:: JORDAN19 semantic presentation

begin

definition
let C be Simple_closed_curve;
func Upper_Appr C ->  SetSequence of the carrier of (TOP-REAL 2) means :Def1: :: JORDAN19:def 1
 for i being Element of NAT holds it . i = Upper_Arc (L~ (Cage (C,i)));
existence
 ex b1 being SetSequence of the carrier of (TOP-REAL 2) st
for i being Element of NAT holds b1 . i = Upper_Arc (L~ (Cage (C,i)))
proof
deffunc H1( Element of NAT ) ->  Element of K6( the carrier of (TOP-REAL 2)) = Upper_Arc (L~ (Cage (C,$1)));
thus  ex S being SetSequence of the carrier of (TOP-REAL 2) st
for i being Element of NAT holds S . i = H1(i) from FUNCT_2:sch_4(); ::_thesis: verum
end;
uniqueness
 for b1, b2 being SetSequence of the carrier of (TOP-REAL 2) st ( for i being Element of NAT holds b1 . i = Upper_Arc (L~ (Cage (C,i))) ) & ( for i being Element of NAT holds b2 . i = Upper_Arc (L~ (Cage (C,i))) ) holds
b1 = b2
proof
deffunc H1( Element of NAT ) ->  Element of K6( the carrier of (TOP-REAL 2)) = Upper_Arc (L~ (Cage (C,$1)));
thus  for S1, S2 being SetSequence of the carrier of (TOP-REAL 2) st ( for i being Element of NAT holds S1 . i = H1(i) ) & ( for i being Element of NAT holds S2 . i = H1(i) ) holds
S1 = S2 from BINOP_2:sch_1(); ::_thesis: verum
end;
func Lower_Appr C ->  SetSequence of the carrier of (TOP-REAL 2) means :Def2: :: JORDAN19:def 2
 for i being Element of NAT holds it . i = Lower_Arc (L~ (Cage (C,i)));
existence
 ex b1 being SetSequence of the carrier of (TOP-REAL 2) st
for i being Element of NAT holds b1 . i = Lower_Arc (L~ (Cage (C,i)))
proof
deffunc H1( Element of NAT ) ->  Element of K6( the carrier of (TOP-REAL 2)) = Lower_Arc (L~ (Cage (C,$1)));
thus  ex S being SetSequence of the carrier of (TOP-REAL 2) st
for i being Element of NAT holds S . i = H1(i) from FUNCT_2:sch_4(); ::_thesis: verum
end;
uniqueness
 for b1, b2 being SetSequence of the carrier of (TOP-REAL 2) st ( for i being Element of NAT holds b1 . i = Lower_Arc (L~ (Cage (C,i))) ) & ( for i being Element of NAT holds b2 . i = Lower_Arc (L~ (Cage (C,i))) ) holds
b1 = b2
proof
deffunc H1( Element of NAT ) ->  Element of K6( the carrier of (TOP-REAL 2)) = Lower_Arc (L~ (Cage (C,$1)));
thus  for S1, S2 being SetSequence of the carrier of (TOP-REAL 2) st ( for i being Element of NAT holds S1 . i = H1(i) ) & ( for i being Element of NAT holds S2 . i = H1(i) ) holds
S1 = S2 from BINOP_2:sch_1(); ::_thesis: verum
end;
end;

:: deftheorem Def1 defines Upper_Appr JORDAN19:def_1_:_
 for C being Simple_closed_curve
for b2 being SetSequence of the carrier of (TOP-REAL 2) holds
( b2 = Upper_Appr C iff for i being Element of NAT holds b2 . i = Upper_Arc (L~ (Cage (C,i))) );

:: deftheorem Def2 defines Lower_Appr JORDAN19:def_2_:_
 for C being Simple_closed_curve
for b2 being SetSequence of the carrier of (TOP-REAL 2) holds
( b2 = Lower_Appr C iff for i being Element of NAT holds b2 . i = Lower_Arc (L~ (Cage (C,i))) );

definition
let C be Simple_closed_curve;
func North_Arc C ->  Subset of (TOP-REAL 2) equals :: JORDAN19:def 3
 Lim_inf (Upper_Appr C);
coherence
 Lim_inf (Upper_Appr C) is Subset of (TOP-REAL 2) ;
func South_Arc C ->  Subset of (TOP-REAL 2) equals :: JORDAN19:def 4
 Lim_inf (Lower_Appr C);
coherence
 Lim_inf (Lower_Appr C) is Subset of (TOP-REAL 2) ;
end;

:: deftheorem  defines North_Arc JORDAN19:def_3_:_
 for C being Simple_closed_curve holds North_Arc C = Lim_inf (Upper_Appr C);

:: deftheorem  defines South_Arc JORDAN19:def_4_:_
 for C being Simple_closed_curve holds South_Arc C = Lim_inf (Lower_Appr C);

Lm1: now__::_thesis:_for_G_being_Go-board
for_j_being_Element_of_NAT_st_1_<=_j_&_j_<=_width_G_holds_
[(Center_G),j]_in_Indices_G
let G be Go-board; ::_thesis: for j being Element of NAT st 1 <= j & j <= width G holds
[(Center G),j] in Indices G
let j be Element of NAT ; ::_thesis: ( 1 <= j & j <= width G implies [(Center G),j] in Indices G )
assume that
A1: 1 <= j and
A2: j <= width G ; ::_thesis: [(Center G),j] in Indices G
 0 + 1 <= ((len G) div 2) + 1 by XREAL_1:6;
then A3: 0 + 1 <= Center G by JORDAN1A:def_1;
 Center G <= len G by JORDAN1B:13;
hence  [(Center G),j] in Indices G by A1, A2, A3, MATRIX_1:36; ::_thesis: verum
end;

Lm2: now__::_thesis:_for_D_being_non_empty_Subset_of_(TOP-REAL_2)
for_n,_i_being_Element_of_NAT_st_[i,(width_(Gauge_(D,n)))]_in_Indices_(Gauge_(D,n))_holds_
((Gauge_(D,n))_*_(i,(width_(Gauge_(D,n)))))_`2_=_(S-bound_D)_+_((((N-bound_D)_-_(S-bound_D))_/_(2_|^_n))_*_((width_(Gauge_(D,n)))_-_2))
let D be non empty Subset of (TOP-REAL 2); ::_thesis: for n, i being Element of NAT st [i,(width (Gauge (D,n)))] in Indices (Gauge (D,n)) holds
((Gauge (D,n)) * (i,(width (Gauge (D,n))))) `2 = (S-bound D) + ((((N-bound D) - (S-bound D)) / (2 |^ n)) * ((width (Gauge (D,n))) - 2))
let n, i be Element of NAT ; ::_thesis: ( [i,(width (Gauge (D,n)))] in Indices (Gauge (D,n)) implies ((Gauge (D,n)) * (i,(width (Gauge (D,n))))) `2 = (S-bound D) + ((((N-bound D) - (S-bound D)) / (2 |^ n)) * ((width (Gauge (D,n))) - 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);
assume  [i,(width (Gauge (D,n)))] in Indices (Gauge (D,n)) ; ::_thesis: ((Gauge (D,n)) * (i,(width (Gauge (D,n))))) `2 = (S-bound D) + ((((N-bound D) - (S-bound D)) / (2 |^ n)) * ((width (Gauge (D,n))) - 2))
hence ((Gauge (D,n)) * (i,(width (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)) * ((width (Gauge (D,n))) - 2)))]| `2 by JORDAN8:def_1
.= (S-bound D) + ((((N-bound D) - (S-bound D)) / (2 |^ n)) * ((width (Gauge (D,n))) - 2)) by EUCLID:52 ;
::_thesis: verum
end;

theorem Th1: :: JORDAN19:1
for n, m being Element of NAT st n <= m & n <> 0 holds
(n + 1) / n >= (m + 1) / m
proof
let n, m be Element of NAT ; ::_thesis: ( n <= m & n <> 0 implies (n + 1) / n >= (m + 1) / m )
assume that
A1: n <= m and
A2: n <> 0 ; ::_thesis: (n + 1) / n >= (m + 1) / m
A3: n > 0 by A2;
A4: 1 / n >= 1 / m by A1, A2, XREAL_1:85;
A5: (n + 1) / n = (n / n) + (1 / n)
.= 1 + (1 / n) by A2, XCMPLX_1:60 ;
(m + 1) / m = (m / m) + (1 / m)
.= 1 + (1 / m) by A1, A3, XCMPLX_1:60 ;
hence  (n + 1) / n >= (m + 1) / m by A4, A5, XREAL_1:7; ::_thesis: verum
end;

theorem Th2: :: JORDAN19:2
for n being Element of NAT
for E being compact non horizontal non vertical Subset of (TOP-REAL 2)
for m, j being Element of NAT st 1 <= m & m <= n & 1 <= j & j <= width (Gauge (E,n)) holds
LSeg (((Gauge (E,n)) * ((Center (Gauge (E,n))),(width (Gauge (E,n))))),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) c= LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),(width (Gauge (E,m))))),((Gauge (E,n)) * ((Center (Gauge (E,n))),j)))
proof
let n be Element of NAT ; ::_thesis: for E being compact non horizontal non vertical Subset of (TOP-REAL 2)
for m, j being Element of NAT st 1 <= m & m <= n & 1 <= j & j <= width (Gauge (E,n)) holds
LSeg (((Gauge (E,n)) * ((Center (Gauge (E,n))),(width (Gauge (E,n))))),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) c= LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),(width (Gauge (E,m))))),((Gauge (E,n)) * ((Center (Gauge (E,n))),j)))
let E be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for m, j being Element of NAT st 1 <= m & m <= n & 1 <= j & j <= width (Gauge (E,n)) holds
LSeg (((Gauge (E,n)) * ((Center (Gauge (E,n))),(width (Gauge (E,n))))),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) c= LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),(width (Gauge (E,m))))),((Gauge (E,n)) * ((Center (Gauge (E,n))),j)))
let m, j be Element of NAT ; ::_thesis: ( 1 <= m & m <= n & 1 <= j & j <= width (Gauge (E,n)) implies LSeg (((Gauge (E,n)) * ((Center (Gauge (E,n))),(width (Gauge (E,n))))),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) c= LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),(width (Gauge (E,m))))),((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))),(width (Gauge (E,n))))),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) c= LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),(width (Gauge (E,m))))),((Gauge (E,n)) * ((Center (Gauge (E,n))),j)))
A5: width (Gauge (E,m)) = len (Gauge (E,m)) by JORDAN8:def_1
.= (2 |^ m) + 3 by JORDAN8:def_1 ;
A6: width (Gauge (E,n)) = len (Gauge (E,n)) by JORDAN8:def_1
.= (2 |^ n) + 3 by JORDAN8:def_1 ;
A7: now__::_thesis:_for_t_being_Element_of_NAT_st_width_(Gauge_(E,n))_>=_t_&_t_>=_j_holds_
(Gauge_(E,n))_*_((Center_(Gauge_(E,n))),t)_in_LSeg_(((Gauge_(E,m))_*_((Center_(Gauge_(E,m))),(width_(Gauge_(E,m))))),((Gauge_(E,n))_*_((Center_(Gauge_(E,n))),j)))
let t be Element of NAT ; ::_thesis: ( width (Gauge (E,n)) >= t & t >= j implies (Gauge (E,n)) * ((Center (Gauge (E,n))),t) in LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),(width (Gauge (E,m))))),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) )
assume that
A8: width (Gauge (E,n)) >= t and
A9: t >= j ; ::_thesis: (Gauge (E,n)) * ((Center (Gauge (E,n))),t) in LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),(width (Gauge (E,m))))),((Gauge (E,n)) * ((Center (Gauge (E,n))),j)))
A10: len (Gauge (E,m)) = width (Gauge (E,m)) by JORDAN8:def_1;
A11: len (Gauge (E,n)) = width (Gauge (E,n)) by JORDAN8:def_1;
A12: 0 < (N-bound E) - (S-bound E) by SPRECT_1:32, XREAL_1:50;
A13: t >= 1 by A3, A9, XXREAL_0:2;
A14: 0 < 2 |^ m by NEWTON:83;
A15: 1 <= len (Gauge (E,m)) by GOBRD11:34;
then A16: ((Gauge (E,m)) * ((Center (Gauge (E,m))),(width (Gauge (E,m))))) `1 = ((Gauge (E,n)) * ((Center (Gauge (E,n))),t)) `1 by A1, A2, A8, A10, A11, A13, JORDAN1A:36;
A17: ((Gauge (E,n)) * ((Center (Gauge (E,n))),t)) `1 = ((Gauge (E,n)) * ((Center (Gauge (E,n))),j)) `1 by A1, A2, A3, A4, A8, A11, A13, JORDAN1A:36;
 [(Center (Gauge (E,n))),t] in Indices (Gauge (E,n)) by A8, A13, Lm1;
then A18: ((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 ;
 [(Center (Gauge (E,m))),(width (Gauge (E,m)))] in Indices (Gauge (E,m)) by A10, A15, Lm1;
then A19: ((Gauge (E,m)) * ((Center (Gauge (E,m))),(width (Gauge (E,m))))) `2 = (S-bound E) + ((((N-bound E) - (S-bound E)) / (2 |^ m)) * ((width (Gauge (E,m))) - 2)) by Lm2;
A20: ((2 |^ m) + 1) / (2 |^ m) >= ((2 |^ n) + 1) / (2 |^ n) by A2, A14, Th1, PREPOWER:93;
 t - 2 <= ((2 |^ n) + 3) - 2 by A6, A8, XREAL_1:9;
then  (t - 2) / (2 |^ n) <= ((2 |^ n) + 1) / (2 |^ n) by XREAL_1:72;
then  (t - 2) / (2 |^ n) <= ((width (Gauge (E,m))) - 2) / (2 |^ m) by A5, A20, XXREAL_0:2;
then  ((N-bound E) - (S-bound E)) * ((t - 2) / (2 |^ n)) <= ((N-bound E) - (S-bound E)) * (((width (Gauge (E,m))) - 2) / (2 |^ m)) by A12, XREAL_1:64;
then A21: (S-bound E) + ((((N-bound E) - (S-bound E)) / (2 |^ m)) * ((width (Gauge (E,m))) - 2)) >= (S-bound E) + ((((N-bound E) - (S-bound E)) / (2 |^ n)) * (t - 2)) by XREAL_1:6;
A22: 1 <= Center (Gauge (E,n)) by JORDAN1B:11;
 Center (Gauge (E,n)) <= len (Gauge (E,n)) by JORDAN1B:13;
then  ((Gauge (E,n)) * ((Center (Gauge (E,n))),t)) `2 >= ((Gauge (E,n)) * ((Center (Gauge (E,n))),j)) `2 by A3, A8, A9, A22, SPRECT_3:12;
hence  (Gauge (E,n)) * ((Center (Gauge (E,n))),t) in LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),(width (Gauge (E,m))))),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) by A16, A17, A18, A19, A21, GOBOARD7:7; ::_thesis: verum
end;
then A23: (Gauge (E,n)) * ((Center (Gauge (E,n))),(width (Gauge (E,n)))) in LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),(width (Gauge (E,m))))),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) by A4;
 (Gauge (E,n)) * ((Center (Gauge (E,n))),j) in LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),(width (Gauge (E,m))))),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) by A4, A7;
hence  LSeg (((Gauge (E,n)) * ((Center (Gauge (E,n))),(width (Gauge (E,n))))),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) c= LSeg (((Gauge (E,m)) * ((Center (Gauge (E,m))),(width (Gauge (E,m))))),((Gauge (E,n)) * ((Center (Gauge (E,n))),j))) by A23, TOPREAL1:6; ::_thesis: verum
end;

theorem Th3: :: JORDAN19:3
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i, j being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Cage (C,n)) holds
LSeg (((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))) meets L~ (Upper_Seq (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 i, j being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Cage (C,n)) holds
LSeg (((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))) meets L~ (Upper_Seq (C,n))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for i, j being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Cage (C,n)) holds
LSeg (((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))) meets L~ (Upper_Seq (C,n))
let i, j be Element of NAT ; ::_thesis: ( 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Cage (C,n)) implies LSeg (((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))) meets L~ (Upper_Seq (C,n)) )
set Gij = (Gauge (C,n)) * (i,j);
assume that
A1: 1 <= i and
A2: i <= len (Gauge (C,n)) and
A3: 1 <= j and
A4: j <= width (Gauge (C,n)) and
A5: (Gauge (C,n)) * (i,j) in L~ (Cage (C,n)) ; ::_thesis: LSeg (((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))) meets L~ (Upper_Seq (C,n))
set NE = SW-corner (L~ (Cage (C,n)));
set v1 = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)));
set wG = width (Gauge (C,n));
set lG = len (Gauge (C,n));
set Gv1 = <*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))));
set v = (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(SW-corner (L~ (Cage (C,n))))*>;
set h = mid ((Upper_Seq (C,n)),2,(len (Upper_Seq (C,n))));
A6: L~ (Cage (C,n)) = (L~ (Lower_Seq (C,n))) \/ (L~ (Upper_Seq (C,n))) by JORDAN1E:13;
A7: len (Upper_Seq (C,n)) >= 3 by JORDAN1E:15;
A8: len (Lower_Seq (C,n)) >= 3 by JORDAN1E:15;
A9: len (Upper_Seq (C,n)) >= 2 by A7, XXREAL_0:2;
A10: len (Upper_Seq (C,n)) >= 1 by A7, XXREAL_0:2;
A11: len (Lower_Seq (C,n)) >= 1 by A8, XXREAL_0:2;
A12: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
then  width (Gauge (C,n)) >= 4 by JORDAN8:10;
then A13: 1 <= width (Gauge (C,n)) by XXREAL_0:2;
A14: ((Gauge (C,n)) * (i,(width (Gauge (C,n))))) `2 = N-bound (L~ (Cage (C,n))) by A1, A2, A12, JORDAN1A:70;
set Ema = E-max (L~ (Cage (C,n)));
now__::_thesis:_LSeg_(((Gauge_(C,n))_*_(i,(width_(Gauge_(C,n))))),((Gauge_(C,n))_*_(i,j)))_meets_L~_(Upper_Seq_(C,n))
percases ( ( (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) & i = len (Gauge (C,n)) ) or ( (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) & (Gauge (C,n)) * (i,j) <> (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) & W-min (L~ (Cage (C,n))) <> SW-corner (L~ (Cage (C,n))) & i < len (Gauge (C,n)) ) or ( (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) & (Gauge (C,n)) * (i,j) <> (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) & W-min (L~ (Cage (C,n))) = SW-corner (L~ (Cage (C,n))) & i < len (Gauge (C,n)) ) or (Gauge (C,n)) * (i,j) in L~ (Upper_Seq (C,n)) or ( (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) & (Gauge (C,n)) * (i,j) = (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) ) ) by A2, A5, A6, XBOOLE_0:def_3, XXREAL_0:1;
supposeA15: ( (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) & i = len (Gauge (C,n)) ) ; ::_thesis: LSeg (((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))) meets L~ (Upper_Seq (C,n))
set G11 = (Gauge (C,n)) * ((len (Gauge (C,n))),(width (Gauge (C,n))));
A16: ((Gauge (C,n)) * ((len (Gauge (C,n))),(width (Gauge (C,n))))) `1 = E-bound (L~ (Cage (C,n))) by A1, A12, A15, JORDAN1A:71;
A17: (E-max (L~ (Cage (C,n)))) `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52;
A18: N-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * ((len (Gauge (C,n))),(width (Gauge (C,n))))) `2 by A1, A12, A15, JORDAN1A:70;
 E-max (L~ (Cage (C,n))) in L~ (Cage (C,n)) by SPRECT_1:14;
then A19: ((Gauge (C,n)) * ((len (Gauge (C,n))),(width (Gauge (C,n))))) `2 >= (E-max (L~ (Cage (C,n)))) `2 by A18, PSCOMP_1:24;
A20: ((Gauge (C,n)) * (i,j)) `1 = E-bound (L~ (Cage (C,n))) by A3, A4, A12, A15, JORDAN1A:71;
then  (Gauge (C,n)) * (i,j) in E-most (L~ (Cage (C,n))) by A5, SPRECT_2:13;
then  (E-max (L~ (Cage (C,n)))) `2 >= ((Gauge (C,n)) * (i,j)) `2 by PSCOMP_1:47;
then A21: E-max (L~ (Cage (C,n))) in LSeg (((Gauge (C,n)) * ((len (Gauge (C,n))),(width (Gauge (C,n))))),((Gauge (C,n)) * ((len (Gauge (C,n))),j))) by A15, A16, A17, A19, A20, GOBOARD7:7;
A22: rng (Upper_Seq (C,n)) c= L~ (Upper_Seq (C,n)) by A7, SPPOL_2:18, XXREAL_0:2;
 (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) = E-max (L~ (Cage (C,n))) by JORDAN1F:7;
then  E-max (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) by REVROT_1:3;
hence  LSeg (((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))) meets L~ (Upper_Seq (C,n)) by A15, A21, A22, XBOOLE_0:3; ::_thesis: verum
end;
supposeA23: ( (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) & (Gauge (C,n)) * (i,j) <> (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) & W-min (L~ (Cage (C,n))) <> SW-corner (L~ (Cage (C,n))) & i < len (Gauge (C,n)) ) ; ::_thesis: LSeg (((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))) meets L~ (Upper_Seq (C,n))
then A24: not L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))) is empty by JORDAN1E:3;
then A25: 0 + 1 <= len (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by NAT_1:13;
then A26: 1 in dom (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by FINSEQ_3:25;
A27: len (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) in dom (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by A25, FINSEQ_3:25;
A28: len (Lower_Seq (C,n)) in dom (Lower_Seq (C,n)) by A11, FINSEQ_3:25;
A29: (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) /. (len (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) = (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) . (len (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) by A27, PARTFUN1:def_6
.= (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) by A23, JORDAN1B:4
.= (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by A28, PARTFUN1:def_6
.= W-min (L~ (Cage (C,n))) by JORDAN1F:8 ;
then A30: (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) /. (len (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))))) = W-min (L~ (Cage (C,n))) by A24, SPRECT_3:1;
A31: (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) /. 1 = (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) . 1 by A26, PARTFUN1:def_6
.= (Gauge (C,n)) * (i,j) by A23, JORDAN3:23 ;
then A32: ((L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) ^ <*(SW-corner (L~ (Cage (C,n))))*>) /. 1 = (Gauge (C,n)) * (i,j) by A25, BOOLMARK:7;
A33: 1 + (len (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) >= 1 + 1 by A25, XREAL_1:7;
len ((<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(SW-corner (L~ (Cage (C,n))))*>) = (len (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))))) + 1 by FINSEQ_2:16
.= (1 + (len (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))))) + 1 by FINSEQ_5:8 ;
then  2 < len ((<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(SW-corner (L~ (Cage (C,n))))*>) by A33, NAT_1:13;
then A34: 2 < len (Rev ((<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(SW-corner (L~ (Cage (C,n))))*>)) by FINSEQ_5:def_3;
 S-bound (L~ (Cage (C,n))) < N-bound (L~ (Cage (C,n))) by SPRECT_1:32;
then  SW-corner (L~ (Cage (C,n))) <> (Gauge (C,n)) * (i,(width (Gauge (C,n)))) by A14, EUCLID:52;
then  not SW-corner (L~ (Cage (C,n))) in {((Gauge (C,n)) * (i,(width (Gauge (C,n)))))} by TARSKI:def_1;
then A35: not SW-corner (L~ (Cage (C,n))) in rng <*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> by FINSEQ_1:39;
 len (Cage (C,n)) > 4 by GOBOARD7:34;
then A36: rng (Cage (C,n)) c= L~ (Cage (C,n)) by SPPOL_2:18, XXREAL_0:2;
A37: not SW-corner (L~ (Cage (C,n))) in rng (Cage (C,n))
proof
assume A38: SW-corner (L~ (Cage (C,n))) in rng (Cage (C,n)) ; ::_thesis: contradiction
A39: (SW-corner (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52;
A40: (SW-corner (L~ (Cage (C,n)))) `2 = S-bound (L~ (Cage (C,n))) by EUCLID:52;
then  (SW-corner (L~ (Cage (C,n)))) `2 <= N-bound (L~ (Cage (C,n))) by SPRECT_1:22;
then  SW-corner (L~ (Cage (C,n))) in  {  p where p is Point of (TOP-REAL 2) : ( p `1 = W-bound (L~ (Cage (C,n))) & p `2 <= N-bound (L~ (Cage (C,n))) & p `2 >= S-bound (L~ (Cage (C,n))) )  }  by A39, A40;
then  SW-corner (L~ (Cage (C,n))) in LSeg ((SW-corner (L~ (Cage (C,n)))),(NW-corner (L~ (Cage (C,n))))) by SPRECT_1:26;
then  SW-corner (L~ (Cage (C,n))) in (LSeg ((SW-corner (L~ (Cage (C,n)))),(NW-corner (L~ (Cage (C,n)))))) /\ (L~ (Cage (C,n))) by A36, A38, XBOOLE_0:def_4;
then A41: (SW-corner (L~ (Cage (C,n)))) `2 >= (W-min (L~ (Cage (C,n)))) `2 by PSCOMP_1:31;
 (W-min (L~ (Cage (C,n)))) `2 >= (SW-corner (L~ (Cage (C,n)))) `2 by PSCOMP_1:30;
then A42: (W-min (L~ (Cage (C,n)))) `2 = (SW-corner (L~ (Cage (C,n)))) `2 by A41, XXREAL_0:1;
 (W-min (L~ (Cage (C,n)))) `1 = (SW-corner (L~ (Cage (C,n)))) `1 by PSCOMP_1:29;
hence  contradiction by A23, A42, TOPREAL3:6; ::_thesis: verum
end;
now__::_thesis:_not_SW-corner_(L~_(Cage_(C,n)))_in_rng_(L_Cut_((Lower_Seq_(C,n)),((Gauge_(C,n))_*_(i,j))))
percases ( (Gauge (C,n)) * (i,j) <> (Lower_Seq (C,n)) . ((Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1) or (Gauge (C,n)) * (i,j) = (Lower_Seq (C,n)) . ((Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1) ) ;
suppose (Gauge (C,n)) * (i,j) <> (Lower_Seq (C,n)) . ((Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1) ; ::_thesis: not SW-corner (L~ (Cage (C,n))) in rng (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))
then  L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))) = <*((Gauge (C,n)) * (i,j))*> ^ (mid ((Lower_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1),(len (Lower_Seq (C,n))))) by JORDAN3:def_3;
then  rng (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) = (rng <*((Gauge (C,n)) * (i,j))*>) \/ (rng (mid ((Lower_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1),(len (Lower_Seq (C,n)))))) by FINSEQ_1:31;
then A43: rng (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) = {((Gauge (C,n)) * (i,j))} \/ (rng (mid ((Lower_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1),(len (Lower_Seq (C,n)))))) by FINSEQ_1:38;
 not SW-corner (L~ (Cage (C,n))) in L~ (Cage (C,n))
proof
assume  SW-corner (L~ (Cage (C,n))) in L~ (Cage (C,n)) ; ::_thesis: contradiction
then consider i being Element of NAT  such that
A44: 1 <= i and
A45: i + 1 <= len (Cage (C,n)) and
A46: SW-corner (L~ (Cage (C,n))) in LSeg (((Cage (C,n)) /. i),((Cage (C,n)) /. (i + 1))) by SPPOL_2:14;
percases ( ((Cage (C,n)) /. i) `1 = ((Cage (C,n)) /. (i + 1)) `1 or ((Cage (C,n)) /. i) `2 = ((Cage (C,n)) /. (i + 1)) `2 ) by A44, A45, TOPREAL1:def_5;
supposeA47: ((Cage (C,n)) /. i) `1 = ((Cage (C,n)) /. (i + 1)) `1 ; ::_thesis: contradiction
then A48: (SW-corner (L~ (Cage (C,n)))) `1 = ((Cage (C,n)) /. i) `1 by A46, GOBOARD7:5;
A49: (SW-corner (L~ (Cage (C,n)))) `2 = S-bound (L~ (Cage (C,n))) by EUCLID:52;
A50: i < len (Cage (C,n)) by A45, NAT_1:13;
then A51: ((Cage (C,n)) /. i) `2 >= (SW-corner (L~ (Cage (C,n)))) `2 by A44, A49, JORDAN5D:11;
A52: 1 <= i + 1 by NAT_1:11;
then A53: ((Cage (C,n)) /. (i + 1)) `2 >= (SW-corner (L~ (Cage (C,n)))) `2 by A45, A49, JORDAN5D:11;
A54: i in dom (Cage (C,n)) by A44, A50, FINSEQ_3:25;
A55: i + 1 in dom (Cage (C,n)) by A45, A52, FINSEQ_3:25;
 ( ((Cage (C,n)) /. i) `2 <= ((Cage (C,n)) /. (i + 1)) `2 or ((Cage (C,n)) /. i) `2 >= ((Cage (C,n)) /. (i + 1)) `2 ) ;
then  ( (SW-corner (L~ (Cage (C,n)))) `2 >= ((Cage (C,n)) /. (i + 1)) `2 or (SW-corner (L~ (Cage (C,n)))) `2 >= ((Cage (C,n)) /. i) `2 ) by A46, TOPREAL1:4;
then  ( (SW-corner (L~ (Cage (C,n)))) `2 = ((Cage (C,n)) /. (i + 1)) `2 or (SW-corner (L~ (Cage (C,n)))) `2 = ((Cage (C,n)) /. i) `2 ) by A51, A53, XXREAL_0:1;
then  ( SW-corner (L~ (Cage (C,n))) = (Cage (C,n)) /. (i + 1) or SW-corner (L~ (Cage (C,n))) = (Cage (C,n)) /. i ) by A47, A48, TOPREAL3:6;
hence  contradiction by A37, A54, A55, PARTFUN2:2; ::_thesis: verum
end;
supposeA56: ((Cage (C,n)) /. i) `2 = ((Cage (C,n)) /. (i + 1)) `2 ; ::_thesis: contradiction
then A57: (SW-corner (L~ (Cage (C,n)))) `2 = ((Cage (C,n)) /. i) `2 by A46, GOBOARD7:6;
A58: (SW-corner (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52;
A59: i < len (Cage (C,n)) by A45, NAT_1:13;
then A60: ((Cage (C,n)) /. i) `1 >= (SW-corner (L~ (Cage (C,n)))) `1 by A44, A58, JORDAN5D:12;
A61: 1 <= i + 1 by NAT_1:11;
then A62: ((Cage (C,n)) /. (i + 1)) `1 >= (SW-corner (L~ (Cage (C,n)))) `1 by A45, A58, JORDAN5D:12;
A63: i in dom (Cage (C,n)) by A44, A59, FINSEQ_3:25;
A64: i + 1 in dom (Cage (C,n)) by A45, A61, FINSEQ_3:25;
 ( ((Cage (C,n)) /. i) `1 <= ((Cage (C,n)) /. (i + 1)) `1 or ((Cage (C,n)) /. i) `1 >= ((Cage (C,n)) /. (i + 1)) `1 ) ;
then  ( (SW-corner (L~ (Cage (C,n)))) `1 >= ((Cage (C,n)) /. (i + 1)) `1 or (SW-corner (L~ (Cage (C,n)))) `1 >= ((Cage (C,n)) /. i) `1 ) by A46, TOPREAL1:3;
then  ( (SW-corner (L~ (Cage (C,n)))) `1 = ((Cage (C,n)) /. (i + 1)) `1 or (SW-corner (L~ (Cage (C,n)))) `1 = ((Cage (C,n)) /. i) `1 ) by A60, A62, XXREAL_0:1;
then  ( SW-corner (L~ (Cage (C,n))) = (Cage (C,n)) /. (i + 1) or SW-corner (L~ (Cage (C,n))) = (Cage (C,n)) /. i ) by A56, A57, TOPREAL3:6;
hence  contradiction by A37, A63, A64, PARTFUN2:2; ::_thesis: verum
end;
end;
end;
then A65: not SW-corner (L~ (Cage (C,n))) in {((Gauge (C,n)) * (i,j))} by A5, TARSKI:def_1;
A66: rng (mid ((Lower_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1),(len (Lower_Seq (C,n))))) c= rng (Lower_Seq (C,n)) by FINSEQ_6:119;
 rng (Lower_Seq (C,n)) c= rng (Cage (C,n)) by JORDAN1G:39;
then  rng (mid ((Lower_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1),(len (Lower_Seq (C,n))))) c= rng (Cage (C,n)) by A66, XBOOLE_1:1;
then  not SW-corner (L~ (Cage (C,n))) in rng (mid ((Lower_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1),(len (Lower_Seq (C,n))))) by A37;
hence  not SW-corner (L~ (Cage (C,n))) in rng (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by A43, A65, XBOOLE_0:def_3; ::_thesis: verum
end;
suppose (Gauge (C,n)) * (i,j) = (Lower_Seq (C,n)) . ((Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1) ; ::_thesis: not SW-corner (L~ (Cage (C,n))) in rng (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))
then  L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))) = mid ((Lower_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1),(len (Lower_Seq (C,n)))) by JORDAN3:def_3;
then A67: rng (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) c= rng (Lower_Seq (C,n)) by FINSEQ_6:119;
 rng (Lower_Seq (C,n)) c= rng (Cage (C,n)) by JORDAN1G:39;
then  rng (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) c= rng (Cage (C,n)) by A67, XBOOLE_1:1;
hence  not SW-corner (L~ (Cage (C,n))) in rng (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by A37; ::_thesis: verum
end;
end;
end;
then  not SW-corner (L~ (Cage (C,n))) in (rng <*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*>) \/ (rng (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) by A35, XBOOLE_0:def_3;
then  not SW-corner (L~ (Cage (C,n))) in rng (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) by FINSEQ_1:31;
then  rng (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) misses {(SW-corner (L~ (Cage (C,n))))} by ZFMISC_1:50;
then A68: rng (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) misses rng <*(SW-corner (L~ (Cage (C,n))))*> by FINSEQ_1:38;
A69: not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in L~ (Lower_Seq (C,n)) by A1, A23, JORDAN1G:45;
 rng (Lower_Seq (C,n)) c= L~ (Lower_Seq (C,n)) by A8, SPPOL_2:18, XXREAL_0:2;
then A70: not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in rng (Lower_Seq (C,n)) by A1, A23, JORDAN1G:45;
 not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in {((Gauge (C,n)) * (i,j))} by A23, A69, TARSKI:def_1;
then A71: not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in rng <*((Gauge (C,n)) * (i,j))*> by FINSEQ_1:38;
set ci = mid ((Lower_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1),(len (Lower_Seq (C,n))));
now__::_thesis:_not_(Gauge_(C,n))_*_(i,(width_(Gauge_(C,n))))_in_rng_(L_Cut_((Lower_Seq_(C,n)),((Gauge_(C,n))_*_(i,j))))
percases ( (Gauge (C,n)) * (i,j) <> (Lower_Seq (C,n)) . ((Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1) or (Gauge (C,n)) * (i,j) = (Lower_Seq (C,n)) . ((Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1) ) ;
supposeA72: (Gauge (C,n)) * (i,j) <> (Lower_Seq (C,n)) . ((Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1) ; ::_thesis: not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in rng (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))
 rng (mid ((Lower_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1),(len (Lower_Seq (C,n))))) c= rng (Lower_Seq (C,n)) by FINSEQ_6:119;
then  not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in rng (mid ((Lower_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1),(len (Lower_Seq (C,n))))) by A70;
then  not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in (rng <*((Gauge (C,n)) * (i,j))*>) \/ (rng (mid ((Lower_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1),(len (Lower_Seq (C,n)))))) by A71, XBOOLE_0:def_3;
then  not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in rng (<*((Gauge (C,n)) * (i,j))*> ^ (mid ((Lower_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1),(len (Lower_Seq (C,n)))))) by FINSEQ_1:31;
hence  not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in rng (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by A72, JORDAN3:def_3; ::_thesis: verum
end;
suppose (Gauge (C,n)) * (i,j) = (Lower_Seq (C,n)) . ((Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1) ; ::_thesis: not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in rng (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))
then  L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))) = mid ((Lower_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1),(len (Lower_Seq (C,n)))) by JORDAN3:def_3;
then  rng (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) c= rng (Lower_Seq (C,n)) by FINSEQ_6:119;
hence  not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in rng (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by A70; ::_thesis: verum
end;
end;
end;
then  {((Gauge (C,n)) * (i,(width (Gauge (C,n)))))} misses rng (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by ZFMISC_1:50;
then A73: rng <*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> misses rng (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by FINSEQ_1:38;
A74: <*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> is one-to-one by FINSEQ_3:93;
A75: L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))) is being_S-Seq by A23, JORDAN3:34;
then A76: <*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) is one-to-one by A73, A74, FINSEQ_3:91;
 <*(SW-corner (L~ (Cage (C,n))))*> is one-to-one by FINSEQ_3:93;
then A77: (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(SW-corner (L~ (Cage (C,n))))*> is one-to-one by A68, A76, FINSEQ_3:91;
(<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. (len <*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*>)) `1 = (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. 1) `1 by FINSEQ_1:39
.= ((Gauge (C,n)) * (i,(width (Gauge (C,n))))) `1 by FINSEQ_4:16
.= ((Gauge (C,n)) * (i,1)) `1 by A1, A2, A13, GOBOARD5:2
.= ((L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) /. 1) `1 by A1, A2, A3, A4, A31, GOBOARD5:2 ;
then A78: <*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) is special by A75, GOBOARD2:8;
((<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) /. (len (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))))) `1 = (SW-corner (L~ (Cage (C,n)))) `1 by A30, PSCOMP_1:29
.= (<*(SW-corner (L~ (Cage (C,n))))*> /. 1) `1 by FINSEQ_4:16 ;
then  (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(SW-corner (L~ (Cage (C,n))))*> is special by A78, GOBOARD2:8;
then A79: Rev ((<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(SW-corner (L~ (Cage (C,n))))*>) is special by SPPOL_2:40;
A80: len (Upper_Seq (C,n)) >= 2 + 1 by JORDAN1E:15;
then A81: len (Upper_Seq (C,n)) > 2 by NAT_1:13;
 len (Upper_Seq (C,n)) > 1 by A80, XXREAL_0:2;
then A82: mid ((Upper_Seq (C,n)),2,(len (Upper_Seq (C,n)))) is S-Sequence_in_R2 by A81, JORDAN3:6;
then A83: 2 <= len (mid ((Upper_Seq (C,n)),2,(len (Upper_Seq (C,n))))) by TOPREAL1:def_8;
 3 <= len (Upper_Seq (C,n)) by JORDAN1E:15;
then  2 <= len (Upper_Seq (C,n)) by XXREAL_0:2;
then A84: 2 in dom (Upper_Seq (C,n)) by FINSEQ_3:25;
A85: len (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by FINSEQ_5:6;
then A86: mid ((Upper_Seq (C,n)),2,(len (Upper_Seq (C,n)))) is_in_the_area_of Cage (C,n) by A84, JORDAN1E:17, SPRECT_2:22;
 (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) = E-max (L~ (Cage (C,n))) by JORDAN1F:7;
then  ((Upper_Seq (C,n)) /. (len (Upper_Seq (C,n)))) `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52;
then A87: ((mid ((Upper_Seq (C,n)),2,(len (Upper_Seq (C,n))))) /. (len (mid ((Upper_Seq (C,n)),2,(len (Upper_Seq (C,n))))))) `1 = E-bound (L~ (Cage (C,n))) by A84, A85, SPRECT_2:9;
 ((Upper_Seq (C,n)) /. (1 + 1)) `1 = W-bound (L~ (Cage (C,n))) by JORDAN1G:31;
then  ((mid ((Upper_Seq (C,n)),2,(len (Upper_Seq (C,n))))) /. 1) `1 = W-bound (L~ (Cage (C,n))) by A84, A85, SPRECT_2:8;
then A88: mid ((Upper_Seq (C,n)),2,(len (Upper_Seq (C,n)))) is_a_h.c._for Cage (C,n) by A86, A87, SPRECT_2:def_2;
now__::_thesis:_for_m_being_Element_of_NAT_st_m_in_dom_<*((Gauge_(C,n))_*_(i,(width_(Gauge_(C,n)))))*>_holds_
(_W-bound_(L~_(Cage_(C,n)))_<=_(<*((Gauge_(C,n))_*_(i,(width_(Gauge_(C,n)))))*>_/._m)_`1_&_(<*((Gauge_(C,n))_*_(i,(width_(Gauge_(C,n)))))*>_/._m)_`1_<=_E-bound_(L~_(Cage_(C,n)))_&_S-bound_(L~_(Cage_(C,n)))_<=_(<*((Gauge_(C,n))_*_(i,(width_(Gauge_(C,n)))))*>_/._m)_`2_&_(<*((Gauge_(C,n))_*_(i,(width_(Gauge_(C,n)))))*>_/._m)_`2_<=_N-bound_(L~_(Cage_(C,n)))_)
let m be Element of NAT ; ::_thesis: ( m in dom <*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> implies ( W-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. m) `1 & (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. m) `2 & (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) )
assume A89: m in dom <*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ; ::_thesis: ( W-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. m) `1 & (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. m) `2 & (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) )
then  m in Seg 1 by FINSEQ_1:38;
then  m = 1 by FINSEQ_1:2, TARSKI:def_1;
then  <*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> . m = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) by FINSEQ_1:40;
then A90: <*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. m = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) by A89, PARTFUN1:def_6;
 ((Gauge (C,n)) * (1,(width (Gauge (C,n))))) `1 <= ((Gauge (C,n)) * (i,(width (Gauge (C,n))))) `1 by A1, A2, A13, SPRECT_3:13;
hence  W-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. m) `1 by A12, A13, A90, JORDAN1A:73; ::_thesis: ( (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. m) `2 & (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) )
 ((Gauge (C,n)) * (i,(width (Gauge (C,n))))) `1 <= ((Gauge (C,n)) * ((len (Gauge (C,n))),(width (Gauge (C,n))))) `1 by A1, A2, A13, SPRECT_3:13;
hence  (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. m) `1 <= E-bound (L~ (Cage (C,n))) by A12, A13, A90, JORDAN1A:71; ::_thesis: ( S-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. m) `2 & (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) )
 (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. m) `2 = N-bound (L~ (Cage (C,n))) by A1, A2, A12, A90, JORDAN1A:70;
hence  S-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. m) `2 by SPRECT_1:22; ::_thesis: (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. m) `2 <= N-bound (L~ (Cage (C,n)))
thus  (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) by A1, A2, A12, A90, JORDAN1A:70; ::_thesis: verum
end;
then A91: <*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> is_in_the_area_of Cage (C,n) by SPRECT_2:def_1;
 <*((Gauge (C,n)) * (i,j))*> is_in_the_area_of Cage (C,n) by A23, JORDAN1E:18, SPRECT_3:46;
then  L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))) is_in_the_area_of Cage (C,n) by A23, JORDAN1E:18, SPRECT_3:56;
then A92: <*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) is_in_the_area_of Cage (C,n) by A91, SPRECT_2:24;
 <*(SW-corner (L~ (Cage (C,n))))*> is_in_the_area_of Cage (C,n) by SPRECT_2:28;
then  (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(SW-corner (L~ (Cage (C,n))))*> is_in_the_area_of Cage (C,n) by A92, SPRECT_2:24;
then A93: Rev ((<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(SW-corner (L~ (Cage (C,n))))*>) is_in_the_area_of Cage (C,n) by SPRECT_3:51;
 (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(SW-corner (L~ (Cage (C,n))))*> = <*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ ((L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) ^ <*(SW-corner (L~ (Cage (C,n))))*>) by FINSEQ_1:32;
then  ((<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(SW-corner (L~ (Cage (C,n))))*>) /. 1 = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) by FINSEQ_5:15;
then  (((<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(SW-corner (L~ (Cage (C,n))))*>) /. 1) `2 = N-bound (L~ (Cage (C,n))) by A1, A2, A12, JORDAN1A:70;
then  ((Rev ((<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(SW-corner (L~ (Cage (C,n))))*>)) /. (len ((<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(SW-corner (L~ (Cage (C,n))))*>))) `2 = N-bound (L~ (Cage (C,n))) by FINSEQ_5:65;
then A94: ((Rev ((<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(SW-corner (L~ (Cage (C,n))))*>)) /. (len (Rev ((<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(SW-corner (L~ (Cage (C,n))))*>)))) `2 = N-bound (L~ (Cage (C,n))) by FINSEQ_5:def_3;
 len ((<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(SW-corner (L~ (Cage (C,n))))*>) = (len (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))))) + 1 by FINSEQ_2:16;
then  ((<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(SW-corner (L~ (Cage (C,n))))*>) /. (len ((<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(SW-corner (L~ (Cage (C,n))))*>)) = SW-corner (L~ (Cage (C,n))) by FINSEQ_4:67;
then  (((<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(SW-corner (L~ (Cage (C,n))))*>) /. (len ((<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(SW-corner (L~ (Cage (C,n))))*>))) `2 = S-bound (L~ (Cage (C,n))) by EUCLID:52;
then  ((Rev ((<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(SW-corner (L~ (Cage (C,n))))*>)) /. 1) `2 = S-bound (L~ (Cage (C,n))) by FINSEQ_5:65;
then  Rev ((<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(SW-corner (L~ (Cage (C,n))))*>) is_a_v.c._for Cage (C,n) by A93, A94, SPRECT_2:def_3;
then  L~ (mid ((Upper_Seq (C,n)),2,(len (Upper_Seq (C,n))))) meets L~ (Rev ((<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(SW-corner (L~ (Cage (C,n))))*>)) by A34, A77, A79, A82, A83, A88, SPRECT_2:29;
then  L~ (mid ((Upper_Seq (C,n)),2,(len (Upper_Seq (C,n))))) meets L~ ((<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(SW-corner (L~ (Cage (C,n))))*>) by SPPOL_2:22;
then consider x being set  such that
A95: x in L~ (mid ((Upper_Seq (C,n)),2,(len (Upper_Seq (C,n))))) and
A96: x in L~ ((<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(SW-corner (L~ (Cage (C,n))))*>) by XBOOLE_0:3;
A97: L~ (mid ((Upper_Seq (C,n)),2,(len (Upper_Seq (C,n))))) c= L~ (Upper_Seq (C,n)) by A9, A10, JORDAN4:35;
A98: L~ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) c= L~ (Lower_Seq (C,n)) by A23, JORDAN3:42;
L~ ((<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(SW-corner (L~ (Cage (C,n))))*>) = L~ (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ ((L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) ^ <*(SW-corner (L~ (Cage (C,n))))*>)) by FINSEQ_1:32
.= (LSeg (((Gauge (C,n)) * (i,(width (Gauge (C,n))))),(((L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) ^ <*(SW-corner (L~ (Cage (C,n))))*>) /. 1))) \/ (L~ ((L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) ^ <*(SW-corner (L~ (Cage (C,n))))*>)) by SPPOL_2:20
.= (LSeg (((Gauge (C,n)) * (i,(width (Gauge (C,n))))),(((L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) ^ <*(SW-corner (L~ (Cage (C,n))))*>) /. 1))) \/ ((L~ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) \/ (LSeg (((L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) /. (len (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))))),(SW-corner (L~ (Cage (C,n))))))) by A24, SPPOL_2:19 ;
then A99: ( x in LSeg (((Gauge (C,n)) * (i,(width (Gauge (C,n))))),(((L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) ^ <*(SW-corner (L~ (Cage (C,n))))*>) /. 1)) or x in (L~ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) \/ (LSeg (((L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) /. (len (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))))),(SW-corner (L~ (Cage (C,n)))))) ) by A96, XBOOLE_0:def_3;
 (Upper_Seq (C,n)) /. 1 = W-min (L~ (Cage (C,n))) by JORDAN1F:5;
then A100: not W-min (L~ (Cage (C,n))) in L~ (mid ((Upper_Seq (C,n)),2,(len (Upper_Seq (C,n))))) by A81, JORDAN5B:16;
now__::_thesis:_L~_(Upper_Seq_(C,n))_meets_L~_<*((Gauge_(C,n))_*_(i,(width_(Gauge_(C,n))))),((Gauge_(C,n))_*_(i,j))*>
percases ( x in LSeg (((Gauge (C,n)) * (i,(width (Gauge (C,n))))),(((L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) ^ <*(SW-corner (L~ (Cage (C,n))))*>) /. 1)) or x in L~ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) or x in LSeg (((L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) /. (len (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))))),(SW-corner (L~ (Cage (C,n))))) ) by A99, XBOOLE_0:def_3;
suppose x in LSeg (((Gauge (C,n)) * (i,(width (Gauge (C,n))))),(((L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) ^ <*(SW-corner (L~ (Cage (C,n))))*>) /. 1)) ; ::_thesis: L~ (Upper_Seq (C,n)) meets L~ <*((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))*>
then  x in L~ <*((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))*> by A32, SPPOL_2:21;
hence  L~ (Upper_Seq (C,n)) meets L~ <*((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))*> by A95, A97, XBOOLE_0:3; ::_thesis: verum
end;
supposeA101: x in L~ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) ; ::_thesis: L~ (Upper_Seq (C,n)) meets L~ <*((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))*>
then  x in (L~ (Lower_Seq (C,n))) /\ (L~ (Upper_Seq (C,n))) by A95, A97, A98, XBOOLE_0:def_4;
then  x in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16;
then A102: x = E-max (L~ (Cage (C,n))) by A95, A100, TARSKI:def_2;
 1 in dom (Lower_Seq (C,n)) by A11, FINSEQ_3:25;
then (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;
then  x = (Gauge (C,n)) * (i,j) by A23, A101, A102, JORDAN1E:7;
then  x in LSeg (((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))) by RLTOPSP1:68;
then  x in L~ <*((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))*> by SPPOL_2:21;
hence  L~ (Upper_Seq (C,n)) meets L~ <*((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))*> by A95, A97, XBOOLE_0:3; ::_thesis: verum
end;
supposeA103: x in LSeg (((L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) /. (len (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))))),(SW-corner (L~ (Cage (C,n))))) ; ::_thesis: L~ (Upper_Seq (C,n)) meets L~ <*((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))*>
 x in L~ (Cage (C,n)) by A6, A95, A97, XBOOLE_0:def_3;
then  x in (LSeg ((W-min (L~ (Cage (C,n)))),(SW-corner (L~ (Cage (C,n)))))) /\ (L~ (Cage (C,n))) by A29, A103, XBOOLE_0:def_4;
then  x in {(W-min (L~ (Cage (C,n))))} by PSCOMP_1:35;
hence  L~ (Upper_Seq (C,n)) meets L~ <*((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))*> by A95, A100, TARSKI:def_1; ::_thesis: verum
end;
end;
end;
then  L~ <*((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))*> meets L~ (Upper_Seq (C,n)) ;
hence  LSeg (((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))) meets L~ (Upper_Seq (C,n)) by SPPOL_2:21; ::_thesis: verum
end;
supposeA104: ( (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) & (Gauge (C,n)) * (i,j) <> (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) & W-min (L~ (Cage (C,n))) = SW-corner (L~ (Cage (C,n))) & i < len (Gauge (C,n)) ) ; ::_thesis: LSeg (((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))) meets L~ (Upper_Seq (C,n))
then A105: not L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))) is empty by JORDAN1E:3;
then A106: 0 + 1 <= len (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by NAT_1:13;
then A107: 1 in dom (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by FINSEQ_3:25;
set v = <*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))));
A108: len (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) in dom (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by A106, FINSEQ_3:25;
A109: len (Lower_Seq (C,n)) in dom (Lower_Seq (C,n)) by A11, FINSEQ_3:25;
(L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) /. (len (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) = (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) . (len (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) by A108, PARTFUN1:def_6
.= (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) by A104, JORDAN1B:4
.= (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by A109, PARTFUN1:def_6
.= W-min (L~ (Cage (C,n))) by JORDAN1F:8 ;
then A110: (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) /. (len (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))))) = W-min (L~ (Cage (C,n))) by A105, SPRECT_3:1;
A111: (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) /. 1 = (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) . 1 by A107, PARTFUN1:def_6
.= (Gauge (C,n)) * (i,j) by A104, JORDAN3:23 ;
 1 + (len (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) >= 1 + 1 by A106, XREAL_1:7;
then  2 <= len (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) by FINSEQ_5:8;
then A112: 2 <= len (Rev (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))))) by FINSEQ_5:def_3;
A113: not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in L~ (Lower_Seq (C,n)) by A1, A104, JORDAN1G:45;
 rng (Lower_Seq (C,n)) c= L~ (Lower_Seq (C,n)) by A8, SPPOL_2:18, XXREAL_0:2;
then A114: not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in rng (Lower_Seq (C,n)) by A1, A104, JORDAN1G:45;
 not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in {((Gauge (C,n)) * (i,j))} by A104, A113, TARSKI:def_1;
then A115: not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in rng <*((Gauge (C,n)) * (i,j))*> by FINSEQ_1:38;
set ci = mid ((Lower_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1),(len (Lower_Seq (C,n))));
now__::_thesis:_not_(Gauge_(C,n))_*_(i,(width_(Gauge_(C,n))))_in_rng_(L_Cut_((Lower_Seq_(C,n)),((Gauge_(C,n))_*_(i,j))))
percases ( (Gauge (C,n)) * (i,j) <> (Lower_Seq (C,n)) . ((Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1) or (Gauge (C,n)) * (i,j) = (Lower_Seq (C,n)) . ((Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1) ) ;
supposeA116: (Gauge (C,n)) * (i,j) <> (Lower_Seq (C,n)) . ((Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1) ; ::_thesis: not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in rng (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))
 rng (mid ((Lower_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1),(len (Lower_Seq (C,n))))) c= rng (Lower_Seq (C,n)) by FINSEQ_6:119;
then  not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in rng (mid ((Lower_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1),(len (Lower_Seq (C,n))))) by A114;
then  not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in (rng <*((Gauge (C,n)) * (i,j))*>) \/ (rng (mid ((Lower_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1),(len (Lower_Seq (C,n)))))) by A115, XBOOLE_0:def_3;
then  not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in rng (<*((Gauge (C,n)) * (i,j))*> ^ (mid ((Lower_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1),(len (Lower_Seq (C,n)))))) by FINSEQ_1:31;
hence  not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in rng (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by A116, JORDAN3:def_3; ::_thesis: verum
end;
suppose (Gauge (C,n)) * (i,j) = (Lower_Seq (C,n)) . ((Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1) ; ::_thesis: not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in rng (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))
then  L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))) = mid ((Lower_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1),(len (Lower_Seq (C,n)))) by JORDAN3:def_3;
then  rng (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) c= rng (Lower_Seq (C,n)) by FINSEQ_6:119;
hence  not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in rng (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by A114; ::_thesis: verum
end;
end;
end;
then  {((Gauge (C,n)) * (i,(width (Gauge (C,n)))))} misses rng (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by ZFMISC_1:50;
then A117: rng <*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> misses rng (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by FINSEQ_1:38;
A118: <*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> is one-to-one by FINSEQ_3:93;
A119: L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))) is being_S-Seq by A104, JORDAN3:34;
then A120: <*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) is one-to-one by A117, A118, FINSEQ_3:91;
(<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. (len <*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*>)) `1 = (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. 1) `1 by FINSEQ_1:39
.= ((Gauge (C,n)) * (i,(width (Gauge (C,n))))) `1 by FINSEQ_4:16
.= ((Gauge (C,n)) * (i,1)) `1 by A1, A2, A13, GOBOARD5:2
.= ((L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) /. 1) `1 by A1, A2, A3, A4, A111, GOBOARD5:2 ;
then  <*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) is special by A119, GOBOARD2:8;
then A121: Rev (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) is special by SPPOL_2:40;
A122: len (Upper_Seq (C,n)) >= 2 + 1 by JORDAN1E:15;
then A123: len (Upper_Seq (C,n)) > 2 by NAT_1:13;
 len (Upper_Seq (C,n)) > 1 by A122, XXREAL_0:2;
then A124: mid ((Upper_Seq (C,n)),2,(len (Upper_Seq (C,n)))) is S-Sequence_in_R2 by A123, JORDAN3:6;
then A125: 2 <= len (mid ((Upper_Seq (C,n)),2,(len (Upper_Seq (C,n))))) by TOPREAL1:def_8;
 3 <= len (Upper_Seq (C,n)) by JORDAN1E:15;
then  2 <= len (Upper_Seq (C,n)) by XXREAL_0:2;
then A126: 2 in dom (Upper_Seq (C,n)) by FINSEQ_3:25;
A127: len (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by FINSEQ_5:6;
then A128: mid ((Upper_Seq (C,n)),2,(len (Upper_Seq (C,n)))) is_in_the_area_of Cage (C,n) by A126, JORDAN1E:17, SPRECT_2:22;
 (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) = E-max (L~ (Cage (C,n))) by JORDAN1F:7;
then  ((Upper_Seq (C,n)) /. (len (Upper_Seq (C,n)))) `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52;
then A129: ((mid ((Upper_Seq (C,n)),2,(len (Upper_Seq (C,n))))) /. (len (mid ((Upper_Seq (C,n)),2,(len (Upper_Seq (C,n))))))) `1 = E-bound (L~ (Cage (C,n))) by A126, A127, SPRECT_2:9;
 ((Upper_Seq (C,n)) /. (1 + 1)) `1 = W-bound (L~ (Cage (C,n))) by JORDAN1G:31;
then  ((mid ((Upper_Seq (C,n)),2,(len (Upper_Seq (C,n))))) /. 1) `1 = W-bound (L~ (Cage (C,n))) by A126, A127, SPRECT_2:8;
then A130: mid ((Upper_Seq (C,n)),2,(len (Upper_Seq (C,n)))) is_a_h.c._for Cage (C,n) by A128, A129, SPRECT_2:def_2;
now__::_thesis:_for_m_being_Element_of_NAT_st_m_in_dom_<*((Gauge_(C,n))_*_(i,(width_(Gauge_(C,n)))))*>_holds_
(_W-bound_(L~_(Cage_(C,n)))_<=_(<*((Gauge_(C,n))_*_(i,(width_(Gauge_(C,n)))))*>_/._m)_`1_&_(<*((Gauge_(C,n))_*_(i,(width_(Gauge_(C,n)))))*>_/._m)_`1_<=_E-bound_(L~_(Cage_(C,n)))_&_S-bound_(L~_(Cage_(C,n)))_<=_(<*((Gauge_(C,n))_*_(i,(width_(Gauge_(C,n)))))*>_/._m)_`2_&_(<*((Gauge_(C,n))_*_(i,(width_(Gauge_(C,n)))))*>_/._m)_`2_<=_N-bound_(L~_(Cage_(C,n)))_)
let m be Element of NAT ; ::_thesis: ( m in dom <*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> implies ( W-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. m) `1 & (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. m) `2 & (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) )
assume A131: m in dom <*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ; ::_thesis: ( W-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. m) `1 & (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. m) `2 & (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) )
then  m in Seg 1 by FINSEQ_1:38;
then  m = 1 by FINSEQ_1:2, TARSKI:def_1;
then  <*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> . m = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) by FINSEQ_1:40;
then A132: <*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. m = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) by A131, PARTFUN1:def_6;
 ((Gauge (C,n)) * (1,(width (Gauge (C,n))))) `1 <= ((Gauge (C,n)) * (i,(width (Gauge (C,n))))) `1 by A1, A2, A13, SPRECT_3:13;
hence  W-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. m) `1 by A12, A13, A132, JORDAN1A:73; ::_thesis: ( (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. m) `2 & (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) )
 ((Gauge (C,n)) * (i,(width (Gauge (C,n))))) `1 <= ((Gauge (C,n)) * ((len (Gauge (C,n))),(width (Gauge (C,n))))) `1 by A1, A2, A13, SPRECT_3:13;
hence  (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. m) `1 <= E-bound (L~ (Cage (C,n))) by A12, A13, A132, JORDAN1A:71; ::_thesis: ( S-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. m) `2 & (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) )
 (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. m) `2 = N-bound (L~ (Cage (C,n))) by A1, A2, A12, A132, JORDAN1A:70;
hence  S-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. m) `2 by SPRECT_1:22; ::_thesis: (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. m) `2 <= N-bound (L~ (Cage (C,n)))
thus  (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) by A1, A2, A12, A132, JORDAN1A:70; ::_thesis: verum
end;
then A133: <*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> is_in_the_area_of Cage (C,n) by SPRECT_2:def_1;
 <*((Gauge (C,n)) * (i,j))*> is_in_the_area_of Cage (C,n) by A104, JORDAN1E:18, SPRECT_3:46;
then  L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))) is_in_the_area_of Cage (C,n) by A104, JORDAN1E:18, SPRECT_3:56;
then  <*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) is_in_the_area_of Cage (C,n) by A133, SPRECT_2:24;
then A134: Rev (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) is_in_the_area_of Cage (C,n) by SPRECT_3:51;
 (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) /. 1 = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) by FINSEQ_5:15;
then  ((<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) /. 1) `2 = N-bound (L~ (Cage (C,n))) by A1, A2, A12, JORDAN1A:70;
then  ((Rev (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))))) /. (len (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))))) `2 = N-bound (L~ (Cage (C,n))) by FINSEQ_5:65;
then A135: ((Rev (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))))) /. (len (Rev (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))))))) `2 = N-bound (L~ (Cage (C,n))) by FINSEQ_5:def_3;
 ((<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) /. (len (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))))) `2 = S-bound (L~ (Cage (C,n))) by A104, A110, EUCLID:52;
then  ((Rev (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))))) /. 1) `2 = S-bound (L~ (Cage (C,n))) by FINSEQ_5:65;
then  Rev (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) is_a_v.c._for Cage (C,n) by A134, A135, SPRECT_2:def_3;
then  L~ (mid ((Upper_Seq (C,n)),2,(len (Upper_Seq (C,n))))) meets L~ (Rev (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))))) by A112, A120, A121, A124, A125, A130, SPRECT_2:29;
then  L~ (mid ((Upper_Seq (C,n)),2,(len (Upper_Seq (C,n))))) meets L~ (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) by SPPOL_2:22;
then consider x being set  such that
A136: x in L~ (mid ((Upper_Seq (C,n)),2,(len (Upper_Seq (C,n))))) and
A137: x in L~ (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) by XBOOLE_0:3;
A138: L~ (mid ((Upper_Seq (C,n)),2,(len (Upper_Seq (C,n))))) c= L~ (Upper_Seq (C,n)) by A9, A10, JORDAN4:35;
A139: L~ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) c= L~ (Lower_Seq (C,n)) by A104, JORDAN3:42;
A140: L~ (<*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ^ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) = (LSeg (((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) /. 1))) \/ (L~ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))))) by A105, SPPOL_2:20;
 (Upper_Seq (C,n)) /. 1 = W-min (L~ (Cage (C,n))) by JORDAN1F:5;
then A141: not W-min (L~ (Cage (C,n))) in L~ (mid ((Upper_Seq (C,n)),2,(len (Upper_Seq (C,n))))) by A123, JORDAN5B:16;
now__::_thesis:_L~_(Upper_Seq_(C,n))_meets_L~_<*((Gauge_(C,n))_*_(i,(width_(Gauge_(C,n))))),((Gauge_(C,n))_*_(i,j))*>
percases ( x in LSeg (((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) /. 1)) or x in L~ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) ) by A137, A140, XBOOLE_0:def_3;
suppose x in LSeg (((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) /. 1)) ; ::_thesis: L~ (Upper_Seq (C,n)) meets L~ <*((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))*>
then  x in L~ <*((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))*> by A111, SPPOL_2:21;
hence  L~ (Upper_Seq (C,n)) meets L~ <*((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))*> by A136, A138, XBOOLE_0:3; ::_thesis: verum
end;
supposeA142: x in L~ (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) ; ::_thesis: L~ (Upper_Seq (C,n)) meets L~ <*((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))*>
then  x in (L~ (Lower_Seq (C,n))) /\ (L~ (Upper_Seq (C,n))) by A136, A138, A139, XBOOLE_0:def_4;
then  x in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16;
then A143: x = E-max (L~ (Cage (C,n))) by A136, A141, TARSKI:def_2;
 1 in dom (Lower_Seq (C,n)) by A11, FINSEQ_3:25;
then (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;
then  x = (Gauge (C,n)) * (i,j) by A104, A142, A143, JORDAN1E:7;
then  x in LSeg (((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))) by RLTOPSP1:68;
then  x in L~ <*((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))*> by SPPOL_2:21;
hence  L~ (Upper_Seq (C,n)) meets L~ <*((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))*> by A136, A138, XBOOLE_0:3; ::_thesis: verum
end;
end;
end;
then  L~ <*((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))*> meets L~ (Upper_Seq (C,n)) ;
hence  LSeg (((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))) meets L~ (Upper_Seq (C,n)) by SPPOL_2:21; ::_thesis: verum
end;
supposeA144: (Gauge (C,n)) * (i,j) in L~ (Upper_Seq (C,n)) ; ::_thesis: LSeg (((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))) meets L~ (Upper_Seq (C,n))
 (Gauge (C,n)) * (i,j) in LSeg (((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))) by RLTOPSP1:68;
hence  LSeg (((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))) meets L~ (Upper_Seq (C,n)) by A144, XBOOLE_0:3; ::_thesis: verum
end;
supposeA145: ( (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) & (Gauge (C,n)) * (i,j) = (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) ) ; ::_thesis: LSeg (((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))) meets L~ (Upper_Seq (C,n))
 len (Lower_Seq (C,n)) in dom (Lower_Seq (C,n)) by A11, FINSEQ_3:25;
then A146: (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by PARTFUN1:def_6
.= W-min (L~ (Cage (C,n))) by JORDAN1F:8 ;
A147: rng (Upper_Seq (C,n)) c= L~ (Upper_Seq (C,n)) by A7, SPPOL_2:18, XXREAL_0:2;
A148: W-min (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) by JORDAN1J:5;
 (Gauge (C,n)) * (i,j) in LSeg (((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))) by RLTOPSP1:68;
hence  LSeg (((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))) meets L~ (Upper_Seq (C,n)) by A145, A146, A147, A148, XBOOLE_0:3; ::_thesis: verum
end;
end;
end;
hence  LSeg (((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))) meets L~ (Upper_Seq (C,n)) ; ::_thesis: verum
end;

theorem Th4: :: JORDAN19:4
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for n being Element of NAT st n > 0 holds
for i, j being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Cage (C,n)) holds
LSeg (((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))) meets Upper_Arc (L~ (Cage (C,n)))
proof
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for n being Element of NAT st n > 0 holds
for i, j being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Cage (C,n)) holds
LSeg (((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))) meets Upper_Arc (L~ (Cage (C,n)))
let n be Element of NAT ; ::_thesis: ( n > 0 implies for i, j being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Cage (C,n)) holds
LSeg (((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))) meets Upper_Arc (L~ (Cage (C,n))) )
assume A1: n > 0 ; ::_thesis: for i, j being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Cage (C,n)) holds
LSeg (((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))) meets Upper_Arc (L~ (Cage (C,n)))
let i, j be Element of NAT ; ::_thesis: ( 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Cage (C,n)) implies LSeg (((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))) meets Upper_Arc (L~ (Cage (C,n))) )
assume that
A2: 1 <= i and
A3: i <= len (Gauge (C,n)) and
A4: 1 <= j and
A5: j <= width (Gauge (C,n)) and
A6: (Gauge (C,n)) * (i,j) in L~ (Cage (C,n)) ; ::_thesis: LSeg (((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))) meets Upper_Arc (L~ (Cage (C,n)))
 L~ (Upper_Seq (C,n)) = Upper_Arc (L~ (Cage (C,n))) by A1, JORDAN1G:55;
hence  LSeg (((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * (i,j))) meets Upper_Arc (L~ (Cage (C,n))) by A2, A3, A4, A5, A6, Th3; ::_thesis: verum
end;

theorem :: JORDAN19:5
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for j being Element of NAT st (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j) in Lower_Arc (L~ (Cage (C,(n + 1)))) & 1 <= j & j <= width (Gauge (C,(n + 1))) holds
LSeg (((Gauge (C,1)) * ((Center (Gauge (C,1))),(width (Gauge (C,1))))),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))) meets Upper_Arc (L~ (Cage (C,(n + 1))))
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for j being Element of NAT st (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j) in Lower_Arc (L~ (Cage (C,(n + 1)))) & 1 <= j & j <= width (Gauge (C,(n + 1))) holds
LSeg (((Gauge (C,1)) * ((Center (Gauge (C,1))),(width (Gauge (C,1))))),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))) meets Upper_Arc (L~ (Cage (C,(n + 1))))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for j being Element of NAT st (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j) in Lower_Arc (L~ (Cage (C,(n + 1)))) & 1 <= j & j <= width (Gauge (C,(n + 1))) holds
LSeg (((Gauge (C,1)) * ((Center (Gauge (C,1))),(width (Gauge (C,1))))),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))) meets Upper_Arc (L~ (Cage (C,(n + 1))))
let j be Element of NAT ; ::_thesis: ( (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j) in Lower_Arc (L~ (Cage (C,(n + 1)))) & 1 <= j & j <= width (Gauge (C,(n + 1))) implies LSeg (((Gauge (C,1)) * ((Center (Gauge (C,1))),(width (Gauge (C,1))))),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))) meets Upper_Arc (L~ (Cage (C,(n + 1)))) )
assume that
A1: (Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j) in Lower_Arc (L~ (Cage (C,(n + 1)))) and
A2: 1 <= j and
A3: j <= width (Gauge (C,(n + 1))) ; ::_thesis: LSeg (((Gauge (C,1)) * ((Center (Gauge (C,1))),(width (Gauge (C,1))))),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))) meets Upper_Arc (L~ (Cage (C,(n + 1))))
set in1 = Center (Gauge (C,(n + 1)));
A4: n + 1 >= 0 + 1 by NAT_1:11;
A5: 1 <= Center (Gauge (C,(n + 1))) by JORDAN1B:11;
A6: Center (Gauge (C,(n + 1))) <= len (Gauge (C,(n + 1))) by JORDAN1B:13;
A7: LSeg (((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),(width (Gauge (C,(n + 1)))))),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))) c= LSeg (((Gauge (C,1)) * ((Center (Gauge (C,1))),(width (Gauge (C,1))))),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))) by A2, A3, A4, Th2;
 Lower_Arc (L~ (Cage (C,(n + 1)))) c= L~ (Cage (C,(n + 1))) by JORDAN6:61;
hence  LSeg (((Gauge (C,1)) * ((Center (Gauge (C,1))),(width (Gauge (C,1))))),((Gauge (C,(n + 1))) * ((Center (Gauge (C,(n + 1)))),j))) meets Upper_Arc (L~ (Cage (C,(n + 1)))) by A1, A2, A3, A5, A6, A7, Th4, XBOOLE_1:63; ::_thesis: verum
end;

theorem Th6: :: JORDAN19:6
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for f being FinSequence of (TOP-REAL 2)
for k being Element of NAT st 1 <= k & k + 1 <= len f & f is_sequence_on Gauge (C,n) & not dist ((f /. k),(f /. (k + 1))) = ((N-bound C) - (S-bound C)) / (2 |^ n) holds
dist ((f /. k),(f /. (k + 1))) = ((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)
for f being FinSequence of (TOP-REAL 2)
for k being Element of NAT st 1 <= k & k + 1 <= len f & f is_sequence_on Gauge (C,n) & not dist ((f /. k),(f /. (k + 1))) = ((N-bound C) - (S-bound C)) / (2 |^ n) holds
dist ((f /. k),(f /. (k + 1))) = ((E-bound C) - (W-bound C)) / (2 |^ n)
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for f being FinSequence of (TOP-REAL 2)
for k being Element of NAT st 1 <= k & k + 1 <= len f & f is_sequence_on Gauge (C,n) & not dist ((f /. k),(f /. (k + 1))) = ((N-bound C) - (S-bound C)) / (2 |^ n) holds
dist ((f /. k),(f /. (k + 1))) = ((E-bound C) - (W-bound C)) / (2 |^ n)
let f be FinSequence of (TOP-REAL 2); ::_thesis: for k being Element of NAT st 1 <= k & k + 1 <= len f & f is_sequence_on Gauge (C,n) & not dist ((f /. k),(f /. (k + 1))) = ((N-bound C) - (S-bound C)) / (2 |^ n) holds
dist ((f /. k),(f /. (k + 1))) = ((E-bound C) - (W-bound C)) / (2 |^ n)
let k be Element of NAT ; ::_thesis: ( 1 <= k & k + 1 <= len f & f is_sequence_on Gauge (C,n) & not dist ((f /. k),(f /. (k + 1))) = ((N-bound C) - (S-bound C)) / (2 |^ n) implies dist ((f /. k),(f /. (k + 1))) = ((E-bound C) - (W-bound C)) / (2 |^ n) )
assume that
A1: 1 <= k and
A2: k + 1 <= len f ; ::_thesis: ( not f is_sequence_on Gauge (C,n) or dist ((f /. k),(f /. (k + 1))) = ((N-bound C) - (S-bound C)) / (2 |^ n) or dist ((f /. k),(f /. (k + 1))) = ((E-bound C) - (W-bound C)) / (2 |^ n) )
assume  f is_sequence_on Gauge (C,n) ; ::_thesis: ( dist ((f /. k),(f /. (k + 1))) = ((N-bound C) - (S-bound C)) / (2 |^ n) or dist ((f /. k),(f /. (k + 1))) = ((E-bound C) - (W-bound C)) / (2 |^ n) )
then consider i1, j1, i2, j2 being Element of NAT  such that
A3: [i1,j1] in Indices (Gauge (C,n)) and
A4: f /. k = (Gauge (C,n)) * (i1,j1) and
A5: [i2,j2] in Indices (Gauge (C,n)) and
A6: f /. (k + 1) = (Gauge (C,n)) * (i2,j2) and
A7: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A1, A2, JORDAN8:3;
percases ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A7;
suppose ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: ( dist ((f /. k),(f /. (k + 1))) = ((N-bound C) - (S-bound C)) / (2 |^ n) or dist ((f /. k),(f /. (k + 1))) = ((E-bound C) - (W-bound C)) / (2 |^ n) )
hence  ( dist ((f /. k),(f /. (k + 1))) = ((N-bound C) - (S-bound C)) / (2 |^ n) or dist ((f /. k),(f /. (k + 1))) = ((E-bound C) - (W-bound C)) / (2 |^ n) ) by A3, A4, A5, A6, GOBRD14:9; ::_thesis: verum
end;
suppose ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: ( dist ((f /. k),(f /. (k + 1))) = ((N-bound C) - (S-bound C)) / (2 |^ n) or dist ((f /. k),(f /. (k + 1))) = ((E-bound C) - (W-bound C)) / (2 |^ n) )
hence  ( dist ((f /. k),(f /. (k + 1))) = ((N-bound C) - (S-bound C)) / (2 |^ n) or dist ((f /. k),(f /. (k + 1))) = ((E-bound C) - (W-bound C)) / (2 |^ n) ) by A3, A4, A5, A6, GOBRD14:10; ::_thesis: verum
end;
suppose ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: ( dist ((f /. k),(f /. (k + 1))) = ((N-bound C) - (S-bound C)) / (2 |^ n) or dist ((f /. k),(f /. (k + 1))) = ((E-bound C) - (W-bound C)) / (2 |^ n) )
hence  ( dist ((f /. k),(f /. (k + 1))) = ((N-bound C) - (S-bound C)) / (2 |^ n) or dist ((f /. k),(f /. (k + 1))) = ((E-bound C) - (W-bound C)) / (2 |^ n) ) by A3, A4, A5, A6, GOBRD14:10; ::_thesis: verum
end;
suppose ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: ( dist ((f /. k),(f /. (k + 1))) = ((N-bound C) - (S-bound C)) / (2 |^ n) or dist ((f /. k),(f /. (k + 1))) = ((E-bound C) - (W-bound C)) / (2 |^ n) )
hence  ( dist ((f /. k),(f /. (k + 1))) = ((N-bound C) - (S-bound C)) / (2 |^ n) or dist ((f /. k),(f /. (k + 1))) = ((E-bound C) - (W-bound C)) / (2 |^ n) ) by A3, A4, A5, A6, GOBRD14:9; ::_thesis: verum
end;
end;
end;

theorem :: JORDAN19:7
for M being symmetric triangle MetrStruct
for r being real number
for p, q, x being Element of M st p in Ball (x,r) & q in Ball (x,r) holds
dist (p,q) < 2 * r
proof
let M be symmetric triangle MetrStruct ; ::_thesis: for r being real number
for p, q, x being Element of M st p in Ball (x,r) & q in Ball (x,r) holds
dist (p,q) < 2 * r
let r be real number ; ::_thesis: for p, q, x being Element of M st p in Ball (x,r) & q in Ball (x,r) holds
dist (p,q) < 2 * r
let p, q, x be Element of M; ::_thesis: ( p in Ball (x,r) & q in Ball (x,r) implies dist (p,q) < 2 * r )
assume that
A1: p in Ball (x,r) and
A2: q in Ball (x,r) ; ::_thesis: dist (p,q) < 2 * r
A3: dist (p,x) < r by A1, METRIC_1:11;
A4: dist (x,q) < r by A2, METRIC_1:11;
A5: dist (p,q) <= (dist (p,x)) + (dist (x,q)) by METRIC_1:4;
 (dist (p,x)) + (dist (x,q)) < r + r by A3, A4, XREAL_1:8;
hence  dist (p,q) < 2 * r by A5, XXREAL_0:2; ::_thesis: verum
end;

theorem :: JORDAN19:8
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds N-bound C < 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) holds N-bound C < N-bound (L~ (Cage (C,n)))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: N-bound C < N-bound (L~ (Cage (C,n)))
A1: 2 |^ n > 0 by NEWTON:83;
 N-bound C > (S-bound C) + 0 by SPRECT_1:32;
then  (N-bound C) - (S-bound C) > 0 by XREAL_1:20;
then A2: ((N-bound C) - (S-bound C)) / (2 |^ n) > (N-bound C) - (N-bound C) by A1, XREAL_1:139;
 N-bound (L~ (Cage (C,n))) = (N-bound C) + (((N-bound C) - (S-bound C)) / (2 |^ n)) by JORDAN10:6;
hence  N-bound C < N-bound (L~ (Cage (C,n))) by A2, XREAL_1:19; ::_thesis: verum
end;

theorem Th9: :: JORDAN19:9
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds E-bound C < 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) holds E-bound C < E-bound (L~ (Cage (C,n)))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: E-bound C < E-bound (L~ (Cage (C,n)))
A1: 2 |^ n > 0 by NEWTON:83;
 E-bound C > (W-bound C) + 0 by SPRECT_1:31;
then  (E-bound C) - (W-bound C) > 0 by XREAL_1:20;
then A2: ((E-bound C) - (W-bound C)) / (2 |^ n) > (E-bound C) - (E-bound C) by A1, XREAL_1:139;
 E-bound (L~ (Cage (C,n))) = (E-bound C) + (((E-bound C) - (W-bound C)) / (2 |^ n)) by JORDAN1A:64;
hence  E-bound C < E-bound (L~ (Cage (C,n))) by A2, XREAL_1:19; ::_thesis: verum
end;

theorem :: JORDAN19:10
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
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
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: S-bound (L~ (Cage (C,n))) < S-bound C
A1: 2 |^ n > 0 by NEWTON:83;
 N-bound C > (S-bound C) + 0 by SPRECT_1:32;
then  (N-bound C) - (S-bound C) > 0 by XREAL_1:20;
then A2: ((N-bound C) - (S-bound C)) / (2 |^ n) > (S-bound C) - (S-bound C) by A1, XREAL_1:139;
 S-bound (L~ (Cage (C,n))) = (S-bound C) - (((N-bound C) - (S-bound C)) / (2 |^ n)) by JORDAN1A:63;
hence  S-bound (L~ (Cage (C,n))) < S-bound C by A2, XREAL_1:11; ::_thesis: verum
end;

theorem Th11: :: JORDAN19:11
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
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
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: W-bound (L~ (Cage (C,n))) < W-bound C
A1: 2 |^ n > 0 by NEWTON:83;
 E-bound C > (W-bound C) + 0 by SPRECT_1:31;
then  (E-bound C) - (W-bound C) > 0 by XREAL_1:20;
then A2: ((E-bound C) - (W-bound C)) / (2 |^ n) > (W-bound C) - (W-bound C) by A1, XREAL_1:139;
 W-bound (L~ (Cage (C,n))) = (W-bound C) - (((E-bound C) - (W-bound C)) / (2 |^ n)) by JORDAN1A:62;
hence  W-bound (L~ (Cage (C,n))) < W-bound C by A2, XREAL_1:11; ::_thesis: verum
end;

theorem Th12: :: JORDAN19:12
for n being Element of NAT
for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= k & k <= j & j <= width (Gauge (C,n)) & (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j))} holds
LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) meets Upper_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= k & k <= j & j <= width (Gauge (C,n)) & (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j))} holds
LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) meets Upper_Arc C
let C be Simple_closed_curve; ::_thesis: for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= k & k <= j & j <= width (Gauge (C,n)) & (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j))} holds
LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) meets Upper_Arc C
let i, j, k be Element of NAT ; ::_thesis: ( 1 < i & i < len (Gauge (C,n)) & 1 <= k & k <= j & j <= width (Gauge (C,n)) & (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j))} implies LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) meets Upper_Arc C )
set Ga = Gauge (C,n);
set US = Upper_Seq (C,n);
set LS = Lower_Seq (C,n);
set UA = Upper_Arc C;
set Wmin = W-min (L~ (Cage (C,n)));
set Emax = E-max (L~ (Cage (C,n)));
set Wbo = W-bound (L~ (Cage (C,n)));
set Ebo = E-bound (L~ (Cage (C,n)));
set Gik = (Gauge (C,n)) * (i,k);
set Gij = (Gauge (C,n)) * (i,j);
assume that
A1: 1 < i and
A2: i < len (Gauge (C,n)) and
A3: 1 <= k and
A4: k <= j and
A5: j <= width (Gauge (C,n)) and
A6: (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k))} and
A7: (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j))} and
A8: LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) misses Upper_Arc C ; ::_thesis: contradiction
 (Gauge (C,n)) * (i,j) in {((Gauge (C,n)) * (i,j))} by TARSKI:def_1;
then A9: (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) by A7, XBOOLE_0:def_4;
 (Gauge (C,n)) * (i,k) in {((Gauge (C,n)) * (i,k))} by TARSKI:def_1;
then A10: (Gauge (C,n)) * (i,k) in L~ (Upper_Seq (C,n)) by A6, XBOOLE_0:def_4;
then A11: j <> k by A1, A2, A3, A5, A9, JORDAN1J:57;
A12: 1 <= j by A3, A4, XXREAL_0:2;
A13: k <= width (Gauge (C,n)) by A4, A5, XXREAL_0:2;
A14: [i,j] in Indices (Gauge (C,n)) by A1, A2, A5, A12, MATRIX_1:36;
A15: [i,k] in Indices (Gauge (C,n)) by A1, A2, A3, A13, MATRIX_1:36;
set do = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)));
set go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,k)));
A16: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
A17: len (Upper_Seq (C,n)) >= 3 by JORDAN1E:15;
then  len (Upper_Seq (C,n)) >= 1 by XXREAL_0:2;
then  1 in dom (Upper_Seq (C,n)) by FINSEQ_3:25;
then A18: (Upper_Seq (C,n)) . 1 = (Upper_Seq (C,n)) /. 1 by PARTFUN1:def_6
.= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ;
A19: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * (1,k)) `1 by A3, A13, A16, JORDAN1A:73 ;
 len (Gauge (C,n)) >= 4 by JORDAN8:10;
then A20: len (Gauge (C,n)) >= 1 by XXREAL_0:2;
then A21: [1,k] in Indices (Gauge (C,n)) by A3, A13, MATRIX_1:36;
then A22: (Gauge (C,n)) * (i,k) <> (Upper_Seq (C,n)) . 1 by A1, A15, A18, A19, JORDAN1G:7;
then reconsider go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,k))) as being_S-Seq FinSequence of (TOP-REAL 2) by A10, JORDAN3:35;
A23: len (Lower_Seq (C,n)) >= 1 + 2 by JORDAN1E:15;
then A24: len (Lower_Seq (C,n)) >= 1 by XXREAL_0:2;
then A25: 1 in dom (Lower_Seq (C,n)) by FINSEQ_3:25;
 len (Lower_Seq (C,n)) in dom (Lower_Seq (C,n)) by A24, FINSEQ_3:25;
then A26: (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by PARTFUN1:def_6
.= W-min (L~ (Cage (C,n))) by JORDAN1F:8 ;
A27: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * (1,k)) `1 by A3, A13, A16, JORDAN1A:73 ;
A28: [i,j] in Indices (Gauge (C,n)) by A1, A2, A5, A12, MATRIX_1:36;
then A29: (Gauge (C,n)) * (i,j) <> (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) by A1, A21, A26, A27, JORDAN1G:7;
then reconsider do = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))) as being_S-Seq FinSequence of (TOP-REAL 2) by A9, JORDAN3:34;
A30: [(len (Gauge (C,n))),k] in Indices (Gauge (C,n)) by A3, A13, A20, MATRIX_1:36;
A31: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by A25, PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;
(E-max (L~ (Cage (C,n)))) `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A3, A13, A16, JORDAN1A:71 ;
then A32: (Gauge (C,n)) * (i,j) <> (Lower_Seq (C,n)) . 1 by A2, A28, A30, A31, JORDAN1G:7;
A33: len go >= 1 + 1 by TOPREAL1:def_8;
A34: (Gauge (C,n)) * (i,k) in rng (Upper_Seq (C,n)) by A1, A2, A3, A10, A13, JORDAN1G:4, JORDAN1J:40;
then A35: go is_sequence_on Gauge (C,n) by JORDAN1G:4, JORDAN1J:38;
A36: len do >= 1 + 1 by TOPREAL1:def_8;
A37: (Gauge (C,n)) * (i,j) in rng (Lower_Seq (C,n)) by A1, A2, A5, A9, A12, JORDAN1G:5, JORDAN1J:40;
then A38: do is_sequence_on Gauge (C,n) by JORDAN1G:5, JORDAN1J:39;
reconsider go = go as non constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A33, A35, JGRAPH_1:12, JORDAN8:5;
reconsider do = do as non constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A36, A38, JGRAPH_1:12, JORDAN8:5;
A39: len go > 1 by A33, NAT_1:13;
then A40: len go in dom go by FINSEQ_3:25;
then A41: go /. (len go) = go . (len go) by PARTFUN1:def_6
.= (Gauge (C,n)) * (i,k) by A10, JORDAN3:24 ;
 len do >= 1 by A36, XXREAL_0:2;
then  1 in dom do by FINSEQ_3:25;
then A42: do /. 1 = do . 1 by PARTFUN1:def_6
.= (Gauge (C,n)) * (i,j) by A9, JORDAN3:23 ;
reconsider m = (len go) - 1 as Element of NAT by A40, FINSEQ_3:26;
A43: m + 1 = len go ;
then A44: (len go) -' 1 = m by NAT_D:34;
A45: LSeg (go,m) c= L~ go by TOPREAL3:19;
A46: L~ go c= L~ (Upper_Seq (C,n)) by A10, JORDAN3:41;
then  LSeg (go,m) c= L~ (Upper_Seq (C,n)) by A45, XBOOLE_1:1;
then A47: (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) c= {((Gauge (C,n)) * (i,k))} by A6, XBOOLE_1:26;
 m >= 1 by A33, XREAL_1:19;
then A48: LSeg (go,m) = LSeg ((go /. m),((Gauge (C,n)) * (i,k))) by A41, A43, TOPREAL1:def_3;
 {((Gauge (C,n)) * (i,k))} c= (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i,k))} or x in (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) )
assume  x in {((Gauge (C,n)) * (i,k))} ; ::_thesis: x in (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))))
then A49: x = (Gauge (C,n)) * (i,k) by TARSKI:def_1;
A50: (Gauge (C,n)) * (i,k) in LSeg (go,m) by A48, RLTOPSP1:68;
 (Gauge (C,n)) * (i,k) in LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) by RLTOPSP1:68;
hence  x in (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) by A49, A50, XBOOLE_0:def_4; ::_thesis: verum
end;
then A51: (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) = {((Gauge (C,n)) * (i,k))} by A47, XBOOLE_0:def_10;
A52: LSeg (do,1) c= L~ do by TOPREAL3:19;
A53: L~ do c= L~ (Lower_Seq (C,n)) by A9, JORDAN3:42;
then  LSeg (do,1) c= L~ (Lower_Seq (C,n)) by A52, XBOOLE_1:1;
then A54: (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) c= {((Gauge (C,n)) * (i,j))} by A7, XBOOLE_1:26;
A55: LSeg (do,1) = LSeg (((Gauge (C,n)) * (i,j)),(do /. (1 + 1))) by A36, A42, TOPREAL1:def_3;
 {((Gauge (C,n)) * (i,j))} c= (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i,j))} or x in (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) )
assume  x in {((Gauge (C,n)) * (i,j))} ; ::_thesis: x in (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))))
then A56: x = (Gauge (C,n)) * (i,j) by TARSKI:def_1;
A57: (Gauge (C,n)) * (i,j) in LSeg (do,1) by A55, RLTOPSP1:68;
 (Gauge (C,n)) * (i,j) in LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) by RLTOPSP1:68;
hence  x in (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) by A56, A57, XBOOLE_0:def_4; ::_thesis: verum
end;
then A58: (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) /\ (LSeg (do,1)) = {((Gauge (C,n)) * (i,j))} by A54, XBOOLE_0:def_10;
A59: go /. 1 = (Upper_Seq (C,n)) /. 1 by A10, SPRECT_3:22
.= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ;
then A60: go /. 1 = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by JORDAN1F:8
.= do /. (len do) by A9, JORDAN1J:35 ;
A61: rng go c= L~ go by A33, SPPOL_2:18;
A62: rng do c= L~ do by A36, SPPOL_2:18;
A63: {(go /. 1)} c= (L~ go) /\ (L~ do)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(go /. 1)} or x in (L~ go) /\ (L~ do) )
assume  x in {(go /. 1)} ; ::_thesis: x in (L~ go) /\ (L~ do)
then A64: x = go /. 1 by TARSKI:def_1;
then A65: x in rng go by FINSEQ_6:42;
 x in rng do by A60, A64, REVROT_1:3;
hence  x in (L~ go) /\ (L~ do) by A61, A62, A65, XBOOLE_0:def_4; ::_thesis: verum
end;
A66: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by A25, PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;
A67: [(len (Gauge (C,n))),j] in Indices (Gauge (C,n)) by A5, A12, A20, MATRIX_1:36;
 (L~ go) /\ (L~ do) c= {(go /. 1)}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ go) /\ (L~ do) or x in {(go /. 1)} )
assume A68: x in (L~ go) /\ (L~ do) ; ::_thesis: x in {(go /. 1)}
then A69: x in L~ go by XBOOLE_0:def_4;
A70: x in L~ do by A68, XBOOLE_0:def_4;
then  x in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A46, A53, A69, XBOOLE_0:def_4;
then  x in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16;
then A71: ( x = W-min (L~ (Cage (C,n))) or x = E-max (L~ (Cage (C,n))) ) by TARSKI:def_2;
now__::_thesis:_not_x_=_E-max_(L~_(Cage_(C,n)))
assume  x = E-max (L~ (Cage (C,n))) ; ::_thesis: contradiction
then A72: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i,j) by A9, A66, A70, JORDAN1E:7;
 ((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 = E-bound (L~ (Cage (C,n))) by A5, A12, A16, JORDAN1A:71;
then  (E-max (L~ (Cage (C,n)))) `1 <> E-bound (L~ (Cage (C,n))) by A2, A14, A67, A72, JORDAN1G:7;
hence  contradiction by EUCLID:52; ::_thesis: verum
end;
hence  x in {(go /. 1)} by A59, A71, TARSKI:def_1; ::_thesis: verum
end;
then A73: (L~ go) /\ (L~ do) = {(go /. 1)} by A63, XBOOLE_0:def_10;
set W2 = go /. 2;
A74: 2 in dom go by A33, FINSEQ_3:25;
A75: now__::_thesis:_not_((Gauge_(C,n))_*_(i,k))_`1_=_W-bound_(L~_(Cage_(C,n)))
assume  ((Gauge (C,n)) * (i,k)) `1 = W-bound (L~ (Cage (C,n))) ; ::_thesis: contradiction
then  ((Gauge (C,n)) * (1,k)) `1 = ((Gauge (C,n)) * (i,k)) `1 by A3, A13, A16, JORDAN1A:73;
hence  contradiction by A1, A15, A21, JORDAN1G:7; ::_thesis: verum
end;
go = mid ((Upper_Seq (C,n)),1,(((Gauge (C,n)) * (i,k)) .. (Upper_Seq (C,n)))) by A34, JORDAN1G:49
.= (Upper_Seq (C,n)) | (((Gauge (C,n)) * (i,k)) .. (Upper_Seq (C,n))) by A34, FINSEQ_4:21, FINSEQ_6:116 ;
then A76: go /. 2 = (Upper_Seq (C,n)) /. 2 by A74, FINSEQ_4:70;
A77: W-min (L~ (Cage (C,n))) in rng go by A59, FINSEQ_6:42;
set pion = <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*>;
A78: now__::_thesis:_for_n_being_Element_of_NAT_st_n_in_dom_<*((Gauge_(C,n))_*_(i,k)),((Gauge_(C,n))_*_(i,j))*>_holds_
ex_i,_j_being_Element_of_NAT_st_
(_[i,j]_in_Indices_(Gauge_(C,n))_&_<*((Gauge_(C,n))_*_(i,k)),((Gauge_(C,n))_*_(i,j))*>_/._n_=_(Gauge_(C,n))_*_(i,j)_)
let n be Element of NAT ; ::_thesis: ( n in dom <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> implies ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. n = (Gauge (C,n)) * (i,j) ) )
assume  n in dom <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. n = (Gauge (C,n)) * (i,j) )
then  n in Seg 2 by FINSEQ_1:89;
then  ( n = 1 or n = 2 ) by FINSEQ_1:2, TARSKI:def_2;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. n = (Gauge (C,n)) * (i,j) ) by A14, A15, FINSEQ_4:17; ::_thesis: verum
end;
A79: (Gauge (C,n)) * (i,k) <> (Gauge (C,n)) * (i,j) by A11, A14, A15, GOBOARD1:5;
A80: ((Gauge (C,n)) * (i,k)) `1 = ((Gauge (C,n)) * (i,1)) `1 by A1, A2, A3, A13, GOBOARD5:2
.= ((Gauge (C,n)) * (i,j)) `1 by A1, A2, A5, A12, GOBOARD5:2 ;
then  LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) is vertical by SPPOL_1:16;
then  <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> is being_S-Seq by A79, JORDAN1B:7;
then consider pion1 being FinSequence of (TOP-REAL 2) such that
A81: pion1 is_sequence_on Gauge (C,n) and
A82: pion1 is being_S-Seq and
A83: L~ <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> = L~ pion1 and
A84: <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. 1 = pion1 /. 1 and
A85: <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. (len <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*>) = pion1 /. (len pion1) and
A86: len <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> <= len pion1 by A78, GOBOARD3:2;
reconsider pion1 = pion1 as being_S-Seq FinSequence of (TOP-REAL 2) by A82;
set godo = (go ^' pion1) ^' do;
A87: 1 + 1 <= len (Cage (C,n)) by GOBOARD7:34, XXREAL_0:2;
A88: 1 + 1 <= len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by GOBOARD7:34, XXREAL_0:2;
 len (go ^' pion1) >= len go by TOPREAL8:7;
then A89: len (go ^' pion1) >= 1 + 1 by A33, XXREAL_0:2;
then A90: len (go ^' pion1) > 1 + 0 by NAT_1:13;
A91: len ((go ^' pion1) ^' do) >= len (go ^' pion1) by TOPREAL8:7;
then A92: 1 + 1 <= len ((go ^' pion1) ^' do) by A89, XXREAL_0:2;
A93: Upper_Seq (C,n) is_sequence_on Gauge (C,n) by JORDAN1G:4;
A94: go /. (len go) = pion1 /. 1 by A41, A84, FINSEQ_4:17;
then A95: go ^' pion1 is_sequence_on Gauge (C,n) by A35, A81, TOPREAL8:12;
A96: (go ^' pion1) /. (len (go ^' pion1)) = <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. (len <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*>) by A85, GRAPH_2:54
.= <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. 2 by FINSEQ_1:44
.= do /. 1 by A42, FINSEQ_4:17 ;
then A97: (go ^' pion1) ^' do is_sequence_on Gauge (C,n) by A38, A95, TOPREAL8:12;
 LSeg (pion1,1) c= L~ <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> by A83, TOPREAL3:19;
then  LSeg (pion1,1) c= LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) by SPPOL_2:21;
then A98: (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) c= {((Gauge (C,n)) * (i,k))} by A44, A51, XBOOLE_1:27;
A99: len pion1 >= 1 + 1 by A86, FINSEQ_1:44;
 {((Gauge (C,n)) * (i,k))} c= (LSeg (go,m)) /\ (LSeg (pion1,1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i,k))} or x in (LSeg (go,m)) /\ (LSeg (pion1,1)) )
assume  x in {((Gauge (C,n)) * (i,k))} ; ::_thesis: x in (LSeg (go,m)) /\ (LSeg (pion1,1))
then A100: x = (Gauge (C,n)) * (i,k) by TARSKI:def_1;
A101: (Gauge (C,n)) * (i,k) in LSeg (go,m) by A48, RLTOPSP1:68;
 (Gauge (C,n)) * (i,k) in LSeg (pion1,1) by A41, A94, A99, TOPREAL1:21;
hence  x in (LSeg (go,m)) /\ (LSeg (pion1,1)) by A100, A101, XBOOLE_0:def_4; ::_thesis: verum
end;
then  (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) = {(go /. (len go))} by A41, A44, A98, XBOOLE_0:def_10;
then A102: go ^' pion1 is unfolded by A94, TOPREAL8:34;
 len pion1 >= 2 + 0 by A86, FINSEQ_1:44;
then A103: (len pion1) - 2 >= 0 by XREAL_1:19;
 ((len (go ^' pion1)) + 1) - 1 = ((len go) + (len pion1)) - 1 by GRAPH_2:13;
then (len (go ^' pion1)) - 1 = (len go) + ((len pion1) - 2)
.= (len go) + ((len pion1) -' 2) by A103, XREAL_0:def_2 ;
then A104: (len (go ^' pion1)) -' 1 = (len go) + ((len pion1) -' 2) by XREAL_0:def_2;
A105: (len pion1) - 1 >= 1 by A99, XREAL_1:19;
then A106: (len pion1) -' 1 = (len pion1) - 1 by XREAL_0:def_2;
A107: ((len pion1) -' 2) + 1 = ((len pion1) - 2) + 1 by A103, XREAL_0:def_2
.= (len pion1) -' 1 by A105, XREAL_0:def_2 ;
 ((len pion1) - 1) + 1 <= len pion1 ;
then A108: (len pion1) -' 1 < len pion1 by A106, NAT_1:13;
 LSeg (pion1,((len pion1) -' 1)) c= L~ <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> by A83, TOPREAL3:19;
then  LSeg (pion1,((len pion1) -' 1)) c= LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) by SPPOL_2:21;
then A109: (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) c= {((Gauge (C,n)) * (i,j))} by A58, XBOOLE_1:27;
 {((Gauge (C,n)) * (i,j))} c= (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i,j))} or x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) )
assume  x in {((Gauge (C,n)) * (i,j))} ; ::_thesis: x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1))
then A110: x = (Gauge (C,n)) * (i,j) by TARSKI:def_1;
A111: (Gauge (C,n)) * (i,j) in LSeg (do,1) by A55, RLTOPSP1:68;
pion1 /. (((len pion1) -' 1) + 1) = <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. 2 by A85, A106, FINSEQ_1:44
.= (Gauge (C,n)) * (i,j) by FINSEQ_4:17 ;
then  (Gauge (C,n)) * (i,j) in LSeg (pion1,((len pion1) -' 1)) by A105, A106, TOPREAL1:21;
hence  x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) by A110, A111, XBOOLE_0:def_4; ::_thesis: verum
end;
then  (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) = {((Gauge (C,n)) * (i,j))} by A109, XBOOLE_0:def_10;
then A112: (LSeg ((go ^' pion1),((len go) + ((len pion1) -' 2)))) /\ (LSeg (do,1)) = {((go ^' pion1) /. (len (go ^' pion1)))} by A42, A94, A96, A107, A108, TOPREAL8:31;
A113: not go ^' pion1 is trivial by A89, NAT_D:60;
A114: rng pion1 c= L~ pion1 by A99, SPPOL_2:18;
A115: {(pion1 /. 1)} c= (L~ go) /\ (L~ pion1)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(pion1 /. 1)} or x in (L~ go) /\ (L~ pion1) )
assume  x in {(pion1 /. 1)} ; ::_thesis: x in (L~ go) /\ (L~ pion1)
then A116: x = pion1 /. 1 by TARSKI:def_1;
then A117: x in rng go by A94, REVROT_1:3;
 x in rng pion1 by A116, FINSEQ_6:42;
hence  x in (L~ go) /\ (L~ pion1) by A61, A114, A117, XBOOLE_0:def_4; ::_thesis: verum
end;
 (L~ go) /\ (L~ pion1) c= {(pion1 /. 1)}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ go) /\ (L~ pion1) or x in {(pion1 /. 1)} )
assume A118: x in (L~ go) /\ (L~ pion1) ; ::_thesis: x in {(pion1 /. 1)}
then A119: x in L~ go by XBOOLE_0:def_4;
 x in L~ pion1 by A118, XBOOLE_0:def_4;
then  x in (L~ pion1) /\ (L~ (Upper_Seq (C,n))) by A46, A119, XBOOLE_0:def_4;
hence  x in {(pion1 /. 1)} by A6, A41, A83, A94, SPPOL_2:21; ::_thesis: verum
end;
then A120: (L~ go) /\ (L~ pion1) = {(pion1 /. 1)} by A115, XBOOLE_0:def_10;
then A121: go ^' pion1 is s.n.c. by A94, JORDAN1J:54;
 (rng go) /\ (rng pion1) c= {(pion1 /. 1)} by A61, A114, A120, XBOOLE_1:27;
then A122: go ^' pion1 is one-to-one by JORDAN1J:55;
A123: <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. (len <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*>) = <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. 2 by FINSEQ_1:44
.= do /. 1 by A42, FINSEQ_4:17 ;
A124: {(pion1 /. (len pion1))} c= (L~ do) /\ (L~ pion1)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(pion1 /. (len pion1))} or x in (L~ do) /\ (L~ pion1) )
assume  x in {(pion1 /. (len pion1))} ; ::_thesis: x in (L~ do) /\ (L~ pion1)
then A125: x = pion1 /. (len pion1) by TARSKI:def_1;
then A126: x in rng do by A85, A123, FINSEQ_6:42;
 x in rng pion1 by A125, REVROT_1:3;
hence  x in (L~ do) /\ (L~ pion1) by A62, A114, A126, XBOOLE_0:def_4; ::_thesis: verum
end;
 (L~ do) /\ (L~ pion1) c= {(pion1 /. (len pion1))}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ do) /\ (L~ pion1) or x in {(pion1 /. (len pion1))} )
assume A127: x in (L~ do) /\ (L~ pion1) ; ::_thesis: x in {(pion1 /. (len pion1))}
then A128: x in L~ do by XBOOLE_0:def_4;
 x in L~ pion1 by A127, XBOOLE_0:def_4;
then  x in (L~ pion1) /\ (L~ (Lower_Seq (C,n))) by A53, A128, XBOOLE_0:def_4;
hence  x in {(pion1 /. (len pion1))} by A7, A42, A83, A85, A123, SPPOL_2:21; ::_thesis: verum
end;
then A129: (L~ do) /\ (L~ pion1) = {(pion1 /. (len pion1))} by A124, XBOOLE_0:def_10;
A130: (L~ (go ^' pion1)) /\ (L~ do) = ((L~ go) \/ (L~ pion1)) /\ (L~ do) by A94, TOPREAL8:35
.= {(go /. 1)} \/ {(do /. 1)} by A73, A85, A123, A129, XBOOLE_1:23
.= {((go ^' pion1) /. 1)} \/ {(do /. 1)} by GRAPH_2:53
.= {((go ^' pion1) /. 1),(do /. 1)} by ENUMSET1:1 ;
 do /. (len do) = (go ^' pion1) /. 1 by A60, GRAPH_2:53;
then reconsider godo = (go ^' pion1) ^' do as non constant standard special_circular_sequence by A92, A96, A97, A102, A104, A112, A113, A121, A122, A130, JORDAN8:4, JORDAN8:5, TOPREAL8:11, TOPREAL8:33, TOPREAL8:34;
A131: Upper_Arc C is_an_arc_of W-min C, E-max C by JORDAN6:def_8;
then A132: Upper_Arc C is connected by JORDAN6:10;
A133: W-min C in Upper_Arc C by A131, TOPREAL1:1;
A134: E-max C in Upper_Arc C by A131, TOPREAL1:1;
set ff = Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))));
 W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
then A135: (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 = W-min (L~ (Cage (C,n))) by FINSEQ_6:92;
A136: L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33;
then  (W-max (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A135, SPRECT_5:22;
then  (N-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A135, A136, SPRECT_5:23, XXREAL_0:2;
then  (N-max (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A135, A136, SPRECT_5:24, XXREAL_0:2;
then A137: (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A135, A136, SPRECT_5:25, XXREAL_0:2;
A138: now__::_thesis:_not_((Gauge_(C,n))_*_(i,k))_.._(Upper_Seq_(C,n))_<=_1
assume A139: ((Gauge (C,n)) * (i,k)) .. (Upper_Seq (C,n)) <= 1 ; ::_thesis: contradiction
 ((Gauge (C,n)) * (i,k)) .. (Upper_Seq (C,n)) >= 1 by A34, FINSEQ_4:21;
then  ((Gauge (C,n)) * (i,k)) .. (Upper_Seq (C,n)) = 1 by A139, XXREAL_0:1;
then  (Gauge (C,n)) * (i,k) = (Upper_Seq (C,n)) /. 1 by A34, FINSEQ_5:38;
hence  contradiction by A18, A22, JORDAN1F:5; ::_thesis: verum
end;
A140: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def_1;
then A141: Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))) is_sequence_on Gauge (C,n) by REVROT_1:34;
A142: (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) c= RightComp godo by A92, A97, JORDAN9:27;
A143: L~ godo = (L~ (go ^' pion1)) \/ (L~ do) by A96, TOPREAL8:35
.= ((L~ go) \/ (L~ pion1)) \/ (L~ do) by A94, TOPREAL8:35 ;
A144: L~ (Cage (C,n)) = (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) by JORDAN1E:13;
then A145: L~ (Upper_Seq (C,n)) c= L~ (Cage (C,n)) by XBOOLE_1:7;
A146: L~ (Lower_Seq (C,n)) c= L~ (Cage (C,n)) by A144, XBOOLE_1:7;
A147: L~ go c= L~ (Cage (C,n)) by A46, A145, XBOOLE_1:1;
A148: L~ do c= L~ (Cage (C,n)) by A53, A146, XBOOLE_1:1;
A149: W-min C in C by SPRECT_1:13;
A150: L~ <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> = LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) by SPPOL_2:21;
A151: now__::_thesis:_not_W-min_C_in_L~_godo
assume  W-min C in L~ godo ; ::_thesis: contradiction
then A152: ( W-min C in (L~ go) \/ (L~ pion1) or W-min C in L~ do ) by A143, XBOOLE_0:def_3;
percases ( W-min C in L~ go or W-min C in L~ pion1 or W-min C in L~ do ) by A152, XBOOLE_0:def_3;
suppose W-min C in L~ go ; ::_thesis: contradiction
then  C meets L~ (Cage (C,n)) by A147, A149, XBOOLE_0:3;
hence  contradiction by JORDAN10:5; ::_thesis: verum
end;
suppose W-min C in L~ pion1 ; ::_thesis: contradiction
hence  contradiction by A8, A83, A133, A150, XBOOLE_0:3; ::_thesis: verum
end;
suppose W-min C in L~ do ; ::_thesis: contradiction
then  C meets L~ (Cage (C,n)) by A148, A149, XBOOLE_0:3;
hence  contradiction by JORDAN10:5; ::_thesis: verum
end;
end;
end;
right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1) = right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(GoB (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) by A88, JORDAN1H:23
.= right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(GoB (Cage (C,n)))) by REVROT_1:28
.= right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(Gauge (C,n))) by JORDAN1H:44
.= right_cell (((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n))))),1,(Gauge (C,n))) by A137, A141, JORDAN1J:53
.= right_cell ((Upper_Seq (C,n)),1,(Gauge (C,n))) by JORDAN1E:def_1
.= right_cell ((R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,k)))),1,(Gauge (C,n))) by A34, A93, A138, JORDAN1J:52
.= right_cell ((go ^' pion1),1,(Gauge (C,n))) by A39, A95, JORDAN1J:51
.= right_cell (godo,1,(Gauge (C,n))) by A90, A97, JORDAN1J:51 ;
then  W-min C in right_cell (godo,1,(Gauge (C,n))) by JORDAN1I:6;
then A153: W-min C in (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) by A151, XBOOLE_0:def_5;
A154: godo /. 1 = (go ^' pion1) /. 1 by GRAPH_2:53
.= W-min (L~ (Cage (C,n))) by A59, GRAPH_2:53 ;
A155: len (Upper_Seq (C,n)) >= 2 by A17, XXREAL_0:2;
A156: godo /. 2 = (go ^' pion1) /. 2 by A89, GRAPH_2:57
.= (Upper_Seq (C,n)) /. 2 by A33, A76, GRAPH_2:57
.= ((Upper_Seq (C,n)) ^' (Lower_Seq (C,n))) /. 2 by A155, GRAPH_2:57
.= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 2 by JORDAN1E:11 ;
A157: (L~ go) \/ (L~ do) is compact by COMPTS_1:10;
 W-min (L~ (Cage (C,n))) in (L~ go) \/ (L~ do) by A61, A77, XBOOLE_0:def_3;
then A158: W-min ((L~ go) \/ (L~ do)) = W-min (L~ (Cage (C,n))) by A147, A148, A157, JORDAN1J:21, XBOOLE_1:8;
A159: (W-min ((L~ go) \/ (L~ do))) `1 = W-bound ((L~ go) \/ (L~ do)) by EUCLID:52;
A160: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52;
 W-bound (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) = ((Gauge (C,n)) * (i,k)) `1 by A80, SPRECT_1:54;
then A161: W-bound (L~ pion1) = ((Gauge (C,n)) * (i,k)) `1 by A83, SPPOL_2:21;
 ((Gauge (C,n)) * (i,k)) `1 >= W-bound (L~ (Cage (C,n))) by A10, A145, PSCOMP_1:24;
then  ((Gauge (C,n)) * (i,k)) `1 > W-bound (L~ (Cage (C,n))) by A75, XXREAL_0:1;
then  W-min (((L~ go) \/ (L~ do)) \/ (L~ pion1)) = W-min ((L~ go) \/ (L~ do)) by A157, A158, A159, A160, A161, JORDAN1J:33;
then A162: W-min (L~ godo) = W-min (L~ (Cage (C,n))) by A143, A158, XBOOLE_1:4;
A163: rng godo c= L~ godo by A89, A91, SPPOL_2:18, XXREAL_0:2;
 2 in dom godo by A92, FINSEQ_3:25;
then A164: godo /. 2 in rng godo by PARTFUN2:2;
 godo /. 2 in W-most (L~ (Cage (C,n))) by A156, JORDAN1I:25;
then (godo /. 2) `1 = (W-min (L~ godo)) `1 by A162, PSCOMP_1:31
.= W-bound (L~ godo) by EUCLID:52 ;
then  godo /. 2 in W-most (L~ godo) by A163, A164, SPRECT_2:12;
then  (Rotate (godo,(W-min (L~ godo)))) /. 2 in W-most (L~ godo) by A154, A162, FINSEQ_6:89;
then reconsider godo = godo as non constant standard clockwise_oriented special_circular_sequence by JORDAN1I:25;
 len (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by FINSEQ_5:6;
then A165: (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) = (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) by PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:7 ;
A166: east_halfline (E-max C) misses L~ go
proof
assume  east_halfline (E-max C) meets L~ go ; ::_thesis: contradiction
then consider p being set  such that
A167: p in east_halfline (E-max C) and
A168: p in L~ go by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A167;
 p in L~ (Upper_Seq (C,n)) by A46, A168;
then  p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A145, A167, XBOOLE_0:def_4;
then A169: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;
then A170: p = E-max (L~ (Cage (C,n))) by A46, A168, JORDAN1J:46;
then  E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i,k) by A10, A165, A168, JORDAN1J:43;
then  ((Gauge (C,n)) * (i,k)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A3, A13, A16, A169, A170, JORDAN1A:71;
hence  contradiction by A2, A15, A30, JORDAN1G:7; ::_thesis: verum
end;
now__::_thesis:_not_east_halfline_(E-max_C)_meets_L~_godo
assume  east_halfline (E-max C) meets L~ godo ; ::_thesis: contradiction
then A171: ( east_halfline (E-max C) meets (L~ go) \/ (L~ pion1) or east_halfline (E-max C) meets L~ do ) by A143, XBOOLE_1:70;
percases ( east_halfline (E-max C) meets L~ go or east_halfline (E-max C) meets L~ pion1 or east_halfline (E-max C) meets L~ do ) by A171, XBOOLE_1:70;
suppose east_halfline (E-max C) meets L~ go ; ::_thesis: contradiction
hence  contradiction by A166; ::_thesis: verum
end;
suppose east_halfline (E-max C) meets L~ pion1 ; ::_thesis: contradiction
then consider p being set  such that
A172: p in east_halfline (E-max C) and
A173: p in L~ pion1 by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A172;
A174: p `1 = ((Gauge (C,n)) * (i,k)) `1 by A80, A83, A150, A173, GOBOARD7:5;
 i + 1 <= len (Gauge (C,n)) by A2, NAT_1:13;
then  (i + 1) - 1 <= (len (Gauge (C,n))) - 1 by XREAL_1:9;
then A175: i <= (len (Gauge (C,n))) -' 1 by XREAL_0:def_2;
 (len (Gauge (C,n))) -' 1 <= len (Gauge (C,n)) by NAT_D:35;
then  p `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1 by A1, A3, A13, A16, A20, A174, A175, JORDAN1A:18;
then  p `1 <= E-bound C by A20, JORDAN8:12;
then A176: p `1 <= (E-max C) `1 by EUCLID:52;
 p `1 >= (E-max C) `1 by A172, TOPREAL1:def_11;
then A177: p `1 = (E-max C) `1 by A176, XXREAL_0:1;
 p `2 = (E-max C) `2 by A172, TOPREAL1:def_11;
then  p = E-max C by A177, TOPREAL3:6;
hence  contradiction by A8, A83, A134, A150, A173, XBOOLE_0:3; ::_thesis: verum
end;
suppose east_halfline (E-max C) meets L~ do ; ::_thesis: contradiction
then consider p being set  such that
A178: p in east_halfline (E-max C) and
A179: p in L~ do by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A178;
 p in L~ (Lower_Seq (C,n)) by A53, A179;
then  p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A146, A178, XBOOLE_0:def_4;
then A180: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;
A181: (E-max C) `2 = p `2 by A178, TOPREAL1:def_11;
set RC = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))));
A182: E-max C in right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by JORDAN1I:7;
A183: 1 + 1 <= len (Lower_Seq (C,n)) by A23, XXREAL_0:2;
 Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) by JORDAN1G:18;
then A184: LSeg ((Lower_Seq (C,n)),1) = LSeg ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by A183, SPPOL_2:9;
A185: L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33;
A186: len (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = len (Cage (C,n)) by REVROT_1:14;
A187: GoB (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = GoB (Cage (C,n)) by REVROT_1:28
.= Gauge (C,n) by JORDAN1H:44 ;
A188: E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
A189: Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))) is_sequence_on Gauge (C,n) by A140, REVROT_1:34;
A190: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = E-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))))) by A185, A188, FINSEQ_6:92;
consider ii, jj being Element of NAT  such that
A191: [ii,(jj + 1)] in Indices (Gauge (C,n)) and
A192: [ii,jj] in Indices (Gauge (C,n)) and
A193: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = (Gauge (C,n)) * (ii,(jj + 1)) and
A194: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1) = (Gauge (C,n)) * (ii,jj) by A87, A185, A186, A188, A189, FINSEQ_6:92, JORDAN1I:23;
consider jj2 being Element of NAT  such that
A195: 1 <= jj2 and
A196: jj2 <= width (Gauge (C,n)) and
A197: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * ((len (Gauge (C,n))),jj2) by JORDAN1D:25;
A198: len (Gauge (C,n)) >= 4 by JORDAN8:10;
then  len (Gauge (C,n)) >= 1 by XXREAL_0:2;
then  [(len (Gauge (C,n))),jj2] in Indices (Gauge (C,n)) by A195, A196, MATRIX_1:36;
then A199: ii = len (Gauge (C,n)) by A185, A190, A191, A193, A197, GOBOARD1:5;
A200: 1 <= ii by A191, MATRIX_1:38;
A201: ii <= len (Gauge (C,n)) by A191, MATRIX_1:38;
A202: 1 <= jj + 1 by A191, MATRIX_1:38;
A203: jj + 1 <= width (Gauge (C,n)) by A191, MATRIX_1:38;
A204: 1 <= ii by A192, MATRIX_1:38;
A205: ii <= len (Gauge (C,n)) by A192, MATRIX_1:38;
A206: 1 <= jj by A192, MATRIX_1:38;
A207: jj <= width (Gauge (C,n)) by A192, MATRIX_1:38;
A208: ii + 1 <> ii ;
 (jj + 1) + 1 <> jj ;
then A209: right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) = cell ((Gauge (C,n)),(ii -' 1),jj) by A87, A186, A187, A191, A192, A193, A194, A208, GOBOARD5:def_6;
A210: (ii -' 1) + 1 = ii by A200, XREAL_1:235;
 ii - 1 >= 4 - 1 by A198, A199, XREAL_1:9;
then A211: ii - 1 >= 1 by XXREAL_0:2;
then A212: 1 <= ii -' 1 by XREAL_0:def_2;
A213: ((Gauge (C,n)) * ((ii -' 1),jj)) `2 <= p `2 by A181, A182, A201, A203, A206, A209, A210, A211, JORDAN9:17;
A214: p `2 <= ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 by A181, A182, A201, A203, A206, A209, A210, A211, JORDAN9:17;
A215: ii -' 1 < len (Gauge (C,n)) by A201, A210, NAT_1:13;
then A216: ((Gauge (C,n)) * ((ii -' 1),jj)) `2 = ((Gauge (C,n)) * (1,jj)) `2 by A206, A207, A212, GOBOARD5:1
.= ((Gauge (C,n)) * (ii,jj)) `2 by A204, A205, A206, A207, GOBOARD5:1 ;
A217: ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 = ((Gauge (C,n)) * (1,(jj + 1))) `2 by A202, A203, A212, A215, GOBOARD5:1
.= ((Gauge (C,n)) * (ii,(jj + 1))) `2 by A200, A201, A202, A203, GOBOARD5:1 ;
A218: ((Gauge (C,n)) * ((len (Gauge (C,n))),jj)) `1 = E-bound (L~ (Cage (C,n))) by A16, A206, A207, JORDAN1A:71;
 E-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * ((len (Gauge (C,n))),(jj + 1))) `1 by A16, A202, A203, JORDAN1A:71;
then  p in LSeg (((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1),((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1))) by A180, A193, A194, A199, A213, A214, A216, A217, A218, GOBOARD7:7;
then A219: p in LSeg ((Lower_Seq (C,n)),1) by A87, A184, A186, TOPREAL1:def_3;
A220: p in LSeg (do,(Index (p,do))) by A179, JORDAN3:9;
A221: do = mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n)))) by A37, JORDAN1J:37;
A222: 1 <= ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) by A37, FINSEQ_4:21;
A223: ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) <= len (Lower_Seq (C,n)) by A37, FINSEQ_4:21;
 ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) <> len (Lower_Seq (C,n)) by A29, A37, FINSEQ_4:19;
then A224: ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) < len (Lower_Seq (C,n)) by A223, XXREAL_0:1;
A225: 1 <= Index (p,do) by A179, JORDAN3:8;
A226: Index (p,do) < len do by A179, JORDAN3:8;
A227: (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1 = ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) by A32, A37, JORDAN1J:56;
consider t being Nat such that
A228: t in dom (Lower_Seq (C,n)) and
A229: (Lower_Seq (C,n)) . t = (Gauge (C,n)) * (i,j) by A37, FINSEQ_2:10;
A230: 1 <= t by A228, FINSEQ_3:25;
A231: t <= len (Lower_Seq (C,n)) by A228, FINSEQ_3:25;
 1 < t by A32, A229, A230, XXREAL_0:1;
then  (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1 = t by A229, A231, JORDAN3:12;
then A232: len (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) = (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) by A9, A229, JORDAN3:26;
set tt = ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1;
A233: 1 <= Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n))) by A9, JORDAN3:8;
 0 + (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) < len (Lower_Seq (C,n)) by A9, JORDAN3:8;
then A234: (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) > 0 by XREAL_1:20;
 Index (p,do) < (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) by A226, A232, XREAL_0:def_2;
then  (Index (p,do)) + 1 <= (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) by NAT_1:13;
then  Index (p,do) <= ((len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n))))) - 1 by XREAL_1:19;
then  Index (p,do) <= ((len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n))))) - 1 by A234, XREAL_0:def_2;
then  Index (p,do) <= (len (Lower_Seq (C,n))) - (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))) by A227;
then  Index (p,do) <= (len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))) by XREAL_0:def_2;
then  Index (p,do) < ((len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) + 1 by NAT_1:13;
then A235: LSeg ((mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n))))),(Index (p,do))) = LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1)) by A222, A224, A225, JORDAN4:19;
A236: 1 + 1 <= ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) by A227, A233, XREAL_1:7;
then  (Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))) >= (1 + 1) + 1 by A225, XREAL_1:7;
then  ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) - 1 >= ((1 + 1) + 1) - 1 by XREAL_1:9;
then A237: ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1 >= 1 + 1 by XREAL_0:def_2;
A238: 2 in dom (Lower_Seq (C,n)) by A183, FINSEQ_3:25;
now__::_thesis:_contradiction
percases ( ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1 > 1 + 1 or ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ) by A237, XXREAL_0:1;
suppose ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1 > 1 + 1 ; ::_thesis: contradiction
then  LSeg ((Lower_Seq (C,n)),1) misses LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1)) by TOPREAL1:def_7;
hence  contradiction by A219, A220, A221, A235, XBOOLE_0:3; ::_thesis: verum
end;
supposeA239: ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ; ::_thesis: contradiction
then  (LSeg ((Lower_Seq (C,n)),1)) /\ (LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1))) = {((Lower_Seq (C,n)) /. 2)} by A23, TOPREAL1:def_6;
then  p in {((Lower_Seq (C,n)) /. 2)} by A219, A220, A221, A235, XBOOLE_0:def_4;
then A240: p = (Lower_Seq (C,n)) /. 2 by TARSKI:def_1;
then A241: p .. (Lower_Seq (C,n)) = 2 by A238, FINSEQ_5:41;
 1 + 1 = ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) - 1 by A239, XREAL_0:def_2;
then  (1 + 1) + 1 = (Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))) ;
then A242: ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) = 2 by A225, A236, JORDAN1E:6;
 p in rng (Lower_Seq (C,n)) by A238, A240, PARTFUN2:2;
then  p = (Gauge (C,n)) * (i,j) by A37, A241, A242, FINSEQ_5:9;
then  ((Gauge (C,n)) * (i,j)) `1 = E-bound (L~ (Cage (C,n))) by A240, JORDAN1G:32;
then  ((Gauge (C,n)) * (i,j)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 by A5, A12, A16, JORDAN1A:71;
hence  contradiction by A2, A14, A67, JORDAN1G:7; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
end;
end;
then  east_halfline (E-max C) c= (L~ godo) ` by SUBSET_1:23;
then consider W being Subset of (TOP-REAL 2) such that
A243: W is_a_component_of (L~ godo) ` and
A244: east_halfline (E-max C) c= W by GOBOARD9:3;
 not W is bounded by A244, JORDAN2C:121, RLTOPSP1:42;
then  W is_outside_component_of L~ godo by A243, JORDAN2C:def_3;
then  W c= UBD (L~ godo) by JORDAN2C:23;
then A245: east_halfline (E-max C) c= UBD (L~ godo) by A244, XBOOLE_1:1;
 E-max C in east_halfline (E-max C) by TOPREAL1:38;
then  E-max C in UBD (L~ godo) by A245;
then  E-max C in LeftComp godo by GOBRD14:36;
then  Upper_Arc C meets L~ godo by A132, A133, A134, A142, A153, JORDAN1J:36;
then A246: ( Upper_Arc C meets (L~ go) \/ (L~ pion1) or Upper_Arc C meets L~ do ) by A143, XBOOLE_1:70;
A247: Upper_Arc C c= C by JORDAN6:61;
percases ( Upper_Arc C meets L~ go or Upper_Arc C meets L~ pion1 or Upper_Arc C meets L~ do ) by A246, XBOOLE_1:70;
suppose Upper_Arc C meets L~ go ; ::_thesis: contradiction
then  Upper_Arc C meets L~ (Cage (C,n)) by A46, A145, XBOOLE_1:1, XBOOLE_1:63;
hence  contradiction by A247, JORDAN10:5, XBOOLE_1:63; ::_thesis: verum
end;
suppose Upper_Arc C meets L~ pion1 ; ::_thesis: contradiction
hence  contradiction by A8, A83, A150; ::_thesis: verum
end;
suppose Upper_Arc C meets L~ do ; ::_thesis: contradiction
then  Upper_Arc C meets L~ (Cage (C,n)) by A53, A146, XBOOLE_1:1, XBOOLE_1:63;
hence  contradiction by A247, JORDAN10:5, XBOOLE_1:63; ::_thesis: verum
end;
end;
end;

theorem Th13: :: JORDAN19:13
for n being Element of NAT
for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= k & k <= j & j <= width (Gauge (C,n)) & (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j))} holds
LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) meets Lower_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= k & k <= j & j <= width (Gauge (C,n)) & (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j))} holds
LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) meets Lower_Arc C
let C be Simple_closed_curve; ::_thesis: for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= k & k <= j & j <= width (Gauge (C,n)) & (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j))} holds
LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) meets Lower_Arc C
let i, j, k be Element of NAT ; ::_thesis: ( 1 < i & i < len (Gauge (C,n)) & 1 <= k & k <= j & j <= width (Gauge (C,n)) & (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j))} implies LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) meets Lower_Arc C )
set Ga = Gauge (C,n);
set US = Upper_Seq (C,n);
set LS = Lower_Seq (C,n);
set LA = Lower_Arc C;
set Wmin = W-min (L~ (Cage (C,n)));
set Emax = E-max (L~ (Cage (C,n)));
set Wbo = W-bound (L~ (Cage (C,n)));
set Ebo = E-bound (L~ (Cage (C,n)));
set Gik = (Gauge (C,n)) * (i,k);
set Gij = (Gauge (C,n)) * (i,j);
assume that
A1: 1 < i and
A2: i < len (Gauge (C,n)) and
A3: 1 <= k and
A4: k <= j and
A5: j <= width (Gauge (C,n)) and
A6: (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,k))} and
A7: (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,j))} and
A8: LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) misses Lower_Arc C ; ::_thesis: contradiction
 (Gauge (C,n)) * (i,j) in {((Gauge (C,n)) * (i,j))} by TARSKI:def_1;
then A9: (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) by A7, XBOOLE_0:def_4;
 (Gauge (C,n)) * (i,k) in {((Gauge (C,n)) * (i,k))} by TARSKI:def_1;
then A10: (Gauge (C,n)) * (i,k) in L~ (Upper_Seq (C,n)) by A6, XBOOLE_0:def_4;
then A11: j <> k by A1, A2, A3, A5, A9, JORDAN1J:57;
A12: 1 <= j by A3, A4, XXREAL_0:2;
A13: k <= width (Gauge (C,n)) by A4, A5, XXREAL_0:2;
A14: [i,j] in Indices (Gauge (C,n)) by A1, A2, A5, A12, MATRIX_1:36;
A15: [i,k] in Indices (Gauge (C,n)) by A1, A2, A3, A13, MATRIX_1:36;
set do = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)));
set go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,k)));
A16: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
A17: len (Upper_Seq (C,n)) >= 3 by JORDAN1E:15;
then  len (Upper_Seq (C,n)) >= 1 by XXREAL_0:2;
then  1 in dom (Upper_Seq (C,n)) by FINSEQ_3:25;
then A18: (Upper_Seq (C,n)) . 1 = (Upper_Seq (C,n)) /. 1 by PARTFUN1:def_6
.= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ;
A19: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * (1,k)) `1 by A3, A13, A16, JORDAN1A:73 ;
 len (Gauge (C,n)) >= 4 by JORDAN8:10;
then A20: len (Gauge (C,n)) >= 1 by XXREAL_0:2;
then A21: [1,k] in Indices (Gauge (C,n)) by A3, A13, MATRIX_1:36;
then A22: (Gauge (C,n)) * (i,k) <> (Upper_Seq (C,n)) . 1 by A1, A15, A18, A19, JORDAN1G:7;
then reconsider go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,k))) as being_S-Seq FinSequence of (TOP-REAL 2) by A10, JORDAN3:35;
A23: len (Lower_Seq (C,n)) >= 1 + 2 by JORDAN1E:15;
then A24: len (Lower_Seq (C,n)) >= 1 by XXREAL_0:2;
then A25: 1 in dom (Lower_Seq (C,n)) by FINSEQ_3:25;
 len (Lower_Seq (C,n)) in dom (Lower_Seq (C,n)) by A24, FINSEQ_3:25;
then A26: (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by PARTFUN1:def_6
.= W-min (L~ (Cage (C,n))) by JORDAN1F:8 ;
A27: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * (1,k)) `1 by A3, A13, A16, JORDAN1A:73 ;
A28: [i,j] in Indices (Gauge (C,n)) by A1, A2, A5, A12, MATRIX_1:36;
then A29: (Gauge (C,n)) * (i,j) <> (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) by A1, A21, A26, A27, JORDAN1G:7;
then reconsider do = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j))) as being_S-Seq FinSequence of (TOP-REAL 2) by A9, JORDAN3:34;
A30: [(len (Gauge (C,n))),k] in Indices (Gauge (C,n)) by A3, A13, A20, MATRIX_1:36;
A31: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by A25, PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;
(E-max (L~ (Cage (C,n)))) `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A3, A13, A16, JORDAN1A:71 ;
then A32: (Gauge (C,n)) * (i,j) <> (Lower_Seq (C,n)) . 1 by A2, A28, A30, A31, JORDAN1G:7;
A33: len go >= 1 + 1 by TOPREAL1:def_8;
A34: (Gauge (C,n)) * (i,k) in rng (Upper_Seq (C,n)) by A1, A2, A3, A10, A13, JORDAN1G:4, JORDAN1J:40;
then A35: go is_sequence_on Gauge (C,n) by JORDAN1G:4, JORDAN1J:38;
A36: len do >= 1 + 1 by TOPREAL1:def_8;
A37: (Gauge (C,n)) * (i,j) in rng (Lower_Seq (C,n)) by A1, A2, A5, A9, A12, JORDAN1G:5, JORDAN1J:40;
then A38: do is_sequence_on Gauge (C,n) by JORDAN1G:5, JORDAN1J:39;
reconsider go = go as non constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A33, A35, JGRAPH_1:12, JORDAN8:5;
reconsider do = do as non constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A36, A38, JGRAPH_1:12, JORDAN8:5;
A39: len go > 1 by A33, NAT_1:13;
then A40: len go in dom go by FINSEQ_3:25;
then A41: go /. (len go) = go . (len go) by PARTFUN1:def_6
.= (Gauge (C,n)) * (i,k) by A10, JORDAN3:24 ;
 len do >= 1 by A36, XXREAL_0:2;
then  1 in dom do by FINSEQ_3:25;
then A42: do /. 1 = do . 1 by PARTFUN1:def_6
.= (Gauge (C,n)) * (i,j) by A9, JORDAN3:23 ;
reconsider m = (len go) - 1 as Element of NAT by A40, FINSEQ_3:26;
A43: m + 1 = len go ;
then A44: (len go) -' 1 = m by NAT_D:34;
A45: LSeg (go,m) c= L~ go by TOPREAL3:19;
A46: L~ go c= L~ (Upper_Seq (C,n)) by A10, JORDAN3:41;
then  LSeg (go,m) c= L~ (Upper_Seq (C,n)) by A45, XBOOLE_1:1;
then A47: (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) c= {((Gauge (C,n)) * (i,k))} by A6, XBOOLE_1:26;
 m >= 1 by A33, XREAL_1:19;
then A48: LSeg (go,m) = LSeg ((go /. m),((Gauge (C,n)) * (i,k))) by A41, A43, TOPREAL1:def_3;
 {((Gauge (C,n)) * (i,k))} c= (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i,k))} or x in (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) )
assume  x in {((Gauge (C,n)) * (i,k))} ; ::_thesis: x in (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))))
then A49: x = (Gauge (C,n)) * (i,k) by TARSKI:def_1;
A50: (Gauge (C,n)) * (i,k) in LSeg (go,m) by A48, RLTOPSP1:68;
 (Gauge (C,n)) * (i,k) in LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) by RLTOPSP1:68;
hence  x in (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) by A49, A50, XBOOLE_0:def_4; ::_thesis: verum
end;
then A51: (LSeg (go,m)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) = {((Gauge (C,n)) * (i,k))} by A47, XBOOLE_0:def_10;
A52: LSeg (do,1) c= L~ do by TOPREAL3:19;
A53: L~ do c= L~ (Lower_Seq (C,n)) by A9, JORDAN3:42;
then  LSeg (do,1) c= L~ (Lower_Seq (C,n)) by A52, XBOOLE_1:1;
then A54: (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) c= {((Gauge (C,n)) * (i,j))} by A7, XBOOLE_1:26;
A55: LSeg (do,1) = LSeg (((Gauge (C,n)) * (i,j)),(do /. (1 + 1))) by A36, A42, TOPREAL1:def_3;
 {((Gauge (C,n)) * (i,j))} c= (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i,j))} or x in (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) )
assume  x in {((Gauge (C,n)) * (i,j))} ; ::_thesis: x in (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))))
then A56: x = (Gauge (C,n)) * (i,j) by TARSKI:def_1;
A57: (Gauge (C,n)) * (i,j) in LSeg (do,1) by A55, RLTOPSP1:68;
 (Gauge (C,n)) * (i,j) in LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) by RLTOPSP1:68;
hence  x in (LSeg (do,1)) /\ (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) by A56, A57, XBOOLE_0:def_4; ::_thesis: verum
end;
then A58: (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) /\ (LSeg (do,1)) = {((Gauge (C,n)) * (i,j))} by A54, XBOOLE_0:def_10;
A59: go /. 1 = (Upper_Seq (C,n)) /. 1 by A10, SPRECT_3:22
.= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ;
then A60: go /. 1 = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by JORDAN1F:8
.= do /. (len do) by A9, JORDAN1J:35 ;
A61: rng go c= L~ go by A33, SPPOL_2:18;
A62: rng do c= L~ do by A36, SPPOL_2:18;
A63: {(go /. 1)} c= (L~ go) /\ (L~ do)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(go /. 1)} or x in (L~ go) /\ (L~ do) )
assume  x in {(go /. 1)} ; ::_thesis: x in (L~ go) /\ (L~ do)
then A64: x = go /. 1 by TARSKI:def_1;
then A65: x in rng go by FINSEQ_6:42;
 x in rng do by A60, A64, REVROT_1:3;
hence  x in (L~ go) /\ (L~ do) by A61, A62, A65, XBOOLE_0:def_4; ::_thesis: verum
end;
A66: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by A25, PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;
A67: [(len (Gauge (C,n))),j] in Indices (Gauge (C,n)) by A5, A12, A20, MATRIX_1:36;
 (L~ go) /\ (L~ do) c= {(go /. 1)}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ go) /\ (L~ do) or x in {(go /. 1)} )
assume A68: x in (L~ go) /\ (L~ do) ; ::_thesis: x in {(go /. 1)}
then A69: x in L~ go by XBOOLE_0:def_4;
A70: x in L~ do by A68, XBOOLE_0:def_4;
then  x in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A46, A53, A69, XBOOLE_0:def_4;
then  x in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16;
then A71: ( x = W-min (L~ (Cage (C,n))) or x = E-max (L~ (Cage (C,n))) ) by TARSKI:def_2;
now__::_thesis:_not_x_=_E-max_(L~_(Cage_(C,n)))
assume  x = E-max (L~ (Cage (C,n))) ; ::_thesis: contradiction
then A72: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i,j) by A9, A66, A70, JORDAN1E:7;
 ((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 = E-bound (L~ (Cage (C,n))) by A5, A12, A16, JORDAN1A:71;
then  (E-max (L~ (Cage (C,n)))) `1 <> E-bound (L~ (Cage (C,n))) by A2, A14, A67, A72, JORDAN1G:7;
hence  contradiction by EUCLID:52; ::_thesis: verum
end;
hence  x in {(go /. 1)} by A59, A71, TARSKI:def_1; ::_thesis: verum
end;
then A73: (L~ go) /\ (L~ do) = {(go /. 1)} by A63, XBOOLE_0:def_10;
set W2 = go /. 2;
A74: 2 in dom go by A33, FINSEQ_3:25;
A75: now__::_thesis:_not_((Gauge_(C,n))_*_(i,k))_`1_=_W-bound_(L~_(Cage_(C,n)))
assume  ((Gauge (C,n)) * (i,k)) `1 = W-bound (L~ (Cage (C,n))) ; ::_thesis: contradiction
then  ((Gauge (C,n)) * (1,k)) `1 = ((Gauge (C,n)) * (i,k)) `1 by A3, A13, A16, JORDAN1A:73;
hence  contradiction by A1, A15, A21, JORDAN1G:7; ::_thesis: verum
end;
go = mid ((Upper_Seq (C,n)),1,(((Gauge (C,n)) * (i,k)) .. (Upper_Seq (C,n)))) by A34, JORDAN1G:49
.= (Upper_Seq (C,n)) | (((Gauge (C,n)) * (i,k)) .. (Upper_Seq (C,n))) by A34, FINSEQ_4:21, FINSEQ_6:116 ;
then A76: go /. 2 = (Upper_Seq (C,n)) /. 2 by A74, FINSEQ_4:70;
A77: W-min (L~ (Cage (C,n))) in rng go by A59, FINSEQ_6:42;
set pion = <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*>;
A78: now__::_thesis:_for_n_being_Element_of_NAT_st_n_in_dom_<*((Gauge_(C,n))_*_(i,k)),((Gauge_(C,n))_*_(i,j))*>_holds_
ex_i,_j_being_Element_of_NAT_st_
(_[i,j]_in_Indices_(Gauge_(C,n))_&_<*((Gauge_(C,n))_*_(i,k)),((Gauge_(C,n))_*_(i,j))*>_/._n_=_(Gauge_(C,n))_*_(i,j)_)
let n be Element of NAT ; ::_thesis: ( n in dom <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> implies ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. n = (Gauge (C,n)) * (i,j) ) )
assume  n in dom <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. n = (Gauge (C,n)) * (i,j) )
then  n in Seg 2 by FINSEQ_1:89;
then  ( n = 1 or n = 2 ) by FINSEQ_1:2, TARSKI:def_2;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. n = (Gauge (C,n)) * (i,j) ) by A14, A15, FINSEQ_4:17; ::_thesis: verum
end;
A79: (Gauge (C,n)) * (i,k) <> (Gauge (C,n)) * (i,j) by A11, A14, A15, GOBOARD1:5;
A80: ((Gauge (C,n)) * (i,k)) `1 = ((Gauge (C,n)) * (i,1)) `1 by A1, A2, A3, A13, GOBOARD5:2
.= ((Gauge (C,n)) * (i,j)) `1 by A1, A2, A5, A12, GOBOARD5:2 ;
then  LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) is vertical by SPPOL_1:16;
then  <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> is being_S-Seq by A79, JORDAN1B:7;
then consider pion1 being FinSequence of (TOP-REAL 2) such that
A81: pion1 is_sequence_on Gauge (C,n) and
A82: pion1 is being_S-Seq and
A83: L~ <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> = L~ pion1 and
A84: <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. 1 = pion1 /. 1 and
A85: <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. (len <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*>) = pion1 /. (len pion1) and
A86: len <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> <= len pion1 by A78, GOBOARD3:2;
reconsider pion1 = pion1 as being_S-Seq FinSequence of (TOP-REAL 2) by A82;
set godo = (go ^' pion1) ^' do;
A87: 1 + 1 <= len (Cage (C,n)) by GOBOARD7:34, XXREAL_0:2;
A88: 1 + 1 <= len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by GOBOARD7:34, XXREAL_0:2;
 len (go ^' pion1) >= len go by TOPREAL8:7;
then A89: len (go ^' pion1) >= 1 + 1 by A33, XXREAL_0:2;
then A90: len (go ^' pion1) > 1 + 0 by NAT_1:13;
A91: len ((go ^' pion1) ^' do) >= len (go ^' pion1) by TOPREAL8:7;
then A92: 1 + 1 <= len ((go ^' pion1) ^' do) by A89, XXREAL_0:2;
A93: Upper_Seq (C,n) is_sequence_on Gauge (C,n) by JORDAN1G:4;
A94: go /. (len go) = pion1 /. 1 by A41, A84, FINSEQ_4:17;
then A95: go ^' pion1 is_sequence_on Gauge (C,n) by A35, A81, TOPREAL8:12;
A96: (go ^' pion1) /. (len (go ^' pion1)) = <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. (len <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*>) by A85, GRAPH_2:54
.= <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. 2 by FINSEQ_1:44
.= do /. 1 by A42, FINSEQ_4:17 ;
then A97: (go ^' pion1) ^' do is_sequence_on Gauge (C,n) by A38, A95, TOPREAL8:12;
 LSeg (pion1,1) c= L~ <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> by A83, TOPREAL3:19;
then  LSeg (pion1,1) c= LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) by SPPOL_2:21;
then A98: (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) c= {((Gauge (C,n)) * (i,k))} by A44, A51, XBOOLE_1:27;
A99: len pion1 >= 1 + 1 by A86, FINSEQ_1:44;
 {((Gauge (C,n)) * (i,k))} c= (LSeg (go,m)) /\ (LSeg (pion1,1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i,k))} or x in (LSeg (go,m)) /\ (LSeg (pion1,1)) )
assume  x in {((Gauge (C,n)) * (i,k))} ; ::_thesis: x in (LSeg (go,m)) /\ (LSeg (pion1,1))
then A100: x = (Gauge (C,n)) * (i,k) by TARSKI:def_1;
A101: (Gauge (C,n)) * (i,k) in LSeg (go,m) by A48, RLTOPSP1:68;
 (Gauge (C,n)) * (i,k) in LSeg (pion1,1) by A41, A94, A99, TOPREAL1:21;
hence  x in (LSeg (go,m)) /\ (LSeg (pion1,1)) by A100, A101, XBOOLE_0:def_4; ::_thesis: verum
end;
then  (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) = {(go /. (len go))} by A41, A44, A98, XBOOLE_0:def_10;
then A102: go ^' pion1 is unfolded by A94, TOPREAL8:34;
 len pion1 >= 2 + 0 by A86, FINSEQ_1:44;
then A103: (len pion1) - 2 >= 0 by XREAL_1:19;
 ((len (go ^' pion1)) + 1) - 1 = ((len go) + (len pion1)) - 1 by GRAPH_2:13;
then (len (go ^' pion1)) - 1 = (len go) + ((len pion1) - 2)
.= (len go) + ((len pion1) -' 2) by A103, XREAL_0:def_2 ;
then A104: (len (go ^' pion1)) -' 1 = (len go) + ((len pion1) -' 2) by XREAL_0:def_2;
A105: (len pion1) - 1 >= 1 by A99, XREAL_1:19;
then A106: (len pion1) -' 1 = (len pion1) - 1 by XREAL_0:def_2;
A107: ((len pion1) -' 2) + 1 = ((len pion1) - 2) + 1 by A103, XREAL_0:def_2
.= (len pion1) -' 1 by A105, XREAL_0:def_2 ;
 ((len pion1) - 1) + 1 <= len pion1 ;
then A108: (len pion1) -' 1 < len pion1 by A106, NAT_1:13;
 LSeg (pion1,((len pion1) -' 1)) c= L~ <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> by A83, TOPREAL3:19;
then  LSeg (pion1,((len pion1) -' 1)) c= LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) by SPPOL_2:21;
then A109: (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) c= {((Gauge (C,n)) * (i,j))} by A58, XBOOLE_1:27;
 {((Gauge (C,n)) * (i,j))} c= (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i,j))} or x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) )
assume  x in {((Gauge (C,n)) * (i,j))} ; ::_thesis: x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1))
then A110: x = (Gauge (C,n)) * (i,j) by TARSKI:def_1;
A111: (Gauge (C,n)) * (i,j) in LSeg (do,1) by A55, RLTOPSP1:68;
pion1 /. (((len pion1) -' 1) + 1) = <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. 2 by A85, A106, FINSEQ_1:44
.= (Gauge (C,n)) * (i,j) by FINSEQ_4:17 ;
then  (Gauge (C,n)) * (i,j) in LSeg (pion1,((len pion1) -' 1)) by A105, A106, TOPREAL1:21;
hence  x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) by A110, A111, XBOOLE_0:def_4; ::_thesis: verum
end;
then  (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) = {((Gauge (C,n)) * (i,j))} by A109, XBOOLE_0:def_10;
then A112: (LSeg ((go ^' pion1),((len go) + ((len pion1) -' 2)))) /\ (LSeg (do,1)) = {((go ^' pion1) /. (len (go ^' pion1)))} by A42, A94, A96, A107, A108, TOPREAL8:31;
A113: not go ^' pion1 is trivial by A89, NAT_D:60;
A114: rng pion1 c= L~ pion1 by A99, SPPOL_2:18;
A115: {(pion1 /. 1)} c= (L~ go) /\ (L~ pion1)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(pion1 /. 1)} or x in (L~ go) /\ (L~ pion1) )
assume  x in {(pion1 /. 1)} ; ::_thesis: x in (L~ go) /\ (L~ pion1)
then A116: x = pion1 /. 1 by TARSKI:def_1;
then A117: x in rng go by A94, REVROT_1:3;
 x in rng pion1 by A116, FINSEQ_6:42;
hence  x in (L~ go) /\ (L~ pion1) by A61, A114, A117, XBOOLE_0:def_4; ::_thesis: verum
end;
 (L~ go) /\ (L~ pion1) c= {(pion1 /. 1)}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ go) /\ (L~ pion1) or x in {(pion1 /. 1)} )
assume A118: x in (L~ go) /\ (L~ pion1) ; ::_thesis: x in {(pion1 /. 1)}
then A119: x in L~ go by XBOOLE_0:def_4;
 x in L~ pion1 by A118, XBOOLE_0:def_4;
then  x in (L~ pion1) /\ (L~ (Upper_Seq (C,n))) by A46, A119, XBOOLE_0:def_4;
hence  x in {(pion1 /. 1)} by A6, A41, A83, A94, SPPOL_2:21; ::_thesis: verum
end;
then A120: (L~ go) /\ (L~ pion1) = {(pion1 /. 1)} by A115, XBOOLE_0:def_10;
then A121: go ^' pion1 is s.n.c. by A94, JORDAN1J:54;
 (rng go) /\ (rng pion1) c= {(pion1 /. 1)} by A61, A114, A120, XBOOLE_1:27;
then A122: go ^' pion1 is one-to-one by JORDAN1J:55;
A123: <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. (len <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*>) = <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> /. 2 by FINSEQ_1:44
.= do /. 1 by A42, FINSEQ_4:17 ;
A124: {(pion1 /. (len pion1))} c= (L~ do) /\ (L~ pion1)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(pion1 /. (len pion1))} or x in (L~ do) /\ (L~ pion1) )
assume  x in {(pion1 /. (len pion1))} ; ::_thesis: x in (L~ do) /\ (L~ pion1)
then A125: x = pion1 /. (len pion1) by TARSKI:def_1;
then A126: x in rng do by A85, A123, FINSEQ_6:42;
 x in rng pion1 by A125, REVROT_1:3;
hence  x in (L~ do) /\ (L~ pion1) by A62, A114, A126, XBOOLE_0:def_4; ::_thesis: verum
end;
 (L~ do) /\ (L~ pion1) c= {(pion1 /. (len pion1))}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ do) /\ (L~ pion1) or x in {(pion1 /. (len pion1))} )
assume A127: x in (L~ do) /\ (L~ pion1) ; ::_thesis: x in {(pion1 /. (len pion1))}
then A128: x in L~ do by XBOOLE_0:def_4;
 x in L~ pion1 by A127, XBOOLE_0:def_4;
then  x in (L~ pion1) /\ (L~ (Lower_Seq (C,n))) by A53, A128, XBOOLE_0:def_4;
hence  x in {(pion1 /. (len pion1))} by A7, A42, A83, A85, A123, SPPOL_2:21; ::_thesis: verum
end;
then A129: (L~ do) /\ (L~ pion1) = {(pion1 /. (len pion1))} by A124, XBOOLE_0:def_10;
A130: (L~ (go ^' pion1)) /\ (L~ do) = ((L~ go) \/ (L~ pion1)) /\ (L~ do) by A94, TOPREAL8:35
.= {(go /. 1)} \/ {(do /. 1)} by A73, A85, A123, A129, XBOOLE_1:23
.= {((go ^' pion1) /. 1)} \/ {(do /. 1)} by GRAPH_2:53
.= {((go ^' pion1) /. 1),(do /. 1)} by ENUMSET1:1 ;
 do /. (len do) = (go ^' pion1) /. 1 by A60, GRAPH_2:53;
then reconsider godo = (go ^' pion1) ^' do as non constant standard special_circular_sequence by A92, A96, A97, A102, A104, A112, A113, A121, A122, A130, JORDAN8:4, JORDAN8:5, TOPREAL8:11, TOPREAL8:33, TOPREAL8:34;
A131: Lower_Arc C is_an_arc_of E-max C, W-min C by JORDAN6:def_9;
then A132: Lower_Arc C is connected by JORDAN6:10;
A133: W-min C in Lower_Arc C by A131, TOPREAL1:1;
A134: E-max C in Lower_Arc C by A131, TOPREAL1:1;
set ff = Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))));
 W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
then A135: (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 = W-min (L~ (Cage (C,n))) by FINSEQ_6:92;
A136: L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33;
then  (W-max (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A135, SPRECT_5:22;
then  (N-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A135, A136, SPRECT_5:23, XXREAL_0:2;
then  (N-max (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A135, A136, SPRECT_5:24, XXREAL_0:2;
then A137: (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A135, A136, SPRECT_5:25, XXREAL_0:2;
A138: now__::_thesis:_not_((Gauge_(C,n))_*_(i,k))_.._(Upper_Seq_(C,n))_<=_1
assume A139: ((Gauge (C,n)) * (i,k)) .. (Upper_Seq (C,n)) <= 1 ; ::_thesis: contradiction
 ((Gauge (C,n)) * (i,k)) .. (Upper_Seq (C,n)) >= 1 by A34, FINSEQ_4:21;
then  ((Gauge (C,n)) * (i,k)) .. (Upper_Seq (C,n)) = 1 by A139, XXREAL_0:1;
then  (Gauge (C,n)) * (i,k) = (Upper_Seq (C,n)) /. 1 by A34, FINSEQ_5:38;
hence  contradiction by A18, A22, JORDAN1F:5; ::_thesis: verum
end;
A140: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def_1;
then A141: Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))) is_sequence_on Gauge (C,n) by REVROT_1:34;
A142: (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) c= RightComp godo by A92, A97, JORDAN9:27;
A143: L~ godo = (L~ (go ^' pion1)) \/ (L~ do) by A96, TOPREAL8:35
.= ((L~ go) \/ (L~ pion1)) \/ (L~ do) by A94, TOPREAL8:35 ;
A144: L~ (Cage (C,n)) = (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) by JORDAN1E:13;
then A145: L~ (Upper_Seq (C,n)) c= L~ (Cage (C,n)) by XBOOLE_1:7;
A146: L~ (Lower_Seq (C,n)) c= L~ (Cage (C,n)) by A144, XBOOLE_1:7;
A147: L~ go c= L~ (Cage (C,n)) by A46, A145, XBOOLE_1:1;
A148: L~ do c= L~ (Cage (C,n)) by A53, A146, XBOOLE_1:1;
A149: W-min C in C by SPRECT_1:13;
A150: L~ <*((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))*> = LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j))) by SPPOL_2:21;
A151: now__::_thesis:_not_W-min_C_in_L~_godo
assume  W-min C in L~ godo ; ::_thesis: contradiction
then A152: ( W-min C in (L~ go) \/ (L~ pion1) or W-min C in L~ do ) by A143, XBOOLE_0:def_3;
percases ( W-min C in L~ go or W-min C in L~ pion1 or W-min C in L~ do ) by A152, XBOOLE_0:def_3;
suppose W-min C in L~ go ; ::_thesis: contradiction
then  C meets L~ (Cage (C,n)) by A147, A149, XBOOLE_0:3;
hence  contradiction by JORDAN10:5; ::_thesis: verum
end;
suppose W-min C in L~ pion1 ; ::_thesis: contradiction
hence  contradiction by A8, A83, A133, A150, XBOOLE_0:3; ::_thesis: verum
end;
suppose W-min C in L~ do ; ::_thesis: contradiction
then  C meets L~ (Cage (C,n)) by A148, A149, XBOOLE_0:3;
hence  contradiction by JORDAN10:5; ::_thesis: verum
end;
end;
end;
right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1) = right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(GoB (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) by A88, JORDAN1H:23
.= right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(GoB (Cage (C,n)))) by REVROT_1:28
.= right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(Gauge (C,n))) by JORDAN1H:44
.= right_cell (((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n))))),1,(Gauge (C,n))) by A137, A141, JORDAN1J:53
.= right_cell ((Upper_Seq (C,n)),1,(Gauge (C,n))) by JORDAN1E:def_1
.= right_cell ((R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,k)))),1,(Gauge (C,n))) by A34, A93, A138, JORDAN1J:52
.= right_cell ((go ^' pion1),1,(Gauge (C,n))) by A39, A95, JORDAN1J:51
.= right_cell (godo,1,(Gauge (C,n))) by A90, A97, JORDAN1J:51 ;
then  W-min C in right_cell (godo,1,(Gauge (C,n))) by JORDAN1I:6;
then A153: W-min C in (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) by A151, XBOOLE_0:def_5;
A154: godo /. 1 = (go ^' pion1) /. 1 by GRAPH_2:53
.= W-min (L~ (Cage (C,n))) by A59, GRAPH_2:53 ;
A155: len (Upper_Seq (C,n)) >= 2 by A17, XXREAL_0:2;
A156: godo /. 2 = (go ^' pion1) /. 2 by A89, GRAPH_2:57
.= (Upper_Seq (C,n)) /. 2 by A33, A76, GRAPH_2:57
.= ((Upper_Seq (C,n)) ^' (Lower_Seq (C,n))) /. 2 by A155, GRAPH_2:57
.= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 2 by JORDAN1E:11 ;
A157: (L~ go) \/ (L~ do) is compact by COMPTS_1:10;
 W-min (L~ (Cage (C,n))) in (L~ go) \/ (L~ do) by A61, A77, XBOOLE_0:def_3;
then A158: W-min ((L~ go) \/ (L~ do)) = W-min (L~ (Cage (C,n))) by A147, A148, A157, JORDAN1J:21, XBOOLE_1:8;
A159: (W-min ((L~ go) \/ (L~ do))) `1 = W-bound ((L~ go) \/ (L~ do)) by EUCLID:52;
A160: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52;
 W-bound (LSeg (((Gauge (C,n)) * (i,k)),((Gauge (C,n)) * (i,j)))) = ((Gauge (C,n)) * (i,k)) `1 by A80, SPRECT_1:54;
then A161: W-bound (L~ pion1) = ((Gauge (C,n)) * (i,k)) `1 by A83, SPPOL_2:21;
 ((Gauge (C,n)) * (i,k)) `1 >= W-bound (L~ (Cage (C,n))) by A10, A145, PSCOMP_1:24;
then  ((Gauge (C,n)) * (i,k)) `1 > W-bound (L~ (Cage (C,n))) by A75, XXREAL_0:1;
then  W-min (((L~ go) \/ (L~ do)) \/ (L~ pion1)) = W-min ((L~ go) \/ (L~ do)) by A157, A158, A159, A160, A161, JORDAN1J:33;
then A162: W-min (L~ godo) = W-min (L~ (Cage (C,n))) by A143, A158, XBOOLE_1:4;
A163: rng godo c= L~ godo by A89, A91, SPPOL_2:18, XXREAL_0:2;
 2 in dom godo by A92, FINSEQ_3:25;
then A164: godo /. 2 in rng godo by PARTFUN2:2;
 godo /. 2 in W-most (L~ (Cage (C,n))) by A156, JORDAN1I:25;
then (godo /. 2) `1 = (W-min (L~ godo)) `1 by A162, PSCOMP_1:31
.= W-bound (L~ godo) by EUCLID:52 ;
then  godo /. 2 in W-most (L~ godo) by A163, A164, SPRECT_2:12;
then  (Rotate (godo,(W-min (L~ godo)))) /. 2 in W-most (L~ godo) by A154, A162, FINSEQ_6:89;
then reconsider godo = godo as non constant standard clockwise_oriented special_circular_sequence by JORDAN1I:25;
 len (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by FINSEQ_5:6;
then A165: (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) = (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) by PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:7 ;
A166: east_halfline (E-max C) misses L~ go
proof
assume  east_halfline (E-max C) meets L~ go ; ::_thesis: contradiction
then consider p being set  such that
A167: p in east_halfline (E-max C) and
A168: p in L~ go by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A167;
 p in L~ (Upper_Seq (C,n)) by A46, A168;
then  p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A145, A167, XBOOLE_0:def_4;
then A169: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;
then A170: p = E-max (L~ (Cage (C,n))) by A46, A168, JORDAN1J:46;
then  E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i,k) by A10, A165, A168, JORDAN1J:43;
then  ((Gauge (C,n)) * (i,k)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A3, A13, A16, A169, A170, JORDAN1A:71;
hence  contradiction by A2, A15, A30, JORDAN1G:7; ::_thesis: verum
end;
now__::_thesis:_not_east_halfline_(E-max_C)_meets_L~_godo
assume  east_halfline (E-max C) meets L~ godo ; ::_thesis: contradiction
then A171: ( east_halfline (E-max C) meets (L~ go) \/ (L~ pion1) or east_halfline (E-max C) meets L~ do ) by A143, XBOOLE_1:70;
percases ( east_halfline (E-max C) meets L~ go or east_halfline (E-max C) meets L~ pion1 or east_halfline (E-max C) meets L~ do ) by A171, XBOOLE_1:70;
suppose east_halfline (E-max C) meets L~ go ; ::_thesis: contradiction
hence  contradiction by A166; ::_thesis: verum
end;
suppose east_halfline (E-max C) meets L~ pion1 ; ::_thesis: contradiction
then consider p being set  such that
A172: p in east_halfline (E-max C) and
A173: p in L~ pion1 by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A172;
A174: p `1 = ((Gauge (C,n)) * (i,k)) `1 by A80, A83, A150, A173, GOBOARD7:5;
 i + 1 <= len (Gauge (C,n)) by A2, NAT_1:13;
then  (i + 1) - 1 <= (len (Gauge (C,n))) - 1 by XREAL_1:9;
then A175: i <= (len (Gauge (C,n))) -' 1 by XREAL_0:def_2;
 (len (Gauge (C,n))) -' 1 <= len (Gauge (C,n)) by NAT_D:35;
then  p `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1 by A1, A3, A13, A16, A20, A174, A175, JORDAN1A:18;
then  p `1 <= E-bound C by A20, JORDAN8:12;
then A176: p `1 <= (E-max C) `1 by EUCLID:52;
 p `1 >= (E-max C) `1 by A172, TOPREAL1:def_11;
then A177: p `1 = (E-max C) `1 by A176, XXREAL_0:1;
 p `2 = (E-max C) `2 by A172, TOPREAL1:def_11;
then  p = E-max C by A177, TOPREAL3:6;
hence  contradiction by A8, A83, A134, A150, A173, XBOOLE_0:3; ::_thesis: verum
end;
suppose east_halfline (E-max C) meets L~ do ; ::_thesis: contradiction
then consider p being set  such that
A178: p in east_halfline (E-max C) and
A179: p in L~ do by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A178;
 p in L~ (Lower_Seq (C,n)) by A53, A179;
then  p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A146, A178, XBOOLE_0:def_4;
then A180: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;
A181: (E-max C) `2 = p `2 by A178, TOPREAL1:def_11;
set RC = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))));
A182: E-max C in right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by JORDAN1I:7;
A183: 1 + 1 <= len (Lower_Seq (C,n)) by A23, XXREAL_0:2;
 Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) by JORDAN1G:18;
then A184: LSeg ((Lower_Seq (C,n)),1) = LSeg ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by A183, SPPOL_2:9;
A185: L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33;
A186: len (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = len (Cage (C,n)) by REVROT_1:14;
A187: GoB (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = GoB (Cage (C,n)) by REVROT_1:28
.= Gauge (C,n) by JORDAN1H:44 ;
A188: E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
A189: Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))) is_sequence_on Gauge (C,n) by A140, REVROT_1:34;
A190: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = E-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))))) by A185, A188, FINSEQ_6:92;
consider ii, jj being Element of NAT  such that
A191: [ii,(jj + 1)] in Indices (Gauge (C,n)) and
A192: [ii,jj] in Indices (Gauge (C,n)) and
A193: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = (Gauge (C,n)) * (ii,(jj + 1)) and
A194: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1) = (Gauge (C,n)) * (ii,jj) by A87, A185, A186, A188, A189, FINSEQ_6:92, JORDAN1I:23;
consider jj2 being Element of NAT  such that
A195: 1 <= jj2 and
A196: jj2 <= width (Gauge (C,n)) and
A197: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * ((len (Gauge (C,n))),jj2) by JORDAN1D:25;
A198: len (Gauge (C,n)) >= 4 by JORDAN8:10;
then  len (Gauge (C,n)) >= 1 by XXREAL_0:2;
then  [(len (Gauge (C,n))),jj2] in Indices (Gauge (C,n)) by A195, A196, MATRIX_1:36;
then A199: ii = len (Gauge (C,n)) by A185, A190, A191, A193, A197, GOBOARD1:5;
A200: 1 <= ii by A191, MATRIX_1:38;
A201: ii <= len (Gauge (C,n)) by A191, MATRIX_1:38;
A202: 1 <= jj + 1 by A191, MATRIX_1:38;
A203: jj + 1 <= width (Gauge (C,n)) by A191, MATRIX_1:38;
A204: 1 <= ii by A192, MATRIX_1:38;
A205: ii <= len (Gauge (C,n)) by A192, MATRIX_1:38;
A206: 1 <= jj by A192, MATRIX_1:38;
A207: jj <= width (Gauge (C,n)) by A192, MATRIX_1:38;
A208: ii + 1 <> ii ;
 (jj + 1) + 1 <> jj ;
then A209: right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) = cell ((Gauge (C,n)),(ii -' 1),jj) by A87, A186, A187, A191, A192, A193, A194, A208, GOBOARD5:def_6;
A210: (ii -' 1) + 1 = ii by A200, XREAL_1:235;
 ii - 1 >= 4 - 1 by A198, A199, XREAL_1:9;
then A211: ii - 1 >= 1 by XXREAL_0:2;
then A212: 1 <= ii -' 1 by XREAL_0:def_2;
A213: ((Gauge (C,n)) * ((ii -' 1),jj)) `2 <= p `2 by A181, A182, A201, A203, A206, A209, A210, A211, JORDAN9:17;
A214: p `2 <= ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 by A181, A182, A201, A203, A206, A209, A210, A211, JORDAN9:17;
A215: ii -' 1 < len (Gauge (C,n)) by A201, A210, NAT_1:13;
then A216: ((Gauge (C,n)) * ((ii -' 1),jj)) `2 = ((Gauge (C,n)) * (1,jj)) `2 by A206, A207, A212, GOBOARD5:1
.= ((Gauge (C,n)) * (ii,jj)) `2 by A204, A205, A206, A207, GOBOARD5:1 ;
A217: ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 = ((Gauge (C,n)) * (1,(jj + 1))) `2 by A202, A203, A212, A215, GOBOARD5:1
.= ((Gauge (C,n)) * (ii,(jj + 1))) `2 by A200, A201, A202, A203, GOBOARD5:1 ;
A218: ((Gauge (C,n)) * ((len (Gauge (C,n))),jj)) `1 = E-bound (L~ (Cage (C,n))) by A16, A206, A207, JORDAN1A:71;
 E-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * ((len (Gauge (C,n))),(jj + 1))) `1 by A16, A202, A203, JORDAN1A:71;
then  p in LSeg (((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1),((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1))) by A180, A193, A194, A199, A213, A214, A216, A217, A218, GOBOARD7:7;
then A219: p in LSeg ((Lower_Seq (C,n)),1) by A87, A184, A186, TOPREAL1:def_3;
A220: p in LSeg (do,(Index (p,do))) by A179, JORDAN3:9;
A221: do = mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n)))) by A37, JORDAN1J:37;
A222: 1 <= ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) by A37, FINSEQ_4:21;
A223: ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) <= len (Lower_Seq (C,n)) by A37, FINSEQ_4:21;
 ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) <> len (Lower_Seq (C,n)) by A29, A37, FINSEQ_4:19;
then A224: ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) < len (Lower_Seq (C,n)) by A223, XXREAL_0:1;
A225: 1 <= Index (p,do) by A179, JORDAN3:8;
A226: Index (p,do) < len do by A179, JORDAN3:8;
A227: (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1 = ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) by A32, A37, JORDAN1J:56;
consider t being Nat such that
A228: t in dom (Lower_Seq (C,n)) and
A229: (Lower_Seq (C,n)) . t = (Gauge (C,n)) * (i,j) by A37, FINSEQ_2:10;
A230: 1 <= t by A228, FINSEQ_3:25;
A231: t <= len (Lower_Seq (C,n)) by A228, FINSEQ_3:25;
 1 < t by A32, A229, A230, XXREAL_0:1;
then  (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) + 1 = t by A229, A231, JORDAN3:12;
then A232: len (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i,j)))) = (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) by A9, A229, JORDAN3:26;
set tt = ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1;
A233: 1 <= Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n))) by A9, JORDAN3:8;
 0 + (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) < len (Lower_Seq (C,n)) by A9, JORDAN3:8;
then A234: (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) > 0 by XREAL_1:20;
 Index (p,do) < (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) by A226, A232, XREAL_0:def_2;
then  (Index (p,do)) + 1 <= (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n)))) by NAT_1:13;
then  Index (p,do) <= ((len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n))))) - 1 by XREAL_1:19;
then  Index (p,do) <= ((len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i,j)),(Lower_Seq (C,n))))) - 1 by A234, XREAL_0:def_2;
then  Index (p,do) <= (len (Lower_Seq (C,n))) - (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))) by A227;
then  Index (p,do) <= (len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))) by XREAL_0:def_2;
then  Index (p,do) < ((len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) + 1 by NAT_1:13;
then A235: LSeg ((mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n))))),(Index (p,do))) = LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1)) by A222, A224, A225, JORDAN4:19;
A236: 1 + 1 <= ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) by A227, A233, XREAL_1:7;
then  (Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))) >= (1 + 1) + 1 by A225, XREAL_1:7;
then  ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) - 1 >= ((1 + 1) + 1) - 1 by XREAL_1:9;
then A237: ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1 >= 1 + 1 by XREAL_0:def_2;
A238: 2 in dom (Lower_Seq (C,n)) by A183, FINSEQ_3:25;
now__::_thesis:_contradiction
percases ( ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1 > 1 + 1 or ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ) by A237, XXREAL_0:1;
suppose ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1 > 1 + 1 ; ::_thesis: contradiction
then  LSeg ((Lower_Seq (C,n)),1) misses LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1)) by TOPREAL1:def_7;
hence  contradiction by A219, A220, A221, A235, XBOOLE_0:3; ::_thesis: verum
end;
supposeA239: ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ; ::_thesis: contradiction
then  (LSeg ((Lower_Seq (C,n)),1)) /\ (LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) -' 1))) = {((Lower_Seq (C,n)) /. 2)} by A23, TOPREAL1:def_6;
then  p in {((Lower_Seq (C,n)) /. 2)} by A219, A220, A221, A235, XBOOLE_0:def_4;
then A240: p = (Lower_Seq (C,n)) /. 2 by TARSKI:def_1;
then A241: p .. (Lower_Seq (C,n)) = 2 by A238, FINSEQ_5:41;
 1 + 1 = ((Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)))) - 1 by A239, XREAL_0:def_2;
then  (1 + 1) + 1 = (Index (p,do)) + (((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n))) ;
then A242: ((Gauge (C,n)) * (i,j)) .. (Lower_Seq (C,n)) = 2 by A225, A236, JORDAN1E:6;
 p in rng (Lower_Seq (C,n)) by A238, A240, PARTFUN2:2;
then  p = (Gauge (C,n)) * (i,j) by A37, A241, A242, FINSEQ_5:9;
then  ((Gauge (C,n)) * (i,j)) `1 = E-bound (L~ (Cage (C,n))) by A240, JORDAN1G:32;
then  ((Gauge (C,n)) * (i,j)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 by A5, A12, A16, JORDAN1A:71;
hence  contradiction by A2, A14, A67, JORDAN1G:7; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
end;
end;
then  east_halfline (E-max C) c= (L~ godo) ` by SUBSET_1:23;
then consider W being Subset of (TOP-REAL 2) such that
A243: W is_a_component_of (L~ godo) ` and
A244: east_halfline (E-max C) c= W by GOBOARD9:3;
 not W is bounded by A244, JORDAN2C:121, RLTOPSP1:42;
then  W is_outside_component_of L~ godo by A243, JORDAN2C:def_3;
then  W c= UBD (L~ godo) by JORDAN2C:23;
then A245: east_halfline (E-max C) c= UBD (L~ godo) by A244, XBOOLE_1:1;
 E-max C in east_halfline (E-max C) by TOPREAL1:38;
then  E-max C in UBD (L~ godo) by A245;
then  E-max C in LeftComp godo by GOBRD14:36;
then  Lower_Arc C meets L~ godo by A132, A133, A134, A142, A153, JORDAN1J:36;
then A246: ( Lower_Arc C meets (L~ go) \/ (L~ pion1) or Lower_Arc C meets L~ do ) by A143, XBOOLE_1:70;
A247: Lower_Arc C c= C by JORDAN6:61;
percases ( Lower_Arc C meets L~ go or Lower_Arc C meets L~ pion1 or Lower_Arc C meets L~ do ) by A246, XBOOLE_1:70;
suppose Lower_Arc C meets L~ go ; ::_thesis: contradiction
then  Lower_Arc C meets L~ (Cage (C,n)) by A46, A145, XBOOLE_1:1, XBOOLE_1:63;
hence  contradiction by A247, JORDAN10:5, XBOOLE_1:63; ::_thesis: verum
end;
suppose Lower_Arc C meets L~ pion1 ; ::_thesis: contradiction
hence  contradiction by A8, A83, A150; ::_thesis: verum
end;
suppose Lower_Arc C meets L~ do ; ::_thesis: contradiction
then  Lower_Arc C meets L~ (Cage (C,n)) by A53, A146, XBOOLE_1:1, XBOOLE_1:63;
hence  contradiction by A247, JORDAN10:5, XBOOLE_1:63; ::_thesis: verum
end;
end;
end;

theorem :: JORDAN19:14
for n being Element of NAT
for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Lower_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Upper_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,j))} holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Lower_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Upper_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,j))} holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C
let C be Simple_closed_curve; ::_thesis: for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Lower_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Upper_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,j))} holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C
let i, j, k be Element of NAT ; ::_thesis: ( 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Lower_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Upper_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,j))} implies LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C )
assume that
A1: 1 < i and
A2: i < len (Gauge (C,n)) and
A3: 1 <= j and
A4: j <= k and
A5: k <= width (Gauge (C,n)) and
A6: n > 0 and
A7: (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Lower_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,k))} and
A8: (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Upper_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,j))} ; ::_thesis: LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C
A9: L~ (Lower_Seq (C,n)) = Lower_Arc (L~ (Cage (C,n))) by A6, JORDAN1G:56;
 L~ (Upper_Seq (C,n)) = Upper_Arc (L~ (Cage (C,n))) by A6, JORDAN1G:55;
hence  LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C by A1, A2, A3, A4, A5, A7, A8, A9, Th12; ::_thesis: verum
end;

theorem :: JORDAN19:15
for n being Element of NAT
for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Lower_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Upper_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,j))} holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Lower_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Upper_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,j))} holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C
let C be Simple_closed_curve; ::_thesis: for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Lower_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Upper_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,j))} holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C
let i, j, k be Element of NAT ; ::_thesis: ( 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Lower_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,k))} & (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Upper_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,j))} implies LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C )
assume that
A1: 1 < i and
A2: i < len (Gauge (C,n)) and
A3: 1 <= j and
A4: j <= k and
A5: k <= width (Gauge (C,n)) and
A6: n > 0 and
A7: (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Lower_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,k))} and
A8: (LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k)))) /\ (Upper_Arc (L~ (Cage (C,n)))) = {((Gauge (C,n)) * (i,j))} ; ::_thesis: LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C
A9: L~ (Lower_Seq (C,n)) = Lower_Arc (L~ (Cage (C,n))) by A6, JORDAN1G:56;
 L~ (Upper_Seq (C,n)) = Upper_Arc (L~ (Cage (C,n))) by A6, JORDAN1G:55;
hence  LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C by A1, A2, A3, A4, A5, A7, A8, A9, Th13; ::_thesis: verum
end;

theorem Th16: :: JORDAN19:16
for n being Element of NAT
for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Lower_Seq (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Upper_Seq (C,n)) holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Lower_Seq (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Upper_Seq (C,n)) holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C
let C be Simple_closed_curve; ::_thesis: for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Lower_Seq (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Upper_Seq (C,n)) holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C
let i, j, k be Element of NAT ; ::_thesis: ( 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Lower_Seq (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Upper_Seq (C,n)) implies LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C )
assume that
A1: 1 < i and
A2: i < len (Gauge (C,n)) and
A3: 1 <= j and
A4: j <= k and
A5: k <= width (Gauge (C,n)) and
A6: (Gauge (C,n)) * (i,k) in L~ (Lower_Seq (C,n)) and
A7: (Gauge (C,n)) * (i,j) in L~ (Upper_Seq (C,n)) ; ::_thesis: LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C
consider j1, k1 being Element of NAT  such that
A8: j <= j1 and
A9: j1 <= k1 and
A10: k1 <= k and
A11: (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,j1))} and
A12: (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))} by A1, A2, A3, A4, A5, A6, A7, JORDAN15:17;
A13: 1 <= j1 by A3, A8, XXREAL_0:2;
 k1 <= width (Gauge (C,n)) by A5, A10, XXREAL_0:2;
then  LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1))) meets Upper_Arc C by A1, A2, A9, A11, A12, A13, Th12;
hence  LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C by A1, A2, A3, A5, A8, A9, A10, JORDAN15:5, XBOOLE_1:63; ::_thesis: verum
end;

theorem Th17: :: JORDAN19:17
for n being Element of NAT
for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Lower_Seq (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Upper_Seq (C,n)) holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Lower_Seq (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Upper_Seq (C,n)) holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C
let C be Simple_closed_curve; ::_thesis: for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Lower_Seq (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Upper_Seq (C,n)) holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C
let i, j, k be Element of NAT ; ::_thesis: ( 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,k) in L~ (Lower_Seq (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Upper_Seq (C,n)) implies LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C )
assume that
A1: 1 < i and
A2: i < len (Gauge (C,n)) and
A3: 1 <= j and
A4: j <= k and
A5: k <= width (Gauge (C,n)) and
A6: (Gauge (C,n)) * (i,k) in L~ (Lower_Seq (C,n)) and
A7: (Gauge (C,n)) * (i,j) in L~ (Upper_Seq (C,n)) ; ::_thesis: LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C
consider j1, k1 being Element of NAT  such that
A8: j <= j1 and
A9: j1 <= k1 and
A10: k1 <= k and
A11: (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i,j1))} and
A12: (LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1)))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i,k1))} by A1, A2, A3, A4, A5, A6, A7, JORDAN15:17;
A13: 1 <= j1 by A3, A8, XXREAL_0:2;
 k1 <= width (Gauge (C,n)) by A5, A10, XXREAL_0:2;
then  LSeg (((Gauge (C,n)) * (i,j1)),((Gauge (C,n)) * (i,k1))) meets Lower_Arc C by A1, A2, A9, A11, A12, A13, Th13;
hence  LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C by A1, A2, A3, A5, A8, A9, A10, JORDAN15:5, XBOOLE_1:63; ::_thesis: verum
end;

theorem Th18: :: JORDAN19:18
for n being Element of NAT
for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (Gauge (C,n)) * (i,k) in Lower_Arc (L~ (Cage (C,n))) & (Gauge (C,n)) * (i,j) in Upper_Arc (L~ (Cage (C,n))) holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (Gauge (C,n)) * (i,k) in Lower_Arc (L~ (Cage (C,n))) & (Gauge (C,n)) * (i,j) in Upper_Arc (L~ (Cage (C,n))) holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C
let C be Simple_closed_curve; ::_thesis: for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (Gauge (C,n)) * (i,k) in Lower_Arc (L~ (Cage (C,n))) & (Gauge (C,n)) * (i,j) in Upper_Arc (L~ (Cage (C,n))) holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C
let i, j, k be Element of NAT ; ::_thesis: ( 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (Gauge (C,n)) * (i,k) in Lower_Arc (L~ (Cage (C,n))) & (Gauge (C,n)) * (i,j) in Upper_Arc (L~ (Cage (C,n))) implies LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C )
assume that
A1: 1 < i and
A2: i < len (Gauge (C,n)) and
A3: 1 <= j and
A4: j <= k and
A5: k <= width (Gauge (C,n)) and
A6: n > 0 and
A7: (Gauge (C,n)) * (i,k) in Lower_Arc (L~ (Cage (C,n))) and
A8: (Gauge (C,n)) * (i,j) in Upper_Arc (L~ (Cage (C,n))) ; ::_thesis: LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C
A9: L~ (Lower_Seq (C,n)) = Lower_Arc (L~ (Cage (C,n))) by A6, JORDAN1G:56;
 L~ (Upper_Seq (C,n)) = Upper_Arc (L~ (Cage (C,n))) by A6, JORDAN1G:55;
hence  LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Upper_Arc C by A1, A2, A3, A4, A5, A7, A8, A9, Th16; ::_thesis: verum
end;

theorem Th19: :: JORDAN19:19
for n being Element of NAT
for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (Gauge (C,n)) * (i,k) in Lower_Arc (L~ (Cage (C,n))) & (Gauge (C,n)) * (i,j) in Upper_Arc (L~ (Cage (C,n))) holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (Gauge (C,n)) * (i,k) in Lower_Arc (L~ (Cage (C,n))) & (Gauge (C,n)) * (i,j) in Upper_Arc (L~ (Cage (C,n))) holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C
let C be Simple_closed_curve; ::_thesis: for i, j, k being Element of NAT st 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (Gauge (C,n)) * (i,k) in Lower_Arc (L~ (Cage (C,n))) & (Gauge (C,n)) * (i,j) in Upper_Arc (L~ (Cage (C,n))) holds
LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C
let i, j, k be Element of NAT ; ::_thesis: ( 1 < i & i < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & n > 0 & (Gauge (C,n)) * (i,k) in Lower_Arc (L~ (Cage (C,n))) & (Gauge (C,n)) * (i,j) in Upper_Arc (L~ (Cage (C,n))) implies LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C )
assume that
A1: 1 < i and
A2: i < len (Gauge (C,n)) and
A3: 1 <= j and
A4: j <= k and
A5: k <= width (Gauge (C,n)) and
A6: n > 0 and
A7: (Gauge (C,n)) * (i,k) in Lower_Arc (L~ (Cage (C,n))) and
A8: (Gauge (C,n)) * (i,j) in Upper_Arc (L~ (Cage (C,n))) ; ::_thesis: LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C
A9: L~ (Lower_Seq (C,n)) = Lower_Arc (L~ (Cage (C,n))) by A6, JORDAN1G:56;
 L~ (Upper_Seq (C,n)) = Upper_Arc (L~ (Cage (C,n))) by A6, JORDAN1G:55;
hence  LSeg (((Gauge (C,n)) * (i,j)),((Gauge (C,n)) * (i,k))) meets Lower_Arc C by A1, A2, A3, A4, A5, A7, A8, A9, Th17; ::_thesis: verum
end;

theorem Th20: :: JORDAN19:20
for n being Element of NAT
for C being Simple_closed_curve
for i1, i2, j, k being Element of NAT st 1 < i1 & i1 <= i2 & i2 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} holds
(LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Upper_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i1, i2, j, k being Element of NAT st 1 < i1 & i1 <= i2 & i2 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} holds
(LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Upper_Arc C
let C be Simple_closed_curve; ::_thesis: for i1, i2, j, k being Element of NAT st 1 < i1 & i1 <= i2 & i2 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} holds
(LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Upper_Arc C
let i1, i2, j, k be Element of NAT ; ::_thesis: ( 1 < i1 & i1 <= i2 & i2 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} implies (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Upper_Arc C )
set G = Gauge (C,n);
set pio = LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)));
set poz = LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)));
set US = Upper_Seq (C,n);
set LS = Lower_Seq (C,n);
assume that
A1: 1 < i1 and
A2: i1 <= i2 and
A3: i2 < len (Gauge (C,n)) and
A4: 1 <= j and
A5: j <= k and
A6: k <= width (Gauge (C,n)) and
A7: ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} and
A8: ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} and
A9: (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) misses Upper_Arc C ; ::_thesis: contradiction
set UA = Upper_Arc C;
set Wmin = W-min (L~ (Cage (C,n)));
set Emax = E-max (L~ (Cage (C,n)));
set Wbo = W-bound (L~ (Cage (C,n)));
set Ebo = E-bound (L~ (Cage (C,n)));
set Gik = (Gauge (C,n)) * (i2,k);
set Gij = (Gauge (C,n)) * (i1,j);
set Gi1k = (Gauge (C,n)) * (i1,k);
A10: i1 < len (Gauge (C,n)) by A2, A3, XXREAL_0:2;
A11: 1 < i2 by A1, A2, XXREAL_0:2;
A12: L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> = (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) by TOPREAL3:16;
 (Gauge (C,n)) * (i2,k) in {((Gauge (C,n)) * (i2,k))} by TARSKI:def_1;
then A13: (Gauge (C,n)) * (i2,k) in L~ (Lower_Seq (C,n)) by A8, XBOOLE_0:def_4;
 (Gauge (C,n)) * (i1,j) in {((Gauge (C,n)) * (i1,j))} by TARSKI:def_1;
then A14: (Gauge (C,n)) * (i1,j) in L~ (Upper_Seq (C,n)) by A7, XBOOLE_0:def_4;
A15: j <= width (Gauge (C,n)) by A5, A6, XXREAL_0:2;
A16: 1 <= k by A4, A5, XXREAL_0:2;
A17: [i1,j] in Indices (Gauge (C,n)) by A1, A4, A10, A15, MATRIX_1:36;
A18: [i2,k] in Indices (Gauge (C,n)) by A3, A6, A11, A16, MATRIX_1:36;
A19: [i1,k] in Indices (Gauge (C,n)) by A1, A6, A10, A16, MATRIX_1:36;
set go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i1,j)));
set do = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i2,k)));
A20: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
A21: len (Upper_Seq (C,n)) >= 3 by JORDAN1E:15;
then  len (Upper_Seq (C,n)) >= 1 by XXREAL_0:2;
then  1 in dom (Upper_Seq (C,n)) by FINSEQ_3:25;
then A22: (Upper_Seq (C,n)) . 1 = (Upper_Seq (C,n)) /. 1 by PARTFUN1:def_6
.= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ;
A23: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * (1,k)) `1 by A6, A16, A20, JORDAN1A:73 ;
 len (Gauge (C,n)) >= 4 by JORDAN8:10;
then A24: len (Gauge (C,n)) >= 1 by XXREAL_0:2;
then A25: [1,k] in Indices (Gauge (C,n)) by A6, A16, MATRIX_1:36;
then A26: (Gauge (C,n)) * (i1,j) <> (Upper_Seq (C,n)) . 1 by A1, A17, A22, A23, JORDAN1G:7;
then reconsider go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i1,j))) as being_S-Seq FinSequence of (TOP-REAL 2) by A14, JORDAN3:35;
A27: [1,j] in Indices (Gauge (C,n)) by A4, A15, A24, MATRIX_1:36;
A28: len (Lower_Seq (C,n)) >= 1 + 2 by JORDAN1E:15;
then A29: len (Lower_Seq (C,n)) >= 1 by XXREAL_0:2;
then A30: 1 in dom (Lower_Seq (C,n)) by FINSEQ_3:25;
 len (Lower_Seq (C,n)) in dom (Lower_Seq (C,n)) by A29, FINSEQ_3:25;
then A31: (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by PARTFUN1:def_6
.= W-min (L~ (Cage (C,n))) by JORDAN1F:8 ;
(W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * (1,k)) `1 by A6, A16, A20, JORDAN1A:73 ;
then A32: (Gauge (C,n)) * (i2,k) <> (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) by A1, A2, A18, A25, A31, JORDAN1G:7;
then reconsider do = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i2,k))) as being_S-Seq FinSequence of (TOP-REAL 2) by A13, JORDAN3:34;
A33: [(len (Gauge (C,n))),k] in Indices (Gauge (C,n)) by A6, A16, A24, MATRIX_1:36;
A34: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by A30, PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;
(E-max (L~ (Cage (C,n)))) `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A6, A16, A20, JORDAN1A:71 ;
then A35: (Gauge (C,n)) * (i2,k) <> (Lower_Seq (C,n)) . 1 by A3, A18, A33, A34, JORDAN1G:7;
A36: len go >= 1 + 1 by TOPREAL1:def_8;
A37: (Gauge (C,n)) * (i1,j) in rng (Upper_Seq (C,n)) by A1, A4, A10, A14, A15, JORDAN1G:4, JORDAN1J:40;
then A38: go is_sequence_on Gauge (C,n) by JORDAN1G:4, JORDAN1J:38;
A39: len do >= 1 + 1 by TOPREAL1:def_8;
A40: (Gauge (C,n)) * (i2,k) in rng (Lower_Seq (C,n)) by A3, A6, A11, A13, A16, JORDAN1G:5, JORDAN1J:40;
then A41: do is_sequence_on Gauge (C,n) by JORDAN1G:5, JORDAN1J:39;
reconsider go = go as non constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A36, A38, JGRAPH_1:12, JORDAN8:5;
reconsider do = do as non constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A39, A41, JGRAPH_1:12, JORDAN8:5;
A42: len go > 1 by A36, NAT_1:13;
then A43: len go in dom go by FINSEQ_3:25;
then A44: go /. (len go) = go . (len go) by PARTFUN1:def_6
.= (Gauge (C,n)) * (i1,j) by A14, JORDAN3:24 ;
 len do >= 1 by A39, XXREAL_0:2;
then  1 in dom do by FINSEQ_3:25;
then A45: do /. 1 = do . 1 by PARTFUN1:def_6
.= (Gauge (C,n)) * (i2,k) by A13, JORDAN3:23 ;
reconsider m = (len go) - 1 as Element of NAT by A43, FINSEQ_3:26;
A46: m + 1 = len go ;
then A47: (len go) -' 1 = m by NAT_D:34;
A48: LSeg (go,m) c= L~ go by TOPREAL3:19;
A49: L~ go c= L~ (Upper_Seq (C,n)) by A14, JORDAN3:41;
then  LSeg (go,m) c= L~ (Upper_Seq (C,n)) by A48, XBOOLE_1:1;
then A50: (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) c= {((Gauge (C,n)) * (i1,j))} by A7, A12, XBOOLE_1:26;
 m >= 1 by A36, XREAL_1:19;
then A51: LSeg (go,m) = LSeg ((go /. m),((Gauge (C,n)) * (i1,j))) by A44, A46, TOPREAL1:def_3;
 {((Gauge (C,n)) * (i1,j))} c= (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i1,j))} or x in (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) )
assume  x in {((Gauge (C,n)) * (i1,j))} ; ::_thesis: x in (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>)
then A52: x = (Gauge (C,n)) * (i1,j) by TARSKI:def_1;
A53: (Gauge (C,n)) * (i1,j) in LSeg (go,m) by A51, RLTOPSP1:68;
 (Gauge (C,n)) * (i1,j) in LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) by RLTOPSP1:68;
then  (Gauge (C,n)) * (i1,j) in (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) by XBOOLE_0:def_3;
then  (Gauge (C,n)) * (i1,j) in L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> by SPRECT_1:8;
hence  x in (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) by A52, A53, XBOOLE_0:def_4; ::_thesis: verum
end;
then A54: (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) = {((Gauge (C,n)) * (i1,j))} by A50, XBOOLE_0:def_10;
A55: LSeg (do,1) c= L~ do by TOPREAL3:19;
A56: L~ do c= L~ (Lower_Seq (C,n)) by A13, JORDAN3:42;
then  LSeg (do,1) c= L~ (Lower_Seq (C,n)) by A55, XBOOLE_1:1;
then A57: (LSeg (do,1)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) c= {((Gauge (C,n)) * (i2,k))} by A8, A12, XBOOLE_1:26;
A58: LSeg (do,1) = LSeg (((Gauge (C,n)) * (i2,k)),(do /. (1 + 1))) by A39, A45, TOPREAL1:def_3;
 {((Gauge (C,n)) * (i2,k))} c= (LSeg (do,1)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i2,k))} or x in (LSeg (do,1)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) )
assume  x in {((Gauge (C,n)) * (i2,k))} ; ::_thesis: x in (LSeg (do,1)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>)
then A59: x = (Gauge (C,n)) * (i2,k) by TARSKI:def_1;
A60: (Gauge (C,n)) * (i2,k) in LSeg (do,1) by A58, RLTOPSP1:68;
 (Gauge (C,n)) * (i2,k) in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) by RLTOPSP1:68;
then  (Gauge (C,n)) * (i2,k) in (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) by XBOOLE_0:def_3;
then  (Gauge (C,n)) * (i2,k) in L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> by SPRECT_1:8;
hence  x in (LSeg (do,1)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) by A59, A60, XBOOLE_0:def_4; ::_thesis: verum
end;
then A61: (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) /\ (LSeg (do,1)) = {((Gauge (C,n)) * (i2,k))} by A57, XBOOLE_0:def_10;
A62: go /. 1 = (Upper_Seq (C,n)) /. 1 by A14, SPRECT_3:22
.= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ;
then A63: go /. 1 = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by JORDAN1F:8
.= do /. (len do) by A13, JORDAN1J:35 ;
A64: rng go c= L~ go by A36, SPPOL_2:18;
A65: rng do c= L~ do by A39, SPPOL_2:18;
A66: {(go /. 1)} c= (L~ go) /\ (L~ do)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(go /. 1)} or x in (L~ go) /\ (L~ do) )
assume  x in {(go /. 1)} ; ::_thesis: x in (L~ go) /\ (L~ do)
then A67: x = go /. 1 by TARSKI:def_1;
then A68: x in rng go by FINSEQ_6:42;
 x in rng do by A63, A67, REVROT_1:3;
hence  x in (L~ go) /\ (L~ do) by A64, A65, A68, XBOOLE_0:def_4; ::_thesis: verum
end;
A69: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by A30, PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;
A70: [(len (Gauge (C,n))),j] in Indices (Gauge (C,n)) by A4, A15, A24, MATRIX_1:36;
 (L~ go) /\ (L~ do) c= {(go /. 1)}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ go) /\ (L~ do) or x in {(go /. 1)} )
assume A71: x in (L~ go) /\ (L~ do) ; ::_thesis: x in {(go /. 1)}
then A72: x in L~ go by XBOOLE_0:def_4;
A73: x in L~ do by A71, XBOOLE_0:def_4;
then  x in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A49, A56, A72, XBOOLE_0:def_4;
then  x in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16;
then A74: ( x = W-min (L~ (Cage (C,n))) or x = E-max (L~ (Cage (C,n))) ) by TARSKI:def_2;
now__::_thesis:_not_x_=_E-max_(L~_(Cage_(C,n)))
assume  x = E-max (L~ (Cage (C,n))) ; ::_thesis: contradiction
then A75: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i2,k) by A13, A69, A73, JORDAN1E:7;
 ((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 = E-bound (L~ (Cage (C,n))) by A4, A15, A20, JORDAN1A:71;
then  (E-max (L~ (Cage (C,n)))) `1 <> E-bound (L~ (Cage (C,n))) by A3, A18, A70, A75, JORDAN1G:7;
hence  contradiction by EUCLID:52; ::_thesis: verum
end;
hence  x in {(go /. 1)} by A62, A74, TARSKI:def_1; ::_thesis: verum
end;
then A76: (L~ go) /\ (L~ do) = {(go /. 1)} by A66, XBOOLE_0:def_10;
set W2 = go /. 2;
A77: 2 in dom go by A36, FINSEQ_3:25;
A78: now__::_thesis:_not_((Gauge_(C,n))_*_(i1,j))_`1_=_W-bound_(L~_(Cage_(C,n)))
assume  ((Gauge (C,n)) * (i1,j)) `1 = W-bound (L~ (Cage (C,n))) ; ::_thesis: contradiction
then  ((Gauge (C,n)) * (1,j)) `1 = ((Gauge (C,n)) * (i1,j)) `1 by A4, A15, A20, JORDAN1A:73;
hence  contradiction by A1, A17, A27, JORDAN1G:7; ::_thesis: verum
end;
go = mid ((Upper_Seq (C,n)),1,(((Gauge (C,n)) * (i1,j)) .. (Upper_Seq (C,n)))) by A37, JORDAN1G:49
.= (Upper_Seq (C,n)) | (((Gauge (C,n)) * (i1,j)) .. (Upper_Seq (C,n))) by A37, FINSEQ_4:21, FINSEQ_6:116 ;
then A79: go /. 2 = (Upper_Seq (C,n)) /. 2 by A77, FINSEQ_4:70;
A80: W-min (L~ (Cage (C,n))) in rng go by A62, FINSEQ_6:42;
set pion = <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>;
A81: now__::_thesis:_for_n_being_Element_of_NAT_st_n_in_dom_<*((Gauge_(C,n))_*_(i1,j)),((Gauge_(C,n))_*_(i1,k)),((Gauge_(C,n))_*_(i2,k))*>_holds_
ex_i,_j_being_Element_of_NAT_st_
(_[i,j]_in_Indices_(Gauge_(C,n))_&_<*((Gauge_(C,n))_*_(i1,j)),((Gauge_(C,n))_*_(i1,k)),((Gauge_(C,n))_*_(i2,k))*>_/._n_=_(Gauge_(C,n))_*_(i,j)_)
let n be Element of NAT ; ::_thesis: ( n in dom <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> implies ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. n = (Gauge (C,n)) * (i,j) ) )
assume  n in dom <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. n = (Gauge (C,n)) * (i,j) )
then  n in {1,2,3} by FINSEQ_1:89, FINSEQ_3:1;
then  ( n = 1 or n = 2 or n = 3 ) by ENUMSET1:def_1;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. n = (Gauge (C,n)) * (i,j) ) by A17, A18, A19, FINSEQ_4:18; ::_thesis: verum
end;
A82: ((Gauge (C,n)) * (i1,k)) `1 = ((Gauge (C,n)) * (i1,1)) `1 by A1, A6, A10, A16, GOBOARD5:2
.= ((Gauge (C,n)) * (i1,j)) `1 by A1, A4, A10, A15, GOBOARD5:2 ;
((Gauge (C,n)) * (i1,k)) `2 = ((Gauge (C,n)) * (1,k)) `2 by A1, A6, A10, A16, GOBOARD5:1
.= ((Gauge (C,n)) * (i2,k)) `2 by A3, A6, A11, A16, GOBOARD5:1 ;
then A83: (Gauge (C,n)) * (i1,k) = |[(((Gauge (C,n)) * (i1,j)) `1),(((Gauge (C,n)) * (i2,k)) `2)]| by A82, EUCLID:53;
A84: (Gauge (C,n)) * (i1,k) in LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) by RLTOPSP1:68;
A85: (Gauge (C,n)) * (i1,k) in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) by RLTOPSP1:68;
now__::_thesis:_contradiction
percases ( ( ((Gauge (C,n)) * (i2,k)) `1 <> ((Gauge (C,n)) * (i1,j)) `1 & ((Gauge (C,n)) * (i2,k)) `2 <> ((Gauge (C,n)) * (i1,j)) `2 ) or ((Gauge (C,n)) * (i2,k)) `1 = ((Gauge (C,n)) * (i1,j)) `1 or ((Gauge (C,n)) * (i2,k)) `2 = ((Gauge (C,n)) * (i1,j)) `2 ) ;
suppose ( ((Gauge (C,n)) * (i2,k)) `1 <> ((Gauge (C,n)) * (i1,j)) `1 & ((Gauge (C,n)) * (i2,k)) `2 <> ((Gauge (C,n)) * (i1,j)) `2 ) ; ::_thesis: contradiction
then  <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> is being_S-Seq by A83, TOPREAL3:34;
then consider pion1 being FinSequence of (TOP-REAL 2) such that
A86: pion1 is_sequence_on Gauge (C,n) and
A87: pion1 is being_S-Seq and
A88: L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> = L~ pion1 and
A89: <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. 1 = pion1 /. 1 and
A90: <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. (len <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) = pion1 /. (len pion1) and
A91: len <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> <= len pion1 by A81, GOBOARD3:2;
reconsider pion1 = pion1 as being_S-Seq FinSequence of (TOP-REAL 2) by A87;
set godo = (go ^' pion1) ^' do;
A92: ((Gauge (C,n)) * (i1,k)) `1 = ((Gauge (C,n)) * (i1,1)) `1 by A1, A6, A10, A16, GOBOARD5:2
.= ((Gauge (C,n)) * (i1,j)) `1 by A1, A4, A10, A15, GOBOARD5:2 ;
A93: ((Gauge (C,n)) * (i1,k)) `1 <= ((Gauge (C,n)) * (i2,k)) `1 by A1, A2, A3, A6, A16, JORDAN1A:18;
then A94: W-bound (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) = ((Gauge (C,n)) * (i1,k)) `1 by SPRECT_1:54;
A95: W-bound (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) = ((Gauge (C,n)) * (i1,j)) `1 by A92, SPRECT_1:54;
W-bound ((LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))))) = min ((W-bound (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))),(W-bound (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))))) by SPRECT_1:47
.= ((Gauge (C,n)) * (i1,j)) `1 by A92, A94, A95 ;
then A96: W-bound (L~ pion1) = ((Gauge (C,n)) * (i1,j)) `1 by A88, TOPREAL3:16;
A97: 1 + 1 <= len (Cage (C,n)) by GOBOARD7:34, XXREAL_0:2;
A98: 1 + 1 <= len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by GOBOARD7:34, XXREAL_0:2;
 len (go ^' pion1) >= len go by TOPREAL8:7;
then A99: len (go ^' pion1) >= 1 + 1 by A36, XXREAL_0:2;
then A100: len (go ^' pion1) > 1 + 0 by NAT_1:13;
A101: len ((go ^' pion1) ^' do) >= len (go ^' pion1) by TOPREAL8:7;
then A102: 1 + 1 <= len ((go ^' pion1) ^' do) by A99, XXREAL_0:2;
A103: Upper_Seq (C,n) is_sequence_on Gauge (C,n) by JORDAN1G:4;
A104: go /. (len go) = pion1 /. 1 by A44, A89, FINSEQ_4:18;
then A105: go ^' pion1 is_sequence_on Gauge (C,n) by A38, A86, TOPREAL8:12;
A106: (go ^' pion1) /. (len (go ^' pion1)) = <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. (len <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) by A90, GRAPH_2:54
.= <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. 3 by FINSEQ_1:45
.= do /. 1 by A45, FINSEQ_4:18 ;
then A107: (go ^' pion1) ^' do is_sequence_on Gauge (C,n) by A41, A105, TOPREAL8:12;
 LSeg (pion1,1) c= L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> by A88, TOPREAL3:19;
then A108: (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) c= {((Gauge (C,n)) * (i1,j))} by A47, A54, XBOOLE_1:27;
 len pion1 >= 2 + 1 by A91, FINSEQ_1:45;
then A109: len pion1 > 1 + 1 by NAT_1:13;
 {((Gauge (C,n)) * (i1,j))} c= (LSeg (go,m)) /\ (LSeg (pion1,1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i1,j))} or x in (LSeg (go,m)) /\ (LSeg (pion1,1)) )
assume  x in {((Gauge (C,n)) * (i1,j))} ; ::_thesis: x in (LSeg (go,m)) /\ (LSeg (pion1,1))
then A110: x = (Gauge (C,n)) * (i1,j) by TARSKI:def_1;
A111: (Gauge (C,n)) * (i1,j) in LSeg (go,m) by A51, RLTOPSP1:68;
 (Gauge (C,n)) * (i1,j) in LSeg (pion1,1) by A44, A104, A109, TOPREAL1:21;
hence  x in (LSeg (go,m)) /\ (LSeg (pion1,1)) by A110, A111, XBOOLE_0:def_4; ::_thesis: verum
end;
then  (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) = {(go /. (len go))} by A44, A47, A108, XBOOLE_0:def_10;
then A112: go ^' pion1 is unfolded by A104, TOPREAL8:34;
 len pion1 >= 2 + 1 by A91, FINSEQ_1:45;
then A113: (len pion1) - 2 >= 0 by XREAL_1:19;
 ((len (go ^' pion1)) + 1) - 1 = ((len go) + (len pion1)) - 1 by GRAPH_2:13;
then (len (go ^' pion1)) - 1 = (len go) + ((len pion1) - 2)
.= (len go) + ((len pion1) -' 2) by A113, XREAL_0:def_2 ;
then A114: (len (go ^' pion1)) -' 1 = (len go) + ((len pion1) -' 2) by XREAL_0:def_2;
A115: (len pion1) - 1 >= 1 by A109, XREAL_1:19;
then A116: (len pion1) -' 1 = (len pion1) - 1 by XREAL_0:def_2;
A117: ((len pion1) -' 2) + 1 = ((len pion1) - 2) + 1 by A113, XREAL_0:def_2
.= (len pion1) -' 1 by A115, XREAL_0:def_2 ;
 ((len pion1) - 1) + 1 <= len pion1 ;
then A118: (len pion1) -' 1 < len pion1 by A116, NAT_1:13;
 LSeg (pion1,((len pion1) -' 1)) c= L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> by A88, TOPREAL3:19;
then A119: (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) c= {((Gauge (C,n)) * (i2,k))} by A61, XBOOLE_1:27;
 {((Gauge (C,n)) * (i2,k))} c= (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i2,k))} or x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) )
assume  x in {((Gauge (C,n)) * (i2,k))} ; ::_thesis: x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1))
then A120: x = (Gauge (C,n)) * (i2,k) by TARSKI:def_1;
A121: (Gauge (C,n)) * (i2,k) in LSeg (do,1) by A58, RLTOPSP1:68;
pion1 /. (((len pion1) -' 1) + 1) = <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. 3 by A90, A116, FINSEQ_1:45
.= (Gauge (C,n)) * (i2,k) by FINSEQ_4:18 ;
then  (Gauge (C,n)) * (i2,k) in LSeg (pion1,((len pion1) -' 1)) by A115, A116, TOPREAL1:21;
hence  x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) by A120, A121, XBOOLE_0:def_4; ::_thesis: verum
end;
then  (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) = {((Gauge (C,n)) * (i2,k))} by A119, XBOOLE_0:def_10;
then A122: (LSeg ((go ^' pion1),((len go) + ((len pion1) -' 2)))) /\ (LSeg (do,1)) = {((go ^' pion1) /. (len (go ^' pion1)))} by A45, A104, A106, A117, A118, TOPREAL8:31;
A123: not go ^' pion1 is trivial by A99, NAT_D:60;
A124: rng pion1 c= L~ pion1 by A109, SPPOL_2:18;
A125: {(pion1 /. 1)} c= (L~ go) /\ (L~ pion1)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(pion1 /. 1)} or x in (L~ go) /\ (L~ pion1) )
assume  x in {(pion1 /. 1)} ; ::_thesis: x in (L~ go) /\ (L~ pion1)
then A126: x = pion1 /. 1 by TARSKI:def_1;
then A127: x in rng go by A104, REVROT_1:3;
 x in rng pion1 by A126, FINSEQ_6:42;
hence  x in (L~ go) /\ (L~ pion1) by A64, A124, A127, XBOOLE_0:def_4; ::_thesis: verum
end;
 (L~ go) /\ (L~ pion1) c= {(pion1 /. 1)}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ go) /\ (L~ pion1) or x in {(pion1 /. 1)} )
assume A128: x in (L~ go) /\ (L~ pion1) ; ::_thesis: x in {(pion1 /. 1)}
then A129: x in L~ go by XBOOLE_0:def_4;
 x in L~ pion1 by A128, XBOOLE_0:def_4;
hence  x in {(pion1 /. 1)} by A7, A12, A44, A49, A88, A104, A129, XBOOLE_0:def_4; ::_thesis: verum
end;
then A130: (L~ go) /\ (L~ pion1) = {(pion1 /. 1)} by A125, XBOOLE_0:def_10;
then A131: go ^' pion1 is s.n.c. by A104, JORDAN1J:54;
 (rng go) /\ (rng pion1) c= {(pion1 /. 1)} by A64, A124, A130, XBOOLE_1:27;
then A132: go ^' pion1 is one-to-one by JORDAN1J:55;
A133: <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. (len <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) = <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. 3 by FINSEQ_1:45
.= do /. 1 by A45, FINSEQ_4:18 ;
A134: {(pion1 /. (len pion1))} c= (L~ do) /\ (L~ pion1)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(pion1 /. (len pion1))} or x in (L~ do) /\ (L~ pion1) )
assume  x in {(pion1 /. (len pion1))} ; ::_thesis: x in (L~ do) /\ (L~ pion1)
then A135: x = pion1 /. (len pion1) by TARSKI:def_1;
then A136: x in rng do by A90, A133, FINSEQ_6:42;
 x in rng pion1 by A135, REVROT_1:3;
hence  x in (L~ do) /\ (L~ pion1) by A65, A124, A136, XBOOLE_0:def_4; ::_thesis: verum
end;
 (L~ do) /\ (L~ pion1) c= {(pion1 /. (len pion1))}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ do) /\ (L~ pion1) or x in {(pion1 /. (len pion1))} )
assume A137: x in (L~ do) /\ (L~ pion1) ; ::_thesis: x in {(pion1 /. (len pion1))}
then A138: x in L~ do by XBOOLE_0:def_4;
 x in L~ pion1 by A137, XBOOLE_0:def_4;
hence  x in {(pion1 /. (len pion1))} by A8, A12, A45, A56, A88, A90, A133, A138, XBOOLE_0:def_4; ::_thesis: verum
end;
then A139: (L~ do) /\ (L~ pion1) = {(pion1 /. (len pion1))} by A134, XBOOLE_0:def_10;
A140: (L~ (go ^' pion1)) /\ (L~ do) = ((L~ go) \/ (L~ pion1)) /\ (L~ do) by A104, TOPREAL8:35
.= {(go /. 1)} \/ {(do /. 1)} by A76, A90, A133, A139, XBOOLE_1:23
.= {((go ^' pion1) /. 1)} \/ {(do /. 1)} by GRAPH_2:53
.= {((go ^' pion1) /. 1),(do /. 1)} by ENUMSET1:1 ;
 do /. (len do) = (go ^' pion1) /. 1 by A63, GRAPH_2:53;
then reconsider godo = (go ^' pion1) ^' do as non constant standard special_circular_sequence by A102, A106, A107, A112, A114, A122, A123, A131, A132, A140, JORDAN8:4, JORDAN8:5, TOPREAL8:11, TOPREAL8:33, TOPREAL8:34;
A141: Upper_Arc C is_an_arc_of W-min C, E-max C by JORDAN6:def_8;
then A142: Upper_Arc C is connected by JORDAN6:10;
A143: W-min C in Upper_Arc C by A141, TOPREAL1:1;
A144: E-max C in Upper_Arc C by A141, TOPREAL1:1;
set ff = Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))));
 W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
then A145: (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 = W-min (L~ (Cage (C,n))) by FINSEQ_6:92;
A146: L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33;
then  (W-max (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A145, SPRECT_5:22;
then  (N-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A145, A146, SPRECT_5:23, XXREAL_0:2;
then  (N-max (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A145, A146, SPRECT_5:24, XXREAL_0:2;
then A147: (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A145, A146, SPRECT_5:25, XXREAL_0:2;
A148: now__::_thesis:_not_((Gauge_(C,n))_*_(i1,j))_.._(Upper_Seq_(C,n))_<=_1
assume A149: ((Gauge (C,n)) * (i1,j)) .. (Upper_Seq (C,n)) <= 1 ; ::_thesis: contradiction
 ((Gauge (C,n)) * (i1,j)) .. (Upper_Seq (C,n)) >= 1 by A37, FINSEQ_4:21;
then  ((Gauge (C,n)) * (i1,j)) .. (Upper_Seq (C,n)) = 1 by A149, XXREAL_0:1;
then  (Gauge (C,n)) * (i1,j) = (Upper_Seq (C,n)) /. 1 by A37, FINSEQ_5:38;
hence  contradiction by A22, A26, JORDAN1F:5; ::_thesis: verum
end;
A150: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def_1;
then A151: Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))) is_sequence_on Gauge (C,n) by REVROT_1:34;
A152: (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) c= RightComp godo by A102, A107, JORDAN9:27;
A153: L~ godo = (L~ (go ^' pion1)) \/ (L~ do) by A106, TOPREAL8:35
.= ((L~ go) \/ (L~ pion1)) \/ (L~ do) by A104, TOPREAL8:35 ;
A154: L~ (Cage (C,n)) = (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) by JORDAN1E:13;
then A155: L~ (Upper_Seq (C,n)) c= L~ (Cage (C,n)) by XBOOLE_1:7;
A156: L~ (Lower_Seq (C,n)) c= L~ (Cage (C,n)) by A154, XBOOLE_1:7;
A157: L~ go c= L~ (Cage (C,n)) by A49, A155, XBOOLE_1:1;
A158: L~ do c= L~ (Cage (C,n)) by A56, A156, XBOOLE_1:1;
A159: W-min C in C by SPRECT_1:13;
A160: now__::_thesis:_not_W-min_C_in_L~_godo
assume  W-min C in L~ godo ; ::_thesis: contradiction
then A161: ( W-min C in (L~ go) \/ (L~ pion1) or W-min C in L~ do ) by A153, XBOOLE_0:def_3;
percases ( W-min C in L~ go or W-min C in L~ pion1 or W-min C in L~ do ) by A161, XBOOLE_0:def_3;
suppose W-min C in L~ go ; ::_thesis: contradiction
then  C meets L~ (Cage (C,n)) by A157, A159, XBOOLE_0:3;
hence  contradiction by JORDAN10:5; ::_thesis: verum
end;
suppose W-min C in L~ pion1 ; ::_thesis: contradiction
hence  contradiction by A9, A12, A88, A143, XBOOLE_0:3; ::_thesis: verum
end;
suppose W-min C in L~ do ; ::_thesis: contradiction
then  C meets L~ (Cage (C,n)) by A158, A159, XBOOLE_0:3;
hence  contradiction by JORDAN10:5; ::_thesis: verum
end;
end;
end;
right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1) = right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(GoB (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) by A98, JORDAN1H:23
.= right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(GoB (Cage (C,n)))) by REVROT_1:28
.= right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(Gauge (C,n))) by JORDAN1H:44
.= right_cell (((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n))))),1,(Gauge (C,n))) by A147, A151, JORDAN1J:53
.= right_cell ((Upper_Seq (C,n)),1,(Gauge (C,n))) by JORDAN1E:def_1
.= right_cell ((R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i1,j)))),1,(Gauge (C,n))) by A37, A103, A148, JORDAN1J:52
.= right_cell ((go ^' pion1),1,(Gauge (C,n))) by A42, A105, JORDAN1J:51
.= right_cell (godo,1,(Gauge (C,n))) by A100, A107, JORDAN1J:51 ;
then  W-min C in right_cell (godo,1,(Gauge (C,n))) by JORDAN1I:6;
then A162: W-min C in (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) by A160, XBOOLE_0:def_5;
A163: godo /. 1 = (go ^' pion1) /. 1 by GRAPH_2:53
.= W-min (L~ (Cage (C,n))) by A62, GRAPH_2:53 ;
A164: len (Upper_Seq (C,n)) >= 2 by A21, XXREAL_0:2;
A165: godo /. 2 = (go ^' pion1) /. 2 by A99, GRAPH_2:57
.= (Upper_Seq (C,n)) /. 2 by A36, A79, GRAPH_2:57
.= ((Upper_Seq (C,n)) ^' (Lower_Seq (C,n))) /. 2 by A164, GRAPH_2:57
.= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 2 by JORDAN1E:11 ;
A166: (L~ go) \/ (L~ do) is compact by COMPTS_1:10;
 W-min (L~ (Cage (C,n))) in (L~ go) \/ (L~ do) by A64, A80, XBOOLE_0:def_3;
then A167: W-min ((L~ go) \/ (L~ do)) = W-min (L~ (Cage (C,n))) by A157, A158, A166, JORDAN1J:21, XBOOLE_1:8;
A168: (W-min ((L~ go) \/ (L~ do))) `1 = W-bound ((L~ go) \/ (L~ do)) by EUCLID:52;
A169: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52;
 ((Gauge (C,n)) * (i1,j)) `1 >= W-bound (L~ (Cage (C,n))) by A14, A155, PSCOMP_1:24;
then  ((Gauge (C,n)) * (i1,j)) `1 > W-bound (L~ (Cage (C,n))) by A78, XXREAL_0:1;
then  W-min (((L~ go) \/ (L~ do)) \/ (L~ pion1)) = W-min ((L~ go) \/ (L~ do)) by A96, A166, A167, A168, A169, JORDAN1J:33;
then A170: W-min (L~ godo) = W-min (L~ (Cage (C,n))) by A153, A167, XBOOLE_1:4;
A171: rng godo c= L~ godo by A99, A101, SPPOL_2:18, XXREAL_0:2;
 2 in dom godo by A102, FINSEQ_3:25;
then A172: godo /. 2 in rng godo by PARTFUN2:2;
 godo /. 2 in W-most (L~ (Cage (C,n))) by A165, JORDAN1I:25;
then (godo /. 2) `1 = (W-min (L~ godo)) `1 by A170, PSCOMP_1:31
.= W-bound (L~ godo) by EUCLID:52 ;
then  godo /. 2 in W-most (L~ godo) by A171, A172, SPRECT_2:12;
then  (Rotate (godo,(W-min (L~ godo)))) /. 2 in W-most (L~ godo) by A163, A170, FINSEQ_6:89;
then reconsider godo = godo as non constant standard clockwise_oriented special_circular_sequence by JORDAN1I:25;
 len (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by FINSEQ_5:6;
then A173: (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) = (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) by PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:7 ;
A174: east_halfline (E-max C) misses L~ go
proof
assume  east_halfline (E-max C) meets L~ go ; ::_thesis: contradiction
then consider p being set  such that
A175: p in east_halfline (E-max C) and
A176: p in L~ go by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A175;
 p in L~ (Upper_Seq (C,n)) by A49, A176;
then  p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A155, A175, XBOOLE_0:def_4;
then A177: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;
then A178: p = E-max (L~ (Cage (C,n))) by A49, A176, JORDAN1J:46;
then  E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i1,j) by A14, A173, A176, JORDAN1J:43;
then  ((Gauge (C,n)) * (i1,j)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A6, A16, A20, A177, A178, JORDAN1A:71;
hence  contradiction by A2, A3, A17, A33, JORDAN1G:7; ::_thesis: verum
end;
now__::_thesis:_not_east_halfline_(E-max_C)_meets_L~_godo
assume  east_halfline (E-max C) meets L~ godo ; ::_thesis: contradiction
then A179: ( east_halfline (E-max C) meets (L~ go) \/ (L~ pion1) or east_halfline (E-max C) meets L~ do ) by A153, XBOOLE_1:70;
percases ( east_halfline (E-max C) meets L~ go or east_halfline (E-max C) meets L~ pion1 or east_halfline (E-max C) meets L~ do ) by A179, XBOOLE_1:70;
suppose east_halfline (E-max C) meets L~ go ; ::_thesis: contradiction
hence  contradiction by A174; ::_thesis: verum
end;
suppose east_halfline (E-max C) meets L~ pion1 ; ::_thesis: contradiction
then consider p being set  such that
A180: p in east_halfline (E-max C) and
A181: p in L~ pion1 by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A180;
A182: now__::_thesis:_p_`1_<=_((Gauge_(C,n))_*_(i2,k))_`1
percases ( p in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) or p in LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) ) by A12, A88, A181, XBOOLE_0:def_3;
suppose p in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) ; ::_thesis: p `1 <= ((Gauge (C,n)) * (i2,k)) `1
hence  p `1 <= ((Gauge (C,n)) * (i2,k)) `1 by A93, TOPREAL1:3; ::_thesis: verum
end;
suppose p in LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) ; ::_thesis: p `1 <= ((Gauge (C,n)) * (i2,k)) `1
hence  p `1 <= ((Gauge (C,n)) * (i2,k)) `1 by A92, A93, GOBOARD7:5; ::_thesis: verum
end;
end;
end;
 i2 + 1 <= len (Gauge (C,n)) by A3, NAT_1:13;
then  i2 <= (len (Gauge (C,n))) - 1 by XREAL_1:19;
then A183: i2 <= (len (Gauge (C,n))) -' 1 by XREAL_0:def_2;
 (len (Gauge (C,n))) -' 1 <= len (Gauge (C,n)) by NAT_D:35;
then  ((Gauge (C,n)) * (i2,k)) `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1 by A6, A11, A16, A20, A24, A183, JORDAN1A:18;
then  p `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1 by A182, XXREAL_0:2;
then  p `1 <= E-bound C by A24, JORDAN8:12;
then A184: p `1 <= (E-max C) `1 by EUCLID:52;
 p `1 >= (E-max C) `1 by A180, TOPREAL1:def_11;
then A185: p `1 = (E-max C) `1 by A184, XXREAL_0:1;
 p `2 = (E-max C) `2 by A180, TOPREAL1:def_11;
then  p = E-max C by A185, TOPREAL3:6;
hence  contradiction by A9, A12, A88, A144, A181, XBOOLE_0:3; ::_thesis: verum
end;
suppose east_halfline (E-max C) meets L~ do ; ::_thesis: contradiction
then consider p being set  such that
A186: p in east_halfline (E-max C) and
A187: p in L~ do by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A186;
 p in L~ (Lower_Seq (C,n)) by A56, A187;
then  p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A156, A186, XBOOLE_0:def_4;
then A188: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;
A189: (E-max C) `2 = p `2 by A186, TOPREAL1:def_11;
set RC = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))));
A190: E-max C in right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by JORDAN1I:7;
A191: 1 + 1 <= len (Lower_Seq (C,n)) by A28, XXREAL_0:2;
 Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) by JORDAN1G:18;
then A192: LSeg ((Lower_Seq (C,n)),1) = LSeg ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by A191, SPPOL_2:9;
A193: L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33;
A194: len (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = len (Cage (C,n)) by REVROT_1:14;
A195: GoB (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = GoB (Cage (C,n)) by REVROT_1:28
.= Gauge (C,n) by JORDAN1H:44 ;
A196: E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
A197: Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))) is_sequence_on Gauge (C,n) by A150, REVROT_1:34;
A198: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = E-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))))) by A193, A196, FINSEQ_6:92;
consider ii, jj being Element of NAT  such that
A199: [ii,(jj + 1)] in Indices (Gauge (C,n)) and
A200: [ii,jj] in Indices (Gauge (C,n)) and
A201: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = (Gauge (C,n)) * (ii,(jj + 1)) and
A202: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1) = (Gauge (C,n)) * (ii,jj) by A97, A193, A194, A196, A197, FINSEQ_6:92, JORDAN1I:23;
consider jj2 being Element of NAT  such that
A203: 1 <= jj2 and
A204: jj2 <= width (Gauge (C,n)) and
A205: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * ((len (Gauge (C,n))),jj2) by JORDAN1D:25;
A206: len (Gauge (C,n)) >= 4 by JORDAN8:10;
then  len (Gauge (C,n)) >= 1 by XXREAL_0:2;
then  [(len (Gauge (C,n))),jj2] in Indices (Gauge (C,n)) by A203, A204, MATRIX_1:36;
then A207: ii = len (Gauge (C,n)) by A193, A198, A199, A201, A205, GOBOARD1:5;
A208: 1 <= ii by A199, MATRIX_1:38;
A209: ii <= len (Gauge (C,n)) by A199, MATRIX_1:38;
A210: 1 <= jj + 1 by A199, MATRIX_1:38;
A211: jj + 1 <= width (Gauge (C,n)) by A199, MATRIX_1:38;
A212: 1 <= ii by A200, MATRIX_1:38;
A213: ii <= len (Gauge (C,n)) by A200, MATRIX_1:38;
A214: 1 <= jj by A200, MATRIX_1:38;
A215: jj <= width (Gauge (C,n)) by A200, MATRIX_1:38;
A216: ii + 1 <> ii ;
 (jj + 1) + 1 <> jj ;
then A217: right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) = cell ((Gauge (C,n)),(ii -' 1),jj) by A97, A194, A195, A199, A200, A201, A202, A216, GOBOARD5:def_6;
A218: (ii -' 1) + 1 = ii by A208, XREAL_1:235;
 ii - 1 >= 4 - 1 by A206, A207, XREAL_1:9;
then A219: ii - 1 >= 1 by XXREAL_0:2;
then A220: 1 <= ii -' 1 by XREAL_0:def_2;
A221: ((Gauge (C,n)) * ((ii -' 1),jj)) `2 <= p `2 by A189, A190, A209, A211, A214, A217, A218, A219, JORDAN9:17;
A222: p `2 <= ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 by A189, A190, A209, A211, A214, A217, A218, A219, JORDAN9:17;
A223: ii -' 1 < len (Gauge (C,n)) by A209, A218, NAT_1:13;
then A224: ((Gauge (C,n)) * ((ii -' 1),jj)) `2 = ((Gauge (C,n)) * (1,jj)) `2 by A214, A215, A220, GOBOARD5:1
.= ((Gauge (C,n)) * (ii,jj)) `2 by A212, A213, A214, A215, GOBOARD5:1 ;
A225: ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 = ((Gauge (C,n)) * (1,(jj + 1))) `2 by A210, A211, A220, A223, GOBOARD5:1
.= ((Gauge (C,n)) * (ii,(jj + 1))) `2 by A208, A209, A210, A211, GOBOARD5:1 ;
A226: ((Gauge (C,n)) * ((len (Gauge (C,n))),jj)) `1 = E-bound (L~ (Cage (C,n))) by A20, A214, A215, JORDAN1A:71;
 E-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * ((len (Gauge (C,n))),(jj + 1))) `1 by A20, A210, A211, JORDAN1A:71;
then  p in LSeg (((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1),((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1))) by A188, A201, A202, A207, A221, A222, A224, A225, A226, GOBOARD7:7;
then A227: p in LSeg ((Lower_Seq (C,n)),1) by A97, A192, A194, TOPREAL1:def_3;
A228: p in LSeg (do,(Index (p,do))) by A187, JORDAN3:9;
A229: do = mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n)))) by A40, JORDAN1J:37;
A230: 1 <= ((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)) by A40, FINSEQ_4:21;
A231: ((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)) <= len (Lower_Seq (C,n)) by A40, FINSEQ_4:21;
 ((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)) <> len (Lower_Seq (C,n)) by A32, A40, FINSEQ_4:19;
then A232: ((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)) < len (Lower_Seq (C,n)) by A231, XXREAL_0:1;
A233: 1 <= Index (p,do) by A187, JORDAN3:8;
A234: Index (p,do) < len do by A187, JORDAN3:8;
A235: (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n)))) + 1 = ((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)) by A35, A40, JORDAN1J:56;
consider t being Nat such that
A236: t in dom (Lower_Seq (C,n)) and
A237: (Lower_Seq (C,n)) . t = (Gauge (C,n)) * (i2,k) by A40, FINSEQ_2:10;
A238: 1 <= t by A236, FINSEQ_3:25;
A239: t <= len (Lower_Seq (C,n)) by A236, FINSEQ_3:25;
 1 < t by A35, A237, A238, XXREAL_0:1;
then  (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n)))) + 1 = t by A237, A239, JORDAN3:12;
then A240: len (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i2,k)))) = (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n)))) by A13, A237, JORDAN3:26;
set tt = ((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1;
A241: 1 <= Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n))) by A13, JORDAN3:8;
 0 + (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n)))) < len (Lower_Seq (C,n)) by A13, JORDAN3:8;
then A242: (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n)))) > 0 by XREAL_1:20;
 Index (p,do) < (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n)))) by A234, A240, XREAL_0:def_2;
then  (Index (p,do)) + 1 <= (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n)))) by NAT_1:13;
then  Index (p,do) <= ((len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n))))) - 1 by XREAL_1:19;
then  Index (p,do) <= ((len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n))))) - 1 by A242, XREAL_0:def_2;
then  Index (p,do) <= (len (Lower_Seq (C,n))) - (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n))) by A235;
then  Index (p,do) <= (len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n))) by XREAL_0:def_2;
then  Index (p,do) < ((len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) + 1 by NAT_1:13;
then A243: LSeg ((mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n))))),(Index (p,do))) = LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1)) by A230, A232, A233, JORDAN4:19;
A244: 1 + 1 <= ((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)) by A235, A241, XREAL_1:7;
then  (Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n))) >= (1 + 1) + 1 by A233, XREAL_1:7;
then  ((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) - 1 >= ((1 + 1) + 1) - 1 by XREAL_1:9;
then A245: ((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1 >= 1 + 1 by XREAL_0:def_2;
A246: 2 in dom (Lower_Seq (C,n)) by A191, FINSEQ_3:25;
now__::_thesis:_contradiction
percases ( ((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1 > 1 + 1 or ((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ) by A245, XXREAL_0:1;
suppose ((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1 > 1 + 1 ; ::_thesis: contradiction
then  LSeg ((Lower_Seq (C,n)),1) misses LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1)) by TOPREAL1:def_7;
hence  contradiction by A227, A228, A229, A243, XBOOLE_0:3; ::_thesis: verum
end;
supposeA247: ((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ; ::_thesis: contradiction
then  (LSeg ((Lower_Seq (C,n)),1)) /\ (LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1))) = {((Lower_Seq (C,n)) /. 2)} by A28, TOPREAL1:def_6;
then  p in {((Lower_Seq (C,n)) /. 2)} by A227, A228, A229, A243, XBOOLE_0:def_4;
then A248: p = (Lower_Seq (C,n)) /. 2 by TARSKI:def_1;
then A249: p .. (Lower_Seq (C,n)) = 2 by A246, FINSEQ_5:41;
 1 + 1 = ((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) - 1 by A247, XREAL_0:def_2;
then  (1 + 1) + 1 = (Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n))) ;
then A250: ((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)) = 2 by A233, A244, JORDAN1E:6;
 p in rng (Lower_Seq (C,n)) by A246, A248, PARTFUN2:2;
then  p = (Gauge (C,n)) * (i2,k) by A40, A249, A250, FINSEQ_5:9;
then  ((Gauge (C,n)) * (i2,k)) `1 = E-bound (L~ (Cage (C,n))) by A248, JORDAN1G:32;
then  ((Gauge (C,n)) * (i2,k)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 by A4, A15, A20, JORDAN1A:71;
hence  contradiction by A3, A18, A70, JORDAN1G:7; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
end;
end;
then  east_halfline (E-max C) c= (L~ godo) ` by SUBSET_1:23;
then consider W being Subset of (TOP-REAL 2) such that
A251: W is_a_component_of (L~ godo) ` and
A252: east_halfline (E-max C) c= W by GOBOARD9:3;
 not W is bounded by A252, JORDAN2C:121, RLTOPSP1:42;
then  W is_outside_component_of L~ godo by A251, JORDAN2C:def_3;
then  W c= UBD (L~ godo) by JORDAN2C:23;
then A253: east_halfline (E-max C) c= UBD (L~ godo) by A252, XBOOLE_1:1;
 E-max C in east_halfline (E-max C) by TOPREAL1:38;
then  E-max C in UBD (L~ godo) by A253;
then  E-max C in LeftComp godo by GOBRD14:36;
then  Upper_Arc C meets L~ godo by A142, A143, A144, A152, A162, JORDAN1J:36;
then A254: ( Upper_Arc C meets (L~ go) \/ (L~ pion1) or Upper_Arc C meets L~ do ) by A153, XBOOLE_1:70;
A255: Upper_Arc C c= C by JORDAN6:61;
now__::_thesis:_contradiction
percases ( Upper_Arc C meets L~ go or Upper_Arc C meets L~ pion1 or Upper_Arc C meets L~ do ) by A254, XBOOLE_1:70;
suppose Upper_Arc C meets L~ go ; ::_thesis: contradiction
then  Upper_Arc C meets L~ (Cage (C,n)) by A49, A155, XBOOLE_1:1, XBOOLE_1:63;
hence  contradiction by A255, JORDAN10:5, XBOOLE_1:63; ::_thesis: verum
end;
suppose Upper_Arc C meets L~ pion1 ; ::_thesis: contradiction
hence  contradiction by A9, A12, A88; ::_thesis: verum
end;
suppose Upper_Arc C meets L~ do ; ::_thesis: contradiction
then  Upper_Arc C meets L~ (Cage (C,n)) by A56, A156, XBOOLE_1:1, XBOOLE_1:63;
hence  contradiction by A255, JORDAN10:5, XBOOLE_1:63; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
suppose ((Gauge (C,n)) * (i2,k)) `1 = ((Gauge (C,n)) * (i1,j)) `1 ; ::_thesis: contradiction
then A256: i1 = i2 by A17, A18, JORDAN1G:7;
then  LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) = {((Gauge (C,n)) * (i1,k))} by RLTOPSP1:70;
then  LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) c= LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) by A84, ZFMISC_1:31;
then  (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) = LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) by XBOOLE_1:12;
hence  contradiction by A1, A3, A4, A5, A6, A7, A8, A9, A256, Th12; ::_thesis: verum
end;
suppose ((Gauge (C,n)) * (i2,k)) `2 = ((Gauge (C,n)) * (i1,j)) `2 ; ::_thesis: contradiction
then A257: j = k by A17, A18, JORDAN1G:6;
then  LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) = {((Gauge (C,n)) * (i1,k))} by RLTOPSP1:70;
then  LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) c= LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) by A85, ZFMISC_1:31;
then  (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) = LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) by XBOOLE_1:12;
hence  contradiction by A1, A2, A3, A4, A6, A7, A8, A9, A257, JORDAN15:37; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;

theorem Th21: :: JORDAN19:21
for n being Element of NAT
for C being Simple_closed_curve
for i1, i2, j, k being Element of NAT st 1 < i1 & i1 <= i2 & i2 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} holds
(LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Lower_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i1, i2, j, k being Element of NAT st 1 < i1 & i1 <= i2 & i2 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} holds
(LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Lower_Arc C
let C be Simple_closed_curve; ::_thesis: for i1, i2, j, k being Element of NAT st 1 < i1 & i1 <= i2 & i2 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} holds
(LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Lower_Arc C
let i1, i2, j, k be Element of NAT ; ::_thesis: ( 1 < i1 & i1 <= i2 & i2 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} implies (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Lower_Arc C )
set G = Gauge (C,n);
set pio = LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)));
set poz = LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)));
set US = Upper_Seq (C,n);
set LS = Lower_Seq (C,n);
assume that
A1: 1 < i1 and
A2: i1 <= i2 and
A3: i2 < len (Gauge (C,n)) and
A4: 1 <= j and
A5: j <= k and
A6: k <= width (Gauge (C,n)) and
A7: ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} and
A8: ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} and
A9: (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) misses Lower_Arc C ; ::_thesis: contradiction
set UA = Lower_Arc C;
set Wmin = W-min (L~ (Cage (C,n)));
set Emax = E-max (L~ (Cage (C,n)));
set Wbo = W-bound (L~ (Cage (C,n)));
set Ebo = E-bound (L~ (Cage (C,n)));
set Gik = (Gauge (C,n)) * (i2,k);
set Gij = (Gauge (C,n)) * (i1,j);
set Gi1k = (Gauge (C,n)) * (i1,k);
A10: i1 < len (Gauge (C,n)) by A2, A3, XXREAL_0:2;
A11: 1 < i2 by A1, A2, XXREAL_0:2;
A12: L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> = (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) by TOPREAL3:16;
 (Gauge (C,n)) * (i2,k) in {((Gauge (C,n)) * (i2,k))} by TARSKI:def_1;
then A13: (Gauge (C,n)) * (i2,k) in L~ (Lower_Seq (C,n)) by A8, XBOOLE_0:def_4;
 (Gauge (C,n)) * (i1,j) in {((Gauge (C,n)) * (i1,j))} by TARSKI:def_1;
then A14: (Gauge (C,n)) * (i1,j) in L~ (Upper_Seq (C,n)) by A7, XBOOLE_0:def_4;
A15: j <= width (Gauge (C,n)) by A5, A6, XXREAL_0:2;
A16: 1 <= k by A4, A5, XXREAL_0:2;
A17: [i1,j] in Indices (Gauge (C,n)) by A1, A4, A10, A15, MATRIX_1:36;
A18: [i2,k] in Indices (Gauge (C,n)) by A3, A6, A11, A16, MATRIX_1:36;
A19: [i1,k] in Indices (Gauge (C,n)) by A1, A6, A10, A16, MATRIX_1:36;
set go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i1,j)));
set do = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i2,k)));
A20: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
A21: len (Upper_Seq (C,n)) >= 3 by JORDAN1E:15;
then  len (Upper_Seq (C,n)) >= 1 by XXREAL_0:2;
then  1 in dom (Upper_Seq (C,n)) by FINSEQ_3:25;
then A22: (Upper_Seq (C,n)) . 1 = (Upper_Seq (C,n)) /. 1 by PARTFUN1:def_6
.= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ;
A23: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * (1,k)) `1 by A6, A16, A20, JORDAN1A:73 ;
 len (Gauge (C,n)) >= 4 by JORDAN8:10;
then A24: len (Gauge (C,n)) >= 1 by XXREAL_0:2;
then A25: [1,k] in Indices (Gauge (C,n)) by A6, A16, MATRIX_1:36;
then A26: (Gauge (C,n)) * (i1,j) <> (Upper_Seq (C,n)) . 1 by A1, A17, A22, A23, JORDAN1G:7;
then reconsider go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i1,j))) as being_S-Seq FinSequence of (TOP-REAL 2) by A14, JORDAN3:35;
A27: [1,j] in Indices (Gauge (C,n)) by A4, A15, A24, MATRIX_1:36;
A28: len (Lower_Seq (C,n)) >= 1 + 2 by JORDAN1E:15;
then A29: len (Lower_Seq (C,n)) >= 1 by XXREAL_0:2;
then A30: 1 in dom (Lower_Seq (C,n)) by FINSEQ_3:25;
 len (Lower_Seq (C,n)) in dom (Lower_Seq (C,n)) by A29, FINSEQ_3:25;
then A31: (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by PARTFUN1:def_6
.= W-min (L~ (Cage (C,n))) by JORDAN1F:8 ;
(W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * (1,k)) `1 by A6, A16, A20, JORDAN1A:73 ;
then A32: (Gauge (C,n)) * (i2,k) <> (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) by A1, A2, A18, A25, A31, JORDAN1G:7;
then reconsider do = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i2,k))) as being_S-Seq FinSequence of (TOP-REAL 2) by A13, JORDAN3:34;
A33: [(len (Gauge (C,n))),k] in Indices (Gauge (C,n)) by A6, A16, A24, MATRIX_1:36;
A34: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by A30, PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;
(E-max (L~ (Cage (C,n)))) `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A6, A16, A20, JORDAN1A:71 ;
then A35: (Gauge (C,n)) * (i2,k) <> (Lower_Seq (C,n)) . 1 by A3, A18, A33, A34, JORDAN1G:7;
A36: len go >= 1 + 1 by TOPREAL1:def_8;
A37: (Gauge (C,n)) * (i1,j) in rng (Upper_Seq (C,n)) by A1, A4, A10, A14, A15, JORDAN1G:4, JORDAN1J:40;
then A38: go is_sequence_on Gauge (C,n) by JORDAN1G:4, JORDAN1J:38;
A39: len do >= 1 + 1 by TOPREAL1:def_8;
A40: (Gauge (C,n)) * (i2,k) in rng (Lower_Seq (C,n)) by A3, A6, A11, A13, A16, JORDAN1G:5, JORDAN1J:40;
then A41: do is_sequence_on Gauge (C,n) by JORDAN1G:5, JORDAN1J:39;
reconsider go = go as non constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A36, A38, JGRAPH_1:12, JORDAN8:5;
reconsider do = do as non constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A39, A41, JGRAPH_1:12, JORDAN8:5;
A42: len go > 1 by A36, NAT_1:13;
then A43: len go in dom go by FINSEQ_3:25;
then A44: go /. (len go) = go . (len go) by PARTFUN1:def_6
.= (Gauge (C,n)) * (i1,j) by A14, JORDAN3:24 ;
 len do >= 1 by A39, XXREAL_0:2;
then  1 in dom do by FINSEQ_3:25;
then A45: do /. 1 = do . 1 by PARTFUN1:def_6
.= (Gauge (C,n)) * (i2,k) by A13, JORDAN3:23 ;
reconsider m = (len go) - 1 as Element of NAT by A43, FINSEQ_3:26;
A46: m + 1 = len go ;
then A47: (len go) -' 1 = m by NAT_D:34;
A48: LSeg (go,m) c= L~ go by TOPREAL3:19;
A49: L~ go c= L~ (Upper_Seq (C,n)) by A14, JORDAN3:41;
then  LSeg (go,m) c= L~ (Upper_Seq (C,n)) by A48, XBOOLE_1:1;
then A50: (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) c= {((Gauge (C,n)) * (i1,j))} by A7, A12, XBOOLE_1:26;
 m >= 1 by A36, XREAL_1:19;
then A51: LSeg (go,m) = LSeg ((go /. m),((Gauge (C,n)) * (i1,j))) by A44, A46, TOPREAL1:def_3;
 {((Gauge (C,n)) * (i1,j))} c= (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i1,j))} or x in (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) )
assume  x in {((Gauge (C,n)) * (i1,j))} ; ::_thesis: x in (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>)
then A52: x = (Gauge (C,n)) * (i1,j) by TARSKI:def_1;
A53: (Gauge (C,n)) * (i1,j) in LSeg (go,m) by A51, RLTOPSP1:68;
 (Gauge (C,n)) * (i1,j) in LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) by RLTOPSP1:68;
then  (Gauge (C,n)) * (i1,j) in (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) by XBOOLE_0:def_3;
then  (Gauge (C,n)) * (i1,j) in L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> by SPRECT_1:8;
hence  x in (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) by A52, A53, XBOOLE_0:def_4; ::_thesis: verum
end;
then A54: (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) = {((Gauge (C,n)) * (i1,j))} by A50, XBOOLE_0:def_10;
A55: LSeg (do,1) c= L~ do by TOPREAL3:19;
A56: L~ do c= L~ (Lower_Seq (C,n)) by A13, JORDAN3:42;
then  LSeg (do,1) c= L~ (Lower_Seq (C,n)) by A55, XBOOLE_1:1;
then A57: (LSeg (do,1)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) c= {((Gauge (C,n)) * (i2,k))} by A8, A12, XBOOLE_1:26;
A58: LSeg (do,1) = LSeg (((Gauge (C,n)) * (i2,k)),(do /. (1 + 1))) by A39, A45, TOPREAL1:def_3;
 {((Gauge (C,n)) * (i2,k))} c= (LSeg (do,1)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i2,k))} or x in (LSeg (do,1)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) )
assume  x in {((Gauge (C,n)) * (i2,k))} ; ::_thesis: x in (LSeg (do,1)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>)
then A59: x = (Gauge (C,n)) * (i2,k) by TARSKI:def_1;
A60: (Gauge (C,n)) * (i2,k) in LSeg (do,1) by A58, RLTOPSP1:68;
 (Gauge (C,n)) * (i2,k) in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) by RLTOPSP1:68;
then  (Gauge (C,n)) * (i2,k) in (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) by XBOOLE_0:def_3;
then  (Gauge (C,n)) * (i2,k) in L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> by SPRECT_1:8;
hence  x in (LSeg (do,1)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) by A59, A60, XBOOLE_0:def_4; ::_thesis: verum
end;
then A61: (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) /\ (LSeg (do,1)) = {((Gauge (C,n)) * (i2,k))} by A57, XBOOLE_0:def_10;
A62: go /. 1 = (Upper_Seq (C,n)) /. 1 by A14, SPRECT_3:22
.= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ;
then A63: go /. 1 = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by JORDAN1F:8
.= do /. (len do) by A13, JORDAN1J:35 ;
A64: rng go c= L~ go by A36, SPPOL_2:18;
A65: rng do c= L~ do by A39, SPPOL_2:18;
A66: {(go /. 1)} c= (L~ go) /\ (L~ do)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(go /. 1)} or x in (L~ go) /\ (L~ do) )
assume  x in {(go /. 1)} ; ::_thesis: x in (L~ go) /\ (L~ do)
then A67: x = go /. 1 by TARSKI:def_1;
then A68: x in rng go by FINSEQ_6:42;
 x in rng do by A63, A67, REVROT_1:3;
hence  x in (L~ go) /\ (L~ do) by A64, A65, A68, XBOOLE_0:def_4; ::_thesis: verum
end;
A69: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by A30, PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;
A70: [(len (Gauge (C,n))),j] in Indices (Gauge (C,n)) by A4, A15, A24, MATRIX_1:36;
 (L~ go) /\ (L~ do) c= {(go /. 1)}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ go) /\ (L~ do) or x in {(go /. 1)} )
assume A71: x in (L~ go) /\ (L~ do) ; ::_thesis: x in {(go /. 1)}
then A72: x in L~ go by XBOOLE_0:def_4;
A73: x in L~ do by A71, XBOOLE_0:def_4;
then  x in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A49, A56, A72, XBOOLE_0:def_4;
then  x in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16;
then A74: ( x = W-min (L~ (Cage (C,n))) or x = E-max (L~ (Cage (C,n))) ) by TARSKI:def_2;
now__::_thesis:_not_x_=_E-max_(L~_(Cage_(C,n)))
assume  x = E-max (L~ (Cage (C,n))) ; ::_thesis: contradiction
then A75: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i2,k) by A13, A69, A73, JORDAN1E:7;
 ((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 = E-bound (L~ (Cage (C,n))) by A4, A15, A20, JORDAN1A:71;
then  (E-max (L~ (Cage (C,n)))) `1 <> E-bound (L~ (Cage (C,n))) by A3, A18, A70, A75, JORDAN1G:7;
hence  contradiction by EUCLID:52; ::_thesis: verum
end;
hence  x in {(go /. 1)} by A62, A74, TARSKI:def_1; ::_thesis: verum
end;
then A76: (L~ go) /\ (L~ do) = {(go /. 1)} by A66, XBOOLE_0:def_10;
set W2 = go /. 2;
A77: 2 in dom go by A36, FINSEQ_3:25;
A78: now__::_thesis:_not_((Gauge_(C,n))_*_(i1,j))_`1_=_W-bound_(L~_(Cage_(C,n)))
assume  ((Gauge (C,n)) * (i1,j)) `1 = W-bound (L~ (Cage (C,n))) ; ::_thesis: contradiction
then  ((Gauge (C,n)) * (1,j)) `1 = ((Gauge (C,n)) * (i1,j)) `1 by A4, A15, A20, JORDAN1A:73;
hence  contradiction by A1, A17, A27, JORDAN1G:7; ::_thesis: verum
end;
go = mid ((Upper_Seq (C,n)),1,(((Gauge (C,n)) * (i1,j)) .. (Upper_Seq (C,n)))) by A37, JORDAN1G:49
.= (Upper_Seq (C,n)) | (((Gauge (C,n)) * (i1,j)) .. (Upper_Seq (C,n))) by A37, FINSEQ_4:21, FINSEQ_6:116 ;
then A79: go /. 2 = (Upper_Seq (C,n)) /. 2 by A77, FINSEQ_4:70;
A80: W-min (L~ (Cage (C,n))) in rng go by A62, FINSEQ_6:42;
set pion = <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>;
A81: now__::_thesis:_for_n_being_Element_of_NAT_st_n_in_dom_<*((Gauge_(C,n))_*_(i1,j)),((Gauge_(C,n))_*_(i1,k)),((Gauge_(C,n))_*_(i2,k))*>_holds_
ex_i,_j_being_Element_of_NAT_st_
(_[i,j]_in_Indices_(Gauge_(C,n))_&_<*((Gauge_(C,n))_*_(i1,j)),((Gauge_(C,n))_*_(i1,k)),((Gauge_(C,n))_*_(i2,k))*>_/._n_=_(Gauge_(C,n))_*_(i,j)_)
let n be Element of NAT ; ::_thesis: ( n in dom <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> implies ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. n = (Gauge (C,n)) * (i,j) ) )
assume  n in dom <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. n = (Gauge (C,n)) * (i,j) )
then  n in {1,2,3} by FINSEQ_1:89, FINSEQ_3:1;
then  ( n = 1 or n = 2 or n = 3 ) by ENUMSET1:def_1;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. n = (Gauge (C,n)) * (i,j) ) by A17, A18, A19, FINSEQ_4:18; ::_thesis: verum
end;
A82: ((Gauge (C,n)) * (i1,k)) `1 = ((Gauge (C,n)) * (i1,1)) `1 by A1, A6, A10, A16, GOBOARD5:2
.= ((Gauge (C,n)) * (i1,j)) `1 by A1, A4, A10, A15, GOBOARD5:2 ;
((Gauge (C,n)) * (i1,k)) `2 = ((Gauge (C,n)) * (1,k)) `2 by A1, A6, A10, A16, GOBOARD5:1
.= ((Gauge (C,n)) * (i2,k)) `2 by A3, A6, A11, A16, GOBOARD5:1 ;
then A83: (Gauge (C,n)) * (i1,k) = |[(((Gauge (C,n)) * (i1,j)) `1),(((Gauge (C,n)) * (i2,k)) `2)]| by A82, EUCLID:53;
A84: (Gauge (C,n)) * (i1,k) in LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) by RLTOPSP1:68;
A85: (Gauge (C,n)) * (i1,k) in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) by RLTOPSP1:68;
now__::_thesis:_contradiction
percases ( ( ((Gauge (C,n)) * (i2,k)) `1 <> ((Gauge (C,n)) * (i1,j)) `1 & ((Gauge (C,n)) * (i2,k)) `2 <> ((Gauge (C,n)) * (i1,j)) `2 ) or ((Gauge (C,n)) * (i2,k)) `1 = ((Gauge (C,n)) * (i1,j)) `1 or ((Gauge (C,n)) * (i2,k)) `2 = ((Gauge (C,n)) * (i1,j)) `2 ) ;
suppose ( ((Gauge (C,n)) * (i2,k)) `1 <> ((Gauge (C,n)) * (i1,j)) `1 & ((Gauge (C,n)) * (i2,k)) `2 <> ((Gauge (C,n)) * (i1,j)) `2 ) ; ::_thesis: contradiction
then  <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> is being_S-Seq by A83, TOPREAL3:34;
then consider pion1 being FinSequence of (TOP-REAL 2) such that
A86: pion1 is_sequence_on Gauge (C,n) and
A87: pion1 is being_S-Seq and
A88: L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> = L~ pion1 and
A89: <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. 1 = pion1 /. 1 and
A90: <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. (len <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) = pion1 /. (len pion1) and
A91: len <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> <= len pion1 by A81, GOBOARD3:2;
reconsider pion1 = pion1 as being_S-Seq FinSequence of (TOP-REAL 2) by A87;
set godo = (go ^' pion1) ^' do;
A92: ((Gauge (C,n)) * (i1,k)) `1 = ((Gauge (C,n)) * (i1,1)) `1 by A1, A6, A10, A16, GOBOARD5:2
.= ((Gauge (C,n)) * (i1,j)) `1 by A1, A4, A10, A15, GOBOARD5:2 ;
A93: ((Gauge (C,n)) * (i1,k)) `1 <= ((Gauge (C,n)) * (i2,k)) `1 by A1, A2, A3, A6, A16, JORDAN1A:18;
then A94: W-bound (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) = ((Gauge (C,n)) * (i1,k)) `1 by SPRECT_1:54;
A95: W-bound (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) = ((Gauge (C,n)) * (i1,j)) `1 by A92, SPRECT_1:54;
W-bound ((LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))))) = min ((W-bound (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))),(W-bound (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))))) by SPRECT_1:47
.= ((Gauge (C,n)) * (i1,j)) `1 by A92, A94, A95 ;
then A96: W-bound (L~ pion1) = ((Gauge (C,n)) * (i1,j)) `1 by A88, TOPREAL3:16;
A97: 1 + 1 <= len (Cage (C,n)) by GOBOARD7:34, XXREAL_0:2;
A98: 1 + 1 <= len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by GOBOARD7:34, XXREAL_0:2;
 len (go ^' pion1) >= len go by TOPREAL8:7;
then A99: len (go ^' pion1) >= 1 + 1 by A36, XXREAL_0:2;
then A100: len (go ^' pion1) > 1 + 0 by NAT_1:13;
A101: len ((go ^' pion1) ^' do) >= len (go ^' pion1) by TOPREAL8:7;
then A102: 1 + 1 <= len ((go ^' pion1) ^' do) by A99, XXREAL_0:2;
A103: Upper_Seq (C,n) is_sequence_on Gauge (C,n) by JORDAN1G:4;
A104: go /. (len go) = pion1 /. 1 by A44, A89, FINSEQ_4:18;
then A105: go ^' pion1 is_sequence_on Gauge (C,n) by A38, A86, TOPREAL8:12;
A106: (go ^' pion1) /. (len (go ^' pion1)) = <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. (len <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) by A90, GRAPH_2:54
.= <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. 3 by FINSEQ_1:45
.= do /. 1 by A45, FINSEQ_4:18 ;
then A107: (go ^' pion1) ^' do is_sequence_on Gauge (C,n) by A41, A105, TOPREAL8:12;
 LSeg (pion1,1) c= L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> by A88, TOPREAL3:19;
then A108: (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) c= {((Gauge (C,n)) * (i1,j))} by A47, A54, XBOOLE_1:27;
 len pion1 >= 2 + 1 by A91, FINSEQ_1:45;
then A109: len pion1 > 1 + 1 by NAT_1:13;
 {((Gauge (C,n)) * (i1,j))} c= (LSeg (go,m)) /\ (LSeg (pion1,1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i1,j))} or x in (LSeg (go,m)) /\ (LSeg (pion1,1)) )
assume  x in {((Gauge (C,n)) * (i1,j))} ; ::_thesis: x in (LSeg (go,m)) /\ (LSeg (pion1,1))
then A110: x = (Gauge (C,n)) * (i1,j) by TARSKI:def_1;
A111: (Gauge (C,n)) * (i1,j) in LSeg (go,m) by A51, RLTOPSP1:68;
 (Gauge (C,n)) * (i1,j) in LSeg (pion1,1) by A44, A104, A109, TOPREAL1:21;
hence  x in (LSeg (go,m)) /\ (LSeg (pion1,1)) by A110, A111, XBOOLE_0:def_4; ::_thesis: verum
end;
then  (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) = {(go /. (len go))} by A44, A47, A108, XBOOLE_0:def_10;
then A112: go ^' pion1 is unfolded by A104, TOPREAL8:34;
 len pion1 >= 2 + 1 by A91, FINSEQ_1:45;
then A113: (len pion1) - 2 >= 0 by XREAL_1:19;
 ((len (go ^' pion1)) + 1) - 1 = ((len go) + (len pion1)) - 1 by GRAPH_2:13;
then (len (go ^' pion1)) - 1 = (len go) + ((len pion1) - 2)
.= (len go) + ((len pion1) -' 2) by A113, XREAL_0:def_2 ;
then A114: (len (go ^' pion1)) -' 1 = (len go) + ((len pion1) -' 2) by XREAL_0:def_2;
A115: (len pion1) - 1 >= 1 by A109, XREAL_1:19;
then A116: (len pion1) -' 1 = (len pion1) - 1 by XREAL_0:def_2;
A117: ((len pion1) -' 2) + 1 = ((len pion1) - 2) + 1 by A113, XREAL_0:def_2
.= (len pion1) -' 1 by A115, XREAL_0:def_2 ;
 ((len pion1) - 1) + 1 <= len pion1 ;
then A118: (len pion1) -' 1 < len pion1 by A116, NAT_1:13;
 LSeg (pion1,((len pion1) -' 1)) c= L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> by A88, TOPREAL3:19;
then A119: (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) c= {((Gauge (C,n)) * (i2,k))} by A61, XBOOLE_1:27;
 {((Gauge (C,n)) * (i2,k))} c= (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i2,k))} or x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) )
assume  x in {((Gauge (C,n)) * (i2,k))} ; ::_thesis: x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1))
then A120: x = (Gauge (C,n)) * (i2,k) by TARSKI:def_1;
A121: (Gauge (C,n)) * (i2,k) in LSeg (do,1) by A58, RLTOPSP1:68;
pion1 /. (((len pion1) -' 1) + 1) = <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. 3 by A90, A116, FINSEQ_1:45
.= (Gauge (C,n)) * (i2,k) by FINSEQ_4:18 ;
then  (Gauge (C,n)) * (i2,k) in LSeg (pion1,((len pion1) -' 1)) by A115, A116, TOPREAL1:21;
hence  x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) by A120, A121, XBOOLE_0:def_4; ::_thesis: verum
end;
then  (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) = {((Gauge (C,n)) * (i2,k))} by A119, XBOOLE_0:def_10;
then A122: (LSeg ((go ^' pion1),((len go) + ((len pion1) -' 2)))) /\ (LSeg (do,1)) = {((go ^' pion1) /. (len (go ^' pion1)))} by A45, A104, A106, A117, A118, TOPREAL8:31;
A123: not go ^' pion1 is trivial by A99, NAT_D:60;
A124: rng pion1 c= L~ pion1 by A109, SPPOL_2:18;
A125: {(pion1 /. 1)} c= (L~ go) /\ (L~ pion1)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(pion1 /. 1)} or x in (L~ go) /\ (L~ pion1) )
assume  x in {(pion1 /. 1)} ; ::_thesis: x in (L~ go) /\ (L~ pion1)
then A126: x = pion1 /. 1 by TARSKI:def_1;
then A127: x in rng go by A104, REVROT_1:3;
 x in rng pion1 by A126, FINSEQ_6:42;
hence  x in (L~ go) /\ (L~ pion1) by A64, A124, A127, XBOOLE_0:def_4; ::_thesis: verum
end;
 (L~ go) /\ (L~ pion1) c= {(pion1 /. 1)}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ go) /\ (L~ pion1) or x in {(pion1 /. 1)} )
assume A128: x in (L~ go) /\ (L~ pion1) ; ::_thesis: x in {(pion1 /. 1)}
then A129: x in L~ go by XBOOLE_0:def_4;
 x in L~ pion1 by A128, XBOOLE_0:def_4;
hence  x in {(pion1 /. 1)} by A7, A12, A44, A49, A88, A104, A129, XBOOLE_0:def_4; ::_thesis: verum
end;
then A130: (L~ go) /\ (L~ pion1) = {(pion1 /. 1)} by A125, XBOOLE_0:def_10;
then A131: go ^' pion1 is s.n.c. by A104, JORDAN1J:54;
 (rng go) /\ (rng pion1) c= {(pion1 /. 1)} by A64, A124, A130, XBOOLE_1:27;
then A132: go ^' pion1 is one-to-one by JORDAN1J:55;
A133: <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. (len <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) = <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. 3 by FINSEQ_1:45
.= do /. 1 by A45, FINSEQ_4:18 ;
A134: {(pion1 /. (len pion1))} c= (L~ do) /\ (L~ pion1)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(pion1 /. (len pion1))} or x in (L~ do) /\ (L~ pion1) )
assume  x in {(pion1 /. (len pion1))} ; ::_thesis: x in (L~ do) /\ (L~ pion1)
then A135: x = pion1 /. (len pion1) by TARSKI:def_1;
then A136: x in rng do by A90, A133, FINSEQ_6:42;
 x in rng pion1 by A135, REVROT_1:3;
hence  x in (L~ do) /\ (L~ pion1) by A65, A124, A136, XBOOLE_0:def_4; ::_thesis: verum
end;
 (L~ do) /\ (L~ pion1) c= {(pion1 /. (len pion1))}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ do) /\ (L~ pion1) or x in {(pion1 /. (len pion1))} )
assume A137: x in (L~ do) /\ (L~ pion1) ; ::_thesis: x in {(pion1 /. (len pion1))}
then A138: x in L~ do by XBOOLE_0:def_4;
 x in L~ pion1 by A137, XBOOLE_0:def_4;
hence  x in {(pion1 /. (len pion1))} by A8, A12, A45, A56, A88, A90, A133, A138, XBOOLE_0:def_4; ::_thesis: verum
end;
then A139: (L~ do) /\ (L~ pion1) = {(pion1 /. (len pion1))} by A134, XBOOLE_0:def_10;
A140: (L~ (go ^' pion1)) /\ (L~ do) = ((L~ go) \/ (L~ pion1)) /\ (L~ do) by A104, TOPREAL8:35
.= {(go /. 1)} \/ {(do /. 1)} by A76, A90, A133, A139, XBOOLE_1:23
.= {((go ^' pion1) /. 1)} \/ {(do /. 1)} by GRAPH_2:53
.= {((go ^' pion1) /. 1),(do /. 1)} by ENUMSET1:1 ;
 do /. (len do) = (go ^' pion1) /. 1 by A63, GRAPH_2:53;
then reconsider godo = (go ^' pion1) ^' do as non constant standard special_circular_sequence by A102, A106, A107, A112, A114, A122, A123, A131, A132, A140, JORDAN8:4, JORDAN8:5, TOPREAL8:11, TOPREAL8:33, TOPREAL8:34;
A141: Lower_Arc C is_an_arc_of E-max C, W-min C by JORDAN6:def_9;
then A142: Lower_Arc C is connected by JORDAN6:10;
A143: W-min C in Lower_Arc C by A141, TOPREAL1:1;
A144: E-max C in Lower_Arc C by A141, TOPREAL1:1;
set ff = Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))));
 W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
then A145: (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 = W-min (L~ (Cage (C,n))) by FINSEQ_6:92;
A146: L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33;
then  (W-max (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A145, SPRECT_5:22;
then  (N-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A145, A146, SPRECT_5:23, XXREAL_0:2;
then  (N-max (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A145, A146, SPRECT_5:24, XXREAL_0:2;
then A147: (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A145, A146, SPRECT_5:25, XXREAL_0:2;
A148: now__::_thesis:_not_((Gauge_(C,n))_*_(i1,j))_.._(Upper_Seq_(C,n))_<=_1
assume A149: ((Gauge (C,n)) * (i1,j)) .. (Upper_Seq (C,n)) <= 1 ; ::_thesis: contradiction
 ((Gauge (C,n)) * (i1,j)) .. (Upper_Seq (C,n)) >= 1 by A37, FINSEQ_4:21;
then  ((Gauge (C,n)) * (i1,j)) .. (Upper_Seq (C,n)) = 1 by A149, XXREAL_0:1;
then  (Gauge (C,n)) * (i1,j) = (Upper_Seq (C,n)) /. 1 by A37, FINSEQ_5:38;
hence  contradiction by A22, A26, JORDAN1F:5; ::_thesis: verum
end;
A150: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def_1;
then A151: Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))) is_sequence_on Gauge (C,n) by REVROT_1:34;
A152: (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) c= RightComp godo by A102, A107, JORDAN9:27;
A153: L~ godo = (L~ (go ^' pion1)) \/ (L~ do) by A106, TOPREAL8:35
.= ((L~ go) \/ (L~ pion1)) \/ (L~ do) by A104, TOPREAL8:35 ;
A154: L~ (Cage (C,n)) = (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) by JORDAN1E:13;
then A155: L~ (Upper_Seq (C,n)) c= L~ (Cage (C,n)) by XBOOLE_1:7;
A156: L~ (Lower_Seq (C,n)) c= L~ (Cage (C,n)) by A154, XBOOLE_1:7;
A157: L~ go c= L~ (Cage (C,n)) by A49, A155, XBOOLE_1:1;
A158: L~ do c= L~ (Cage (C,n)) by A56, A156, XBOOLE_1:1;
A159: W-min C in C by SPRECT_1:13;
A160: now__::_thesis:_not_W-min_C_in_L~_godo
assume  W-min C in L~ godo ; ::_thesis: contradiction
then A161: ( W-min C in (L~ go) \/ (L~ pion1) or W-min C in L~ do ) by A153, XBOOLE_0:def_3;
percases ( W-min C in L~ go or W-min C in L~ pion1 or W-min C in L~ do ) by A161, XBOOLE_0:def_3;
suppose W-min C in L~ go ; ::_thesis: contradiction
then  C meets L~ (Cage (C,n)) by A157, A159, XBOOLE_0:3;
hence  contradiction by JORDAN10:5; ::_thesis: verum
end;
suppose W-min C in L~ pion1 ; ::_thesis: contradiction
hence  contradiction by A9, A12, A88, A143, XBOOLE_0:3; ::_thesis: verum
end;
suppose W-min C in L~ do ; ::_thesis: contradiction
then  C meets L~ (Cage (C,n)) by A158, A159, XBOOLE_0:3;
hence  contradiction by JORDAN10:5; ::_thesis: verum
end;
end;
end;
right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1) = right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(GoB (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) by A98, JORDAN1H:23
.= right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(GoB (Cage (C,n)))) by REVROT_1:28
.= right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(Gauge (C,n))) by JORDAN1H:44
.= right_cell (((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n))))),1,(Gauge (C,n))) by A147, A151, JORDAN1J:53
.= right_cell ((Upper_Seq (C,n)),1,(Gauge (C,n))) by JORDAN1E:def_1
.= right_cell ((R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i1,j)))),1,(Gauge (C,n))) by A37, A103, A148, JORDAN1J:52
.= right_cell ((go ^' pion1),1,(Gauge (C,n))) by A42, A105, JORDAN1J:51
.= right_cell (godo,1,(Gauge (C,n))) by A100, A107, JORDAN1J:51 ;
then  W-min C in right_cell (godo,1,(Gauge (C,n))) by JORDAN1I:6;
then A162: W-min C in (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) by A160, XBOOLE_0:def_5;
A163: godo /. 1 = (go ^' pion1) /. 1 by GRAPH_2:53
.= W-min (L~ (Cage (C,n))) by A62, GRAPH_2:53 ;
A164: len (Upper_Seq (C,n)) >= 2 by A21, XXREAL_0:2;
A165: godo /. 2 = (go ^' pion1) /. 2 by A99, GRAPH_2:57
.= (Upper_Seq (C,n)) /. 2 by A36, A79, GRAPH_2:57
.= ((Upper_Seq (C,n)) ^' (Lower_Seq (C,n))) /. 2 by A164, GRAPH_2:57
.= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 2 by JORDAN1E:11 ;
A166: (L~ go) \/ (L~ do) is compact by COMPTS_1:10;
 W-min (L~ (Cage (C,n))) in (L~ go) \/ (L~ do) by A64, A80, XBOOLE_0:def_3;
then A167: W-min ((L~ go) \/ (L~ do)) = W-min (L~ (Cage (C,n))) by A157, A158, A166, JORDAN1J:21, XBOOLE_1:8;
A168: (W-min ((L~ go) \/ (L~ do))) `1 = W-bound ((L~ go) \/ (L~ do)) by EUCLID:52;
A169: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52;
 ((Gauge (C,n)) * (i1,j)) `1 >= W-bound (L~ (Cage (C,n))) by A14, A155, PSCOMP_1:24;
then  ((Gauge (C,n)) * (i1,j)) `1 > W-bound (L~ (Cage (C,n))) by A78, XXREAL_0:1;
then  W-min (((L~ go) \/ (L~ do)) \/ (L~ pion1)) = W-min ((L~ go) \/ (L~ do)) by A96, A166, A167, A168, A169, JORDAN1J:33;
then A170: W-min (L~ godo) = W-min (L~ (Cage (C,n))) by A153, A167, XBOOLE_1:4;
A171: rng godo c= L~ godo by A99, A101, SPPOL_2:18, XXREAL_0:2;
 2 in dom godo by A102, FINSEQ_3:25;
then A172: godo /. 2 in rng godo by PARTFUN2:2;
 godo /. 2 in W-most (L~ (Cage (C,n))) by A165, JORDAN1I:25;
then (godo /. 2) `1 = (W-min (L~ godo)) `1 by A170, PSCOMP_1:31
.= W-bound (L~ godo) by EUCLID:52 ;
then  godo /. 2 in W-most (L~ godo) by A171, A172, SPRECT_2:12;
then  (Rotate (godo,(W-min (L~ godo)))) /. 2 in W-most (L~ godo) by A163, A170, FINSEQ_6:89;
then reconsider godo = godo as non constant standard clockwise_oriented special_circular_sequence by JORDAN1I:25;
 len (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by FINSEQ_5:6;
then A173: (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) = (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) by PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:7 ;
A174: east_halfline (E-max C) misses L~ go
proof
assume  east_halfline (E-max C) meets L~ go ; ::_thesis: contradiction
then consider p being set  such that
A175: p in east_halfline (E-max C) and
A176: p in L~ go by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A175;
 p in L~ (Upper_Seq (C,n)) by A49, A176;
then  p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A155, A175, XBOOLE_0:def_4;
then A177: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;
then A178: p = E-max (L~ (Cage (C,n))) by A49, A176, JORDAN1J:46;
then  E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i1,j) by A14, A173, A176, JORDAN1J:43;
then  ((Gauge (C,n)) * (i1,j)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A6, A16, A20, A177, A178, JORDAN1A:71;
hence  contradiction by A2, A3, A17, A33, JORDAN1G:7; ::_thesis: verum
end;
now__::_thesis:_not_east_halfline_(E-max_C)_meets_L~_godo
assume  east_halfline (E-max C) meets L~ godo ; ::_thesis: contradiction
then A179: ( east_halfline (E-max C) meets (L~ go) \/ (L~ pion1) or east_halfline (E-max C) meets L~ do ) by A153, XBOOLE_1:70;
percases ( east_halfline (E-max C) meets L~ go or east_halfline (E-max C) meets L~ pion1 or east_halfline (E-max C) meets L~ do ) by A179, XBOOLE_1:70;
suppose east_halfline (E-max C) meets L~ go ; ::_thesis: contradiction
hence  contradiction by A174; ::_thesis: verum
end;
suppose east_halfline (E-max C) meets L~ pion1 ; ::_thesis: contradiction
then consider p being set  such that
A180: p in east_halfline (E-max C) and
A181: p in L~ pion1 by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A180;
A182: now__::_thesis:_p_`1_<=_((Gauge_(C,n))_*_(i2,k))_`1
percases ( p in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) or p in LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) ) by A12, A88, A181, XBOOLE_0:def_3;
suppose p in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) ; ::_thesis: p `1 <= ((Gauge (C,n)) * (i2,k)) `1
hence  p `1 <= ((Gauge (C,n)) * (i2,k)) `1 by A93, TOPREAL1:3; ::_thesis: verum
end;
suppose p in LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) ; ::_thesis: p `1 <= ((Gauge (C,n)) * (i2,k)) `1
hence  p `1 <= ((Gauge (C,n)) * (i2,k)) `1 by A92, A93, GOBOARD7:5; ::_thesis: verum
end;
end;
end;
 i2 + 1 <= len (Gauge (C,n)) by A3, NAT_1:13;
then  i2 <= (len (Gauge (C,n))) - 1 by XREAL_1:19;
then A183: i2 <= (len (Gauge (C,n))) -' 1 by XREAL_0:def_2;
 (len (Gauge (C,n))) -' 1 <= len (Gauge (C,n)) by NAT_D:35;
then  ((Gauge (C,n)) * (i2,k)) `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1 by A6, A11, A16, A20, A24, A183, JORDAN1A:18;
then  p `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1 by A182, XXREAL_0:2;
then  p `1 <= E-bound C by A24, JORDAN8:12;
then A184: p `1 <= (E-max C) `1 by EUCLID:52;
 p `1 >= (E-max C) `1 by A180, TOPREAL1:def_11;
then A185: p `1 = (E-max C) `1 by A184, XXREAL_0:1;
 p `2 = (E-max C) `2 by A180, TOPREAL1:def_11;
then  p = E-max C by A185, TOPREAL3:6;
hence  contradiction by A9, A12, A88, A144, A181, XBOOLE_0:3; ::_thesis: verum
end;
suppose east_halfline (E-max C) meets L~ do ; ::_thesis: contradiction
then consider p being set  such that
A186: p in east_halfline (E-max C) and
A187: p in L~ do by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A186;
 p in L~ (Lower_Seq (C,n)) by A56, A187;
then  p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A156, A186, XBOOLE_0:def_4;
then A188: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;
A189: (E-max C) `2 = p `2 by A186, TOPREAL1:def_11;
set RC = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))));
A190: E-max C in right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by JORDAN1I:7;
A191: 1 + 1 <= len (Lower_Seq (C,n)) by A28, XXREAL_0:2;
 Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) by JORDAN1G:18;
then A192: LSeg ((Lower_Seq (C,n)),1) = LSeg ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by A191, SPPOL_2:9;
A193: L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33;
A194: len (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = len (Cage (C,n)) by REVROT_1:14;
A195: GoB (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = GoB (Cage (C,n)) by REVROT_1:28
.= Gauge (C,n) by JORDAN1H:44 ;
A196: E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
A197: Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))) is_sequence_on Gauge (C,n) by A150, REVROT_1:34;
A198: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = E-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))))) by A193, A196, FINSEQ_6:92;
consider ii, jj being Element of NAT  such that
A199: [ii,(jj + 1)] in Indices (Gauge (C,n)) and
A200: [ii,jj] in Indices (Gauge (C,n)) and
A201: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = (Gauge (C,n)) * (ii,(jj + 1)) and
A202: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1) = (Gauge (C,n)) * (ii,jj) by A97, A193, A194, A196, A197, FINSEQ_6:92, JORDAN1I:23;
consider jj2 being Element of NAT  such that
A203: 1 <= jj2 and
A204: jj2 <= width (Gauge (C,n)) and
A205: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * ((len (Gauge (C,n))),jj2) by JORDAN1D:25;
A206: len (Gauge (C,n)) >= 4 by JORDAN8:10;
then  len (Gauge (C,n)) >= 1 by XXREAL_0:2;
then  [(len (Gauge (C,n))),jj2] in Indices (Gauge (C,n)) by A203, A204, MATRIX_1:36;
then A207: ii = len (Gauge (C,n)) by A193, A198, A199, A201, A205, GOBOARD1:5;
A208: 1 <= ii by A199, MATRIX_1:38;
A209: ii <= len (Gauge (C,n)) by A199, MATRIX_1:38;
A210: 1 <= jj + 1 by A199, MATRIX_1:38;
A211: jj + 1 <= width (Gauge (C,n)) by A199, MATRIX_1:38;
A212: 1 <= ii by A200, MATRIX_1:38;
A213: ii <= len (Gauge (C,n)) by A200, MATRIX_1:38;
A214: 1 <= jj by A200, MATRIX_1:38;
A215: jj <= width (Gauge (C,n)) by A200, MATRIX_1:38;
A216: ii + 1 <> ii ;
 (jj + 1) + 1 <> jj ;
then A217: right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) = cell ((Gauge (C,n)),(ii -' 1),jj) by A97, A194, A195, A199, A200, A201, A202, A216, GOBOARD5:def_6;
A218: (ii -' 1) + 1 = ii by A208, XREAL_1:235;
 ii - 1 >= 4 - 1 by A206, A207, XREAL_1:9;
then A219: ii - 1 >= 1 by XXREAL_0:2;
then A220: 1 <= ii -' 1 by XREAL_0:def_2;
A221: ((Gauge (C,n)) * ((ii -' 1),jj)) `2 <= p `2 by A189, A190, A209, A211, A214, A217, A218, A219, JORDAN9:17;
A222: p `2 <= ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 by A189, A190, A209, A211, A214, A217, A218, A219, JORDAN9:17;
A223: ii -' 1 < len (Gauge (C,n)) by A209, A218, NAT_1:13;
then A224: ((Gauge (C,n)) * ((ii -' 1),jj)) `2 = ((Gauge (C,n)) * (1,jj)) `2 by A214, A215, A220, GOBOARD5:1
.= ((Gauge (C,n)) * (ii,jj)) `2 by A212, A213, A214, A215, GOBOARD5:1 ;
A225: ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 = ((Gauge (C,n)) * (1,(jj + 1))) `2 by A210, A211, A220, A223, GOBOARD5:1
.= ((Gauge (C,n)) * (ii,(jj + 1))) `2 by A208, A209, A210, A211, GOBOARD5:1 ;
A226: ((Gauge (C,n)) * ((len (Gauge (C,n))),jj)) `1 = E-bound (L~ (Cage (C,n))) by A20, A214, A215, JORDAN1A:71;
 E-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * ((len (Gauge (C,n))),(jj + 1))) `1 by A20, A210, A211, JORDAN1A:71;
then  p in LSeg (((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1),((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1))) by A188, A201, A202, A207, A221, A222, A224, A225, A226, GOBOARD7:7;
then A227: p in LSeg ((Lower_Seq (C,n)),1) by A97, A192, A194, TOPREAL1:def_3;
A228: p in LSeg (do,(Index (p,do))) by A187, JORDAN3:9;
A229: do = mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n)))) by A40, JORDAN1J:37;
A230: 1 <= ((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)) by A40, FINSEQ_4:21;
A231: ((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)) <= len (Lower_Seq (C,n)) by A40, FINSEQ_4:21;
 ((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)) <> len (Lower_Seq (C,n)) by A32, A40, FINSEQ_4:19;
then A232: ((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)) < len (Lower_Seq (C,n)) by A231, XXREAL_0:1;
A233: 1 <= Index (p,do) by A187, JORDAN3:8;
A234: Index (p,do) < len do by A187, JORDAN3:8;
A235: (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n)))) + 1 = ((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)) by A35, A40, JORDAN1J:56;
consider t being Nat such that
A236: t in dom (Lower_Seq (C,n)) and
A237: (Lower_Seq (C,n)) . t = (Gauge (C,n)) * (i2,k) by A40, FINSEQ_2:10;
A238: 1 <= t by A236, FINSEQ_3:25;
A239: t <= len (Lower_Seq (C,n)) by A236, FINSEQ_3:25;
 1 < t by A35, A237, A238, XXREAL_0:1;
then  (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n)))) + 1 = t by A237, A239, JORDAN3:12;
then A240: len (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i2,k)))) = (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n)))) by A13, A237, JORDAN3:26;
set tt = ((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1;
A241: 1 <= Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n))) by A13, JORDAN3:8;
 0 + (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n)))) < len (Lower_Seq (C,n)) by A13, JORDAN3:8;
then A242: (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n)))) > 0 by XREAL_1:20;
 Index (p,do) < (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n)))) by A234, A240, XREAL_0:def_2;
then  (Index (p,do)) + 1 <= (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n)))) by NAT_1:13;
then  Index (p,do) <= ((len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n))))) - 1 by XREAL_1:19;
then  Index (p,do) <= ((len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n))))) - 1 by A242, XREAL_0:def_2;
then  Index (p,do) <= (len (Lower_Seq (C,n))) - (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n))) by A235;
then  Index (p,do) <= (len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n))) by XREAL_0:def_2;
then  Index (p,do) < ((len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) + 1 by NAT_1:13;
then A243: LSeg ((mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n))))),(Index (p,do))) = LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1)) by A230, A232, A233, JORDAN4:19;
A244: 1 + 1 <= ((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)) by A235, A241, XREAL_1:7;
then  (Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n))) >= (1 + 1) + 1 by A233, XREAL_1:7;
then  ((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) - 1 >= ((1 + 1) + 1) - 1 by XREAL_1:9;
then A245: ((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1 >= 1 + 1 by XREAL_0:def_2;
A246: 2 in dom (Lower_Seq (C,n)) by A191, FINSEQ_3:25;
now__::_thesis:_contradiction
percases ( ((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1 > 1 + 1 or ((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ) by A245, XXREAL_0:1;
suppose ((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1 > 1 + 1 ; ::_thesis: contradiction
then  LSeg ((Lower_Seq (C,n)),1) misses LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1)) by TOPREAL1:def_7;
hence  contradiction by A227, A228, A229, A243, XBOOLE_0:3; ::_thesis: verum
end;
supposeA247: ((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ; ::_thesis: contradiction
then  (LSeg ((Lower_Seq (C,n)),1)) /\ (LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1))) = {((Lower_Seq (C,n)) /. 2)} by A28, TOPREAL1:def_6;
then  p in {((Lower_Seq (C,n)) /. 2)} by A227, A228, A229, A243, XBOOLE_0:def_4;
then A248: p = (Lower_Seq (C,n)) /. 2 by TARSKI:def_1;
then A249: p .. (Lower_Seq (C,n)) = 2 by A246, FINSEQ_5:41;
 1 + 1 = ((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) - 1 by A247, XREAL_0:def_2;
then  (1 + 1) + 1 = (Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n))) ;
then A250: ((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)) = 2 by A233, A244, JORDAN1E:6;
 p in rng (Lower_Seq (C,n)) by A246, A248, PARTFUN2:2;
then  p = (Gauge (C,n)) * (i2,k) by A40, A249, A250, FINSEQ_5:9;
then  ((Gauge (C,n)) * (i2,k)) `1 = E-bound (L~ (Cage (C,n))) by A248, JORDAN1G:32;
then  ((Gauge (C,n)) * (i2,k)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 by A4, A15, A20, JORDAN1A:71;
hence  contradiction by A3, A18, A70, JORDAN1G:7; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
end;
end;
then  east_halfline (E-max C) c= (L~ godo) ` by SUBSET_1:23;
then consider W being Subset of (TOP-REAL 2) such that
A251: W is_a_component_of (L~ godo) ` and
A252: east_halfline (E-max C) c= W by GOBOARD9:3;
 not W is bounded by A252, JORDAN2C:121, RLTOPSP1:42;
then  W is_outside_component_of L~ godo by A251, JORDAN2C:def_3;
then  W c= UBD (L~ godo) by JORDAN2C:23;
then A253: east_halfline (E-max C) c= UBD (L~ godo) by A252, XBOOLE_1:1;
 E-max C in east_halfline (E-max C) by TOPREAL1:38;
then  E-max C in UBD (L~ godo) by A253;
then  E-max C in LeftComp godo by GOBRD14:36;
then  Lower_Arc C meets L~ godo by A142, A143, A144, A152, A162, JORDAN1J:36;
then A254: ( Lower_Arc C meets (L~ go) \/ (L~ pion1) or Lower_Arc C meets L~ do ) by A153, XBOOLE_1:70;
A255: Lower_Arc C c= C by JORDAN6:61;
now__::_thesis:_contradiction
percases ( Lower_Arc C meets L~ go or Lower_Arc C meets L~ pion1 or Lower_Arc C meets L~ do ) by A254, XBOOLE_1:70;
suppose Lower_Arc C meets L~ go ; ::_thesis: contradiction
then  Lower_Arc C meets L~ (Cage (C,n)) by A49, A155, XBOOLE_1:1, XBOOLE_1:63;
hence  contradiction by A255, JORDAN10:5, XBOOLE_1:63; ::_thesis: verum
end;
suppose Lower_Arc C meets L~ pion1 ; ::_thesis: contradiction
hence  contradiction by A9, A12, A88; ::_thesis: verum
end;
suppose Lower_Arc C meets L~ do ; ::_thesis: contradiction
then  Lower_Arc C meets L~ (Cage (C,n)) by A56, A156, XBOOLE_1:1, XBOOLE_1:63;
hence  contradiction by A255, JORDAN10:5, XBOOLE_1:63; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
suppose ((Gauge (C,n)) * (i2,k)) `1 = ((Gauge (C,n)) * (i1,j)) `1 ; ::_thesis: contradiction
then A256: i1 = i2 by A17, A18, JORDAN1G:7;
then  LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) = {((Gauge (C,n)) * (i1,k))} by RLTOPSP1:70;
then  LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) c= LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) by A84, ZFMISC_1:31;
then  (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) = LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) by XBOOLE_1:12;
hence  contradiction by A1, A3, A4, A5, A6, A7, A8, A9, A256, Th13; ::_thesis: verum
end;
suppose ((Gauge (C,n)) * (i2,k)) `2 = ((Gauge (C,n)) * (i1,j)) `2 ; ::_thesis: contradiction
then A257: j = k by A17, A18, JORDAN1G:6;
then  LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) = {((Gauge (C,n)) * (i1,k))} by RLTOPSP1:70;
then  LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) c= LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) by A85, ZFMISC_1:31;
then  (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) = LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) by XBOOLE_1:12;
hence  contradiction by A1, A2, A3, A4, A6, A7, A8, A9, A257, JORDAN15:36; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;

theorem Th22: :: JORDAN19:22
for n being Element of NAT
for C being Simple_closed_curve
for i1, i2, j, k being Element of NAT st 1 < i2 & i2 <= i1 & i1 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} holds
(LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Upper_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i1, i2, j, k being Element of NAT st 1 < i2 & i2 <= i1 & i1 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} holds
(LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Upper_Arc C
let C be Simple_closed_curve; ::_thesis: for i1, i2, j, k being Element of NAT st 1 < i2 & i2 <= i1 & i1 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} holds
(LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Upper_Arc C
let i1, i2, j, k be Element of NAT ; ::_thesis: ( 1 < i2 & i2 <= i1 & i1 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} implies (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Upper_Arc C )
set G = Gauge (C,n);
set pio = LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)));
set poz = LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)));
set US = Upper_Seq (C,n);
set LS = Lower_Seq (C,n);
assume that
A1: 1 < i2 and
A2: i2 <= i1 and
A3: i1 < len (Gauge (C,n)) and
A4: 1 <= j and
A5: j <= k and
A6: k <= width (Gauge (C,n)) and
A7: ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} and
A8: ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} and
A9: (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) misses Upper_Arc C ; ::_thesis: contradiction
set UA = Upper_Arc C;
set Wmin = W-min (L~ (Cage (C,n)));
set Emax = E-max (L~ (Cage (C,n)));
set Wbo = W-bound (L~ (Cage (C,n)));
set Ebo = E-bound (L~ (Cage (C,n)));
set Gik = (Gauge (C,n)) * (i2,k);
set Gij = (Gauge (C,n)) * (i1,j);
set Gi1k = (Gauge (C,n)) * (i1,k);
A10: 1 < i1 by A1, A2, XXREAL_0:2;
A11: i2 < len (Gauge (C,n)) by A2, A3, XXREAL_0:2;
A12: L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> = (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) by TOPREAL3:16;
 (Gauge (C,n)) * (i2,k) in {((Gauge (C,n)) * (i2,k))} by TARSKI:def_1;
then A13: (Gauge (C,n)) * (i2,k) in L~ (Lower_Seq (C,n)) by A8, XBOOLE_0:def_4;
 (Gauge (C,n)) * (i1,j) in {((Gauge (C,n)) * (i1,j))} by TARSKI:def_1;
then A14: (Gauge (C,n)) * (i1,j) in L~ (Upper_Seq (C,n)) by A7, XBOOLE_0:def_4;
A15: j <= width (Gauge (C,n)) by A5, A6, XXREAL_0:2;
A16: 1 <= k by A4, A5, XXREAL_0:2;
A17: [i1,j] in Indices (Gauge (C,n)) by A3, A4, A10, A15, MATRIX_1:36;
A18: [i2,k] in Indices (Gauge (C,n)) by A1, A6, A11, A16, MATRIX_1:36;
A19: [i1,k] in Indices (Gauge (C,n)) by A3, A6, A10, A16, MATRIX_1:36;
set go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i1,j)));
set do = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i2,k)));
A20: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
A21: len (Upper_Seq (C,n)) >= 3 by JORDAN1E:15;
then  len (Upper_Seq (C,n)) >= 1 by XXREAL_0:2;
then  1 in dom (Upper_Seq (C,n)) by FINSEQ_3:25;
then A22: (Upper_Seq (C,n)) . 1 = (Upper_Seq (C,n)) /. 1 by PARTFUN1:def_6
.= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ;
A23: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * (1,k)) `1 by A6, A16, A20, JORDAN1A:73 ;
 len (Gauge (C,n)) >= 4 by JORDAN8:10;
then A24: len (Gauge (C,n)) >= 1 by XXREAL_0:2;
then A25: [1,k] in Indices (Gauge (C,n)) by A6, A16, MATRIX_1:36;
then A26: (Gauge (C,n)) * (i1,j) <> (Upper_Seq (C,n)) . 1 by A1, A2, A17, A22, A23, JORDAN1G:7;
then reconsider go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i1,j))) as being_S-Seq FinSequence of (TOP-REAL 2) by A14, JORDAN3:35;
A27: len (Lower_Seq (C,n)) >= 1 + 2 by JORDAN1E:15;
then A28: len (Lower_Seq (C,n)) >= 1 by XXREAL_0:2;
then A29: 1 in dom (Lower_Seq (C,n)) by FINSEQ_3:25;
 len (Lower_Seq (C,n)) in dom (Lower_Seq (C,n)) by A28, FINSEQ_3:25;
then A30: (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by PARTFUN1:def_6
.= W-min (L~ (Cage (C,n))) by JORDAN1F:8 ;
(W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * (1,k)) `1 by A6, A16, A20, JORDAN1A:73 ;
then A31: (Gauge (C,n)) * (i2,k) <> (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) by A1, A18, A25, A30, JORDAN1G:7;
then reconsider do = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i2,k))) as being_S-Seq FinSequence of (TOP-REAL 2) by A13, JORDAN3:34;
A32: [(len (Gauge (C,n))),k] in Indices (Gauge (C,n)) by A6, A16, A24, MATRIX_1:36;
A33: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by A29, PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;
(E-max (L~ (Cage (C,n)))) `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A6, A16, A20, JORDAN1A:71 ;
then A34: (Gauge (C,n)) * (i2,k) <> (Lower_Seq (C,n)) . 1 by A2, A3, A18, A32, A33, JORDAN1G:7;
A35: len go >= 1 + 1 by TOPREAL1:def_8;
A36: (Gauge (C,n)) * (i1,j) in rng (Upper_Seq (C,n)) by A3, A4, A10, A14, A15, JORDAN1G:4, JORDAN1J:40;
then A37: go is_sequence_on Gauge (C,n) by JORDAN1G:4, JORDAN1J:38;
A38: len do >= 1 + 1 by TOPREAL1:def_8;
A39: (Gauge (C,n)) * (i2,k) in rng (Lower_Seq (C,n)) by A1, A6, A11, A13, A16, JORDAN1G:5, JORDAN1J:40;
then A40: do is_sequence_on Gauge (C,n) by JORDAN1G:5, JORDAN1J:39;
reconsider go = go as non constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A35, A37, JGRAPH_1:12, JORDAN8:5;
reconsider do = do as non constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A38, A40, JGRAPH_1:12, JORDAN8:5;
A41: len go > 1 by A35, NAT_1:13;
then A42: len go in dom go by FINSEQ_3:25;
then A43: go /. (len go) = go . (len go) by PARTFUN1:def_6
.= (Gauge (C,n)) * (i1,j) by A14, JORDAN3:24 ;
 len do >= 1 by A38, XXREAL_0:2;
then  1 in dom do by FINSEQ_3:25;
then A44: do /. 1 = do . 1 by PARTFUN1:def_6
.= (Gauge (C,n)) * (i2,k) by A13, JORDAN3:23 ;
reconsider m = (len go) - 1 as Element of NAT by A42, FINSEQ_3:26;
A45: m + 1 = len go ;
then A46: (len go) -' 1 = m by NAT_D:34;
A47: LSeg (go,m) c= L~ go by TOPREAL3:19;
A48: L~ go c= L~ (Upper_Seq (C,n)) by A14, JORDAN3:41;
then  LSeg (go,m) c= L~ (Upper_Seq (C,n)) by A47, XBOOLE_1:1;
then A49: (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) c= {((Gauge (C,n)) * (i1,j))} by A7, A12, XBOOLE_1:26;
 m >= 1 by A35, XREAL_1:19;
then A50: LSeg (go,m) = LSeg ((go /. m),((Gauge (C,n)) * (i1,j))) by A43, A45, TOPREAL1:def_3;
 {((Gauge (C,n)) * (i1,j))} c= (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i1,j))} or x in (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) )
assume  x in {((Gauge (C,n)) * (i1,j))} ; ::_thesis: x in (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>)
then A51: x = (Gauge (C,n)) * (i1,j) by TARSKI:def_1;
A52: (Gauge (C,n)) * (i1,j) in LSeg (go,m) by A50, RLTOPSP1:68;
 (Gauge (C,n)) * (i1,j) in LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) by RLTOPSP1:68;
then  (Gauge (C,n)) * (i1,j) in (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) by XBOOLE_0:def_3;
then  (Gauge (C,n)) * (i1,j) in L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> by SPRECT_1:8;
hence  x in (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) by A51, A52, XBOOLE_0:def_4; ::_thesis: verum
end;
then A53: (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) = {((Gauge (C,n)) * (i1,j))} by A49, XBOOLE_0:def_10;
A54: LSeg (do,1) c= L~ do by TOPREAL3:19;
A55: L~ do c= L~ (Lower_Seq (C,n)) by A13, JORDAN3:42;
then  LSeg (do,1) c= L~ (Lower_Seq (C,n)) by A54, XBOOLE_1:1;
then A56: (LSeg (do,1)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) c= {((Gauge (C,n)) * (i2,k))} by A8, A12, XBOOLE_1:26;
A57: LSeg (do,1) = LSeg (((Gauge (C,n)) * (i2,k)),(do /. (1 + 1))) by A38, A44, TOPREAL1:def_3;
 {((Gauge (C,n)) * (i2,k))} c= (LSeg (do,1)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i2,k))} or x in (LSeg (do,1)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) )
assume  x in {((Gauge (C,n)) * (i2,k))} ; ::_thesis: x in (LSeg (do,1)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>)
then A58: x = (Gauge (C,n)) * (i2,k) by TARSKI:def_1;
A59: (Gauge (C,n)) * (i2,k) in LSeg (do,1) by A57, RLTOPSP1:68;
 (Gauge (C,n)) * (i2,k) in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) by RLTOPSP1:68;
then  (Gauge (C,n)) * (i2,k) in (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) by XBOOLE_0:def_3;
then  (Gauge (C,n)) * (i2,k) in L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> by SPRECT_1:8;
hence  x in (LSeg (do,1)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) by A58, A59, XBOOLE_0:def_4; ::_thesis: verum
end;
then A60: (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) /\ (LSeg (do,1)) = {((Gauge (C,n)) * (i2,k))} by A56, XBOOLE_0:def_10;
A61: go /. 1 = (Upper_Seq (C,n)) /. 1 by A14, SPRECT_3:22
.= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ;
then A62: go /. 1 = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by JORDAN1F:8
.= do /. (len do) by A13, JORDAN1J:35 ;
A63: rng go c= L~ go by A35, SPPOL_2:18;
A64: rng do c= L~ do by A38, SPPOL_2:18;
A65: {(go /. 1)} c= (L~ go) /\ (L~ do)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(go /. 1)} or x in (L~ go) /\ (L~ do) )
assume  x in {(go /. 1)} ; ::_thesis: x in (L~ go) /\ (L~ do)
then A66: x = go /. 1 by TARSKI:def_1;
then A67: x in rng go by FINSEQ_6:42;
 x in rng do by A62, A66, REVROT_1:3;
hence  x in (L~ go) /\ (L~ do) by A63, A64, A67, XBOOLE_0:def_4; ::_thesis: verum
end;
A68: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by A29, PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;
A69: [(len (Gauge (C,n))),j] in Indices (Gauge (C,n)) by A4, A15, A24, MATRIX_1:36;
 (L~ go) /\ (L~ do) c= {(go /. 1)}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ go) /\ (L~ do) or x in {(go /. 1)} )
assume A70: x in (L~ go) /\ (L~ do) ; ::_thesis: x in {(go /. 1)}
then A71: x in L~ go by XBOOLE_0:def_4;
A72: x in L~ do by A70, XBOOLE_0:def_4;
then  x in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A48, A55, A71, XBOOLE_0:def_4;
then  x in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16;
then A73: ( x = W-min (L~ (Cage (C,n))) or x = E-max (L~ (Cage (C,n))) ) by TARSKI:def_2;
now__::_thesis:_not_x_=_E-max_(L~_(Cage_(C,n)))
assume  x = E-max (L~ (Cage (C,n))) ; ::_thesis: contradiction
then A74: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i2,k) by A13, A68, A72, JORDAN1E:7;
 ((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 = E-bound (L~ (Cage (C,n))) by A4, A15, A20, JORDAN1A:71;
then  (E-max (L~ (Cage (C,n)))) `1 <> E-bound (L~ (Cage (C,n))) by A2, A3, A18, A69, A74, JORDAN1G:7;
hence  contradiction by EUCLID:52; ::_thesis: verum
end;
hence  x in {(go /. 1)} by A61, A73, TARSKI:def_1; ::_thesis: verum
end;
then A75: (L~ go) /\ (L~ do) = {(go /. 1)} by A65, XBOOLE_0:def_10;
set W2 = go /. 2;
A76: 2 in dom go by A35, FINSEQ_3:25;
A77: now__::_thesis:_not_((Gauge_(C,n))_*_(i2,k))_`1_=_W-bound_(L~_(Cage_(C,n)))
assume  ((Gauge (C,n)) * (i2,k)) `1 = W-bound (L~ (Cage (C,n))) ; ::_thesis: contradiction
then  ((Gauge (C,n)) * (1,k)) `1 = ((Gauge (C,n)) * (i2,k)) `1 by A6, A16, A20, JORDAN1A:73;
hence  contradiction by A1, A18, A25, JORDAN1G:7; ::_thesis: verum
end;
go = mid ((Upper_Seq (C,n)),1,(((Gauge (C,n)) * (i1,j)) .. (Upper_Seq (C,n)))) by A36, JORDAN1G:49
.= (Upper_Seq (C,n)) | (((Gauge (C,n)) * (i1,j)) .. (Upper_Seq (C,n))) by A36, FINSEQ_4:21, FINSEQ_6:116 ;
then A78: go /. 2 = (Upper_Seq (C,n)) /. 2 by A76, FINSEQ_4:70;
A79: W-min (L~ (Cage (C,n))) in rng go by A61, FINSEQ_6:42;
set pion = <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>;
A80: now__::_thesis:_for_n_being_Element_of_NAT_st_n_in_dom_<*((Gauge_(C,n))_*_(i1,j)),((Gauge_(C,n))_*_(i1,k)),((Gauge_(C,n))_*_(i2,k))*>_holds_
ex_i,_j_being_Element_of_NAT_st_
(_[i,j]_in_Indices_(Gauge_(C,n))_&_<*((Gauge_(C,n))_*_(i1,j)),((Gauge_(C,n))_*_(i1,k)),((Gauge_(C,n))_*_(i2,k))*>_/._n_=_(Gauge_(C,n))_*_(i,j)_)
let n be Element of NAT ; ::_thesis: ( n in dom <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> implies ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. n = (Gauge (C,n)) * (i,j) ) )
assume  n in dom <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. n = (Gauge (C,n)) * (i,j) )
then  n in {1,2,3} by FINSEQ_1:89, FINSEQ_3:1;
then  ( n = 1 or n = 2 or n = 3 ) by ENUMSET1:def_1;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. n = (Gauge (C,n)) * (i,j) ) by A17, A18, A19, FINSEQ_4:18; ::_thesis: verum
end;
A81: ((Gauge (C,n)) * (i1,k)) `1 = ((Gauge (C,n)) * (i1,1)) `1 by A3, A6, A10, A16, GOBOARD5:2
.= ((Gauge (C,n)) * (i1,j)) `1 by A3, A4, A10, A15, GOBOARD5:2 ;
((Gauge (C,n)) * (i1,k)) `2 = ((Gauge (C,n)) * (1,k)) `2 by A3, A6, A10, A16, GOBOARD5:1
.= ((Gauge (C,n)) * (i2,k)) `2 by A1, A6, A11, A16, GOBOARD5:1 ;
then A82: (Gauge (C,n)) * (i1,k) = |[(((Gauge (C,n)) * (i1,j)) `1),(((Gauge (C,n)) * (i2,k)) `2)]| by A81, EUCLID:53;
A83: (Gauge (C,n)) * (i1,k) in LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) by RLTOPSP1:68;
A84: (Gauge (C,n)) * (i1,k) in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) by RLTOPSP1:68;
now__::_thesis:_contradiction
percases ( ( ((Gauge (C,n)) * (i2,k)) `1 <> ((Gauge (C,n)) * (i1,j)) `1 & ((Gauge (C,n)) * (i2,k)) `2 <> ((Gauge (C,n)) * (i1,j)) `2 ) or ((Gauge (C,n)) * (i2,k)) `1 = ((Gauge (C,n)) * (i1,j)) `1 or ((Gauge (C,n)) * (i2,k)) `2 = ((Gauge (C,n)) * (i1,j)) `2 ) ;
suppose ( ((Gauge (C,n)) * (i2,k)) `1 <> ((Gauge (C,n)) * (i1,j)) `1 & ((Gauge (C,n)) * (i2,k)) `2 <> ((Gauge (C,n)) * (i1,j)) `2 ) ; ::_thesis: contradiction
then  <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> is being_S-Seq by A82, TOPREAL3:34;
then consider pion1 being FinSequence of (TOP-REAL 2) such that
A85: pion1 is_sequence_on Gauge (C,n) and
A86: pion1 is being_S-Seq and
A87: L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> = L~ pion1 and
A88: <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. 1 = pion1 /. 1 and
A89: <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. (len <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) = pion1 /. (len pion1) and
A90: len <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> <= len pion1 by A80, GOBOARD3:2;
reconsider pion1 = pion1 as being_S-Seq FinSequence of (TOP-REAL 2) by A86;
set godo = (go ^' pion1) ^' do;
A91: ((Gauge (C,n)) * (i1,k)) `1 = ((Gauge (C,n)) * (i1,1)) `1 by A3, A6, A10, A16, GOBOARD5:2
.= ((Gauge (C,n)) * (i1,j)) `1 by A3, A4, A10, A15, GOBOARD5:2 ;
A92: ((Gauge (C,n)) * (i2,k)) `1 <= ((Gauge (C,n)) * (i1,k)) `1 by A1, A2, A3, A6, A16, JORDAN1A:18;
then A93: W-bound (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) = ((Gauge (C,n)) * (i2,k)) `1 by SPRECT_1:54;
A94: W-bound (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) = ((Gauge (C,n)) * (i1,j)) `1 by A91, SPRECT_1:54;
W-bound ((LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))))) = min ((W-bound (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))),(W-bound (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))))) by SPRECT_1:47
.= ((Gauge (C,n)) * (i2,k)) `1 by A91, A92, A93, A94, XXREAL_0:def_9 ;
then A95: W-bound (L~ pion1) = ((Gauge (C,n)) * (i2,k)) `1 by A87, TOPREAL3:16;
A96: 1 + 1 <= len (Cage (C,n)) by GOBOARD7:34, XXREAL_0:2;
A97: 1 + 1 <= len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by GOBOARD7:34, XXREAL_0:2;
 len (go ^' pion1) >= len go by TOPREAL8:7;
then A98: len (go ^' pion1) >= 1 + 1 by A35, XXREAL_0:2;
then A99: len (go ^' pion1) > 1 + 0 by NAT_1:13;
A100: len ((go ^' pion1) ^' do) >= len (go ^' pion1) by TOPREAL8:7;
then A101: 1 + 1 <= len ((go ^' pion1) ^' do) by A98, XXREAL_0:2;
A102: Upper_Seq (C,n) is_sequence_on Gauge (C,n) by JORDAN1G:4;
A103: go /. (len go) = pion1 /. 1 by A43, A88, FINSEQ_4:18;
then A104: go ^' pion1 is_sequence_on Gauge (C,n) by A37, A85, TOPREAL8:12;
A105: (go ^' pion1) /. (len (go ^' pion1)) = <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. (len <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) by A89, GRAPH_2:54
.= <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. 3 by FINSEQ_1:45
.= do /. 1 by A44, FINSEQ_4:18 ;
then A106: (go ^' pion1) ^' do is_sequence_on Gauge (C,n) by A40, A104, TOPREAL8:12;
 LSeg (pion1,1) c= L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> by A87, TOPREAL3:19;
then A107: (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) c= {((Gauge (C,n)) * (i1,j))} by A46, A53, XBOOLE_1:27;
 len pion1 >= 2 + 1 by A90, FINSEQ_1:45;
then A108: len pion1 > 1 + 1 by NAT_1:13;
 {((Gauge (C,n)) * (i1,j))} c= (LSeg (go,m)) /\ (LSeg (pion1,1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i1,j))} or x in (LSeg (go,m)) /\ (LSeg (pion1,1)) )
assume  x in {((Gauge (C,n)) * (i1,j))} ; ::_thesis: x in (LSeg (go,m)) /\ (LSeg (pion1,1))
then A109: x = (Gauge (C,n)) * (i1,j) by TARSKI:def_1;
A110: (Gauge (C,n)) * (i1,j) in LSeg (go,m) by A50, RLTOPSP1:68;
 (Gauge (C,n)) * (i1,j) in LSeg (pion1,1) by A43, A103, A108, TOPREAL1:21;
hence  x in (LSeg (go,m)) /\ (LSeg (pion1,1)) by A109, A110, XBOOLE_0:def_4; ::_thesis: verum
end;
then  (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) = {(go /. (len go))} by A43, A46, A107, XBOOLE_0:def_10;
then A111: go ^' pion1 is unfolded by A103, TOPREAL8:34;
 len pion1 >= 2 + 1 by A90, FINSEQ_1:45;
then A112: (len pion1) - 2 >= 0 by XREAL_1:19;
 ((len (go ^' pion1)) + 1) - 1 = ((len go) + (len pion1)) - 1 by GRAPH_2:13;
then (len (go ^' pion1)) - 1 = (len go) + ((len pion1) - 2)
.= (len go) + ((len pion1) -' 2) by A112, XREAL_0:def_2 ;
then A113: (len (go ^' pion1)) -' 1 = (len go) + ((len pion1) -' 2) by XREAL_0:def_2;
A114: (len pion1) - 1 >= 1 by A108, XREAL_1:19;
then A115: (len pion1) -' 1 = (len pion1) - 1 by XREAL_0:def_2;
A116: ((len pion1) -' 2) + 1 = ((len pion1) - 2) + 1 by A112, XREAL_0:def_2
.= (len pion1) -' 1 by A114, XREAL_0:def_2 ;
 ((len pion1) - 1) + 1 <= len pion1 ;
then A117: (len pion1) -' 1 < len pion1 by A115, NAT_1:13;
 LSeg (pion1,((len pion1) -' 1)) c= L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> by A87, TOPREAL3:19;
then A118: (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) c= {((Gauge (C,n)) * (i2,k))} by A60, XBOOLE_1:27;
 {((Gauge (C,n)) * (i2,k))} c= (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i2,k))} or x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) )
assume  x in {((Gauge (C,n)) * (i2,k))} ; ::_thesis: x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1))
then A119: x = (Gauge (C,n)) * (i2,k) by TARSKI:def_1;
A120: (Gauge (C,n)) * (i2,k) in LSeg (do,1) by A57, RLTOPSP1:68;
pion1 /. (((len pion1) -' 1) + 1) = <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. 3 by A89, A115, FINSEQ_1:45
.= (Gauge (C,n)) * (i2,k) by FINSEQ_4:18 ;
then  (Gauge (C,n)) * (i2,k) in LSeg (pion1,((len pion1) -' 1)) by A114, A115, TOPREAL1:21;
hence  x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) by A119, A120, XBOOLE_0:def_4; ::_thesis: verum
end;
then  (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) = {((Gauge (C,n)) * (i2,k))} by A118, XBOOLE_0:def_10;
then A121: (LSeg ((go ^' pion1),((len go) + ((len pion1) -' 2)))) /\ (LSeg (do,1)) = {((go ^' pion1) /. (len (go ^' pion1)))} by A44, A103, A105, A116, A117, TOPREAL8:31;
A122: not go ^' pion1 is trivial by A98, NAT_D:60;
A123: rng pion1 c= L~ pion1 by A108, SPPOL_2:18;
A124: {(pion1 /. 1)} c= (L~ go) /\ (L~ pion1)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(pion1 /. 1)} or x in (L~ go) /\ (L~ pion1) )
assume  x in {(pion1 /. 1)} ; ::_thesis: x in (L~ go) /\ (L~ pion1)
then A125: x = pion1 /. 1 by TARSKI:def_1;
then A126: x in rng go by A103, REVROT_1:3;
 x in rng pion1 by A125, FINSEQ_6:42;
hence  x in (L~ go) /\ (L~ pion1) by A63, A123, A126, XBOOLE_0:def_4; ::_thesis: verum
end;
 (L~ go) /\ (L~ pion1) c= {(pion1 /. 1)}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ go) /\ (L~ pion1) or x in {(pion1 /. 1)} )
assume A127: x in (L~ go) /\ (L~ pion1) ; ::_thesis: x in {(pion1 /. 1)}
then A128: x in L~ go by XBOOLE_0:def_4;
 x in L~ pion1 by A127, XBOOLE_0:def_4;
hence  x in {(pion1 /. 1)} by A7, A12, A43, A48, A87, A103, A128, XBOOLE_0:def_4; ::_thesis: verum
end;
then A129: (L~ go) /\ (L~ pion1) = {(pion1 /. 1)} by A124, XBOOLE_0:def_10;
then A130: go ^' pion1 is s.n.c. by A103, JORDAN1J:54;
 (rng go) /\ (rng pion1) c= {(pion1 /. 1)} by A63, A123, A129, XBOOLE_1:27;
then A131: go ^' pion1 is one-to-one by JORDAN1J:55;
A132: <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. (len <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) = <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. 3 by FINSEQ_1:45
.= do /. 1 by A44, FINSEQ_4:18 ;
A133: {(pion1 /. (len pion1))} c= (L~ do) /\ (L~ pion1)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(pion1 /. (len pion1))} or x in (L~ do) /\ (L~ pion1) )
assume  x in {(pion1 /. (len pion1))} ; ::_thesis: x in (L~ do) /\ (L~ pion1)
then A134: x = pion1 /. (len pion1) by TARSKI:def_1;
then A135: x in rng do by A89, A132, FINSEQ_6:42;
 x in rng pion1 by A134, REVROT_1:3;
hence  x in (L~ do) /\ (L~ pion1) by A64, A123, A135, XBOOLE_0:def_4; ::_thesis: verum
end;
 (L~ do) /\ (L~ pion1) c= {(pion1 /. (len pion1))}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ do) /\ (L~ pion1) or x in {(pion1 /. (len pion1))} )
assume A136: x in (L~ do) /\ (L~ pion1) ; ::_thesis: x in {(pion1 /. (len pion1))}
then A137: x in L~ do by XBOOLE_0:def_4;
 x in L~ pion1 by A136, XBOOLE_0:def_4;
hence  x in {(pion1 /. (len pion1))} by A8, A12, A44, A55, A87, A89, A132, A137, XBOOLE_0:def_4; ::_thesis: verum
end;
then A138: (L~ do) /\ (L~ pion1) = {(pion1 /. (len pion1))} by A133, XBOOLE_0:def_10;
A139: (L~ (go ^' pion1)) /\ (L~ do) = ((L~ go) \/ (L~ pion1)) /\ (L~ do) by A103, TOPREAL8:35
.= {(go /. 1)} \/ {(do /. 1)} by A75, A89, A132, A138, XBOOLE_1:23
.= {((go ^' pion1) /. 1)} \/ {(do /. 1)} by GRAPH_2:53
.= {((go ^' pion1) /. 1),(do /. 1)} by ENUMSET1:1 ;
 do /. (len do) = (go ^' pion1) /. 1 by A62, GRAPH_2:53;
then reconsider godo = (go ^' pion1) ^' do as non constant standard special_circular_sequence by A101, A105, A106, A111, A113, A121, A122, A130, A131, A139, JORDAN8:4, JORDAN8:5, TOPREAL8:11, TOPREAL8:33, TOPREAL8:34;
A140: Upper_Arc C is_an_arc_of W-min C, E-max C by JORDAN6:def_8;
then A141: Upper_Arc C is connected by JORDAN6:10;
A142: W-min C in Upper_Arc C by A140, TOPREAL1:1;
A143: E-max C in Upper_Arc C by A140, TOPREAL1:1;
set ff = Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))));
 W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
then A144: (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 = W-min (L~ (Cage (C,n))) by FINSEQ_6:92;
A145: L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33;
then  (W-max (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A144, SPRECT_5:22;
then  (N-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A144, A145, SPRECT_5:23, XXREAL_0:2;
then  (N-max (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A144, A145, SPRECT_5:24, XXREAL_0:2;
then A146: (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A144, A145, SPRECT_5:25, XXREAL_0:2;
A147: now__::_thesis:_not_((Gauge_(C,n))_*_(i1,j))_.._(Upper_Seq_(C,n))_<=_1
assume A148: ((Gauge (C,n)) * (i1,j)) .. (Upper_Seq (C,n)) <= 1 ; ::_thesis: contradiction
 ((Gauge (C,n)) * (i1,j)) .. (Upper_Seq (C,n)) >= 1 by A36, FINSEQ_4:21;
then  ((Gauge (C,n)) * (i1,j)) .. (Upper_Seq (C,n)) = 1 by A148, XXREAL_0:1;
then  (Gauge (C,n)) * (i1,j) = (Upper_Seq (C,n)) /. 1 by A36, FINSEQ_5:38;
hence  contradiction by A22, A26, JORDAN1F:5; ::_thesis: verum
end;
A149: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def_1;
then A150: Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))) is_sequence_on Gauge (C,n) by REVROT_1:34;
A151: (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) c= RightComp godo by A101, A106, JORDAN9:27;
A152: L~ godo = (L~ (go ^' pion1)) \/ (L~ do) by A105, TOPREAL8:35
.= ((L~ go) \/ (L~ pion1)) \/ (L~ do) by A103, TOPREAL8:35 ;
A153: L~ (Cage (C,n)) = (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) by JORDAN1E:13;
then A154: L~ (Upper_Seq (C,n)) c= L~ (Cage (C,n)) by XBOOLE_1:7;
A155: L~ (Lower_Seq (C,n)) c= L~ (Cage (C,n)) by A153, XBOOLE_1:7;
A156: L~ go c= L~ (Cage (C,n)) by A48, A154, XBOOLE_1:1;
A157: L~ do c= L~ (Cage (C,n)) by A55, A155, XBOOLE_1:1;
A158: W-min C in C by SPRECT_1:13;
A159: now__::_thesis:_not_W-min_C_in_L~_godo
assume  W-min C in L~ godo ; ::_thesis: contradiction
then A160: ( W-min C in (L~ go) \/ (L~ pion1) or W-min C in L~ do ) by A152, XBOOLE_0:def_3;
percases ( W-min C in L~ go or W-min C in L~ pion1 or W-min C in L~ do ) by A160, XBOOLE_0:def_3;
suppose W-min C in L~ go ; ::_thesis: contradiction
then  C meets L~ (Cage (C,n)) by A156, A158, XBOOLE_0:3;
hence  contradiction by JORDAN10:5; ::_thesis: verum
end;
suppose W-min C in L~ pion1 ; ::_thesis: contradiction
hence  contradiction by A9, A12, A87, A142, XBOOLE_0:3; ::_thesis: verum
end;
suppose W-min C in L~ do ; ::_thesis: contradiction
then  C meets L~ (Cage (C,n)) by A157, A158, XBOOLE_0:3;
hence  contradiction by JORDAN10:5; ::_thesis: verum
end;
end;
end;
right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1) = right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(GoB (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) by A97, JORDAN1H:23
.= right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(GoB (Cage (C,n)))) by REVROT_1:28
.= right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(Gauge (C,n))) by JORDAN1H:44
.= right_cell (((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n))))),1,(Gauge (C,n))) by A146, A150, JORDAN1J:53
.= right_cell ((Upper_Seq (C,n)),1,(Gauge (C,n))) by JORDAN1E:def_1
.= right_cell ((R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i1,j)))),1,(Gauge (C,n))) by A36, A102, A147, JORDAN1J:52
.= right_cell ((go ^' pion1),1,(Gauge (C,n))) by A41, A104, JORDAN1J:51
.= right_cell (godo,1,(Gauge (C,n))) by A99, A106, JORDAN1J:51 ;
then  W-min C in right_cell (godo,1,(Gauge (C,n))) by JORDAN1I:6;
then A161: W-min C in (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) by A159, XBOOLE_0:def_5;
A162: godo /. 1 = (go ^' pion1) /. 1 by GRAPH_2:53
.= W-min (L~ (Cage (C,n))) by A61, GRAPH_2:53 ;
A163: len (Upper_Seq (C,n)) >= 2 by A21, XXREAL_0:2;
A164: godo /. 2 = (go ^' pion1) /. 2 by A98, GRAPH_2:57
.= (Upper_Seq (C,n)) /. 2 by A35, A78, GRAPH_2:57
.= ((Upper_Seq (C,n)) ^' (Lower_Seq (C,n))) /. 2 by A163, GRAPH_2:57
.= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 2 by JORDAN1E:11 ;
A165: (L~ go) \/ (L~ do) is compact by COMPTS_1:10;
 W-min (L~ (Cage (C,n))) in (L~ go) \/ (L~ do) by A63, A79, XBOOLE_0:def_3;
then A166: W-min ((L~ go) \/ (L~ do)) = W-min (L~ (Cage (C,n))) by A156, A157, A165, JORDAN1J:21, XBOOLE_1:8;
A167: (W-min ((L~ go) \/ (L~ do))) `1 = W-bound ((L~ go) \/ (L~ do)) by EUCLID:52;
A168: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52;
 ((Gauge (C,n)) * (i2,k)) `1 >= W-bound (L~ (Cage (C,n))) by A13, A155, PSCOMP_1:24;
then  ((Gauge (C,n)) * (i2,k)) `1 > W-bound (L~ (Cage (C,n))) by A77, XXREAL_0:1;
then  W-min (((L~ go) \/ (L~ do)) \/ (L~ pion1)) = W-min ((L~ go) \/ (L~ do)) by A95, A165, A166, A167, A168, JORDAN1J:33;
then A169: W-min (L~ godo) = W-min (L~ (Cage (C,n))) by A152, A166, XBOOLE_1:4;
A170: rng godo c= L~ godo by A98, A100, SPPOL_2:18, XXREAL_0:2;
 2 in dom godo by A101, FINSEQ_3:25;
then A171: godo /. 2 in rng godo by PARTFUN2:2;
 godo /. 2 in W-most (L~ (Cage (C,n))) by A164, JORDAN1I:25;
then (godo /. 2) `1 = (W-min (L~ godo)) `1 by A169, PSCOMP_1:31
.= W-bound (L~ godo) by EUCLID:52 ;
then  godo /. 2 in W-most (L~ godo) by A170, A171, SPRECT_2:12;
then  (Rotate (godo,(W-min (L~ godo)))) /. 2 in W-most (L~ godo) by A162, A169, FINSEQ_6:89;
then reconsider godo = godo as non constant standard clockwise_oriented special_circular_sequence by JORDAN1I:25;
 len (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by FINSEQ_5:6;
then A172: (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) = (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) by PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:7 ;
A173: east_halfline (E-max C) misses L~ go
proof
assume  east_halfline (E-max C) meets L~ go ; ::_thesis: contradiction
then consider p being set  such that
A174: p in east_halfline (E-max C) and
A175: p in L~ go by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A174;
 p in L~ (Upper_Seq (C,n)) by A48, A175;
then  p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A154, A174, XBOOLE_0:def_4;
then A176: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;
then A177: p = E-max (L~ (Cage (C,n))) by A48, A175, JORDAN1J:46;
then  E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i1,j) by A14, A172, A175, JORDAN1J:43;
then  ((Gauge (C,n)) * (i1,j)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A6, A16, A20, A176, A177, JORDAN1A:71;
hence  contradiction by A3, A17, A32, JORDAN1G:7; ::_thesis: verum
end;
now__::_thesis:_not_east_halfline_(E-max_C)_meets_L~_godo
assume  east_halfline (E-max C) meets L~ godo ; ::_thesis: contradiction
then A178: ( east_halfline (E-max C) meets (L~ go) \/ (L~ pion1) or east_halfline (E-max C) meets L~ do ) by A152, XBOOLE_1:70;
percases ( east_halfline (E-max C) meets L~ go or east_halfline (E-max C) meets L~ pion1 or east_halfline (E-max C) meets L~ do ) by A178, XBOOLE_1:70;
suppose east_halfline (E-max C) meets L~ go ; ::_thesis: contradiction
hence  contradiction by A173; ::_thesis: verum
end;
suppose east_halfline (E-max C) meets L~ pion1 ; ::_thesis: contradiction
then consider p being set  such that
A179: p in east_halfline (E-max C) and
A180: p in L~ pion1 by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A179;
A181: now__::_thesis:_p_`1_<=_((Gauge_(C,n))_*_(i1,j))_`1
percases ( p in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) or p in LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) ) by A12, A87, A180, XBOOLE_0:def_3;
suppose p in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) ; ::_thesis: p `1 <= ((Gauge (C,n)) * (i1,j)) `1
hence  p `1 <= ((Gauge (C,n)) * (i1,j)) `1 by A91, A92, TOPREAL1:3; ::_thesis: verum
end;
suppose p in LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) ; ::_thesis: p `1 <= ((Gauge (C,n)) * (i1,j)) `1
hence  p `1 <= ((Gauge (C,n)) * (i1,j)) `1 by A91, GOBOARD7:5; ::_thesis: verum
end;
end;
end;
 i1 + 1 <= len (Gauge (C,n)) by A3, NAT_1:13;
then  i1 <= (len (Gauge (C,n))) - 1 by XREAL_1:19;
then A182: i1 <= (len (Gauge (C,n))) -' 1 by XREAL_0:def_2;
 (len (Gauge (C,n))) -' 1 <= len (Gauge (C,n)) by NAT_D:35;
then  ((Gauge (C,n)) * (i1,j)) `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1 by A4, A10, A15, A20, A24, A182, JORDAN1A:18;
then  p `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1 by A181, XXREAL_0:2;
then  p `1 <= E-bound C by A24, JORDAN8:12;
then A183: p `1 <= (E-max C) `1 by EUCLID:52;
 p `1 >= (E-max C) `1 by A179, TOPREAL1:def_11;
then A184: p `1 = (E-max C) `1 by A183, XXREAL_0:1;
 p `2 = (E-max C) `2 by A179, TOPREAL1:def_11;
then  p = E-max C by A184, TOPREAL3:6;
hence  contradiction by A9, A12, A87, A143, A180, XBOOLE_0:3; ::_thesis: verum
end;
suppose east_halfline (E-max C) meets L~ do ; ::_thesis: contradiction
then consider p being set  such that
A185: p in east_halfline (E-max C) and
A186: p in L~ do by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A185;
 p in L~ (Lower_Seq (C,n)) by A55, A186;
then  p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A155, A185, XBOOLE_0:def_4;
then A187: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;
A188: (E-max C) `2 = p `2 by A185, TOPREAL1:def_11;
set RC = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))));
A189: E-max C in right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by JORDAN1I:7;
A190: 1 + 1 <= len (Lower_Seq (C,n)) by A27, XXREAL_0:2;
 Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) by JORDAN1G:18;
then A191: LSeg ((Lower_Seq (C,n)),1) = LSeg ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by A190, SPPOL_2:9;
A192: L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33;
A193: len (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = len (Cage (C,n)) by REVROT_1:14;
A194: GoB (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = GoB (Cage (C,n)) by REVROT_1:28
.= Gauge (C,n) by JORDAN1H:44 ;
A195: E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
A196: Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))) is_sequence_on Gauge (C,n) by A149, REVROT_1:34;
A197: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = E-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))))) by A192, A195, FINSEQ_6:92;
consider ii, jj being Element of NAT  such that
A198: [ii,(jj + 1)] in Indices (Gauge (C,n)) and
A199: [ii,jj] in Indices (Gauge (C,n)) and
A200: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = (Gauge (C,n)) * (ii,(jj + 1)) and
A201: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1) = (Gauge (C,n)) * (ii,jj) by A96, A192, A193, A195, A196, FINSEQ_6:92, JORDAN1I:23;
consider jj2 being Element of NAT  such that
A202: 1 <= jj2 and
A203: jj2 <= width (Gauge (C,n)) and
A204: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * ((len (Gauge (C,n))),jj2) by JORDAN1D:25;
A205: len (Gauge (C,n)) >= 4 by JORDAN8:10;
then  len (Gauge (C,n)) >= 1 by XXREAL_0:2;
then  [(len (Gauge (C,n))),jj2] in Indices (Gauge (C,n)) by A202, A203, MATRIX_1:36;
then A206: ii = len (Gauge (C,n)) by A192, A197, A198, A200, A204, GOBOARD1:5;
A207: 1 <= ii by A198, MATRIX_1:38;
A208: ii <= len (Gauge (C,n)) by A198, MATRIX_1:38;
A209: 1 <= jj + 1 by A198, MATRIX_1:38;
A210: jj + 1 <= width (Gauge (C,n)) by A198, MATRIX_1:38;
A211: 1 <= ii by A199, MATRIX_1:38;
A212: ii <= len (Gauge (C,n)) by A199, MATRIX_1:38;
A213: 1 <= jj by A199, MATRIX_1:38;
A214: jj <= width (Gauge (C,n)) by A199, MATRIX_1:38;
A215: ii + 1 <> ii ;
 (jj + 1) + 1 <> jj ;
then A216: right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) = cell ((Gauge (C,n)),(ii -' 1),jj) by A96, A193, A194, A198, A199, A200, A201, A215, GOBOARD5:def_6;
A217: (ii -' 1) + 1 = ii by A207, XREAL_1:235;
 ii - 1 >= 4 - 1 by A205, A206, XREAL_1:9;
then A218: ii - 1 >= 1 by XXREAL_0:2;
then A219: 1 <= ii -' 1 by XREAL_0:def_2;
A220: ((Gauge (C,n)) * ((ii -' 1),jj)) `2 <= p `2 by A188, A189, A208, A210, A213, A216, A217, A218, JORDAN9:17;
A221: p `2 <= ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 by A188, A189, A208, A210, A213, A216, A217, A218, JORDAN9:17;
A222: ii -' 1 < len (Gauge (C,n)) by A208, A217, NAT_1:13;
then A223: ((Gauge (C,n)) * ((ii -' 1),jj)) `2 = ((Gauge (C,n)) * (1,jj)) `2 by A213, A214, A219, GOBOARD5:1
.= ((Gauge (C,n)) * (ii,jj)) `2 by A211, A212, A213, A214, GOBOARD5:1 ;
A224: ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 = ((Gauge (C,n)) * (1,(jj + 1))) `2 by A209, A210, A219, A222, GOBOARD5:1
.= ((Gauge (C,n)) * (ii,(jj + 1))) `2 by A207, A208, A209, A210, GOBOARD5:1 ;
A225: ((Gauge (C,n)) * ((len (Gauge (C,n))),jj)) `1 = E-bound (L~ (Cage (C,n))) by A20, A213, A214, JORDAN1A:71;
 E-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * ((len (Gauge (C,n))),(jj + 1))) `1 by A20, A209, A210, JORDAN1A:71;
then  p in LSeg (((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1),((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1))) by A187, A200, A201, A206, A220, A221, A223, A224, A225, GOBOARD7:7;
then A226: p in LSeg ((Lower_Seq (C,n)),1) by A96, A191, A193, TOPREAL1:def_3;
A227: p in LSeg (do,(Index (p,do))) by A186, JORDAN3:9;
A228: do = mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n)))) by A39, JORDAN1J:37;
A229: 1 <= ((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)) by A39, FINSEQ_4:21;
A230: ((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)) <= len (Lower_Seq (C,n)) by A39, FINSEQ_4:21;
 ((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)) <> len (Lower_Seq (C,n)) by A31, A39, FINSEQ_4:19;
then A231: ((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)) < len (Lower_Seq (C,n)) by A230, XXREAL_0:1;
A232: 1 <= Index (p,do) by A186, JORDAN3:8;
A233: Index (p,do) < len do by A186, JORDAN3:8;
A234: (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n)))) + 1 = ((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)) by A34, A39, JORDAN1J:56;
consider t being Nat such that
A235: t in dom (Lower_Seq (C,n)) and
A236: (Lower_Seq (C,n)) . t = (Gauge (C,n)) * (i2,k) by A39, FINSEQ_2:10;
A237: 1 <= t by A235, FINSEQ_3:25;
A238: t <= len (Lower_Seq (C,n)) by A235, FINSEQ_3:25;
 1 < t by A34, A236, A237, XXREAL_0:1;
then  (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n)))) + 1 = t by A236, A238, JORDAN3:12;
then A239: len (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i2,k)))) = (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n)))) by A13, A236, JORDAN3:26;
set tt = ((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1;
A240: 1 <= Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n))) by A13, JORDAN3:8;
 0 + (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n)))) < len (Lower_Seq (C,n)) by A13, JORDAN3:8;
then A241: (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n)))) > 0 by XREAL_1:20;
 Index (p,do) < (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n)))) by A233, A239, XREAL_0:def_2;
then  (Index (p,do)) + 1 <= (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n)))) by NAT_1:13;
then  Index (p,do) <= ((len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n))))) - 1 by XREAL_1:19;
then  Index (p,do) <= ((len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n))))) - 1 by A241, XREAL_0:def_2;
then  Index (p,do) <= (len (Lower_Seq (C,n))) - (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n))) by A234;
then  Index (p,do) <= (len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n))) by XREAL_0:def_2;
then  Index (p,do) < ((len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) + 1 by NAT_1:13;
then A242: LSeg ((mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n))))),(Index (p,do))) = LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1)) by A229, A231, A232, JORDAN4:19;
A243: 1 + 1 <= ((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)) by A234, A240, XREAL_1:7;
then  (Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n))) >= (1 + 1) + 1 by A232, XREAL_1:7;
then  ((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) - 1 >= ((1 + 1) + 1) - 1 by XREAL_1:9;
then A244: ((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1 >= 1 + 1 by XREAL_0:def_2;
A245: 2 in dom (Lower_Seq (C,n)) by A190, FINSEQ_3:25;
now__::_thesis:_contradiction
percases ( ((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1 > 1 + 1 or ((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ) by A244, XXREAL_0:1;
suppose ((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1 > 1 + 1 ; ::_thesis: contradiction
then  LSeg ((Lower_Seq (C,n)),1) misses LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1)) by TOPREAL1:def_7;
hence  contradiction by A226, A227, A228, A242, XBOOLE_0:3; ::_thesis: verum
end;
supposeA246: ((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ; ::_thesis: contradiction
then  (LSeg ((Lower_Seq (C,n)),1)) /\ (LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1))) = {((Lower_Seq (C,n)) /. 2)} by A27, TOPREAL1:def_6;
then  p in {((Lower_Seq (C,n)) /. 2)} by A226, A227, A228, A242, XBOOLE_0:def_4;
then A247: p = (Lower_Seq (C,n)) /. 2 by TARSKI:def_1;
then A248: p .. (Lower_Seq (C,n)) = 2 by A245, FINSEQ_5:41;
 1 + 1 = ((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) - 1 by A246, XREAL_0:def_2;
then  (1 + 1) + 1 = (Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n))) ;
then A249: ((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)) = 2 by A232, A243, JORDAN1E:6;
 p in rng (Lower_Seq (C,n)) by A245, A247, PARTFUN2:2;
then  p = (Gauge (C,n)) * (i2,k) by A39, A248, A249, FINSEQ_5:9;
then  ((Gauge (C,n)) * (i2,k)) `1 = E-bound (L~ (Cage (C,n))) by A247, JORDAN1G:32;
then  ((Gauge (C,n)) * (i2,k)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 by A4, A15, A20, JORDAN1A:71;
hence  contradiction by A2, A3, A18, A69, JORDAN1G:7; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
end;
end;
then  east_halfline (E-max C) c= (L~ godo) ` by SUBSET_1:23;
then consider W being Subset of (TOP-REAL 2) such that
A250: W is_a_component_of (L~ godo) ` and
A251: east_halfline (E-max C) c= W by GOBOARD9:3;
 not W is bounded by A251, JORDAN2C:121, RLTOPSP1:42;
then  W is_outside_component_of L~ godo by A250, JORDAN2C:def_3;
then  W c= UBD (L~ godo) by JORDAN2C:23;
then A252: east_halfline (E-max C) c= UBD (L~ godo) by A251, XBOOLE_1:1;
 E-max C in east_halfline (E-max C) by TOPREAL1:38;
then  E-max C in UBD (L~ godo) by A252;
then  E-max C in LeftComp godo by GOBRD14:36;
then  Upper_Arc C meets L~ godo by A141, A142, A143, A151, A161, JORDAN1J:36;
then A253: ( Upper_Arc C meets (L~ go) \/ (L~ pion1) or Upper_Arc C meets L~ do ) by A152, XBOOLE_1:70;
A254: Upper_Arc C c= C by JORDAN6:61;
now__::_thesis:_contradiction
percases ( Upper_Arc C meets L~ go or Upper_Arc C meets L~ pion1 or Upper_Arc C meets L~ do ) by A253, XBOOLE_1:70;
suppose Upper_Arc C meets L~ go ; ::_thesis: contradiction
then  Upper_Arc C meets L~ (Cage (C,n)) by A48, A154, XBOOLE_1:1, XBOOLE_1:63;
hence  contradiction by A254, JORDAN10:5, XBOOLE_1:63; ::_thesis: verum
end;
suppose Upper_Arc C meets L~ pion1 ; ::_thesis: contradiction
hence  contradiction by A9, A12, A87; ::_thesis: verum
end;
suppose Upper_Arc C meets L~ do ; ::_thesis: contradiction
then  Upper_Arc C meets L~ (Cage (C,n)) by A55, A155, XBOOLE_1:1, XBOOLE_1:63;
hence  contradiction by A254, JORDAN10:5, XBOOLE_1:63; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
suppose ((Gauge (C,n)) * (i2,k)) `1 = ((Gauge (C,n)) * (i1,j)) `1 ; ::_thesis: contradiction
then A255: i1 = i2 by A17, A18, JORDAN1G:7;
then  LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) = {((Gauge (C,n)) * (i1,k))} by RLTOPSP1:70;
then  LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) c= LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) by A83, ZFMISC_1:31;
then  (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) = LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) by XBOOLE_1:12;
hence  contradiction by A1, A3, A4, A5, A6, A7, A8, A9, A255, Th12; ::_thesis: verum
end;
suppose ((Gauge (C,n)) * (i2,k)) `2 = ((Gauge (C,n)) * (i1,j)) `2 ; ::_thesis: contradiction
then A256: j = k by A17, A18, JORDAN1G:6;
then  LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) = {((Gauge (C,n)) * (i1,k))} by RLTOPSP1:70;
then  LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) c= LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) by A84, ZFMISC_1:31;
then  (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) = LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) by XBOOLE_1:12;
hence  contradiction by A1, A2, A3, A4, A6, A7, A8, A9, A256, JORDAN15:29; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;

theorem Th23: :: JORDAN19:23
for n being Element of NAT
for C being Simple_closed_curve
for i1, i2, j, k being Element of NAT st 1 < i2 & i2 <= i1 & i1 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} holds
(LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Lower_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i1, i2, j, k being Element of NAT st 1 < i2 & i2 <= i1 & i1 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} holds
(LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Lower_Arc C
let C be Simple_closed_curve; ::_thesis: for i1, i2, j, k being Element of NAT st 1 < i2 & i2 <= i1 & i1 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} holds
(LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Lower_Arc C
let i1, i2, j, k be Element of NAT ; ::_thesis: ( 1 < i2 & i2 <= i1 & i1 < len (Gauge (C,n)) & 1 <= j & j <= k & k <= width (Gauge (C,n)) & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} & ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} implies (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) meets Lower_Arc C )
set G = Gauge (C,n);
set pio = LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)));
set poz = LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)));
set US = Upper_Seq (C,n);
set LS = Lower_Seq (C,n);
assume that
A1: 1 < i2 and
A2: i2 <= i1 and
A3: i1 < len (Gauge (C,n)) and
A4: 1 <= j and
A5: j <= k and
A6: k <= width (Gauge (C,n)) and
A7: ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Upper_Seq (C,n))) = {((Gauge (C,n)) * (i1,j))} and
A8: ((LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))) /\ (L~ (Lower_Seq (C,n))) = {((Gauge (C,n)) * (i2,k))} and
A9: (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) misses Lower_Arc C ; ::_thesis: contradiction
set UA = Lower_Arc C;
set Wmin = W-min (L~ (Cage (C,n)));
set Emax = E-max (L~ (Cage (C,n)));
set Wbo = W-bound (L~ (Cage (C,n)));
set Ebo = E-bound (L~ (Cage (C,n)));
set Gik = (Gauge (C,n)) * (i2,k);
set Gij = (Gauge (C,n)) * (i1,j);
set Gi1k = (Gauge (C,n)) * (i1,k);
A10: 1 < i1 by A1, A2, XXREAL_0:2;
A11: i2 < len (Gauge (C,n)) by A2, A3, XXREAL_0:2;
A12: L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> = (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) by TOPREAL3:16;
 (Gauge (C,n)) * (i2,k) in {((Gauge (C,n)) * (i2,k))} by TARSKI:def_1;
then A13: (Gauge (C,n)) * (i2,k) in L~ (Lower_Seq (C,n)) by A8, XBOOLE_0:def_4;
 (Gauge (C,n)) * (i1,j) in {((Gauge (C,n)) * (i1,j))} by TARSKI:def_1;
then A14: (Gauge (C,n)) * (i1,j) in L~ (Upper_Seq (C,n)) by A7, XBOOLE_0:def_4;
A15: j <= width (Gauge (C,n)) by A5, A6, XXREAL_0:2;
A16: 1 <= k by A4, A5, XXREAL_0:2;
A17: [i1,j] in Indices (Gauge (C,n)) by A3, A4, A10, A15, MATRIX_1:36;
A18: [i2,k] in Indices (Gauge (C,n)) by A1, A6, A11, A16, MATRIX_1:36;
A19: [i1,k] in Indices (Gauge (C,n)) by A3, A6, A10, A16, MATRIX_1:36;
set go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i1,j)));
set do = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i2,k)));
A20: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
A21: len (Upper_Seq (C,n)) >= 3 by JORDAN1E:15;
then  len (Upper_Seq (C,n)) >= 1 by XXREAL_0:2;
then  1 in dom (Upper_Seq (C,n)) by FINSEQ_3:25;
then A22: (Upper_Seq (C,n)) . 1 = (Upper_Seq (C,n)) /. 1 by PARTFUN1:def_6
.= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ;
A23: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * (1,k)) `1 by A6, A16, A20, JORDAN1A:73 ;
 len (Gauge (C,n)) >= 4 by JORDAN8:10;
then A24: len (Gauge (C,n)) >= 1 by XXREAL_0:2;
then A25: [1,k] in Indices (Gauge (C,n)) by A6, A16, MATRIX_1:36;
then A26: (Gauge (C,n)) * (i1,j) <> (Upper_Seq (C,n)) . 1 by A1, A2, A17, A22, A23, JORDAN1G:7;
then reconsider go = R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i1,j))) as being_S-Seq FinSequence of (TOP-REAL 2) by A14, JORDAN3:35;
A27: len (Lower_Seq (C,n)) >= 1 + 2 by JORDAN1E:15;
then A28: len (Lower_Seq (C,n)) >= 1 by XXREAL_0:2;
then A29: 1 in dom (Lower_Seq (C,n)) by FINSEQ_3:25;
 len (Lower_Seq (C,n)) in dom (Lower_Seq (C,n)) by A28, FINSEQ_3:25;
then A30: (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by PARTFUN1:def_6
.= W-min (L~ (Cage (C,n))) by JORDAN1F:8 ;
(W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * (1,k)) `1 by A6, A16, A20, JORDAN1A:73 ;
then A31: (Gauge (C,n)) * (i2,k) <> (Lower_Seq (C,n)) . (len (Lower_Seq (C,n))) by A1, A18, A25, A30, JORDAN1G:7;
then reconsider do = L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i2,k))) as being_S-Seq FinSequence of (TOP-REAL 2) by A13, JORDAN3:34;
A32: [(len (Gauge (C,n))),k] in Indices (Gauge (C,n)) by A6, A16, A24, MATRIX_1:36;
A33: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by A29, PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;
(E-max (L~ (Cage (C,n)))) `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A6, A16, A20, JORDAN1A:71 ;
then A34: (Gauge (C,n)) * (i2,k) <> (Lower_Seq (C,n)) . 1 by A2, A3, A18, A32, A33, JORDAN1G:7;
A35: len go >= 1 + 1 by TOPREAL1:def_8;
A36: (Gauge (C,n)) * (i1,j) in rng (Upper_Seq (C,n)) by A3, A4, A10, A14, A15, JORDAN1G:4, JORDAN1J:40;
then A37: go is_sequence_on Gauge (C,n) by JORDAN1G:4, JORDAN1J:38;
A38: len do >= 1 + 1 by TOPREAL1:def_8;
A39: (Gauge (C,n)) * (i2,k) in rng (Lower_Seq (C,n)) by A1, A6, A11, A13, A16, JORDAN1G:5, JORDAN1J:40;
then A40: do is_sequence_on Gauge (C,n) by JORDAN1G:5, JORDAN1J:39;
reconsider go = go as non constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A35, A37, JGRAPH_1:12, JORDAN8:5;
reconsider do = do as non constant being_S-Seq s.c.c. FinSequence of (TOP-REAL 2) by A38, A40, JGRAPH_1:12, JORDAN8:5;
A41: len go > 1 by A35, NAT_1:13;
then A42: len go in dom go by FINSEQ_3:25;
then A43: go /. (len go) = go . (len go) by PARTFUN1:def_6
.= (Gauge (C,n)) * (i1,j) by A14, JORDAN3:24 ;
 len do >= 1 by A38, XXREAL_0:2;
then  1 in dom do by FINSEQ_3:25;
then A44: do /. 1 = do . 1 by PARTFUN1:def_6
.= (Gauge (C,n)) * (i2,k) by A13, JORDAN3:23 ;
reconsider m = (len go) - 1 as Element of NAT by A42, FINSEQ_3:26;
A45: m + 1 = len go ;
then A46: (len go) -' 1 = m by NAT_D:34;
A47: LSeg (go,m) c= L~ go by TOPREAL3:19;
A48: L~ go c= L~ (Upper_Seq (C,n)) by A14, JORDAN3:41;
then  LSeg (go,m) c= L~ (Upper_Seq (C,n)) by A47, XBOOLE_1:1;
then A49: (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) c= {((Gauge (C,n)) * (i1,j))} by A7, A12, XBOOLE_1:26;
 m >= 1 by A35, XREAL_1:19;
then A50: LSeg (go,m) = LSeg ((go /. m),((Gauge (C,n)) * (i1,j))) by A43, A45, TOPREAL1:def_3;
 {((Gauge (C,n)) * (i1,j))} c= (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i1,j))} or x in (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) )
assume  x in {((Gauge (C,n)) * (i1,j))} ; ::_thesis: x in (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>)
then A51: x = (Gauge (C,n)) * (i1,j) by TARSKI:def_1;
A52: (Gauge (C,n)) * (i1,j) in LSeg (go,m) by A50, RLTOPSP1:68;
 (Gauge (C,n)) * (i1,j) in LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) by RLTOPSP1:68;
then  (Gauge (C,n)) * (i1,j) in (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) by XBOOLE_0:def_3;
then  (Gauge (C,n)) * (i1,j) in L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> by SPRECT_1:8;
hence  x in (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) by A51, A52, XBOOLE_0:def_4; ::_thesis: verum
end;
then A53: (LSeg (go,m)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) = {((Gauge (C,n)) * (i1,j))} by A49, XBOOLE_0:def_10;
A54: LSeg (do,1) c= L~ do by TOPREAL3:19;
A55: L~ do c= L~ (Lower_Seq (C,n)) by A13, JORDAN3:42;
then  LSeg (do,1) c= L~ (Lower_Seq (C,n)) by A54, XBOOLE_1:1;
then A56: (LSeg (do,1)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) c= {((Gauge (C,n)) * (i2,k))} by A8, A12, XBOOLE_1:26;
A57: LSeg (do,1) = LSeg (((Gauge (C,n)) * (i2,k)),(do /. (1 + 1))) by A38, A44, TOPREAL1:def_3;
 {((Gauge (C,n)) * (i2,k))} c= (LSeg (do,1)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i2,k))} or x in (LSeg (do,1)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) )
assume  x in {((Gauge (C,n)) * (i2,k))} ; ::_thesis: x in (LSeg (do,1)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>)
then A58: x = (Gauge (C,n)) * (i2,k) by TARSKI:def_1;
A59: (Gauge (C,n)) * (i2,k) in LSeg (do,1) by A57, RLTOPSP1:68;
 (Gauge (C,n)) * (i2,k) in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) by RLTOPSP1:68;
then  (Gauge (C,n)) * (i2,k) in (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) by XBOOLE_0:def_3;
then  (Gauge (C,n)) * (i2,k) in L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> by SPRECT_1:8;
hence  x in (LSeg (do,1)) /\ (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) by A58, A59, XBOOLE_0:def_4; ::_thesis: verum
end;
then A60: (L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) /\ (LSeg (do,1)) = {((Gauge (C,n)) * (i2,k))} by A56, XBOOLE_0:def_10;
A61: go /. 1 = (Upper_Seq (C,n)) /. 1 by A14, SPRECT_3:22
.= W-min (L~ (Cage (C,n))) by JORDAN1F:5 ;
then A62: go /. 1 = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by JORDAN1F:8
.= do /. (len do) by A13, JORDAN1J:35 ;
A63: rng go c= L~ go by A35, SPPOL_2:18;
A64: rng do c= L~ do by A38, SPPOL_2:18;
A65: {(go /. 1)} c= (L~ go) /\ (L~ do)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(go /. 1)} or x in (L~ go) /\ (L~ do) )
assume  x in {(go /. 1)} ; ::_thesis: x in (L~ go) /\ (L~ do)
then A66: x = go /. 1 by TARSKI:def_1;
then A67: x in rng go by FINSEQ_6:42;
 x in rng do by A62, A66, REVROT_1:3;
hence  x in (L~ go) /\ (L~ do) by A63, A64, A67, XBOOLE_0:def_4; ::_thesis: verum
end;
A68: (Lower_Seq (C,n)) . 1 = (Lower_Seq (C,n)) /. 1 by A29, PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;
A69: [(len (Gauge (C,n))),j] in Indices (Gauge (C,n)) by A4, A15, A24, MATRIX_1:36;
 (L~ go) /\ (L~ do) c= {(go /. 1)}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ go) /\ (L~ do) or x in {(go /. 1)} )
assume A70: x in (L~ go) /\ (L~ do) ; ::_thesis: x in {(go /. 1)}
then A71: x in L~ go by XBOOLE_0:def_4;
A72: x in L~ do by A70, XBOOLE_0:def_4;
then  x in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A48, A55, A71, XBOOLE_0:def_4;
then  x in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16;
then A73: ( x = W-min (L~ (Cage (C,n))) or x = E-max (L~ (Cage (C,n))) ) by TARSKI:def_2;
now__::_thesis:_not_x_=_E-max_(L~_(Cage_(C,n)))
assume  x = E-max (L~ (Cage (C,n))) ; ::_thesis: contradiction
then A74: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i2,k) by A13, A68, A72, JORDAN1E:7;
 ((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 = E-bound (L~ (Cage (C,n))) by A4, A15, A20, JORDAN1A:71;
then  (E-max (L~ (Cage (C,n)))) `1 <> E-bound (L~ (Cage (C,n))) by A2, A3, A18, A69, A74, JORDAN1G:7;
hence  contradiction by EUCLID:52; ::_thesis: verum
end;
hence  x in {(go /. 1)} by A61, A73, TARSKI:def_1; ::_thesis: verum
end;
then A75: (L~ go) /\ (L~ do) = {(go /. 1)} by A65, XBOOLE_0:def_10;
set W2 = go /. 2;
A76: 2 in dom go by A35, FINSEQ_3:25;
A77: now__::_thesis:_not_((Gauge_(C,n))_*_(i2,k))_`1_=_W-bound_(L~_(Cage_(C,n)))
assume  ((Gauge (C,n)) * (i2,k)) `1 = W-bound (L~ (Cage (C,n))) ; ::_thesis: contradiction
then  ((Gauge (C,n)) * (1,k)) `1 = ((Gauge (C,n)) * (i2,k)) `1 by A6, A16, A20, JORDAN1A:73;
hence  contradiction by A1, A18, A25, JORDAN1G:7; ::_thesis: verum
end;
go = mid ((Upper_Seq (C,n)),1,(((Gauge (C,n)) * (i1,j)) .. (Upper_Seq (C,n)))) by A36, JORDAN1G:49
.= (Upper_Seq (C,n)) | (((Gauge (C,n)) * (i1,j)) .. (Upper_Seq (C,n))) by A36, FINSEQ_4:21, FINSEQ_6:116 ;
then A78: go /. 2 = (Upper_Seq (C,n)) /. 2 by A76, FINSEQ_4:70;
A79: W-min (L~ (Cage (C,n))) in rng go by A61, FINSEQ_6:42;
set pion = <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>;
A80: now__::_thesis:_for_n_being_Element_of_NAT_st_n_in_dom_<*((Gauge_(C,n))_*_(i1,j)),((Gauge_(C,n))_*_(i1,k)),((Gauge_(C,n))_*_(i2,k))*>_holds_
ex_i,_j_being_Element_of_NAT_st_
(_[i,j]_in_Indices_(Gauge_(C,n))_&_<*((Gauge_(C,n))_*_(i1,j)),((Gauge_(C,n))_*_(i1,k)),((Gauge_(C,n))_*_(i2,k))*>_/._n_=_(Gauge_(C,n))_*_(i,j)_)
let n be Element of NAT ; ::_thesis: ( n in dom <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> implies ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. n = (Gauge (C,n)) * (i,j) ) )
assume  n in dom <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. n = (Gauge (C,n)) * (i,j) )
then  n in {1,2,3} by FINSEQ_1:89, FINSEQ_3:1;
then  ( n = 1 or n = 2 or n = 3 ) by ENUMSET1:def_1;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. n = (Gauge (C,n)) * (i,j) ) by A17, A18, A19, FINSEQ_4:18; ::_thesis: verum
end;
A81: ((Gauge (C,n)) * (i1,k)) `1 = ((Gauge (C,n)) * (i1,1)) `1 by A3, A6, A10, A16, GOBOARD5:2
.= ((Gauge (C,n)) * (i1,j)) `1 by A3, A4, A10, A15, GOBOARD5:2 ;
((Gauge (C,n)) * (i1,k)) `2 = ((Gauge (C,n)) * (1,k)) `2 by A3, A6, A10, A16, GOBOARD5:1
.= ((Gauge (C,n)) * (i2,k)) `2 by A1, A6, A11, A16, GOBOARD5:1 ;
then A82: (Gauge (C,n)) * (i1,k) = |[(((Gauge (C,n)) * (i1,j)) `1),(((Gauge (C,n)) * (i2,k)) `2)]| by A81, EUCLID:53;
A83: (Gauge (C,n)) * (i1,k) in LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) by RLTOPSP1:68;
A84: (Gauge (C,n)) * (i1,k) in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) by RLTOPSP1:68;
now__::_thesis:_contradiction
percases ( ( ((Gauge (C,n)) * (i2,k)) `1 <> ((Gauge (C,n)) * (i1,j)) `1 & ((Gauge (C,n)) * (i2,k)) `2 <> ((Gauge (C,n)) * (i1,j)) `2 ) or ((Gauge (C,n)) * (i2,k)) `1 = ((Gauge (C,n)) * (i1,j)) `1 or ((Gauge (C,n)) * (i2,k)) `2 = ((Gauge (C,n)) * (i1,j)) `2 ) ;
suppose ( ((Gauge (C,n)) * (i2,k)) `1 <> ((Gauge (C,n)) * (i1,j)) `1 & ((Gauge (C,n)) * (i2,k)) `2 <> ((Gauge (C,n)) * (i1,j)) `2 ) ; ::_thesis: contradiction
then  <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> is being_S-Seq by A82, TOPREAL3:34;
then consider pion1 being FinSequence of (TOP-REAL 2) such that
A85: pion1 is_sequence_on Gauge (C,n) and
A86: pion1 is being_S-Seq and
A87: L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> = L~ pion1 and
A88: <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. 1 = pion1 /. 1 and
A89: <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. (len <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) = pion1 /. (len pion1) and
A90: len <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> <= len pion1 by A80, GOBOARD3:2;
reconsider pion1 = pion1 as being_S-Seq FinSequence of (TOP-REAL 2) by A86;
set godo = (go ^' pion1) ^' do;
A91: ((Gauge (C,n)) * (i1,k)) `1 = ((Gauge (C,n)) * (i1,1)) `1 by A3, A6, A10, A16, GOBOARD5:2
.= ((Gauge (C,n)) * (i1,j)) `1 by A3, A4, A10, A15, GOBOARD5:2 ;
A92: ((Gauge (C,n)) * (i2,k)) `1 <= ((Gauge (C,n)) * (i1,k)) `1 by A1, A2, A3, A6, A16, JORDAN1A:18;
then A93: W-bound (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) = ((Gauge (C,n)) * (i2,k)) `1 by SPRECT_1:54;
A94: W-bound (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) = ((Gauge (C,n)) * (i1,j)) `1 by A91, SPRECT_1:54;
W-bound ((LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))))) = min ((W-bound (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))))),(W-bound (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))))) by SPRECT_1:47
.= ((Gauge (C,n)) * (i2,k)) `1 by A91, A92, A93, A94, XXREAL_0:def_9 ;
then A95: W-bound (L~ pion1) = ((Gauge (C,n)) * (i2,k)) `1 by A87, TOPREAL3:16;
A96: 1 + 1 <= len (Cage (C,n)) by GOBOARD7:34, XXREAL_0:2;
A97: 1 + 1 <= len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by GOBOARD7:34, XXREAL_0:2;
 len (go ^' pion1) >= len go by TOPREAL8:7;
then A98: len (go ^' pion1) >= 1 + 1 by A35, XXREAL_0:2;
then A99: len (go ^' pion1) > 1 + 0 by NAT_1:13;
A100: len ((go ^' pion1) ^' do) >= len (go ^' pion1) by TOPREAL8:7;
then A101: 1 + 1 <= len ((go ^' pion1) ^' do) by A98, XXREAL_0:2;
A102: Upper_Seq (C,n) is_sequence_on Gauge (C,n) by JORDAN1G:4;
A103: go /. (len go) = pion1 /. 1 by A43, A88, FINSEQ_4:18;
then A104: go ^' pion1 is_sequence_on Gauge (C,n) by A37, A85, TOPREAL8:12;
A105: (go ^' pion1) /. (len (go ^' pion1)) = <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. (len <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) by A89, GRAPH_2:54
.= <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. 3 by FINSEQ_1:45
.= do /. 1 by A44, FINSEQ_4:18 ;
then A106: (go ^' pion1) ^' do is_sequence_on Gauge (C,n) by A40, A104, TOPREAL8:12;
 LSeg (pion1,1) c= L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> by A87, TOPREAL3:19;
then A107: (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) c= {((Gauge (C,n)) * (i1,j))} by A46, A53, XBOOLE_1:27;
 len pion1 >= 2 + 1 by A90, FINSEQ_1:45;
then A108: len pion1 > 1 + 1 by NAT_1:13;
 {((Gauge (C,n)) * (i1,j))} c= (LSeg (go,m)) /\ (LSeg (pion1,1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i1,j))} or x in (LSeg (go,m)) /\ (LSeg (pion1,1)) )
assume  x in {((Gauge (C,n)) * (i1,j))} ; ::_thesis: x in (LSeg (go,m)) /\ (LSeg (pion1,1))
then A109: x = (Gauge (C,n)) * (i1,j) by TARSKI:def_1;
A110: (Gauge (C,n)) * (i1,j) in LSeg (go,m) by A50, RLTOPSP1:68;
 (Gauge (C,n)) * (i1,j) in LSeg (pion1,1) by A43, A103, A108, TOPREAL1:21;
hence  x in (LSeg (go,m)) /\ (LSeg (pion1,1)) by A109, A110, XBOOLE_0:def_4; ::_thesis: verum
end;
then  (LSeg (go,((len go) -' 1))) /\ (LSeg (pion1,1)) = {(go /. (len go))} by A43, A46, A107, XBOOLE_0:def_10;
then A111: go ^' pion1 is unfolded by A103, TOPREAL8:34;
 len pion1 >= 2 + 1 by A90, FINSEQ_1:45;
then A112: (len pion1) - 2 >= 0 by XREAL_1:19;
 ((len (go ^' pion1)) + 1) - 1 = ((len go) + (len pion1)) - 1 by GRAPH_2:13;
then (len (go ^' pion1)) - 1 = (len go) + ((len pion1) - 2)
.= (len go) + ((len pion1) -' 2) by A112, XREAL_0:def_2 ;
then A113: (len (go ^' pion1)) -' 1 = (len go) + ((len pion1) -' 2) by XREAL_0:def_2;
A114: (len pion1) - 1 >= 1 by A108, XREAL_1:19;
then A115: (len pion1) -' 1 = (len pion1) - 1 by XREAL_0:def_2;
A116: ((len pion1) -' 2) + 1 = ((len pion1) - 2) + 1 by A112, XREAL_0:def_2
.= (len pion1) -' 1 by A114, XREAL_0:def_2 ;
 ((len pion1) - 1) + 1 <= len pion1 ;
then A117: (len pion1) -' 1 < len pion1 by A115, NAT_1:13;
 LSeg (pion1,((len pion1) -' 1)) c= L~ <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> by A87, TOPREAL3:19;
then A118: (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) c= {((Gauge (C,n)) * (i2,k))} by A60, XBOOLE_1:27;
 {((Gauge (C,n)) * (i2,k))} c= (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,n)) * (i2,k))} or x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) )
assume  x in {((Gauge (C,n)) * (i2,k))} ; ::_thesis: x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1))
then A119: x = (Gauge (C,n)) * (i2,k) by TARSKI:def_1;
A120: (Gauge (C,n)) * (i2,k) in LSeg (do,1) by A57, RLTOPSP1:68;
pion1 /. (((len pion1) -' 1) + 1) = <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. 3 by A89, A115, FINSEQ_1:45
.= (Gauge (C,n)) * (i2,k) by FINSEQ_4:18 ;
then  (Gauge (C,n)) * (i2,k) in LSeg (pion1,((len pion1) -' 1)) by A114, A115, TOPREAL1:21;
hence  x in (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) by A119, A120, XBOOLE_0:def_4; ::_thesis: verum
end;
then  (LSeg (pion1,((len pion1) -' 1))) /\ (LSeg (do,1)) = {((Gauge (C,n)) * (i2,k))} by A118, XBOOLE_0:def_10;
then A121: (LSeg ((go ^' pion1),((len go) + ((len pion1) -' 2)))) /\ (LSeg (do,1)) = {((go ^' pion1) /. (len (go ^' pion1)))} by A44, A103, A105, A116, A117, TOPREAL8:31;
A122: not go ^' pion1 is trivial by A98, NAT_D:60;
A123: rng pion1 c= L~ pion1 by A108, SPPOL_2:18;
A124: {(pion1 /. 1)} c= (L~ go) /\ (L~ pion1)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(pion1 /. 1)} or x in (L~ go) /\ (L~ pion1) )
assume  x in {(pion1 /. 1)} ; ::_thesis: x in (L~ go) /\ (L~ pion1)
then A125: x = pion1 /. 1 by TARSKI:def_1;
then A126: x in rng go by A103, REVROT_1:3;
 x in rng pion1 by A125, FINSEQ_6:42;
hence  x in (L~ go) /\ (L~ pion1) by A63, A123, A126, XBOOLE_0:def_4; ::_thesis: verum
end;
 (L~ go) /\ (L~ pion1) c= {(pion1 /. 1)}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ go) /\ (L~ pion1) or x in {(pion1 /. 1)} )
assume A127: x in (L~ go) /\ (L~ pion1) ; ::_thesis: x in {(pion1 /. 1)}
then A128: x in L~ go by XBOOLE_0:def_4;
 x in L~ pion1 by A127, XBOOLE_0:def_4;
hence  x in {(pion1 /. 1)} by A7, A12, A43, A48, A87, A103, A128, XBOOLE_0:def_4; ::_thesis: verum
end;
then A129: (L~ go) /\ (L~ pion1) = {(pion1 /. 1)} by A124, XBOOLE_0:def_10;
then A130: go ^' pion1 is s.n.c. by A103, JORDAN1J:54;
 (rng go) /\ (rng pion1) c= {(pion1 /. 1)} by A63, A123, A129, XBOOLE_1:27;
then A131: go ^' pion1 is one-to-one by JORDAN1J:55;
A132: <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. (len <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*>) = <*((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))*> /. 3 by FINSEQ_1:45
.= do /. 1 by A44, FINSEQ_4:18 ;
A133: {(pion1 /. (len pion1))} c= (L~ do) /\ (L~ pion1)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(pion1 /. (len pion1))} or x in (L~ do) /\ (L~ pion1) )
assume  x in {(pion1 /. (len pion1))} ; ::_thesis: x in (L~ do) /\ (L~ pion1)
then A134: x = pion1 /. (len pion1) by TARSKI:def_1;
then A135: x in rng do by A89, A132, FINSEQ_6:42;
 x in rng pion1 by A134, REVROT_1:3;
hence  x in (L~ do) /\ (L~ pion1) by A64, A123, A135, XBOOLE_0:def_4; ::_thesis: verum
end;
 (L~ do) /\ (L~ pion1) c= {(pion1 /. (len pion1))}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ do) /\ (L~ pion1) or x in {(pion1 /. (len pion1))} )
assume A136: x in (L~ do) /\ (L~ pion1) ; ::_thesis: x in {(pion1 /. (len pion1))}
then A137: x in L~ do by XBOOLE_0:def_4;
 x in L~ pion1 by A136, XBOOLE_0:def_4;
hence  x in {(pion1 /. (len pion1))} by A8, A12, A44, A55, A87, A89, A132, A137, XBOOLE_0:def_4; ::_thesis: verum
end;
then A138: (L~ do) /\ (L~ pion1) = {(pion1 /. (len pion1))} by A133, XBOOLE_0:def_10;
A139: (L~ (go ^' pion1)) /\ (L~ do) = ((L~ go) \/ (L~ pion1)) /\ (L~ do) by A103, TOPREAL8:35
.= {(go /. 1)} \/ {(do /. 1)} by A75, A89, A132, A138, XBOOLE_1:23
.= {((go ^' pion1) /. 1)} \/ {(do /. 1)} by GRAPH_2:53
.= {((go ^' pion1) /. 1),(do /. 1)} by ENUMSET1:1 ;
 do /. (len do) = (go ^' pion1) /. 1 by A62, GRAPH_2:53;
then reconsider godo = (go ^' pion1) ^' do as non constant standard special_circular_sequence by A101, A105, A106, A111, A113, A121, A122, A130, A131, A139, JORDAN8:4, JORDAN8:5, TOPREAL8:11, TOPREAL8:33, TOPREAL8:34;
A140: Lower_Arc C is_an_arc_of E-max C, W-min C by JORDAN6:def_9;
then A141: Lower_Arc C is connected by JORDAN6:10;
A142: W-min C in Lower_Arc C by A140, TOPREAL1:1;
A143: E-max C in Lower_Arc C by A140, TOPREAL1:1;
set ff = Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))));
 W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
then A144: (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 = W-min (L~ (Cage (C,n))) by FINSEQ_6:92;
A145: L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33;
then  (W-max (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A144, SPRECT_5:22;
then  (N-min (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A144, A145, SPRECT_5:23, XXREAL_0:2;
then  (N-max (L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A144, A145, SPRECT_5:24, XXREAL_0:2;
then A146: (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) > 1 by A144, A145, SPRECT_5:25, XXREAL_0:2;
A147: now__::_thesis:_not_((Gauge_(C,n))_*_(i1,j))_.._(Upper_Seq_(C,n))_<=_1
assume A148: ((Gauge (C,n)) * (i1,j)) .. (Upper_Seq (C,n)) <= 1 ; ::_thesis: contradiction
 ((Gauge (C,n)) * (i1,j)) .. (Upper_Seq (C,n)) >= 1 by A36, FINSEQ_4:21;
then  ((Gauge (C,n)) * (i1,j)) .. (Upper_Seq (C,n)) = 1 by A148, XXREAL_0:1;
then  (Gauge (C,n)) * (i1,j) = (Upper_Seq (C,n)) /. 1 by A36, FINSEQ_5:38;
hence  contradiction by A22, A26, JORDAN1F:5; ::_thesis: verum
end;
A149: Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def_1;
then A150: Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))) is_sequence_on Gauge (C,n) by REVROT_1:34;
A151: (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) c= RightComp godo by A101, A106, JORDAN9:27;
A152: L~ godo = (L~ (go ^' pion1)) \/ (L~ do) by A105, TOPREAL8:35
.= ((L~ go) \/ (L~ pion1)) \/ (L~ do) by A103, TOPREAL8:35 ;
A153: L~ (Cage (C,n)) = (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) by JORDAN1E:13;
then A154: L~ (Upper_Seq (C,n)) c= L~ (Cage (C,n)) by XBOOLE_1:7;
A155: L~ (Lower_Seq (C,n)) c= L~ (Cage (C,n)) by A153, XBOOLE_1:7;
A156: L~ go c= L~ (Cage (C,n)) by A48, A154, XBOOLE_1:1;
A157: L~ do c= L~ (Cage (C,n)) by A55, A155, XBOOLE_1:1;
A158: W-min C in C by SPRECT_1:13;
A159: now__::_thesis:_not_W-min_C_in_L~_godo
assume  W-min C in L~ godo ; ::_thesis: contradiction
then A160: ( W-min C in (L~ go) \/ (L~ pion1) or W-min C in L~ do ) by A152, XBOOLE_0:def_3;
percases ( W-min C in L~ go or W-min C in L~ pion1 or W-min C in L~ do ) by A160, XBOOLE_0:def_3;
suppose W-min C in L~ go ; ::_thesis: contradiction
then  C meets L~ (Cage (C,n)) by A156, A158, XBOOLE_0:3;
hence  contradiction by JORDAN10:5; ::_thesis: verum
end;
suppose W-min C in L~ pion1 ; ::_thesis: contradiction
hence  contradiction by A9, A12, A87, A142, XBOOLE_0:3; ::_thesis: verum
end;
suppose W-min C in L~ do ; ::_thesis: contradiction
then  C meets L~ (Cage (C,n)) by A157, A158, XBOOLE_0:3;
hence  contradiction by JORDAN10:5; ::_thesis: verum
end;
end;
end;
right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1) = right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(GoB (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) by A97, JORDAN1H:23
.= right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(GoB (Cage (C,n)))) by REVROT_1:28
.= right_cell ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))),1,(Gauge (C,n))) by JORDAN1H:44
.= right_cell (((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n))))),1,(Gauge (C,n))) by A146, A150, JORDAN1J:53
.= right_cell ((Upper_Seq (C,n)),1,(Gauge (C,n))) by JORDAN1E:def_1
.= right_cell ((R_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i1,j)))),1,(Gauge (C,n))) by A36, A102, A147, JORDAN1J:52
.= right_cell ((go ^' pion1),1,(Gauge (C,n))) by A41, A104, JORDAN1J:51
.= right_cell (godo,1,(Gauge (C,n))) by A99, A106, JORDAN1J:51 ;
then  W-min C in right_cell (godo,1,(Gauge (C,n))) by JORDAN1I:6;
then A161: W-min C in (right_cell (godo,1,(Gauge (C,n)))) \ (L~ godo) by A159, XBOOLE_0:def_5;
A162: godo /. 1 = (go ^' pion1) /. 1 by GRAPH_2:53
.= W-min (L~ (Cage (C,n))) by A61, GRAPH_2:53 ;
A163: len (Upper_Seq (C,n)) >= 2 by A21, XXREAL_0:2;
A164: godo /. 2 = (go ^' pion1) /. 2 by A98, GRAPH_2:57
.= (Upper_Seq (C,n)) /. 2 by A35, A78, GRAPH_2:57
.= ((Upper_Seq (C,n)) ^' (Lower_Seq (C,n))) /. 2 by A163, GRAPH_2:57
.= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 2 by JORDAN1E:11 ;
A165: (L~ go) \/ (L~ do) is compact by COMPTS_1:10;
 W-min (L~ (Cage (C,n))) in (L~ go) \/ (L~ do) by A63, A79, XBOOLE_0:def_3;
then A166: W-min ((L~ go) \/ (L~ do)) = W-min (L~ (Cage (C,n))) by A156, A157, A165, JORDAN1J:21, XBOOLE_1:8;
A167: (W-min ((L~ go) \/ (L~ do))) `1 = W-bound ((L~ go) \/ (L~ do)) by EUCLID:52;
A168: (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52;
 ((Gauge (C,n)) * (i2,k)) `1 >= W-bound (L~ (Cage (C,n))) by A13, A155, PSCOMP_1:24;
then  ((Gauge (C,n)) * (i2,k)) `1 > W-bound (L~ (Cage (C,n))) by A77, XXREAL_0:1;
then  W-min (((L~ go) \/ (L~ do)) \/ (L~ pion1)) = W-min ((L~ go) \/ (L~ do)) by A95, A165, A166, A167, A168, JORDAN1J:33;
then A169: W-min (L~ godo) = W-min (L~ (Cage (C,n))) by A152, A166, XBOOLE_1:4;
A170: rng godo c= L~ godo by A98, A100, SPPOL_2:18, XXREAL_0:2;
 2 in dom godo by A101, FINSEQ_3:25;
then A171: godo /. 2 in rng godo by PARTFUN2:2;
 godo /. 2 in W-most (L~ (Cage (C,n))) by A164, JORDAN1I:25;
then (godo /. 2) `1 = (W-min (L~ godo)) `1 by A169, PSCOMP_1:31
.= W-bound (L~ godo) by EUCLID:52 ;
then  godo /. 2 in W-most (L~ godo) by A170, A171, SPRECT_2:12;
then  (Rotate (godo,(W-min (L~ godo)))) /. 2 in W-most (L~ godo) by A162, A169, FINSEQ_6:89;
then reconsider godo = godo as non constant standard clockwise_oriented special_circular_sequence by JORDAN1I:25;
 len (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by FINSEQ_5:6;
then A172: (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) = (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) by PARTFUN1:def_6
.= E-max (L~ (Cage (C,n))) by JORDAN1F:7 ;
A173: east_halfline (E-max C) misses L~ go
proof
assume  east_halfline (E-max C) meets L~ go ; ::_thesis: contradiction
then consider p being set  such that
A174: p in east_halfline (E-max C) and
A175: p in L~ go by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A174;
 p in L~ (Upper_Seq (C,n)) by A48, A175;
then  p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A154, A174, XBOOLE_0:def_4;
then A176: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;
then A177: p = E-max (L~ (Cage (C,n))) by A48, A175, JORDAN1J:46;
then  E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i1,j) by A14, A172, A175, JORDAN1J:43;
then  ((Gauge (C,n)) * (i1,j)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),k)) `1 by A6, A16, A20, A176, A177, JORDAN1A:71;
hence  contradiction by A3, A17, A32, JORDAN1G:7; ::_thesis: verum
end;
now__::_thesis:_not_east_halfline_(E-max_C)_meets_L~_godo
assume  east_halfline (E-max C) meets L~ godo ; ::_thesis: contradiction
then A178: ( east_halfline (E-max C) meets (L~ go) \/ (L~ pion1) or east_halfline (E-max C) meets L~ do ) by A152, XBOOLE_1:70;
percases ( east_halfline (E-max C) meets L~ go or east_halfline (E-max C) meets L~ pion1 or east_halfline (E-max C) meets L~ do ) by A178, XBOOLE_1:70;
suppose east_halfline (E-max C) meets L~ go ; ::_thesis: contradiction
hence  contradiction by A173; ::_thesis: verum
end;
suppose east_halfline (E-max C) meets L~ pion1 ; ::_thesis: contradiction
then consider p being set  such that
A179: p in east_halfline (E-max C) and
A180: p in L~ pion1 by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A179;
A181: now__::_thesis:_p_`1_<=_((Gauge_(C,n))_*_(i1,j))_`1
percases ( p in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) or p in LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) ) by A12, A87, A180, XBOOLE_0:def_3;
suppose p in LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) ; ::_thesis: p `1 <= ((Gauge (C,n)) * (i1,j)) `1
hence  p `1 <= ((Gauge (C,n)) * (i1,j)) `1 by A91, A92, TOPREAL1:3; ::_thesis: verum
end;
suppose p in LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) ; ::_thesis: p `1 <= ((Gauge (C,n)) * (i1,j)) `1
hence  p `1 <= ((Gauge (C,n)) * (i1,j)) `1 by A91, GOBOARD7:5; ::_thesis: verum
end;
end;
end;
 i1 + 1 <= len (Gauge (C,n)) by A3, NAT_1:13;
then  i1 <= (len (Gauge (C,n))) - 1 by XREAL_1:19;
then A182: i1 <= (len (Gauge (C,n))) -' 1 by XREAL_0:def_2;
 (len (Gauge (C,n))) -' 1 <= len (Gauge (C,n)) by NAT_D:35;
then  ((Gauge (C,n)) * (i1,j)) `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1 by A4, A10, A15, A20, A24, A182, JORDAN1A:18;
then  p `1 <= ((Gauge (C,n)) * (((len (Gauge (C,n))) -' 1),1)) `1 by A181, XXREAL_0:2;
then  p `1 <= E-bound C by A24, JORDAN8:12;
then A183: p `1 <= (E-max C) `1 by EUCLID:52;
 p `1 >= (E-max C) `1 by A179, TOPREAL1:def_11;
then A184: p `1 = (E-max C) `1 by A183, XXREAL_0:1;
 p `2 = (E-max C) `2 by A179, TOPREAL1:def_11;
then  p = E-max C by A184, TOPREAL3:6;
hence  contradiction by A9, A12, A87, A143, A180, XBOOLE_0:3; ::_thesis: verum
end;
suppose east_halfline (E-max C) meets L~ do ; ::_thesis: contradiction
then consider p being set  such that
A185: p in east_halfline (E-max C) and
A186: p in L~ do by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A185;
 p in L~ (Lower_Seq (C,n)) by A55, A186;
then  p in (east_halfline (E-max C)) /\ (L~ (Cage (C,n))) by A155, A185, XBOOLE_0:def_4;
then A187: p `1 = E-bound (L~ (Cage (C,n))) by JORDAN1A:83, PSCOMP_1:50;
A188: (E-max C) `2 = p `2 by A185, TOPREAL1:def_11;
set RC = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))));
A189: E-max C in right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by JORDAN1I:7;
A190: 1 + 1 <= len (Lower_Seq (C,n)) by A27, XXREAL_0:2;
 Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) by JORDAN1G:18;
then A191: LSeg ((Lower_Seq (C,n)),1) = LSeg ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) by A190, SPPOL_2:9;
A192: L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33;
A193: len (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = len (Cage (C,n)) by REVROT_1:14;
A194: GoB (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = GoB (Cage (C,n)) by REVROT_1:28
.= Gauge (C,n) by JORDAN1H:44 ;
A195: E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
A196: Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))) is_sequence_on Gauge (C,n) by A149, REVROT_1:34;
A197: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = E-max (L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))))) by A192, A195, FINSEQ_6:92;
consider ii, jj being Element of NAT  such that
A198: [ii,(jj + 1)] in Indices (Gauge (C,n)) and
A199: [ii,jj] in Indices (Gauge (C,n)) and
A200: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = (Gauge (C,n)) * (ii,(jj + 1)) and
A201: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1) = (Gauge (C,n)) * (ii,jj) by A96, A192, A193, A195, A196, FINSEQ_6:92, JORDAN1I:23;
consider jj2 being Element of NAT  such that
A202: 1 <= jj2 and
A203: jj2 <= width (Gauge (C,n)) and
A204: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * ((len (Gauge (C,n))),jj2) by JORDAN1D:25;
A205: len (Gauge (C,n)) >= 4 by JORDAN8:10;
then  len (Gauge (C,n)) >= 1 by XXREAL_0:2;
then  [(len (Gauge (C,n))),jj2] in Indices (Gauge (C,n)) by A202, A203, MATRIX_1:36;
then A206: ii = len (Gauge (C,n)) by A192, A197, A198, A200, A204, GOBOARD1:5;
A207: 1 <= ii by A198, MATRIX_1:38;
A208: ii <= len (Gauge (C,n)) by A198, MATRIX_1:38;
A209: 1 <= jj + 1 by A198, MATRIX_1:38;
A210: jj + 1 <= width (Gauge (C,n)) by A198, MATRIX_1:38;
A211: 1 <= ii by A199, MATRIX_1:38;
A212: ii <= len (Gauge (C,n)) by A199, MATRIX_1:38;
A213: 1 <= jj by A199, MATRIX_1:38;
A214: jj <= width (Gauge (C,n)) by A199, MATRIX_1:38;
A215: ii + 1 <> ii ;
 (jj + 1) + 1 <> jj ;
then A216: right_cell ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))),1) = cell ((Gauge (C,n)),(ii -' 1),jj) by A96, A193, A194, A198, A199, A200, A201, A215, GOBOARD5:def_6;
A217: (ii -' 1) + 1 = ii by A207, XREAL_1:235;
 ii - 1 >= 4 - 1 by A205, A206, XREAL_1:9;
then A218: ii - 1 >= 1 by XXREAL_0:2;
then A219: 1 <= ii -' 1 by XREAL_0:def_2;
A220: ((Gauge (C,n)) * ((ii -' 1),jj)) `2 <= p `2 by A188, A189, A208, A210, A213, A216, A217, A218, JORDAN9:17;
A221: p `2 <= ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 by A188, A189, A208, A210, A213, A216, A217, A218, JORDAN9:17;
A222: ii -' 1 < len (Gauge (C,n)) by A208, A217, NAT_1:13;
then A223: ((Gauge (C,n)) * ((ii -' 1),jj)) `2 = ((Gauge (C,n)) * (1,jj)) `2 by A213, A214, A219, GOBOARD5:1
.= ((Gauge (C,n)) * (ii,jj)) `2 by A211, A212, A213, A214, GOBOARD5:1 ;
A224: ((Gauge (C,n)) * ((ii -' 1),(jj + 1))) `2 = ((Gauge (C,n)) * (1,(jj + 1))) `2 by A209, A210, A219, A222, GOBOARD5:1
.= ((Gauge (C,n)) * (ii,(jj + 1))) `2 by A207, A208, A209, A210, GOBOARD5:1 ;
A225: ((Gauge (C,n)) * ((len (Gauge (C,n))),jj)) `1 = E-bound (L~ (Cage (C,n))) by A20, A213, A214, JORDAN1A:71;
 E-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * ((len (Gauge (C,n))),(jj + 1))) `1 by A20, A209, A210, JORDAN1A:71;
then  p in LSeg (((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1),((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. (1 + 1))) by A187, A200, A201, A206, A220, A221, A223, A224, A225, GOBOARD7:7;
then A226: p in LSeg ((Lower_Seq (C,n)),1) by A96, A191, A193, TOPREAL1:def_3;
A227: p in LSeg (do,(Index (p,do))) by A186, JORDAN3:9;
A228: do = mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n)))) by A39, JORDAN1J:37;
A229: 1 <= ((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)) by A39, FINSEQ_4:21;
A230: ((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)) <= len (Lower_Seq (C,n)) by A39, FINSEQ_4:21;
 ((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)) <> len (Lower_Seq (C,n)) by A31, A39, FINSEQ_4:19;
then A231: ((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)) < len (Lower_Seq (C,n)) by A230, XXREAL_0:1;
A232: 1 <= Index (p,do) by A186, JORDAN3:8;
A233: Index (p,do) < len do by A186, JORDAN3:8;
A234: (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n)))) + 1 = ((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)) by A34, A39, JORDAN1J:56;
consider t being Nat such that
A235: t in dom (Lower_Seq (C,n)) and
A236: (Lower_Seq (C,n)) . t = (Gauge (C,n)) * (i2,k) by A39, FINSEQ_2:10;
A237: 1 <= t by A235, FINSEQ_3:25;
A238: t <= len (Lower_Seq (C,n)) by A235, FINSEQ_3:25;
 1 < t by A34, A236, A237, XXREAL_0:1;
then  (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n)))) + 1 = t by A236, A238, JORDAN3:12;
then A239: len (L_Cut ((Lower_Seq (C,n)),((Gauge (C,n)) * (i2,k)))) = (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n)))) by A13, A236, JORDAN3:26;
set tt = ((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1;
A240: 1 <= Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n))) by A13, JORDAN3:8;
 0 + (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n)))) < len (Lower_Seq (C,n)) by A13, JORDAN3:8;
then A241: (len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n)))) > 0 by XREAL_1:20;
 Index (p,do) < (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n)))) by A233, A239, XREAL_0:def_2;
then  (Index (p,do)) + 1 <= (len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n)))) by NAT_1:13;
then  Index (p,do) <= ((len (Lower_Seq (C,n))) -' (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n))))) - 1 by XREAL_1:19;
then  Index (p,do) <= ((len (Lower_Seq (C,n))) - (Index (((Gauge (C,n)) * (i2,k)),(Lower_Seq (C,n))))) - 1 by A241, XREAL_0:def_2;
then  Index (p,do) <= (len (Lower_Seq (C,n))) - (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n))) by A234;
then  Index (p,do) <= (len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n))) by XREAL_0:def_2;
then  Index (p,do) < ((len (Lower_Seq (C,n))) -' (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) + 1 by NAT_1:13;
then A242: LSeg ((mid ((Lower_Seq (C,n)),(((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n))),(len (Lower_Seq (C,n))))),(Index (p,do))) = LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1)) by A229, A231, A232, JORDAN4:19;
A243: 1 + 1 <= ((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)) by A234, A240, XREAL_1:7;
then  (Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n))) >= (1 + 1) + 1 by A232, XREAL_1:7;
then  ((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) - 1 >= ((1 + 1) + 1) - 1 by XREAL_1:9;
then A244: ((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1 >= 1 + 1 by XREAL_0:def_2;
A245: 2 in dom (Lower_Seq (C,n)) by A190, FINSEQ_3:25;
now__::_thesis:_contradiction
percases ( ((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1 > 1 + 1 or ((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ) by A244, XXREAL_0:1;
suppose ((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1 > 1 + 1 ; ::_thesis: contradiction
then  LSeg ((Lower_Seq (C,n)),1) misses LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1)) by TOPREAL1:def_7;
hence  contradiction by A226, A227, A228, A242, XBOOLE_0:3; ::_thesis: verum
end;
supposeA246: ((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1 = 1 + 1 ; ::_thesis: contradiction
then  (LSeg ((Lower_Seq (C,n)),1)) /\ (LSeg ((Lower_Seq (C,n)),(((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) -' 1))) = {((Lower_Seq (C,n)) /. 2)} by A27, TOPREAL1:def_6;
then  p in {((Lower_Seq (C,n)) /. 2)} by A226, A227, A228, A242, XBOOLE_0:def_4;
then A247: p = (Lower_Seq (C,n)) /. 2 by TARSKI:def_1;
then A248: p .. (Lower_Seq (C,n)) = 2 by A245, FINSEQ_5:41;
 1 + 1 = ((Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)))) - 1 by A246, XREAL_0:def_2;
then  (1 + 1) + 1 = (Index (p,do)) + (((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n))) ;
then A249: ((Gauge (C,n)) * (i2,k)) .. (Lower_Seq (C,n)) = 2 by A232, A243, JORDAN1E:6;
 p in rng (Lower_Seq (C,n)) by A245, A247, PARTFUN2:2;
then  p = (Gauge (C,n)) * (i2,k) by A39, A248, A249, FINSEQ_5:9;
then  ((Gauge (C,n)) * (i2,k)) `1 = E-bound (L~ (Cage (C,n))) by A247, JORDAN1G:32;
then  ((Gauge (C,n)) * (i2,k)) `1 = ((Gauge (C,n)) * ((len (Gauge (C,n))),j)) `1 by A4, A15, A20, JORDAN1A:71;
hence  contradiction by A2, A3, A18, A69, JORDAN1G:7; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
end;
end;
then  east_halfline (E-max C) c= (L~ godo) ` by SUBSET_1:23;
then consider W being Subset of (TOP-REAL 2) such that
A250: W is_a_component_of (L~ godo) ` and
A251: east_halfline (E-max C) c= W by GOBOARD9:3;
 not W is bounded by A251, JORDAN2C:121, RLTOPSP1:42;
then  W is_outside_component_of L~ godo by A250, JORDAN2C:def_3;
then  W c= UBD (L~ godo) by JORDAN2C:23;
then A252: east_halfline (E-max C) c= UBD (L~ godo) by A251, XBOOLE_1:1;
 E-max C in east_halfline (E-max C) by TOPREAL1:38;
then  E-max C in UBD (L~ godo) by A252;
then  E-max C in LeftComp godo by GOBRD14:36;
then  Lower_Arc C meets L~ godo by A141, A142, A143, A151, A161, JORDAN1J:36;
then A253: ( Lower_Arc C meets (L~ go) \/ (L~ pion1) or Lower_Arc C meets L~ do ) by A152, XBOOLE_1:70;
A254: Lower_Arc C c= C by JORDAN6:61;
now__::_thesis:_contradiction
percases ( Lower_Arc C meets L~ go or Lower_Arc C meets L~ pion1 or Lower_Arc C meets L~ do ) by A253, XBOOLE_1:70;
suppose Lower_Arc C meets L~ go ; ::_thesis: contradiction
then  Lower_Arc C meets L~ (Cage (C,n)) by A48, A154, XBOOLE_1:1, XBOOLE_1:63;
hence  contradiction by A254, JORDAN10:5, XBOOLE_1:63; ::_thesis: verum
end;
suppose Lower_Arc C meets L~ pion1 ; ::_thesis: contradiction
hence  contradiction by A9, A12, A87; ::_thesis: verum
end;
suppose Lower_Arc C meets L~ do ; ::_thesis: contradiction
then  Lower_Arc C meets L~ (Cage (C,n)) by A55, A155, XBOOLE_1:1, XBOOLE_1:63;
hence  contradiction by A254, JORDAN10:5, XBOOLE_1:63; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
suppose ((Gauge (C,n)) * (i2,k)) `1 = ((Gauge (C,n)) * (i1,j)) `1 ; ::_thesis: contradiction
then A255: i1 = i2 by A17, A18, JORDAN1G:7;
then  LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) = {((Gauge (C,n)) * (i1,k))} by RLTOPSP1:70;
then  LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) c= LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) by A83, ZFMISC_1:31;
then  (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) = LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) by XBOOLE_1:12;
hence  contradiction by A1, A3, A4, A5, A6, A7, A8, A9, A255, Th13; ::_thesis: verum
end;
suppose ((Gauge (C,n)) * (i2,k)) `2 = ((Gauge (C,n)) * (i1,j)) `2 ; ::_thesis: contradiction
then A256: j = k by A17, A18, JORDAN1G:6;
then  LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) = {((Gauge (C,n)) * (i1,k))} by RLTOPSP1:70;
then  LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k))) c= LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) by A84, ZFMISC_1:31;
then  (LSeg (((Gauge (C,n)) * (i1,j)),((Gauge (C,n)) * (i1,k)))) \/ (LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k)))) = LSeg (((Gauge (C,n)) * (i1,k)),((Gauge (C,n)) * (i2,k))) by XBOOLE_1:12;
hence  contradiction by A1, A2, A3, A4, A6, A7, A8, A9, A256, JORDAN15:28; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;

theorem Th24: :: JORDAN19:24
for n being Element of NAT
for C being Simple_closed_curve
for i1, i2, j, k being Element of NAT st 1 < i1 & i1 < len (Gauge (C,(n + 1))) & 1 < i2 & i2 < len (Gauge (C,(n + 1))) & 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (i1,k) in Lower_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * (i2,j) in Upper_Arc (L~ (Cage (C,(n + 1)))) holds
(LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Lower_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i1, i2, j, k being Element of NAT st 1 < i1 & i1 < len (Gauge (C,(n + 1))) & 1 < i2 & i2 < len (Gauge (C,(n + 1))) & 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (i1,k) in Lower_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * (i2,j) in Upper_Arc (L~ (Cage (C,(n + 1)))) holds
(LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Lower_Arc C
let C be Simple_closed_curve; ::_thesis: for i1, i2, j, k being Element of NAT st 1 < i1 & i1 < len (Gauge (C,(n + 1))) & 1 < i2 & i2 < len (Gauge (C,(n + 1))) & 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (i1,k) in Lower_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * (i2,j) in Upper_Arc (L~ (Cage (C,(n + 1)))) holds
(LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Lower_Arc C
let i1, i2, j, k be Element of NAT ; ::_thesis: ( 1 < i1 & i1 < len (Gauge (C,(n + 1))) & 1 < i2 & i2 < len (Gauge (C,(n + 1))) & 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (i1,k) in Lower_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * (i2,j) in Upper_Arc (L~ (Cage (C,(n + 1)))) implies (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Lower_Arc C )
set G = Gauge (C,(n + 1));
assume that
A1: 1 < i1 and
A2: i1 < len (Gauge (C,(n + 1))) and
A3: 1 < i2 and
A4: i2 < len (Gauge (C,(n + 1))) and
A5: 1 <= j and
A6: j <= k and
A7: k <= width (Gauge (C,(n + 1))) and
A8: (Gauge (C,(n + 1))) * (i1,k) in Lower_Arc (L~ (Cage (C,(n + 1)))) and
A9: (Gauge (C,(n + 1))) * (i2,j) in Upper_Arc (L~ (Cage (C,(n + 1)))) ; ::_thesis: (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Lower_Arc C
A10: Lower_Arc (L~ (Cage (C,(n + 1)))) = L~ (Lower_Seq (C,(n + 1))) by JORDAN1G:56;
A11: Upper_Arc (L~ (Cage (C,(n + 1)))) = L~ (Upper_Seq (C,(n + 1))) by JORDAN1G:55;
A12: j <= width (Gauge (C,(n + 1))) by A6, A7, XXREAL_0:2;
then A13: [i2,j] in Indices (Gauge (C,(n + 1))) by A3, A4, A5, MATRIX_1:36;
A14: 1 <= k by A5, A6, XXREAL_0:2;
then A15: [i2,k] in Indices (Gauge (C,(n + 1))) by A3, A4, A7, MATRIX_1:36;
((Gauge (C,(n + 1))) * (i2,j)) `1 = ((Gauge (C,(n + 1))) * (i2,1)) `1 by A3, A4, A5, A12, GOBOARD5:2
.= ((Gauge (C,(n + 1))) * (i2,k)) `1 by A3, A4, A7, A14, GOBOARD5:2 ;
then A16: LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k))) is vertical by SPPOL_1:16;
((Gauge (C,(n + 1))) * (i2,k)) `2 = ((Gauge (C,(n + 1))) * (1,k)) `2 by A3, A4, A7, A14, GOBOARD5:1
.= ((Gauge (C,(n + 1))) * (i1,k)) `2 by A1, A2, A7, A14, GOBOARD5:1 ;
then A17: LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) is horizontal by SPPOL_1:15;
A18: [i2,k] in Indices (Gauge (C,(n + 1))) by A3, A4, A7, A14, MATRIX_1:36;
A19: [i1,k] in Indices (Gauge (C,(n + 1))) by A1, A2, A7, A14, MATRIX_1:36;
now__::_thesis:_(LSeg_(((Gauge_(C,(n_+_1)))_*_(i2,j)),((Gauge_(C,(n_+_1)))_*_(i2,k))))_\/_(LSeg_(((Gauge_(C,(n_+_1)))_*_(i2,k)),((Gauge_(C,(n_+_1)))_*_(i1,k))))_meets_Lower_Arc_C
percases ( LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k))) meets Lower_Arc (L~ (Cage (C,(n + 1)))) or ( LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) meets Upper_Arc (L~ (Cage (C,(n + 1)))) & i2 <= i1 ) or ( LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) meets Upper_Arc (L~ (Cage (C,(n + 1)))) & i1 < i2 ) or ( LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k))) misses Lower_Arc (L~ (Cage (C,(n + 1)))) & LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) misses Upper_Arc (L~ (Cage (C,(n + 1)))) ) ) ;
supposeA20: LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k))) meets Lower_Arc (L~ (Cage (C,(n + 1)))) ; ::_thesis: (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Lower_Arc C
then consider m being Element of NAT  such that
A21: j <= m and
A22: m <= k and
A23: ((Gauge (C,(n + 1))) * (i2,m)) `2 = lower_bound (proj2 .: ((LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))))) by A6, A10, A13, A15, JORDAN1F:1, JORDAN1G:5;
A24: 1 <= m by A5, A21, XXREAL_0:2;
A25: m <= width (Gauge (C,(n + 1))) by A7, A22, XXREAL_0:2;
set X = (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1))));
A26: ((Gauge (C,(n + 1))) * (i2,m)) `1 = ((Gauge (C,(n + 1))) * (i2,1)) `1 by A3, A4, A24, A25, GOBOARD5:2;
then A27: |[(((Gauge (C,(n + 1))) * (i2,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))))))]| = (Gauge (C,(n + 1))) * (i2,m) by A23, EUCLID:53;
then A28: ((Gauge (C,(n + 1))) * (i2,j)) `1 = |[(((Gauge (C,(n + 1))) * (i2,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))))))]| `1 by A3, A4, A5, A12, A26, GOBOARD5:2;
 ex x being set st
( x in LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k))) & x in L~ (Lower_Seq (C,(n + 1))) ) by A10, A20, XBOOLE_0:3;
then reconsider X1 = (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))) as non empty compact Subset of (TOP-REAL 2) by XBOOLE_0:def_4;
consider pp being set  such that
A29: pp in S-most X1 by XBOOLE_0:def_1;
reconsider pp = pp as Point of (TOP-REAL 2) by A29;
A30: pp in (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))) by A29, XBOOLE_0:def_4;
then A31: pp in L~ (Lower_Seq (C,(n + 1))) by XBOOLE_0:def_4;
 pp in LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k))) by A30, XBOOLE_0:def_4;
then A32: pp `1 = |[(((Gauge (C,(n + 1))) * (i2,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))))))]| `1 by A16, A28, SPPOL_1:41;
|[(((Gauge (C,(n + 1))) * (i2,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))))))]| `2 = S-bound ((LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1))))) by A23, A27, SPRECT_1:44
.= (S-min ((LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))))) `2 by EUCLID:52
.= pp `2 by A29, PSCOMP_1:55 ;
then  (Gauge (C,(n + 1))) * (i2,m) in Lower_Arc (L~ (Cage (C,(n + 1)))) by A10, A27, A31, A32, TOPREAL3:6;
then  LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,m))) meets Lower_Arc C by A3, A4, A5, A9, A21, A25, Th19;
then  LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k))) meets Lower_Arc C by A3, A4, A5, A7, A21, A22, JORDAN15:5, XBOOLE_1:63;
hence  (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Lower_Arc C by XBOOLE_1:70; ::_thesis: verum
end;
supposeA33: ( LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) meets Upper_Arc (L~ (Cage (C,(n + 1)))) & i2 <= i1 ) ; ::_thesis: (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Lower_Arc C
then consider m being Element of NAT  such that
A34: i2 <= m and
A35: m <= i1 and
A36: ((Gauge (C,(n + 1))) * (m,k)) `1 = upper_bound (proj1 .: ((LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))))) by A11, A18, A19, JORDAN1F:4, JORDAN1G:4;
A37: 1 < m by A3, A34, XXREAL_0:2;
A38: m < len (Gauge (C,(n + 1))) by A2, A35, XXREAL_0:2;
set X = (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) /\ (L~ (Upper_Seq (C,(n + 1))));
A39: ((Gauge (C,(n + 1))) * (m,k)) `2 = ((Gauge (C,(n + 1))) * (1,k)) `2 by A7, A14, A37, A38, GOBOARD5:1;
then A40: |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) /\ (L~ (Upper_Seq (C,(n + 1))))))),(((Gauge (C,(n + 1))) * (1,k)) `2)]| = (Gauge (C,(n + 1))) * (m,k) by A36, EUCLID:53;
then A41: ((Gauge (C,(n + 1))) * (i2,k)) `2 = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) /\ (L~ (Upper_Seq (C,(n + 1))))))),(((Gauge (C,(n + 1))) * (1,k)) `2)]| `2 by A3, A4, A7, A14, A39, GOBOARD5:1;
 ex x being set st
( x in LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) & x in L~ (Upper_Seq (C,(n + 1))) ) by A11, A33, XBOOLE_0:3;
then reconsider X1 = (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))) as non empty compact Subset of (TOP-REAL 2) by XBOOLE_0:def_4;
consider pp being set  such that
A42: pp in E-most X1 by XBOOLE_0:def_1;
reconsider pp = pp as Point of (TOP-REAL 2) by A42;
A43: pp in (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))) by A42, XBOOLE_0:def_4;
then A44: pp in L~ (Upper_Seq (C,(n + 1))) by XBOOLE_0:def_4;
 pp in LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) by A43, XBOOLE_0:def_4;
then A45: pp `2 = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) /\ (L~ (Upper_Seq (C,(n + 1))))))),(((Gauge (C,(n + 1))) * (1,k)) `2)]| `2 by A17, A41, SPPOL_1:40;
|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) /\ (L~ (Upper_Seq (C,(n + 1))))))),(((Gauge (C,(n + 1))) * (1,k)) `2)]| `1 = E-bound ((LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) /\ (L~ (Upper_Seq (C,(n + 1))))) by A36, A40, SPRECT_1:46
.= (E-min ((LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))))) `1 by EUCLID:52
.= pp `1 by A42, PSCOMP_1:47 ;
then  (Gauge (C,(n + 1))) * (m,k) in Upper_Arc (L~ (Cage (C,(n + 1)))) by A11, A40, A44, A45, TOPREAL3:6;
then  LSeg (((Gauge (C,(n + 1))) * (m,k)),((Gauge (C,(n + 1))) * (i1,k))) meets Lower_Arc C by A2, A7, A8, A14, A35, A37, JORDAN15:40;
then  LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) meets Lower_Arc C by A2, A3, A7, A14, A34, A35, JORDAN15:6, XBOOLE_1:63;
hence  (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Lower_Arc C by XBOOLE_1:70; ::_thesis: verum
end;
supposeA46: ( LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) meets Upper_Arc (L~ (Cage (C,(n + 1)))) & i1 < i2 ) ; ::_thesis: (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Lower_Arc C
then consider m being Element of NAT  such that
A47: i1 <= m and
A48: m <= i2 and
A49: ((Gauge (C,(n + 1))) * (m,k)) `1 = lower_bound (proj1 .: ((LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))))) by A11, A18, A19, JORDAN1F:3, JORDAN1G:4;
A50: 1 < m by A1, A47, XXREAL_0:2;
A51: m < len (Gauge (C,(n + 1))) by A4, A48, XXREAL_0:2;
set X = (LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1))));
A52: ((Gauge (C,(n + 1))) * (m,k)) `2 = ((Gauge (C,(n + 1))) * (1,k)) `2 by A7, A14, A50, A51, GOBOARD5:1;
then A53: |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1))))))),(((Gauge (C,(n + 1))) * (1,k)) `2)]| = (Gauge (C,(n + 1))) * (m,k) by A49, EUCLID:53;
then A54: ((Gauge (C,(n + 1))) * (i1,k)) `2 = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1))))))),(((Gauge (C,(n + 1))) * (1,k)) `2)]| `2 by A1, A2, A7, A14, A52, GOBOARD5:1;
 ex x being set st
( x in LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k))) & x in L~ (Upper_Seq (C,(n + 1))) ) by A11, A46, XBOOLE_0:3;
then reconsider X1 = (LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))) as non empty compact Subset of (TOP-REAL 2) by XBOOLE_0:def_4;
consider pp being set  such that
A55: pp in W-most X1 by XBOOLE_0:def_1;
reconsider pp = pp as Point of (TOP-REAL 2) by A55;
A56: pp in (LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))) by A55, XBOOLE_0:def_4;
then A57: pp in L~ (Upper_Seq (C,(n + 1))) by XBOOLE_0:def_4;
 pp in LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k))) by A56, XBOOLE_0:def_4;
then A58: pp `2 = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1))))))),(((Gauge (C,(n + 1))) * (1,k)) `2)]| `2 by A17, A54, SPPOL_1:40;
|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1))))))),(((Gauge (C,(n + 1))) * (1,k)) `2)]| `1 = W-bound ((LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1))))) by A49, A53, SPRECT_1:43
.= (W-min ((LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))))) `1 by EUCLID:52
.= pp `1 by A55, PSCOMP_1:31 ;
then  (Gauge (C,(n + 1))) * (m,k) in Upper_Arc (L~ (Cage (C,(n + 1)))) by A11, A53, A57, A58, TOPREAL3:6;
then  LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (m,k))) meets Lower_Arc C by A1, A7, A8, A14, A47, A51, JORDAN15:32;
then  LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k))) meets Lower_Arc C by A1, A4, A7, A14, A47, A48, JORDAN15:6, XBOOLE_1:63;
hence  (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Lower_Arc C by XBOOLE_1:70; ::_thesis: verum
end;
supposeA59: ( LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k))) misses Lower_Arc (L~ (Cage (C,(n + 1)))) & LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) misses Upper_Arc (L~ (Cage (C,(n + 1)))) ) ; ::_thesis: (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Lower_Arc C
consider j1 being Element of NAT  such that
A60: j <= j1 and
A61: j1 <= k and
A62: (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))) = {((Gauge (C,(n + 1))) * (i2,j1))} by A3, A4, A5, A6, A7, A9, A11, JORDAN15:15;
 (Gauge (C,(n + 1))) * (i2,j1) in (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))) by A62, TARSKI:def_1;
then A63: (Gauge (C,(n + 1))) * (i2,j1) in L~ (Upper_Seq (C,(n + 1))) by XBOOLE_0:def_4;
A64: 1 <= j1 by A5, A60, XXREAL_0:2;
now__::_thesis:_(LSeg_(((Gauge_(C,(n_+_1)))_*_(i2,j)),((Gauge_(C,(n_+_1)))_*_(i2,k))))_\/_(LSeg_(((Gauge_(C,(n_+_1)))_*_(i2,k)),((Gauge_(C,(n_+_1)))_*_(i1,k))))_meets_Lower_Arc_C
percases ( i2 <= i1 or i1 < i2 ) ;
suppose i2 <= i1 ; ::_thesis: (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Lower_Arc C
then consider i3 being Element of NAT  such that
A65: i2 <= i3 and
A66: i3 <= i1 and
A67: (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))) = {((Gauge (C,(n + 1))) * (i3,k))} by A2, A3, A7, A8, A10, A14, JORDAN15:19;
A68: i3 < len (Gauge (C,(n + 1))) by A2, A66, XXREAL_0:2;
 (Gauge (C,(n + 1))) * (i3,k) in (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))) by A67, TARSKI:def_1;
then A69: (Gauge (C,(n + 1))) * (i3,k) in L~ (Lower_Seq (C,(n + 1))) by XBOOLE_0:def_4;
A70: LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k))) c= LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k))) by A3, A4, A5, A7, A60, A61, JORDAN15:5;
A71: LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))) c= LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) by A2, A3, A7, A14, A65, A66, JORDAN15:6;
then A72: (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) c= (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) by A70, XBOOLE_1:13;
A73: ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) = {((Gauge (C,(n + 1))) * (i3,k))}
proof
thus  ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) c= {((Gauge (C,(n + 1))) * (i3,k))} :: according to XBOOLE_0:def_10 ::_thesis: {((Gauge (C,(n + 1))) * (i3,k))} c= ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) or x in {((Gauge (C,(n + 1))) * (i3,k))} )
assume A74: x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) ; ::_thesis: x in {((Gauge (C,(n + 1))) * (i3,k))}
then  x in (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) by XBOOLE_0:def_4;
then A75: ( x in LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k))) or x in LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))) ) by XBOOLE_0:def_3;
 x in L~ (Lower_Seq (C,(n + 1))) by A74, XBOOLE_0:def_4;
hence  x in {((Gauge (C,(n + 1))) * (i3,k))} by A10, A59, A67, A70, A75, XBOOLE_0:3, XBOOLE_0:def_4; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,(n + 1))) * (i3,k))} or x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) )
assume  x in {((Gauge (C,(n + 1))) * (i3,k))} ; ::_thesis: x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1))))
then A76: x = (Gauge (C,(n + 1))) * (i3,k) by TARSKI:def_1;
 (Gauge (C,(n + 1))) * (i3,k) in LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))) by RLTOPSP1:68;
then  (Gauge (C,(n + 1))) * (i3,k) in (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) by XBOOLE_0:def_3;
hence  x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) by A69, A76, XBOOLE_0:def_4; ::_thesis: verum
end;
 ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) = {((Gauge (C,(n + 1))) * (i2,j1))}
proof
thus  ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) c= {((Gauge (C,(n + 1))) * (i2,j1))} :: according to XBOOLE_0:def_10 ::_thesis: {((Gauge (C,(n + 1))) * (i2,j1))} c= ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) or x in {((Gauge (C,(n + 1))) * (i2,j1))} )
assume A77: x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) ; ::_thesis: x in {((Gauge (C,(n + 1))) * (i2,j1))}
then  x in (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) by XBOOLE_0:def_4;
then A78: ( x in LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k))) or x in LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))) ) by XBOOLE_0:def_3;
 x in L~ (Upper_Seq (C,(n + 1))) by A77, XBOOLE_0:def_4;
hence  x in {((Gauge (C,(n + 1))) * (i2,j1))} by A11, A59, A62, A71, A78, XBOOLE_0:3, XBOOLE_0:def_4; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,(n + 1))) * (i2,j1))} or x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) )
assume  x in {((Gauge (C,(n + 1))) * (i2,j1))} ; ::_thesis: x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1))))
then A79: x = (Gauge (C,(n + 1))) * (i2,j1) by TARSKI:def_1;
 (Gauge (C,(n + 1))) * (i2,j1) in LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k))) by RLTOPSP1:68;
then  (Gauge (C,(n + 1))) * (i2,j1) in (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) by XBOOLE_0:def_3;
hence  x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) by A63, A79, XBOOLE_0:def_4; ::_thesis: verum
end;
hence  (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Lower_Arc C by A3, A7, A61, A64, A65, A68, A72, A73, Th21, XBOOLE_1:63; ::_thesis: verum
end;
suppose i1 < i2 ; ::_thesis: (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Lower_Arc C
then consider i3 being Element of NAT  such that
A80: i1 <= i3 and
A81: i3 <= i2 and
A82: (LSeg (((Gauge (C,(n + 1))) * (i3,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))) = {((Gauge (C,(n + 1))) * (i3,k))} by A1, A4, A7, A8, A10, A14, JORDAN15:12;
A83: 1 < i3 by A1, A80, XXREAL_0:2;
 (Gauge (C,(n + 1))) * (i3,k) in (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))) by A82, TARSKI:def_1;
then A84: (Gauge (C,(n + 1))) * (i3,k) in L~ (Lower_Seq (C,(n + 1))) by XBOOLE_0:def_4;
A85: LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k))) c= LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k))) by A3, A4, A5, A7, A60, A61, JORDAN15:5;
A86: LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))) c= LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) by A1, A4, A7, A14, A80, A81, JORDAN15:6;
then A87: (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) c= (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) by A85, XBOOLE_1:13;
A88: ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) = {((Gauge (C,(n + 1))) * (i3,k))}
proof
thus  ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) c= {((Gauge (C,(n + 1))) * (i3,k))} :: according to XBOOLE_0:def_10 ::_thesis: {((Gauge (C,(n + 1))) * (i3,k))} c= ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) or x in {((Gauge (C,(n + 1))) * (i3,k))} )
assume A89: x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) ; ::_thesis: x in {((Gauge (C,(n + 1))) * (i3,k))}
then  x in (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) by XBOOLE_0:def_4;
then A90: ( x in LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k))) or x in LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))) ) by XBOOLE_0:def_3;
 x in L~ (Lower_Seq (C,(n + 1))) by A89, XBOOLE_0:def_4;
hence  x in {((Gauge (C,(n + 1))) * (i3,k))} by A10, A59, A82, A85, A90, XBOOLE_0:3, XBOOLE_0:def_4; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,(n + 1))) * (i3,k))} or x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) )
assume  x in {((Gauge (C,(n + 1))) * (i3,k))} ; ::_thesis: x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1))))
then A91: x = (Gauge (C,(n + 1))) * (i3,k) by TARSKI:def_1;
 (Gauge (C,(n + 1))) * (i3,k) in LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))) by RLTOPSP1:68;
then  (Gauge (C,(n + 1))) * (i3,k) in (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) by XBOOLE_0:def_3;
hence  x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) by A84, A91, XBOOLE_0:def_4; ::_thesis: verum
end;
 ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) = {((Gauge (C,(n + 1))) * (i2,j1))}
proof
thus  ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) c= {((Gauge (C,(n + 1))) * (i2,j1))} :: according to XBOOLE_0:def_10 ::_thesis: {((Gauge (C,(n + 1))) * (i2,j1))} c= ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) or x in {((Gauge (C,(n + 1))) * (i2,j1))} )
assume A92: x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) ; ::_thesis: x in {((Gauge (C,(n + 1))) * (i2,j1))}
then  x in (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) by XBOOLE_0:def_4;
then A93: ( x in LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k))) or x in LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))) ) by XBOOLE_0:def_3;
 x in L~ (Upper_Seq (C,(n + 1))) by A92, XBOOLE_0:def_4;
hence  x in {((Gauge (C,(n + 1))) * (i2,j1))} by A11, A59, A62, A86, A93, XBOOLE_0:3, XBOOLE_0:def_4; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,(n + 1))) * (i2,j1))} or x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) )
assume  x in {((Gauge (C,(n + 1))) * (i2,j1))} ; ::_thesis: x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1))))
then A94: x = (Gauge (C,(n + 1))) * (i2,j1) by TARSKI:def_1;
 (Gauge (C,(n + 1))) * (i2,j1) in LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k))) by RLTOPSP1:68;
then  (Gauge (C,(n + 1))) * (i2,j1) in (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) by XBOOLE_0:def_3;
hence  x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) by A63, A94, XBOOLE_0:def_4; ::_thesis: verum
end;
hence  (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Lower_Arc C by A4, A7, A61, A64, A81, A83, A87, A88, Th23, XBOOLE_1:63; ::_thesis: verum
end;
end;
end;
hence  (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Lower_Arc C ; ::_thesis: verum
end;
end;
end;
hence  (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Lower_Arc C ; ::_thesis: verum
end;

theorem Th25: :: JORDAN19:25
for n being Element of NAT
for C being Simple_closed_curve
for i1, i2, j, k being Element of NAT st 1 < i1 & i1 < len (Gauge (C,(n + 1))) & 1 < i2 & i2 < len (Gauge (C,(n + 1))) & 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (i1,k) in Lower_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * (i2,j) in Upper_Arc (L~ (Cage (C,(n + 1)))) holds
(LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Upper_Arc C
proof
let n be Element of NAT ; ::_thesis: for C being Simple_closed_curve
for i1, i2, j, k being Element of NAT st 1 < i1 & i1 < len (Gauge (C,(n + 1))) & 1 < i2 & i2 < len (Gauge (C,(n + 1))) & 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (i1,k) in Lower_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * (i2,j) in Upper_Arc (L~ (Cage (C,(n + 1)))) holds
(LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Upper_Arc C
let C be Simple_closed_curve; ::_thesis: for i1, i2, j, k being Element of NAT st 1 < i1 & i1 < len (Gauge (C,(n + 1))) & 1 < i2 & i2 < len (Gauge (C,(n + 1))) & 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (i1,k) in Lower_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * (i2,j) in Upper_Arc (L~ (Cage (C,(n + 1)))) holds
(LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Upper_Arc C
let i1, i2, j, k be Element of NAT ; ::_thesis: ( 1 < i1 & i1 < len (Gauge (C,(n + 1))) & 1 < i2 & i2 < len (Gauge (C,(n + 1))) & 1 <= j & j <= k & k <= width (Gauge (C,(n + 1))) & (Gauge (C,(n + 1))) * (i1,k) in Lower_Arc (L~ (Cage (C,(n + 1)))) & (Gauge (C,(n + 1))) * (i2,j) in Upper_Arc (L~ (Cage (C,(n + 1)))) implies (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Upper_Arc C )
set G = Gauge (C,(n + 1));
assume that
A1: 1 < i1 and
A2: i1 < len (Gauge (C,(n + 1))) and
A3: 1 < i2 and
A4: i2 < len (Gauge (C,(n + 1))) and
A5: 1 <= j and
A6: j <= k and
A7: k <= width (Gauge (C,(n + 1))) and
A8: (Gauge (C,(n + 1))) * (i1,k) in Lower_Arc (L~ (Cage (C,(n + 1)))) and
A9: (Gauge (C,(n + 1))) * (i2,j) in Upper_Arc (L~ (Cage (C,(n + 1)))) ; ::_thesis: (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Upper_Arc C
A10: Lower_Arc (L~ (Cage (C,(n + 1)))) = L~ (Lower_Seq (C,(n + 1))) by JORDAN1G:56;
A11: Upper_Arc (L~ (Cage (C,(n + 1)))) = L~ (Upper_Seq (C,(n + 1))) by JORDAN1G:55;
A12: j <= width (Gauge (C,(n + 1))) by A6, A7, XXREAL_0:2;
then A13: [i2,j] in Indices (Gauge (C,(n + 1))) by A3, A4, A5, MATRIX_1:36;
A14: 1 <= k by A5, A6, XXREAL_0:2;
then A15: [i2,k] in Indices (Gauge (C,(n + 1))) by A3, A4, A7, MATRIX_1:36;
((Gauge (C,(n + 1))) * (i2,j)) `1 = ((Gauge (C,(n + 1))) * (i2,1)) `1 by A3, A4, A5, A12, GOBOARD5:2
.= ((Gauge (C,(n + 1))) * (i2,k)) `1 by A3, A4, A7, A14, GOBOARD5:2 ;
then A16: LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k))) is vertical by SPPOL_1:16;
((Gauge (C,(n + 1))) * (i2,k)) `2 = ((Gauge (C,(n + 1))) * (1,k)) `2 by A3, A4, A7, A14, GOBOARD5:1
.= ((Gauge (C,(n + 1))) * (i1,k)) `2 by A1, A2, A7, A14, GOBOARD5:1 ;
then A17: LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) is horizontal by SPPOL_1:15;
A18: [i2,k] in Indices (Gauge (C,(n + 1))) by A3, A4, A7, A14, MATRIX_1:36;
A19: [i1,k] in Indices (Gauge (C,(n + 1))) by A1, A2, A7, A14, MATRIX_1:36;
now__::_thesis:_(LSeg_(((Gauge_(C,(n_+_1)))_*_(i2,j)),((Gauge_(C,(n_+_1)))_*_(i2,k))))_\/_(LSeg_(((Gauge_(C,(n_+_1)))_*_(i2,k)),((Gauge_(C,(n_+_1)))_*_(i1,k))))_meets_Upper_Arc_C
percases ( LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k))) meets Lower_Arc (L~ (Cage (C,(n + 1)))) or ( LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) meets Upper_Arc (L~ (Cage (C,(n + 1)))) & i2 <= i1 ) or ( LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) meets Upper_Arc (L~ (Cage (C,(n + 1)))) & i1 < i2 ) or ( LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k))) misses Lower_Arc (L~ (Cage (C,(n + 1)))) & LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) misses Upper_Arc (L~ (Cage (C,(n + 1)))) ) ) ;
supposeA20: LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k))) meets Lower_Arc (L~ (Cage (C,(n + 1)))) ; ::_thesis: (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Upper_Arc C
then consider m being Element of NAT  such that
A21: j <= m and
A22: m <= k and
A23: ((Gauge (C,(n + 1))) * (i2,m)) `2 = lower_bound (proj2 .: ((LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))))) by A6, A10, A13, A15, JORDAN1F:1, JORDAN1G:5;
A24: 1 <= m by A5, A21, XXREAL_0:2;
A25: m <= width (Gauge (C,(n + 1))) by A7, A22, XXREAL_0:2;
set X = (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1))));
A26: ((Gauge (C,(n + 1))) * (i2,m)) `1 = ((Gauge (C,(n + 1))) * (i2,1)) `1 by A3, A4, A24, A25, GOBOARD5:2;
then A27: |[(((Gauge (C,(n + 1))) * (i2,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))))))]| = (Gauge (C,(n + 1))) * (i2,m) by A23, EUCLID:53;
then A28: ((Gauge (C,(n + 1))) * (i2,j)) `1 = |[(((Gauge (C,(n + 1))) * (i2,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))))))]| `1 by A3, A4, A5, A12, A26, GOBOARD5:2;
 ex x being set st
( x in LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k))) & x in L~ (Lower_Seq (C,(n + 1))) ) by A10, A20, XBOOLE_0:3;
then reconsider X1 = (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))) as non empty compact Subset of (TOP-REAL 2) by XBOOLE_0:def_4;
consider pp being set  such that
A29: pp in S-most X1 by XBOOLE_0:def_1;
reconsider pp = pp as Point of (TOP-REAL 2) by A29;
A30: pp in (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))) by A29, XBOOLE_0:def_4;
then A31: pp in L~ (Lower_Seq (C,(n + 1))) by XBOOLE_0:def_4;
 pp in LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k))) by A30, XBOOLE_0:def_4;
then A32: pp `1 = |[(((Gauge (C,(n + 1))) * (i2,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))))))]| `1 by A16, A28, SPPOL_1:41;
|[(((Gauge (C,(n + 1))) * (i2,1)) `1),(lower_bound (proj2 .: ((LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))))))]| `2 = S-bound ((LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1))))) by A23, A27, SPRECT_1:44
.= (S-min ((LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))))) `2 by EUCLID:52
.= pp `2 by A29, PSCOMP_1:55 ;
then  (Gauge (C,(n + 1))) * (i2,m) in Lower_Arc (L~ (Cage (C,(n + 1)))) by A10, A27, A31, A32, TOPREAL3:6;
then  LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,m))) meets Upper_Arc C by A3, A4, A5, A9, A21, A25, Th18;
then  LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k))) meets Upper_Arc C by A3, A4, A5, A7, A21, A22, JORDAN15:5, XBOOLE_1:63;
hence  (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Upper_Arc C by XBOOLE_1:70; ::_thesis: verum
end;
supposeA33: ( LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) meets Upper_Arc (L~ (Cage (C,(n + 1)))) & i2 <= i1 ) ; ::_thesis: (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Upper_Arc C
then consider m being Element of NAT  such that
A34: i2 <= m and
A35: m <= i1 and
A36: ((Gauge (C,(n + 1))) * (m,k)) `1 = upper_bound (proj1 .: ((LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))))) by A11, A18, A19, JORDAN1F:4, JORDAN1G:4;
A37: 1 < m by A3, A34, XXREAL_0:2;
A38: m < len (Gauge (C,(n + 1))) by A2, A35, XXREAL_0:2;
set X = (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) /\ (L~ (Upper_Seq (C,(n + 1))));
A39: ((Gauge (C,(n + 1))) * (m,k)) `2 = ((Gauge (C,(n + 1))) * (1,k)) `2 by A7, A14, A37, A38, GOBOARD5:1;
then A40: |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) /\ (L~ (Upper_Seq (C,(n + 1))))))),(((Gauge (C,(n + 1))) * (1,k)) `2)]| = (Gauge (C,(n + 1))) * (m,k) by A36, EUCLID:53;
then A41: ((Gauge (C,(n + 1))) * (i2,k)) `2 = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) /\ (L~ (Upper_Seq (C,(n + 1))))))),(((Gauge (C,(n + 1))) * (1,k)) `2)]| `2 by A3, A4, A7, A14, A39, GOBOARD5:1;
 ex x being set st
( x in LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) & x in L~ (Upper_Seq (C,(n + 1))) ) by A11, A33, XBOOLE_0:3;
then reconsider X1 = (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))) as non empty compact Subset of (TOP-REAL 2) by XBOOLE_0:def_4;
consider pp being set  such that
A42: pp in E-most X1 by XBOOLE_0:def_1;
reconsider pp = pp as Point of (TOP-REAL 2) by A42;
A43: pp in (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))) by A42, XBOOLE_0:def_4;
then A44: pp in L~ (Upper_Seq (C,(n + 1))) by XBOOLE_0:def_4;
 pp in LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) by A43, XBOOLE_0:def_4;
then A45: pp `2 = |[(upper_bound (proj1 .: ((LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) /\ (L~ (Upper_Seq (C,(n + 1))))))),(((Gauge (C,(n + 1))) * (1,k)) `2)]| `2 by A17, A41, SPPOL_1:40;
|[(upper_bound (proj1 .: ((LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) /\ (L~ (Upper_Seq (C,(n + 1))))))),(((Gauge (C,(n + 1))) * (1,k)) `2)]| `1 = E-bound ((LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) /\ (L~ (Upper_Seq (C,(n + 1))))) by A36, A40, SPRECT_1:46
.= (E-min ((LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))))) `1 by EUCLID:52
.= pp `1 by A42, PSCOMP_1:47 ;
then  (Gauge (C,(n + 1))) * (m,k) in Upper_Arc (L~ (Cage (C,(n + 1)))) by A11, A40, A44, A45, TOPREAL3:6;
then  LSeg (((Gauge (C,(n + 1))) * (m,k)),((Gauge (C,(n + 1))) * (i1,k))) meets Upper_Arc C by A2, A7, A8, A14, A35, A37, JORDAN15:41;
then  LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) meets Upper_Arc C by A2, A3, A7, A14, A34, A35, JORDAN15:6, XBOOLE_1:63;
hence  (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Upper_Arc C by XBOOLE_1:70; ::_thesis: verum
end;
supposeA46: ( LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) meets Upper_Arc (L~ (Cage (C,(n + 1)))) & i1 < i2 ) ; ::_thesis: (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Upper_Arc C
then consider m being Element of NAT  such that
A47: i1 <= m and
A48: m <= i2 and
A49: ((Gauge (C,(n + 1))) * (m,k)) `1 = lower_bound (proj1 .: ((LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))))) by A11, A18, A19, JORDAN1F:3, JORDAN1G:4;
A50: 1 < m by A1, A47, XXREAL_0:2;
A51: m < len (Gauge (C,(n + 1))) by A4, A48, XXREAL_0:2;
set X = (LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1))));
A52: ((Gauge (C,(n + 1))) * (m,k)) `2 = ((Gauge (C,(n + 1))) * (1,k)) `2 by A7, A14, A50, A51, GOBOARD5:1;
then A53: |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1))))))),(((Gauge (C,(n + 1))) * (1,k)) `2)]| = (Gauge (C,(n + 1))) * (m,k) by A49, EUCLID:53;
then A54: ((Gauge (C,(n + 1))) * (i1,k)) `2 = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1))))))),(((Gauge (C,(n + 1))) * (1,k)) `2)]| `2 by A1, A2, A7, A14, A52, GOBOARD5:1;
 ex x being set st
( x in LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k))) & x in L~ (Upper_Seq (C,(n + 1))) ) by A11, A46, XBOOLE_0:3;
then reconsider X1 = (LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))) as non empty compact Subset of (TOP-REAL 2) by XBOOLE_0:def_4;
consider pp being set  such that
A55: pp in W-most X1 by XBOOLE_0:def_1;
reconsider pp = pp as Point of (TOP-REAL 2) by A55;
A56: pp in (LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))) by A55, XBOOLE_0:def_4;
then A57: pp in L~ (Upper_Seq (C,(n + 1))) by XBOOLE_0:def_4;
 pp in LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k))) by A56, XBOOLE_0:def_4;
then A58: pp `2 = |[(lower_bound (proj1 .: ((LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1))))))),(((Gauge (C,(n + 1))) * (1,k)) `2)]| `2 by A17, A54, SPPOL_1:40;
|[(lower_bound (proj1 .: ((LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1))))))),(((Gauge (C,(n + 1))) * (1,k)) `2)]| `1 = W-bound ((LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1))))) by A49, A53, SPRECT_1:43
.= (W-min ((LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))))) `1 by EUCLID:52
.= pp `1 by A55, PSCOMP_1:31 ;
then  (Gauge (C,(n + 1))) * (m,k) in Upper_Arc (L~ (Cage (C,(n + 1)))) by A11, A53, A57, A58, TOPREAL3:6;
then  LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (m,k))) meets Upper_Arc C by A1, A7, A8, A14, A47, A51, JORDAN15:33;
then  LSeg (((Gauge (C,(n + 1))) * (i1,k)),((Gauge (C,(n + 1))) * (i2,k))) meets Upper_Arc C by A1, A4, A7, A14, A47, A48, JORDAN15:6, XBOOLE_1:63;
hence  (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Upper_Arc C by XBOOLE_1:70; ::_thesis: verum
end;
supposeA59: ( LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k))) misses Lower_Arc (L~ (Cage (C,(n + 1)))) & LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) misses Upper_Arc (L~ (Cage (C,(n + 1)))) ) ; ::_thesis: (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Upper_Arc C
consider j1 being Element of NAT  such that
A60: j <= j1 and
A61: j1 <= k and
A62: (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))) = {((Gauge (C,(n + 1))) * (i2,j1))} by A3, A4, A5, A6, A7, A9, A11, JORDAN15:15;
 (Gauge (C,(n + 1))) * (i2,j1) in (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Upper_Seq (C,(n + 1)))) by A62, TARSKI:def_1;
then A63: (Gauge (C,(n + 1))) * (i2,j1) in L~ (Upper_Seq (C,(n + 1))) by XBOOLE_0:def_4;
A64: 1 <= j1 by A5, A60, XXREAL_0:2;
now__::_thesis:_(LSeg_(((Gauge_(C,(n_+_1)))_*_(i2,j)),((Gauge_(C,(n_+_1)))_*_(i2,k))))_\/_(LSeg_(((Gauge_(C,(n_+_1)))_*_(i2,k)),((Gauge_(C,(n_+_1)))_*_(i1,k))))_meets_Upper_Arc_C
percases ( i2 <= i1 or i1 < i2 ) ;
suppose i2 <= i1 ; ::_thesis: (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Upper_Arc C
then consider i3 being Element of NAT  such that
A65: i2 <= i3 and
A66: i3 <= i1 and
A67: (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))) = {((Gauge (C,(n + 1))) * (i3,k))} by A2, A3, A7, A8, A10, A14, JORDAN15:19;
A68: i3 < len (Gauge (C,(n + 1))) by A2, A66, XXREAL_0:2;
 (Gauge (C,(n + 1))) * (i3,k) in (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))) by A67, TARSKI:def_1;
then A69: (Gauge (C,(n + 1))) * (i3,k) in L~ (Lower_Seq (C,(n + 1))) by XBOOLE_0:def_4;
A70: LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k))) c= LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k))) by A3, A4, A5, A7, A60, A61, JORDAN15:5;
A71: LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))) c= LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) by A2, A3, A7, A14, A65, A66, JORDAN15:6;
then A72: (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) c= (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) by A70, XBOOLE_1:13;
A73: ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) = {((Gauge (C,(n + 1))) * (i3,k))}
proof
thus  ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) c= {((Gauge (C,(n + 1))) * (i3,k))} :: according to XBOOLE_0:def_10 ::_thesis: {((Gauge (C,(n + 1))) * (i3,k))} c= ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) or x in {((Gauge (C,(n + 1))) * (i3,k))} )
assume A74: x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) ; ::_thesis: x in {((Gauge (C,(n + 1))) * (i3,k))}
then  x in (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) by XBOOLE_0:def_4;
then A75: ( x in LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k))) or x in LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))) ) by XBOOLE_0:def_3;
 x in L~ (Lower_Seq (C,(n + 1))) by A74, XBOOLE_0:def_4;
hence  x in {((Gauge (C,(n + 1))) * (i3,k))} by A10, A59, A67, A70, A75, XBOOLE_0:3, XBOOLE_0:def_4; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,(n + 1))) * (i3,k))} or x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) )
assume  x in {((Gauge (C,(n + 1))) * (i3,k))} ; ::_thesis: x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1))))
then A76: x = (Gauge (C,(n + 1))) * (i3,k) by TARSKI:def_1;
 (Gauge (C,(n + 1))) * (i3,k) in LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))) by RLTOPSP1:68;
then  (Gauge (C,(n + 1))) * (i3,k) in (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) by XBOOLE_0:def_3;
hence  x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) by A69, A76, XBOOLE_0:def_4; ::_thesis: verum
end;
 ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) = {((Gauge (C,(n + 1))) * (i2,j1))}
proof
thus  ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) c= {((Gauge (C,(n + 1))) * (i2,j1))} :: according to XBOOLE_0:def_10 ::_thesis: {((Gauge (C,(n + 1))) * (i2,j1))} c= ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) or x in {((Gauge (C,(n + 1))) * (i2,j1))} )
assume A77: x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) ; ::_thesis: x in {((Gauge (C,(n + 1))) * (i2,j1))}
then  x in (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) by XBOOLE_0:def_4;
then A78: ( x in LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k))) or x in LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))) ) by XBOOLE_0:def_3;
 x in L~ (Upper_Seq (C,(n + 1))) by A77, XBOOLE_0:def_4;
hence  x in {((Gauge (C,(n + 1))) * (i2,j1))} by A11, A59, A62, A71, A78, XBOOLE_0:3, XBOOLE_0:def_4; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,(n + 1))) * (i2,j1))} or x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) )
assume  x in {((Gauge (C,(n + 1))) * (i2,j1))} ; ::_thesis: x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1))))
then A79: x = (Gauge (C,(n + 1))) * (i2,j1) by TARSKI:def_1;
 (Gauge (C,(n + 1))) * (i2,j1) in LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k))) by RLTOPSP1:68;
then  (Gauge (C,(n + 1))) * (i2,j1) in (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) by XBOOLE_0:def_3;
hence  x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) by A63, A79, XBOOLE_0:def_4; ::_thesis: verum
end;
hence  (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Upper_Arc C by A3, A7, A61, A64, A65, A68, A72, A73, Th20, XBOOLE_1:63; ::_thesis: verum
end;
suppose i1 < i2 ; ::_thesis: (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Upper_Arc C
then consider i3 being Element of NAT  such that
A80: i1 <= i3 and
A81: i3 <= i2 and
A82: (LSeg (((Gauge (C,(n + 1))) * (i3,k)),((Gauge (C,(n + 1))) * (i2,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))) = {((Gauge (C,(n + 1))) * (i3,k))} by A1, A4, A7, A8, A10, A14, JORDAN15:12;
A83: 1 < i3 by A1, A80, XXREAL_0:2;
 (Gauge (C,(n + 1))) * (i3,k) in (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) /\ (L~ (Lower_Seq (C,(n + 1)))) by A82, TARSKI:def_1;
then A84: (Gauge (C,(n + 1))) * (i3,k) in L~ (Lower_Seq (C,(n + 1))) by XBOOLE_0:def_4;
A85: LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k))) c= LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k))) by A3, A4, A5, A7, A60, A61, JORDAN15:5;
A86: LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))) c= LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k))) by A1, A4, A7, A14, A80, A81, JORDAN15:6;
then A87: (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) c= (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) by A85, XBOOLE_1:13;
A88: ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) = {((Gauge (C,(n + 1))) * (i3,k))}
proof
thus  ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) c= {((Gauge (C,(n + 1))) * (i3,k))} :: according to XBOOLE_0:def_10 ::_thesis: {((Gauge (C,(n + 1))) * (i3,k))} c= ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) or x in {((Gauge (C,(n + 1))) * (i3,k))} )
assume A89: x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) ; ::_thesis: x in {((Gauge (C,(n + 1))) * (i3,k))}
then  x in (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) by XBOOLE_0:def_4;
then A90: ( x in LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k))) or x in LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))) ) by XBOOLE_0:def_3;
 x in L~ (Lower_Seq (C,(n + 1))) by A89, XBOOLE_0:def_4;
hence  x in {((Gauge (C,(n + 1))) * (i3,k))} by A10, A59, A82, A85, A90, XBOOLE_0:3, XBOOLE_0:def_4; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,(n + 1))) * (i3,k))} or x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) )
assume  x in {((Gauge (C,(n + 1))) * (i3,k))} ; ::_thesis: x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1))))
then A91: x = (Gauge (C,(n + 1))) * (i3,k) by TARSKI:def_1;
 (Gauge (C,(n + 1))) * (i3,k) in LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))) by RLTOPSP1:68;
then  (Gauge (C,(n + 1))) * (i3,k) in (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) by XBOOLE_0:def_3;
hence  x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Lower_Seq (C,(n + 1)))) by A84, A91, XBOOLE_0:def_4; ::_thesis: verum
end;
 ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) = {((Gauge (C,(n + 1))) * (i2,j1))}
proof
thus  ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) c= {((Gauge (C,(n + 1))) * (i2,j1))} :: according to XBOOLE_0:def_10 ::_thesis: {((Gauge (C,(n + 1))) * (i2,j1))} c= ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) or x in {((Gauge (C,(n + 1))) * (i2,j1))} )
assume A92: x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) ; ::_thesis: x in {((Gauge (C,(n + 1))) * (i2,j1))}
then  x in (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) by XBOOLE_0:def_4;
then A93: ( x in LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k))) or x in LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))) ) by XBOOLE_0:def_3;
 x in L~ (Upper_Seq (C,(n + 1))) by A92, XBOOLE_0:def_4;
hence  x in {((Gauge (C,(n + 1))) * (i2,j1))} by A11, A59, A62, A86, A93, XBOOLE_0:3, XBOOLE_0:def_4; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {((Gauge (C,(n + 1))) * (i2,j1))} or x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) )
assume  x in {((Gauge (C,(n + 1))) * (i2,j1))} ; ::_thesis: x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1))))
then A94: x = (Gauge (C,(n + 1))) * (i2,j1) by TARSKI:def_1;
 (Gauge (C,(n + 1))) * (i2,j1) in LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k))) by RLTOPSP1:68;
then  (Gauge (C,(n + 1))) * (i2,j1) in (LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k)))) by XBOOLE_0:def_3;
hence  x in ((LSeg (((Gauge (C,(n + 1))) * (i2,j1)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i3,k))))) /\ (L~ (Upper_Seq (C,(n + 1)))) by A63, A94, XBOOLE_0:def_4; ::_thesis: verum
end;
hence  (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Upper_Arc C by A4, A7, A61, A64, A81, A83, A87, A88, Th22, XBOOLE_1:63; ::_thesis: verum
end;
end;
end;
hence  (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Upper_Arc C ; ::_thesis: verum
end;
end;
end;
hence  (LSeg (((Gauge (C,(n + 1))) * (i2,j)),((Gauge (C,(n + 1))) * (i2,k)))) \/ (LSeg (((Gauge (C,(n + 1))) * (i2,k)),((Gauge (C,(n + 1))) * (i1,k)))) meets Upper_Arc C ; ::_thesis: verum
end;

theorem Th26: :: JORDAN19:26
for C being Simple_closed_curve
for p being Point of (TOP-REAL 2) st W-bound C < p `1 & p `1 < E-bound C & p in North_Arc C holds
not p in South_Arc C
proof
let C be Simple_closed_curve; ::_thesis: for p being Point of (TOP-REAL 2) st W-bound C < p `1 & p `1 < E-bound C & p in North_Arc C holds
not p in South_Arc C
let p be Point of (TOP-REAL 2); ::_thesis: ( W-bound C < p `1 & p `1 < E-bound C & p in North_Arc C implies not p in South_Arc C )
reconsider p9 = p as Point of (Euclid 2) by EUCLID:22;
assume that
A1: W-bound C < p `1 and
A2: p `1 < E-bound C and
A3: p in North_Arc C and
A4: p in South_Arc C ; ::_thesis: contradiction
set s = min (((p `1) - (W-bound C)),((E-bound C) - (p `1)));
A5: W-bound C = (W-bound C) + 0 ;
A6: p `1 = (p `1) + 0 ;
A7: (p `1) - (W-bound C) > 0 by A1, A5, XREAL_1:20;
 (E-bound C) - (p `1) > 0 by A2, A6, XREAL_1:20;
then A8: min (((p `1) - (W-bound C)),((E-bound C) - (p `1))) > 0 by A7, XXREAL_0:15;
now__::_thesis:_for_r_being_real_number_st_0_<_r_&_r_<_min_(((p_`1)_-_(W-bound_C)),((E-bound_C)_-_(p_`1)))_holds_
Ball_(p9,r)_meets_Upper_Arc_C
let r be real number ; ::_thesis: ( 0 < r & r < min (((p `1) - (W-bound C)),((E-bound C) - (p `1))) implies Ball (p9,r) meets Upper_Arc C )
reconsider rr = r as Real by XREAL_0:def_1;
assume that
A9: 0 < r and
A10: r < min (((p `1) - (W-bound C)),((E-bound C) - (p `1))) ; ::_thesis: Ball (p9,r) meets Upper_Arc C
A11: r / 8 > 0 by A9, XREAL_1:139;
reconsider G = Ball (p9,(r / 8)) as a_neighborhood of p by A9, GOBOARD6:2, XREAL_1:139;
consider k1 being Element of NAT  such that
A12: for m being Element of NAT st m > k1 holds
(Upper_Appr C) . m meets G by A3, KURATO_2:def_1;
consider k2 being Element of NAT  such that
A13: for m being Element of NAT st m > k2 holds
(Lower_Appr C) . m meets G by A4, KURATO_2:def_1;
set k = max (k1,k2);
A14: max (k1,k2) >= k1 by XXREAL_0:25;
set z9 = max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)));
set z = max ((max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))),(r / 8));
 (max ((max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))),(r / 8))) / (r / 8) >= 1 by A11, XREAL_1:181, XXREAL_0:25;
then  log (2,((max ((max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))),(r / 8))) / (r / 8))) >= log (2,1) by PRE_FF:10;
then  log (2,((max ((max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))),(r / 8))) / (r / 8))) >= 0 by POWER:51;
then reconsider m9 = [\(log (2,((max ((max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))),(r / 8))) / (r / 8))))/] as Element of NAT by INT_1:53;
A15: 2 to_power (m9 + 1) > 0 by POWER:34;
set N = 2 to_power (m9 + 1);
 log (2,((max ((max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))),(r / 8))) / (r / 8))) < (m9 + 1) * 1 by INT_1:29;
then  log (2,((max ((max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))),(r / 8))) / (r / 8))) < (m9 + 1) * (log (2,2)) by POWER:52;
then  log (2,((max ((max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))),(r / 8))) / (r / 8))) < log (2,(2 to_power (m9 + 1))) by POWER:55;
then  (max ((max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))),(r / 8))) / (r / 8) < 2 to_power (m9 + 1) by A15, PRE_FF:10;
then  ((max ((max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))),(r / 8))) / (r / 8)) * (r / 8) < (2 to_power (m9 + 1)) * (r / 8) by A11, XREAL_1:68;
then  max ((max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))),(r / 8)) < (2 to_power (m9 + 1)) * (r / 8) by A11, XCMPLX_1:87;
then  (max ((max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))),(r / 8))) / (2 to_power (m9 + 1)) < ((2 to_power (m9 + 1)) * (r / 8)) / (2 to_power (m9 + 1)) by A15, XREAL_1:74;
then  (max ((max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))),(r / 8))) / (2 to_power (m9 + 1)) < ((r / 8) / (2 to_power (m9 + 1))) * (2 to_power (m9 + 1)) ;
then A16: (max ((max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))),(r / 8))) / (2 to_power (m9 + 1)) < r / 8 by A15, XCMPLX_1:87;
 (max ((max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))),(r / 8))) / (2 to_power (m9 + 1)) >= (max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))) / (2 to_power (m9 + 1)) by A15, XREAL_1:72, XXREAL_0:25;
then A17: (max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))) / (2 to_power (m9 + 1)) < r / 8 by A16, XXREAL_0:2;
set m = (max ((max (k1,k2)),m9)) + 1;
A18: len (Gauge (C,((max ((max (k1,k2)),m9)) + 1))) = width (Gauge (C,((max ((max (k1,k2)),m9)) + 1))) by JORDAN8:def_1;
 max ((max (k1,k2)),m9) >= max (k1,k2) by XXREAL_0:25;
then  max ((max (k1,k2)),m9) >= k1 by A14, XXREAL_0:2;
then  (max ((max (k1,k2)),m9)) + 1 > k1 by NAT_1:13;
then  (Upper_Appr C) . ((max ((max (k1,k2)),m9)) + 1) meets G by A12;
then  Upper_Arc (L~ (Cage (C,((max ((max (k1,k2)),m9)) + 1)))) meets G by Def1;
then consider p1 being set  such that
A19: p1 in Upper_Arc (L~ (Cage (C,((max ((max (k1,k2)),m9)) + 1)))) and
A20: p1 in G by XBOOLE_0:3;
reconsider p1 = p1 as Point of (TOP-REAL 2) by A19;
reconsider p19 = p1 as Point of (Euclid 2) by EUCLID:22;
set f = Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1));
A21: Upper_Arc (L~ (Cage (C,((max ((max (k1,k2)),m9)) + 1)))) = L~ (Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) by JORDAN1G:55;
then consider i1 being Element of NAT  such that
A22: 1 <= i1 and
A23: i1 + 1 <= len (Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) and
A24: p1 in LSeg (((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1),((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. (i1 + 1))) by A19, SPPOL_2:14;
reconsider c1 = (Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1 as Point of (Euclid 2) by EUCLID:22;
reconsider c2 = (Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. (i1 + 1) as Point of (Euclid 2) by EUCLID:22;
A25: Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1)) is_sequence_on Gauge (C,((max ((max (k1,k2)),m9)) + 1)) by JORDAN1G:4;
 i1 < len (Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) by A23, NAT_1:13;
then  i1 in Seg (len (Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1)))) by A22, FINSEQ_1:1;
then A26: i1 in dom (Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) by FINSEQ_1:def_3;
then consider ii1, jj1 being Element of NAT  such that
A27: [ii1,jj1] in Indices (Gauge (C,((max ((max (k1,k2)),m9)) + 1))) and
A28: (Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1 = (Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,jj1) by A25, GOBOARD1:def_9;
A29: N-bound C > (S-bound C) + 0 by TOPREAL5:16;
A30: E-bound C > (W-bound C) + 0 by TOPREAL5:17;
A31: (N-bound C) - (S-bound C) > 0 by A29, XREAL_1:20;
A32: (E-bound C) - (W-bound C) > 0 by A30, XREAL_1:20;
A33: 2 |^ (m9 + 1) > 0 by A15, POWER:41;
 max ((max (k1,k2)),m9) >= m9 by XXREAL_0:25;
then  (max ((max (k1,k2)),m9)) + 1 > m9 by NAT_1:13;
then  (max ((max (k1,k2)),m9)) + 1 >= m9 + 1 by NAT_1:13;
then A34: 2 |^ ((max ((max (k1,k2)),m9)) + 1) >= 2 |^ (m9 + 1) by PREPOWER:93;
then A35: ((N-bound C) - (S-bound C)) / (2 |^ ((max ((max (k1,k2)),m9)) + 1)) <= ((N-bound C) - (S-bound C)) / (2 |^ (m9 + 1)) by A31, A33, XREAL_1:118;
A36: ((E-bound C) - (W-bound C)) / (2 |^ ((max ((max (k1,k2)),m9)) + 1)) <= ((E-bound C) - (W-bound C)) / (2 |^ (m9 + 1)) by A32, A33, A34, XREAL_1:118;
A37: ((N-bound C) - (S-bound C)) / (2 to_power (m9 + 1)) <= (max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))) / (2 to_power (m9 + 1)) by A15, XREAL_1:72, XXREAL_0:25;
A38: ((E-bound C) - (W-bound C)) / (2 to_power (m9 + 1)) <= (max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))) / (2 to_power (m9 + 1)) by A15, XREAL_1:72, XXREAL_0:25;
A39: ((N-bound C) - (S-bound C)) / (2 |^ (m9 + 1)) <= (max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))) / (2 to_power (m9 + 1)) by A37, POWER:41;
A40: ((E-bound C) - (W-bound C)) / (2 |^ (m9 + 1)) <= (max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))) / (2 to_power (m9 + 1)) by A38, POWER:41;
A41: ((N-bound C) - (S-bound C)) / (2 |^ ((max ((max (k1,k2)),m9)) + 1)) <= (max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))) / (2 to_power (m9 + 1)) by A35, A39, XXREAL_0:2;
A42: ((E-bound C) - (W-bound C)) / (2 |^ ((max ((max (k1,k2)),m9)) + 1)) <= (max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))) / (2 to_power (m9 + 1)) by A36, A40, XXREAL_0:2;
then  dist (((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1),((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. (i1 + 1))) <= (max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))) / (2 to_power (m9 + 1)) by A22, A23, A25, A41, Th6;
then  dist (((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1),((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. (i1 + 1))) < r / 8 by A17, XXREAL_0:2;
then  dist (c1,c2) < r / 8 by TOPREAL6:def_1;
then A43: |.(((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) - ((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. (i1 + 1))).| < r / 8 by SPPOL_1:39;
 |.(p1 - ((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1)).| <= |.(((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) - ((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. (i1 + 1))).| by A24, JGRAPH_1:36;
then A44: |.(p1 - ((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1)).| < r / 8 by A43, XXREAL_0:2;
 dist (p19,p9) < r / 8 by A20, METRIC_1:11;
then  |.(p - p1).| < r / 8 by SPPOL_1:39;
then A45: |.(p - p1).| + |.(p1 - ((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1)).| < (r / (2 * 4)) + (r / (2 * 4)) by A44, XREAL_1:8;
 |.(p - ((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1)).| <= |.(p - p1).| + |.(p1 - ((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1)).| by TOPRNS_1:34;
then A46: |.(p - ((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1)).| < r / 4 by A45, XXREAL_0:2;
then A47: dist (p9,c1) < r / 4 by SPPOL_1:39;
then A48: (Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1 in Ball (p9,(r / 4)) by METRIC_1:11;
A49: (Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1 in Upper_Arc (L~ (Cage (C,((max ((max (k1,k2)),m9)) + 1)))) by A21, A26, SPPOL_2:44;
A50: max (k1,k2) >= k2 by XXREAL_0:25;
 max ((max (k1,k2)),m9) >= max (k1,k2) by XXREAL_0:25;
then  max ((max (k1,k2)),m9) >= k2 by A50, XXREAL_0:2;
then  (max ((max (k1,k2)),m9)) + 1 > k2 by NAT_1:13;
then  (Lower_Appr C) . ((max ((max (k1,k2)),m9)) + 1) meets G by A13;
then  Lower_Arc (L~ (Cage (C,((max ((max (k1,k2)),m9)) + 1)))) meets G by Def2;
then consider p2 being set  such that
A51: p2 in Lower_Arc (L~ (Cage (C,((max ((max (k1,k2)),m9)) + 1)))) and
A52: p2 in G by XBOOLE_0:3;
reconsider p2 = p2 as Point of (TOP-REAL 2) by A51;
reconsider p29 = p2 as Point of (Euclid 2) by EUCLID:22;
set g = Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1));
A53: Lower_Arc (L~ (Cage (C,((max ((max (k1,k2)),m9)) + 1)))) = L~ (Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) by JORDAN1G:56;
then consider i2 being Element of NAT  such that
A54: 1 <= i2 and
A55: i2 + 1 <= len (Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) and
A56: p2 in LSeg (((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2),((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. (i2 + 1))) by A51, SPPOL_2:14;
reconsider d1 = (Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2 as Point of (Euclid 2) by EUCLID:22;
reconsider d2 = (Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. (i2 + 1) as Point of (Euclid 2) by EUCLID:22;
A57: Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1)) is_sequence_on Gauge (C,((max ((max (k1,k2)),m9)) + 1)) by JORDAN1G:5;
 i2 < len (Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) by A55, NAT_1:13;
then  i2 in Seg (len (Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1)))) by A54, FINSEQ_1:1;
then A58: i2 in dom (Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) by FINSEQ_1:def_3;
then consider ii2, jj2 being Element of NAT  such that
A59: [ii2,jj2] in Indices (Gauge (C,((max ((max (k1,k2)),m9)) + 1))) and
A60: (Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2 = (Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,jj2) by A57, GOBOARD1:def_9;
 dist (((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2),((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. (i2 + 1))) <= (max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))) / (2 to_power (m9 + 1)) by A41, A42, A54, A55, A57, Th6;
then  dist (((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2),((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. (i2 + 1))) < r / 8 by A17, XXREAL_0:2;
then  dist (d1,d2) < r / 8 by TOPREAL6:def_1;
then A61: |.(((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) - ((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. (i2 + 1))).| < r / 8 by SPPOL_1:39;
 |.(p2 - ((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2)).| <= |.(((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) - ((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. (i2 + 1))).| by A56, JGRAPH_1:36;
then A62: |.(p2 - ((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2)).| < r / 8 by A61, XXREAL_0:2;
 dist (p29,p9) < r / 8 by A52, METRIC_1:11;
then  |.(p - p2).| < r / 8 by SPPOL_1:39;
then A63: |.(p - p2).| + |.(p2 - ((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2)).| < (r / (2 * 4)) + (r / (2 * 4)) by A62, XREAL_1:8;
 |.(p - ((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2)).| <= |.(p - p2).| + |.(p2 - ((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2)).| by TOPRNS_1:34;
then A64: |.(p - ((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2)).| < r / 4 by A63, XXREAL_0:2;
then A65: dist (p9,d1) < r / 4 by SPPOL_1:39;
then A66: (Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2 in Ball (p9,(r / 4)) by METRIC_1:11;
A67: (Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2 in Lower_Arc (L~ (Cage (C,((max ((max (k1,k2)),m9)) + 1)))) by A53, A58, SPPOL_2:44;
set Gij = (Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,jj1);
set Gji = (Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,jj2);
reconsider Gij9 = (Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,jj1), Gji9 = (Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,jj2) as Point of (Euclid 2) by EUCLID:22;
A68: 1 <= ii1 by A27, MATRIX_1:38;
A69: ii1 <= len (Gauge (C,((max ((max (k1,k2)),m9)) + 1))) by A27, MATRIX_1:38;
A70: 1 <= jj1 by A27, MATRIX_1:38;
A71: jj1 <= width (Gauge (C,((max ((max (k1,k2)),m9)) + 1))) by A27, MATRIX_1:38;
A72: 1 <= ii2 by A59, MATRIX_1:38;
A73: ii2 <= len (Gauge (C,((max ((max (k1,k2)),m9)) + 1))) by A59, MATRIX_1:38;
A74: 1 <= jj2 by A59, MATRIX_1:38;
A75: jj2 <= width (Gauge (C,((max ((max (k1,k2)),m9)) + 1))) by A59, MATRIX_1:38;
A76: len (Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) >= 3 by JORDAN1E:15;
A77: len (Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) >= 3 by JORDAN1E:15;
A78: len (Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) >= 1 by A76, XXREAL_0:2;
A79: len (Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) >= 1 by A77, XXREAL_0:2;
A80: len (Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) in Seg (len (Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1)))) by A78, FINSEQ_1:1;
A81: len (Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) in Seg (len (Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1)))) by A79, FINSEQ_1:1;
A82: len (Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) in dom (Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) by A80, FINSEQ_1:def_3;
A83: len (Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) in dom (Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) by A81, FINSEQ_1:def_3;
A84: r / 4 < r by A9, XREAL_1:223;
A85: r / 2 < r by A9, XREAL_1:216;
A86: min (((p `1) - (W-bound C)),((E-bound C) - (p `1))) <= (p `1) - (W-bound C) by XXREAL_0:17;
A87: min (((p `1) - (W-bound C)),((E-bound C) - (p `1))) <= (E-bound C) - (p `1) by XXREAL_0:17;
A88: now__::_thesis:_not_1_>=_ii1
assume  1 >= ii1 ; ::_thesis: contradiction
then A89: ii1 = 1 by A68, XXREAL_0:1;
 dist (p9,c1) < r by A47, A84, XXREAL_0:2;
then  dist (p9,c1) < min (((p `1) - (W-bound C)),((E-bound C) - (p `1))) by A10, XXREAL_0:2;
then A90: dist (p9,c1) < (p `1) - (W-bound C) by A86, XXREAL_0:2;
A91: (p `1) - (((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) `1) <= abs ((p `1) - (((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) `1)) by ABSVALUE:4;
 abs ((p `1) - (((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) `1)) <= |.(p - ((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1)).| by JGRAPH_1:34;
then  (p `1) - (((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) `1) <= |.(p - ((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1)).| by A91, XXREAL_0:2;
then  (p `1) - (W-bound (L~ (Cage (C,((max ((max (k1,k2)),m9)) + 1))))) <= |.(p - ((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1)).| by A18, A28, A70, A71, A89, JORDAN1A:73;
then  (p `1) - (W-bound (L~ (Cage (C,((max ((max (k1,k2)),m9)) + 1))))) <= dist (p9,c1) by SPPOL_1:39;
then  (p `1) - (W-bound (L~ (Cage (C,((max ((max (k1,k2)),m9)) + 1))))) < (p `1) - (W-bound C) by A90, XXREAL_0:2;
then  W-bound (L~ (Cage (C,((max ((max (k1,k2)),m9)) + 1)))) > W-bound C by XREAL_1:13;
hence  contradiction by Th11; ::_thesis: verum
end;
A92: now__::_thesis:_not_ii1_>=_len_(Gauge_(C,((max_((max_(k1,k2)),m9))_+_1)))
assume  ii1 >= len (Gauge (C,((max ((max (k1,k2)),m9)) + 1))) ; ::_thesis: contradiction
then A93: ii1 = len (Gauge (C,((max ((max (k1,k2)),m9)) + 1))) by A69, XXREAL_0:1;
 ((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * ((len (Gauge (C,((max ((max (k1,k2)),m9)) + 1)))),jj1)) `1 = E-bound (L~ (Cage (C,((max ((max (k1,k2)),m9)) + 1)))) by A18, A70, A71, JORDAN1A:71;
then (Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1 = E-max (L~ (Cage (C,((max ((max (k1,k2)),m9)) + 1)))) by A21, A26, A28, A93, JORDAN1J:46, SPPOL_2:44
.= (Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. (len (Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1)))) by JORDAN1F:7 ;
then  i1 = len (Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) by A26, A82, PARTFUN2:10;
hence  contradiction by A23, NAT_1:13; ::_thesis: verum
end;
A94: now__::_thesis:_not_ii2_<=_1
assume  ii2 <= 1 ; ::_thesis: contradiction
then A95: ii2 = 1 by A72, XXREAL_0:1;
 ((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (1,jj2)) `1 = W-bound (L~ (Cage (C,((max ((max (k1,k2)),m9)) + 1)))) by A18, A74, A75, JORDAN1A:73;
then (Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2 = W-min (L~ (Cage (C,((max ((max (k1,k2)),m9)) + 1)))) by A53, A58, A60, A95, JORDAN1J:47, SPPOL_2:44
.= (Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. (len (Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1)))) by JORDAN1F:8 ;
then  i2 = len (Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) by A58, A83, PARTFUN2:10;
hence  contradiction by A55, NAT_1:13; ::_thesis: verum
end;
A96: now__::_thesis:_not_ii2_>=_len_(Gauge_(C,((max_((max_(k1,k2)),m9))_+_1)))
assume  ii2 >= len (Gauge (C,((max ((max (k1,k2)),m9)) + 1))) ; ::_thesis: contradiction
then A97: ii2 = len (Gauge (C,((max ((max (k1,k2)),m9)) + 1))) by A73, XXREAL_0:1;
 dist (p9,d1) < r by A65, A84, XXREAL_0:2;
then  dist (p9,d1) < min (((p `1) - (W-bound C)),((E-bound C) - (p `1))) by A10, XXREAL_0:2;
then A98: dist (p9,d1) < (E-bound C) - (p `1) by A87, XXREAL_0:2;
A99: (((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) `1) - (p `1) <= abs ((((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) `1) - (p `1)) by ABSVALUE:4;
 abs ((((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) `1) - (p `1)) <= |.(((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) - p).| by JGRAPH_1:34;
then  abs ((((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) `1) - (p `1)) <= |.(p - ((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2)).| by TOPRNS_1:27;
then  (((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) `1) - (p `1) <= |.(p - ((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2)).| by A99, XXREAL_0:2;
then  (E-bound (L~ (Cage (C,((max ((max (k1,k2)),m9)) + 1))))) - (p `1) <= |.(p - ((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2)).| by A18, A60, A74, A75, A97, JORDAN1A:71;
then  (E-bound (L~ (Cage (C,((max ((max (k1,k2)),m9)) + 1))))) - (p `1) <= dist (p9,d1) by SPPOL_1:39;
then  (E-bound (L~ (Cage (C,((max ((max (k1,k2)),m9)) + 1))))) - (p `1) < (E-bound C) - (p `1) by A98, XXREAL_0:2;
then  E-bound (L~ (Cage (C,((max ((max (k1,k2)),m9)) + 1)))) < E-bound C by XREAL_1:13;
hence  contradiction by Th9; ::_thesis: verum
end;
A100: Ball (p9,(rr / 4)) c= Ball (p9,rr) by A84, PCOMPS_1:1;
A101: ((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,jj1)) `1 = ((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,1)) `1 by A70, A71, A72, A73, GOBOARD5:2
.= ((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) `1 by A60, A72, A73, A74, A75, GOBOARD5:2 ;
A102: ((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,jj1)) `2 = ((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (1,jj1)) `2 by A70, A71, A72, A73, GOBOARD5:1
.= ((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) `2 by A28, A68, A69, A70, A71, GOBOARD5:1 ;
A103: ((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,jj2)) `1 = ((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,1)) `1 by A68, A69, A74, A75, GOBOARD5:2
.= ((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) `1 by A28, A68, A69, A70, A71, GOBOARD5:2 ;
A104: ((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,jj2)) `2 = ((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (1,jj2)) `2 by A68, A69, A74, A75, GOBOARD5:1
.= ((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) `2 by A60, A72, A73, A74, A75, GOBOARD5:1 ;
A105: abs ((((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) `1) - (p `1)) <= |.(((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) - p).| by JGRAPH_1:34;
A106: abs ((((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) `2) - (p `2)) <= |.(((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) - p).| by JGRAPH_1:34;
A107: abs ((((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) `1) - (p `1)) <= |.(p - ((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2)).| by A105, TOPRNS_1:27;
A108: abs ((((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) `2) - (p `2)) <= |.(p - ((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1)).| by A106, TOPRNS_1:27;
A109: abs ((((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) `1) - (p `1)) <= r / 4 by A64, A107, XXREAL_0:2;
 abs ((((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) `2) - (p `2)) <= r / 4 by A46, A108, XXREAL_0:2;
then  (abs ((((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) `1) - (p `1))) + (abs ((((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) `2) - (p `2))) <= (r / (2 * 2)) + (r / (2 * 2)) by A109, XREAL_1:7;
then A110: (abs ((((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) `1) - (p `1))) + (abs ((((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) `2) - (p `2))) < r by A85, XXREAL_0:2;
A111: abs ((((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) `1) - (p `1)) <= |.(((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) - p).| by JGRAPH_1:34;
A112: abs ((((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) `2) - (p `2)) <= |.(((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) - p).| by JGRAPH_1:34;
A113: abs ((((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) `1) - (p `1)) <= |.(p - ((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1)).| by A111, TOPRNS_1:27;
A114: abs ((((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) `2) - (p `2)) <= |.(p - ((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2)).| by A112, TOPRNS_1:27;
A115: abs ((((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) `1) - (p `1)) <= r / 4 by A46, A113, XXREAL_0:2;
 abs ((((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) `2) - (p `2)) <= r / 4 by A64, A114, XXREAL_0:2;
then  (abs ((((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) `1) - (p `1))) + (abs ((((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) `2) - (p `2))) <= (r / (2 * 2)) + (r / (2 * 2)) by A115, XREAL_1:7;
then A116: (abs ((((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) `1) - (p `1))) + (abs ((((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) `2) - (p `2))) < r by A85, XXREAL_0:2;
 |.(((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,jj1)) - p).| <= (abs ((((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) `1) - (p `1))) + (abs ((((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) `2) - (p `2))) by A101, A102, JGRAPH_1:32;
then  |.(((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,jj1)) - p).| < r by A110, XXREAL_0:2;
then  dist (Gij9,p9) < r by SPPOL_1:39;
then A117: (Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,jj1) in Ball (p9,r) by METRIC_1:11;
 |.(((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,jj2)) - p).| <= (abs ((((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) `1) - (p `1))) + (abs ((((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) `2) - (p `2))) by A103, A104, JGRAPH_1:32;
then  |.(((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,jj2)) - p).| < r by A116, XXREAL_0:2;
then  dist (Gji9,p9) < r by SPPOL_1:39;
then A118: (Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,jj2) in Ball (p9,r) by METRIC_1:11;
A119: LSeg (((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2),((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,jj1))) c= Ball (p9,rr) by A66, A100, A117, TOPREAL3:21;
A120: LSeg (((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,jj1)),((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1)) c= Ball (p9,rr) by A48, A100, A117, TOPREAL3:21;
A121: LSeg (((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2),((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,jj2))) c= Ball (p9,rr) by A66, A100, A118, TOPREAL3:21;
A122: LSeg (((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,jj2)),((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1)) c= Ball (p9,rr) by A48, A100, A118, TOPREAL3:21;
now__::_thesis:_Ball_(p9,r)_meets_Upper_Arc_C
percases ( jj2 <= jj1 or jj1 < jj2 ) ;
supposeA123: jj2 <= jj1 ; ::_thesis: Ball (p9,r) meets Upper_Arc C
 (LSeg (((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2),((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,jj1)))) \/ (LSeg (((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,jj1)),((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1))) c= Ball (p9,r)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (LSeg (((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2),((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,jj1)))) \/ (LSeg (((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,jj1)),((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1))) or x in Ball (p9,r) )
assume A124: x in (LSeg (((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2),((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,jj1)))) \/ (LSeg (((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,jj1)),((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1))) ; ::_thesis: x in Ball (p9,r)
then reconsider x9 = x as Point of (TOP-REAL 2) ;
now__::_thesis:_x9_in_Ball_(p9,r)
percases ( x9 in LSeg (((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2),((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,jj1))) or x9 in LSeg (((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,jj1)),((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1)) ) by A124, XBOOLE_0:def_3;
suppose x9 in LSeg (((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2),((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,jj1))) ; ::_thesis: x9 in Ball (p9,r)
hence  x9 in Ball (p9,r) by A119; ::_thesis: verum
end;
suppose x9 in LSeg (((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,jj1)),((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1)) ; ::_thesis: x9 in Ball (p9,r)
hence  x9 in Ball (p9,r) by A120; ::_thesis: verum
end;
end;
end;
hence  x in Ball (p9,r) ; ::_thesis: verum
end;
hence  Ball (p9,r) meets Upper_Arc C by A28, A49, A60, A67, A71, A74, A88, A92, A94, A96, A123, JORDAN15:48, XBOOLE_1:63; ::_thesis: verum
end;
supposeA125: jj1 < jj2 ; ::_thesis: Ball (p9,r) meets Upper_Arc C
 (LSeg (((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1),((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,jj2)))) \/ (LSeg (((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,jj2)),((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2))) c= Ball (p9,r)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (LSeg (((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1),((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,jj2)))) \/ (LSeg (((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,jj2)),((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2))) or x in Ball (p9,r) )
assume A126: x in (LSeg (((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1),((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,jj2)))) \/ (LSeg (((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,jj2)),((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2))) ; ::_thesis: x in Ball (p9,r)
then reconsider x9 = x as Point of (TOP-REAL 2) ;
now__::_thesis:_x9_in_Ball_(p9,r)
percases ( x9 in LSeg (((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1),((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,jj2))) or x9 in LSeg (((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,jj2)),((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2)) ) by A126, XBOOLE_0:def_3;
suppose x9 in LSeg (((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1),((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,jj2))) ; ::_thesis: x9 in Ball (p9,r)
hence  x9 in Ball (p9,r) by A122; ::_thesis: verum
end;
suppose x9 in LSeg (((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,jj2)),((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2)) ; ::_thesis: x9 in Ball (p9,r)
hence  x9 in Ball (p9,r) by A121; ::_thesis: verum
end;
end;
end;
hence  x in Ball (p9,r) ; ::_thesis: verum
end;
hence  Ball (p9,r) meets Upper_Arc C by A28, A49, A60, A67, A70, A75, A88, A92, A94, A96, A125, Th25, XBOOLE_1:63; ::_thesis: verum
end;
end;
end;
hence  Ball (p9,r) meets Upper_Arc C ; ::_thesis: verum
end;
then  p in Cl (Upper_Arc C) by A8, GOBOARD6:93;
then A127: p in Upper_Arc C by PRE_TOPC:22;
now__::_thesis:_for_r_being_real_number_st_0_<_r_&_r_<_min_(((p_`1)_-_(W-bound_C)),((E-bound_C)_-_(p_`1)))_holds_
Ball_(p9,r)_meets_Lower_Arc_C
let r be real number ; ::_thesis: ( 0 < r & r < min (((p `1) - (W-bound C)),((E-bound C) - (p `1))) implies Ball (p9,r) meets Lower_Arc C )
reconsider rr = r as Real by XREAL_0:def_1;
assume that
A128: 0 < r and
A129: r < min (((p `1) - (W-bound C)),((E-bound C) - (p `1))) ; ::_thesis: Ball (p9,r) meets Lower_Arc C
A130: r / 8 > 0 by A128, XREAL_1:139;
reconsider G = Ball (p9,(r / 8)) as a_neighborhood of p by A128, GOBOARD6:2, XREAL_1:139;
consider k1 being Element of NAT  such that
A131: for m being Element of NAT st m > k1 holds
(Upper_Appr C) . m meets G by A3, KURATO_2:def_1;
consider k2 being Element of NAT  such that
A132: for m being Element of NAT st m > k2 holds
(Lower_Appr C) . m meets G by A4, KURATO_2:def_1;
set k = max (k1,k2);
A133: max (k1,k2) >= k1 by XXREAL_0:25;
set z9 = max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)));
set z = max ((max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))),(r / 8));
 (max ((max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))),(r / 8))) / (r / 8) >= 1 by A130, XREAL_1:181, XXREAL_0:25;
then  log (2,((max ((max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))),(r / 8))) / (r / 8))) >= log (2,1) by PRE_FF:10;
then  log (2,((max ((max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))),(r / 8))) / (r / 8))) >= 0 by POWER:51;
then reconsider m9 = [\(log (2,((max ((max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))),(r / 8))) / (r / 8))))/] as Element of NAT by INT_1:53;
A134: 2 to_power (m9 + 1) > 0 by POWER:34;
set N = 2 to_power (m9 + 1);
 log (2,((max ((max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))),(r / 8))) / (r / 8))) < (m9 + 1) * 1 by INT_1:29;
then  log (2,((max ((max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))),(r / 8))) / (r / 8))) < (m9 + 1) * (log (2,2)) by POWER:52;
then  log (2,((max ((max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))),(r / 8))) / (r / 8))) < log (2,(2 to_power (m9 + 1))) by POWER:55;
then  (max ((max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))),(r / 8))) / (r / 8) < 2 to_power (m9 + 1) by A134, PRE_FF:10;
then  ((max ((max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))),(r / 8))) / (r / 8)) * (r / 8) < (2 to_power (m9 + 1)) * (r / 8) by A130, XREAL_1:68;
then  max ((max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))),(r / 8)) < (2 to_power (m9 + 1)) * (r / 8) by A130, XCMPLX_1:87;
then  (max ((max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))),(r / 8))) / (2 to_power (m9 + 1)) < ((2 to_power (m9 + 1)) * (r / 8)) / (2 to_power (m9 + 1)) by A134, XREAL_1:74;
then  (max ((max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))),(r / 8))) / (2 to_power (m9 + 1)) < ((r / 8) / (2 to_power (m9 + 1))) * (2 to_power (m9 + 1)) ;
then A135: (max ((max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))),(r / 8))) / (2 to_power (m9 + 1)) < r / 8 by A134, XCMPLX_1:87;
 (max ((max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))),(r / 8))) / (2 to_power (m9 + 1)) >= (max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))) / (2 to_power (m9 + 1)) by A134, XREAL_1:72, XXREAL_0:25;
then A136: (max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))) / (2 to_power (m9 + 1)) < r / 8 by A135, XXREAL_0:2;
set m = (max ((max (k1,k2)),m9)) + 1;
A137: len (Gauge (C,((max ((max (k1,k2)),m9)) + 1))) = width (Gauge (C,((max ((max (k1,k2)),m9)) + 1))) by JORDAN8:def_1;
 max ((max (k1,k2)),m9) >= max (k1,k2) by XXREAL_0:25;
then  max ((max (k1,k2)),m9) >= k1 by A133, XXREAL_0:2;
then  (max ((max (k1,k2)),m9)) + 1 > k1 by NAT_1:13;
then  (Upper_Appr C) . ((max ((max (k1,k2)),m9)) + 1) meets G by A131;
then  Upper_Arc (L~ (Cage (C,((max ((max (k1,k2)),m9)) + 1)))) meets G by Def1;
then consider p1 being set  such that
A138: p1 in Upper_Arc (L~ (Cage (C,((max ((max (k1,k2)),m9)) + 1)))) and
A139: p1 in G by XBOOLE_0:3;
reconsider p1 = p1 as Point of (TOP-REAL 2) by A138;
reconsider p19 = p1 as Point of (Euclid 2) by EUCLID:22;
set f = Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1));
A140: Upper_Arc (L~ (Cage (C,((max ((max (k1,k2)),m9)) + 1)))) = L~ (Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) by JORDAN1G:55;
then consider i1 being Element of NAT  such that
A141: 1 <= i1 and
A142: i1 + 1 <= len (Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) and
A143: p1 in LSeg (((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1),((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. (i1 + 1))) by A138, SPPOL_2:14;
reconsider c1 = (Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1 as Point of (Euclid 2) by EUCLID:22;
reconsider c2 = (Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. (i1 + 1) as Point of (Euclid 2) by EUCLID:22;
A144: Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1)) is_sequence_on Gauge (C,((max ((max (k1,k2)),m9)) + 1)) by JORDAN1G:4;
 i1 < len (Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) by A142, NAT_1:13;
then  i1 in Seg (len (Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1)))) by A141, FINSEQ_1:1;
then A145: i1 in dom (Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) by FINSEQ_1:def_3;
then consider ii1, jj1 being Element of NAT  such that
A146: [ii1,jj1] in Indices (Gauge (C,((max ((max (k1,k2)),m9)) + 1))) and
A147: (Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1 = (Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,jj1) by A144, GOBOARD1:def_9;
A148: N-bound C > (S-bound C) + 0 by TOPREAL5:16;
A149: E-bound C > (W-bound C) + 0 by TOPREAL5:17;
A150: (N-bound C) - (S-bound C) > 0 by A148, XREAL_1:20;
A151: (E-bound C) - (W-bound C) > 0 by A149, XREAL_1:20;
A152: 2 |^ (m9 + 1) > 0 by A134, POWER:41;
 max ((max (k1,k2)),m9) >= m9 by XXREAL_0:25;
then  (max ((max (k1,k2)),m9)) + 1 > m9 by NAT_1:13;
then  (max ((max (k1,k2)),m9)) + 1 >= m9 + 1 by NAT_1:13;
then A153: 2 |^ ((max ((max (k1,k2)),m9)) + 1) >= 2 |^ (m9 + 1) by PREPOWER:93;
then A154: ((N-bound C) - (S-bound C)) / (2 |^ ((max ((max (k1,k2)),m9)) + 1)) <= ((N-bound C) - (S-bound C)) / (2 |^ (m9 + 1)) by A150, A152, XREAL_1:118;
A155: ((E-bound C) - (W-bound C)) / (2 |^ ((max ((max (k1,k2)),m9)) + 1)) <= ((E-bound C) - (W-bound C)) / (2 |^ (m9 + 1)) by A151, A152, A153, XREAL_1:118;
A156: ((N-bound C) - (S-bound C)) / (2 to_power (m9 + 1)) <= (max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))) / (2 to_power (m9 + 1)) by A134, XREAL_1:72, XXREAL_0:25;
A157: ((E-bound C) - (W-bound C)) / (2 to_power (m9 + 1)) <= (max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))) / (2 to_power (m9 + 1)) by A134, XREAL_1:72, XXREAL_0:25;
A158: ((N-bound C) - (S-bound C)) / (2 |^ (m9 + 1)) <= (max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))) / (2 to_power (m9 + 1)) by A156, POWER:41;
A159: ((E-bound C) - (W-bound C)) / (2 |^ (m9 + 1)) <= (max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))) / (2 to_power (m9 + 1)) by A157, POWER:41;
A160: ((N-bound C) - (S-bound C)) / (2 |^ ((max ((max (k1,k2)),m9)) + 1)) <= (max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))) / (2 to_power (m9 + 1)) by A154, A158, XXREAL_0:2;
A161: ((E-bound C) - (W-bound C)) / (2 |^ ((max ((max (k1,k2)),m9)) + 1)) <= (max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))) / (2 to_power (m9 + 1)) by A155, A159, XXREAL_0:2;
then  dist (((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1),((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. (i1 + 1))) <= (max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))) / (2 to_power (m9 + 1)) by A141, A142, A144, A160, Th6;
then  dist (((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1),((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. (i1 + 1))) < r / 8 by A136, XXREAL_0:2;
then  dist (c1,c2) < r / 8 by TOPREAL6:def_1;
then A162: |.(((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) - ((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. (i1 + 1))).| < r / 8 by SPPOL_1:39;
 |.(p1 - ((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1)).| <= |.(((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) - ((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. (i1 + 1))).| by A143, JGRAPH_1:36;
then A163: |.(p1 - ((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1)).| < r / 8 by A162, XXREAL_0:2;
 dist (p19,p9) < r / 8 by A139, METRIC_1:11;
then  |.(p - p1).| < r / 8 by SPPOL_1:39;
then A164: |.(p - p1).| + |.(p1 - ((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1)).| < (r / (2 * 4)) + (r / (2 * 4)) by A163, XREAL_1:8;
 |.(p - ((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1)).| <= |.(p - p1).| + |.(p1 - ((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1)).| by TOPRNS_1:34;
then A165: |.(p - ((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1)).| < r / 4 by A164, XXREAL_0:2;
then A166: dist (p9,c1) < r / 4 by SPPOL_1:39;
then A167: (Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1 in Ball (p9,(r / 4)) by METRIC_1:11;
A168: (Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1 in Upper_Arc (L~ (Cage (C,((max ((max (k1,k2)),m9)) + 1)))) by A140, A145, SPPOL_2:44;
A169: max (k1,k2) >= k2 by XXREAL_0:25;
 max ((max (k1,k2)),m9) >= max (k1,k2) by XXREAL_0:25;
then  max ((max (k1,k2)),m9) >= k2 by A169, XXREAL_0:2;
then  (max ((max (k1,k2)),m9)) + 1 > k2 by NAT_1:13;
then  (Lower_Appr C) . ((max ((max (k1,k2)),m9)) + 1) meets G by A132;
then  Lower_Arc (L~ (Cage (C,((max ((max (k1,k2)),m9)) + 1)))) meets G by Def2;
then consider p2 being set  such that
A170: p2 in Lower_Arc (L~ (Cage (C,((max ((max (k1,k2)),m9)) + 1)))) and
A171: p2 in G by XBOOLE_0:3;
reconsider p2 = p2 as Point of (TOP-REAL 2) by A170;
reconsider p29 = p2 as Point of (Euclid 2) by EUCLID:22;
set g = Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1));
A172: Lower_Arc (L~ (Cage (C,((max ((max (k1,k2)),m9)) + 1)))) = L~ (Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) by JORDAN1G:56;
then consider i2 being Element of NAT  such that
A173: 1 <= i2 and
A174: i2 + 1 <= len (Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) and
A175: p2 in LSeg (((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2),((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. (i2 + 1))) by A170, SPPOL_2:14;
reconsider d1 = (Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2 as Point of (Euclid 2) by EUCLID:22;
reconsider d2 = (Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. (i2 + 1) as Point of (Euclid 2) by EUCLID:22;
A176: Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1)) is_sequence_on Gauge (C,((max ((max (k1,k2)),m9)) + 1)) by JORDAN1G:5;
 i2 < len (Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) by A174, NAT_1:13;
then  i2 in Seg (len (Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1)))) by A173, FINSEQ_1:1;
then A177: i2 in dom (Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) by FINSEQ_1:def_3;
then consider ii2, jj2 being Element of NAT  such that
A178: [ii2,jj2] in Indices (Gauge (C,((max ((max (k1,k2)),m9)) + 1))) and
A179: (Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2 = (Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,jj2) by A176, GOBOARD1:def_9;
 dist (((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2),((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. (i2 + 1))) <= (max (((N-bound C) - (S-bound C)),((E-bound C) - (W-bound C)))) / (2 to_power (m9 + 1)) by A160, A161, A173, A174, A176, Th6;
then  dist (((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2),((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. (i2 + 1))) < r / 8 by A136, XXREAL_0:2;
then  dist (d1,d2) < r / 8 by TOPREAL6:def_1;
then A180: |.(((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) - ((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. (i2 + 1))).| < r / 8 by SPPOL_1:39;
 |.(p2 - ((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2)).| <= |.(((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) - ((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. (i2 + 1))).| by A175, JGRAPH_1:36;
then A181: |.(p2 - ((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2)).| < r / 8 by A180, XXREAL_0:2;
 dist (p29,p9) < r / 8 by A171, METRIC_1:11;
then  |.(p - p2).| < r / 8 by SPPOL_1:39;
then A182: |.(p - p2).| + |.(p2 - ((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2)).| < (r / (2 * 4)) + (r / (2 * 4)) by A181, XREAL_1:8;
 |.(p - ((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2)).| <= |.(p - p2).| + |.(p2 - ((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2)).| by TOPRNS_1:34;
then A183: |.(p - ((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2)).| < r / 4 by A182, XXREAL_0:2;
then A184: dist (p9,d1) < r / 4 by SPPOL_1:39;
then A185: (Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2 in Ball (p9,(r / 4)) by METRIC_1:11;
A186: (Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2 in Lower_Arc (L~ (Cage (C,((max ((max (k1,k2)),m9)) + 1)))) by A172, A177, SPPOL_2:44;
set Gij = (Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,jj1);
set Gji = (Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,jj2);
reconsider Gij9 = (Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,jj1), Gji9 = (Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,jj2) as Point of (Euclid 2) by EUCLID:22;
A187: 1 <= ii1 by A146, MATRIX_1:38;
A188: ii1 <= len (Gauge (C,((max ((max (k1,k2)),m9)) + 1))) by A146, MATRIX_1:38;
A189: 1 <= jj1 by A146, MATRIX_1:38;
A190: jj1 <= width (Gauge (C,((max ((max (k1,k2)),m9)) + 1))) by A146, MATRIX_1:38;
A191: 1 <= ii2 by A178, MATRIX_1:38;
A192: ii2 <= len (Gauge (C,((max ((max (k1,k2)),m9)) + 1))) by A178, MATRIX_1:38;
A193: 1 <= jj2 by A178, MATRIX_1:38;
A194: jj2 <= width (Gauge (C,((max ((max (k1,k2)),m9)) + 1))) by A178, MATRIX_1:38;
A195: len (Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) >= 3 by JORDAN1E:15;
A196: len (Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) >= 3 by JORDAN1E:15;
A197: len (Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) >= 1 by A195, XXREAL_0:2;
A198: len (Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) >= 1 by A196, XXREAL_0:2;
A199: len (Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) in Seg (len (Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1)))) by A197, FINSEQ_1:1;
A200: len (Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) in Seg (len (Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1)))) by A198, FINSEQ_1:1;
A201: len (Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) in dom (Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) by A199, FINSEQ_1:def_3;
A202: len (Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) in dom (Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) by A200, FINSEQ_1:def_3;
A203: r / 4 < r by A128, XREAL_1:223;
A204: r / 2 < r by A128, XREAL_1:216;
A205: min (((p `1) - (W-bound C)),((E-bound C) - (p `1))) <= (p `1) - (W-bound C) by XXREAL_0:17;
A206: min (((p `1) - (W-bound C)),((E-bound C) - (p `1))) <= (E-bound C) - (p `1) by XXREAL_0:17;
A207: now__::_thesis:_not_1_>=_ii1
assume  1 >= ii1 ; ::_thesis: contradiction
then A208: ii1 = 1 by A187, XXREAL_0:1;
 dist (p9,c1) < r by A166, A203, XXREAL_0:2;
then  dist (p9,c1) < min (((p `1) - (W-bound C)),((E-bound C) - (p `1))) by A129, XXREAL_0:2;
then A209: dist (p9,c1) < (p `1) - (W-bound C) by A205, XXREAL_0:2;
A210: (p `1) - (((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) `1) <= abs ((p `1) - (((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) `1)) by ABSVALUE:4;
 abs ((p `1) - (((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) `1)) <= |.(p - ((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1)).| by JGRAPH_1:34;
then  (p `1) - (((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) `1) <= |.(p - ((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1)).| by A210, XXREAL_0:2;
then  (p `1) - (W-bound (L~ (Cage (C,((max ((max (k1,k2)),m9)) + 1))))) <= |.(p - ((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1)).| by A137, A147, A189, A190, A208, JORDAN1A:73;
then  (p `1) - (W-bound (L~ (Cage (C,((max ((max (k1,k2)),m9)) + 1))))) <= dist (p9,c1) by SPPOL_1:39;
then  (p `1) - (W-bound (L~ (Cage (C,((max ((max (k1,k2)),m9)) + 1))))) < (p `1) - (W-bound C) by A209, XXREAL_0:2;
then  W-bound (L~ (Cage (C,((max ((max (k1,k2)),m9)) + 1)))) > W-bound C by XREAL_1:13;
hence  contradiction by Th11; ::_thesis: verum
end;
A211: now__::_thesis:_not_ii1_>=_len_(Gauge_(C,((max_((max_(k1,k2)),m9))_+_1)))
assume  ii1 >= len (Gauge (C,((max ((max (k1,k2)),m9)) + 1))) ; ::_thesis: contradiction
then A212: ii1 = len (Gauge (C,((max ((max (k1,k2)),m9)) + 1))) by A188, XXREAL_0:1;
 ((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * ((len (Gauge (C,((max ((max (k1,k2)),m9)) + 1)))),jj1)) `1 = E-bound (L~ (Cage (C,((max ((max (k1,k2)),m9)) + 1)))) by A137, A189, A190, JORDAN1A:71;
then (Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1 = E-max (L~ (Cage (C,((max ((max (k1,k2)),m9)) + 1)))) by A140, A145, A147, A212, JORDAN1J:46, SPPOL_2:44
.= (Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. (len (Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1)))) by JORDAN1F:7 ;
then  i1 = len (Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) by A145, A201, PARTFUN2:10;
hence  contradiction by A142, NAT_1:13; ::_thesis: verum
end;
A213: now__::_thesis:_not_ii2_<=_1
assume  ii2 <= 1 ; ::_thesis: contradiction
then A214: ii2 = 1 by A191, XXREAL_0:1;
 ((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (1,jj2)) `1 = W-bound (L~ (Cage (C,((max ((max (k1,k2)),m9)) + 1)))) by A137, A193, A194, JORDAN1A:73;
then (Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2 = W-min (L~ (Cage (C,((max ((max (k1,k2)),m9)) + 1)))) by A172, A177, A179, A214, JORDAN1J:47, SPPOL_2:44
.= (Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. (len (Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1)))) by JORDAN1F:8 ;
then  i2 = len (Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) by A177, A202, PARTFUN2:10;
hence  contradiction by A174, NAT_1:13; ::_thesis: verum
end;
A215: now__::_thesis:_not_ii2_>=_len_(Gauge_(C,((max_((max_(k1,k2)),m9))_+_1)))
assume  ii2 >= len (Gauge (C,((max ((max (k1,k2)),m9)) + 1))) ; ::_thesis: contradiction
then A216: ii2 = len (Gauge (C,((max ((max (k1,k2)),m9)) + 1))) by A192, XXREAL_0:1;
 dist (p9,d1) < r by A184, A203, XXREAL_0:2;
then  dist (p9,d1) < min (((p `1) - (W-bound C)),((E-bound C) - (p `1))) by A129, XXREAL_0:2;
then A217: dist (p9,d1) < (E-bound C) - (p `1) by A206, XXREAL_0:2;
A218: (((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) `1) - (p `1) <= abs ((((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) `1) - (p `1)) by ABSVALUE:4;
 abs ((((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) `1) - (p `1)) <= |.(((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) - p).| by JGRAPH_1:34;
then  abs ((((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) `1) - (p `1)) <= |.(p - ((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2)).| by TOPRNS_1:27;
then  (((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) `1) - (p `1) <= |.(p - ((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2)).| by A218, XXREAL_0:2;
then  (E-bound (L~ (Cage (C,((max ((max (k1,k2)),m9)) + 1))))) - (p `1) <= |.(p - ((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2)).| by A137, A179, A193, A194, A216, JORDAN1A:71;
then  (E-bound (L~ (Cage (C,((max ((max (k1,k2)),m9)) + 1))))) - (p `1) <= dist (p9,d1) by SPPOL_1:39;
then  (E-bound (L~ (Cage (C,((max ((max (k1,k2)),m9)) + 1))))) - (p `1) < (E-bound C) - (p `1) by A217, XXREAL_0:2;
then  E-bound (L~ (Cage (C,((max ((max (k1,k2)),m9)) + 1)))) < E-bound C by XREAL_1:13;
hence  contradiction by Th9; ::_thesis: verum
end;
A219: Ball (p9,(rr / 4)) c= Ball (p9,rr) by A203, PCOMPS_1:1;
A220: ((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,jj1)) `1 = ((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,1)) `1 by A189, A190, A191, A192, GOBOARD5:2
.= ((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) `1 by A179, A191, A192, A193, A194, GOBOARD5:2 ;
A221: ((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,jj1)) `2 = ((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (1,jj1)) `2 by A189, A190, A191, A192, GOBOARD5:1
.= ((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) `2 by A147, A187, A188, A189, A190, GOBOARD5:1 ;
A222: ((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,jj2)) `1 = ((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,1)) `1 by A187, A188, A193, A194, GOBOARD5:2
.= ((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) `1 by A147, A187, A188, A189, A190, GOBOARD5:2 ;
A223: ((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,jj2)) `2 = ((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (1,jj2)) `2 by A187, A188, A193, A194, GOBOARD5:1
.= ((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) `2 by A179, A191, A192, A193, A194, GOBOARD5:1 ;
A224: abs ((((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) `1) - (p `1)) <= |.(((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) - p).| by JGRAPH_1:34;
A225: abs ((((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) `2) - (p `2)) <= |.(((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) - p).| by JGRAPH_1:34;
A226: abs ((((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) `1) - (p `1)) <= |.(p - ((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2)).| by A224, TOPRNS_1:27;
A227: abs ((((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) `2) - (p `2)) <= |.(p - ((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1)).| by A225, TOPRNS_1:27;
A228: abs ((((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) `1) - (p `1)) <= r / 4 by A183, A226, XXREAL_0:2;
 abs ((((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) `2) - (p `2)) <= r / 4 by A165, A227, XXREAL_0:2;
then  (abs ((((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) `1) - (p `1))) + (abs ((((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) `2) - (p `2))) <= (r / (2 * 2)) + (r / (2 * 2)) by A228, XREAL_1:7;
then A229: (abs ((((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) `1) - (p `1))) + (abs ((((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) `2) - (p `2))) < r by A204, XXREAL_0:2;
A230: abs ((((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) `1) - (p `1)) <= |.(((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) - p).| by JGRAPH_1:34;
A231: abs ((((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) `2) - (p `2)) <= |.(((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) - p).| by JGRAPH_1:34;
A232: abs ((((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) `1) - (p `1)) <= |.(p - ((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1)).| by A230, TOPRNS_1:27;
A233: abs ((((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) `2) - (p `2)) <= |.(p - ((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2)).| by A231, TOPRNS_1:27;
A234: abs ((((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) `1) - (p `1)) <= r / 4 by A165, A232, XXREAL_0:2;
 abs ((((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) `2) - (p `2)) <= r / 4 by A183, A233, XXREAL_0:2;
then  (abs ((((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) `1) - (p `1))) + (abs ((((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) `2) - (p `2))) <= (r / (2 * 2)) + (r / (2 * 2)) by A234, XREAL_1:7;
then A235: (abs ((((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) `1) - (p `1))) + (abs ((((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) `2) - (p `2))) < r by A204, XXREAL_0:2;
 |.(((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,jj1)) - p).| <= (abs ((((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) `1) - (p `1))) + (abs ((((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) `2) - (p `2))) by A220, A221, JGRAPH_1:32;
then  |.(((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,jj1)) - p).| < r by A229, XXREAL_0:2;
then  dist (Gij9,p9) < r by SPPOL_1:39;
then A236: (Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,jj1) in Ball (p9,r) by METRIC_1:11;
 |.(((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,jj2)) - p).| <= (abs ((((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1) `1) - (p `1))) + (abs ((((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2) `2) - (p `2))) by A222, A223, JGRAPH_1:32;
then  |.(((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,jj2)) - p).| < r by A235, XXREAL_0:2;
then  dist (Gji9,p9) < r by SPPOL_1:39;
then A237: (Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,jj2) in Ball (p9,r) by METRIC_1:11;
A238: LSeg (((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2),((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,jj1))) c= Ball (p9,rr) by A185, A219, A236, TOPREAL3:21;
A239: LSeg (((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,jj1)),((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1)) c= Ball (p9,rr) by A167, A219, A236, TOPREAL3:21;
A240: LSeg (((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2),((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,jj2))) c= Ball (p9,rr) by A185, A219, A237, TOPREAL3:21;
A241: LSeg (((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,jj2)),((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1)) c= Ball (p9,rr) by A167, A219, A237, TOPREAL3:21;
now__::_thesis:_Ball_(p9,r)_meets_Lower_Arc_C
percases ( jj2 <= jj1 or jj1 < jj2 ) ;
supposeA242: jj2 <= jj1 ; ::_thesis: Ball (p9,r) meets Lower_Arc C
 (LSeg (((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2),((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,jj1)))) \/ (LSeg (((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,jj1)),((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1))) c= Ball (p9,r)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (LSeg (((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2),((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,jj1)))) \/ (LSeg (((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,jj1)),((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1))) or x in Ball (p9,r) )
assume A243: x in (LSeg (((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2),((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,jj1)))) \/ (LSeg (((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,jj1)),((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1))) ; ::_thesis: x in Ball (p9,r)
then reconsider x9 = x as Point of (TOP-REAL 2) ;
now__::_thesis:_x9_in_Ball_(p9,r)
percases ( x9 in LSeg (((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2),((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,jj1))) or x9 in LSeg (((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,jj1)),((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1)) ) by A243, XBOOLE_0:def_3;
suppose x9 in LSeg (((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2),((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,jj1))) ; ::_thesis: x9 in Ball (p9,r)
hence  x9 in Ball (p9,r) by A238; ::_thesis: verum
end;
suppose x9 in LSeg (((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii2,jj1)),((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1)) ; ::_thesis: x9 in Ball (p9,r)
hence  x9 in Ball (p9,r) by A239; ::_thesis: verum
end;
end;
end;
hence  x in Ball (p9,r) ; ::_thesis: verum
end;
hence  Ball (p9,r) meets Lower_Arc C by A147, A168, A179, A186, A190, A193, A207, A211, A213, A215, A242, JORDAN15:49, XBOOLE_1:63; ::_thesis: verum
end;
supposeA244: jj1 < jj2 ; ::_thesis: Ball (p9,r) meets Lower_Arc C
 (LSeg (((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1),((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,jj2)))) \/ (LSeg (((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,jj2)),((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2))) c= Ball (p9,r)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (LSeg (((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1),((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,jj2)))) \/ (LSeg (((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,jj2)),((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2))) or x in Ball (p9,r) )
assume A245: x in (LSeg (((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1),((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,jj2)))) \/ (LSeg (((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,jj2)),((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2))) ; ::_thesis: x in Ball (p9,r)
then reconsider x9 = x as Point of (TOP-REAL 2) ;
now__::_thesis:_x9_in_Ball_(p9,r)
percases ( x9 in LSeg (((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1),((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,jj2))) or x9 in LSeg (((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,jj2)),((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2)) ) by A245, XBOOLE_0:def_3;
suppose x9 in LSeg (((Upper_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i1),((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,jj2))) ; ::_thesis: x9 in Ball (p9,r)
hence  x9 in Ball (p9,r) by A241; ::_thesis: verum
end;
suppose x9 in LSeg (((Gauge (C,((max ((max (k1,k2)),m9)) + 1))) * (ii1,jj2)),((Lower_Seq (C,((max ((max (k1,k2)),m9)) + 1))) /. i2)) ; ::_thesis: x9 in Ball (p9,r)
hence  x9 in Ball (p9,r) by A240; ::_thesis: verum
end;
end;
end;
hence  x in Ball (p9,r) ; ::_thesis: verum
end;
hence  Ball (p9,r) meets Lower_Arc C by A147, A168, A179, A186, A189, A194, A207, A211, A213, A215, A244, Th24, XBOOLE_1:63; ::_thesis: verum
end;
end;
end;
hence  Ball (p9,r) meets Lower_Arc C ; ::_thesis: verum
end;
then  p in Cl (Lower_Arc C) by A8, GOBOARD6:93;
then  p in Lower_Arc C by PRE_TOPC:22;
then  p in (Upper_Arc C) /\ (Lower_Arc C) by A127, XBOOLE_0:def_4;
then  p in {(W-min C),(E-max C)} by JORDAN6:50;
then  ( p = W-min C or p = E-max C ) by TARSKI:def_2;
hence  contradiction by A1, A2, EUCLID:52; ::_thesis: verum
end;

theorem :: JORDAN19:27
for C being Simple_closed_curve
for p being Point of (TOP-REAL 2) st p `1 = ((W-bound C) + (E-bound C)) / 2 & p in North_Arc C holds
not p in South_Arc C
proof
let C be Simple_closed_curve; ::_thesis: for p being Point of (TOP-REAL 2) st p `1 = ((W-bound C) + (E-bound C)) / 2 & p in North_Arc C holds
not p in South_Arc C
let p be Point of (TOP-REAL 2); ::_thesis: ( p `1 = ((W-bound C) + (E-bound C)) / 2 & p in North_Arc C implies not p in South_Arc C )
A1: W-bound C < E-bound C by SPRECT_1:31;
assume A2: p `1 = ((W-bound C) + (E-bound C)) / 2 ; ::_thesis: ( not p in North_Arc C or not p in South_Arc C )
then A3: W-bound C < p `1 by A1, XREAL_1:226;
 p `1 < E-bound C by A1, A2, XREAL_1:226;
hence  ( not p in North_Arc C or not p in South_Arc C ) by A3, Th26; ::_thesis: verum
end;