55: :: JORDAN1A:55
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) ex k, t being Element of NAT st
( 1 <= k & k < len (Cage (C,n)) & 1 <= t & t <= width (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * (1,t) )
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) ex k, t being Element of NAT st
( 1 <= k & k < len (Cage (C,n)) & 1 <= t & t <= width (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * (1,t) )
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ex k, t being Element of NAT st
( 1 <= k & k < len (Cage (C,n)) & 1 <= t & t <= width (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * (1,t) )
consider k, t being Element of NAT  such that
A1: 1 <= k and
A2: k <= len (Cage (C,n)) and
A3: ( 1 <= t & t <= width (Gauge (C,n)) ) and
A4: (Cage (C,n)) /. k = (Gauge (C,n)) * (1,t) by Lm15;
percases ( k < len (Cage (C,n)) or k = len (Cage (C,n)) ) by A2, XXREAL_0:1;
suppose k < len (Cage (C,n)) ; ::_thesis: ex k, t being Element of NAT st
( 1 <= k & k < len (Cage (C,n)) & 1 <= t & t <= width (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * (1,t) )
hence  ex k, t being Element of NAT st
( 1 <= k & k < len (Cage (C,n)) & 1 <= t & t <= width (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * (1,t) ) by A1, A3, A4; ::_thesis: verum
end;
supposeA5: k = len (Cage (C,n)) ; ::_thesis: ex k, t being Element of NAT st
( 1 <= k & k < len (Cage (C,n)) & 1 <= t & t <= width (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * (1,t) )
take  1 ; ::_thesis: ex t being Element of NAT st
( 1 <= 1 & 1 < len (Cage (C,n)) & 1 <= t & t <= width (Gauge (C,n)) & (Cage (C,n)) /. 1 = (Gauge (C,n)) * (1,t) )
take  t ; ::_thesis: ( 1 <= 1 & 1 < len (Cage (C,n)) & 1 <= t & t <= width (Gauge (C,n)) & (Cage (C,n)) /. 1 = (Gauge (C,n)) * (1,t) )
thus  ( 1 <= 1 & 1 < len (Cage (C,n)) ) by GOBOARD7:34, XXREAL_0:2; ::_thesis: ( 1 <= t & t <= width (Gauge (C,n)) & (Cage (C,n)) /. 1 = (Gauge (C,n)) * (1,t) )
thus  ( 1 <= t & t <= width (Gauge (C,n)) ) by A3; ::_thesis: (Cage (C,n)) /. 1 = (Gauge (C,n)) * (1,t)
thus  (Cage (C,n)) /. 1 = (Gauge (C,n)) * (1,t) by A4, A5, FINSEQ_6:def_1; ::_thesis: verum
end;
end;
end;

theorem Th56: :: JORDAN1A:56
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) ex k, t being Element of NAT st
( 1 <= k & k < len (Cage (C,n)) & 1 <= t & t <= len (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * (t,1) )
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) ex k, t being Element of NAT st
( 1 <= k & k < len (Cage (C,n)) & 1 <= t & t <= len (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * (t,1) )
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ex k, t being Element of NAT st
( 1 <= k & k < len (Cage (C,n)) & 1 <= t & t <= len (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * (t,1) )
consider k, t being Element of NAT  such that
A1: 1 <= k and
A2: k <= len (Cage (C,n)) and
A3: ( 1 <= t & t <= len (Gauge (C,n)) ) and
A4: (Cage (C,n)) /. k = (Gauge (C,n)) * (t,1) by Lm16;
percases ( k < len (Cage (C,n)) or k = len (Cage (C,n)) ) by A2, XXREAL_0:1;
suppose k < len (Cage (C,n)) ; ::_thesis: ex k, t being Element of NAT st
( 1 <= k & k < len (Cage (C,n)) & 1 <= t & t <= len (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * (t,1) )
hence  ex k, t being Element of NAT st
( 1 <= k & k < len (Cage (C,n)) & 1 <= t & t <= len (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * (t,1) ) by A1, A3, A4; ::_thesis: verum
end;
supposeA5: k = len (Cage (C,n)) ; ::_thesis: ex k, t being Element of NAT st
( 1 <= k & k < len (Cage (C,n)) & 1 <= t & t <= len (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * (t,1) )
take  1 ; ::_thesis: ex t being Element of NAT st
( 1 <= 1 & 1 < len (Cage (C,n)) & 1 <= t & t <= len (Gauge (C,n)) & (Cage (C,n)) /. 1 = (Gauge (C,n)) * (t,1) )
take  t ; ::_thesis: ( 1 <= 1 & 1 < len (Cage (C,n)) & 1 <= t & t <= len (Gauge (C,n)) & (Cage (C,n)) /. 1 = (Gauge (C,n)) * (t,1) )
thus  ( 1 <= 1 & 1 < len (Cage (C,n)) ) by GOBOARD7:34, XXREAL_0:2; ::_thesis: ( 1 <= t & t <= len (Gauge (C,n)) & (Cage (C,n)) /. 1 = (Gauge (C,n)) * (t,1) )
thus  ( 1 <= t & t <= len (Gauge (C,n)) ) by A3; ::_thesis: (Cage (C,n)) /. 1 = (Gauge (C,n)) * (t,1)
thus  (Cage (C,n)) /. 1 = (Gauge (C,n)) * (t,1) by A4, A5, FINSEQ_6:def_1; ::_thesis: verum
end;
end;
end;

theorem Th57: :: JORDAN1A:57
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) ex k, t being Element of NAT st
( 1 <= k & k < len (Cage (C,n)) & 1 <= t & t <= width (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * ((len (Gauge (C,n))),t) )
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) ex k, t being Element of NAT st
( 1 <= k & k < len (Cage (C,n)) & 1 <= t & t <= width (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * ((len (Gauge (C,n))),t) )
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ex k, t being Element of NAT st
( 1 <= k & k < len (Cage (C,n)) & 1 <= t & t <= width (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * ((len (Gauge (C,n))),t) )
consider k, t being Element of NAT  such that
A1: 1 <= k and
A2: k <= len (Cage (C,n)) and
A3: ( 1 <= t & t <= width (Gauge (C,n)) ) and
A4: (Cage (C,n)) /. k = (Gauge (C,n)) * ((len (Gauge (C,n))),t) by Lm17;
percases ( k < len (Cage (C,n)) or k = len (Cage (C,n)) ) by A2, XXREAL_0:1;
suppose k < len (Cage (C,n)) ; ::_thesis: ex k, t being Element of NAT st
( 1 <= k & k < len (Cage (C,n)) & 1 <= t & t <= width (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * ((len (Gauge (C,n))),t) )
hence  ex k, t being Element of NAT st
( 1 <= k & k < len (Cage (C,n)) & 1 <= t & t <= width (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * ((len (Gauge (C,n))),t) ) by A1, A3, A4; ::_thesis: verum
end;
supposeA5: k = len (Cage (C,n)) ; ::_thesis: ex k, t being Element of NAT st
( 1 <= k & k < len (Cage (C,n)) & 1 <= t & t <= width (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * ((len (Gauge (C,n))),t) )
take  1 ; ::_thesis: ex t being Element of NAT st
( 1 <= 1 & 1 < len (Cage (C,n)) & 1 <= t & t <= width (Gauge (C,n)) & (Cage (C,n)) /. 1 = (Gauge (C,n)) * ((len (Gauge (C,n))),t) )
take  t ; ::_thesis: ( 1 <= 1 & 1 < len (Cage (C,n)) & 1 <= t & t <= width (Gauge (C,n)) & (Cage (C,n)) /. 1 = (Gauge (C,n)) * ((len (Gauge (C,n))),t) )
thus  ( 1 <= 1 & 1 < len (Cage (C,n)) ) by GOBOARD7:34, XXREAL_0:2; ::_thesis: ( 1 <= t & t <= width (Gauge (C,n)) & (Cage (C,n)) /. 1 = (Gauge (C,n)) * ((len (Gauge (C,n))),t) )
thus  ( 1 <= t & t <= width (Gauge (C,n)) ) by A3; ::_thesis: (Cage (C,n)) /. 1 = (Gauge (C,n)) * ((len (Gauge (C,n))),t)
thus  (Cage (C,n)) /. 1 = (Gauge (C,n)) * ((len (Gauge (C,n))),t) by A4, A5, FINSEQ_6:def_1; ::_thesis: verum
end;
end;
end;

theorem Th58: :: JORDAN1A:58
for k, n, t being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= k & k <= len (Cage (C,n)) & 1 <= t & t <= len (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * (t,(width (Gauge (C,n)))) holds
(Cage (C,n)) /. k in N-most (L~ (Cage (C,n)))
proof
let k, n, t be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= k & k <= len (Cage (C,n)) & 1 <= t & t <= len (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * (t,(width (Gauge (C,n)))) holds
(Cage (C,n)) /. k in N-most (L~ (Cage (C,n)))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( 1 <= k & k <= len (Cage (C,n)) & 1 <= t & t <= len (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * (t,(width (Gauge (C,n)))) implies (Cage (C,n)) /. k in N-most (L~ (Cage (C,n))) )
assume that
A1: ( 1 <= k & k <= len (Cage (C,n)) ) and
A2: ( 1 <= t & t <= len (Gauge (C,n)) ) and
A3: (Cage (C,n)) /. k = (Gauge (C,n)) * (t,(width (Gauge (C,n)))) ; ::_thesis: (Cage (C,n)) /. k in N-most (L~ (Cage (C,n)))
 Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def_1;
then A4: ((Gauge (C,n)) * (t,(width (Gauge (C,n))))) `2 >= N-bound (L~ (Cage (C,n))) by A2, Th20;
 len (Cage (C,n)) >= 2 by GOBOARD7:34, XXREAL_0:2;
then A5: (Cage (C,n)) /. k in L~ (Cage (C,n)) by A1, TOPREAL3:39;
then  N-bound (L~ (Cage (C,n))) >= ((Cage (C,n)) /. k) `2 by PSCOMP_1:24;
hence  (Cage (C,n)) /. k in N-most (L~ (Cage (C,n))) by A3, A5, A4, SPRECT_2:10, XXREAL_0:1; ::_thesis: verum
end;

theorem Th59: :: JORDAN1A:59
for k, n, t being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= k & k <= len (Cage (C,n)) & 1 <= t & t <= width (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * (1,t) holds
(Cage (C,n)) /. k in W-most (L~ (Cage (C,n)))
proof
let k, n, t be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= k & k <= len (Cage (C,n)) & 1 <= t & t <= width (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * (1,t) holds
(Cage (C,n)) /. k in W-most (L~ (Cage (C,n)))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( 1 <= k & k <= len (Cage (C,n)) & 1 <= t & t <= width (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * (1,t) implies (Cage (C,n)) /. k in W-most (L~ (Cage (C,n))) )
assume that
A1: ( 1 <= k & k <= len (Cage (C,n)) ) and
A2: ( 1 <= t & t <= width (Gauge (C,n)) ) and
A3: (Cage (C,n)) /. k = (Gauge (C,n)) * (1,t) ; ::_thesis: (Cage (C,n)) /. k in W-most (L~ (Cage (C,n)))
 Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def_1;
then A4: ((Gauge (C,n)) * (1,t)) `1 <= W-bound (L~ (Cage (C,n))) by A2, Th21;
 len (Cage (C,n)) >= 2 by GOBOARD7:34, XXREAL_0:2;
then A5: (Cage (C,n)) /. k in L~ (Cage (C,n)) by A1, TOPREAL3:39;
then  W-bound (L~ (Cage (C,n))) <= ((Cage (C,n)) /. k) `1 by PSCOMP_1:24;
hence  (Cage (C,n)) /. k in W-most (L~ (Cage (C,n))) by A3, A5, A4, SPRECT_2:12, XXREAL_0:1; ::_thesis: verum
end;

theorem Th60: :: JORDAN1A:60
for k, n, t being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= k & k <= len (Cage (C,n)) & 1 <= t & t <= len (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * (t,1) holds
(Cage (C,n)) /. k in S-most (L~ (Cage (C,n)))
proof
let k, n, t be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= k & k <= len (Cage (C,n)) & 1 <= t & t <= len (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * (t,1) holds
(Cage (C,n)) /. k in S-most (L~ (Cage (C,n)))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( 1 <= k & k <= len (Cage (C,n)) & 1 <= t & t <= len (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * (t,1) implies (Cage (C,n)) /. k in S-most (L~ (Cage (C,n))) )
assume that
A1: ( 1 <= k & k <= len (Cage (C,n)) ) and
A2: ( 1 <= t & t <= len (Gauge (C,n)) ) and
A3: (Cage (C,n)) /. k = (Gauge (C,n)) * (t,1) ; ::_thesis: (Cage (C,n)) /. k in S-most (L~ (Cage (C,n)))
 Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def_1;
then A4: ((Gauge (C,n)) * (t,1)) `2 <= S-bound (L~ (Cage (C,n))) by A2, Th22;
 len (Cage (C,n)) >= 2 by GOBOARD7:34, XXREAL_0:2;
then A5: (Cage (C,n)) /. k in L~ (Cage (C,n)) by A1, TOPREAL3:39;
then  S-bound (L~ (Cage (C,n))) <= ((Cage (C,n)) /. k) `2 by PSCOMP_1:24;
hence  (Cage (C,n)) /. k in S-most (L~ (Cage (C,n))) by A3, A5, A4, SPRECT_2:11, XXREAL_0:1; ::_thesis: verum
end;

theorem Th61: :: JORDAN1A:61
for k, n, t being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= k & k <= len (Cage (C,n)) & 1 <= t & t <= width (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * ((len (Gauge (C,n))),t) holds
(Cage (C,n)) /. k in E-most (L~ (Cage (C,n)))
proof
let k, n, t be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= k & k <= len (Cage (C,n)) & 1 <= t & t <= width (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * ((len (Gauge (C,n))),t) holds
(Cage (C,n)) /. k in E-most (L~ (Cage (C,n)))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( 1 <= k & k <= len (Cage (C,n)) & 1 <= t & t <= width (Gauge (C,n)) & (Cage (C,n)) /. k = (Gauge (C,n)) * ((len (Gauge (C,n))),t) implies (Cage (C,n)) /. k in E-most (L~ (Cage (C,n))) )
assume that
A1: ( 1 <= k & k <= len (Cage (C,n)) ) and
A2: ( 1 <= t & t <= width (Gauge (C,n)) ) and
A3: (Cage (C,n)) /. k = (Gauge (C,n)) * ((len (Gauge (C,n))),t) ; ::_thesis: (Cage (C,n)) /. k in E-most (L~ (Cage (C,n)))
 Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def_1;
then A4: ((Gauge (C,n)) * ((len (Gauge (C,n))),t)) `1 >= E-bound (L~ (Cage (C,n))) by A2, Th23;
 len (Cage (C,n)) >= 2 by GOBOARD7:34, XXREAL_0:2;
then A5: (Cage (C,n)) /. k in L~ (Cage (C,n)) by A1, TOPREAL3:39;
then  E-bound (L~ (Cage (C,n))) >= ((Cage (C,n)) /. k) `1 by PSCOMP_1:24;
hence  (Cage (C,n)) /. k in E-most (L~ (Cage (C,n))) by A3, A5, A4, SPRECT_2:13, XXREAL_0:1; ::_thesis: verum
end;

theorem Th62: :: JORDAN1A:62
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds W-bound (L~ (Cage (C,n))) = (W-bound C) - (((E-bound C) - (W-bound C)) / (2 |^ n))
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds W-bound (L~ (Cage (C,n))) = (W-bound C) - (((E-bound C) - (W-bound C)) / (2 |^ n))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: W-bound (L~ (Cage (C,n))) = (W-bound C) - (((E-bound C) - (W-bound C)) / (2 |^ n))
set a = N-bound C;
set s = S-bound C;
set w = W-bound C;
set e = E-bound C;
set f = Cage (C,n);
set G = Gauge (C,n);
consider p, q being Element of NAT  such that
A1: ( 1 <= p & p < len (Cage (C,n)) ) and
A2: ( 1 <= q & q <= width (Gauge (C,n)) ) and
A3: (Cage (C,n)) /. p = (Gauge (C,n)) * (1,q) by Th55;
 (Cage (C,n)) /. p in W-most (L~ (Cage (C,n))) by A1, A2, A3, Th59;
then A4: ((Cage (C,n)) /. p) `1 = (W-min (L~ (Cage (C,n)))) `1 by PSCOMP_1:31;
 4 <= len (Gauge (C,n)) by JORDAN8:10;
then  1 <= len (Gauge (C,n)) by XXREAL_0:2;
then A5: [1,q] in Indices (Gauge (C,n)) by A2, MATRIX_1:36;
thus  W-bound (L~ (Cage (C,n))) = (W-min (L~ (Cage (C,n)))) `1 by EUCLID:52
.= |[((W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * (1 - 2))),((S-bound C) + ((((N-bound C) - (S-bound C)) / (2 |^ n)) * (q - 2)))]| `1 by A3, A4, A5, JORDAN8:def_1
.= (W-bound C) - (((E-bound C) - (W-bound C)) / (2 |^ n)) by EUCLID:52 ; ::_thesis: verum
end;

theorem Th63: :: JORDAN1A:63
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds S-bound (L~ (Cage (C,n))) = (S-bound C) - (((N-bound C) - (S-bound C)) / (2 |^ n))
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds S-bound (L~ (Cage (C,n))) = (S-bound C) - (((N-bound C) - (S-bound C)) / (2 |^ n))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: S-bound (L~ (Cage (C,n))) = (S-bound C) - (((N-bound C) - (S-bound C)) / (2 |^ n))
set a = N-bound C;
set s = S-bound C;
set w = W-bound C;
set e = E-bound C;
set f = Cage (C,n);
set G = Gauge (C,n);
consider p, q being Element of NAT  such that
A1: ( 1 <= p & p < len (Cage (C,n)) ) and
A2: ( 1 <= q & q <= len (Gauge (C,n)) ) and
A3: (Cage (C,n)) /. p = (Gauge (C,n)) * (q,1) by Th56;
 (Cage (C,n)) /. p in S-most (L~ (Cage (C,n))) by A1, A2, A3, Th60;
then A4: ((Cage (C,n)) /. p) `2 = (S-min (L~ (Cage (C,n)))) `2 by PSCOMP_1:55;
 4 <= len (Gauge (C,n)) by JORDAN8:10;
then  ( len (Gauge (C,n)) = width (Gauge (C,n)) & 1 <= len (Gauge (C,n)) ) by JORDAN8:def_1, XXREAL_0:2;
then A5: [q,1] in Indices (Gauge (C,n)) by A2, MATRIX_1:36;
thus  S-bound (L~ (Cage (C,n))) = (S-min (L~ (Cage (C,n)))) `2 by EUCLID:52
.= |[((W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * (q - 2))),((S-bound C) + ((((N-bound C) - (S-bound C)) / (2 |^ n)) * (1 - 2)))]| `2 by A3, A4, A5, JORDAN8:def_1
.= (S-bound C) - (((N-bound C) - (S-bound C)) / (2 |^ n)) by EUCLID:52 ; ::_thesis: verum
end;

theorem Th64: :: JORDAN1A:64
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds E-bound (L~ (Cage (C,n))) = (E-bound C) + (((E-bound C) - (W-bound C)) / (2 |^ n))
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds E-bound (L~ (Cage (C,n))) = (E-bound C) + (((E-bound C) - (W-bound C)) / (2 |^ n))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: E-bound (L~ (Cage (C,n))) = (E-bound C) + (((E-bound C) - (W-bound C)) / (2 |^ n))
set a = N-bound C;
set s = S-bound C;
set w = W-bound C;
set e = E-bound C;
set f = Cage (C,n);
set G = Gauge (C,n);
consider p, q being Element of NAT  such that
A1: ( 1 <= p & p < len (Cage (C,n)) ) and
A2: ( 1 <= q & q <= width (Gauge (C,n)) ) and
A3: (Cage (C,n)) /. p = (Gauge (C,n)) * ((len (Gauge (C,n))),q) by Th57;
 (Cage (C,n)) /. p in E-most (L~ (Cage (C,n))) by A1, A2, A3, Th61;
then A4: ((Cage (C,n)) /. p) `1 = (E-min (L~ (Cage (C,n)))) `1 by PSCOMP_1:47;
 4 <= len (Gauge (C,n)) by JORDAN8:10;
then  1 <= len (Gauge (C,n)) by XXREAL_0:2;
then A5: [(len (Gauge (C,n))),q] in Indices (Gauge (C,n)) by A2, MATRIX_1:36;
thus  E-bound (L~ (Cage (C,n))) = (E-min (L~ (Cage (C,n)))) `1 by EUCLID:52
.= |[((W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * ((len (Gauge (C,n))) - 2))),((S-bound C) + ((((N-bound C) - (S-bound C)) / (2 |^ n)) * (q - 2)))]| `1 by A3, A4, A5, JORDAN8:def_1
.= (W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * ((len (Gauge (C,n))) - 2)) by EUCLID:52
.= (E-bound C) + (((E-bound C) - (W-bound C)) / (2 |^ n)) by Lm10 ; ::_thesis: verum
end;

theorem :: JORDAN1A:65
for n, m being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (N-bound (L~ (Cage (C,n)))) + (S-bound (L~ (Cage (C,n)))) = (N-bound (L~ (Cage (C,m)))) + (S-bound (L~ (Cage (C,m))))
proof
let n, m be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (N-bound (L~ (Cage (C,n)))) + (S-bound (L~ (Cage (C,n)))) = (N-bound (L~ (Cage (C,m)))) + (S-bound (L~ (Cage (C,m))))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: (N-bound (L~ (Cage (C,n)))) + (S-bound (L~ (Cage (C,n)))) = (N-bound (L~ (Cage (C,m)))) + (S-bound (L~ (Cage (C,m))))
thus (N-bound (L~ (Cage (C,n)))) + (S-bound (L~ (Cage (C,n)))) = ((N-bound C) + (((N-bound C) - (S-bound C)) / (2 |^ n))) + (S-bound (L~ (Cage (C,n)))) by JORDAN10:6
.= ((N-bound C) + (((N-bound C) - (S-bound C)) / (2 |^ n))) + ((S-bound C) - (((N-bound C) - (S-bound C)) / (2 |^ n))) by Th63
.= ((N-bound C) + (((N-bound C) - (S-bound C)) / (2 |^ m))) + ((S-bound C) - (((N-bound C) - (S-bound C)) / (2 |^ m)))
.= ((N-bound C) + (((N-bound C) - (S-bound C)) / (2 |^ m))) + (S-bound (L~ (Cage (C,m)))) by Th63
.= (N-bound (L~ (Cage (C,m)))) + (S-bound (L~ (Cage (C,m)))) by JORDAN10:6 ; ::_thesis: verum
end;

theorem :: JORDAN1A:66
for n, m being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (E-bound (L~ (Cage (C,n)))) + (W-bound (L~ (Cage (C,n)))) = (E-bound (L~ (Cage (C,m)))) + (W-bound (L~ (Cage (C,m))))
proof
let n, m be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (E-bound (L~ (Cage (C,n)))) + (W-bound (L~ (Cage (C,n)))) = (E-bound (L~ (Cage (C,m)))) + (W-bound (L~ (Cage (C,m))))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: (E-bound (L~ (Cage (C,n)))) + (W-bound (L~ (Cage (C,n)))) = (E-bound (L~ (Cage (C,m)))) + (W-bound (L~ (Cage (C,m))))
thus (E-bound (L~ (Cage (C,n)))) + (W-bound (L~ (Cage (C,n)))) = ((E-bound C) + (((E-bound C) - (W-bound C)) / (2 |^ n))) + (W-bound (L~ (Cage (C,n)))) by Th64
.= ((E-bound C) + (((E-bound C) - (W-bound C)) / (2 |^ n))) + ((W-bound C) - (((E-bound C) - (W-bound C)) / (2 |^ n))) by Th62
.= ((E-bound C) + (((E-bound C) - (W-bound C)) / (2 |^ m))) + ((W-bound C) - (((E-bound C) - (W-bound C)) / (2 |^ m)))
.= ((E-bound C) + (((E-bound C) - (W-bound C)) / (2 |^ m))) + (W-bound (L~ (Cage (C,m)))) by Th62
.= (E-bound (L~ (Cage (C,m)))) + (W-bound (L~ (Cage (C,m)))) by Th64 ; ::_thesis: verum
end;

theorem :: JORDAN1A:67
for i, j being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) st i < j holds
E-bound (L~ (Cage (C,j))) < E-bound (L~ (Cage (C,i)))
proof
let i, j be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) st i < j holds
E-bound (L~ (Cage (C,j))) < E-bound (L~ (Cage (C,i)))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( i < j implies E-bound (L~ (Cage (C,j))) < E-bound (L~ (Cage (C,i))) )
assume A1: i < j ; ::_thesis: E-bound (L~ (Cage (C,j))) < E-bound (L~ (Cage (C,i)))
defpred S1[ Element of NAT ] means ( i < $1 implies E-bound (L~ (Cage (C,$1))) < E-bound (L~ (Cage (C,i))) );
A2: for n being Element of NAT st S1[n] holds
S1[n + 1]
proof
let n be Element of NAT ; ::_thesis: ( S1[n] implies S1[n + 1] )
assume A3: S1[n] ; ::_thesis: S1[n + 1]
set j = n + 1;
set a = E-bound C;
set s = W-bound C;
A4: ((E-bound C) + (((E-bound C) - (W-bound C)) / (2 |^ (n + 1)))) - ((E-bound C) + (((E-bound C) - (W-bound C)) / (2 |^ n))) = (0 + (((E-bound C) - (W-bound C)) / (2 |^ (n + 1)))) - (((E-bound C) - (W-bound C)) / (2 |^ n))
.= (((E-bound C) - (W-bound C)) / ((2 |^ n) * 2)) - ((((E-bound C) - (W-bound C)) / (2 |^ n)) / (2 / 2)) by NEWTON:6
.= (((E-bound C) - (W-bound C)) / ((2 |^ n) * 2)) - ((((E-bound C) - (W-bound C)) * 2) / ((2 |^ n) * 2)) by XCMPLX_1:84
.= (((E-bound C) - (W-bound C)) - ((2 * (E-bound C)) - (2 * (W-bound C)))) / ((2 |^ n) * 2) by XCMPLX_1:120
.= ((W-bound C) - (E-bound C)) / ((2 |^ n) * 2) ;
 2 |^ n > 0 by NEWTON:83;
then A5: (2 |^ n) * 2 > 0 * 2 by XREAL_1:68;
A6: ( E-bound (L~ (Cage (C,n))) = (E-bound C) + (((E-bound C) - (W-bound C)) / (2 |^ n)) & E-bound (L~ (Cage (C,(n + 1)))) = (E-bound C) + (((E-bound C) - (W-bound C)) / (2 |^ (n + 1))) ) by Th64;
 (W-bound C) - (E-bound C) < 0 by SPRECT_1:31, XREAL_1:49;
then  0 > ((E-bound C) + (((E-bound C) - (W-bound C)) / (2 |^ (n + 1)))) - ((E-bound C) + (((E-bound C) - (W-bound C)) / (2 |^ n))) by A5, A4, XREAL_1:141;
then A7: E-bound (L~ (Cage (C,(n + 1)))) < E-bound (L~ (Cage (C,n))) by A6, XREAL_1:48;
assume  i < n + 1 ; ::_thesis: E-bound (L~ (Cage (C,(n + 1)))) < E-bound (L~ (Cage (C,i)))
then  i <= n by NAT_1:13;
hence  E-bound (L~ (Cage (C,(n + 1)))) < E-bound (L~ (Cage (C,i))) by A3, A7, XXREAL_0:1, XXREAL_0:2; ::_thesis: verum
end;
A8: S1[ 0 ] ;
 for n being Element of NAT holds S1[n] from NAT_1:sch_1(A8, A2);
hence  E-bound (L~ (Cage (C,j))) < E-bound (L~ (Cage (C,i))) by A1; ::_thesis: verum
end;

theorem :: JORDAN1A:68
for i, j being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) st i < j holds
W-bound (L~ (Cage (C,i))) < W-bound (L~ (Cage (C,j)))
proof
let i, j be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) st i < j holds
W-bound (L~ (Cage (C,i))) < W-bound (L~ (Cage (C,j)))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( i < j implies W-bound (L~ (Cage (C,i))) < W-bound (L~ (Cage (C,j))) )
assume A1: i < j ; ::_thesis: W-bound (L~ (Cage (C,i))) < W-bound (L~ (Cage (C,j)))
defpred S1[ Element of NAT ] means ( i < $1 implies W-bound (L~ (Cage (C,i))) < W-bound (L~ (Cage (C,$1))) );
A2: for n being Element of NAT st S1[n] holds
S1[n + 1]
proof
let n be Element of NAT ; ::_thesis: ( S1[n] implies S1[n + 1] )
assume A3: S1[n] ; ::_thesis: S1[n + 1]
set j = n + 1;
set a = E-bound C;
set s = W-bound C;
A4: ((W-bound C) - (((E-bound C) - (W-bound C)) / (2 |^ (n + 1)))) - ((W-bound C) - (((E-bound C) - (W-bound C)) / (2 |^ n))) = (((W-bound C) + (- (((E-bound C) - (W-bound C)) / (2 |^ (n + 1))))) - (W-bound C)) + (((E-bound C) - (W-bound C)) / (2 |^ n))
.= (- (((E-bound C) - (W-bound C)) / ((2 |^ n) * 2))) + (((E-bound C) - (W-bound C)) / (2 |^ n)) by NEWTON:6
.= ((- ((E-bound C) - (W-bound C))) / ((2 |^ n) * 2)) + (((E-bound C) - (W-bound C)) / (2 |^ n)) by XCMPLX_1:187
.= ((- ((E-bound C) - (W-bound C))) / ((2 |^ n) * 2)) + ((((E-bound C) - (W-bound C)) * 2) / ((2 |^ n) * 2)) by XCMPLX_1:91
.= ((- ((E-bound C) - (W-bound C))) + (((E-bound C) - (W-bound C)) * 2)) / ((2 |^ n) * 2) by XCMPLX_1:62
.= ((E-bound C) - (W-bound C)) / ((2 |^ n) * 2) ;
 2 |^ n > 0 by NEWTON:83;
then A5: (2 |^ n) * 2 > 0 * 2 by XREAL_1:68;
A6: ( W-bound (L~ (Cage (C,n))) = (W-bound C) - (((E-bound C) - (W-bound C)) / (2 |^ n)) & W-bound (L~ (Cage (C,(n + 1)))) = (W-bound C) - (((E-bound C) - (W-bound C)) / (2 |^ (n + 1))) ) by Th62;
 (E-bound C) - (W-bound C) > 0 by SPRECT_1:31, XREAL_1:50;
then  0 < ((W-bound C) - (((E-bound C) - (W-bound C)) / (2 |^ (n + 1)))) - ((W-bound C) - (((E-bound C) - (W-bound C)) / (2 |^ n))) by A5, A4, XREAL_1:139;
then A7: W-bound (L~ (Cage (C,n))) < W-bound (L~ (Cage (C,(n + 1)))) by A6, XREAL_1:47;
assume  i < n + 1 ; ::_thesis: W-bound (L~ (Cage (C,i))) < W-bound (L~ (Cage (C,(n + 1))))
then  i <= n by NAT_1:13;
hence  W-bound (L~ (Cage (C,i))) < W-bound (L~ (Cage (C,(n + 1)))) by A3, A7, XXREAL_0:1, XXREAL_0:2; ::_thesis: verum
end;
A8: S1[ 0 ] ;
 for n being Element of NAT holds S1[n] from NAT_1:sch_1(A8, A2);
hence  W-bound (L~ (Cage (C,i))) < W-bound (L~ (Cage (C,j))) by A1; ::_thesis: verum
end;

theorem :: JORDAN1A:69
for i, j being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) st i < j holds
S-bound (L~ (Cage (C,i))) < S-bound (L~ (Cage (C,j)))
proof
let i, j be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) st i < j holds
S-bound (L~ (Cage (C,i))) < S-bound (L~ (Cage (C,j)))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( i < j implies S-bound (L~ (Cage (C,i))) < S-bound (L~ (Cage (C,j))) )
assume A1: i < j ; ::_thesis: S-bound (L~ (Cage (C,i))) < S-bound (L~ (Cage (C,j)))
defpred S1[ Element of NAT ] means ( i < $1 implies S-bound (L~ (Cage (C,i))) < S-bound (L~ (Cage (C,$1))) );
A2: for n being Element of NAT st S1[n] holds
S1[n + 1]
proof
let n be Element of NAT ; ::_thesis: ( S1[n] implies S1[n + 1] )
assume A3: S1[n] ; ::_thesis: S1[n + 1]
set j = n + 1;
set a = N-bound C;
set s = S-bound C;
A4: ((S-bound C) - (((N-bound C) - (S-bound C)) / (2 |^ (n + 1)))) - ((S-bound C) - (((N-bound C) - (S-bound C)) / (2 |^ n))) = (((S-bound C) + (- (((N-bound C) - (S-bound C)) / (2 |^ (n + 1))))) - (S-bound C)) + (((N-bound C) - (S-bound C)) / (2 |^ n))
.= (- (((N-bound C) - (S-bound C)) / ((2 |^ n) * 2))) + (((N-bound C) - (S-bound C)) / (2 |^ n)) by NEWTON:6
.= ((- ((N-bound C) - (S-bound C))) / ((2 |^ n) * 2)) + (((N-bound C) - (S-bound C)) / (2 |^ n)) by XCMPLX_1:187
.= ((- ((N-bound C) - (S-bound C))) / ((2 |^ n) * 2)) + ((((N-bound C) - (S-bound C)) * 2) / ((2 |^ n) * 2)) by XCMPLX_1:91
.= ((- ((N-bound C) - (S-bound C))) + (((N-bound C) - (S-bound C)) * 2)) / ((2 |^ n) * 2) by XCMPLX_1:62
.= ((N-bound C) - (S-bound C)) / ((2 |^ n) * 2) ;
 2 |^ n > 0 by NEWTON:83;
then A5: (2 |^ n) * 2 > 0 * 2 by XREAL_1:68;
A6: ( S-bound (L~ (Cage (C,n))) = (S-bound C) - (((N-bound C) - (S-bound C)) / (2 |^ n)) & S-bound (L~ (Cage (C,(n + 1)))) = (S-bound C) - (((N-bound C) - (S-bound C)) / (2 |^ (n + 1))) ) by Th63;
 (N-bound C) - (S-bound C) > 0 by SPRECT_1:32, XREAL_1:50;
then  0 < ((S-bound C) - (((N-bound C) - (S-bound C)) / (2 |^ (n + 1)))) - ((S-bound C) - (((N-bound C) - (S-bound C)) / (2 |^ n))) by A5, A4, XREAL_1:139;
then A7: S-bound (L~ (Cage (C,n))) < S-bound (L~ (Cage (C,(n + 1)))) by A6, XREAL_1:47;
assume  i < n + 1 ; ::_thesis: S-bound (L~ (Cage (C,i))) < S-bound (L~ (Cage (C,(n + 1))))
then  i <= n by NAT_1:13;
hence  S-bound (L~ (Cage (C,i))) < S-bound (L~ (Cage (C,(n + 1)))) by A3, A7, XXREAL_0:1, XXREAL_0:2; ::_thesis: verum
end;
A8: S1[ 0 ] ;
 for n being Element of NAT holds S1[n] from NAT_1:sch_1(A8, A2);
hence  S-bound (L~ (Cage (C,i))) < S-bound (L~ (Cage (C,j))) by A1; ::_thesis: verum
end;

theorem :: JORDAN1A:70
for i, n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= i & i <= len (Gauge (C,n)) holds
N-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * (i,(len (Gauge (C,n))))) `2
proof
let i, n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= i & i <= len (Gauge (C,n)) holds
N-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * (i,(len (Gauge (C,n))))) `2
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( 1 <= i & i <= len (Gauge (C,n)) implies N-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * (i,(len (Gauge (C,n))))) `2 )
set a = N-bound C;
set s = S-bound C;
set w = W-bound C;
set e = E-bound C;
set f = Cage (C,n);
set G = Gauge (C,n);
A1: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
assume A2: ( 1 <= i & i <= len (Gauge (C,n)) ) ; ::_thesis: N-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * (i,(len (Gauge (C,n))))) `2
then  1 <= len (Gauge (C,n)) by XXREAL_0:2;
then A3: [i,(len (Gauge (C,n)))] in Indices (Gauge (C,n)) by A2, A1, MATRIX_1:36;
thus  N-bound (L~ (Cage (C,n))) = (N-bound C) + (((N-bound C) - (S-bound C)) / (2 |^ n)) by JORDAN10:6
.= (S-bound C) + ((((N-bound C) - (S-bound C)) / (2 |^ n)) * ((len (Gauge (C,n))) - 2)) by Lm10
.= |[((W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * (i - 2))),((S-bound C) + ((((N-bound C) - (S-bound C)) / (2 |^ n)) * ((len (Gauge (C,n))) - 2)))]| `2 by EUCLID:52
.= ((Gauge (C,n)) * (i,(len (Gauge (C,n))))) `2 by A3, JORDAN8:def_1 ; ::_thesis: verum
end;

theorem :: JORDAN1A:71
for i, n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= i & i <= len (Gauge (C,n)) holds
E-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * ((len (Gauge (C,n))),i)) `1
proof
let i, n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= i & i <= len (Gauge (C,n)) holds
E-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * ((len (Gauge (C,n))),i)) `1
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( 1 <= i & i <= len (Gauge (C,n)) implies E-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * ((len (Gauge (C,n))),i)) `1 )
set a = N-bound C;
set s = S-bound C;
set w = W-bound C;
set e = E-bound C;
set f = Cage (C,n);
set G = Gauge (C,n);
A1: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
assume A2: ( 1 <= i & i <= len (Gauge (C,n)) ) ; ::_thesis: E-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * ((len (Gauge (C,n))),i)) `1
then  1 <= len (Gauge (C,n)) by XXREAL_0:2;
then A3: [(len (Gauge (C,n))),i] in Indices (Gauge (C,n)) by A2, A1, MATRIX_1:36;
thus  E-bound (L~ (Cage (C,n))) = (E-bound C) + (((E-bound C) - (W-bound C)) / (2 |^ n)) by Th64
.= (W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * ((len (Gauge (C,n))) - 2)) by Lm10
.= |[((W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * ((len (Gauge (C,n))) - 2))),((S-bound C) + ((((N-bound C) - (S-bound C)) / (2 |^ n)) * (i - 2)))]| `1 by EUCLID:52
.= ((Gauge (C,n)) * ((len (Gauge (C,n))),i)) `1 by A3, JORDAN8:def_1 ; ::_thesis: verum
end;

theorem :: JORDAN1A:72
for i, n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= i & i <= len (Gauge (C,n)) holds
S-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * (i,1)) `2
proof
let i, n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= i & i <= len (Gauge (C,n)) holds
S-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * (i,1)) `2
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( 1 <= i & i <= len (Gauge (C,n)) implies S-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * (i,1)) `2 )
set a = N-bound C;
set s = S-bound C;
set w = W-bound C;
set e = E-bound C;
set f = Cage (C,n);
set G = Gauge (C,n);
A1: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
assume A2: ( 1 <= i & i <= len (Gauge (C,n)) ) ; ::_thesis: S-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * (i,1)) `2
then  1 <= len (Gauge (C,n)) by XXREAL_0:2;
then A3: [i,1] in Indices (Gauge (C,n)) by A2, A1, MATRIX_1:36;
thus  S-bound (L~ (Cage (C,n))) = (S-bound C) - (((N-bound C) - (S-bound C)) / (2 |^ n)) by Th63
.= |[((W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * (i - 2))),((S-bound C) + ((((N-bound C) - (S-bound C)) / (2 |^ n)) * (1 - 2)))]| `2 by EUCLID:52
.= ((Gauge (C,n)) * (i,1)) `2 by A3, JORDAN8:def_1 ; ::_thesis: verum
end;

theorem :: JORDAN1A:73
for i, n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= i & i <= len (Gauge (C,n)) holds
W-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * (1,i)) `1
proof
let i, n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) st 1 <= i & i <= len (Gauge (C,n)) holds
W-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * (1,i)) `1
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( 1 <= i & i <= len (Gauge (C,n)) implies W-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * (1,i)) `1 )
set a = N-bound C;
set s = S-bound C;
set w = W-bound C;
set e = E-bound C;
set f = Cage (C,n);
set G = Gauge (C,n);
A1: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
assume A2: ( 1 <= i & i <= len (Gauge (C,n)) ) ; ::_thesis: W-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * (1,i)) `1
then  1 <= len (Gauge (C,n)) by XXREAL_0:2;
then A3: [1,i] in Indices (Gauge (C,n)) by A2, A1, MATRIX_1:36;
thus  W-bound (L~ (Cage (C,n))) = (W-bound C) - (((E-bound C) - (W-bound C)) / (2 |^ n)) by Th62
.= |[((W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * (1 - 2))),((S-bound C) + ((((N-bound C) - (S-bound C)) / (2 |^ n)) * (i - 2)))]| `1 by EUCLID:52
.= ((Gauge (C,n)) * (1,i)) `1 by A3, JORDAN8:def_1 ; ::_thesis: verum
end;

theorem Th74: :: JORDAN1A:74
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x, p being Point of (TOP-REAL 2) st x in C & p in (north_halfline x) /\ (L~ (Cage (C,n))) holds
p `2 > x `2
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x, p being Point of (TOP-REAL 2) st x in C & p in (north_halfline x) /\ (L~ (Cage (C,n))) holds
p `2 > x `2
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for x, p being Point of (TOP-REAL 2) st x in C & p in (north_halfline x) /\ (L~ (Cage (C,n))) holds
p `2 > x `2
let x, p be Point of (TOP-REAL 2); ::_thesis: ( x in C & p in (north_halfline x) /\ (L~ (Cage (C,n))) implies p `2 > x `2 )
set f = Cage (C,n);
assume A1: x in C ; ::_thesis: ( not p in (north_halfline x) /\ (L~ (Cage (C,n))) or p `2 > x `2 )
assume A2: p in (north_halfline x) /\ (L~ (Cage (C,n))) ; ::_thesis: p `2 > x `2
then A3: p in north_halfline x by XBOOLE_0:def_4;
then A4: p `1 = x `1 by TOPREAL1:def_10;
assume A5: p `2 <= x `2 ; ::_thesis: contradiction
 p `2 >= x `2 by A3, TOPREAL1:def_10;
then  p `2 = x `2 by A5, XXREAL_0:1;
then A6: p = x by A4, TOPREAL3:6;
 p in L~ (Cage (C,n)) by A2, XBOOLE_0:def_4;
then  x in C /\ (L~ (Cage (C,n))) by A1, A6, XBOOLE_0:def_4;
then  C meets L~ (Cage (C,n)) by XBOOLE_0:4;
hence  contradiction by JORDAN10:5; ::_thesis: verum
end;

theorem Th75: :: JORDAN1A:75
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x, p being Point of (TOP-REAL 2) st x in C & p in (east_halfline x) /\ (L~ (Cage (C,n))) holds
p `1 > x `1
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x, p being Point of (TOP-REAL 2) st x in C & p in (east_halfline x) /\ (L~ (Cage (C,n))) holds
p `1 > x `1
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for x, p being Point of (TOP-REAL 2) st x in C & p in (east_halfline x) /\ (L~ (Cage (C,n))) holds
p `1 > x `1
let x, p be Point of (TOP-REAL 2); ::_thesis: ( x in C & p in (east_halfline x) /\ (L~ (Cage (C,n))) implies p `1 > x `1 )
set f = Cage (C,n);
assume A1: x in C ; ::_thesis: ( not p in (east_halfline x) /\ (L~ (Cage (C,n))) or p `1 > x `1 )
assume A2: p in (east_halfline x) /\ (L~ (Cage (C,n))) ; ::_thesis: p `1 > x `1
then A3: p in east_halfline x by XBOOLE_0:def_4;
then A4: p `2 = x `2 by TOPREAL1:def_11;
assume A5: p `1 <= x `1 ; ::_thesis: contradiction
 p `1 >= x `1 by A3, TOPREAL1:def_11;
then  p `1 = x `1 by A5, XXREAL_0:1;
then A6: p = x by A4, TOPREAL3:6;
 p in L~ (Cage (C,n)) by A2, XBOOLE_0:def_4;
then  x in C /\ (L~ (Cage (C,n))) by A1, A6, XBOOLE_0:def_4;
then  C meets L~ (Cage (C,n)) by XBOOLE_0:4;
hence  contradiction by JORDAN10:5; ::_thesis: verum
end;

theorem Th76: :: JORDAN1A:76
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x, p being Point of (TOP-REAL 2) st x in C & p in (south_halfline x) /\ (L~ (Cage (C,n))) holds
p `2 < x `2
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x, p being Point of (TOP-REAL 2) st x in C & p in (south_halfline x) /\ (L~ (Cage (C,n))) holds
p `2 < x `2
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for x, p being Point of (TOP-REAL 2) st x in C & p in (south_halfline x) /\ (L~ (Cage (C,n))) holds
p `2 < x `2
let x, p be Point of (TOP-REAL 2); ::_thesis: ( x in C & p in (south_halfline x) /\ (L~ (Cage (C,n))) implies p `2 < x `2 )
set f = Cage (C,n);
assume A1: x in C ; ::_thesis: ( not p in (south_halfline x) /\ (L~ (Cage (C,n))) or p `2 < x `2 )
assume A2: p in (south_halfline x) /\ (L~ (Cage (C,n))) ; ::_thesis: p `2 < x `2
then A3: p in south_halfline x by XBOOLE_0:def_4;
then A4: p `1 = x `1 by TOPREAL1:def_12;
assume A5: p `2 >= x `2 ; ::_thesis: contradiction
 p `2 <= x `2 by A3, TOPREAL1:def_12;
then  p `2 = x `2 by A5, XXREAL_0:1;
then A6: p = x by A4, TOPREAL3:6;
 p in L~ (Cage (C,n)) by A2, XBOOLE_0:def_4;
then  x in C /\ (L~ (Cage (C,n))) by A1, A6, XBOOLE_0:def_4;
then  C meets L~ (Cage (C,n)) by XBOOLE_0:4;
hence  contradiction by JORDAN10:5; ::_thesis: verum
end;

theorem Th77: :: JORDAN1A:77
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x, p being Point of (TOP-REAL 2) st x in C & p in (west_halfline x) /\ (L~ (Cage (C,n))) holds
p `1 < x `1
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x, p being Point of (TOP-REAL 2) st x in C & p in (west_halfline x) /\ (L~ (Cage (C,n))) holds
p `1 < x `1
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for x, p being Point of (TOP-REAL 2) st x in C & p in (west_halfline x) /\ (L~ (Cage (C,n))) holds
p `1 < x `1
let x, p be Point of (TOP-REAL 2); ::_thesis: ( x in C & p in (west_halfline x) /\ (L~ (Cage (C,n))) implies p `1 < x `1 )
set f = Cage (C,n);
assume A1: x in C ; ::_thesis: ( not p in (west_halfline x) /\ (L~ (Cage (C,n))) or p `1 < x `1 )
assume A2: p in (west_halfline x) /\ (L~ (Cage (C,n))) ; ::_thesis: p `1 < x `1
then A3: p in west_halfline x by XBOOLE_0:def_4;
then A4: p `2 = x `2 by TOPREAL1:def_13;
assume A5: p `1 >= x `1 ; ::_thesis: contradiction
 p `1 <= x `1 by A3, TOPREAL1:def_13;
then  p `1 = x `1 by A5, XXREAL_0:1;
then A6: p = x by A4, TOPREAL3:6;
 p in L~ (Cage (C,n)) by A2, XBOOLE_0:def_4;
then  x in C /\ (L~ (Cage (C,n))) by A1, A6, XBOOLE_0:def_4;
then  C meets L~ (Cage (C,n)) by XBOOLE_0:4;
hence  contradiction by JORDAN10:5; ::_thesis: verum
end;

theorem Th78: :: JORDAN1A:78
for i, n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x, p being Point of (TOP-REAL 2) st x in N-most C & p in north_halfline x & 1 <= i & i < len (Cage (C,n)) & p in LSeg ((Cage (C,n)),i) holds
LSeg ((Cage (C,n)),i) is horizontal
proof
let i, n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x, p being Point of (TOP-REAL 2) st x in N-most C & p in north_halfline x & 1 <= i & i < len (Cage (C,n)) & p in LSeg ((Cage (C,n)),i) holds
LSeg ((Cage (C,n)),i) is horizontal
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for x, p being Point of (TOP-REAL 2) st x in N-most C & p in north_halfline x & 1 <= i & i < len (Cage (C,n)) & p in LSeg ((Cage (C,n)),i) holds
LSeg ((Cage (C,n)),i) is horizontal
let x, p be Point of (TOP-REAL 2); ::_thesis: ( x in N-most C & p in north_halfline x & 1 <= i & i < len (Cage (C,n)) & p in LSeg ((Cage (C,n)),i) implies LSeg ((Cage (C,n)),i) is horizontal )
set G = Gauge (C,n);
set f = Cage (C,n);
assume that
A1: x in N-most C and
A2: p in north_halfline x and
A3: 1 <= i and
A4: i < len (Cage (C,n)) and
A5: p in LSeg ((Cage (C,n)),i) ; ::_thesis: LSeg ((Cage (C,n)),i) is horizontal
assume A6: not LSeg ((Cage (C,n)),i) is horizontal ; ::_thesis: contradiction
A7: i + 1 <= len (Cage (C,n)) by A4, NAT_1:13;
then A8: LSeg ((Cage (C,n)),i) = LSeg (((Cage (C,n)) /. i),((Cage (C,n)) /. (i + 1))) by A3, TOPREAL1:def_3;
 1 <= i + 1 by A3, NAT_1:13;
then  i + 1 in Seg (len (Cage (C,n))) by A7, FINSEQ_1:1;
then A9: i + 1 in dom (Cage (C,n)) by FINSEQ_1:def_3;
 i in Seg (len (Cage (C,n))) by A3, A4, FINSEQ_1:1;
then A10: i in dom (Cage (C,n)) by FINSEQ_1:def_3;
A11: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
then A12: (len (Gauge (C,n))) -' 1 <= width (Gauge (C,n)) by NAT_D:35;
A13: x in C by A1, XBOOLE_0:def_4;
 p in L~ (Cage (C,n)) by A5, SPPOL_2:17;
then A14: p in (north_halfline x) /\ (L~ (Cage (C,n))) by A2, XBOOLE_0:def_4;
A15: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def_1;
A16: x `1 = p `1 by A2, TOPREAL1:def_10
.= ((Cage (C,n)) /. i) `1 by A5, A8, A6, SPPOL_1:19, SPPOL_1:41 ;
A17: x `1 = p `1 by A2, TOPREAL1:def_10
.= ((Cage (C,n)) /. (i + 1)) `1 by A5, A8, A6, SPPOL_1:19, SPPOL_1:41 ;
percases ( ((Cage (C,n)) /. i) `2 <= ((Cage (C,n)) /. (i + 1)) `2 or ((Cage (C,n)) /. i) `2 >= ((Cage (C,n)) /. (i + 1)) `2 ) ;
supposeA18: ((Cage (C,n)) /. i) `2 <= ((Cage (C,n)) /. (i + 1)) `2 ; ::_thesis: contradiction
then  p `2 <= ((Cage (C,n)) /. (i + 1)) `2 by A5, A8, TOPREAL1:4;
then A19: ((Cage (C,n)) /. (i + 1)) `2 > x `2 by A13, A14, Th74, XXREAL_0:2;
consider i1, i2 being Element of NAT  such that
A20: [i1,i2] in Indices (Gauge (C,n)) and
A21: (Cage (C,n)) /. (i + 1) = (Gauge (C,n)) * (i1,i2) by A15, A9, GOBOARD1:def_9;
A22: 1 <= i2 by A20, MATRIX_1:38;
 i2 <= width (Gauge (C,n)) by A20, MATRIX_1:38;
then A23: i2 <= len (Gauge (C,n)) by JORDAN8:def_1;
A24: ( 1 <= i1 & i1 <= len (Gauge (C,n)) ) by A20, MATRIX_1:38;
consider j1, j2 being Element of NAT  such that
A25: [j1,j2] in Indices (Gauge (C,n)) and
A26: (Cage (C,n)) /. i = (Gauge (C,n)) * (j1,j2) by A15, A10, GOBOARD1:def_9;
A27: ( 1 <= j1 & j1 <= len (Gauge (C,n)) ) by A25, MATRIX_1:38;
now__::_thesis:_not_((Cage_(C,n))_/._i)_`2_=_((Cage_(C,n))_/._(i_+_1))_`2
assume  ((Cage (C,n)) /. i) `2 = ((Cage (C,n)) /. (i + 1)) `2 ; ::_thesis: contradiction
then A28: (Cage (C,n)) /. i = (Cage (C,n)) /. (i + 1) by A17, A16, TOPREAL3:6;
then A29: i2 = j2 by A20, A21, A25, A26, GOBOARD1:5;
 ( i1 = j1 & (abs (i1 - j1)) + (abs (i2 - j2)) = 1 ) by A15, A10, A9, A20, A21, A25, A26, A28, GOBOARD1:5, GOBOARD1:def_9;
then 1 = 0 + (abs (i2 - j2)) by GOBOARD7:2
.= 0 + 0 by A29, GOBOARD7:2 ;
hence  contradiction ; ::_thesis: verum
end;
then A30: ((Cage (C,n)) /. i) `2 < ((Cage (C,n)) /. (i + 1)) `2 by A18, XXREAL_0:1;
A31: 1 <= j2 by A25, MATRIX_1:38;
 j2 <= width (Gauge (C,n)) by A25, MATRIX_1:38;
then  i2 > j2 by A21, A22, A24, A26, A27, A30, Th19;
then  len (Gauge (C,n)) > j2 by A23, XXREAL_0:2;
then A32: (len (Gauge (C,n))) -' 1 >= j2 by NAT_D:49;
x `2 = (N-min C) `2 by A1, PSCOMP_1:39
.= N-bound C by EUCLID:52
.= ((Gauge (C,n)) * (i1,((len (Gauge (C,n))) -' 1))) `2 by A24, JORDAN8:14 ;
then  x `2 >= ((Cage (C,n)) /. i) `2 by A12, A24, A26, A31, A27, A32, Th19;
then  x in L~ (Cage (C,n)) by A8, A17, A16, A19, GOBOARD7:7, SPPOL_2:17;
then  L~ (Cage (C,n)) meets C by A13, XBOOLE_0:3;
hence  contradiction by JORDAN10:5; ::_thesis: verum
end;
supposeA33: ((Cage (C,n)) /. i) `2 >= ((Cage (C,n)) /. (i + 1)) `2 ; ::_thesis: contradiction
then  p `2 <= ((Cage (C,n)) /. i) `2 by A5, A8, TOPREAL1:4;
then A34: ((Cage (C,n)) /. i) `2 > x `2 by A13, A14, Th74, XXREAL_0:2;
consider i1, i2 being Element of NAT  such that
A35: [i1,i2] in Indices (Gauge (C,n)) and
A36: (Cage (C,n)) /. i = (Gauge (C,n)) * (i1,i2) by A15, A10, GOBOARD1:def_9;
A37: 1 <= i2 by A35, MATRIX_1:38;
consider j1, j2 being Element of NAT  such that
A38: [j1,j2] in Indices (Gauge (C,n)) and
A39: (Cage (C,n)) /. (i + 1) = (Gauge (C,n)) * (j1,j2) by A15, A9, GOBOARD1:def_9;
A40: ( 1 <= j1 & j1 <= len (Gauge (C,n)) ) by A38, MATRIX_1:38;
now__::_thesis:_not_((Cage_(C,n))_/._i)_`2_=_((Cage_(C,n))_/._(i_+_1))_`2
assume  ((Cage (C,n)) /. i) `2 = ((Cage (C,n)) /. (i + 1)) `2 ; ::_thesis: contradiction
then A41: (Cage (C,n)) /. i = (Cage (C,n)) /. (i + 1) by A17, A16, TOPREAL3:6;
then A42: i2 = j2 by A35, A36, A38, A39, GOBOARD1:5;
 ( i1 = j1 & (abs (j1 - i1)) + (abs (j2 - i2)) = 1 ) by A15, A10, A9, A35, A36, A38, A39, A41, GOBOARD1:5, GOBOARD1:def_9;
then 1 = 0 + (abs (i2 - j2)) by A42, GOBOARD7:2
.= 0 + 0 by A42, GOBOARD7:2 ;
hence  contradiction ; ::_thesis: verum
end;
then A43: ((Cage (C,n)) /. (i + 1)) `2 < ((Cage (C,n)) /. i) `2 by A33, XXREAL_0:1;
A44: i2 <= width (Gauge (C,n)) by A35, MATRIX_1:38;
A45: ( 1 <= i1 & i1 <= len (Gauge (C,n)) ) by A35, MATRIX_1:38;
A46: 1 <= j2 by A38, MATRIX_1:38;
 j2 <= width (Gauge (C,n)) by A38, MATRIX_1:38;
then  i2 > j2 by A36, A37, A45, A39, A40, A43, Th19;
then  len (Gauge (C,n)) > j2 by A11, A44, XXREAL_0:2;
then A47: (len (Gauge (C,n))) -' 1 >= j2 by NAT_D:49;
x `2 = (N-min C) `2 by A1, PSCOMP_1:39
.= N-bound C by EUCLID:52
.= ((Gauge (C,n)) * (i1,((len (Gauge (C,n))) -' 1))) `2 by A45, JORDAN8:14 ;
then  x `2 >= ((Cage (C,n)) /. (i + 1)) `2 by A12, A45, A39, A46, A40, A47, Th19;
then  x in L~ (Cage (C,n)) by A8, A17, A16, A34, GOBOARD7:7, SPPOL_2:17;
then  x in (L~ (Cage (C,n))) /\ C by A13, XBOOLE_0:def_4;
then  L~ (Cage (C,n)) meets C by XBOOLE_0:4;
hence  contradiction by JORDAN10:5; ::_thesis: verum
end;
end;
end;

theorem Th79: :: JORDAN1A:79
for i, n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x, p being Point of (TOP-REAL 2) st x in E-most C & p in east_halfline x & 1 <= i & i < len (Cage (C,n)) & p in LSeg ((Cage (C,n)),i) holds
LSeg ((Cage (C,n)),i) is vertical
proof
let i, n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x, p being Point of (TOP-REAL 2) st x in E-most C & p in east_halfline x & 1 <= i & i < len (Cage (C,n)) & p in LSeg ((Cage (C,n)),i) holds
LSeg ((Cage (C,n)),i) is vertical
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for x, p being Point of (TOP-REAL 2) st x in E-most C & p in east_halfline x & 1 <= i & i < len (Cage (C,n)) & p in LSeg ((Cage (C,n)),i) holds
LSeg ((Cage (C,n)),i) is vertical
let x, p be Point of (TOP-REAL 2); ::_thesis: ( x in E-most C & p in east_halfline x & 1 <= i & i < len (Cage (C,n)) & p in LSeg ((Cage (C,n)),i) implies LSeg ((Cage (C,n)),i) is vertical )
set G = Gauge (C,n);
set f = Cage (C,n);
assume that
A1: x in E-most C and
A2: p in east_halfline x and
A3: 1 <= i and
A4: i < len (Cage (C,n)) and
A5: p in LSeg ((Cage (C,n)),i) ; ::_thesis: LSeg ((Cage (C,n)),i) is vertical
assume A6: not LSeg ((Cage (C,n)),i) is vertical ; ::_thesis: contradiction
A7: i + 1 <= len (Cage (C,n)) by A4, NAT_1:13;
then A8: LSeg ((Cage (C,n)),i) = LSeg (((Cage (C,n)) /. i),((Cage (C,n)) /. (i + 1))) by A3, TOPREAL1:def_3;
 1 <= i + 1 by A3, NAT_1:13;
then  i + 1 in Seg (len (Cage (C,n))) by A7, FINSEQ_1:1;
then A9: i + 1 in dom (Cage (C,n)) by FINSEQ_1:def_3;
 i in Seg (len (Cage (C,n))) by A3, A4, FINSEQ_1:1;
then A10: i in dom (Cage (C,n)) by FINSEQ_1:def_3;
 p in L~ (Cage (C,n)) by A5, SPPOL_2:17;
then A11: p in (east_halfline x) /\ (L~ (Cage (C,n))) by A2, XBOOLE_0:def_4;
A12: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def_1;
A13: x `2 = p `2 by A2, TOPREAL1:def_11
.= ((Cage (C,n)) /. i) `2 by A5, A8, A6, SPPOL_1:19, SPPOL_1:40 ;
A14: x `2 = p `2 by A2, TOPREAL1:def_11
.= ((Cage (C,n)) /. (i + 1)) `2 by A5, A8, A6, SPPOL_1:19, SPPOL_1:40 ;
A15: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
then A16: (len (Gauge (C,n))) -' 1 <= width (Gauge (C,n)) by NAT_D:35;
A17: x in C by A1, XBOOLE_0:def_4;
percases ( ((Cage (C,n)) /. i) `1 <= ((Cage (C,n)) /. (i + 1)) `1 or ((Cage (C,n)) /. i) `1 >= ((Cage (C,n)) /. (i + 1)) `1 ) ;
supposeA18: ((Cage (C,n)) /. i) `1 <= ((Cage (C,n)) /. (i + 1)) `1 ; ::_thesis: contradiction
then  p `1 <= ((Cage (C,n)) /. (i + 1)) `1 by A5, A8, TOPREAL1:3;
then A19: ((Cage (C,n)) /. (i + 1)) `1 > x `1 by A17, A11, Th75, XXREAL_0:2;
consider i1, i2 being Element of NAT  such that
A20: [i1,i2] in Indices (Gauge (C,n)) and
A21: (Cage (C,n)) /. (i + 1) = (Gauge (C,n)) * (i1,i2) by A12, A9, GOBOARD1:def_9;
A22: ( 1 <= i2 & i2 <= width (Gauge (C,n)) ) by A20, MATRIX_1:38;
consider j1, j2 being Element of NAT  such that
A23: [j1,j2] in Indices (Gauge (C,n)) and
A24: (Cage (C,n)) /. i = (Gauge (C,n)) * (j1,j2) by A12, A10, GOBOARD1:def_9;
A25: ( 1 <= j2 & j2 <= width (Gauge (C,n)) ) by A23, MATRIX_1:38;
A26: i1 <= len (Gauge (C,n)) by A20, MATRIX_1:38;
A27: 1 <= i1 by A20, MATRIX_1:38;
now__::_thesis:_not_((Cage_(C,n))_/._i)_`1_=_((Cage_(C,n))_/._(i_+_1))_`1
assume  ((Cage (C,n)) /. i) `1 = ((Cage (C,n)) /. (i + 1)) `1 ; ::_thesis: contradiction
then A28: (Cage (C,n)) /. i = (Cage (C,n)) /. (i + 1) by A14, A13, TOPREAL3:6;
then A29: i2 = j2 by A20, A21, A23, A24, GOBOARD1:5;
 ( i1 = j1 & (abs (i1 - j1)) + (abs (i2 - j2)) = 1 ) by A12, A10, A9, A20, A21, A23, A24, A28, GOBOARD1:5, GOBOARD1:def_9;
then 1 = 0 + (abs (i2 - j2)) by GOBOARD7:2
.= 0 + 0 by A29, GOBOARD7:2 ;
hence  contradiction ; ::_thesis: verum
end;
then A30: ((Cage (C,n)) /. i) `1 < ((Cage (C,n)) /. (i + 1)) `1 by A18, XXREAL_0:1;
A31: 1 <= j1 by A23, MATRIX_1:38;
 j1 <= len (Gauge (C,n)) by A23, MATRIX_1:38;
then  i1 > j1 by A21, A22, A27, A24, A25, A30, Th18;
then  len (Gauge (C,n)) > j1 by A26, XXREAL_0:2;
then A32: (len (Gauge (C,n))) -' 1 >= j1 by NAT_D:49;
x `1 = (E-min C) `1 by A1, PSCOMP_1:47
.= E-bound C by EUCLID:52
.= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),i2)) `1 by A15, A22, JORDAN8:12 ;
then  x `1 >= ((Cage (C,n)) /. i) `1 by A15, A16, A22, A24, A25, A31, A32, Th18;
then  x in L~ (Cage (C,n)) by A8, A14, A13, A19, GOBOARD7:8, SPPOL_2:17;
then  x in (L~ (Cage (C,n))) /\ C by A17, XBOOLE_0:def_4;
then  L~ (Cage (C,n)) meets C by XBOOLE_0:4;
hence  contradiction by JORDAN10:5; ::_thesis: verum
end;
supposeA33: ((Cage (C,n)) /. i) `1 >= ((Cage (C,n)) /. (i + 1)) `1 ; ::_thesis: contradiction
then  p `1 <= ((Cage (C,n)) /. i) `1 by A5, A8, TOPREAL1:3;
then A34: ((Cage (C,n)) /. i) `1 > x `1 by A17, A11, Th75, XXREAL_0:2;
consider i1, i2 being Element of NAT  such that
A35: [i1,i2] in Indices (Gauge (C,n)) and
A36: (Cage (C,n)) /. i = (Gauge (C,n)) * (i1,i2) by A12, A10, GOBOARD1:def_9;
A37: ( 1 <= i2 & i2 <= width (Gauge (C,n)) ) by A35, MATRIX_1:38;
consider j1, j2 being Element of NAT  such that
A38: [j1,j2] in Indices (Gauge (C,n)) and
A39: (Cage (C,n)) /. (i + 1) = (Gauge (C,n)) * (j1,j2) by A12, A9, GOBOARD1:def_9;
A40: ( 1 <= j2 & j2 <= width (Gauge (C,n)) ) by A38, MATRIX_1:38;
A41: i1 <= len (Gauge (C,n)) by A35, MATRIX_1:38;
A42: 1 <= i1 by A35, MATRIX_1:38;
now__::_thesis:_not_((Cage_(C,n))_/._i)_`1_=_((Cage_(C,n))_/._(i_+_1))_`1
assume  ((Cage (C,n)) /. i) `1 = ((Cage (C,n)) /. (i + 1)) `1 ; ::_thesis: contradiction
then A43: (Cage (C,n)) /. i = (Cage (C,n)) /. (i + 1) by A14, A13, TOPREAL3:6;
then A44: i2 = j2 by A35, A36, A38, A39, GOBOARD1:5;
 ( i1 = j1 & (abs (j1 - i1)) + (abs (j2 - i2)) = 1 ) by A12, A10, A9, A35, A36, A38, A39, A43, GOBOARD1:5, GOBOARD1:def_9;
then 1 = 0 + (abs (i2 - j2)) by A44, GOBOARD7:2
.= 0 + 0 by A44, GOBOARD7:2 ;
hence  contradiction ; ::_thesis: verum
end;
then A45: ((Cage (C,n)) /. (i + 1)) `1 < ((Cage (C,n)) /. i) `1 by A33, XXREAL_0:1;
A46: 1 <= j1 by A38, MATRIX_1:38;
 j1 <= len (Gauge (C,n)) by A38, MATRIX_1:38;
then  i1 > j1 by A36, A37, A42, A39, A40, A45, Th18;
then  len (Gauge (C,n)) > j1 by A41, XXREAL_0:2;
then A47: (len (Gauge (C,n))) -' 1 >= j1 by NAT_D:49;
x `1 = (E-min C) `1 by A1, PSCOMP_1:47
.= E-bound C by EUCLID:52
.= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),i2)) `1 by A15, A37, JORDAN8:12 ;
then  x `1 >= ((Cage (C,n)) /. (i + 1)) `1 by A15, A16, A37, A39, A40, A46, A47, Th18;
then  x in L~ (Cage (C,n)) by A8, A14, A13, A34, GOBOARD7:8, SPPOL_2:17;
then  x in (L~ (Cage (C,n))) /\ C by A17, XBOOLE_0:def_4;
then  L~ (Cage (C,n)) meets C by XBOOLE_0:4;
hence  contradiction by JORDAN10:5; ::_thesis: verum
end;
end;
end;

theorem Th80: :: JORDAN1A:80
for i, n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x, p being Point of (TOP-REAL 2) st x in S-most C & p in south_halfline x & 1 <= i & i < len (Cage (C,n)) & p in LSeg ((Cage (C,n)),i) holds
LSeg ((Cage (C,n)),i) is horizontal
proof
let i, n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x, p being Point of (TOP-REAL 2) st x in S-most C & p in south_halfline x & 1 <= i & i < len (Cage (C,n)) & p in LSeg ((Cage (C,n)),i) holds
LSeg ((Cage (C,n)),i) is horizontal
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for x, p being Point of (TOP-REAL 2) st x in S-most C & p in south_halfline x & 1 <= i & i < len (Cage (C,n)) & p in LSeg ((Cage (C,n)),i) holds
LSeg ((Cage (C,n)),i) is horizontal
let x, p be Point of (TOP-REAL 2); ::_thesis: ( x in S-most C & p in south_halfline x & 1 <= i & i < len (Cage (C,n)) & p in LSeg ((Cage (C,n)),i) implies LSeg ((Cage (C,n)),i) is horizontal )
set G = Gauge (C,n);
set f = Cage (C,n);
assume that
A1: x in S-most C and
A2: p in south_halfline x and
A3: 1 <= i and
A4: i < len (Cage (C,n)) and
A5: p in LSeg ((Cage (C,n)),i) ; ::_thesis: LSeg ((Cage (C,n)),i) is horizontal
assume A6: not LSeg ((Cage (C,n)),i) is horizontal ; ::_thesis: contradiction
A7: i + 1 <= len (Cage (C,n)) by A4, NAT_1:13;
then A8: LSeg ((Cage (C,n)),i) = LSeg (((Cage (C,n)) /. i),((Cage (C,n)) /. (i + 1))) by A3, TOPREAL1:def_3;
 1 <= i + 1 by A3, NAT_1:13;
then  i + 1 in Seg (len (Cage (C,n))) by A7, FINSEQ_1:1;
then A9: i + 1 in dom (Cage (C,n)) by FINSEQ_1:def_3;
 p in L~ (Cage (C,n)) by A5, SPPOL_2:17;
then A10: p in (south_halfline x) /\ (L~ (Cage (C,n))) by A2, XBOOLE_0:def_4;
A11: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def_1;
A12: x `1 = p `1 by A2, TOPREAL1:def_12
.= ((Cage (C,n)) /. i) `1 by A5, A8, A6, SPPOL_1:19, SPPOL_1:41 ;
 i in Seg (len (Cage (C,n))) by A3, A4, FINSEQ_1:1;
then A13: i in dom (Cage (C,n)) by FINSEQ_1:def_3;
A14: x `1 = p `1 by A2, TOPREAL1:def_12
.= ((Cage (C,n)) /. (i + 1)) `1 by A5, A8, A6, SPPOL_1:19, SPPOL_1:41 ;
A15: x in C by A1, XBOOLE_0:def_4;
percases ( ((Cage (C,n)) /. i) `2 <= ((Cage (C,n)) /. (i + 1)) `2 or ((Cage (C,n)) /. i) `2 >= ((Cage (C,n)) /. (i + 1)) `2 ) ;
supposeA16: ((Cage (C,n)) /. i) `2 <= ((Cage (C,n)) /. (i + 1)) `2 ; ::_thesis: contradiction
then  ((Cage (C,n)) /. i) `2 <= p `2 by A5, A8, TOPREAL1:4;
then A17: ((Cage (C,n)) /. i) `2 < x `2 by A15, A10, Th76, XXREAL_0:2;
consider i1, i2 being Element of NAT  such that
A18: [i1,i2] in Indices (Gauge (C,n)) and
A19: (Cage (C,n)) /. i = (Gauge (C,n)) * (i1,i2) by A11, A13, GOBOARD1:def_9;
A20: i2 <= width (Gauge (C,n)) by A18, MATRIX_1:38;
A21: 1 <= i2 by A18, MATRIX_1:38;
A22: ( 1 <= i1 & i1 <= len (Gauge (C,n)) ) by A18, MATRIX_1:38;
A23: x `2 = (S-min C) `2 by A1, PSCOMP_1:55
.= S-bound C by EUCLID:52
.= ((Gauge (C,n)) * (i1,2)) `2 by A22, JORDAN8:13 ;
then  i2 < 1 + 1 by A17, A19, A20, A22, SPRECT_3:12;
then A24: i2 <= 1 by NAT_1:13;
consider j1, j2 being Element of NAT  such that
A25: [j1,j2] in Indices (Gauge (C,n)) and
A26: (Cage (C,n)) /. (i + 1) = (Gauge (C,n)) * (j1,j2) by A11, A9, GOBOARD1:def_9;
A27: j2 <= width (Gauge (C,n)) by A25, MATRIX_1:38;
now__::_thesis:_not_((Cage_(C,n))_/._i)_`2_=_((Cage_(C,n))_/._(i_+_1))_`2
assume  ((Cage (C,n)) /. i) `2 = ((Cage (C,n)) /. (i + 1)) `2 ; ::_thesis: contradiction
then A28: (Cage (C,n)) /. i = (Cage (C,n)) /. (i + 1) by A14, A12, TOPREAL3:6;
then A29: i1 = j1 by A18, A19, A25, A26, GOBOARD1:5;
A30: i2 = j2 by A18, A19, A25, A26, A28, GOBOARD1:5;
 (abs (i1 - j1)) + (abs (i2 - j2)) = 1 by A11, A13, A9, A18, A19, A25, A26, GOBOARD1:def_9;
then 1 = 0 + (abs (i2 - j2)) by A29, GOBOARD7:2
.= 0 + 0 by A30, GOBOARD7:2 ;
hence  contradiction ; ::_thesis: verum
end;
then A31: ((Cage (C,n)) /. i) `2 < ((Cage (C,n)) /. (i + 1)) `2 by A16, XXREAL_0:1;
A32: ( 1 <= j1 & j1 <= len (Gauge (C,n)) ) by A25, MATRIX_1:38;
 1 <= j2 by A25, MATRIX_1:38;
then  i2 < j2 by A19, A20, A22, A26, A32, A31, Th19;
then  1 < j2 by A21, A24, XXREAL_0:1;
then  1 + 1 <= j2 by NAT_1:13;
then  x `2 <= ((Cage (C,n)) /. (i + 1)) `2 by A22, A23, A26, A27, A32, Th19;
then  x in L~ (Cage (C,n)) by A8, A14, A12, A17, GOBOARD7:7, SPPOL_2:17;
then  x in (L~ (Cage (C,n))) /\ C by A15, XBOOLE_0:def_4;
then  L~ (Cage (C,n)) meets C by XBOOLE_0:4;
hence  contradiction by JORDAN10:5; ::_thesis: verum
end;
supposeA33: ((Cage (C,n)) /. i) `2 >= ((Cage (C,n)) /. (i + 1)) `2 ; ::_thesis: contradiction
then  ((Cage (C,n)) /. (i + 1)) `2 <= p `2 by A5, A8, TOPREAL1:4;
then A34: ((Cage (C,n)) /. (i + 1)) `2 < x `2 by A15, A10, Th76, XXREAL_0:2;
consider i1, i2 being Element of NAT  such that
A35: [i1,i2] in Indices (Gauge (C,n)) and
A36: (Cage (C,n)) /. (i + 1) = (Gauge (C,n)) * (i1,i2) by A11, A9, GOBOARD1:def_9;
A37: i2 <= width (Gauge (C,n)) by A35, MATRIX_1:38;
A38: 1 <= i2 by A35, MATRIX_1:38;
A39: ( 1 <= i1 & i1 <= len (Gauge (C,n)) ) by A35, MATRIX_1:38;
A40: x `2 = (S-min C) `2 by A1, PSCOMP_1:55
.= S-bound C by EUCLID:52
.= ((Gauge (C,n)) * (i1,2)) `2 by A39, JORDAN8:13 ;
then  i2 < 1 + 1 by A34, A36, A37, A39, SPRECT_3:12;
then A41: i2 <= 1 by NAT_1:13;
consider j1, j2 being Element of NAT  such that
A42: [j1,j2] in Indices (Gauge (C,n)) and
A43: (Cage (C,n)) /. i = (Gauge (C,n)) * (j1,j2) by A11, A13, GOBOARD1:def_9;
A44: j2 <= width (Gauge (C,n)) by A42, MATRIX_1:38;
now__::_thesis:_not_((Cage_(C,n))_/._i)_`2_=_((Cage_(C,n))_/._(i_+_1))_`2
assume  ((Cage (C,n)) /. i) `2 = ((Cage (C,n)) /. (i + 1)) `2 ; ::_thesis: contradiction
then A45: (Cage (C,n)) /. i = (Cage (C,n)) /. (i + 1) by A14, A12, TOPREAL3:6;
then A46: i1 = j1 by A35, A36, A42, A43, GOBOARD1:5;
A47: i2 = j2 by A35, A36, A42, A43, A45, GOBOARD1:5;
 (abs (j1 - i1)) + (abs (j2 - i2)) = 1 by A11, A13, A9, A35, A36, A42, A43, GOBOARD1:def_9;
then 1 = 0 + (abs (i2 - j2)) by A46, A47, GOBOARD7:2
.= 0 + 0 by A47, GOBOARD7:2 ;
hence  contradiction ; ::_thesis: verum
end;
then A48: ((Cage (C,n)) /. (i + 1)) `2 < ((Cage (C,n)) /. i) `2 by A33, XXREAL_0:1;
A49: ( 1 <= j1 & j1 <= len (Gauge (C,n)) ) by A42, MATRIX_1:38;
 1 <= j2 by A42, MATRIX_1:38;
then  i2 < j2 by A36, A37, A39, A43, A49, A48, Th19;
then  1 < j2 by A38, A41, XXREAL_0:1;
then  1 + 1 <= j2 by NAT_1:13;
then  x `2 <= ((Cage (C,n)) /. i) `2 by A39, A40, A43, A44, A49, Th19;
then  x in L~ (Cage (C,n)) by A8, A14, A12, A34, GOBOARD7:7, SPPOL_2:17;
then  x in (L~ (Cage (C,n))) /\ C by A15, XBOOLE_0:def_4;
then  L~ (Cage (C,n)) meets C by XBOOLE_0:4;
hence  contradiction by JORDAN10:5; ::_thesis: verum
end;
end;
end;

theorem Th81: :: JORDAN1A:81
for i, n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x, p being Point of (TOP-REAL 2) st x in W-most C & p in west_halfline x & 1 <= i & i < len (Cage (C,n)) & p in LSeg ((Cage (C,n)),i) holds
LSeg ((Cage (C,n)),i) is vertical
proof
let i, n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x, p being Point of (TOP-REAL 2) st x in W-most C & p in west_halfline x & 1 <= i & i < len (Cage (C,n)) & p in LSeg ((Cage (C,n)),i) holds
LSeg ((Cage (C,n)),i) is vertical
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for x, p being Point of (TOP-REAL 2) st x in W-most C & p in west_halfline x & 1 <= i & i < len (Cage (C,n)) & p in LSeg ((Cage (C,n)),i) holds
LSeg ((Cage (C,n)),i) is vertical
let x, p be Point of (TOP-REAL 2); ::_thesis: ( x in W-most C & p in west_halfline x & 1 <= i & i < len (Cage (C,n)) & p in LSeg ((Cage (C,n)),i) implies LSeg ((Cage (C,n)),i) is vertical )
set G = Gauge (C,n);
set f = Cage (C,n);
assume that
A1: x in W-most C and
A2: p in west_halfline x and
A3: 1 <= i and
A4: i < len (Cage (C,n)) and
A5: p in LSeg ((Cage (C,n)),i) ; ::_thesis: LSeg ((Cage (C,n)),i) is vertical
assume A6: not LSeg ((Cage (C,n)),i) is vertical ; ::_thesis: contradiction
A7: i + 1 <= len (Cage (C,n)) by A4, NAT_1:13;
then A8: LSeg ((Cage (C,n)),i) = LSeg (((Cage (C,n)) /. i),((Cage (C,n)) /. (i + 1))) by A3, TOPREAL1:def_3;
 1 <= i + 1 by A3, NAT_1:13;
then  i + 1 in Seg (len (Cage (C,n))) by A7, FINSEQ_1:1;
then A9: i + 1 in dom (Cage (C,n)) by FINSEQ_1:def_3;
 p in L~ (Cage (C,n)) by A5, SPPOL_2:17;
then A10: p in (west_halfline x) /\ (L~ (Cage (C,n))) by A2, XBOOLE_0:def_4;
A11: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def_1;
A12: x `2 = p `2 by A2, TOPREAL1:def_13
.= ((Cage (C,n)) /. i) `2 by A5, A8, A6, SPPOL_1:19, SPPOL_1:40 ;
 i in Seg (len (Cage (C,n))) by A3, A4, FINSEQ_1:1;
then A13: i in dom (Cage (C,n)) by FINSEQ_1:def_3;
A14: x `2 = p `2 by A2, TOPREAL1:def_13
.= ((Cage (C,n)) /. (i + 1)) `2 by A5, A8, A6, SPPOL_1:19, SPPOL_1:40 ;
A15: x in C by A1, XBOOLE_0:def_4;
percases ( ((Cage (C,n)) /. i) `1 <= ((Cage (C,n)) /. (i + 1)) `1 or ((Cage (C,n)) /. i) `1 >= ((Cage (C,n)) /. (i + 1)) `1 ) ;
supposeA16: ((Cage (C,n)) /. i) `1 <= ((Cage (C,n)) /. (i + 1)) `1 ; ::_thesis: contradiction
consider i1, i2 being Element of NAT  such that
A17: [i1,i2] in Indices (Gauge (C,n)) and
A18: (Cage (C,n)) /. i = (Gauge (C,n)) * (i1,i2) by A11, A13, GOBOARD1:def_9;
A19: 1 <= i2 by A17, MATRIX_1:38;
A20: i2 <= width (Gauge (C,n)) by A17, MATRIX_1:38;
then A21: i2 <= len (Gauge (C,n)) by JORDAN8:def_1;
consider j1, j2 being Element of NAT  such that
A22: [j1,j2] in Indices (Gauge (C,n)) and
A23: (Cage (C,n)) /. (i + 1) = (Gauge (C,n)) * (j1,j2) by A11, A9, GOBOARD1:def_9;
A24: ( 1 <= j2 & j2 <= width (Gauge (C,n)) ) by A22, MATRIX_1:38;
now__::_thesis:_not_((Cage_(C,n))_/._i)_`1_=_((Cage_(C,n))_/._(i_+_1))_`1
assume  ((Cage (C,n)) /. i) `1 = ((Cage (C,n)) /. (i + 1)) `1 ; ::_thesis: contradiction
then A25: (Cage (C,n)) /. i = (Cage (C,n)) /. (i + 1) by A14, A12, TOPREAL3:6;
then A26: i1 = j1 by A17, A18, A22, A23, GOBOARD1:5;
A27: i2 = j2 by A17, A18, A22, A23, A25, GOBOARD1:5;
 (abs (i1 - j1)) + (abs (i2 - j2)) = 1 by A11, A13, A9, A17, A18, A22, A23, GOBOARD1:def_9;
then 1 = 0 + (abs (i2 - j2)) by A26, GOBOARD7:2
.= 0 + 0 by A27, GOBOARD7:2 ;
hence  contradiction ; ::_thesis: verum
end;
then A28: ((Cage (C,n)) /. i) `1 < ((Cage (C,n)) /. (i + 1)) `1 by A16, XXREAL_0:1;
 ((Cage (C,n)) /. i) `1 <= p `1 by A5, A8, A16, TOPREAL1:3;
then A29: ((Cage (C,n)) /. i) `1 < x `1 by A15, A10, Th77, XXREAL_0:2;
A30: 1 <= i1 by A17, MATRIX_1:38;
A31: i1 <= len (Gauge (C,n)) by A17, MATRIX_1:38;
A32: x `1 = (W-min C) `1 by A1, PSCOMP_1:31
.= W-bound C by EUCLID:52
.= ((Gauge (C,n)) * (2,i2)) `1 by A19, A21, JORDAN8:11 ;
then  i1 < 1 + 1 by A29, A18, A19, A20, A31, SPRECT_3:13;
then A33: i1 <= 1 by NAT_1:13;
A34: j1 <= len (Gauge (C,n)) by A22, MATRIX_1:38;
 1 <= j1 by A22, MATRIX_1:38;
then  i1 < j1 by A18, A19, A20, A31, A23, A24, A28, Th18;
then  1 < j1 by A30, A33, XXREAL_0:1;
then  1 + 1 <= j1 by NAT_1:13;
then  x `1 <= ((Cage (C,n)) /. (i + 1)) `1 by A19, A20, A32, A23, A24, A34, Th18;
then  x in L~ (Cage (C,n)) by A8, A14, A12, A29, GOBOARD7:8, SPPOL_2:17;
then  x in (L~ (Cage (C,n))) /\ C by A15, XBOOLE_0:def_4;
then  L~ (Cage (C,n)) meets C by XBOOLE_0:4;
hence  contradiction by JORDAN10:5; ::_thesis: verum
end;
supposeA35: ((Cage (C,n)) /. i) `1 >= ((Cage (C,n)) /. (i + 1)) `1 ; ::_thesis: contradiction
consider i1, i2 being Element of NAT  such that
A36: [i1,i2] in Indices (Gauge (C,n)) and
A37: (Cage (C,n)) /. (i + 1) = (Gauge (C,n)) * (i1,i2) by A11, A9, GOBOARD1:def_9;
A38: 1 <= i2 by A36, MATRIX_1:38;
A39: i2 <= width (Gauge (C,n)) by A36, MATRIX_1:38;
then A40: i2 <= len (Gauge (C,n)) by JORDAN8:def_1;
consider j1, j2 being Element of NAT  such that
A41: [j1,j2] in Indices (Gauge (C,n)) and
A42: (Cage (C,n)) /. i = (Gauge (C,n)) * (j1,j2) by A11, A13, GOBOARD1:def_9;
A43: ( 1 <= j2 & j2 <= width (Gauge (C,n)) ) by A41, MATRIX_1:38;
now__::_thesis:_not_((Cage_(C,n))_/._i)_`1_=_((Cage_(C,n))_/._(i_+_1))_`1
assume  ((Cage (C,n)) /. i) `1 = ((Cage (C,n)) /. (i + 1)) `1 ; ::_thesis: contradiction
then A44: (Cage (C,n)) /. i = (Cage (C,n)) /. (i + 1) by A14, A12, TOPREAL3:6;
then A45: i1 = j1 by A36, A37, A41, A42, GOBOARD1:5;
A46: i2 = j2 by A36, A37, A41, A42, A44, GOBOARD1:5;
 (abs (j1 - i1)) + (abs (j2 - i2)) = 1 by A11, A13, A9, A36, A37, A41, A42, GOBOARD1:def_9;
then 1 = 0 + (abs (i2 - j2)) by A45, A46, GOBOARD7:2
.= 0 + 0 by A46, GOBOARD7:2 ;
hence  contradiction ; ::_thesis: verum
end;
then A47: ((Cage (C,n)) /. (i + 1)) `1 < ((Cage (C,n)) /. i) `1 by A35, XXREAL_0:1;
 ((Cage (C,n)) /. (i + 1)) `1 <= p `1 by A5, A8, A35, TOPREAL1:3;
then A48: ((Cage (C,n)) /. (i + 1)) `1 < x `1 by A15, A10, Th77, XXREAL_0:2;
A49: 1 <= i1 by A36, MATRIX_1:38;
A50: i1 <= len (Gauge (C,n)) by A36, MATRIX_1:38;
A51: x `1 = (W-min C) `1 by A1, PSCOMP_1:31
.= W-bound C by EUCLID:52
.= ((Gauge (C,n)) * (2,i2)) `1 by A38, A40, JORDAN8:11 ;
then  i1 < 1 + 1 by A48, A37, A38, A39, A50, SPRECT_3:13;
then A52: i1 <= 1 by NAT_1:13;
A53: j1 <= len (Gauge (C,n)) by A41, MATRIX_1:38;
 1 <= j1 by A41, MATRIX_1:38;
then  i1 < j1 by A37, A38, A39, A50, A42, A43, A47, Th18;
then  1 < j1 by A49, A52, XXREAL_0:1;
then  1 + 1 <= j1 by NAT_1:13;
then  x `1 <= ((Cage (C,n)) /. i) `1 by A38, A39, A51, A42, A43, A53, Th18;
then  x in L~ (Cage (C,n)) by A8, A14, A12, A48, GOBOARD7:8, SPPOL_2:17;
then  x in (L~ (Cage (C,n))) /\ C by A15, XBOOLE_0:def_4;
then  L~ (Cage (C,n)) meets C by XBOOLE_0:4;
hence  contradiction by JORDAN10:5; ::_thesis: verum
end;
end;
end;

theorem Th82: :: JORDAN1A:82
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x, p being Point of (TOP-REAL 2) st x in N-most C & p in (north_halfline x) /\ (L~ (Cage (C,n))) holds
p `2 = N-bound (L~ (Cage (C,n)))
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x, p being Point of (TOP-REAL 2) st x in N-most C & p in (north_halfline x) /\ (L~ (Cage (C,n))) holds
p `2 = N-bound (L~ (Cage (C,n)))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for x, p being Point of (TOP-REAL 2) st x in N-most C & p in (north_halfline x) /\ (L~ (Cage (C,n))) holds
p `2 = N-bound (L~ (Cage (C,n)))
let x, p be Point of (TOP-REAL 2); ::_thesis: ( x in N-most C & p in (north_halfline x) /\ (L~ (Cage (C,n))) implies p `2 = N-bound (L~ (Cage (C,n))) )
set G = Gauge (C,n);
set f = Cage (C,n);
A1: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def_1;
assume A2: x in N-most C ; ::_thesis: ( not p in (north_halfline x) /\ (L~ (Cage (C,n))) or p `2 = N-bound (L~ (Cage (C,n))) )
then A3: x in C by XBOOLE_0:def_4;
assume A4: p in (north_halfline x) /\ (L~ (Cage (C,n))) ; ::_thesis: p `2 = N-bound (L~ (Cage (C,n)))
then  p in L~ (Cage (C,n)) by XBOOLE_0:def_4;
then consider i being Element of NAT  such that
A5: 1 <= i and
A6: i + 1 <= len (Cage (C,n)) and
A7: p in LSeg ((Cage (C,n)),i) by SPPOL_2:13;
A8: LSeg ((Cage (C,n)),i) = LSeg (((Cage (C,n)) /. i),((Cage (C,n)) /. (i + 1))) by A5, A6, TOPREAL1:def_3;
A9: i < len (Cage (C,n)) by A6, NAT_1:13;
then  i in Seg (len (Cage (C,n))) by A5, FINSEQ_1:1;
then  i in dom (Cage (C,n)) by FINSEQ_1:def_3;
then consider i1, i2 being Element of NAT  such that
A10: [i1,i2] in Indices (Gauge (C,n)) and
A11: (Cage (C,n)) /. i = (Gauge (C,n)) * (i1,i2) by A1, GOBOARD1:def_9;
A12: 1 <= i2 by A10, MATRIX_1:38;
 p in north_halfline x by A4, XBOOLE_0:def_4;
then  LSeg ((Cage (C,n)),i) is horizontal by A2, A5, A7, A9, Th78;
then  ((Cage (C,n)) /. i) `2 = ((Cage (C,n)) /. (i + 1)) `2 by A8, SPPOL_1:15;
then A13: p `2 = ((Cage (C,n)) /. i) `2 by A7, A8, GOBOARD7:6;
A14: i2 <= width (Gauge (C,n)) by A10, MATRIX_1:38;
A15: ( 1 <= i1 & i1 <= len (Gauge (C,n)) ) by A10, MATRIX_1:38;
A16: (len (Gauge (C,n))) -' 1 <= len (Gauge (C,n)) by NAT_D:35;
A17: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
x `2 = (N-min C) `2 by A2, PSCOMP_1:39
.= N-bound C by EUCLID:52
.= ((Gauge (C,n)) * (i1,((len (Gauge (C,n))) -' 1))) `2 by A15, JORDAN8:14 ;
then  i2 > (len (Gauge (C,n))) -' 1 by A3, A4, A11, A17, A12, A15, A13, A16, Th74, SPRECT_3:12;
then  i2 >= ((len (Gauge (C,n))) -' 1) + 1 by NAT_1:13;
then  i2 >= len (Gauge (C,n)) by A12, XREAL_1:235, XXREAL_0:2;
then  i2 = len (Gauge (C,n)) by A17, A14, XXREAL_0:1;
then  (Cage (C,n)) /. i in N-most (L~ (Cage (C,n))) by A5, A9, A11, A17, A15, Th58;
hence  p `2 = N-bound (L~ (Cage (C,n))) by A13, Th3; ::_thesis: verum
end;

theorem Th83: :: JORDAN1A:83
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x, p being Point of (TOP-REAL 2) st x in E-most C & p in (east_halfline x) /\ (L~ (Cage (C,n))) holds
p `1 = E-bound (L~ (Cage (C,n)))
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x, p being Point of (TOP-REAL 2) st x in E-most C & p in (east_halfline x) /\ (L~ (Cage (C,n))) holds
p `1 = E-bound (L~ (Cage (C,n)))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for x, p being Point of (TOP-REAL 2) st x in E-most C & p in (east_halfline x) /\ (L~ (Cage (C,n))) holds
p `1 = E-bound (L~ (Cage (C,n)))
let x, p be Point of (TOP-REAL 2); ::_thesis: ( x in E-most C & p in (east_halfline x) /\ (L~ (Cage (C,n))) implies p `1 = E-bound (L~ (Cage (C,n))) )
set G = Gauge (C,n);
set f = Cage (C,n);
A1: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def_1;
assume A2: x in E-most C ; ::_thesis: ( not p in (east_halfline x) /\ (L~ (Cage (C,n))) or p `1 = E-bound (L~ (Cage (C,n))) )
then A3: x in C by XBOOLE_0:def_4;
A4: (len (Gauge (C,n))) -' 1 <= len (Gauge (C,n)) by NAT_D:35;
A5: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
assume A6: p in (east_halfline x) /\ (L~ (Cage (C,n))) ; ::_thesis: p `1 = E-bound (L~ (Cage (C,n)))
then  p in L~ (Cage (C,n)) by XBOOLE_0:def_4;
then consider i being Element of NAT  such that
A7: 1 <= i and
A8: i + 1 <= len (Cage (C,n)) and
A9: p in LSeg ((Cage (C,n)),i) by SPPOL_2:13;
A10: LSeg ((Cage (C,n)),i) = LSeg (((Cage (C,n)) /. i),((Cage (C,n)) /. (i + 1))) by A7, A8, TOPREAL1:def_3;
A11: i < len (Cage (C,n)) by A8, NAT_1:13;
then  i in Seg (len (Cage (C,n))) by A7, FINSEQ_1:1;
then  i in dom (Cage (C,n)) by FINSEQ_1:def_3;
then consider i1, i2 being Element of NAT  such that
A12: [i1,i2] in Indices (Gauge (C,n)) and
A13: (Cage (C,n)) /. i = (Gauge (C,n)) * (i1,i2) by A1, GOBOARD1:def_9;
A14: ( 1 <= i2 & i2 <= width (Gauge (C,n)) ) by A12, MATRIX_1:38;
 p in east_halfline x by A6, XBOOLE_0:def_4;
then  LSeg ((Cage (C,n)),i) is vertical by A2, A7, A9, A11, Th79;
then  ((Cage (C,n)) /. i) `1 = ((Cage (C,n)) /. (i + 1)) `1 by A10, SPPOL_1:16;
then A15: p `1 = ((Cage (C,n)) /. i) `1 by A9, A10, GOBOARD7:5;
A16: i1 <= len (Gauge (C,n)) by A12, MATRIX_1:38;
A17: 1 <= i1 by A12, MATRIX_1:38;
x `1 = (E-min C) `1 by A2, PSCOMP_1:47
.= E-bound C by EUCLID:52
.= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),i2)) `1 by A5, A14, JORDAN8:12 ;
then  i1 > (len (Gauge (C,n))) -' 1 by A3, A6, A13, A14, A17, A15, A4, Th75, SPRECT_3:13;
then  i1 >= ((len (Gauge (C,n))) -' 1) + 1 by NAT_1:13;
then  i1 >= len (Gauge (C,n)) by A17, XREAL_1:235, XXREAL_0:2;
then  i1 = len (Gauge (C,n)) by A16, XXREAL_0:1;
then  (Cage (C,n)) /. i in E-most (L~ (Cage (C,n))) by A7, A11, A13, A14, Th61;
hence  p `1 = E-bound (L~ (Cage (C,n))) by A15, Th4; ::_thesis: verum
end;

theorem Th84: :: JORDAN1A:84
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x, p being Point of (TOP-REAL 2) st x in S-most C & p in (south_halfline x) /\ (L~ (Cage (C,n))) holds
p `2 = S-bound (L~ (Cage (C,n)))
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x, p being Point of (TOP-REAL 2) st x in S-most C & p in (south_halfline x) /\ (L~ (Cage (C,n))) holds
p `2 = S-bound (L~ (Cage (C,n)))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for x, p being Point of (TOP-REAL 2) st x in S-most C & p in (south_halfline x) /\ (L~ (Cage (C,n))) holds
p `2 = S-bound (L~ (Cage (C,n)))
let x, p be Point of (TOP-REAL 2); ::_thesis: ( x in S-most C & p in (south_halfline x) /\ (L~ (Cage (C,n))) implies p `2 = S-bound (L~ (Cage (C,n))) )
set G = Gauge (C,n);
set f = Cage (C,n);
A1: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def_1;
assume A2: x in S-most C ; ::_thesis: ( not p in (south_halfline x) /\ (L~ (Cage (C,n))) or p `2 = S-bound (L~ (Cage (C,n))) )
then A3: x in C by XBOOLE_0:def_4;
assume A4: p in (south_halfline x) /\ (L~ (Cage (C,n))) ; ::_thesis: p `2 = S-bound (L~ (Cage (C,n)))
then  p in L~ (Cage (C,n)) by XBOOLE_0:def_4;
then consider i being Element of NAT  such that
A5: 1 <= i and
A6: i + 1 <= len (Cage (C,n)) and
A7: p in LSeg ((Cage (C,n)),i) by SPPOL_2:13;
A8: LSeg ((Cage (C,n)),i) = LSeg (((Cage (C,n)) /. i),((Cage (C,n)) /. (i + 1))) by A5, A6, TOPREAL1:def_3;
A9: i < len (Cage (C,n)) by A6, NAT_1:13;
then  i in Seg (len (Cage (C,n))) by A5, FINSEQ_1:1;
then  i in dom (Cage (C,n)) by FINSEQ_1:def_3;
then consider i1, i2 being Element of NAT  such that
A10: [i1,i2] in Indices (Gauge (C,n)) and
A11: (Cage (C,n)) /. i = (Gauge (C,n)) * (i1,i2) by A1, GOBOARD1:def_9;
A12: 1 <= i2 by A10, MATRIX_1:38;
 p in south_halfline x by A4, XBOOLE_0:def_4;
then  LSeg ((Cage (C,n)),i) is horizontal by A2, A5, A7, A9, Th80;
then  ((Cage (C,n)) /. i) `2 = ((Cage (C,n)) /. (i + 1)) `2 by A8, SPPOL_1:15;
then A13: p `2 = ((Cage (C,n)) /. i) `2 by A7, A8, GOBOARD7:6;
A14: i2 <= width (Gauge (C,n)) by A10, MATRIX_1:38;
A15: ( 1 <= i1 & i1 <= len (Gauge (C,n)) ) by A10, MATRIX_1:38;
x `2 = (S-min C) `2 by A2, PSCOMP_1:55
.= S-bound C by EUCLID:52
.= ((Gauge (C,n)) * (i1,2)) `2 by A15, JORDAN8:13 ;
then  i2 < 1 + 1 by A3, A4, A11, A14, A15, A13, Th76, SPRECT_3:12;
then  i2 <= 1 by NAT_1:13;
then  i2 = 1 by A12, XXREAL_0:1;
then  (Cage (C,n)) /. i in S-most (L~ (Cage (C,n))) by A5, A9, A11, A15, Th60;
hence  p `2 = S-bound (L~ (Cage (C,n))) by A13, Th5; ::_thesis: verum
end;

theorem Th85: :: JORDAN1A:85
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x, p being Point of (TOP-REAL 2) st x in W-most C & p in (west_halfline x) /\ (L~ (Cage (C,n))) holds
p `1 = W-bound (L~ (Cage (C,n)))
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x, p being Point of (TOP-REAL 2) st x in W-most C & p in (west_halfline x) /\ (L~ (Cage (C,n))) holds
p `1 = W-bound (L~ (Cage (C,n)))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for x, p being Point of (TOP-REAL 2) st x in W-most C & p in (west_halfline x) /\ (L~ (Cage (C,n))) holds
p `1 = W-bound (L~ (Cage (C,n)))
let x, p be Point of (TOP-REAL 2); ::_thesis: ( x in W-most C & p in (west_halfline x) /\ (L~ (Cage (C,n))) implies p `1 = W-bound (L~ (Cage (C,n))) )
set G = Gauge (C,n);
set f = Cage (C,n);
A1: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def_1;
assume A2: x in W-most C ; ::_thesis: ( not p in (west_halfline x) /\ (L~ (Cage (C,n))) or p `1 = W-bound (L~ (Cage (C,n))) )
then A3: x in C by XBOOLE_0:def_4;
A4: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
assume A5: p in (west_halfline x) /\ (L~ (Cage (C,n))) ; ::_thesis: p `1 = W-bound (L~ (Cage (C,n)))
then  p in L~ (Cage (C,n)) by XBOOLE_0:def_4;
then consider i being Element of NAT  such that
A6: 1 <= i and
A7: i + 1 <= len (Cage (C,n)) and
A8: p in LSeg ((Cage (C,n)),i) by SPPOL_2:13;
A9: LSeg ((Cage (C,n)),i) = LSeg (((Cage (C,n)) /. i),((Cage (C,n)) /. (i + 1))) by A6, A7, TOPREAL1:def_3;
A10: i < len (Cage (C,n)) by A7, NAT_1:13;
then  i in Seg (len (Cage (C,n))) by A6, FINSEQ_1:1;
then  i in dom (Cage (C,n)) by FINSEQ_1:def_3;
then consider i1, i2 being Element of NAT  such that
A11: [i1,i2] in Indices (Gauge (C,n)) and
A12: (Cage (C,n)) /. i = (Gauge (C,n)) * (i1,i2) by A1, GOBOARD1:def_9;
A13: ( 1 <= i2 & i2 <= width (Gauge (C,n)) ) by A11, MATRIX_1:38;
 p in west_halfline x by A5, XBOOLE_0:def_4;
then  LSeg ((Cage (C,n)),i) is vertical by A2, A6, A8, A10, Th81;
then  ((Cage (C,n)) /. i) `1 = ((Cage (C,n)) /. (i + 1)) `1 by A9, SPPOL_1:16;
then A14: p `1 = ((Cage (C,n)) /. i) `1 by A8, A9, GOBOARD7:5;
A15: i1 <= len (Gauge (C,n)) by A11, MATRIX_1:38;
A16: 1 <= i1 by A11, MATRIX_1:38;
x `1 = (W-min C) `1 by A2, PSCOMP_1:31
.= W-bound C by EUCLID:52
.= ((Gauge (C,n)) * (2,i2)) `1 by A4, A13, JORDAN8:11 ;
then  i1 < 1 + 1 by A3, A5, A12, A13, A15, A14, Th77, SPRECT_3:13;
then  i1 <= 1 by NAT_1:13;
then  i1 = 1 by A16, XXREAL_0:1;
then  (Cage (C,n)) /. i in W-most (L~ (Cage (C,n))) by A6, A10, A12, A13, Th59;
hence  p `1 = W-bound (L~ (Cage (C,n))) by A14, Th6; ::_thesis: verum
end;

theorem :: JORDAN1A:86
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x being Point of (TOP-REAL 2) st x in N-most C holds
ex p being Point of (TOP-REAL 2) st (north_halfline x) /\ (L~ (Cage (C,n))) = {p}
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x being Point of (TOP-REAL 2) st x in N-most C holds
ex p being Point of (TOP-REAL 2) st (north_halfline x) /\ (L~ (Cage (C,n))) = {p}
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for x being Point of (TOP-REAL 2) st x in N-most C holds
ex p being Point of (TOP-REAL 2) st (north_halfline x) /\ (L~ (Cage (C,n))) = {p}
let x be Point of (TOP-REAL 2); ::_thesis: ( x in N-most C implies ex p being Point of (TOP-REAL 2) st (north_halfline x) /\ (L~ (Cage (C,n))) = {p} )
set f = Cage (C,n);
assume A1: x in N-most C ; ::_thesis: ex p being Point of (TOP-REAL 2) st (north_halfline x) /\ (L~ (Cage (C,n))) = {p}
then  x in C by XBOOLE_0:def_4;
then  north_halfline x meets L~ (Cage (C,n)) by Th51;
then consider p being set  such that
A2: p in north_halfline x and
A3: p in L~ (Cage (C,n)) by XBOOLE_0:3;
A4: p in (north_halfline x) /\ (L~ (Cage (C,n))) by A2, A3, XBOOLE_0:def_4;
reconsider p = p as Point of (TOP-REAL 2) by A2;
take  p ; ::_thesis: (north_halfline x) /\ (L~ (Cage (C,n))) = {p}
hereby  :: according to TARSKI:def_3,XBOOLE_0:def_10 ::_thesis: {p} c= (north_halfline x) /\ (L~ (Cage (C,n)))
let a be set ; ::_thesis: ( a in (north_halfline x) /\ (L~ (Cage (C,n))) implies a in {p} )
assume A5: a in (north_halfline x) /\ (L~ (Cage (C,n))) ; ::_thesis: a in {p}
then reconsider y = a as Point of (TOP-REAL 2) ;
 y in north_halfline x by A5, XBOOLE_0:def_4;
then A6: y `1 = x `1 by TOPREAL1:def_10
.= p `1 by A2, TOPREAL1:def_10 ;
p `2 = N-bound (L~ (Cage (C,n))) by A1, A4, Th82
.= y `2 by A1, A5, Th82 ;
then  y = p by A6, TOPREAL3:6;
hence  a in {p} by TARSKI:def_1; ::_thesis: verum
end;
thus  {p} c= (north_halfline x) /\ (L~ (Cage (C,n))) by A4, ZFMISC_1:31; ::_thesis: verum
end;

theorem :: JORDAN1A:87
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x being Point of (TOP-REAL 2) st x in E-most C holds
ex p being Point of (TOP-REAL 2) st (east_halfline x) /\ (L~ (Cage (C,n))) = {p}
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x being Point of (TOP-REAL 2) st x in E-most C holds
ex p being Point of (TOP-REAL 2) st (east_halfline x) /\ (L~ (Cage (C,n))) = {p}
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for x being Point of (TOP-REAL 2) st x in E-most C holds
ex p being Point of (TOP-REAL 2) st (east_halfline x) /\ (L~ (Cage (C,n))) = {p}
let x be Point of (TOP-REAL 2); ::_thesis: ( x in E-most C implies ex p being Point of (TOP-REAL 2) st (east_halfline x) /\ (L~ (Cage (C,n))) = {p} )
set f = Cage (C,n);
assume A1: x in E-most C ; ::_thesis: ex p being Point of (TOP-REAL 2) st (east_halfline x) /\ (L~ (Cage (C,n))) = {p}
then  x in C by XBOOLE_0:def_4;
then  east_halfline x meets L~ (Cage (C,n)) by Th52;
then consider p being set  such that
A2: p in east_halfline x and
A3: p in L~ (Cage (C,n)) by XBOOLE_0:3;
A4: p in (east_halfline x) /\ (L~ (Cage (C,n))) by A2, A3, XBOOLE_0:def_4;
reconsider p = p as Point of (TOP-REAL 2) by A2;
take  p ; ::_thesis: (east_halfline x) /\ (L~ (Cage (C,n))) = {p}
hereby  :: according to TARSKI:def_3,XBOOLE_0:def_10 ::_thesis: {p} c= (east_halfline x) /\ (L~ (Cage (C,n)))
let a be set ; ::_thesis: ( a in (east_halfline x) /\ (L~ (Cage (C,n))) implies a in {p} )
assume A5: a in (east_halfline x) /\ (L~ (Cage (C,n))) ; ::_thesis: a in {p}
then reconsider y = a as Point of (TOP-REAL 2) ;
 y in east_halfline x by A5, XBOOLE_0:def_4;
then A6: y `2 = x `2 by TOPREAL1:def_11
.= p `2 by A2, TOPREAL1:def_11 ;
p `1 = E-bound (L~ (Cage (C,n))) by A1, A4, Th83
.= y `1 by A1, A5, Th83 ;
then  y = p by A6, TOPREAL3:6;
hence  a in {p} by TARSKI:def_1; ::_thesis: verum
end;
thus  {p} c= (east_halfline x) /\ (L~ (Cage (C,n))) by A4, ZFMISC_1:31; ::_thesis: verum
end;

theorem :: JORDAN1A:88
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x being Point of (TOP-REAL 2) st x in S-most C holds
ex p being Point of (TOP-REAL 2) st (south_halfline x) /\ (L~ (Cage (C,n))) = {p}
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x being Point of (TOP-REAL 2) st x in S-most C holds
ex p being Point of (TOP-REAL 2) st (south_halfline x) /\ (L~ (Cage (C,n))) = {p}
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for x being Point of (TOP-REAL 2) st x in S-most C holds
ex p being Point of (TOP-REAL 2) st (south_halfline x) /\ (L~ (Cage (C,n))) = {p}
let x be Point of (TOP-REAL 2); ::_thesis: ( x in S-most C implies ex p being Point of (TOP-REAL 2) st (south_halfline x) /\ (L~ (Cage (C,n))) = {p} )
set f = Cage (C,n);
assume A1: x in S-most C ; ::_thesis: ex p being Point of (TOP-REAL 2) st (south_halfline x) /\ (L~ (Cage (C,n))) = {p}
then  x in C by XBOOLE_0:def_4;
then  south_halfline x meets L~ (Cage (C,n)) by Th53;
then consider p being set  such that
A2: p in south_halfline x and
A3: p in L~ (Cage (C,n)) by XBOOLE_0:3;
A4: p in (south_halfline x) /\ (L~ (Cage (C,n))) by A2, A3, XBOOLE_0:def_4;
reconsider p = p as Point of (TOP-REAL 2) by A2;
take  p ; ::_thesis: (south_halfline x) /\ (L~ (Cage (C,n))) = {p}
hereby  :: according to TARSKI:def_3,XBOOLE_0:def_10 ::_thesis: {p} c= (south_halfline x) /\ (L~ (Cage (C,n)))
let a be set ; ::_thesis: ( a in (south_halfline x) /\ (L~ (Cage (C,n))) implies a in {p} )
assume A5: a in (south_halfline x) /\ (L~ (Cage (C,n))) ; ::_thesis: a in {p}
then reconsider y = a as Point of (TOP-REAL 2) ;
 y in south_halfline x by A5, XBOOLE_0:def_4;
then A6: y `1 = x `1 by TOPREAL1:def_12
.= p `1 by A2, TOPREAL1:def_12 ;
p `2 = S-bound (L~ (Cage (C,n))) by A1, A4, Th84
.= y `2 by A1, A5, Th84 ;
then  y = p by A6, TOPREAL3:6;
hence  a in {p} by TARSKI:def_1; ::_thesis: verum
end;
thus  {p} c= (south_halfline x) /\ (L~ (Cage (C,n))) by A4, ZFMISC_1:31; ::_thesis: verum
end;

theorem :: JORDAN1A:89
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x being Point of (TOP-REAL 2) st x in W-most C holds
ex p being Point of (TOP-REAL 2) st (west_halfline x) /\ (L~ (Cage (C,n))) = {p}
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for x being Point of (TOP-REAL 2) st x in W-most C holds
ex p being Point of (TOP-REAL 2) st (west_halfline x) /\ (L~ (Cage (C,n))) = {p}
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for x being Point of (TOP-REAL 2) st x in W-most C holds
ex p being Point of (TOP-REAL 2) st (west_halfline x) /\ (L~ (Cage (C,n))) = {p}
let x be Point of (TOP-REAL 2); ::_thesis: ( x in W-most C implies ex p being Point of (TOP-REAL 2) st (west_halfline x) /\ (L~ (Cage (C,n))) = {p} )
set f = Cage (C,n);
assume A1: x in W-most C ; ::_thesis: ex p being Point of (TOP-REAL 2) st (west_halfline x) /\ (L~ (Cage (C,n))) = {p}
then  x in C by XBOOLE_0:def_4;
then  west_halfline x meets L~ (Cage (C,n)) by Th54;
then consider p being set  such that
A2: p in west_halfline x and
A3: p in L~ (Cage (C,n)) by XBOOLE_0:3;
A4: p in (west_halfline x) /\ (L~ (Cage (C,n))) by A2, A3, XBOOLE_0:def_4;
reconsider p = p as Point of (TOP-REAL 2) by A2;
take  p ; ::_thesis: (west_halfline x) /\ (L~ (Cage (C,n))) = {p}
hereby  :: according to TARSKI:def_3,XBOOLE_0:def_10 ::_thesis: {p} c= (west_halfline x) /\ (L~ (Cage (C,n)))
let a be set ; ::_thesis: ( a in (west_halfline x) /\ (L~ (Cage (C,n))) implies a in {p} )
assume A5: a in (west_halfline x) /\ (L~ (Cage (C,n))) ; ::_thesis: a in {p}
then reconsider y = a as Point of (TOP-REAL 2) ;
 y in west_halfline x by A5, XBOOLE_0:def_4;
then A6: y `2 = x `2 by TOPREAL1:def_13
.= p `2 by A2, TOPREAL1:def_13 ;
p `1 = W-bound (L~ (Cage (C,n))) by A1, A4, Th85
.= y `1 by A1, A5, Th85 ;
then  y = p by A6, TOPREAL3:6;
hence  a in {p} by TARSKI:def_1; ::_thesis: verum
end;
thus  {p} c= (west_halfline x) /\ (L~ (Cage (C,n))) by A4, ZFMISC_1:31; ::_thesis: verum
end;