:: JORDAN9 semantic presentation

begin

       Lm1:  for n being Element of NAT st 1 <= n holds
(n -' 1) + 2 = n + 1
proof
let n be Element of NAT ; ::_thesis: ( 1 <= n implies (n -' 1) + 2 = n + 1 )
assume  1 <= n ; ::_thesis: (n -' 1) + 2 = n + 1
hence (n -' 1) + 2 = (n + 2) -' 1 by NAT_D:38
.= ((n + 1) + 1) - 1 by NAT_D:37
.= n + 1 ;
::_thesis: verum
end;

theorem Th1: :: JORDAN9:1
for T being non empty TopSpace
for B, C1, C2, D being Subset of T st B is connected & C1 is_a_component_of D & C2 is_a_component_of D & B meets C1 & B meets C2 & B c= D holds
C1 = C2
proof
let T be non empty TopSpace; ::_thesis: for B, C1, C2, D being Subset of T st B is connected & C1 is_a_component_of D & C2 is_a_component_of D & B meets C1 & B meets C2 & B c= D holds
C1 = C2
let B, C1, C2, D be Subset of T; ::_thesis: ( B is connected & C1 is_a_component_of D & C2 is_a_component_of D & B meets C1 & B meets C2 & B c= D implies C1 = C2 )
assume that
A1: B is connected and
A2: ( C1 is_a_component_of D & C2 is_a_component_of D ) and
A3: B meets C1 and
A4: ( B meets C2 & B c= D ) ; ::_thesis: C1 = C2
A5: B <> {} by A3, XBOOLE_1:65;
 ( B c= C1 & B c= C2 ) by A1, A2, A3, A4, GOBOARD9:4;
hence  C1 = C2 by A2, A5, GOBOARD9:1, XBOOLE_1:68; ::_thesis: verum
end;

theorem Th2: :: JORDAN9:2
for D being non empty set
for f, g being FinSequence of D st ( for n being Element of NAT holds f | n = g | n ) holds
f = g
proof
let D be non empty set ; ::_thesis: for f, g being FinSequence of D st ( for n being Element of NAT holds f | n = g | n ) holds
f = g
let f, g be FinSequence of D; ::_thesis: ( ( for n being Element of NAT holds f | n = g | n ) implies f = g )
assume A1: for n being Element of NAT holds f | n = g | n ; ::_thesis: f = g
A2: now__::_thesis:_not_len_f_<>_len_g
assume A3: len f <> len g ; ::_thesis: contradiction
percases ( len f < len g or len g < len f ) by A3, XXREAL_0:1;
supposeA4: len f < len g ; ::_thesis: contradiction
A5: g | (len g) = g by FINSEQ_1:58;
 f | (len g) = f by A4, FINSEQ_1:58;
hence  contradiction by A1, A4, A5; ::_thesis: verum
end;
supposeA6: len g < len f ; ::_thesis: contradiction
then  ( f | (len f) = f & g | (len f) = g ) by FINSEQ_1:58;
hence  contradiction by A1, A6; ::_thesis: verum
end;
end;
end;
 ( f | (len f) = f & g | (len g) = g ) by FINSEQ_1:58;
hence  f = g by A1, A2; ::_thesis: verum
end;

theorem Th3: :: JORDAN9:3
for n being Element of NAT
for D being non empty set
for f being FinSequence of D st n in dom f holds
ex k being Element of NAT st
( k in dom (Rev f) & n + k = (len f) + 1 & f /. n = (Rev f) /. k )
proof
let n be Element of NAT ; ::_thesis: for D being non empty set
for f being FinSequence of D st n in dom f holds
ex k being Element of NAT st
( k in dom (Rev f) & n + k = (len f) + 1 & f /. n = (Rev f) /. k )
let D be non empty set ; ::_thesis: for f being FinSequence of D st n in dom f holds
ex k being Element of NAT st
( k in dom (Rev f) & n + k = (len f) + 1 & f /. n = (Rev f) /. k )
let f be FinSequence of D; ::_thesis: ( n in dom f implies ex k being Element of NAT st
( k in dom (Rev f) & n + k = (len f) + 1 & f /. n = (Rev f) /. k ) )
assume A1: n in dom f ; ::_thesis: ex k being Element of NAT st
( k in dom (Rev f) & n + k = (len f) + 1 & f /. n = (Rev f) /. k )
take k = ((len f) + 1) -' n; ::_thesis: ( k in dom (Rev f) & n + k = (len f) + 1 & f /. n = (Rev f) /. k )
 1 <= n by A1, FINSEQ_3:25;
then  k + 1 <= k + n by XREAL_1:6;
then A2: (k + 1) -' 1 <= (k + n) -' 1 by NAT_D:42;
A3: n <= len f by A1, FINSEQ_3:25;
then  n + 1 <= (len f) + 1 by XREAL_1:6;
then A4: 1 <= k by NAT_D:55;
 n <= (len f) + 1 by A3, XREAL_1:145;
then A5: k + n = (len f) + 1 by XREAL_1:235;
then  (k + n) -' 1 = len f by NAT_D:34;
then  k <= len f by A2, NAT_D:34;
then  k in dom f by A4, FINSEQ_3:25;
hence  ( k in dom (Rev f) & n + k = (len f) + 1 & f /. n = (Rev f) /. k ) by A1, A5, FINSEQ_5:57, FINSEQ_5:66; ::_thesis: verum
end;

theorem Th4: :: JORDAN9:4
for n being Element of NAT
for D being non empty set
for f being FinSequence of D st n in dom (Rev f) holds
ex k being Element of NAT st
( k in dom f & n + k = (len f) + 1 & (Rev f) /. n = f /. k )
proof
let n be Element of NAT ; ::_thesis: for D being non empty set
for f being FinSequence of D st n in dom (Rev f) holds
ex k being Element of NAT st
( k in dom f & n + k = (len f) + 1 & (Rev f) /. n = f /. k )
let D be non empty set ; ::_thesis: for f being FinSequence of D st n in dom (Rev f) holds
ex k being Element of NAT st
( k in dom f & n + k = (len f) + 1 & (Rev f) /. n = f /. k )
let f be FinSequence of D; ::_thesis: ( n in dom (Rev f) implies ex k being Element of NAT st
( k in dom f & n + k = (len f) + 1 & (Rev f) /. n = f /. k ) )
assume  n in dom (Rev f) ; ::_thesis: ex k being Element of NAT st
( k in dom f & n + k = (len f) + 1 & (Rev f) /. n = f /. k )
then  n in dom f by FINSEQ_5:57;
then consider k being Element of NAT  such that
A1: ( k in dom (Rev f) & n + k = (len f) + 1 ) and
 f /. n = (Rev f) /. k by Th3;
A2: dom f = dom (Rev f) by FINSEQ_5:57;
then  (Rev f) /. n = f /. k by A1, FINSEQ_5:66;
hence  ex k being Element of NAT st
( k in dom f & n + k = (len f) + 1 & (Rev f) /. n = f /. k ) by A1, A2; ::_thesis: verum
end;

begin


theorem Th5: :: JORDAN9:5
for D being non empty set
for G being Matrix of D
for f being FinSequence of D holds
( f is_sequence_on G iff Rev f is_sequence_on G )
proof
let D be non empty set ; ::_thesis: for G being Matrix of D
for f being FinSequence of D holds
( f is_sequence_on G iff Rev f is_sequence_on G )
let G be Matrix of D; ::_thesis: for f being FinSequence of D holds
( f is_sequence_on G iff Rev f is_sequence_on G )
let f be FinSequence of D; ::_thesis: ( f is_sequence_on G iff Rev f is_sequence_on G )
hereby  ::_thesis: ( Rev f is_sequence_on G implies f is_sequence_on G )
assume A1: f is_sequence_on G ; ::_thesis: Rev f is_sequence_on G
A2: for n being Element of NAT st n in dom (Rev f) & n + 1 in dom (Rev f) holds
for m, k, i, j being Element of NAT st [m,k] in Indices G & [i,j] in Indices G & (Rev f) /. n = G * (m,k) & (Rev f) /. (n + 1) = G * (i,j) holds
(abs (m - i)) + (abs (k - j)) = 1
proof
let n be Element of NAT ; ::_thesis: ( n in dom (Rev f) & n + 1 in dom (Rev f) implies for m, k, i, j being Element of NAT st [m,k] in Indices G & [i,j] in Indices G & (Rev f) /. n = G * (m,k) & (Rev f) /. (n + 1) = G * (i,j) holds
(abs (m - i)) + (abs (k - j)) = 1 )
assume that
A3: n in dom (Rev f) and
A4: n + 1 in dom (Rev f) ; ::_thesis: for m, k, i, j being Element of NAT st [m,k] in Indices G & [i,j] in Indices G & (Rev f) /. n = G * (m,k) & (Rev f) /. (n + 1) = G * (i,j) holds
(abs (m - i)) + (abs (k - j)) = 1
consider l being Element of NAT  such that
A5: l in dom f and
A6: n + l = (len f) + 1 and
A7: (Rev f) /. n = f /. l by A3, Th4;
let m, k, i, j be Element of NAT ; ::_thesis: ( [m,k] in Indices G & [i,j] in Indices G & (Rev f) /. n = G * (m,k) & (Rev f) /. (n + 1) = G * (i,j) implies (abs (m - i)) + (abs (k - j)) = 1 )
assume A8: ( [m,k] in Indices G & [i,j] in Indices G & (Rev f) /. n = G * (m,k) & (Rev f) /. (n + 1) = G * (i,j) ) ; ::_thesis: (abs (m - i)) + (abs (k - j)) = 1
A9: ( abs (i - m) = abs (m - i) & abs (j - k) = abs (k - j) ) by UNIFORM1:11;
consider l9 being Element of NAT  such that
A10: l9 in dom f and
A11: (n + 1) + l9 = (len f) + 1 and
A12: (Rev f) /. (n + 1) = f /. l9 by A4, Th4;
 n + (1 + l9) = n + l by A6, A11;
hence  (abs (m - i)) + (abs (k - j)) = 1 by A1, A8, A5, A7, A10, A12, A9, GOBOARD1:def_9; ::_thesis: verum
end;
 for n being Element of NAT st n in dom (Rev f) holds
ex i, j being Element of NAT st
( [i,j] in Indices G & (Rev f) /. n = G * (i,j) )
proof
let n be Element of NAT ; ::_thesis: ( n in dom (Rev f) implies ex i, j being Element of NAT st
( [i,j] in Indices G & (Rev f) /. n = G * (i,j) ) )
assume  n in dom (Rev f) ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices G & (Rev f) /. n = G * (i,j) )
then consider k being Element of NAT  such that
A13: k in dom f and
 n + k = (len f) + 1 and
A14: (Rev f) /. n = f /. k by Th4;
consider i, j being Element of NAT  such that
A15: ( [i,j] in Indices G & f /. k = G * (i,j) ) by A1, A13, GOBOARD1:def_9;
take  i ; ::_thesis: ex j being Element of NAT st
( [i,j] in Indices G & (Rev f) /. n = G * (i,j) )
take  j ; ::_thesis: ( [i,j] in Indices G & (Rev f) /. n = G * (i,j) )
thus  ( [i,j] in Indices G & (Rev f) /. n = G * (i,j) ) by A14, A15; ::_thesis: verum
end;
hence  Rev f is_sequence_on G by A2, GOBOARD1:def_9; ::_thesis: verum
end;
assume A16: Rev f is_sequence_on G ; ::_thesis: f is_sequence_on G
A17: for n being Element of NAT st n in dom f & n + 1 in dom f holds
for m, k, i, j being Element of NAT st [m,k] in Indices G & [i,j] in Indices G & f /. n = G * (m,k) & f /. (n + 1) = G * (i,j) holds
(abs (m - i)) + (abs (k - j)) = 1
proof
let n be Element of NAT ; ::_thesis: ( n in dom f & n + 1 in dom f implies for m, k, i, j being Element of NAT st [m,k] in Indices G & [i,j] in Indices G & f /. n = G * (m,k) & f /. (n + 1) = G * (i,j) holds
(abs (m - i)) + (abs (k - j)) = 1 )
assume that
A18: n in dom f and
A19: n + 1 in dom f ; ::_thesis: for m, k, i, j being Element of NAT st [m,k] in Indices G & [i,j] in Indices G & f /. n = G * (m,k) & f /. (n + 1) = G * (i,j) holds
(abs (m - i)) + (abs (k - j)) = 1
consider l being Element of NAT  such that
A20: l in dom (Rev f) and
A21: n + l = (len f) + 1 and
A22: f /. n = (Rev f) /. l by A18, Th3;
let m, k, i, j be Element of NAT ; ::_thesis: ( [m,k] in Indices G & [i,j] in Indices G & f /. n = G * (m,k) & f /. (n + 1) = G * (i,j) implies (abs (m - i)) + (abs (k - j)) = 1 )
assume A23: ( [m,k] in Indices G & [i,j] in Indices G & f /. n = G * (m,k) & f /. (n + 1) = G * (i,j) ) ; ::_thesis: (abs (m - i)) + (abs (k - j)) = 1
A24: ( abs (i - m) = abs (m - i) & abs (j - k) = abs (k - j) ) by UNIFORM1:11;
consider l9 being Element of NAT  such that
A25: l9 in dom (Rev f) and
A26: (n + 1) + l9 = (len f) + 1 and
A27: f /. (n + 1) = (Rev f) /. l9 by A19, Th3;
 n + (1 + l9) = n + l by A21, A26;
hence  (abs (m - i)) + (abs (k - j)) = 1 by A16, A23, A20, A22, A25, A27, A24, GOBOARD1:def_9; ::_thesis: verum
end;
 for n being Element of NAT st n in dom f holds
ex i, j being Element of NAT st
( [i,j] in Indices G & f /. n = G * (i,j) )
proof
let n be Element of NAT ; ::_thesis: ( n in dom f implies ex i, j being Element of NAT st
( [i,j] in Indices G & f /. n = G * (i,j) ) )
assume  n in dom f ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices G & f /. n = G * (i,j) )
then consider k being Element of NAT  such that
A28: k in dom (Rev f) and
 n + k = (len f) + 1 and
A29: f /. n = (Rev f) /. k by Th3;
consider i, j being Element of NAT  such that
A30: ( [i,j] in Indices G & (Rev f) /. k = G * (i,j) ) by A16, A28, GOBOARD1:def_9;
take  i ; ::_thesis: ex j being Element of NAT st
( [i,j] in Indices G & f /. n = G * (i,j) )
take  j ; ::_thesis: ( [i,j] in Indices G & f /. n = G * (i,j) )
thus  ( [i,j] in Indices G & f /. n = G * (i,j) ) by A29, A30; ::_thesis: verum
end;
hence  f is_sequence_on G by A17, GOBOARD1:def_9; ::_thesis: verum
end;

theorem Th6: :: JORDAN9:6
for k being Element of NAT
for G being Matrix of (TOP-REAL 2)
for f being FinSequence of (TOP-REAL 2) st f is_sequence_on G & 1 <= k & k <= len f holds
f /. k in Values G
proof
let k be Element of NAT ; ::_thesis: for G being Matrix of (TOP-REAL 2)
for f being FinSequence of (TOP-REAL 2) st f is_sequence_on G & 1 <= k & k <= len f holds
f /. k in Values G
let G be Matrix of (TOP-REAL 2); ::_thesis: for f being FinSequence of (TOP-REAL 2) st f is_sequence_on G & 1 <= k & k <= len f holds
f /. k in Values G
let f be FinSequence of (TOP-REAL 2); ::_thesis: ( f is_sequence_on G & 1 <= k & k <= len f implies f /. k in Values G )
assume that
A1: f is_sequence_on G and
A2: ( 1 <= k & k <= len f ) ; ::_thesis: f /. k in Values G
A3: k in dom f by A2, FINSEQ_3:25;
then  f /. k = f . k by PARTFUN1:def_6;
then A4: f /. k in rng f by A3, FUNCT_1:def_3;
 rng f c= Values G by A1, GOBRD13:8;
hence  f /. k in Values G by A4; ::_thesis: verum
end;

Lm2:  for k being Element of NAT
for G being Go-board
for f being FinSequence of (TOP-REAL 2) st f is_sequence_on G & 1 <= k & k <= len f holds
ex i, j being Element of NAT st
( [i,j] in Indices G & f /. k = G * (i,j) )
proof
let k be Element of NAT ; ::_thesis: for G being Go-board
for f being FinSequence of (TOP-REAL 2) st f is_sequence_on G & 1 <= k & k <= len f holds
ex i, j being Element of NAT st
( [i,j] in Indices G & f /. k = G * (i,j) )
let G be Go-board; ::_thesis: for f being FinSequence of (TOP-REAL 2) st f is_sequence_on G & 1 <= k & k <= len f holds
ex i, j being Element of NAT st
( [i,j] in Indices G & f /. k = G * (i,j) )
let f be FinSequence of (TOP-REAL 2); ::_thesis: ( f is_sequence_on G & 1 <= k & k <= len f implies ex i, j being Element of NAT st
( [i,j] in Indices G & f /. k = G * (i,j) ) )
assume that
A1: f is_sequence_on G and
A2: ( 1 <= k & k <= len f ) ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices G & f /. k = G * (i,j) )
 k in dom f by A2, FINSEQ_3:25;
then consider i, j being Element of NAT  such that
A3: ( [i,j] in Indices G & f /. k = G * (i,j) ) by A1, GOBOARD1:def_9;
take  i ; ::_thesis: ex j being Element of NAT st
( [i,j] in Indices G & f /. k = G * (i,j) )
take  j ; ::_thesis: ( [i,j] in Indices G & f /. k = G * (i,j) )
thus  ( [i,j] in Indices G & f /. k = G * (i,j) ) by A3; ::_thesis: verum
end;

theorem Th7: :: JORDAN9:7
for n being Element of NAT
for f being FinSequence of (TOP-REAL 2)
for x being set st n <= len f & x in L~ (f /^ n) holds
ex i being Element of NAT st
( n + 1 <= i & i + 1 <= len f & x in LSeg (f,i) )
proof
let n be Element of NAT ; ::_thesis: for f being FinSequence of (TOP-REAL 2)
for x being set st n <= len f & x in L~ (f /^ n) holds
ex i being Element of NAT st
( n + 1 <= i & i + 1 <= len f & x in LSeg (f,i) )
let f be FinSequence of (TOP-REAL 2); ::_thesis: for x being set st n <= len f & x in L~ (f /^ n) holds
ex i being Element of NAT st
( n + 1 <= i & i + 1 <= len f & x in LSeg (f,i) )
let x be set ; ::_thesis: ( n <= len f & x in L~ (f /^ n) implies ex i being Element of NAT st
( n + 1 <= i & i + 1 <= len f & x in LSeg (f,i) ) )
assume that
A1: n <= len f and
A2: x in L~ (f /^ n) ; ::_thesis: ex i being Element of NAT st
( n + 1 <= i & i + 1 <= len f & x in LSeg (f,i) )
consider j being Element of NAT  such that
A3: 1 <= j and
A4: j + 1 <= len (f /^ n) and
A5: x in LSeg ((f /^ n),j) by A2, SPPOL_2:13;
 j + 1 <= (len f) - n by A1, A4, RFINSEQ:def_1;
then A6: n + (j + 1) <= len f by XREAL_1:19;
take  n + j ; ::_thesis: ( n + 1 <= n + j & (n + j) + 1 <= len f & x in LSeg (f,(n + j)) )
 j + 1 <= (len f) - n by A1, A4, RFINSEQ:def_1;
hence  ( n + 1 <= n + j & (n + j) + 1 <= len f & x in LSeg (f,(n + j)) ) by A3, A5, A6, SPPOL_2:5, XREAL_1:6; ::_thesis: verum
end;

theorem Th8: :: JORDAN9:8
for k being Element of NAT
for G being Go-board
for f being FinSequence of (TOP-REAL 2) st f is_sequence_on G & 1 <= k & k + 1 <= len f holds
( f /. k in left_cell (f,k,G) & f /. k in right_cell (f,k,G) )
proof
let k be Element of NAT ; ::_thesis: for G being Go-board
for f being FinSequence of (TOP-REAL 2) st f is_sequence_on G & 1 <= k & k + 1 <= len f holds
( f /. k in left_cell (f,k,G) & f /. k in right_cell (f,k,G) )
let G be Go-board; ::_thesis: for f being FinSequence of (TOP-REAL 2) st f is_sequence_on G & 1 <= k & k + 1 <= len f holds
( f /. k in left_cell (f,k,G) & f /. k in right_cell (f,k,G) )
let f be FinSequence of (TOP-REAL 2); ::_thesis: ( f is_sequence_on G & 1 <= k & k + 1 <= len f implies ( f /. k in left_cell (f,k,G) & f /. k in right_cell (f,k,G) ) )
assume that
A1: f is_sequence_on G and
A2: ( 1 <= k & k + 1 <= len f ) ; ::_thesis: ( f /. k in left_cell (f,k,G) & f /. k in right_cell (f,k,G) )
set p = f /. k;
 LSeg (f,k) = LSeg ((f /. k),(f /. (k + 1))) by A2, TOPREAL1:def_3;
then  f /. k in LSeg (f,k) by RLTOPSP1:68;
then  f /. k in (left_cell (f,k,G)) /\ (right_cell (f,k,G)) by A1, A2, GOBRD13:29;
hence  ( f /. k in left_cell (f,k,G) & f /. k in right_cell (f,k,G) ) by XBOOLE_0:def_4; ::_thesis: verum
end;

theorem Th9: :: JORDAN9:9
for k being Element of NAT
for G being Go-board
for f being FinSequence of (TOP-REAL 2) st f is_sequence_on G & 1 <= k & k + 1 <= len f holds
( Int (left_cell (f,k,G)) <> {} & Int (right_cell (f,k,G)) <> {} )
proof
let k be Element of NAT ; ::_thesis: for G being Go-board
for f being FinSequence of (TOP-REAL 2) st f is_sequence_on G & 1 <= k & k + 1 <= len f holds
( Int (left_cell (f,k,G)) <> {} & Int (right_cell (f,k,G)) <> {} )
let G be Go-board; ::_thesis: for f being FinSequence of (TOP-REAL 2) st f is_sequence_on G & 1 <= k & k + 1 <= len f holds
( Int (left_cell (f,k,G)) <> {} & Int (right_cell (f,k,G)) <> {} )
let f be FinSequence of (TOP-REAL 2); ::_thesis: ( f is_sequence_on G & 1 <= k & k + 1 <= len f implies ( Int (left_cell (f,k,G)) <> {} & Int (right_cell (f,k,G)) <> {} ) )
assume A1: ( f is_sequence_on G & 1 <= k & k + 1 <= len f ) ; ::_thesis: ( Int (left_cell (f,k,G)) <> {} & Int (right_cell (f,k,G)) <> {} )
then consider i1, j1, i2, j2 being Element of NAT  such that
A2: [i1,j1] in Indices G and
A3: f /. k = G * (i1,j1) and
A4: [i2,j2] in Indices G and
A5: f /. (k + 1) = G * (i2,j2) and
A6: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by JORDAN8:3;
A7: i2 <= len G by A4, MATRIX_1:38;
A8: i1 <= len G by A2, MATRIX_1:38;
then A9: i1 -' 1 <= len G by NAT_D:44;
A10: j2 <= width G by A4, MATRIX_1:38;
then A11: j2 -' 1 <= width G by NAT_D:44;
A12: ( j1 + 1 > j1 & j2 + 1 > j2 ) by NAT_1:13;
A13: j1 <= width G by A2, MATRIX_1:38;
then A14: j1 -' 1 <= width G by NAT_D:44;
A15: ( i1 + 1 > i1 & i2 + 1 > i2 ) by NAT_1:13;
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 A6;
suppose ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: ( Int (left_cell (f,k,G)) <> {} & Int (right_cell (f,k,G)) <> {} )
then  ( right_cell (f,k,G) = cell (G,i1,j1) & left_cell (f,k,G) = cell (G,(i1 -' 1),j1) ) by A1, A2, A3, A4, A5, A12, GOBRD13:def_2, GOBRD13:def_3;
hence  ( Int (left_cell (f,k,G)) <> {} & Int (right_cell (f,k,G)) <> {} ) by A8, A13, A9, GOBOARD9:14; ::_thesis: verum
end;
suppose ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: ( Int (left_cell (f,k,G)) <> {} & Int (right_cell (f,k,G)) <> {} )
then  ( right_cell (f,k,G) = cell (G,i1,(j1 -' 1)) & left_cell (f,k,G) = cell (G,i1,j1) ) by A1, A2, A3, A4, A5, A15, GOBRD13:def_2, GOBRD13:def_3;
hence  ( Int (left_cell (f,k,G)) <> {} & Int (right_cell (f,k,G)) <> {} ) by A8, A13, A14, GOBOARD9:14; ::_thesis: verum
end;
suppose ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: ( Int (left_cell (f,k,G)) <> {} & Int (right_cell (f,k,G)) <> {} )
then  ( right_cell (f,k,G) = cell (G,i2,j2) & left_cell (f,k,G) = cell (G,i2,(j2 -' 1)) ) by A1, A2, A3, A4, A5, A15, GOBRD13:def_2, GOBRD13:def_3;
hence  ( Int (left_cell (f,k,G)) <> {} & Int (right_cell (f,k,G)) <> {} ) by A7, A10, A11, GOBOARD9:14; ::_thesis: verum
end;
suppose ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: ( Int (left_cell (f,k,G)) <> {} & Int (right_cell (f,k,G)) <> {} )
then  ( right_cell (f,k,G) = cell (G,(i1 -' 1),j2) & left_cell (f,k,G) = cell (G,i1,j2) ) by A1, A2, A3, A4, A5, A12, GOBRD13:def_2, GOBRD13:def_3;
hence  ( Int (left_cell (f,k,G)) <> {} & Int (right_cell (f,k,G)) <> {} ) by A8, A10, A9, GOBOARD9:14; ::_thesis: verum
end;
end;
end;

theorem Th10: :: JORDAN9:10
for k being Element of NAT
for G being Go-board
for f being FinSequence of (TOP-REAL 2) st f is_sequence_on G & 1 <= k & k + 1 <= len f holds
( Int (left_cell (f,k,G)) is convex & Int (right_cell (f,k,G)) is convex )
proof
let k be Element of NAT ; ::_thesis: for G being Go-board
for f being FinSequence of (TOP-REAL 2) st f is_sequence_on G & 1 <= k & k + 1 <= len f holds
( Int (left_cell (f,k,G)) is convex & Int (right_cell (f,k,G)) is convex )
let G be Go-board; ::_thesis: for f being FinSequence of (TOP-REAL 2) st f is_sequence_on G & 1 <= k & k + 1 <= len f holds
( Int (left_cell (f,k,G)) is convex & Int (right_cell (f,k,G)) is convex )
let f be FinSequence of (TOP-REAL 2); ::_thesis: ( f is_sequence_on G & 1 <= k & k + 1 <= len f implies ( Int (left_cell (f,k,G)) is convex & Int (right_cell (f,k,G)) is convex ) )
assume A1: ( f is_sequence_on G & 1 <= k & k + 1 <= len f ) ; ::_thesis: ( Int (left_cell (f,k,G)) is convex & Int (right_cell (f,k,G)) is convex )
then consider i1, j1, i2, j2 being Element of NAT  such that
A2: [i1,j1] in Indices G and
A3: f /. k = G * (i1,j1) and
A4: [i2,j2] in Indices G and
A5: f /. (k + 1) = G * (i2,j2) and
A6: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by JORDAN8:3;
A7: i2 <= len G by A4, MATRIX_1:38;
A8: i1 <= len G by A2, MATRIX_1:38;
then A9: i1 -' 1 <= len G by NAT_D:44;
A10: j2 <= width G by A4, MATRIX_1:38;
then A11: j2 -' 1 <= width G by NAT_D:44;
A12: ( j1 + 1 > j1 & j2 + 1 > j2 ) by NAT_1:13;
A13: j1 <= width G by A2, MATRIX_1:38;
then A14: j1 -' 1 <= width G by NAT_D:44;
A15: ( i1 + 1 > i1 & i2 + 1 > i2 ) by NAT_1:13;
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 A6;
suppose ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: ( Int (left_cell (f,k,G)) is convex & Int (right_cell (f,k,G)) is convex )
then  ( right_cell (f,k,G) = cell (G,i1,j1) & left_cell (f,k,G) = cell (G,(i1 -' 1),j1) ) by A1, A2, A3, A4, A5, A12, GOBRD13:def_2, GOBRD13:def_3;
hence  ( Int (left_cell (f,k,G)) is convex & Int (right_cell (f,k,G)) is convex ) by A8, A13, A9, GOBOARD9:17; ::_thesis: verum
end;
suppose ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: ( Int (left_cell (f,k,G)) is convex & Int (right_cell (f,k,G)) is convex )
then  ( right_cell (f,k,G) = cell (G,i1,(j1 -' 1)) & left_cell (f,k,G) = cell (G,i1,j1) ) by A1, A2, A3, A4, A5, A15, GOBRD13:def_2, GOBRD13:def_3;
hence  ( Int (left_cell (f,k,G)) is convex & Int (right_cell (f,k,G)) is convex ) by A8, A13, A14, GOBOARD9:17; ::_thesis: verum
end;
suppose ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: ( Int (left_cell (f,k,G)) is convex & Int (right_cell (f,k,G)) is convex )
then  ( right_cell (f,k,G) = cell (G,i2,j2) & left_cell (f,k,G) = cell (G,i2,(j2 -' 1)) ) by A1, A2, A3, A4, A5, A15, GOBRD13:def_2, GOBRD13:def_3;
hence  ( Int (left_cell (f,k,G)) is convex & Int (right_cell (f,k,G)) is convex ) by A7, A10, A11, GOBOARD9:17; ::_thesis: verum
end;
suppose ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: ( Int (left_cell (f,k,G)) is convex & Int (right_cell (f,k,G)) is convex )
then  ( right_cell (f,k,G) = cell (G,(i1 -' 1),j2) & left_cell (f,k,G) = cell (G,i1,j2) ) by A1, A2, A3, A4, A5, A12, GOBRD13:def_2, GOBRD13:def_3;
hence  ( Int (left_cell (f,k,G)) is convex & Int (right_cell (f,k,G)) is convex ) by A8, A10, A9, GOBOARD9:17; ::_thesis: verum
end;
end;
end;

theorem Th11: :: JORDAN9:11
for k being Element of NAT
for G being Go-board
for f being FinSequence of (TOP-REAL 2) st f is_sequence_on G & 1 <= k & k + 1 <= len f holds
( Cl (Int (left_cell (f,k,G))) = left_cell (f,k,G) & Cl (Int (right_cell (f,k,G))) = right_cell (f,k,G) )
proof
let k be Element of NAT ; ::_thesis: for G being Go-board
for f being FinSequence of (TOP-REAL 2) st f is_sequence_on G & 1 <= k & k + 1 <= len f holds
( Cl (Int (left_cell (f,k,G))) = left_cell (f,k,G) & Cl (Int (right_cell (f,k,G))) = right_cell (f,k,G) )
let G be Go-board; ::_thesis: for f being FinSequence of (TOP-REAL 2) st f is_sequence_on G & 1 <= k & k + 1 <= len f holds
( Cl (Int (left_cell (f,k,G))) = left_cell (f,k,G) & Cl (Int (right_cell (f,k,G))) = right_cell (f,k,G) )
let f be FinSequence of (TOP-REAL 2); ::_thesis: ( f is_sequence_on G & 1 <= k & k + 1 <= len f implies ( Cl (Int (left_cell (f,k,G))) = left_cell (f,k,G) & Cl (Int (right_cell (f,k,G))) = right_cell (f,k,G) ) )
assume A1: ( f is_sequence_on G & 1 <= k & k + 1 <= len f ) ; ::_thesis: ( Cl (Int (left_cell (f,k,G))) = left_cell (f,k,G) & Cl (Int (right_cell (f,k,G))) = right_cell (f,k,G) )
then consider i1, j1, i2, j2 being Element of NAT  such that
A2: [i1,j1] in Indices G and
A3: f /. k = G * (i1,j1) and
A4: [i2,j2] in Indices G and
A5: f /. (k + 1) = G * (i2,j2) and
A6: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by JORDAN8:3;
A7: i2 <= len G by A4, MATRIX_1:38;
A8: i1 <= len G by A2, MATRIX_1:38;
then A9: i1 -' 1 <= len G by NAT_D:44;
A10: j2 <= width G by A4, MATRIX_1:38;
then A11: j2 -' 1 <= width G by NAT_D:44;
A12: ( j1 + 1 > j1 & j2 + 1 > j2 ) by NAT_1:13;
A13: j1 <= width G by A2, MATRIX_1:38;
then A14: j1 -' 1 <= width G by NAT_D:44;
A15: ( i1 + 1 > i1 & i2 + 1 > i2 ) by NAT_1:13;
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 A6;
suppose ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: ( Cl (Int (left_cell (f,k,G))) = left_cell (f,k,G) & Cl (Int (right_cell (f,k,G))) = right_cell (f,k,G) )
then  ( right_cell (f,k,G) = cell (G,i1,j1) & left_cell (f,k,G) = cell (G,(i1 -' 1),j1) ) by A1, A2, A3, A4, A5, A12, GOBRD13:def_2, GOBRD13:def_3;
hence  ( Cl (Int (left_cell (f,k,G))) = left_cell (f,k,G) & Cl (Int (right_cell (f,k,G))) = right_cell (f,k,G) ) by A8, A13, A9, GOBRD11:35; ::_thesis: verum
end;
suppose ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: ( Cl (Int (left_cell (f,k,G))) = left_cell (f,k,G) & Cl (Int (right_cell (f,k,G))) = right_cell (f,k,G) )
then  ( right_cell (f,k,G) = cell (G,i1,(j1 -' 1)) & left_cell (f,k,G) = cell (G,i1,j1) ) by A1, A2, A3, A4, A5, A15, GOBRD13:def_2, GOBRD13:def_3;
hence  ( Cl (Int (left_cell (f,k,G))) = left_cell (f,k,G) & Cl (Int (right_cell (f,k,G))) = right_cell (f,k,G) ) by A8, A13, A14, GOBRD11:35; ::_thesis: verum
end;
suppose ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: ( Cl (Int (left_cell (f,k,G))) = left_cell (f,k,G) & Cl (Int (right_cell (f,k,G))) = right_cell (f,k,G) )
then  ( right_cell (f,k,G) = cell (G,i2,j2) & left_cell (f,k,G) = cell (G,i2,(j2 -' 1)) ) by A1, A2, A3, A4, A5, A15, GOBRD13:def_2, GOBRD13:def_3;
hence  ( Cl (Int (left_cell (f,k,G))) = left_cell (f,k,G) & Cl (Int (right_cell (f,k,G))) = right_cell (f,k,G) ) by A7, A10, A11, GOBRD11:35; ::_thesis: verum
end;
suppose ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: ( Cl (Int (left_cell (f,k,G))) = left_cell (f,k,G) & Cl (Int (right_cell (f,k,G))) = right_cell (f,k,G) )
then  ( right_cell (f,k,G) = cell (G,(i1 -' 1),j2) & left_cell (f,k,G) = cell (G,i1,j2) ) by A1, A2, A3, A4, A5, A12, GOBRD13:def_2, GOBRD13:def_3;
hence  ( Cl (Int (left_cell (f,k,G))) = left_cell (f,k,G) & Cl (Int (right_cell (f,k,G))) = right_cell (f,k,G) ) by A8, A10, A9, GOBRD11:35; ::_thesis: verum
end;
end;
end;

theorem Th12: :: JORDAN9:12
for k being Element of NAT
for G being Go-board
for f being FinSequence of (TOP-REAL 2) st f is_sequence_on G & LSeg (f,k) is horizontal holds
ex j being Element of NAT st
( 1 <= j & j <= width G & ( for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds
p `2 = (G * (1,j)) `2 ) )
proof
let k be Element of NAT ; ::_thesis: for G being Go-board
for f being FinSequence of (TOP-REAL 2) st f is_sequence_on G & LSeg (f,k) is horizontal holds
ex j being Element of NAT st
( 1 <= j & j <= width G & ( for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds
p `2 = (G * (1,j)) `2 ) )
let G be Go-board; ::_thesis: for f being FinSequence of (TOP-REAL 2) st f is_sequence_on G & LSeg (f,k) is horizontal holds
ex j being Element of NAT st
( 1 <= j & j <= width G & ( for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds
p `2 = (G * (1,j)) `2 ) )
let f be FinSequence of (TOP-REAL 2); ::_thesis: ( f is_sequence_on G & LSeg (f,k) is horizontal implies ex j being Element of NAT st
( 1 <= j & j <= width G & ( for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds
p `2 = (G * (1,j)) `2 ) ) )
assume that
A1: f is_sequence_on G and
A2: LSeg (f,k) is horizontal ; ::_thesis: ex j being Element of NAT st
( 1 <= j & j <= width G & ( for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds
p `2 = (G * (1,j)) `2 ) )
percases ( ( 1 <= k & k + 1 <= len f ) or not 1 <= k or not k + 1 <= len f ) ;
supposeA3: ( 1 <= k & k + 1 <= len f ) ; ::_thesis: ex j being Element of NAT st
( 1 <= j & j <= width G & ( for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds
p `2 = (G * (1,j)) `2 ) )
 k <= k + 1 by NAT_1:11;
then  k <= len f by A3, XXREAL_0:2;
then consider i, j being Element of NAT  such that
A4: [i,j] in Indices G and
A5: f /. k = G * (i,j) by A1, A3, Lm2;
take  j ; ::_thesis: ( 1 <= j & j <= width G & ( for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds
p `2 = (G * (1,j)) `2 ) )
thus A6: ( 1 <= j & j <= width G ) by A4, MATRIX_1:38; ::_thesis: for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds
p `2 = (G * (1,j)) `2
A7: f /. k in LSeg (f,k) by A3, TOPREAL1:21;
let p be Point of (TOP-REAL 2); ::_thesis: ( p in LSeg (f,k) implies p `2 = (G * (1,j)) `2 )
A8: ( 1 <= i & i <= len G ) by A4, MATRIX_1:38;
assume  p in LSeg (f,k) ; ::_thesis: p `2 = (G * (1,j)) `2
hence p `2 = (f /. k) `2 by A2, A7, SPPOL_1:def_2
.= (G * (1,j)) `2 by A5, A6, A8, GOBOARD5:1 ;
::_thesis: verum
end;
supposeA9: ( not 1 <= k or not k + 1 <= len f ) ; ::_thesis: ex j being Element of NAT st
( 1 <= j & j <= width G & ( for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds
p `2 = (G * (1,j)) `2 ) )
take  1 ; ::_thesis: ( 1 <= 1 & 1 <= width G & ( for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds
p `2 = (G * (1,1)) `2 ) )
 width G <> 0 by GOBOARD1:def_3;
hence  ( 1 <= 1 & 1 <= width G ) by NAT_1:14; ::_thesis: for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds
p `2 = (G * (1,1)) `2
thus  for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds
p `2 = (G * (1,1)) `2 by A9, TOPREAL1:def_3; ::_thesis: verum
end;
end;
end;

theorem Th13: :: JORDAN9:13
for k being Element of NAT
for G being Go-board
for f being FinSequence of (TOP-REAL 2) st f is_sequence_on G & LSeg (f,k) is vertical holds
ex i being Element of NAT st
( 1 <= i & i <= len G & ( for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds
p `1 = (G * (i,1)) `1 ) )
proof
let k be Element of NAT ; ::_thesis: for G being Go-board
for f being FinSequence of (TOP-REAL 2) st f is_sequence_on G & LSeg (f,k) is vertical holds
ex i being Element of NAT st
( 1 <= i & i <= len G & ( for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds
p `1 = (G * (i,1)) `1 ) )
let G be Go-board; ::_thesis: for f being FinSequence of (TOP-REAL 2) st f is_sequence_on G & LSeg (f,k) is vertical holds
ex i being Element of NAT st
( 1 <= i & i <= len G & ( for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds
p `1 = (G * (i,1)) `1 ) )
let f be FinSequence of (TOP-REAL 2); ::_thesis: ( f is_sequence_on G & LSeg (f,k) is vertical implies ex i being Element of NAT st
( 1 <= i & i <= len G & ( for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds
p `1 = (G * (i,1)) `1 ) ) )
assume that
A1: f is_sequence_on G and
A2: LSeg (f,k) is vertical ; ::_thesis: ex i being Element of NAT st
( 1 <= i & i <= len G & ( for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds
p `1 = (G * (i,1)) `1 ) )
percases ( ( 1 <= k & k + 1 <= len f ) or not 1 <= k or not k + 1 <= len f ) ;
supposeA3: ( 1 <= k & k + 1 <= len f ) ; ::_thesis: ex i being Element of NAT st
( 1 <= i & i <= len G & ( for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds
p `1 = (G * (i,1)) `1 ) )
 k <= k + 1 by NAT_1:11;
then  k <= len f by A3, XXREAL_0:2;
then consider i, j being Element of NAT  such that
A4: [i,j] in Indices G and
A5: f /. k = G * (i,j) by A1, A3, Lm2;
take  i ; ::_thesis: ( 1 <= i & i <= len G & ( for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds
p `1 = (G * (i,1)) `1 ) )
thus A6: ( 1 <= i & i <= len G ) by A4, MATRIX_1:38; ::_thesis: for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds
p `1 = (G * (i,1)) `1
A7: f /. k in LSeg (f,k) by A3, TOPREAL1:21;
let p be Point of (TOP-REAL 2); ::_thesis: ( p in LSeg (f,k) implies p `1 = (G * (i,1)) `1 )
A8: ( 1 <= j & j <= width G ) by A4, MATRIX_1:38;
assume  p in LSeg (f,k) ; ::_thesis: p `1 = (G * (i,1)) `1
hence p `1 = (f /. k) `1 by A2, A7, SPPOL_1:def_3
.= (G * (i,1)) `1 by A5, A6, A8, GOBOARD5:2 ;
::_thesis: verum
end;
supposeA9: ( not 1 <= k or not k + 1 <= len f ) ; ::_thesis: ex i being Element of NAT st
( 1 <= i & i <= len G & ( for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds
p `1 = (G * (i,1)) `1 ) )
take  1 ; ::_thesis: ( 1 <= 1 & 1 <= len G & ( for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds
p `1 = (G * (1,1)) `1 ) )
 0 <> len G by GOBOARD1:def_3;
hence  ( 1 <= 1 & 1 <= len G ) by NAT_1:14; ::_thesis: for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds
p `1 = (G * (1,1)) `1
thus  for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds
p `1 = (G * (1,1)) `1 by A9, TOPREAL1:def_3; ::_thesis: verum
end;
end;
end;

theorem Th14: :: JORDAN9:14
for i, j being Element of NAT
for G being Go-board
for f being FinSequence of (TOP-REAL 2) st f is_sequence_on G & f is special & i <= len G & j <= width G holds
Int (cell (G,i,j)) misses L~ f
proof
let i, j be Element of NAT ; ::_thesis: for G being Go-board
for f being FinSequence of (TOP-REAL 2) st f is_sequence_on G & f is special & i <= len G & j <= width G holds
Int (cell (G,i,j)) misses L~ f
let G be Go-board; ::_thesis: for f being FinSequence of (TOP-REAL 2) st f is_sequence_on G & f is special & i <= len G & j <= width G holds
Int (cell (G,i,j)) misses L~ f
let f be FinSequence of (TOP-REAL 2); ::_thesis: ( f is_sequence_on G & f is special & i <= len G & j <= width G implies Int (cell (G,i,j)) misses L~ f )
assume that
A1: f is_sequence_on G and
A2: f is special and
A3: i <= len G and
A4: j <= width G ; ::_thesis: Int (cell (G,i,j)) misses L~ f
A5: Int (cell (G,i,j)) = (Int (v_strip (G,i))) /\ (Int (h_strip (G,j))) by TOPS_1:17;
assume  Int (cell (G,i,j)) meets L~ f ; ::_thesis: contradiction
then consider x being set  such that
A6: x in Int (cell (G,i,j)) and
A7: x in L~ f by XBOOLE_0:3;
 L~ f = union  {  (LSeg (f,k)) where k is Element of NAT : ( 1 <= k & k + 1 <= len f )  }  by TOPREAL1:def_4;
then consider X being set  such that
A8: x in X and
A9: X in  {  (LSeg (f,k)) where k is Element of NAT : ( 1 <= k & k + 1 <= len f )  }  by A7, TARSKI:def_4;
consider k being Element of NAT  such that
A10: X = LSeg (f,k) and
 1 <= k and
 k + 1 <= len f by A9;
reconsider p = x as Point of (TOP-REAL 2) by A8, A10;
percases ( LSeg (f,k) is horizontal or LSeg (f,k) is vertical ) by A2, SPPOL_1:19;
suppose LSeg (f,k) is horizontal ; ::_thesis: contradiction
then consider j0 being Element of NAT  such that
A11: 1 <= j0 and
A12: j0 <= width G and
A13: for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds
p `2 = (G * (1,j0)) `2 by A1, Th12;
now__::_thesis:_not_p_in_Int_(h_strip_(G,j))
A14: ( j0 > j implies j0 >= j + 1 ) by NAT_1:13;
assume A15: p in Int (h_strip (G,j)) ; ::_thesis: contradiction
percases ( j0 < j or j0 = j or j0 > j + 1 or j0 = j + 1 ) by A14, XXREAL_0:1;
supposeA16: j0 < j ; ::_thesis: contradiction
 0 <> len G by GOBOARD1:def_3;
then  1 <= len G by NAT_1:14;
then A17: (G * (1,j)) `2 > (G * (1,j0)) `2 by A4, A11, A16, GOBOARD5:4;
 j >= 1 by A11, A16, XXREAL_0:2;
then  p `2 > (G * (1,j)) `2 by A4, A15, GOBOARD6:27;
hence  contradiction by A8, A10, A13, A17; ::_thesis: verum
end;
suppose j0 = j ; ::_thesis: contradiction
then  p `2 > (G * (1,j0)) `2 by A11, A12, A15, GOBOARD6:27;
hence  contradiction by A8, A10, A13; ::_thesis: verum
end;
supposeA18: j0 > j + 1 ; ::_thesis: contradiction
then  j + 1 <= width G by A12, XXREAL_0:2;
then  j < width G by NAT_1:13;
then A19: p `2 < (G * (1,(j + 1))) `2 by A15, GOBOARD6:28;
 0 <> len G by GOBOARD1:def_3;
then A20: 1 <= len G by NAT_1:14;
 j + 1 >= 1 by NAT_1:14;
then  (G * (1,(j + 1))) `2 < (G * (1,j0)) `2 by A12, A18, A20, GOBOARD5:4;
hence  contradiction by A8, A10, A13, A19; ::_thesis: verum
end;
supposeA21: j0 = j + 1 ; ::_thesis: contradiction
then  j < width G by A12, NAT_1:13;
then  p `2 < (G * (1,j0)) `2 by A15, A21, GOBOARD6:28;
hence  contradiction by A8, A10, A13; ::_thesis: verum
end;
end;
end;
hence  contradiction by A6, A5, XBOOLE_0:def_4; ::_thesis: verum
end;
suppose LSeg (f,k) is vertical ; ::_thesis: contradiction
then consider i0 being Element of NAT  such that
A22: 1 <= i0 and
A23: i0 <= len G and
A24: for p being Point of (TOP-REAL 2) st p in LSeg (f,k) holds
p `1 = (G * (i0,1)) `1 by A1, Th13;
now__::_thesis:_not_p_in_Int_(v_strip_(G,i))
A25: ( i0 > i implies i0 >= i + 1 ) by NAT_1:13;
assume A26: p in Int (v_strip (G,i)) ; ::_thesis: contradiction
percases ( i0 < i or i0 = i or i0 > i + 1 or i0 = i + 1 ) by A25, XXREAL_0:1;
supposeA27: i0 < i ; ::_thesis: contradiction
 0 <> width G by GOBOARD1:def_3;
then  1 <= width G by NAT_1:14;
then A28: (G * (i,1)) `1 > (G * (i0,1)) `1 by A3, A22, A27, GOBOARD5:3;
 i >= 1 by A22, A27, XXREAL_0:2;
then  p `1 > (G * (i,1)) `1 by A3, A26, GOBOARD6:29;
hence  contradiction by A8, A10, A24, A28; ::_thesis: verum
end;
suppose i0 = i ; ::_thesis: contradiction
then  p `1 > (G * (i0,1)) `1 by A22, A23, A26, GOBOARD6:29;
hence  contradiction by A8, A10, A24; ::_thesis: verum
end;
supposeA29: i0 > i + 1 ; ::_thesis: contradiction
then  i + 1 <= len G by A23, XXREAL_0:2;
then  i < len G by NAT_1:13;
then A30: p `1 < (G * ((i + 1),1)) `1 by A26, GOBOARD6:30;
 0 <> width G by GOBOARD1:def_3;
then A31: 1 <= width G by NAT_1:14;
 i + 1 >= 1 by NAT_1:14;
then  (G * ((i + 1),1)) `1 < (G * (i0,1)) `1 by A23, A29, A31, GOBOARD5:3;
hence  contradiction by A8, A10, A24, A30; ::_thesis: verum
end;
supposeA32: i0 = i + 1 ; ::_thesis: contradiction
then  i < len G by A23, NAT_1:13;
then  p `1 < (G * (i0,1)) `1 by A26, A32, GOBOARD6:30;
hence  contradiction by A8, A10, A24; ::_thesis: verum
end;
end;
end;
hence  contradiction by A6, A5, XBOOLE_0:def_4; ::_thesis: verum
end;
end;
end;

theorem Th15: :: JORDAN9:15
for k being Element of NAT
for G being Go-board
for f being FinSequence of (TOP-REAL 2) st f is_sequence_on G & f is special & 1 <= k & k + 1 <= len f holds
( Int (left_cell (f,k,G)) misses L~ f & Int (right_cell (f,k,G)) misses L~ f )
proof
let k be Element of NAT ; ::_thesis: for G being Go-board
for f being FinSequence of (TOP-REAL 2) st f is_sequence_on G & f is special & 1 <= k & k + 1 <= len f holds
( Int (left_cell (f,k,G)) misses L~ f & Int (right_cell (f,k,G)) misses L~ f )
let G be Go-board; ::_thesis: for f being FinSequence of (TOP-REAL 2) st f is_sequence_on G & f is special & 1 <= k & k + 1 <= len f holds
( Int (left_cell (f,k,G)) misses L~ f & Int (right_cell (f,k,G)) misses L~ f )
let f be FinSequence of (TOP-REAL 2); ::_thesis: ( f is_sequence_on G & f is special & 1 <= k & k + 1 <= len f implies ( Int (left_cell (f,k,G)) misses L~ f & Int (right_cell (f,k,G)) misses L~ f ) )
assume that
A1: f is_sequence_on G and
A2: f is special and
A3: ( 1 <= k & k + 1 <= len f ) ; ::_thesis: ( Int (left_cell (f,k,G)) misses L~ f & Int (right_cell (f,k,G)) misses L~ f )
consider i1, j1, i2, j2 being Element of NAT  such that
A4: [i1,j1] in Indices G and
A5: f /. k = G * (i1,j1) and
A6: [i2,j2] in Indices G and
A7: f /. (k + 1) = G * (i2,j2) and
A8: ( ( 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, A3, JORDAN8:3;
A9: i2 <= len G by A6, MATRIX_1:38;
A10: i1 <= len G by A4, MATRIX_1:38;
then A11: i1 -' 1 <= len G by NAT_D:44;
A12: j2 <= width G by A6, MATRIX_1:38;
then A13: j2 -' 1 <= width G by NAT_D:44;
A14: ( j1 + 1 > j1 & j2 + 1 > j2 ) by NAT_1:13;
A15: j1 <= width G by A4, MATRIX_1:38;
then A16: j1 -' 1 <= width G by NAT_D:44;
A17: ( i1 + 1 > i1 & i2 + 1 > i2 ) by NAT_1:13;
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 A8;
suppose ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: ( Int (left_cell (f,k,G)) misses L~ f & Int (right_cell (f,k,G)) misses L~ f )
then  ( right_cell (f,k,G) = cell (G,i1,j1) & left_cell (f,k,G) = cell (G,(i1 -' 1),j1) ) by A1, A3, A4, A5, A6, A7, A14, GOBRD13:def_2, GOBRD13:def_3;
hence  ( Int (left_cell (f,k,G)) misses L~ f & Int (right_cell (f,k,G)) misses L~ f ) by A1, A2, A10, A15, A11, Th14; ::_thesis: verum
end;
suppose ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: ( Int (left_cell (f,k,G)) misses L~ f & Int (right_cell (f,k,G)) misses L~ f )
then  ( right_cell (f,k,G) = cell (G,i1,(j1 -' 1)) & left_cell (f,k,G) = cell (G,i1,j1) ) by A1, A3, A4, A5, A6, A7, A17, GOBRD13:def_2, GOBRD13:def_3;
hence  ( Int (left_cell (f,k,G)) misses L~ f & Int (right_cell (f,k,G)) misses L~ f ) by A1, A2, A10, A15, A16, Th14; ::_thesis: verum
end;
suppose ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: ( Int (left_cell (f,k,G)) misses L~ f & Int (right_cell (f,k,G)) misses L~ f )
then  ( right_cell (f,k,G) = cell (G,i2,j2) & left_cell (f,k,G) = cell (G,i2,(j2 -' 1)) ) by A1, A3, A4, A5, A6, A7, A17, GOBRD13:def_2, GOBRD13:def_3;
hence  ( Int (left_cell (f,k,G)) misses L~ f & Int (right_cell (f,k,G)) misses L~ f ) by A1, A2, A9, A12, A13, Th14; ::_thesis: verum
end;
suppose ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: ( Int (left_cell (f,k,G)) misses L~ f & Int (right_cell (f,k,G)) misses L~ f )
then  ( right_cell (f,k,G) = cell (G,(i1 -' 1),j2) & left_cell (f,k,G) = cell (G,i1,j2) ) by A1, A3, A4, A5, A6, A7, A14, GOBRD13:def_2, GOBRD13:def_3;
hence  ( Int (left_cell (f,k,G)) misses L~ f & Int (right_cell (f,k,G)) misses L~ f ) by A1, A2, A10, A12, A11, Th14; ::_thesis: verum
end;
end;
end;

theorem Th16: :: JORDAN9:16
for i, j being Element of NAT
for G being Go-board st 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G holds
( (G * (i,j)) `1 = (G * (i,(j + 1))) `1 & (G * (i,j)) `2 = (G * ((i + 1),j)) `2 & (G * ((i + 1),(j + 1))) `1 = (G * ((i + 1),j)) `1 & (G * ((i + 1),(j + 1))) `2 = (G * (i,(j + 1))) `2 )
proof
let i, j be Element of NAT ; ::_thesis: for G being Go-board st 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G holds
( (G * (i,j)) `1 = (G * (i,(j + 1))) `1 & (G * (i,j)) `2 = (G * ((i + 1),j)) `2 & (G * ((i + 1),(j + 1))) `1 = (G * ((i + 1),j)) `1 & (G * ((i + 1),(j + 1))) `2 = (G * (i,(j + 1))) `2 )
let G be Go-board; ::_thesis: ( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G implies ( (G * (i,j)) `1 = (G * (i,(j + 1))) `1 & (G * (i,j)) `2 = (G * ((i + 1),j)) `2 & (G * ((i + 1),(j + 1))) `1 = (G * ((i + 1),j)) `1 & (G * ((i + 1),(j + 1))) `2 = (G * (i,(j + 1))) `2 ) )
assume that
A1: 1 <= i and
A2: i + 1 <= len G and
A3: 1 <= j and
A4: j + 1 <= width G ; ::_thesis: ( (G * (i,j)) `1 = (G * (i,(j + 1))) `1 & (G * (i,j)) `2 = (G * ((i + 1),j)) `2 & (G * ((i + 1),(j + 1))) `1 = (G * ((i + 1),j)) `1 & (G * ((i + 1),(j + 1))) `2 = (G * (i,(j + 1))) `2 )
A5: j < width G by A4, NAT_1:13;
A6: 1 <= j + 1 by NAT_1:11;
A7: i < len G by A2, NAT_1:13;
hence (G * (i,j)) `1 = (G * (i,1)) `1 by A1, A3, A5, GOBOARD5:2
.= (G * (i,(j + 1))) `1 by A1, A4, A7, A6, GOBOARD5:2 ;
::_thesis: ( (G * (i,j)) `2 = (G * ((i + 1),j)) `2 & (G * ((i + 1),(j + 1))) `1 = (G * ((i + 1),j)) `1 & (G * ((i + 1),(j + 1))) `2 = (G * (i,(j + 1))) `2 )
A8: 1 <= i + 1 by NAT_1:11;
thus (G * (i,j)) `2 = (G * (1,j)) `2 by A1, A3, A7, A5, GOBOARD5:1
.= (G * ((i + 1),j)) `2 by A2, A3, A5, A8, GOBOARD5:1 ; ::_thesis: ( (G * ((i + 1),(j + 1))) `1 = (G * ((i + 1),j)) `1 & (G * ((i + 1),(j + 1))) `2 = (G * (i,(j + 1))) `2 )
thus (G * ((i + 1),(j + 1))) `1 = (G * ((i + 1),1)) `1 by A2, A4, A8, A6, GOBOARD5:2
.= (G * ((i + 1),j)) `1 by A2, A3, A5, A8, GOBOARD5:2 ; ::_thesis: (G * ((i + 1),(j + 1))) `2 = (G * (i,(j + 1))) `2
thus (G * ((i + 1),(j + 1))) `2 = (G * (1,(j + 1))) `2 by A2, A4, A8, A6, GOBOARD5:1
.= (G * (i,(j + 1))) `2 by A1, A4, A7, A6, GOBOARD5:1 ; ::_thesis: verum
end;

theorem Th17: :: JORDAN9:17
for G being Go-board
for p being Point of (TOP-REAL 2)
for i, j being Element of NAT st 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G holds
( p in cell (G,i,j) iff ( (G * (i,j)) `1 <= p `1 & p `1 <= (G * ((i + 1),j)) `1 & (G * (i,j)) `2 <= p `2 & p `2 <= (G * (i,(j + 1))) `2 ) )
proof
let G be Go-board; ::_thesis: for p being Point of (TOP-REAL 2)
for i, j being Element of NAT st 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G holds
( p in cell (G,i,j) iff ( (G * (i,j)) `1 <= p `1 & p `1 <= (G * ((i + 1),j)) `1 & (G * (i,j)) `2 <= p `2 & p `2 <= (G * (i,(j + 1))) `2 ) )
let p be Point of (TOP-REAL 2); ::_thesis: for i, j being Element of NAT st 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G holds
( p in cell (G,i,j) iff ( (G * (i,j)) `1 <= p `1 & p `1 <= (G * ((i + 1),j)) `1 & (G * (i,j)) `2 <= p `2 & p `2 <= (G * (i,(j + 1))) `2 ) )
let i, j be Element of NAT ; ::_thesis: ( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G implies ( p in cell (G,i,j) iff ( (G * (i,j)) `1 <= p `1 & p `1 <= (G * ((i + 1),j)) `1 & (G * (i,j)) `2 <= p `2 & p `2 <= (G * (i,(j + 1))) `2 ) ) )
assume that
A1: 1 <= i and
A2: i + 1 <= len G and
A3: 1 <= j and
A4: j + 1 <= width G ; ::_thesis: ( p in cell (G,i,j) iff ( (G * (i,j)) `1 <= p `1 & p `1 <= (G * ((i + 1),j)) `1 & (G * (i,j)) `2 <= p `2 & p `2 <= (G * (i,(j + 1))) `2 ) )
A5: ( i < len G & j < width G ) by A2, A4, NAT_1:13;
then A6: h_strip (G,j) =  {  |[r,s]| where r, s is Real : ( (G * (i,j)) `2 <= s & s <= (G * (i,(j + 1))) `2 )  }  by A1, A3, GOBOARD5:5;
A7: v_strip (G,i) =  {  |[r,s]| where r, s is Real : ( (G * (i,j)) `1 <= r & r <= (G * ((i + 1),j)) `1 )  }  by A1, A3, A5, GOBOARD5:8;
hereby  ::_thesis: ( (G * (i,j)) `1 <= p `1 & p `1 <= (G * ((i + 1),j)) `1 & (G * (i,j)) `2 <= p `2 & p `2 <= (G * (i,(j + 1))) `2 implies p in cell (G,i,j) )
assume A8: p in cell (G,i,j) ; ::_thesis: ( (G * (i,j)) `1 <= p `1 & p `1 <= (G * ((i + 1),j)) `1 & (G * (i,j)) `2 <= p `2 & p `2 <= (G * (i,(j + 1))) `2 )
then  p in v_strip (G,i) by XBOOLE_0:def_4;
then  ex r, s being Real st
( |[r,s]| = p & (G * (i,j)) `1 <= r & r <= (G * ((i + 1),j)) `1 ) by A7;
hence  ( (G * (i,j)) `1 <= p `1 & p `1 <= (G * ((i + 1),j)) `1 ) by EUCLID:52; ::_thesis: ( (G * (i,j)) `2 <= p `2 & p `2 <= (G * (i,(j + 1))) `2 )
 p in h_strip (G,j) by A8, XBOOLE_0:def_4;
then  ex r, s being Real st
( |[r,s]| = p & (G * (i,j)) `2 <= s & s <= (G * (i,(j + 1))) `2 ) by A6;
hence  ( (G * (i,j)) `2 <= p `2 & p `2 <= (G * (i,(j + 1))) `2 ) by EUCLID:52; ::_thesis: verum
end;
assume that
A9: ( (G * (i,j)) `1 <= p `1 & p `1 <= (G * ((i + 1),j)) `1 ) and
A10: ( (G * (i,j)) `2 <= p `2 & p `2 <= (G * (i,(j + 1))) `2 ) ; ::_thesis: p in cell (G,i,j)
A11: p = |[(p `1),(p `2)]| by EUCLID:53;
then A12: p in h_strip (G,j) by A6, A10;
 p in v_strip (G,i) by A7, A9, A11;
hence  p in cell (G,i,j) by A12, XBOOLE_0:def_4; ::_thesis: verum
end;

theorem :: JORDAN9:18
for i, j being Element of NAT
for G being Go-board st 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G holds
cell (G,i,j) =  {  |[r,s]| where r, s is Real : ( (G * (i,j)) `1 <= r & r <= (G * ((i + 1),j)) `1 & (G * (i,j)) `2 <= s & s <= (G * (i,(j + 1))) `2 )  }
proof
let i, j be Element of NAT ; ::_thesis: for G being Go-board st 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G holds
cell (G,i,j) =  {  |[r,s]| where r, s is Real : ( (G * (i,j)) `1 <= r & r <= (G * ((i + 1),j)) `1 & (G * (i,j)) `2 <= s & s <= (G * (i,(j + 1))) `2 )  }
let G be Go-board; ::_thesis: ( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G implies cell (G,i,j) =  {  |[r,s]| where r, s is Real : ( (G * (i,j)) `1 <= r & r <= (G * ((i + 1),j)) `1 & (G * (i,j)) `2 <= s & s <= (G * (i,(j + 1))) `2 )  }  )
set A =  {  |[r,s]| where r, s is Real : ( (G * (i,j)) `1 <= r & r <= (G * ((i + 1),j)) `1 & (G * (i,j)) `2 <= s & s <= (G * (i,(j + 1))) `2 )  }  ;
assume A1: ( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G ) ; ::_thesis: cell (G,i,j) =  {  |[r,s]| where r, s is Real : ( (G * (i,j)) `1 <= r & r <= (G * ((i + 1),j)) `1 & (G * (i,j)) `2 <= s & s <= (G * (i,(j + 1))) `2 )  }
now__::_thesis:_for_p_being_set_st_p_in_cell_(G,i,j)_holds_
p_in__{__|[r,s]|_where_r,_s_is_Real_:_(_(G_*_(i,j))_`1_<=_r_&_r_<=_(G_*_((i_+_1),j))_`1_&_(G_*_(i,j))_`2_<=_s_&_s_<=_(G_*_(i,(j_+_1)))_`2_)__}_
let p be set ; ::_thesis: ( p in cell (G,i,j) implies p in  {  |[r,s]| where r, s is Real : ( (G * (i,j)) `1 <= r & r <= (G * ((i + 1),j)) `1 & (G * (i,j)) `2 <= s & s <= (G * (i,(j + 1))) `2 )  }  )
assume A2: p in cell (G,i,j) ; ::_thesis: p in  {  |[r,s]| where r, s is Real : ( (G * (i,j)) `1 <= r & r <= (G * ((i + 1),j)) `1 & (G * (i,j)) `2 <= s & s <= (G * (i,(j + 1))) `2 )  }
then reconsider q = p as Point of (TOP-REAL 2) ;
A3: ( (G * (i,j)) `2 <= q `2 & q `2 <= (G * (i,(j + 1))) `2 ) by A1, A2, Th17;
A4: p = |[(q `1),(q `2)]| by EUCLID:53;
 ( (G * (i,j)) `1 <= q `1 & q `1 <= (G * ((i + 1),j)) `1 ) by A1, A2, Th17;
hence  p in  {  |[r,s]| where r, s is Real : ( (G * (i,j)) `1 <= r & r <= (G * ((i + 1),j)) `1 & (G * (i,j)) `2 <= s & s <= (G * (i,(j + 1))) `2 )  }  by A4, A3; ::_thesis: verum
end;
hence  cell (G,i,j) c=  {  |[r,s]| where r, s is Real : ( (G * (i,j)) `1 <= r & r <= (G * ((i + 1),j)) `1 & (G * (i,j)) `2 <= s & s <= (G * (i,(j + 1))) `2 )  }  by TARSKI:def_3; :: according to XBOOLE_0:def_10 ::_thesis:  {  |[r,s]| where r, s is Real : ( (G * (i,j)) `1 <= r & r <= (G * ((i + 1),j)) `1 & (G * (i,j)) `2 <= s & s <= (G * (i,(j + 1))) `2 )  }  c= cell (G,i,j)
now__::_thesis:_for_p_being_set_st_p_in__{__|[r,s]|_where_r,_s_is_Real_:_(_(G_*_(i,j))_`1_<=_r_&_r_<=_(G_*_((i_+_1),j))_`1_&_(G_*_(i,j))_`2_<=_s_&_s_<=_(G_*_(i,(j_+_1)))_`2_)__}__holds_
p_in_cell_(G,i,j)
let p be set ; ::_thesis: ( p in  {  |[r,s]| where r, s is Real : ( (G * (i,j)) `1 <= r & r <= (G * ((i + 1),j)) `1 & (G * (i,j)) `2 <= s & s <= (G * (i,(j + 1))) `2 )  }  implies p in cell (G,i,j) )
assume  p in  {  |[r,s]| where r, s is Real : ( (G * (i,j)) `1 <= r & r <= (G * ((i + 1),j)) `1 & (G * (i,j)) `2 <= s & s <= (G * (i,(j + 1))) `2 )  }  ; ::_thesis: p in cell (G,i,j)
then consider r, s being Real such that
A5: |[r,s]| = p and
A6: ( (G * (i,j)) `1 <= r & r <= (G * ((i + 1),j)) `1 & (G * (i,j)) `2 <= s & s <= (G * (i,(j + 1))) `2 ) ;
reconsider q = p as Point of (TOP-REAL 2) by A5;
 ( r = q `1 & s = q `2 ) by A5, EUCLID:52;
hence  p in cell (G,i,j) by A1, A6, Th17; ::_thesis: verum
end;
hence   {  |[r,s]| where r, s is Real : ( (G * (i,j)) `1 <= r & r <= (G * ((i + 1),j)) `1 & (G * (i,j)) `2 <= s & s <= (G * (i,(j + 1))) `2 )  }  c= cell (G,i,j) by TARSKI:def_3; ::_thesis: verum
end;

theorem Th19: :: JORDAN9:19
for i, j being Element of NAT
for G being Go-board
for p being Point of (TOP-REAL 2) st 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & p in Values G & p in cell (G,i,j) & not p = G * (i,j) & not p = G * (i,(j + 1)) & not p = G * ((i + 1),(j + 1)) holds
p = G * ((i + 1),j)
proof
let i, j be Element of NAT ; ::_thesis: for G being Go-board
for p being Point of (TOP-REAL 2) st 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & p in Values G & p in cell (G,i,j) & not p = G * (i,j) & not p = G * (i,(j + 1)) & not p = G * ((i + 1),(j + 1)) holds
p = G * ((i + 1),j)
let G be Go-board; ::_thesis: for p being Point of (TOP-REAL 2) st 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & p in Values G & p in cell (G,i,j) & not p = G * (i,j) & not p = G * (i,(j + 1)) & not p = G * ((i + 1),(j + 1)) holds
p = G * ((i + 1),j)
let p be Point of (TOP-REAL 2); ::_thesis: ( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & p in Values G & p in cell (G,i,j) & not p = G * (i,j) & not p = G * (i,(j + 1)) & not p = G * ((i + 1),(j + 1)) implies p = G * ((i + 1),j) )
assume that
A1: 1 <= i and
A2: i + 1 <= len G and
A3: 1 <= j and
A4: j + 1 <= width G and
A5: p in Values G and
A6: p in cell (G,i,j) ; ::_thesis: ( p = G * (i,j) or p = G * (i,(j + 1)) or p = G * ((i + 1),(j + 1)) or p = G * ((i + 1),j) )
A7: Values G =  {  (G * (k,l)) where k, l is Element of NAT : [k,l] in Indices G  }  by MATRIX_1:45;
A8: i < len G by A2, NAT_1:13;
A9: j < width G by A4, NAT_1:13;
consider k, l being Element of NAT  such that
A10: p = G * (k,l) and
A11: [k,l] in Indices G by A5, A7;
A12: 1 <= k by A11, MATRIX_1:38;
A13: l <= width G by A11, MATRIX_1:38;
A14: 1 <= l by A11, MATRIX_1:38;
A15: k <= len G by A11, MATRIX_1:38;
A16: 1 <= j + 1 by NAT_1:11;
A17: now__::_thesis:_(_l_<>_j_implies_not_l_<>_j_+_1_)
assume A18: ( l <> j & l <> j + 1 ) ; ::_thesis: contradiction
percases ( l < j or j + 1 < l ) by A18, NAT_1:9;
suppose l < j ; ::_thesis: contradiction
then  (G * (k,l)) `2 < (G * (k,j)) `2 by A9, A12, A15, A14, GOBOARD5:4;
then  (G * (k,l)) `2 < (G * (1,j)) `2 by A3, A9, A12, A15, GOBOARD5:1;
then  (G * (k,l)) `2 < (G * (i,j)) `2 by A1, A3, A8, A9, GOBOARD5:1;
hence  contradiction by A1, A2, A3, A4, A6, A10, Th17; ::_thesis: verum
end;
suppose j + 1 < l ; ::_thesis: contradiction
then  (G * (k,(j + 1))) `2 < (G * (k,l)) `2 by A16, A12, A15, A13, GOBOARD5:4;
then  (G * (1,(j + 1))) `2 < (G * (k,l)) `2 by A4, A16, A12, A15, GOBOARD5:1;
then  (G * (i,(j + 1))) `2 < (G * (k,l)) `2 by A1, A4, A8, A16, GOBOARD5:1;
hence  contradiction by A1, A2, A3, A4, A6, A10, Th17; ::_thesis: verum
end;
end;
end;
A19: 1 <= i + 1 by NAT_1:11;
now__::_thesis:_(_k_<>_i_implies_not_k_<>_i_+_1_)
assume A20: ( k <> i & k <> i + 1 ) ; ::_thesis: contradiction
percases ( k < i or i + 1 < k ) by A20, NAT_1:9;
suppose k < i ; ::_thesis: contradiction
then  (G * (k,l)) `1 < (G * (i,l)) `1 by A8, A12, A14, A13, GOBOARD5:3;
then  (G * (k,l)) `1 < (G * (i,1)) `1 by A1, A8, A14, A13, GOBOARD5:2;
then  (G * (k,l)) `1 < (G * (i,j)) `1 by A1, A3, A8, A9, GOBOARD5:2;
hence  contradiction by A1, A2, A3, A4, A6, A10, Th17; ::_thesis: verum
end;
suppose i + 1 < k ; ::_thesis: contradiction
then  (G * ((i + 1),l)) `1 < (G * (k,l)) `1 by A19, A15, A14, A13, GOBOARD5:3;
then  (G * ((i + 1),1)) `1 < (G * (k,l)) `1 by A2, A19, A14, A13, GOBOARD5:2;
then  (G * ((i + 1),j)) `1 < (G * (k,l)) `1 by A2, A3, A9, A19, GOBOARD5:2;
hence  contradiction by A1, A2, A3, A4, A6, A10, Th17; ::_thesis: verum
end;
end;
end;
hence  ( p = G * (i,j) or p = G * (i,(j + 1)) or p = G * ((i + 1),(j + 1)) or p = G * ((i + 1),j) ) by A10, A17; ::_thesis: verum
end;

theorem Th20: :: JORDAN9:20
for i, j being Element of NAT
for G being Go-board st 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G holds
( G * (i,j) in cell (G,i,j) & G * (i,(j + 1)) in cell (G,i,j) & G * ((i + 1),(j + 1)) in cell (G,i,j) & G * ((i + 1),j) in cell (G,i,j) )
proof
let i, j be Element of NAT ; ::_thesis: for G being Go-board st 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G holds
( G * (i,j) in cell (G,i,j) & G * (i,(j + 1)) in cell (G,i,j) & G * ((i + 1),(j + 1)) in cell (G,i,j) & G * ((i + 1),j) in cell (G,i,j) )
let G be Go-board; ::_thesis: ( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G implies ( G * (i,j) in cell (G,i,j) & G * (i,(j + 1)) in cell (G,i,j) & G * ((i + 1),(j + 1)) in cell (G,i,j) & G * ((i + 1),j) in cell (G,i,j) ) )
assume that
A1: 1 <= i and
A2: i + 1 <= len G and
A3: 1 <= j and
A4: j + 1 <= width G ; ::_thesis: ( G * (i,j) in cell (G,i,j) & G * (i,(j + 1)) in cell (G,i,j) & G * ((i + 1),(j + 1)) in cell (G,i,j) & G * ((i + 1),j) in cell (G,i,j) )
A5: ( i < i + 1 & j < width G ) by A4, NAT_1:13;
then A6: (G * (i,j)) `1 <= (G * ((i + 1),j)) `1 by A1, A2, A3, GOBOARD5:3;
A7: (G * (i,j)) `1 <= (G * ((i + 1),j)) `1 by A1, A2, A3, A5, GOBOARD5:3;
A8: ( j < j + 1 & i < len G ) by A2, NAT_1:13;
then A9: (G * (i,j)) `2 <= (G * (i,(j + 1))) `2 by A1, A3, A4, GOBOARD5:4;
A10: (G * ((i + 1),(j + 1))) `1 = (G * ((i + 1),j)) `1 by A1, A2, A3, A4, Th16;
then A11: (G * (i,j)) `1 <= (G * ((i + 1),(j + 1))) `1 by A1, A2, A3, A5, GOBOARD5:3;
 (G * (i,j)) `2 <= (G * (i,(j + 1))) `2 by A1, A3, A4, A8, GOBOARD5:4;
hence  G * (i,j) in cell (G,i,j) by A1, A2, A3, A4, A6, Th17; ::_thesis: ( G * (i,(j + 1)) in cell (G,i,j) & G * ((i + 1),(j + 1)) in cell (G,i,j) & G * ((i + 1),j) in cell (G,i,j) )
A12: (G * (i,j)) `1 = (G * (i,(j + 1))) `1 by A1, A2, A3, A4, Th16;
then  (G * (i,(j + 1))) `1 <= (G * ((i + 1),j)) `1 by A1, A2, A3, A5, GOBOARD5:3;
hence  G * (i,(j + 1)) in cell (G,i,j) by A1, A2, A3, A4, A12, A9, Th17; ::_thesis: ( G * ((i + 1),(j + 1)) in cell (G,i,j) & G * ((i + 1),j) in cell (G,i,j) )
A13: (G * ((i + 1),(j + 1))) `2 = (G * (i,(j + 1))) `2 by A1, A2, A3, A4, Th16;
then  (G * (i,j)) `2 <= (G * ((i + 1),(j + 1))) `2 by A1, A3, A4, A8, GOBOARD5:4;
hence  G * ((i + 1),(j + 1)) in cell (G,i,j) by A1, A2, A3, A4, A10, A11, A13, Th17; ::_thesis: G * ((i + 1),j) in cell (G,i,j)
A14: (G * (i,j)) `2 = (G * ((i + 1),j)) `2 by A1, A2, A3, A4, Th16;
then  (G * ((i + 1),j)) `2 <= (G * (i,(j + 1))) `2 by A1, A3, A4, A8, GOBOARD5:4;
hence  G * ((i + 1),j) in cell (G,i,j) by A1, A2, A3, A4, A7, A14, Th17; ::_thesis: verum
end;

theorem Th21: :: JORDAN9:21
for i, j being Element of NAT
for G being Go-board
for p being Point of (TOP-REAL 2) st 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & p in Values G & p in cell (G,i,j) holds
p is_extremal_in cell (G,i,j)
proof
let i, j be Element of NAT ; ::_thesis: for G being Go-board
for p being Point of (TOP-REAL 2) st 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & p in Values G & p in cell (G,i,j) holds
p is_extremal_in cell (G,i,j)
let G be Go-board; ::_thesis: for p being Point of (TOP-REAL 2) st 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & p in Values G & p in cell (G,i,j) holds
p is_extremal_in cell (G,i,j)
let p be Point of (TOP-REAL 2); ::_thesis: ( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & p in Values G & p in cell (G,i,j) implies p is_extremal_in cell (G,i,j) )
assume that
A1: ( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G ) and
A2: p in Values G and
A3: p in cell (G,i,j) ; ::_thesis: p is_extremal_in cell (G,i,j)
 for a, b being Point of (TOP-REAL 2) st p in LSeg (a,b) & LSeg (a,b) c= cell (G,i,j) & not p = a holds
p = b
proof
let a, b be Point of (TOP-REAL 2); ::_thesis: ( p in LSeg (a,b) & LSeg (a,b) c= cell (G,i,j) & not p = a implies p = b )
assume that
A4: p in LSeg (a,b) and
A5: LSeg (a,b) c= cell (G,i,j) ; ::_thesis: ( p = a or p = b )
A6: a in LSeg (a,b) by RLTOPSP1:68;
A7: b in LSeg (a,b) by RLTOPSP1:68;
assume that
A8: a <> p and
A9: b <> p ; ::_thesis: contradiction
percases ( p = G * (i,j) or p = G * (i,(j + 1)) or p = G * ((i + 1),(j + 1)) or p = G * ((i + 1),j) ) by A1, A2, A3, Th19;
supposeA10: p = G * (i,j) ; ::_thesis: contradiction
then A11: p `2 <= b `2 by A1, A5, A7, Th17;
A12: p `1 <= a `1 by A1, A5, A6, A10, Th17;
A13: p `1 <= b `1 by A1, A5, A7, A10, Th17;
A14: p `2 <= a `2 by A1, A5, A6, A10, Th17;
now__::_thesis:_contradiction
percases ( ( a `1 <= b `1 & a `2 <= b `2 ) or ( a `1 <= b `1 & b `2 < a `2 ) or ( b `1 < a `1 & a `2 <= b `2 ) or ( b `1 < a `1 & b `2 < a `2 ) ) ;
supposeA15: ( a `1 <= b `1 & a `2 <= b `2 ) ; ::_thesis: contradiction
then  a `2 <= p `2 by A4, TOPREAL1:4;
then A16: a `2 = p `2 by A14, XXREAL_0:1;
 a `1 <= p `1 by A4, A15, TOPREAL1:3;
then  a `1 = p `1 by A12, XXREAL_0:1;
hence  contradiction by A8, A16, TOPREAL3:6; ::_thesis: verum
end;
supposeA17: ( a `1 <= b `1 & b `2 < a `2 ) ; ::_thesis: contradiction
then  b `2 <= p `2 by A4, TOPREAL1:4;
then A18: b `2 = p `2 by A11, XXREAL_0:1;
A19: a `1 <= p `1 by A4, A17, TOPREAL1:3;
then A20: a `1 = p `1 by A12, XXREAL_0:1;
then  a `2 <> p `2 by A8, TOPREAL3:6;
then  LSeg (a,b) is vertical by A4, A6, A12, A19, SPPOL_1:18, XXREAL_0:1;
then  a `1 = b `1 by SPPOL_1:16;
hence  contradiction by A9, A20, A18, TOPREAL3:6; ::_thesis: verum
end;
supposeA21: ( b `1 < a `1 & a `2 <= b `2 ) ; ::_thesis: contradiction
then  a `2 <= p `2 by A4, TOPREAL1:4;
then A22: a `2 = p `2 by A14, XXREAL_0:1;
A23: b `1 <= p `1 by A4, A21, TOPREAL1:3;
then A24: b `1 = p `1 by A13, XXREAL_0:1;
then  b `2 <> p `2 by A9, TOPREAL3:6;
then  LSeg (a,b) is vertical by A4, A7, A13, A23, SPPOL_1:18, XXREAL_0:1;
then  a `1 = b `1 by SPPOL_1:16;
hence  contradiction by A8, A24, A22, TOPREAL3:6; ::_thesis: verum
end;
supposeA25: ( b `1 < a `1 & b `2 < a `2 ) ; ::_thesis: contradiction
then  b `2 <= p `2 by A4, TOPREAL1:4;
then A26: b `2 = p `2 by A11, XXREAL_0:1;
 b `1 <= p `1 by A4, A25, TOPREAL1:3;
then  b `1 = p `1 by A13, XXREAL_0:1;
hence  contradiction by A9, A26, TOPREAL3:6; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
supposeA27: p = G * (i,(j + 1)) ; ::_thesis: contradiction
then A28: b `2 <= p `2 by A1, A5, A7, Th17;
A29: p `1 = (G * (i,j)) `1 by A1, A27, Th16;
then A30: p `1 <= a `1 by A1, A5, A6, Th17;
A31: p `1 <= b `1 by A1, A5, A7, A29, Th17;
A32: a `2 <= p `2 by A1, A5, A6, A27, Th17;
now__::_thesis:_contradiction
percases ( ( a `1 <= b `1 & a `2 <= b `2 ) or ( a `1 <= b `1 & b `2 < a `2 ) or ( b `1 < a `1 & a `2 <= b `2 ) or ( b `1 < a `1 & b `2 < a `2 ) ) ;
supposeA33: ( a `1 <= b `1 & a `2 <= b `2 ) ; ::_thesis: contradiction
then  p `2 <= b `2 by A4, TOPREAL1:4;
then A34: b `2 = p `2 by A28, XXREAL_0:1;
A35: a `1 <= p `1 by A4, A33, TOPREAL1:3;
then A36: a `1 = p `1 by A30, XXREAL_0:1;
then  a `2 <> p `2 by A8, TOPREAL3:6;
then  LSeg (a,b) is vertical by A4, A6, A30, A35, SPPOL_1:18, XXREAL_0:1;
then  a `1 = b `1 by SPPOL_1:16;
hence  contradiction by A9, A36, A34, TOPREAL3:6; ::_thesis: verum
end;
supposeA37: ( a `1 <= b `1 & b `2 < a `2 ) ; ::_thesis: contradiction
then  p `2 <= a `2 by A4, TOPREAL1:4;
then A38: a `2 = p `2 by A32, XXREAL_0:1;
 a `1 <= p `1 by A4, A37, TOPREAL1:3;
then  a `1 = p `1 by A30, XXREAL_0:1;
hence  contradiction by A8, A38, TOPREAL3:6; ::_thesis: verum
end;
supposeA39: ( b `1 < a `1 & a `2 <= b `2 ) ; ::_thesis: contradiction
then  p `2 <= b `2 by A4, TOPREAL1:4;
then A40: b `2 = p `2 by A28, XXREAL_0:1;
 b `1 <= p `1 by A4, A39, TOPREAL1:3;
then  b `1 = p `1 by A31, XXREAL_0:1;
hence  contradiction by A9, A40, TOPREAL3:6; ::_thesis: verum
end;
supposeA41: ( b `1 < a `1 & b `2 < a `2 ) ; ::_thesis: contradiction
then  p `2 <= a `2 by A4, TOPREAL1:4;
then A42: a `2 = p `2 by A32, XXREAL_0:1;
A43: b `1 <= p `1 by A4, A41, TOPREAL1:3;
then A44: b `1 = p `1 by A31, XXREAL_0:1;
then  b `2 <> p `2 by A9, TOPREAL3:6;
then  LSeg (a,b) is vertical by A4, A7, A31, A43, SPPOL_1:18, XXREAL_0:1;
then  a `1 = b `1 by SPPOL_1:16;
hence  contradiction by A8, A44, A42, TOPREAL3:6; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
supposeA45: p = G * ((i + 1),(j + 1)) ; ::_thesis: contradiction
then A46: p `1 = (G * ((i + 1),j)) `1 by A1, Th16;
then A47: a `1 <= p `1 by A1, A5, A6, Th17;
A48: p `2 = (G * (i,(j + 1))) `2 by A1, A45, Th16;
then A49: b `2 <= p `2 by A1, A5, A7, Th17;
A50: b `1 <= p `1 by A1, A5, A7, A46, Th17;
A51: a `2 <= p `2 by A1, A5, A6, A48, Th17;
now__::_thesis:_contradiction
percases ( ( a `1 <= b `1 & a `2 <= b `2 ) or ( a `1 <= b `1 & b `2 < a `2 ) or ( b `1 < a `1 & a `2 <= b `2 ) or ( b `1 < a `1 & b `2 < a `2 ) ) ;
supposeA52: ( a `1 <= b `1 & a `2 <= b `2 ) ; ::_thesis: contradiction
then  p `2 <= b `2 by A4, TOPREAL1:4;
then A53: b `2 = p `2 by A49, XXREAL_0:1;
 p `1 <= b `1 by A4, A52, TOPREAL1:3;
then  b `1 = p `1 by A50, XXREAL_0:1;
hence  contradiction by A9, A53, TOPREAL3:6; ::_thesis: verum
end;
supposeA54: ( a `1 <= b `1 & b `2 < a `2 ) ; ::_thesis: contradiction
then  p `2 <= a `2 by A4, TOPREAL1:4;
then A55: a `2 = p `2 by A51, XXREAL_0:1;
A56: p `1 <= b `1 by A4, A54, TOPREAL1:3;
then A57: b `1 = p `1 by A50, XXREAL_0:1;
then  b `2 <> p `2 by A9, TOPREAL3:6;
then  LSeg (a,b) is vertical by A4, A7, A50, A56, SPPOL_1:18, XXREAL_0:1;
then  a `1 = b `1 by SPPOL_1:16;
hence  contradiction by A8, A57, A55, TOPREAL3:6; ::_thesis: verum
end;
supposeA58: ( b `1 < a `1 & a `2 <= b `2 ) ; ::_thesis: contradiction
then  p `2 <= b `2 by A4, TOPREAL1:4;
then A59: b `2 = p `2 by A49, XXREAL_0:1;
A60: p `1 <= a `1 by A4, A58, TOPREAL1:3;
then A61: a `1 = p `1 by A47, XXREAL_0:1;
then  a `2 <> p `2 by A8, TOPREAL3:6;
then  LSeg (a,b) is vertical by A4, A6, A47, A60, SPPOL_1:18, XXREAL_0:1;
then  a `1 = b `1 by SPPOL_1:16;
hence  contradiction by A9, A61, A59, TOPREAL3:6; ::_thesis: verum
end;
supposeA62: ( b `1 < a `1 & b `2 < a `2 ) ; ::_thesis: contradiction
then  p `2 <= a `2 by A4, TOPREAL1:4;
then A63: a `2 = p `2 by A51, XXREAL_0:1;
 p `1 <= a `1 by A4, A62, TOPREAL1:3;
then  a `1 = p `1 by A47, XXREAL_0:1;
hence  contradiction by A8, A63, TOPREAL3:6; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
supposeA64: p = G * ((i + 1),j) ; ::_thesis: contradiction
then A65: p `2 = (G * (i,j)) `2 by A1, Th16;
then A66: p `2 <= b `2 by A1, A5, A7, Th17;
A67: a `1 <= p `1 by A1, A5, A6, A64, Th17;
A68: b `1 <= p `1 by A1, A5, A7, A64, Th17;
A69: p `2 <= a `2 by A1, A5, A6, A65, Th17;
now__::_thesis:_contradiction
percases ( ( a `1 <= b `1 & a `2 <= b `2 ) or ( a `1 <= b `1 & b `2 < a `2 ) or ( b `1 < a `1 & a `2 <= b `2 ) or ( b `1 < a `1 & b `2 < a `2 ) ) ;
supposeA70: ( a `1 <= b `1 & a `2 <= b `2 ) ; ::_thesis: contradiction
then  a `2 <= p `2 by A4, TOPREAL1:4;
then A71: a `2 = p `2 by A69, XXREAL_0:1;
A72: p `1 <= b `1 by A4, A70, TOPREAL1:3;
then A73: b `1 = p `1 by A68, XXREAL_0:1;
then  b `2 <> p `2 by A9, TOPREAL3:6;
then  LSeg (a,b) is vertical by A4, A7, A68, A72, SPPOL_1:18, XXREAL_0:1;
then  a `1 = b `1 by SPPOL_1:16;
hence  contradiction by A8, A73, A71, TOPREAL3:6; ::_thesis: verum
end;
supposeA74: ( a `1 <= b `1 & b `2 < a `2 ) ; ::_thesis: contradiction
then  b `2 <= p `2 by A4, TOPREAL1:4;
then A75: b `2 = p `2 by A66, XXREAL_0:1;
 p `1 <= b `1 by A4, A74, TOPREAL1:3;
then  b `1 = p `1 by A68, XXREAL_0:1;
hence  contradiction by A9, A75, TOPREAL3:6; ::_thesis: verum
end;
supposeA76: ( b `1 < a `1 & a `2 <= b `2 ) ; ::_thesis: contradiction
then  a `2 <= p `2 by A4, TOPREAL1:4;
then A77: a `2 = p `2 by A69, XXREAL_0:1;
 p `1 <= a `1 by A4, A76, TOPREAL1:3;
then  a `1 = p `1 by A67, XXREAL_0:1;
hence  contradiction by A8, A77, TOPREAL3:6; ::_thesis: verum
end;
supposeA78: ( b `1 < a `1 & b `2 < a `2 ) ; ::_thesis: contradiction
then  b `2 <= p `2 by A4, TOPREAL1:4;
then A79: b `2 = p `2 by A66, XXREAL_0:1;
A80: p `1 <= a `1 by A4, A78, TOPREAL1:3;
then A81: a `1 = p `1 by A67, XXREAL_0:1;
then  a `2 <> p `2 by A8, TOPREAL3:6;
then  LSeg (a,b) is vertical by A4, A6, A67, A80, SPPOL_1:18, XXREAL_0:1;
then  a `1 = b `1 by SPPOL_1:16;
hence  contradiction by A9, A81, A79, TOPREAL3:6; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
end;
end;
hence  p is_extremal_in cell (G,i,j) by A3, SPPOL_1:def_1; ::_thesis: verum
end;

theorem Th22: :: JORDAN9:22
for k being Element of NAT
for G being Go-board
for f being FinSequence of (TOP-REAL 2) st 2 <= len G & 2 <= width G & f is_sequence_on G & 1 <= k & k + 1 <= len f holds
ex i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) )
proof
let k be Element of NAT ; ::_thesis: for G being Go-board
for f being FinSequence of (TOP-REAL 2) st 2 <= len G & 2 <= width G & f is_sequence_on G & 1 <= k & k + 1 <= len f holds
ex i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) )
let G be Go-board; ::_thesis: for f being FinSequence of (TOP-REAL 2) st 2 <= len G & 2 <= width G & f is_sequence_on G & 1 <= k & k + 1 <= len f holds
ex i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) )
let f be FinSequence of (TOP-REAL 2); ::_thesis: ( 2 <= len G & 2 <= width G & f is_sequence_on G & 1 <= k & k + 1 <= len f implies ex i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) ) )
assume that
A1: 2 <= len G and
A2: 2 <= width G and
A3: f is_sequence_on G and
A4: ( 1 <= k & k + 1 <= len f ) ; ::_thesis: ex i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) )
consider i1, j1, i2, j2 being Element of NAT  such that
A5: [i1,j1] in Indices G and
A6: f /. k = G * (i1,j1) and
A7: [i2,j2] in Indices G and
A8: f /. (k + 1) = G * (i2,j2) and
A9: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A3, A4, JORDAN8:3;
A10: LSeg (f,k) = LSeg ((f /. k),(f /. (k + 1))) by A4, TOPREAL1:def_3;
A11: 1 <= i2 by A7, MATRIX_1:38;
A12: 1 <= i1 by A5, MATRIX_1:38;
A13: 1 <= j2 by A7, MATRIX_1:38;
A14: 1 <= j1 by A5, MATRIX_1:38;
A15: i2 <= len G by A7, MATRIX_1:38;
A16: i1 <= len G by A5, MATRIX_1:38;
A17: j2 <= width G by A7, MATRIX_1:38;
A18: j1 <= width G by A5, MATRIX_1:38;
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 A9;
supposeA19: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: ex i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) )
then A20: j1 < width G by A17, XREAL_1:145;
now__::_thesis:_ex_i1,_j1,_i,_j_being_Element_of_NAT_st_
(_1_<=_i_&_i_+_1_<=_len_G_&_1_<=_j_&_j_+_1_<=_width_G_&_LSeg_(f,k)_c=_cell_(G,i,j)_)
percases ( i1 < len G or i1 = len G ) by A16, XXREAL_0:1;
supposeA21: i1 < len G ; ::_thesis: ex i1, j1, i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) )
take i1 = i1; ::_thesis: ex j1, i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) )
take j1 = j1; ::_thesis: ex i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) )
A22: i1 + 1 <= len G by A21, NAT_1:13;
 LSeg (f,k) c= cell (G,i1,j1) by A10, A6, A8, A12, A16, A14, A17, A19, GOBOARD5:19, XREAL_1:145;
hence  ex i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) ) by A12, A14, A17, A19, A22; ::_thesis: verum
end;
supposeA23: i1 = len G ; ::_thesis: ex i19 being Element of NAT ex j1, i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) )
take i19 = i1 -' 1; ::_thesis: ex j1, i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) )
take j1 = j1; ::_thesis: ex i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) )
 ( 2 - 1 <= 2 -' 1 & 2 -' 1 <= i19 ) by A1, A23, NAT_D:42, XREAL_0:def_2;
then A24: 1 <= i19 by XXREAL_0:2;
A25: i19 + 1 = i1 by A12, XREAL_1:235;
then  i19 < len G by A16, NAT_1:13;
then  LSeg (f,k) c= cell (G,i19,j1) by A10, A6, A8, A14, A19, A20, A25, GOBOARD5:18;
hence  ex i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) ) by A16, A14, A17, A19, A24, A25; ::_thesis: verum
end;
end;
end;
hence  ex i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) ) ; ::_thesis: verum
end;
supposeA26: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: ex i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) )
then A27: i1 < len G by A15, XREAL_1:145;
now__::_thesis:_ex_i1,_j1,_i,_j_being_Element_of_NAT_st_
(_1_<=_i_&_i_+_1_<=_len_G_&_1_<=_j_&_j_+_1_<=_width_G_&_LSeg_(f,k)_c=_cell_(G,i,j)_)
percases ( j1 < width G or j1 = width G ) by A18, XXREAL_0:1;
supposeA28: j1 < width G ; ::_thesis: ex i1, j1, i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) )
take i1 = i1; ::_thesis: ex j1, i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) )
take j1 = j1; ::_thesis: ex i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) )
A29: j1 + 1 <= width G by A28, NAT_1:13;
 LSeg (f,k) c= cell (G,i1,j1) by A10, A6, A8, A12, A14, A18, A15, A26, GOBOARD5:22, XREAL_1:145;
hence  ex i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) ) by A12, A14, A15, A26, A29; ::_thesis: verum
end;
supposeA30: j1 = width G ; ::_thesis: ex i1 being Element of NAT ex j19 being Element of NAT ex i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) )
take i1 = i1; ::_thesis: ex j19 being Element of NAT ex i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) )
take j19 = j1 -' 1; ::_thesis: ex i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) )
 ( 2 - 1 <= 2 -' 1 & 2 -' 1 <= j19 ) by A2, A30, NAT_D:42, XREAL_0:def_2;
then A31: 1 <= j19 by XXREAL_0:2;
A32: j19 + 1 = j1 by A14, XREAL_1:235;
then  j19 < width G by A30, NAT_1:13;
then  LSeg (f,k) c= cell (G,i1,j19) by A10, A6, A8, A12, A26, A27, A32, GOBOARD5:21;
hence  ex i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) ) by A12, A18, A15, A26, A31, A32; ::_thesis: verum
end;
end;
end;
hence  ex i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) ) ; ::_thesis: verum
end;
supposeA33: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: ex i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) )
then A34: i2 < len G by A16, XREAL_1:145;
now__::_thesis:_ex_i2,_j1,_i,_j_being_Element_of_NAT_st_
(_1_<=_i_&_i_+_1_<=_len_G_&_1_<=_j_&_j_+_1_<=_width_G_&_LSeg_(f,k)_c=_cell_(G,i,j)_)
percases ( j1 < width G or j1 = width G ) by A18, XXREAL_0:1;
supposeA35: j1 < width G ; ::_thesis: ex i2, j1, i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) )
take i2 = i2; ::_thesis: ex j1, i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) )
take j1 = j1; ::_thesis: ex i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) )
A36: j1 + 1 <= width G by A35, NAT_1:13;
 LSeg (f,k) c= cell (G,i2,j1) by A10, A6, A8, A16, A11, A13, A17, A33, GOBOARD5:22, XREAL_1:145;
hence  ex i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) ) by A16, A14, A11, A33, A36; ::_thesis: verum
end;
supposeA37: j1 = width G ; ::_thesis: ex i2 being Element of NAT ex j19 being Element of NAT ex i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) )
take i2 = i2; ::_thesis: ex j19 being Element of NAT ex i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) )
take j19 = j1 -' 1; ::_thesis: ex i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) )
 ( 2 - 1 <= 2 -' 1 & 2 -' 1 <= j19 ) by A2, A37, NAT_D:42, XREAL_0:def_2;
then A38: 1 <= j19 by XXREAL_0:2;
A39: j19 + 1 = j1 by A14, XREAL_1:235;
then  j19 < width G by A37, NAT_1:13;
then  LSeg (f,k) c= cell (G,i2,j19) by A10, A6, A8, A11, A33, A34, A39, GOBOARD5:21;
hence  ex i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) ) by A16, A18, A11, A33, A38, A39; ::_thesis: verum
end;
end;
end;
hence  ex i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) ) ; ::_thesis: verum
end;
supposeA40: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: ex i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) )
then A41: j2 < width G by A18, XREAL_1:145;
now__::_thesis:_ex_i1,_j2,_i,_j_being_Element_of_NAT_st_
(_1_<=_i_&_i_+_1_<=_len_G_&_1_<=_j_&_j_+_1_<=_width_G_&_LSeg_(f,k)_c=_cell_(G,i,j)_)
percases ( i1 < len G or i1 = len G ) by A16, XXREAL_0:1;
supposeA42: i1 < len G ; ::_thesis: ex i1, j2, i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) )
take i1 = i1; ::_thesis: ex j2, i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) )
take j2 = j2; ::_thesis: ex i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) )
A43: i1 + 1 <= len G by A42, NAT_1:13;
 LSeg (f,k) c= cell (G,i1,j2) by A10, A6, A8, A18, A11, A15, A13, A40, GOBOARD5:19, XREAL_1:145;
hence  ex i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) ) by A12, A18, A13, A40, A43; ::_thesis: verum
end;
supposeA44: i1 = len G ; ::_thesis: ex i19 being Element of NAT ex j2, i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) )
take i19 = i1 -' 1; ::_thesis: ex j2, i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) )
take j2 = j2; ::_thesis: ex i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) )
 ( 2 - 1 <= 2 -' 1 & 2 -' 1 <= i19 ) by A1, A44, NAT_D:42, XREAL_0:def_2;
then A45: 1 <= i19 by XXREAL_0:2;
A46: i19 + 1 = i1 by A12, XREAL_1:235;
then  i19 < len G by A16, NAT_1:13;
then  LSeg (f,k) c= cell (G,i19,j2) by A10, A6, A8, A13, A40, A41, A46, GOBOARD5:18;
hence  ex i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) ) by A16, A18, A13, A40, A45, A46; ::_thesis: verum
end;
end;
end;
hence  ex i, j being Element of NAT st
( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G & LSeg (f,k) c= cell (G,i,j) ) ; ::_thesis: verum
end;
end;
end;

theorem Th23: :: JORDAN9:23
for k being Element of NAT
for G being Go-board
for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st 2 <= len G & 2 <= width G & f is_sequence_on G & 1 <= k & k + 1 <= len f & p in Values G & p in LSeg (f,k) & not p = f /. k holds
p = f /. (k + 1)
proof
let k be Element of NAT ; ::_thesis: for G being Go-board
for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st 2 <= len G & 2 <= width G & f is_sequence_on G & 1 <= k & k + 1 <= len f & p in Values G & p in LSeg (f,k) & not p = f /. k holds
p = f /. (k + 1)
let G be Go-board; ::_thesis: for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st 2 <= len G & 2 <= width G & f is_sequence_on G & 1 <= k & k + 1 <= len f & p in Values G & p in LSeg (f,k) & not p = f /. k holds
p = f /. (k + 1)
let f be FinSequence of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st 2 <= len G & 2 <= width G & f is_sequence_on G & 1 <= k & k + 1 <= len f & p in Values G & p in LSeg (f,k) & not p = f /. k holds
p = f /. (k + 1)
let p be Point of (TOP-REAL 2); ::_thesis: ( 2 <= len G & 2 <= width G & f is_sequence_on G & 1 <= k & k + 1 <= len f & p in Values G & p in LSeg (f,k) & not p = f /. k implies p = f /. (k + 1) )
assume that
A1: ( 2 <= len G & 2 <= width G & f is_sequence_on G ) and
A2: ( 1 <= k & k + 1 <= len f ) and
A3: p in Values G and
A4: p in LSeg (f,k) ; ::_thesis: ( p = f /. k or p = f /. (k + 1) )
A5: LSeg (f,k) = LSeg ((f /. k),(f /. (k + 1))) by A2, TOPREAL1:def_3;
consider i, j being Element of NAT  such that
A6: ( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G ) and
A7: LSeg (f,k) c= cell (G,i,j) by A1, A2, Th22;
 p is_extremal_in cell (G,i,j) by A3, A4, A6, A7, Th21;
hence  ( p = f /. k or p = f /. (k + 1) ) by A4, A7, A5, SPPOL_1:def_1; ::_thesis: verum
end;

theorem :: JORDAN9:24
for i, j, k being Element of NAT
for G being Go-board st [i,j] in Indices G & 1 <= k & k <= width G holds
(G * (i,j)) `1 <= (G * ((len G),k)) `1
proof
let i, j, k be Element of NAT ; ::_thesis: for G being Go-board st [i,j] in Indices G & 1 <= k & k <= width G holds
(G * (i,j)) `1 <= (G * ((len G),k)) `1
let G be Go-board; ::_thesis: ( [i,j] in Indices G & 1 <= k & k <= width G implies (G * (i,j)) `1 <= (G * ((len G),k)) `1 )
assume that
A1: [i,j] in Indices G and
A2: ( 1 <= k & k <= width G ) ; ::_thesis: (G * (i,j)) `1 <= (G * ((len G),k)) `1
A3: 1 <= i by A1, MATRIX_1:38;
A4: i <= len G by A1, MATRIX_1:38;
then A5: ( i < len G or i = len G ) by XXREAL_0:1;
 ( 1 <= j & j <= width G ) by A1, MATRIX_1:38;
then (G * (i,j)) `1 = (G * (i,1)) `1 by A3, A4, GOBOARD5:2
.= (G * (i,k)) `1 by A2, A3, A4, GOBOARD5:2 ;
hence  (G * (i,j)) `1 <= (G * ((len G),k)) `1 by A2, A3, A5, GOBOARD5:3; ::_thesis: verum
end;

theorem :: JORDAN9:25
for i, j, k being Element of NAT
for G being Go-board st [i,j] in Indices G & 1 <= k & k <= len G holds
(G * (i,j)) `2 <= (G * (k,(width G))) `2
proof
let i, j, k be Element of NAT ; ::_thesis: for G being Go-board st [i,j] in Indices G & 1 <= k & k <= len G holds
(G * (i,j)) `2 <= (G * (k,(width G))) `2
let G be Go-board; ::_thesis: ( [i,j] in Indices G & 1 <= k & k <= len G implies (G * (i,j)) `2 <= (G * (k,(width G))) `2 )
assume that
A1: [i,j] in Indices G and
A2: ( 1 <= k & k <= len G ) ; ::_thesis: (G * (i,j)) `2 <= (G * (k,(width G))) `2
A3: 1 <= j by A1, MATRIX_1:38;
A4: j <= width G by A1, MATRIX_1:38;
then A5: ( j < width G or j = width G ) by XXREAL_0:1;
 ( 1 <= i & i <= len G ) by A1, MATRIX_1:38;
then (G * (i,j)) `2 = (G * (1,j)) `2 by A3, A4, GOBOARD5:1
.= (G * (k,j)) `2 by A2, A3, A4, GOBOARD5:1 ;
hence  (G * (i,j)) `2 <= (G * (k,(width G))) `2 by A2, A3, A5, GOBOARD5:4; ::_thesis: verum
end;

theorem Th26: :: JORDAN9:26
for k being Element of NAT
for G being Go-board
for f, g being FinSequence of (TOP-REAL 2) st f is_sequence_on G & f is special & L~ g c= L~ f & 1 <= k & k + 1 <= len f holds
for A being Subset of (TOP-REAL 2) st ( A = (right_cell (f,k,G)) \ (L~ g) or A = (left_cell (f,k,G)) \ (L~ g) ) holds
A is connected
proof
let k be Element of NAT ; ::_thesis: for G being Go-board
for f, g being FinSequence of (TOP-REAL 2) st f is_sequence_on G & f is special & L~ g c= L~ f & 1 <= k & k + 1 <= len f holds
for A being Subset of (TOP-REAL 2) st ( A = (right_cell (f,k,G)) \ (L~ g) or A = (left_cell (f,k,G)) \ (L~ g) ) holds
A is connected
let G be Go-board; ::_thesis: for f, g being FinSequence of (TOP-REAL 2) st f is_sequence_on G & f is special & L~ g c= L~ f & 1 <= k & k + 1 <= len f holds
for A being Subset of (TOP-REAL 2) st ( A = (right_cell (f,k,G)) \ (L~ g) or A = (left_cell (f,k,G)) \ (L~ g) ) holds
A is connected
let f, g be FinSequence of (TOP-REAL 2); ::_thesis: ( f is_sequence_on G & f is special & L~ g c= L~ f & 1 <= k & k + 1 <= len f implies for A being Subset of (TOP-REAL 2) st ( A = (right_cell (f,k,G)) \ (L~ g) or A = (left_cell (f,k,G)) \ (L~ g) ) holds
A is connected )
assume that
A1: f is_sequence_on G and
A2: f is special and
A3: L~ g c= L~ f and
A4: ( 1 <= k & k + 1 <= len f ) ; ::_thesis: for A being Subset of (TOP-REAL 2) st ( A = (right_cell (f,k,G)) \ (L~ g) or A = (left_cell (f,k,G)) \ (L~ g) ) holds
A is connected
let A be Subset of (TOP-REAL 2); ::_thesis: ( ( A = (right_cell (f,k,G)) \ (L~ g) or A = (left_cell (f,k,G)) \ (L~ g) ) implies A is connected )
assume A5: ( A = (right_cell (f,k,G)) \ (L~ g) or A = (left_cell (f,k,G)) \ (L~ g) ) ; ::_thesis: A is connected
percases ( A = (right_cell (f,k,G)) \ (L~ g) or A = (left_cell (f,k,G)) \ (L~ g) ) by A5;
supposeA6: A = (right_cell (f,k,G)) \ (L~ g) ; ::_thesis: A is connected
 Int (right_cell (f,k,G)) misses L~ f by A1, A2, A4, Th15;
then  Int (right_cell (f,k,G)) misses L~ g by A3, XBOOLE_1:63;
then A7: Int (right_cell (f,k,G)) c= (L~ g) ` by SUBSET_1:23;
 A c= right_cell (f,k,G) by A6, XBOOLE_1:36;
then A8: A c= Cl (Int (right_cell (f,k,G))) by A1, A4, Th11;
A9: A = (right_cell (f,k,G)) /\ ((L~ g) `) by A6, SUBSET_1:13;
 ( Int (right_cell (f,k,G)) is convex & Int (right_cell (f,k,G)) c= right_cell (f,k,G) ) by A1, A4, Th10, TOPS_1:16;
hence  A is connected by A9, A7, A8, CONNSP_1:18, XBOOLE_1:19; ::_thesis: verum
end;
supposeA10: A = (left_cell (f,k,G)) \ (L~ g) ; ::_thesis: A is connected
 Int (left_cell (f,k,G)) misses L~ f by A1, A2, A4, Th15;
then  Int (left_cell (f,k,G)) misses L~ g by A3, XBOOLE_1:63;
then A11: Int (left_cell (f,k,G)) c= (L~ g) ` by SUBSET_1:23;
 A c= left_cell (f,k,G) by A10, XBOOLE_1:36;
then A12: A c= Cl (Int (left_cell (f,k,G))) by A1, A4, Th11;
A13: A = (left_cell (f,k,G)) /\ ((L~ g) `) by A10, SUBSET_1:13;
 ( Int (left_cell (f,k,G)) is convex & Int (left_cell (f,k,G)) c= left_cell (f,k,G) ) by A1, A4, Th10, TOPS_1:16;
hence  A is connected by A13, A11, A12, CONNSP_1:18, XBOOLE_1:19; ::_thesis: verum
end;
end;
end;

theorem Th27: :: JORDAN9:27
for G being Go-board
for f being non constant standard special_circular_sequence st f is_sequence_on G holds
for k being Element of NAT st 1 <= k & k + 1 <= len f holds
( (right_cell (f,k,G)) \ (L~ f) c= RightComp f & (left_cell (f,k,G)) \ (L~ f) c= LeftComp f )
proof
let G be Go-board; ::_thesis: for f being non constant standard special_circular_sequence st f is_sequence_on G holds
for k being Element of NAT st 1 <= k & k + 1 <= len f holds
( (right_cell (f,k,G)) \ (L~ f) c= RightComp f & (left_cell (f,k,G)) \ (L~ f) c= LeftComp f )
let f be non constant standard special_circular_sequence; ::_thesis: ( f is_sequence_on G implies for k being Element of NAT st 1 <= k & k + 1 <= len f holds
( (right_cell (f,k,G)) \ (L~ f) c= RightComp f & (left_cell (f,k,G)) \ (L~ f) c= LeftComp f ) )
assume A1: f is_sequence_on G ; ::_thesis: for k being Element of NAT st 1 <= k & k + 1 <= len f holds
( (right_cell (f,k,G)) \ (L~ f) c= RightComp f & (left_cell (f,k,G)) \ (L~ f) c= LeftComp f )
let k be Element of NAT ; ::_thesis: ( 1 <= k & k + 1 <= len f implies ( (right_cell (f,k,G)) \ (L~ f) c= RightComp f & (left_cell (f,k,G)) \ (L~ f) c= LeftComp f ) )
assume A2: ( 1 <= k & k + 1 <= len f ) ; ::_thesis: ( (right_cell (f,k,G)) \ (L~ f) c= RightComp f & (left_cell (f,k,G)) \ (L~ f) c= LeftComp f )
A3: Int (right_cell (f,k,G)) <> {} by A1, A2, Th9;
set rc = (right_cell (f,k,G)) \ (L~ f);
 ((right_cell (f,k,G)) \ (L~ f)) \/ (L~ f) = (right_cell (f,k,G)) \/ (L~ f) by XBOOLE_1:39;
then  ( Int (right_cell (f,k,G)) c= right_cell (f,k,G) & right_cell (f,k,G) c= ((right_cell (f,k,G)) \ (L~ f)) \/ (L~ f) ) by TOPS_1:16, XBOOLE_1:7;
then A4: Int (right_cell (f,k,G)) c= ((right_cell (f,k,G)) \ (L~ f)) \/ (L~ f) by XBOOLE_1:1;
set lc = (left_cell (f,k,G)) \ (L~ f);
 (right_cell (f,k,G)) \ (L~ f) = (right_cell (f,k,G)) /\ ((L~ f) `) by SUBSET_1:13;
then A5: ( RightComp f is_a_component_of (L~ f) ` & (right_cell (f,k,G)) \ (L~ f) c= (L~ f) ` ) by GOBOARD9:def_2, XBOOLE_1:17;
 ( (right_cell (f,k,G)) \ (L~ f) c= right_cell (f,k,G) & right_cell (f,k,G) c= right_cell (f,k) ) by A1, A2, GOBRD13:33, XBOOLE_1:36;
then  (right_cell (f,k,G)) \ (L~ f) c= right_cell (f,k) by XBOOLE_1:1;
then A6: Int ((right_cell (f,k,G)) \ (L~ f)) c= Int (right_cell (f,k)) by TOPS_1:19;
 Int (right_cell (f,k)) c= RightComp f by A2, GOBOARD9:25;
then A7: Int ((right_cell (f,k,G)) \ (L~ f)) c= RightComp f by A6, XBOOLE_1:1;
 Int (right_cell (f,k,G)) misses L~ f by A1, A2, Th15;
then A8: Int (Int (right_cell (f,k,G))) c= Int ((right_cell (f,k,G)) \ (L~ f)) by A4, TOPS_1:19, XBOOLE_1:73;
 Int (right_cell (f,k,G)) c= (right_cell (f,k,G)) \ (L~ f) by A1, A2, A4, Th15, XBOOLE_1:73;
then A9: (right_cell (f,k,G)) \ (L~ f) meets Int ((right_cell (f,k,G)) \ (L~ f)) by A3, A8, XBOOLE_1:68;
 (right_cell (f,k,G)) \ (L~ f) is connected by A1, A2, Th26;
hence  (right_cell (f,k,G)) \ (L~ f) c= RightComp f by A7, A9, A5, GOBOARD9:4; ::_thesis: (left_cell (f,k,G)) \ (L~ f) c= LeftComp f
 (left_cell (f,k,G)) \ (L~ f) = (left_cell (f,k,G)) /\ ((L~ f) `) by SUBSET_1:13;
then A10: ( LeftComp f is_a_component_of (L~ f) ` & (left_cell (f,k,G)) \ (L~ f) c= (L~ f) ` ) by GOBOARD9:def_1, XBOOLE_1:17;
 ((left_cell (f,k,G)) \ (L~ f)) \/ (L~ f) = (left_cell (f,k,G)) \/ (L~ f) by XBOOLE_1:39;
then  ( Int (left_cell (f,k,G)) c= left_cell (f,k,G) & left_cell (f,k,G) c= ((left_cell (f,k,G)) \ (L~ f)) \/ (L~ f) ) by TOPS_1:16, XBOOLE_1:7;
then A11: Int (left_cell (f,k,G)) c= ((left_cell (f,k,G)) \ (L~ f)) \/ (L~ f) by XBOOLE_1:1;
 ( (left_cell (f,k,G)) \ (L~ f) c= left_cell (f,k,G) & left_cell (f,k,G) c= left_cell (f,k) ) by A1, A2, GOBRD13:33, XBOOLE_1:36;
then  (left_cell (f,k,G)) \ (L~ f) c= left_cell (f,k) by XBOOLE_1:1;
then A12: Int ((left_cell (f,k,G)) \ (L~ f)) c= Int (left_cell (f,k)) by TOPS_1:19;
 Int (left_cell (f,k)) c= LeftComp f by A2, GOBOARD9:21;
then A13: Int ((left_cell (f,k,G)) \ (L~ f)) c= LeftComp f by A12, XBOOLE_1:1;
A14: Int (left_cell (f,k,G)) <> {} by A1, A2, Th9;
 Int (left_cell (f,k,G)) misses L~ f by A1, A2, Th15;
then A15: Int (Int (left_cell (f,k,G))) c= Int ((left_cell (f,k,G)) \ (L~ f)) by A11, TOPS_1:19, XBOOLE_1:73;
 Int (left_cell (f,k,G)) c= (left_cell (f,k,G)) \ (L~ f) by A1, A2, A11, Th15, XBOOLE_1:73;
then A16: (left_cell (f,k,G)) \ (L~ f) meets Int ((left_cell (f,k,G)) \ (L~ f)) by A14, A15, XBOOLE_1:68;
 (left_cell (f,k,G)) \ (L~ f) is connected by A1, A2, Th26;
hence  (left_cell (f,k,G)) \ (L~ f) c= LeftComp f by A13, A16, A10, GOBOARD9:4; ::_thesis: verum
end;

begin


theorem Th28: :: JORDAN9:28
for C being non empty compact non horizontal non vertical Subset of (TOP-REAL 2)
for n being Nat ex i being Element of NAT st
( 1 <= i & i + 1 <= len (Gauge (C,n)) & N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) & N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1)) )
proof
let C be non empty compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for n being Nat ex i being Element of NAT st
( 1 <= i & i + 1 <= len (Gauge (C,n)) & N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) & N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1)) )
let n be Nat; ::_thesis: ex i being Element of NAT st
( 1 <= i & i + 1 <= len (Gauge (C,n)) & N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) & N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1)) )
set G = Gauge (C,n);
defpred S1[ Nat] means ( 1 <= $1 & $1 < len (Gauge (C,n)) & ((Gauge (C,n)) * ($1,((width (Gauge (C,n))) -' 1))) `1 < (N-min C) `1 );
A1: for k being Nat st S1[k] holds
k <= len (Gauge (C,n)) ;
A2: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
 (NW-corner C) `1 <= (N-min C) `1 by PSCOMP_1:38;
then A3: W-bound C <= (N-min C) `1 by EUCLID:52;
A4: len (Gauge (C,n)) >= 4 by JORDAN8:10;
then A5: ( (len (Gauge (C,n))) -' 1 <= len (Gauge (C,n)) & 2 <= len (Gauge (C,n)) ) by NAT_D:35, XXREAL_0:2;
A6: 1 < len (Gauge (C,n)) by A4, XXREAL_0:2;
then A7: 1 <= (len (Gauge (C,n))) -' 1 by NAT_D:49;
A8: ( n in NAT & len (Gauge (C,n)) = width (Gauge (C,n)) ) by JORDAN8:def_1, ORDINAL1:def_12;
then  ((Gauge (C,n)) * (2,((width (Gauge (C,n))) -' 1))) `1 = W-bound C by A7, JORDAN8:11, NAT_D:35;
then  ((Gauge (C,n)) * (1,((width (Gauge (C,n))) -' 1))) `1 < W-bound C by A2, A7, A5, GOBOARD5:3;
then  ((Gauge (C,n)) * (1,((width (Gauge (C,n))) -' 1))) `1 < (N-min C) `1 by A3, XXREAL_0:2;
then A9: ex k being Nat st S1[k] by A6;
 ex i being Nat st
( S1[i] & ( for n being Nat st S1[n] holds
n <= i ) ) from NAT_1:sch_6(A1, A9);
then consider i being Nat such that
A10: 1 <= i and
A11: i < len (Gauge (C,n)) and
A12: ((Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1))) `1 < (N-min C) `1 and
A13: for n being Nat st S1[n] holds
n <= i ;
reconsider i = i as Element of NAT by ORDINAL1:def_12;
A14: ( 1 <= i + 1 & i < i + 1 ) by NAT_1:12, NAT_1:13;
A15: (N-min C) `2 = N-bound C by EUCLID:52;
A16: i + 1 <= len (Gauge (C,n)) by A11, NAT_1:13;
then A17: (N-min C) `2 = ((Gauge (C,n)) * ((i + 1),((width (Gauge (C,n))) -' 1))) `2 by A8, A15, JORDAN8:14, NAT_1:12;
now__::_thesis:_not_i_+_1_=_len_(Gauge_(C,n))
assume  i + 1 = len (Gauge (C,n)) ; ::_thesis: contradiction
then  (len (Gauge (C,n))) -' 1 = i by NAT_D:34;
then A18: ((Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1))) `1 = E-bound C by A8, A7, JORDAN8:12, NAT_D:35;
 (NE-corner C) `1 >= (N-min C) `1 by PSCOMP_1:38;
hence  contradiction by A12, A18, EUCLID:52; ::_thesis: verum
end;
then  i + 1 < len (Gauge (C,n)) by A16, XXREAL_0:1;
then A19: (N-min C) `1 <= ((Gauge (C,n)) * ((i + 1),((width (Gauge (C,n))) -' 1))) `1 by A13, A14;
 ((Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1))) `2 = (N-min C) `2 by A8, A10, A11, A15, JORDAN8:14;
then A20: N-min C in LSeg (((Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1))),((Gauge (C,n)) * ((i + 1),((width (Gauge (C,n))) -' 1)))) by A12, A17, A19, GOBOARD7:8;
take  i ; ::_thesis: ( 1 <= i & i + 1 <= len (Gauge (C,n)) & N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) & N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1)) )
thus  ( 1 <= i & i + 1 <= len (Gauge (C,n)) ) by A10, A11, NAT_1:13; ::_thesis: ( N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) & N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1)) )
 LSeg (((Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1))),((Gauge (C,n)) * ((i + 1),((width (Gauge (C,n))) -' 1)))) c= cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) by A2, A7, A10, A11, GOBOARD5:22, NAT_D:35;
hence  N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) by A20; ::_thesis: N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1))
thus  N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1)) by A12; ::_thesis: verum
end;

theorem Th29: :: JORDAN9:29
for C being non empty compact non horizontal non vertical Subset of (TOP-REAL 2)
for n, i1, i2 being Nat st 1 <= i1 & i1 + 1 <= len (Gauge (C,n)) & N-min C in cell ((Gauge (C,n)),i1,((width (Gauge (C,n))) -' 1)) & N-min C <> (Gauge (C,n)) * (i1,((width (Gauge (C,n))) -' 1)) & 1 <= i2 & i2 + 1 <= len (Gauge (C,n)) & N-min C in cell ((Gauge (C,n)),i2,((width (Gauge (C,n))) -' 1)) & N-min C <> (Gauge (C,n)) * (i2,((width (Gauge (C,n))) -' 1)) holds
i1 = i2
proof
let C be non empty compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for n, i1, i2 being Nat st 1 <= i1 & i1 + 1 <= len (Gauge (C,n)) & N-min C in cell ((Gauge (C,n)),i1,((width (Gauge (C,n))) -' 1)) & N-min C <> (Gauge (C,n)) * (i1,((width (Gauge (C,n))) -' 1)) & 1 <= i2 & i2 + 1 <= len (Gauge (C,n)) & N-min C in cell ((Gauge (C,n)),i2,((width (Gauge (C,n))) -' 1)) & N-min C <> (Gauge (C,n)) * (i2,((width (Gauge (C,n))) -' 1)) holds
i1 = i2
let n, i1, i2 be Nat; ::_thesis: ( 1 <= i1 & i1 + 1 <= len (Gauge (C,n)) & N-min C in cell ((Gauge (C,n)),i1,((width (Gauge (C,n))) -' 1)) & N-min C <> (Gauge (C,n)) * (i1,((width (Gauge (C,n))) -' 1)) & 1 <= i2 & i2 + 1 <= len (Gauge (C,n)) & N-min C in cell ((Gauge (C,n)),i2,((width (Gauge (C,n))) -' 1)) & N-min C <> (Gauge (C,n)) * (i2,((width (Gauge (C,n))) -' 1)) implies i1 = i2 )
A1: i1 in NAT by ORDINAL1:def_12;
set G = Gauge (C,n);
set j = (width (Gauge (C,n))) -' 1;
A2: i2 in NAT by ORDINAL1:def_12;
A3: 2 |^ n >= n + 1 by NEWTON:85;
A4: 1 + (n + 3) > 1 + 0 by XREAL_1:6;
A5: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
A6: len (Gauge (C,n)) = (2 |^ n) + 3 by JORDAN8:def_1;
then A7: len (Gauge (C,n)) >= (n + 1) + 3 by A3, XREAL_1:6;
then  len (Gauge (C,n)) > 1 by A4, XXREAL_0:2;
then A8: len (Gauge (C,n)) >= 1 + 1 by NAT_1:13;
then A9: 1 <= (width (Gauge (C,n))) -' 1 by A5, JORDAN5B:2;
A10: ((width (Gauge (C,n))) -' 1) + 1 = len (Gauge (C,n)) by A5, A7, A4, XREAL_1:235, XXREAL_0:2;
then A11: (width (Gauge (C,n))) -' 1 < len (Gauge (C,n)) by NAT_1:13;
assume that
A12: 1 <= i1 and
A13: i1 + 1 <= len (Gauge (C,n)) and
A14: N-min C in cell ((Gauge (C,n)),i1,((width (Gauge (C,n))) -' 1)) and
A15: N-min C <> (Gauge (C,n)) * (i1,((width (Gauge (C,n))) -' 1)) and
A16: 1 <= i2 and
A17: i2 + 1 <= len (Gauge (C,n)) and
A18: N-min C in cell ((Gauge (C,n)),i2,((width (Gauge (C,n))) -' 1)) and
A19: N-min C <> (Gauge (C,n)) * (i2,((width (Gauge (C,n))) -' 1)) and
A20: i1 <> i2 ; ::_thesis: contradiction
A21: cell ((Gauge (C,n)),i1,((width (Gauge (C,n))) -' 1)) meets cell ((Gauge (C,n)),i2,((width (Gauge (C,n))) -' 1)) by A14, A18, XBOOLE_0:3;
A22: i1 < len (Gauge (C,n)) by A13, NAT_1:13;
A23: i2 < len (Gauge (C,n)) by A17, NAT_1:13;
percases ( i1 < i2 or i2 < i1 ) by A20, XXREAL_0:1;
supposeA24: i1 < i2 ; ::_thesis: contradiction
then A25: (i2 -' i1) + i1 = i2 by XREAL_1:235;
then  i2 -' i1 <= 1 by A1, A23, A5, A21, A9, A11, JORDAN8:7;
then  ( i2 -' i1 < 1 or i2 -' i1 = 1 ) by XXREAL_0:1;
then  ( i2 -' i1 = 0 or i2 -' i1 = 1 ) by NAT_1:14;
then  (cell ((Gauge (C,n)),i1,((width (Gauge (C,n))) -' 1))) /\ (cell ((Gauge (C,n)),i2,((width (Gauge (C,n))) -' 1))) = LSeg (((Gauge (C,n)) * (i2,((width (Gauge (C,n))) -' 1))),((Gauge (C,n)) * (i2,(((width (Gauge (C,n))) -' 1) + 1)))) by A1, A22, A5, A8, A11, A24, A25, GOBOARD5:25, JORDAN5B:2;
then A26: N-min C in LSeg (((Gauge (C,n)) * (i2,((width (Gauge (C,n))) -' 1))),((Gauge (C,n)) * (i2,(((width (Gauge (C,n))) -' 1) + 1)))) by A14, A18, XBOOLE_0:def_4;
 1 <= ((width (Gauge (C,n))) -' 1) + 1 by NAT_1:12;
then A27: [i2,(((width (Gauge (C,n))) -' 1) + 1)] in Indices (Gauge (C,n)) by A16, A23, A5, A10, MATRIX_1:36;
set y2 = (S-bound C) + ((((N-bound C) - (S-bound C)) / (2 |^ n)) * (((width (Gauge (C,n))) -' 1) - 1));
set y1 = (S-bound C) + ((((N-bound C) - (S-bound C)) / (2 |^ n)) * (((width (Gauge (C,n))) -' 1) - 2));
set x = (W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * (i2 - 2));
(width (Gauge (C,n))) -' 1 = (((2 |^ n) + 2) + 1) -' 1 by A6, JORDAN8:def_1
.= (2 |^ n) + 2 by NAT_D:34 ;
then A28: (((N-bound C) - (S-bound C)) / (2 |^ n)) * (((width (Gauge (C,n))) -' 1) - 2) = (N-bound C) - (S-bound C) by A3, XCMPLX_1:87;
 [i2,((width (Gauge (C,n))) -' 1)] in Indices (Gauge (C,n)) by A16, A23, A5, A9, A11, MATRIX_1:36;
then A29: (Gauge (C,n)) * (i2,((width (Gauge (C,n))) -' 1)) = |[((W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * (i2 - 2))),((S-bound C) + ((((N-bound C) - (S-bound C)) / (2 |^ n)) * (((width (Gauge (C,n))) -' 1) - 2)))]| by JORDAN8:def_1;
then A30: ((Gauge (C,n)) * (i2,((width (Gauge (C,n))) -' 1))) `1 = (W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * (i2 - 2)) by EUCLID:52;
 (((width (Gauge (C,n))) -' 1) + 1) - (1 + 1) = ((width (Gauge (C,n))) -' 1) - 1 ;
then  (Gauge (C,n)) * (i2,(((width (Gauge (C,n))) -' 1) + 1)) = |[((W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * (i2 - 2))),((S-bound C) + ((((N-bound C) - (S-bound C)) / (2 |^ n)) * (((width (Gauge (C,n))) -' 1) - 1)))]| by A27, JORDAN8:def_1;
then  ((Gauge (C,n)) * (i2,(((width (Gauge (C,n))) -' 1) + 1))) `1 = (W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * (i2 - 2)) by EUCLID:52;
then  LSeg (((Gauge (C,n)) * (i2,((width (Gauge (C,n))) -' 1))),((Gauge (C,n)) * (i2,(((width (Gauge (C,n))) -' 1) + 1)))) is vertical by A30, SPPOL_1:16;
then  (N-min C) `1 = ((Gauge (C,n)) * (i2,((width (Gauge (C,n))) -' 1))) `1 by A26, SPPOL_1:41;
hence  contradiction by A19, A29, A30, A28, EUCLID:52; ::_thesis: verum
end;
supposeA31: i2 < i1 ; ::_thesis: contradiction
then A32: (i1 -' i2) + i2 = i1 by XREAL_1:235;
then  i1 -' i2 <= 1 by A2, A22, A5, A21, A9, A11, JORDAN8:7;
then  ( i1 -' i2 < 1 or i1 -' i2 = 1 ) by XXREAL_0:1;
then  ( i1 -' i2 = 0 or i1 -' i2 = 1 ) by NAT_1:14;
then  (cell ((Gauge (C,n)),i2,((width (Gauge (C,n))) -' 1))) /\ (cell ((Gauge (C,n)),i1,((width (Gauge (C,n))) -' 1))) = LSeg (((Gauge (C,n)) * (i1,((width (Gauge (C,n))) -' 1))),((Gauge (C,n)) * (i1,(((width (Gauge (C,n))) -' 1) + 1)))) by A2, A23, A5, A8, A11, A31, A32, GOBOARD5:25, JORDAN5B:2;
then A33: N-min C in LSeg (((Gauge (C,n)) * (i1,((width (Gauge (C,n))) -' 1))),((Gauge (C,n)) * (i1,(((width (Gauge (C,n))) -' 1) + 1)))) by A14, A18, XBOOLE_0:def_4;
 1 <= ((width (Gauge (C,n))) -' 1) + 1 by NAT_1:12;
then A34: [i1,(((width (Gauge (C,n))) -' 1) + 1)] in Indices (Gauge (C,n)) by A12, A22, A5, A10, MATRIX_1:36;
set y2 = (S-bound C) + ((((N-bound C) - (S-bound C)) / (2 |^ n)) * (((width (Gauge (C,n))) -' 1) - 1));
set y1 = (S-bound C) + ((((N-bound C) - (S-bound C)) / (2 |^ n)) * (((width (Gauge (C,n))) -' 1) - 2));
set x = (W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * (i1 - 2));
(width (Gauge (C,n))) -' 1 = (((2 |^ n) + 2) + 1) -' 1 by A6, JORDAN8:def_1
.= (2 |^ n) + 2 by NAT_D:34 ;
then A35: (((N-bound C) - (S-bound C)) / (2 |^ n)) * (((width (Gauge (C,n))) -' 1) - 2) = (N-bound C) - (S-bound C) by A3, XCMPLX_1:87;
 [i1,((width (Gauge (C,n))) -' 1)] in Indices (Gauge (C,n)) by A12, A22, A5, A9, A11, MATRIX_1:36;
then A36: (Gauge (C,n)) * (i1,((width (Gauge (C,n))) -' 1)) = |[((W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * (i1 - 2))),((S-bound C) + ((((N-bound C) - (S-bound C)) / (2 |^ n)) * (((width (Gauge (C,n))) -' 1) - 2)))]| by JORDAN8:def_1;
then A37: ((Gauge (C,n)) * (i1,((width (Gauge (C,n))) -' 1))) `1 = (W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * (i1 - 2)) by EUCLID:52;
 (((width (Gauge (C,n))) -' 1) + 1) - (1 + 1) = ((width (Gauge (C,n))) -' 1) - 1 ;
then  (Gauge (C,n)) * (i1,(((width (Gauge (C,n))) -' 1) + 1)) = |[((W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * (i1 - 2))),((S-bound C) + ((((N-bound C) - (S-bound C)) / (2 |^ n)) * (((width (Gauge (C,n))) -' 1) - 1)))]| by A34, JORDAN8:def_1;
then  ((Gauge (C,n)) * (i1,(((width (Gauge (C,n))) -' 1) + 1))) `1 = (W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * (i1 - 2)) by EUCLID:52;
then  LSeg (((Gauge (C,n)) * (i1,((width (Gauge (C,n))) -' 1))),((Gauge (C,n)) * (i1,(((width (Gauge (C,n))) -' 1) + 1)))) is vertical by A37, SPPOL_1:16;
then  (N-min C) `1 = ((Gauge (C,n)) * (i1,((width (Gauge (C,n))) -' 1))) `1 by A33, SPPOL_1:41;
hence  contradiction by A15, A36, A37, A35, EUCLID:52; ::_thesis: verum
end;
end;
end;

theorem Th30: :: JORDAN9:30
for C being non empty compact non horizontal non vertical Subset of (TOP-REAL 2)
for n being Nat
for f being non constant standard special_circular_sequence st f is_sequence_on Gauge (C,n) & ( for k being Element of NAT st 1 <= k & k + 1 <= len f holds
( left_cell (f,k,(Gauge (C,n))) misses C & right_cell (f,k,(Gauge (C,n))) meets C ) ) & ex i being Element of NAT st
( 1 <= i & i + 1 <= len (Gauge (C,n)) & f /. 1 = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) & f /. 2 = (Gauge (C,n)) * ((i + 1),(width (Gauge (C,n)))) & N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) & N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1)) ) holds
N-min (L~ f) = f /. 1
proof
let C be non empty compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for n being Nat
for f being non constant standard special_circular_sequence st f is_sequence_on Gauge (C,n) & ( for k being Element of NAT st 1 <= k & k + 1 <= len f holds
( left_cell (f,k,(Gauge (C,n))) misses C & right_cell (f,k,(Gauge (C,n))) meets C ) ) & ex i being Element of NAT st
( 1 <= i & i + 1 <= len (Gauge (C,n)) & f /. 1 = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) & f /. 2 = (Gauge (C,n)) * ((i + 1),(width (Gauge (C,n)))) & N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) & N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1)) ) holds
N-min (L~ f) = f /. 1
let n be Nat; ::_thesis: for f being non constant standard special_circular_sequence st f is_sequence_on Gauge (C,n) & ( for k being Element of NAT st 1 <= k & k + 1 <= len f holds
( left_cell (f,k,(Gauge (C,n))) misses C & right_cell (f,k,(Gauge (C,n))) meets C ) ) & ex i being Element of NAT st
( 1 <= i & i + 1 <= len (Gauge (C,n)) & f /. 1 = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) & f /. 2 = (Gauge (C,n)) * ((i + 1),(width (Gauge (C,n)))) & N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) & N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1)) ) holds
N-min (L~ f) = f /. 1
set G = Gauge (C,n);
let f be non constant standard special_circular_sequence; ::_thesis: ( f is_sequence_on Gauge (C,n) & ( for k being Element of NAT st 1 <= k & k + 1 <= len f holds
( left_cell (f,k,(Gauge (C,n))) misses C & right_cell (f,k,(Gauge (C,n))) meets C ) ) & ex i being Element of NAT st
( 1 <= i & i + 1 <= len (Gauge (C,n)) & f /. 1 = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) & f /. 2 = (Gauge (C,n)) * ((i + 1),(width (Gauge (C,n)))) & N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) & N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1)) ) implies N-min (L~ f) = f /. 1 )
assume that
A1: f is_sequence_on Gauge (C,n) and
A2: for k being Element of NAT st 1 <= k & k + 1 <= len f holds
( left_cell (f,k,(Gauge (C,n))) misses C & right_cell (f,k,(Gauge (C,n))) meets C ) ; ::_thesis: ( for i being Element of NAT holds
( not 1 <= i or not i + 1 <= len (Gauge (C,n)) or not f /. 1 = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) or not f /. 2 = (Gauge (C,n)) * ((i + 1),(width (Gauge (C,n)))) or not N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) or not N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1)) ) or N-min (L~ f) = f /. 1 )
 N-min (L~ f) in rng f by SPRECT_2:39;
then consider m being Nat such that
A3: m in dom f and
A4: f . m = N-min (L~ f) by FINSEQ_2:10;
reconsider m = m as Element of NAT by ORDINAL1:def_12;
consider i, j being Element of NAT  such that
A5: [i,j] in Indices (Gauge (C,n)) and
A6: f /. m = (Gauge (C,n)) * (i,j) by A1, A3, GOBOARD1:def_9;
A7: f /. m = f . m by A3, PARTFUN1:def_6;
A8: (N-min (L~ f)) `2 = N-bound (L~ f) by EUCLID:52;
set W = W-bound C;
set S = S-bound C;
set E = E-bound C;
set N = N-bound C;
given i9 being Element of NAT  such that A9: 1 <= i9 and
A10: i9 + 1 <= len (Gauge (C,n)) and
A11: f /. 1 = (Gauge (C,n)) * (i9,(width (Gauge (C,n)))) and
A12: f /. 2 = (Gauge (C,n)) * ((i9 + 1),(width (Gauge (C,n)))) and
A13: N-min C in cell ((Gauge (C,n)),i9,((width (Gauge (C,n))) -' 1)) and
A14: N-min C <> (Gauge (C,n)) * (i9,((width (Gauge (C,n))) -' 1)) ; ::_thesis: N-min (L~ f) = f /. 1
A15: ( (Gauge (C,n)) * (i9,((len (Gauge (C,n))) -' 1)) = |[(((Gauge (C,n)) * (i9,((len (Gauge (C,n))) -' 1))) `1),(((Gauge (C,n)) * (i9,((len (Gauge (C,n))) -' 1))) `2)]| & (N-min C) `2 = N-bound C ) by EUCLID:52, EUCLID:53;
 (Gauge (C,n)) * (i,j) = |[((W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * (i - 2))),((S-bound C) + ((((N-bound C) - (S-bound C)) / (2 |^ n)) * (j - 2)))]| by A5, JORDAN8:def_1;
then A16: (S-bound C) + ((((N-bound C) - (S-bound C)) / (2 |^ n)) * (j - 2)) = N-bound (L~ f) by A4, A7, A8, A6, EUCLID:52;
 N-bound C > S-bound C by JORDAN8:9;
then  ( 2 |^ n > 0 & (N-bound C) - (S-bound C) > 0 ) by NEWTON:83, XREAL_1:50;
then A17: ((N-bound C) - (S-bound C)) / (2 |^ n) > 0 by XREAL_1:139;
A18: ( (NW-corner (L~ f)) `1 = W-bound (L~ f) & (NE-corner (L~ f)) `1 = E-bound (L~ f) ) by EUCLID:52;
A19: 1 <= i by A5, MATRIX_1:38;
A20: ( (NW-corner (L~ f)) `2 = N-bound (L~ f) & (NE-corner (L~ f)) `2 = N-bound (L~ f) ) by EUCLID:52;
A21: m <= len f by A3, FINSEQ_3:25;
A22: 1 <= j by A5, MATRIX_1:38;
 len (Gauge (C,n)) = (2 |^ n) + 3 by JORDAN8:def_1;
then A23: len (Gauge (C,n)) >= 3 by NAT_1:12;
then A24: 1 < len (Gauge (C,n)) by XXREAL_0:2;
then A25: 1 <= (len (Gauge (C,n))) -' 1 by NAT_D:49;
then A26: (len (Gauge (C,n))) -' 1 < len (Gauge (C,n)) by NAT_D:51;
A27: i <= len (Gauge (C,n)) by A5, MATRIX_1:38;
A28: j <= width (Gauge (C,n)) by A5, MATRIX_1:38;
then A29: ((Gauge (C,n)) * (i,j)) `2 = ((Gauge (C,n)) * (1,j)) `2 by A19, A27, A22, GOBOARD5:1;
A30: len f > 4 by GOBOARD7:34;
 1 in dom f by FINSEQ_5:6;
then A31: f /. 1 in L~ f by A30, GOBOARD1:1, XXREAL_0:2;
then A32: N-bound (L~ f) >= (f /. 1) `2 by PSCOMP_1:24;
A33: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
A34: i9 < len (Gauge (C,n)) by A10, NAT_1:13;
then  ((Gauge (C,n)) * (i9,j)) `2 = ((Gauge (C,n)) * (1,j)) `2 by A9, A22, A28, GOBOARD5:1;
then  ((Gauge (C,n)) * (i,j)) `2 <= ((Gauge (C,n)) * (i9,(len (Gauge (C,n))))) `2 by A9, A34, A33, A22, A28, A29, SPRECT_3:12;
then A35: N-bound (L~ f) = (f /. 1) `2 by A11, A33, A4, A7, A8, A6, A32, XXREAL_0:1;
 [i9,(len (Gauge (C,n)))] in Indices (Gauge (C,n)) by A9, A34, A33, A24, MATRIX_1:36;
then  (Gauge (C,n)) * (i9,(len (Gauge (C,n)))) = |[((W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * (i9 - 2))),((S-bound C) + ((((N-bound C) - (S-bound C)) / (2 |^ n)) * ((len (Gauge (C,n))) - 2)))]| by JORDAN8:def_1;
then  (S-bound C) + ((((N-bound C) - (S-bound C)) / (2 |^ n)) * ((len (Gauge (C,n))) - 2)) = N-bound (L~ f) by A11, A33, A35, EUCLID:52;
then A36: (len (Gauge (C,n))) - 2 = j - 2 by A17, A16, XCMPLX_1:5;
then A37: ((Gauge (C,n)) * (i9,(len (Gauge (C,n))))) `1 = ((Gauge (C,n)) * (i9,1)) `1 by A9, A34, A33, A22, GOBOARD5:2;
 ( W-bound (L~ f) <= (f /. 1) `1 & (f /. 1) `1 <= E-bound (L~ f) ) by A31, PSCOMP_1:24;
then  f /. 1 in LSeg ((NW-corner (L~ f)),(NE-corner (L~ f))) by A35, A18, A20, GOBOARD7:8;
then  f /. 1 in (LSeg ((NW-corner (L~ f)),(NE-corner (L~ f)))) /\ (L~ f) by A31, XBOOLE_0:def_4;
then A38: (N-min (L~ f)) `1 <= (f /. 1) `1 by PSCOMP_1:39;
then A39: i <= i9 by A9, A11, A33, A4, A7, A6, A27, A22, A36, GOBOARD5:3;
then A40: i < len (Gauge (C,n)) by A34, XXREAL_0:2;
then A41: i + 1 <= len (Gauge (C,n)) by NAT_1:13;
A42: ((len (Gauge (C,n))) -' 1) + 1 = len (Gauge (C,n)) by A23, XREAL_1:235, XXREAL_0:2;
then  N-min C in  {  |[r9,s9]| where r9, s9 is Real : ( ((Gauge (C,n)) * (i9,1)) `1 <= r9 & r9 <= ((Gauge (C,n)) * ((i9 + 1),1)) `1 & ((Gauge (C,n)) * (1,((len (Gauge (C,n))) -' 1))) `2 <= s9 & s9 <= ((Gauge (C,n)) * (1,(len (Gauge (C,n))))) `2 )  }  by A9, A13, A34, A33, A25, A26, GOBRD11:32;
then  ex r9, s9 being Real st
( N-min C = |[r9,s9]| & ((Gauge (C,n)) * (i9,1)) `1 <= r9 & r9 <= ((Gauge (C,n)) * ((i9 + 1),1)) `1 & ((Gauge (C,n)) * (1,((len (Gauge (C,n))) -' 1))) `2 <= s9 & s9 <= ((Gauge (C,n)) * (1,(len (Gauge (C,n))))) `2 ) ;
then A43: (f /. 1) `1 <= (N-min C) `1 by A11, A33, A37, EUCLID:52;
then A44: (N-min (L~ f)) `1 <= (N-min C) `1 by A38, XXREAL_0:2;
A45: 1 <= m by A3, FINSEQ_3:25;
A46: n in NAT by ORDINAL1:def_12;
then A47: ((Gauge (C,n)) * (i9,((len (Gauge (C,n))) -' 1))) `2 = N-bound C by A9, A34, JORDAN8:14;
A48: N-min C = |[((N-min C) `1),((N-min C) `2)]| by EUCLID:53;
A49: ( (NW-corner C) `2 = N-bound C & (NE-corner C) `2 = N-bound C ) by EUCLID:52;
A50: ( (NW-corner C) `1 = W-bound C & (NE-corner C) `1 = E-bound C ) by EUCLID:52;
A51: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
 ((Gauge (C,n)) * (i9,((len (Gauge (C,n))) -' 1))) `1 = ((Gauge (C,n)) * (i9,1)) `1 by A9, A34, A33, A25, A26, GOBOARD5:2;
then A52: ((Gauge (C,n)) * (i9,((len (Gauge (C,n))) -' 1))) `1 < (N-min C) `1 by A11, A14, A33, A37, A43, A48, A15, A47, XXREAL_0:1;
A53: ((Gauge (C,n)) * (i,(len (Gauge (C,n))))) `1 = ((Gauge (C,n)) * (i,1)) `1 by A19, A27, A22, A28, A36, GOBOARD5:2;
percases ( m = len f or m < len f ) by A21, XXREAL_0:1;
suppose m = len f ; ::_thesis: N-min (L~ f) = f /. 1
hence  N-min (L~ f) = f /. 1 by A4, A7, FINSEQ_6:def_1; ::_thesis: verum
end;
suppose m < len f ; ::_thesis: N-min (L~ f) = f /. 1
then A54: m + 1 <= len f by NAT_1:13;
then consider i1, j1, i2, j2 being Element of NAT  such that
A55: ( [i1,j1] in Indices (Gauge (C,n)) & f /. m = (Gauge (C,n)) * (i1,j1) ) and
A56: [i2,j2] in Indices (Gauge (C,n)) and
A57: f /. (m + 1) = (Gauge (C,n)) * (i2,j2) and
A58: ( ( 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, A45, JORDAN8:3;
A59: right_cell (f,m,(Gauge (C,n))) meets C by A2, A45, A54;
then consider p being set  such that
A60: p in right_cell (f,m,(Gauge (C,n))) and
A61: p in C by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A60;
A62: ( W-bound C <= p `1 & p `1 <= E-bound C ) by A61, PSCOMP_1:24;
A63: (N-min C) `2 = N-bound C by EUCLID:52;
then A64: p `2 <= (N-min C) `2 by A61, PSCOMP_1:24;
A65: ((Gauge (C,n)) * (1,((len (Gauge (C,n))) -' 1))) `2 < ((Gauge (C,n)) * (1,(len (Gauge (C,n))))) `2 by A51, A24, A25, A26, GOBOARD5:4;
A66: ((Gauge (C,n)) * (1,((len (Gauge (C,n))) -' 1))) `2 = N-bound C by A24, A46, JORDAN8:14;
A67: j2 <= len (Gauge (C,n)) by A51, A56, MATRIX_1:38;
now__::_thesis:_N-min_(L~_f)_=_f_/._1
percases ( ( i = i2 & (len (Gauge (C,n))) + 1 = j2 ) or ( i + 1 = i2 & len (Gauge (C,n)) = j2 ) or ( i = i2 + 1 & len (Gauge (C,n)) = j2 ) or ( i = i2 & len (Gauge (C,n)) = j2 + 1 ) ) by A5, A6, A36, A55, A58, GOBOARD1:5;
suppose ( i = i2 & (len (Gauge (C,n))) + 1 = j2 ) ; ::_thesis: N-min (L~ f) = f /. 1
hence  N-min (L~ f) = f /. 1 by A67, NAT_1:13; ::_thesis: verum
end;
supposeA68: ( i + 1 = i2 & len (Gauge (C,n)) = j2 ) ; ::_thesis: N-min (L~ f) = f /. 1
A69: cell ((Gauge (C,n)),i,((len (Gauge (C,n))) -' 1)) =  {  |[r,s]| where r, s is Real : ( ((Gauge (C,n)) * (i,1)) `1 <= r & r <= ((Gauge (C,n)) * ((i + 1),1)) `1 & ((Gauge (C,n)) * (1,((len (Gauge (C,n))) -' 1))) `2 <= s & s <= ((Gauge (C,n)) * (1,(((len (Gauge (C,n))) -' 1) + 1))) `2 )  }  by A51, A19, A25, A26, A40, GOBRD11:32;
 right_cell (f,m,(Gauge (C,n))) = cell ((Gauge (C,n)),i,((len (Gauge (C,n))) -' 1)) by A1, A45, A5, A6, A36, A54, A56, A57, A68, GOBRD13:24;
then consider r, s being Real such that
A70: p = |[r,s]| and
 ((Gauge (C,n)) * (i,1)) `1 <= r and
A71: r <= ((Gauge (C,n)) * ((i + 1),1)) `1 and
A72: ((Gauge (C,n)) * (1,((len (Gauge (C,n))) -' 1))) `2 <= s and
 s <= ((Gauge (C,n)) * (1,(((len (Gauge (C,n))) -' 1) + 1))) `2 by A60, A69;
 p `2 = s by A70, EUCLID:52;
then  p `2 = N-bound C by A63, A64, A66, A72, XXREAL_0:1;
then  p in LSeg ((NW-corner C),(NE-corner C)) by A50, A49, A62, GOBOARD7:8;
then  p in (LSeg ((NW-corner C),(NE-corner C))) /\ C by A61, XBOOLE_0:def_4;
then A73: (N-min C) `1 <= p `1 by PSCOMP_1:39;
 p `1 = r by A70, EUCLID:52;
then  (N-min C) `1 <= ((Gauge (C,n)) * ((i + 1),1)) `1 by A71, A73, XXREAL_0:2;
then A74: N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) by A33, A4, A7, A6, A36, A53, A42, A44, A48, A63, A66, A65, A69;
 N-min C <> (Gauge (C,n)) * (i,((len (Gauge (C,n))) -' 1)) by A34, A33, A19, A25, A26, A52, A39, SPRECT_3:13;
hence  N-min (L~ f) = f /. 1 by A9, A10, A11, A13, A14, A33, A4, A7, A6, A19, A36, A41, A74, Th29; ::_thesis: verum
end;
suppose ( i = i2 + 1 & len (Gauge (C,n)) = j2 ) ; ::_thesis: N-min (L~ f) = f /. 1
then  ( right_cell (f,m,(Gauge (C,n))) = cell ((Gauge (C,n)),i2,(len (Gauge (C,n)))) & i2 < len (Gauge (C,n)) ) by A1, A45, A5, A6, A27, A36, A54, A56, A57, GOBRD13:26, NAT_1:13;
hence  N-min (L~ f) = f /. 1 by A2, A45, A54, JORDAN8:15; ::_thesis: verum
end;
supposeA75: ( i = i2 & len (Gauge (C,n)) = j2 + 1 ) ; ::_thesis: N-min (L~ f) = f /. 1
then A76: j2 = (len (Gauge (C,n))) -' 1 by NAT_D:34;
then A77: right_cell (f,m,(Gauge (C,n))) = cell ((Gauge (C,n)),(i -' 1),((len (Gauge (C,n))) -' 1)) by A1, A45, A5, A6, A36, A54, A56, A57, A75, GOBRD13:28;
 m -' 1 <= m by NAT_D:35;
then A78: m -' 1 <= len f by A21, XXREAL_0:2;
now__::_thesis:_not_m_=_1
 1 <= i9 + 1 by A9, NAT_1:13;
then A79: ((Gauge (C,n)) * ((i9 + 1),(len (Gauge (C,n))))) `2 = ((Gauge (C,n)) * (1,(len (Gauge (C,n))))) `2 by A10, A33, A24, GOBOARD5:1;
assume A80: m = 1 ; ::_thesis: contradiction
 ((Gauge (C,n)) * (i9,(len (Gauge (C,n))))) `2 = ((Gauge (C,n)) * (1,(len (Gauge (C,n))))) `2 by A9, A34, A33, A24, GOBOARD5:1;
hence  contradiction by A11, A12, A33, A6, A19, A27, A36, A25, A26, A57, A75, A76, A80, A79, GOBOARD5:4; ::_thesis: verum
end;
then  m > 1 by A45, XXREAL_0:1;
then A81: m -' 1 >= 1 by NAT_D:49;
A82: (m -' 1) + 1 = m by A45, XREAL_1:235;
then consider i19, j19, i29, j29 being Element of NAT  such that
A83: [i19,j19] in Indices (Gauge (C,n)) and
A84: f /. (m -' 1) = (Gauge (C,n)) * (i19,j19) and
A85: ( [i29,j29] in Indices (Gauge (C,n)) & f /. m = (Gauge (C,n)) * (i29,j29) & ( ( i19 = i29 & j19 + 1 = j29 ) or ( i19 + 1 = i29 & j19 = j29 ) or ( i19 = i29 + 1 & j19 = j29 ) or ( i19 = i29 & j19 = j29 + 1 ) ) ) by A1, A21, A81, JORDAN8:3;
A86: 1 <= i19 by A83, MATRIX_1:38;
A87: i19 <= len (Gauge (C,n)) by A83, MATRIX_1:38;
now__::_thesis:_contradiction
percases ( ( i19 = i & j19 + 1 = len (Gauge (C,n)) ) or ( i19 + 1 = i & j19 = len (Gauge (C,n)) ) or ( i19 = i + 1 & j19 = len (Gauge (C,n)) ) or ( i19 = i & j19 = (len (Gauge (C,n))) + 1 ) ) by A5, A6, A36, A85, GOBOARD1:5;
supposeA88: ( i19 = i & j19 + 1 = len (Gauge (C,n)) ) ; ::_thesis: contradiction
then  j19 = (len (Gauge (C,n))) -' 1 by NAT_D:34;
then  left_cell (f,(m -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),(i -' 1),((len (Gauge (C,n))) -' 1)) by A1, A21, A5, A6, A36, A81, A82, A83, A84, A88, GOBRD13:21;
hence  contradiction by A2, A21, A59, A77, A81, A82; ::_thesis: verum
end;
supposeA89: ( i19 + 1 = i & j19 = len (Gauge (C,n)) ) ; ::_thesis: contradiction
A90: ( ((Gauge (C,n)) * (i19,j)) `2 = ((Gauge (C,n)) * (1,j)) `2 & ((Gauge (C,n)) * (i,j)) `2 = ((Gauge (C,n)) * (1,j)) `2 ) by A19, A27, A22, A28, A86, A87, GOBOARD5:1;
 m -' 1 in dom f by A81, A78, FINSEQ_3:25;
then A91: f /. (m -' 1) in L~ f by A30, GOBOARD1:1, XXREAL_0:2;
then  ( W-bound (L~ f) <= (f /. (m -' 1)) `1 & (f /. (m -' 1)) `1 <= E-bound (L~ f) ) by PSCOMP_1:24;
then  f /. (m -' 1) in LSeg ((NW-corner (L~ f)),(NE-corner (L~ f))) by A4, A7, A8, A6, A36, A18, A20, A84, A89, A90, GOBOARD7:8;
then A92: f /. (m -' 1) in (LSeg ((NW-corner (L~ f)),(NE-corner (L~ f)))) /\ (L~ f) by A91, XBOOLE_0:def_4;
 i19 < i by A89, NAT_1:13;
then  (f /. (m -' 1)) `1 < (f /. m) `1 by A6, A27, A22, A28, A36, A84, A86, A89, GOBOARD5:3;
hence  contradiction by A4, A7, A92, PSCOMP_1:39; ::_thesis: verum
end;
suppose ( i19 = i + 1 & j19 = len (Gauge (C,n)) ) ; ::_thesis: contradiction
then  right_cell (f,(m -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),i,(len (Gauge (C,n)))) by A1, A21, A5, A6, A36, A81, A82, A83, A84, GOBRD13:26;
hence  contradiction by A2, A21, A27, A81, A82, JORDAN8:15; ::_thesis: verum
end;
suppose ( i19 = i & j19 = (len (Gauge (C,n))) + 1 ) ; ::_thesis: contradiction
then  (len (Gauge (C,n))) + 1 <= (len (Gauge (C,n))) + 0 by A51, A83, MATRIX_1:38;
hence  contradiction by XREAL_1:6; ::_thesis: verum
end;
end;
end;
hence  N-min (L~ f) = f /. 1 ; ::_thesis: verum
end;
end;
end;
hence  N-min (L~ f) = f /. 1 ; ::_thesis: verum
end;
end;
end;

definition
let C be non empty compact non horizontal non vertical Subset of (TOP-REAL 2);
let n be Nat;
assume A1: C is connected ;
func Cage (C,n) ->  non constant standard clockwise_oriented special_circular_sequence means :Def1: :: JORDAN9:def 1
( it is_sequence_on Gauge (C,n) & ex i being Element of NAT st
( 1 <= i & i + 1 <= len (Gauge (C,n)) & it /. 1 = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) & it /. 2 = (Gauge (C,n)) * ((i + 1),(width (Gauge (C,n)))) & N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) & N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1)) ) & ( for k being Element of NAT st 1 <= k & k + 2 <= len it holds
( ( front_left_cell (it,k,(Gauge (C,n))) misses C & front_right_cell (it,k,(Gauge (C,n))) misses C implies it turns_right k, Gauge (C,n) ) & ( front_left_cell (it,k,(Gauge (C,n))) misses C & front_right_cell (it,k,(Gauge (C,n))) meets C implies it goes_straight k, Gauge (C,n) ) & ( front_left_cell (it,k,(Gauge (C,n))) meets C implies it turns_left k, Gauge (C,n) ) ) ) );
existence
 ex b1 being non constant standard clockwise_oriented special_circular_sequence st
( b1 is_sequence_on Gauge (C,n) & ex i being Element of NAT st
( 1 <= i & i + 1 <= len (Gauge (C,n)) & b1 /. 1 = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) & b1 /. 2 = (Gauge (C,n)) * ((i + 1),(width (Gauge (C,n)))) & N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) & N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1)) ) & ( for k being Element of NAT st 1 <= k & k + 2 <= len b1 holds
( ( front_left_cell (b1,k,(Gauge (C,n))) misses C & front_right_cell (b1,k,(Gauge (C,n))) misses C implies b1 turns_right k, Gauge (C,n) ) & ( front_left_cell (b1,k,(Gauge (C,n))) misses C & front_right_cell (b1,k,(Gauge (C,n))) meets C implies b1 goes_straight k, Gauge (C,n) ) & ( front_left_cell (b1,k,(Gauge (C,n))) meets C implies b1 turns_left k, Gauge (C,n) ) ) ) )
proof
set W = W-bound C;
set E = E-bound C;
set S = S-bound C;
set N = N-bound C;
set G = Gauge (C,n);
defpred S1[ Element of NAT , set , set ] means ( ( $1 = 0 implies ex i being Element of NAT st
( 1 <= i & i + 1 <= len (Gauge (C,n)) & N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) & N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1)) & $3 = <*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> ) ) & ( $1 = 1 implies ex i being Element of NAT st
( 1 <= i & i + 1 <= len (Gauge (C,n)) & N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) & N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1)) & $3 = <*((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * ((i + 1),(width (Gauge (C,n)))))*> ) ) & ( $1 > 1 & $2 is FinSequence of (TOP-REAL 2) implies for f being FinSequence of (TOP-REAL 2) st $2 = f holds
( ( len f = $1 implies ( ( f is_sequence_on Gauge (C,n) & right_cell (f,((len f) -' 1),(Gauge (C,n))) meets C implies ( ( front_left_cell (f,((len f) -' 1),(Gauge (C,n))) misses C & front_right_cell (f,((len f) -' 1),(Gauge (C,n))) misses C implies ex i, j being Element of NAT st
( f ^ <*((Gauge (C,n)) * (i,j))*> turns_right (len f) -' 1, Gauge (C,n) & $3 = f ^ <*((Gauge (C,n)) * (i,j))*> ) ) & ( front_left_cell (f,((len f) -' 1),(Gauge (C,n))) misses C & front_right_cell (f,((len f) -' 1),(Gauge (C,n))) meets C implies ex i, j being Element of NAT st
( f ^ <*((Gauge (C,n)) * (i,j))*> goes_straight (len f) -' 1, Gauge (C,n) & $3 = f ^ <*((Gauge (C,n)) * (i,j))*> ) ) & ( front_left_cell (f,((len f) -' 1),(Gauge (C,n))) meets C implies ex i, j being Element of NAT st
( f ^ <*((Gauge (C,n)) * (i,j))*> turns_left (len f) -' 1, Gauge (C,n) & $3 = f ^ <*((Gauge (C,n)) * (i,j))*> ) ) ) ) & ( ( not f is_sequence_on Gauge (C,n) or right_cell (f,((len f) -' 1),(Gauge (C,n))) misses C ) implies $3 = f ^ <*((Gauge (C,n)) * (1,1))*> ) ) ) & ( len f <> $1 implies $3 = {} ) ) ) & ( $1 > 1 & $2 is not FinSequence of (TOP-REAL 2) implies $3 = {} ) );
A2: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
A3: for k being Element of NAT
for x being set ex y being set st S1[k,x,y]
proof
let k be Element of NAT ; ::_thesis: for x being set ex y being set st S1[k,x,y]
let x be set ; ::_thesis: ex y being set st S1[k,x,y]
consider m being Element of NAT  such that
A4: ( 1 <= m & m + 1 <= len (Gauge (C,n)) & N-min C in cell ((Gauge (C,n)),m,((width (Gauge (C,n))) -' 1)) & N-min C <> (Gauge (C,n)) * (m,((width (Gauge (C,n))) -' 1)) ) by Th28;
percases ( k = 0 or k = 1 or ( k > 1 & x is FinSequence of (TOP-REAL 2) ) or ( k > 1 & x is not FinSequence of (TOP-REAL 2) ) ) by NAT_1:25;
supposeA5: k = 0 ; ::_thesis: ex y being set st S1[k,x,y]
take  <*((Gauge (C,n)) * (m,(width (Gauge (C,n)))))*> ; ::_thesis: S1[k,x,<*((Gauge (C,n)) * (m,(width (Gauge (C,n)))))*>]
thus  S1[k,x,<*((Gauge (C,n)) * (m,(width (Gauge (C,n)))))*>] by A4, A5; ::_thesis: verum
end;
supposeA6: k = 1 ; ::_thesis: ex y being set st S1[k,x,y]
take  <*((Gauge (C,n)) * (m,(width (Gauge (C,n))))),((Gauge (C,n)) * ((m + 1),(width (Gauge (C,n)))))*> ; ::_thesis: S1[k,x,<*((Gauge (C,n)) * (m,(width (Gauge (C,n))))),((Gauge (C,n)) * ((m + 1),(width (Gauge (C,n)))))*>]
thus  S1[k,x,<*((Gauge (C,n)) * (m,(width (Gauge (C,n))))),((Gauge (C,n)) * ((m + 1),(width (Gauge (C,n)))))*>] by A4, A6; ::_thesis: verum
end;
supposethat A7: k > 1 and
A8: x is FinSequence of (TOP-REAL 2) ; ::_thesis: ex y being set st S1[k,x,y]
reconsider f = x as FinSequence of (TOP-REAL 2) by A8;
thus  ex y being set st S1[k,x,y] ::_thesis: verum
proof
percases ( len f = k or len f <> k ) ;
supposeA9: len f = k ; ::_thesis: ex y being set st S1[k,x,y]
thus  ex y being set st S1[k,x,y] ::_thesis: verum
proof
percases ( ( f is_sequence_on Gauge (C,n) & right_cell (f,((len f) -' 1),(Gauge (C,n))) meets C ) or not f is_sequence_on Gauge (C,n) or right_cell (f,((len f) -' 1),(Gauge (C,n))) misses C ) ;
supposeA10: ( f is_sequence_on Gauge (C,n) & right_cell (f,((len f) -' 1),(Gauge (C,n))) meets C ) ; ::_thesis: ex y being set st S1[k,x,y]
A11: ((len f) -' 1) + 1 = len f by A7, A9, XREAL_1:235;
then A12: ((len f) -' 1) + (1 + 1) = (len f) + 1 ;
A13: ((len f) -' 1) + 1 in dom f by A7, A9, A11, FINSEQ_3:25;
A14: 1 <= (len f) -' 1 by A7, A9, NAT_D:49;
then consider i1, j1, i2, j2 being Element of NAT  such that
A15: [i1,j1] in Indices (Gauge (C,n)) and
A16: f /. ((len f) -' 1) = (Gauge (C,n)) * (i1,j1) and
A17: [i2,j2] in Indices (Gauge (C,n)) and
A18: f /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i2,j2) and
A19: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A10, A11, JORDAN8:3;
A20: i1 <= len (Gauge (C,n)) by A15, MATRIX_1:38;
A21: 1 <= j2 + 1 by NAT_1:12;
A22: 1 <= i2 by A17, MATRIX_1:38;
A23: j1 <= width (Gauge (C,n)) by A15, MATRIX_1:38;
A24: 1 <= i2 + 1 by NAT_1:12;
A25: 1 <= j2 by A17, MATRIX_1:38;
 (len f) -' 1 <= len f by NAT_D:35;
then A26: (len f) -' 1 in dom f by A14, FINSEQ_3:25;
A27: j2 <= width (Gauge (C,n)) by A17, MATRIX_1:38;
then A28: j2 -' 1 <= width (Gauge (C,n)) by NAT_D:44;
A29: i2 <= len (Gauge (C,n)) by A17, MATRIX_1:38;
then A30: i2 -' 1 <= len (Gauge (C,n)) by NAT_D:44;
thus  ex y being set st S1[k,x,y] ::_thesis: verum
proof
percases ( ( front_left_cell (f,((len f) -' 1),(Gauge (C,n))) misses C & front_right_cell (f,((len f) -' 1),(Gauge (C,n))) misses C ) or ( front_left_cell (f,((len f) -' 1),(Gauge (C,n))) misses C & front_right_cell (f,((len f) -' 1),(Gauge (C,n))) meets C ) or front_left_cell (f,((len f) -' 1),(Gauge (C,n))) meets C ) ;
supposeA31: ( front_left_cell (f,((len f) -' 1),(Gauge (C,n))) misses C & front_right_cell (f,((len f) -' 1),(Gauge (C,n))) misses C ) ; ::_thesis: ex y being set st S1[k,x,y]
thus  ex y being set st S1[k,x,y] ::_thesis: verum
proof
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 A19;
supposeA32: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: ex y being set st S1[k,x,y]
take f1 = f ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*>; ::_thesis: S1[k,x,f1]
now__::_thesis:_ex_i_being_Element_of_NAT_ex_j_being_Element_of_NAT_st_f1_turns_right_(len_f)_-'_1,_Gauge_(C,n)
take i = i2 + 1; ::_thesis: ex j being Element of NAT st f1 turns_right (len f) -' 1, Gauge (C,n)
take j = j2; ::_thesis: f1 turns_right (len f) -' 1, Gauge (C,n)
thus  f1 turns_right (len f) -' 1, Gauge (C,n) ::_thesis: verum
proof
let i19, j19, i29, j29 be Element of NAT ; :: according to GOBRD13:def_6 ::_thesis: ( not [i19,j19] in Indices (Gauge (C,n)) or not [i29,j29] in Indices (Gauge (C,n)) or not f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) or not f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) or ( i19 = i29 & j19 + 1 = j29 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) or ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) or ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 = i29 & j19 = j29 + 1 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) )
assume that
A33: [i19,j19] in Indices (Gauge (C,n)) and
A34: [i29,j29] in Indices (Gauge (C,n)) and
A35: f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) and
A36: f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) ; ::_thesis: ( ( i19 = i29 & j19 + 1 = j29 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) or ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) or ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 = i29 & j19 = j29 + 1 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) )
A37: f /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) by A26, A35, FINSEQ_4:68;
then A38: i1 = i19 by A15, A16, A33, GOBOARD1:5;
A39: j1 = j19 by A15, A16, A33, A37, GOBOARD1:5;
A40: f /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) by A13, A36, FINSEQ_4:68;
then A41: i2 = i29 by A17, A18, A34, GOBOARD1:5;
A42: j2 = j29 by A17, A18, A34, A40, GOBOARD1:5;
percases ( ( i19 = i29 & j19 + 1 = j29 ) or ( i19 + 1 = i29 & j19 = j29 ) or ( i19 = i29 + 1 & j19 = j29 ) or ( i19 = i29 & j19 = j29 + 1 ) ) by A15, A16, A19, A33, A37, A41, A42, GOBOARD1:5;
case ( i19 = i29 & j19 + 1 = j29 ) ; ::_thesis: ( [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) )
now__::_thesis:_not_i2_+_1_>_len_(Gauge_(C,n))
assume  i2 + 1 > len (Gauge (C,n)) ; ::_thesis: contradiction
then A43: (len (Gauge (C,n))) + 1 <= i2 + 1 by NAT_1:13;
 i2 + 1 <= (len (Gauge (C,n))) + 1 by A29, XREAL_1:6;
then  i2 + 1 = (len (Gauge (C,n))) + 1 by A43, XXREAL_0:1;
then  cell ((Gauge (C,n)),(len (Gauge (C,n))),j1) meets C by A10, A14, A11, A15, A16, A17, A18, A32, GOBRD13:22;
hence  contradiction by A2, A23, JORDAN8:16; ::_thesis: verum
end;
hence  [(i29 + 1),j29] in Indices (Gauge (C,n)) by A25, A27, A24, A41, A42, MATRIX_1:36; ::_thesis: f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29)
thus  f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) by A12, A41, A42, FINSEQ_4:67; ::_thesis: verum
end;
case ( i19 + 1 = i29 & j19 = j29 ) ; ::_thesis: ( [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) )
hence  ( [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) by A32, A38, A41; ::_thesis: verum
end;
case ( i19 = i29 + 1 & j19 = j29 ) ; ::_thesis: ( [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) )
hence  ( [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) by A32, A38, A41; ::_thesis: verum
end;
case ( i19 = i29 & j19 = j29 + 1 ) ; ::_thesis: ( [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) )
hence  ( [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) by A32, A39, A42; ::_thesis: verum
end;
end;
end;
end;
hence  S1[k,x,f1] by A7, A9, A10, A31; ::_thesis: verum
end;
supposeA44: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: ex y being set st S1[k,x,y]
take f1 = f ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*>; ::_thesis: S1[k,x,f1]
now__::_thesis:_ex_i_being_Element_of_NAT_ex_j_being_Element_of_NAT_st_f1_turns_right_(len_f)_-'_1,_Gauge_(C,n)
take i = i2; ::_thesis: ex j being Element of NAT st f1 turns_right (len f) -' 1, Gauge (C,n)
take j = j2 -' 1; ::_thesis: f1 turns_right (len f) -' 1, Gauge (C,n)
thus  f1 turns_right (len f) -' 1, Gauge (C,n) ::_thesis: verum
proof
let i19, j19, i29, j29 be Element of NAT ; :: according to GOBRD13:def_6 ::_thesis: ( not [i19,j19] in Indices (Gauge (C,n)) or not [i29,j29] in Indices (Gauge (C,n)) or not f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) or not f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) or ( i19 = i29 & j19 + 1 = j29 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) or ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) or ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 = i29 & j19 = j29 + 1 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) )
assume that
A45: [i19,j19] in Indices (Gauge (C,n)) and
A46: [i29,j29] in Indices (Gauge (C,n)) and
A47: f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) and
A48: f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) ; ::_thesis: ( ( i19 = i29 & j19 + 1 = j29 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) or ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) or ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 = i29 & j19 = j29 + 1 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) )
A49: f /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) by A13, A48, FINSEQ_4:68;
then A50: i2 = i29 by A17, A18, A46, GOBOARD1:5;
A51: f /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) by A26, A47, FINSEQ_4:68;
then A52: i1 = i19 by A15, A16, A45, GOBOARD1:5;
A53: j2 = j29 by A17, A18, A46, A49, GOBOARD1:5;
percases ( ( i19 = i29 & j19 + 1 = j29 ) or ( i19 + 1 = i29 & j19 = j29 ) or ( i19 = i29 + 1 & j19 = j29 ) or ( i19 = i29 & j19 = j29 + 1 ) ) by A15, A16, A19, A45, A51, A50, A53, GOBOARD1:5;
case ( i19 = i29 & j19 + 1 = j29 ) ; ::_thesis: ( [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) )
hence  ( [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) by A44, A52, A50; ::_thesis: verum
end;
case ( i19 + 1 = i29 & j19 = j29 ) ; ::_thesis: ( [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) )
now__::_thesis:_not_j2_-'_1_<_1
assume  j2 -' 1 < 1 ; ::_thesis: contradiction
then  j2 <= 1 by NAT_1:14, NAT_D:36;
then  j2 = 1 by A25, XXREAL_0:1;
then  cell ((Gauge (C,n)),i1,(1 -' 1)) meets C by A10, A14, A11, A15, A16, A17, A18, A44, GOBRD13:24;
then  cell ((Gauge (C,n)),i1,0) meets C by XREAL_1:232;
hence  contradiction by A20, JORDAN8:17; ::_thesis: verum
end;
hence  [i29,(j29 -' 1)] in Indices (Gauge (C,n)) by A22, A29, A28, A50, A53, MATRIX_1:36; ::_thesis: f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1))
thus  f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) by A12, A50, A53, FINSEQ_4:67; ::_thesis: verum
end;
case ( i19 = i29 + 1 & j19 = j29 ) ; ::_thesis: ( [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) )
hence  ( [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) by A44, A52, A50; ::_thesis: verum
end;
case ( i19 = i29 & j19 = j29 + 1 ) ; ::_thesis: ( [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) )
hence  ( [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) by A44, A52, A50; ::_thesis: verum
end;
end;
end;
end;
hence  S1[k,x,f1] by A7, A9, A10, A31; ::_thesis: verum
end;
supposeA54: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: ex y being set st S1[k,x,y]
take f1 = f ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*>; ::_thesis: S1[k,x,f1]
now__::_thesis:_ex_i_being_Element_of_NAT_ex_j_being_Element_of_NAT_st_f1_turns_right_(len_f)_-'_1,_Gauge_(C,n)
take i = i2; ::_thesis: ex j being Element of NAT st f1 turns_right (len f) -' 1, Gauge (C,n)
take j = j2 + 1; ::_thesis: f1 turns_right (len f) -' 1, Gauge (C,n)
thus  f1 turns_right (len f) -' 1, Gauge (C,n) ::_thesis: verum
proof
let i19, j19, i29, j29 be Element of NAT ; :: according to GOBRD13:def_6 ::_thesis: ( not [i19,j19] in Indices (Gauge (C,n)) or not [i29,j29] in Indices (Gauge (C,n)) or not f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) or not f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) or ( i19 = i29 & j19 + 1 = j29 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) or ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) or ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 = i29 & j19 = j29 + 1 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) )
assume that
A55: [i19,j19] in Indices (Gauge (C,n)) and
A56: [i29,j29] in Indices (Gauge (C,n)) and
A57: f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) and
A58: f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) ; ::_thesis: ( ( i19 = i29 & j19 + 1 = j29 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) or ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) or ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 = i29 & j19 = j29 + 1 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) )
A59: f /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) by A13, A58, FINSEQ_4:68;
then A60: i2 = i29 by A17, A18, A56, GOBOARD1:5;
A61: f /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) by A26, A57, FINSEQ_4:68;
then A62: i1 = i19 by A15, A16, A55, GOBOARD1:5;
A63: j2 = j29 by A17, A18, A56, A59, GOBOARD1:5;
percases ( ( i19 = i29 & j19 + 1 = j29 ) or ( i19 + 1 = i29 & j19 = j29 ) or ( i19 = i29 + 1 & j19 = j29 ) or ( i19 = i29 & j19 = j29 + 1 ) ) by A15, A16, A19, A55, A61, A60, A63, GOBOARD1:5;
case ( i19 = i29 & j19 + 1 = j29 ) ; ::_thesis: ( [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) )
hence  ( [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) by A54, A62, A60; ::_thesis: verum
end;
case ( i19 + 1 = i29 & j19 = j29 ) ; ::_thesis: ( [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) )
hence  ( [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) by A54, A62, A60; ::_thesis: verum
end;
case ( i19 = i29 + 1 & j19 = j29 ) ; ::_thesis: ( [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) )
now__::_thesis:_not_j2_+_1_>_len_(Gauge_(C,n))
assume  j2 + 1 > len (Gauge (C,n)) ; ::_thesis: contradiction
then A64: (len (Gauge (C,n))) + 1 <= j2 + 1 by NAT_1:13;
 j2 + 1 <= (len (Gauge (C,n))) + 1 by A2, A27, XREAL_1:6;
then  j2 + 1 = (len (Gauge (C,n))) + 1 by A64, XXREAL_0:1;
then  cell ((Gauge (C,n)),i2,(len (Gauge (C,n)))) meets C by A10, A14, A11, A15, A16, A17, A18, A54, GOBRD13:26;
hence  contradiction by A29, JORDAN8:15; ::_thesis: verum
end;
hence  [i29,(j29 + 1)] in Indices (Gauge (C,n)) by A2, A22, A29, A21, A60, A63, MATRIX_1:36; ::_thesis: f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1))
thus  f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) by A12, A60, A63, FINSEQ_4:67; ::_thesis: verum
end;
case ( i19 = i29 & j19 = j29 + 1 ) ; ::_thesis: ( [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) )
hence  ( [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) by A54, A62, A60; ::_thesis: verum
end;
end;
end;
end;
hence  S1[k,x,f1] by A7, A9, A10, A31; ::_thesis: verum
end;
supposeA65: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: ex y being set st S1[k,x,y]
take f1 = f ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*>; ::_thesis: S1[k,x,f1]
now__::_thesis:_ex_i_being_Element_of_NAT_ex_j_being_Element_of_NAT_st_f1_turns_right_(len_f)_-'_1,_Gauge_(C,n)
take i = i2 -' 1; ::_thesis: ex j being Element of NAT st f1 turns_right (len f) -' 1, Gauge (C,n)
take j = j2; ::_thesis: f1 turns_right (len f) -' 1, Gauge (C,n)
thus  f1 turns_right (len f) -' 1, Gauge (C,n) ::_thesis: verum
proof
let i19, j19, i29, j29 be Element of NAT ; :: according to GOBRD13:def_6 ::_thesis: ( not [i19,j19] in Indices (Gauge (C,n)) or not [i29,j29] in Indices (Gauge (C,n)) or not f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) or not f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) or ( i19 = i29 & j19 + 1 = j29 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) or ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) or ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 = i29 & j19 = j29 + 1 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) )
assume that
A66: [i19,j19] in Indices (Gauge (C,n)) and
A67: [i29,j29] in Indices (Gauge (C,n)) and
A68: f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) and
A69: f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) ; ::_thesis: ( ( i19 = i29 & j19 + 1 = j29 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) or ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) or ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 = i29 & j19 = j29 + 1 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) )
A70: f /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) by A26, A68, FINSEQ_4:68;
then A71: i1 = i19 by A15, A16, A66, GOBOARD1:5;
A72: j1 = j19 by A15, A16, A66, A70, GOBOARD1:5;
A73: f /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) by A13, A69, FINSEQ_4:68;
then A74: i2 = i29 by A17, A18, A67, GOBOARD1:5;
A75: j2 = j29 by A17, A18, A67, A73, GOBOARD1:5;
percases ( ( i19 = i29 & j19 + 1 = j29 ) or ( i19 + 1 = i29 & j19 = j29 ) or ( i19 = i29 + 1 & j19 = j29 ) or ( i19 = i29 & j19 = j29 + 1 ) ) by A15, A16, A19, A66, A70, A74, A75, GOBOARD1:5;
case ( i19 = i29 & j19 + 1 = j29 ) ; ::_thesis: ( [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) )
hence  ( [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) by A65, A72, A75; ::_thesis: verum
end;
case ( i19 + 1 = i29 & j19 = j29 ) ; ::_thesis: ( [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) )
hence  ( [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) by A65, A71, A74; ::_thesis: verum
end;
case ( i19 = i29 + 1 & j19 = j29 ) ; ::_thesis: ( [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) )
hence  ( [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) by A65, A71, A74; ::_thesis: verum
end;
case ( i19 = i29 & j19 = j29 + 1 ) ; ::_thesis: ( [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) )
now__::_thesis:_not_i2_-'_1_<_1
assume  i2 -' 1 < 1 ; ::_thesis: contradiction
then  i2 <= 1 by NAT_1:14, NAT_D:36;
then  i2 = 1 by A22, XXREAL_0:1;
then  cell ((Gauge (C,n)),(1 -' 1),j2) meets C by A10, A14, A11, A15, A16, A17, A18, A65, GOBRD13:28;
then  cell ((Gauge (C,n)),0,j2) meets C by XREAL_1:232;
hence  contradiction by A2, A27, JORDAN8:18; ::_thesis: verum
end;
hence  [(i29 -' 1),j29] in Indices (Gauge (C,n)) by A25, A27, A30, A74, A75, MATRIX_1:36; ::_thesis: f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29)
thus  f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) by A12, A74, A75, FINSEQ_4:67; ::_thesis: verum
end;
end;
end;
end;
hence  S1[k,x,f1] by A7, A9, A10, A31; ::_thesis: verum
end;
end;
end;
end;
supposeA76: ( front_left_cell (f,((len f) -' 1),(Gauge (C,n))) misses C & front_right_cell (f,((len f) -' 1),(Gauge (C,n))) meets C ) ; ::_thesis: ex y being set st S1[k,x,y]
thus  ex y being set st S1[k,x,y] ::_thesis: verum
proof
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 A19;
supposeA77: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: ex y being set st S1[k,x,y]
take f1 = f ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*>; ::_thesis: S1[k,x,f1]
now__::_thesis:_ex_i_being_Element_of_NAT_ex_j_being_Element_of_NAT_st_f1_goes_straight_(len_f)_-'_1,_Gauge_(C,n)
take i = i2; ::_thesis: ex j being Element of NAT st f1 goes_straight (len f) -' 1, Gauge (C,n)
take j = j2 + 1; ::_thesis: f1 goes_straight (len f) -' 1, Gauge (C,n)
thus  f1 goes_straight (len f) -' 1, Gauge (C,n) ::_thesis: verum
proof
let i19, j19, i29, j29 be Element of NAT ; :: according to GOBRD13:def_8 ::_thesis: ( not [i19,j19] in Indices (Gauge (C,n)) or not [i29,j29] in Indices (Gauge (C,n)) or not f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) or not f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) or ( i19 = i29 & j19 + 1 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 + 1 = i29 & j19 = j29 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) or ( i19 = i29 + 1 & j19 = j29 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) or ( i19 = i29 & j19 = j29 + 1 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) )
assume that
A78: [i19,j19] in Indices (Gauge (C,n)) and
A79: [i29,j29] in Indices (Gauge (C,n)) and
A80: f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) and
A81: f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) ; ::_thesis: ( ( i19 = i29 & j19 + 1 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 + 1 = i29 & j19 = j29 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) or ( i19 = i29 + 1 & j19 = j29 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) or ( i19 = i29 & j19 = j29 + 1 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) )
A82: f /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) by A26, A80, FINSEQ_4:68;
then A83: i1 = i19 by A15, A16, A78, GOBOARD1:5;
A84: j1 = j19 by A15, A16, A78, A82, GOBOARD1:5;
A85: f /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) by A13, A81, FINSEQ_4:68;
then A86: i2 = i29 by A17, A18, A79, GOBOARD1:5;
A87: j2 = j29 by A17, A18, A79, A85, GOBOARD1:5;
percases ( ( i19 = i29 & j19 + 1 = j29 ) or ( i19 + 1 = i29 & j19 = j29 ) or ( i19 = i29 + 1 & j19 = j29 ) or ( i19 = i29 & j19 = j29 + 1 ) ) by A15, A16, A19, A78, A82, A86, A87, GOBOARD1:5;
case ( i19 = i29 & j19 + 1 = j29 ) ; ::_thesis: ( [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) )
now__::_thesis:_not_j2_+_1_>_len_(Gauge_(C,n))
assume  j2 + 1 > len (Gauge (C,n)) ; ::_thesis: contradiction
then A88: (len (Gauge (C,n))) + 1 <= j2 + 1 by NAT_1:13;
 j2 + 1 <= (len (Gauge (C,n))) + 1 by A2, A27, XREAL_1:6;
then  j2 + 1 = (len (Gauge (C,n))) + 1 by A88, XXREAL_0:1;
then  cell ((Gauge (C,n)),i1,(len (Gauge (C,n)))) meets C by A10, A14, A11, A15, A16, A17, A18, A76, A77, GOBRD13:35;
hence  contradiction by A20, JORDAN8:15; ::_thesis: verum
end;
hence  [i29,(j29 + 1)] in Indices (Gauge (C,n)) by A2, A22, A29, A21, A86, A87, MATRIX_1:36; ::_thesis: f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1))
thus  f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) by A12, A86, A87, FINSEQ_4:67; ::_thesis: verum
end;
case ( i19 + 1 = i29 & j19 = j29 ) ; ::_thesis: ( [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) )
hence  ( [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) by A77, A83, A86; ::_thesis: verum
end;
case ( i19 = i29 + 1 & j19 = j29 ) ; ::_thesis: ( [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) )
hence  ( [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) by A77, A83, A86; ::_thesis: verum
end;
case ( i19 = i29 & j19 = j29 + 1 ) ; ::_thesis: ( [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) )
hence  ( [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) by A77, A84, A87; ::_thesis: verum
end;
end;
end;
end;
hence  S1[k,x,f1] by A7, A9, A10, A76; ::_thesis: verum
end;
supposeA89: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: ex y being set st S1[k,x,y]
take f1 = f ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*>; ::_thesis: S1[k,x,f1]
now__::_thesis:_ex_i_being_Element_of_NAT_ex_j_being_Element_of_NAT_st_f1_goes_straight_(len_f)_-'_1,_Gauge_(C,n)
take i = i2 + 1; ::_thesis: ex j being Element of NAT st f1 goes_straight (len f) -' 1, Gauge (C,n)
take j = j2; ::_thesis: f1 goes_straight (len f) -' 1, Gauge (C,n)
thus  f1 goes_straight (len f) -' 1, Gauge (C,n) ::_thesis: verum
proof
let i19, j19, i29, j29 be Element of NAT ; :: according to GOBRD13:def_8 ::_thesis: ( not [i19,j19] in Indices (Gauge (C,n)) or not [i29,j29] in Indices (Gauge (C,n)) or not f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) or not f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) or ( i19 = i29 & j19 + 1 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 + 1 = i29 & j19 = j29 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) or ( i19 = i29 + 1 & j19 = j29 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) or ( i19 = i29 & j19 = j29 + 1 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) )
assume that
A90: [i19,j19] in Indices (Gauge (C,n)) and
A91: [i29,j29] in Indices (Gauge (C,n)) and
A92: f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) and
A93: f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) ; ::_thesis: ( ( i19 = i29 & j19 + 1 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 + 1 = i29 & j19 = j29 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) or ( i19 = i29 + 1 & j19 = j29 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) or ( i19 = i29 & j19 = j29 + 1 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) )
A94: f /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) by A13, A93, FINSEQ_4:68;
then A95: i2 = i29 by A17, A18, A91, GOBOARD1:5;
A96: f /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) by A26, A92, FINSEQ_4:68;
then A97: i1 = i19 by A15, A16, A90, GOBOARD1:5;
A98: j2 = j29 by A17, A18, A91, A94, GOBOARD1:5;
percases ( ( i19 = i29 & j19 + 1 = j29 ) or ( i19 + 1 = i29 & j19 = j29 ) or ( i19 = i29 + 1 & j19 = j29 ) or ( i19 = i29 & j19 = j29 + 1 ) ) by A15, A16, A19, A90, A96, A95, A98, GOBOARD1:5;
case ( i19 = i29 & j19 + 1 = j29 ) ; ::_thesis: ( [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) )
hence  ( [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) by A89, A97, A95; ::_thesis: verum
end;
case ( i19 + 1 = i29 & j19 = j29 ) ; ::_thesis: ( [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) )
now__::_thesis:_not_i2_+_1_>_len_(Gauge_(C,n))
assume  i2 + 1 > len (Gauge (C,n)) ; ::_thesis: contradiction
then A99: (len (Gauge (C,n))) + 1 <= i2 + 1 by NAT_1:13;
 i2 + 1 <= (len (Gauge (C,n))) + 1 by A29, XREAL_1:6;
then  i2 + 1 = (len (Gauge (C,n))) + 1 by A99, XXREAL_0:1;
then  cell ((Gauge (C,n)),(len (Gauge (C,n))),(j1 -' 1)) meets C by A10, A14, A11, A15, A16, A17, A18, A76, A89, GOBRD13:37;
hence  contradiction by A2, A23, JORDAN8:16, NAT_D:44; ::_thesis: verum
end;
hence  [(i29 + 1),j29] in Indices (Gauge (C,n)) by A25, A27, A24, A95, A98, MATRIX_1:36; ::_thesis: f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29)
thus  f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) by A12, A95, A98, FINSEQ_4:67; ::_thesis: verum
end;
case ( i19 = i29 + 1 & j19 = j29 ) ; ::_thesis: ( [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) )
hence  ( [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) by A89, A97, A95; ::_thesis: verum
end;
case ( i19 = i29 & j19 = j29 + 1 ) ; ::_thesis: ( [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) )
hence  ( [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) by A89, A97, A95; ::_thesis: verum
end;
end;
end;
end;
hence  S1[k,x,f1] by A7, A9, A10, A76; ::_thesis: verum
end;
supposeA100: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: ex y being set st S1[k,x,y]
take f1 = f ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*>; ::_thesis: S1[k,x,f1]
now__::_thesis:_ex_i_being_Element_of_NAT_ex_j_being_Element_of_NAT_st_f1_goes_straight_(len_f)_-'_1,_Gauge_(C,n)
take i = i2 -' 1; ::_thesis: ex j being Element of NAT st f1 goes_straight (len f) -' 1, Gauge (C,n)
take j = j2; ::_thesis: f1 goes_straight (len f) -' 1, Gauge (C,n)
thus  f1 goes_straight (len f) -' 1, Gauge (C,n) ::_thesis: verum
proof
let i19, j19, i29, j29 be Element of NAT ; :: according to GOBRD13:def_8 ::_thesis: ( not [i19,j19] in Indices (Gauge (C,n)) or not [i29,j29] in Indices (Gauge (C,n)) or not f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) or not f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) or ( i19 = i29 & j19 + 1 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 + 1 = i29 & j19 = j29 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) or ( i19 = i29 + 1 & j19 = j29 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) or ( i19 = i29 & j19 = j29 + 1 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) )
assume that
A101: [i19,j19] in Indices (Gauge (C,n)) and
A102: [i29,j29] in Indices (Gauge (C,n)) and
A103: f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) and
A104: f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) ; ::_thesis: ( ( i19 = i29 & j19 + 1 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 + 1 = i29 & j19 = j29 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) or ( i19 = i29 + 1 & j19 = j29 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) or ( i19 = i29 & j19 = j29 + 1 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) )
A105: f /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) by A13, A104, FINSEQ_4:68;
then A106: i2 = i29 by A17, A18, A102, GOBOARD1:5;
A107: f /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) by A26, A103, FINSEQ_4:68;
then A108: i1 = i19 by A15, A16, A101, GOBOARD1:5;
A109: j2 = j29 by A17, A18, A102, A105, GOBOARD1:5;
percases ( ( i19 = i29 & j19 + 1 = j29 ) or ( i19 + 1 = i29 & j19 = j29 ) or ( i19 = i29 + 1 & j19 = j29 ) or ( i19 = i29 & j19 = j29 + 1 ) ) by A15, A16, A19, A101, A107, A106, A109, GOBOARD1:5;
case ( i19 = i29 & j19 + 1 = j29 ) ; ::_thesis: ( [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) )
hence  ( [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) by A100, A108, A106; ::_thesis: verum
end;
case ( i19 + 1 = i29 & j19 = j29 ) ; ::_thesis: ( [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) )
hence  ( [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) by A100, A108, A106; ::_thesis: verum
end;
case ( i19 = i29 + 1 & j19 = j29 ) ; ::_thesis: ( [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) )
now__::_thesis:_not_i2_-'_1_<_1
assume  i2 -' 1 < 1 ; ::_thesis: contradiction
then  i2 <= 1 by NAT_1:14, NAT_D:36;
then  i2 = 1 by A22, XXREAL_0:1;
then  cell ((Gauge (C,n)),(1 -' 1),j1) meets C by A10, A14, A11, A15, A16, A17, A18, A76, A100, GOBRD13:39;
then  cell ((Gauge (C,n)),0,j1) meets C by XREAL_1:232;
hence  contradiction by A2, A23, JORDAN8:18; ::_thesis: verum
end;
hence  [(i29 -' 1),j29] in Indices (Gauge (C,n)) by A25, A27, A30, A106, A109, MATRIX_1:36; ::_thesis: f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29)
thus  f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) by A12, A106, A109, FINSEQ_4:67; ::_thesis: verum
end;
case ( i19 = i29 & j19 = j29 + 1 ) ; ::_thesis: ( [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) )
hence  ( [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) by A100, A108, A106; ::_thesis: verum
end;
end;
end;
end;
hence  S1[k,x,f1] by A7, A9, A10, A76; ::_thesis: verum
end;
supposeA110: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: ex y being set st S1[k,x,y]
take f1 = f ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*>; ::_thesis: S1[k,x,f1]
now__::_thesis:_ex_i_being_Element_of_NAT_ex_j_being_Element_of_NAT_st_f1_goes_straight_(len_f)_-'_1,_Gauge_(C,n)
take i = i2; ::_thesis: ex j being Element of NAT st f1 goes_straight (len f) -' 1, Gauge (C,n)
take j = j2 -' 1; ::_thesis: f1 goes_straight (len f) -' 1, Gauge (C,n)
thus  f1 goes_straight (len f) -' 1, Gauge (C,n) ::_thesis: verum
proof
let i19, j19, i29, j29 be Element of NAT ; :: according to GOBRD13:def_8 ::_thesis: ( not [i19,j19] in Indices (Gauge (C,n)) or not [i29,j29] in Indices (Gauge (C,n)) or not f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) or not f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) or ( i19 = i29 & j19 + 1 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 + 1 = i29 & j19 = j29 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) or ( i19 = i29 + 1 & j19 = j29 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) or ( i19 = i29 & j19 = j29 + 1 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) )
assume that
A111: [i19,j19] in Indices (Gauge (C,n)) and
A112: [i29,j29] in Indices (Gauge (C,n)) and
A113: f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) and
A114: f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) ; ::_thesis: ( ( i19 = i29 & j19 + 1 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 + 1 = i29 & j19 = j29 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) or ( i19 = i29 + 1 & j19 = j29 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) or ( i19 = i29 & j19 = j29 + 1 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) )
A115: f /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) by A26, A113, FINSEQ_4:68;
then A116: i1 = i19 by A15, A16, A111, GOBOARD1:5;
A117: j1 = j19 by A15, A16, A111, A115, GOBOARD1:5;
A118: f /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) by A13, A114, FINSEQ_4:68;
then A119: i2 = i29 by A17, A18, A112, GOBOARD1:5;
A120: j2 = j29 by A17, A18, A112, A118, GOBOARD1:5;
percases ( ( i19 = i29 & j19 + 1 = j29 ) or ( i19 + 1 = i29 & j19 = j29 ) or ( i19 = i29 + 1 & j19 = j29 ) or ( i19 = i29 & j19 = j29 + 1 ) ) by A15, A16, A19, A111, A115, A119, A120, GOBOARD1:5;
case ( i19 = i29 & j19 + 1 = j29 ) ; ::_thesis: ( [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) )
hence  ( [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) by A110, A117, A120; ::_thesis: verum
end;
case ( i19 + 1 = i29 & j19 = j29 ) ; ::_thesis: ( [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) )
hence  ( [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) by A110, A116, A119; ::_thesis: verum
end;
case ( i19 = i29 + 1 & j19 = j29 ) ; ::_thesis: ( [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) )
hence  ( [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) by A110, A116, A119; ::_thesis: verum
end;
case ( i19 = i29 & j19 = j29 + 1 ) ; ::_thesis: ( [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) )
now__::_thesis:_not_j2_-'_1_<_1
assume  j2 -' 1 < 1 ; ::_thesis: contradiction
then  j2 <= 1 by NAT_1:14, NAT_D:36;
then  j2 = 1 by A25, XXREAL_0:1;
then  cell ((Gauge (C,n)),(i1 -' 1),(1 -' 1)) meets C by A10, A14, A11, A15, A16, A17, A18, A76, A110, GOBRD13:41;
then  cell ((Gauge (C,n)),(i1 -' 1),0) meets C by XREAL_1:232;
hence  contradiction by A20, JORDAN8:17, NAT_D:44; ::_thesis: verum
end;
hence  [i29,(j29 -' 1)] in Indices (Gauge (C,n)) by A22, A29, A28, A119, A120, MATRIX_1:36; ::_thesis: f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1))
thus  f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) by A12, A119, A120, FINSEQ_4:67; ::_thesis: verum
end;
end;
end;
end;
hence  S1[k,x,f1] by A7, A9, A10, A76; ::_thesis: verum
end;
end;
end;
end;
supposeA121: front_left_cell (f,((len f) -' 1),(Gauge (C,n))) meets C ; ::_thesis: ex y being set st S1[k,x,y]
thus  ex y being set st S1[k,x,y] ::_thesis: verum
proof
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 A19;
supposeA122: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: ex y being set st S1[k,x,y]
take f1 = f ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*>; ::_thesis: S1[k,x,f1]
now__::_thesis:_ex_i_being_Element_of_NAT_ex_j_being_Element_of_NAT_st_f1_turns_left_(len_f)_-'_1,_Gauge_(C,n)
take i = i2 -' 1; ::_thesis: ex j being Element of NAT st f1 turns_left (len f) -' 1, Gauge (C,n)
take j = j2; ::_thesis: f1 turns_left (len f) -' 1, Gauge (C,n)
thus  f1 turns_left (len f) -' 1, Gauge (C,n) ::_thesis: verum
proof
let i19, j19, i29, j29 be Element of NAT ; :: according to GOBRD13:def_7 ::_thesis: ( not [i19,j19] in Indices (Gauge (C,n)) or not [i29,j29] in Indices (Gauge (C,n)) or not f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) or not f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) or ( i19 = i29 & j19 + 1 = j29 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) or ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) )
assume that
A123: [i19,j19] in Indices (Gauge (C,n)) and
A124: [i29,j29] in Indices (Gauge (C,n)) and
A125: f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) and
A126: f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) ; ::_thesis: ( ( i19 = i29 & j19 + 1 = j29 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) or ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) )
A127: f /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) by A26, A125, FINSEQ_4:68;
then A128: i1 = i19 by A15, A16, A123, GOBOARD1:5;
A129: j1 = j19 by A15, A16, A123, A127, GOBOARD1:5;
A130: f /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) by A13, A126, FINSEQ_4:68;
then A131: i2 = i29 by A17, A18, A124, GOBOARD1:5;
A132: j2 = j29 by A17, A18, A124, A130, GOBOARD1:5;
percases ( ( i19 = i29 & j19 + 1 = j29 ) or ( i19 + 1 = i29 & j19 = j29 ) or ( i19 = i29 + 1 & j19 = j29 ) or ( i19 = i29 & j19 = j29 + 1 ) ) by A15, A16, A19, A123, A127, A131, A132, GOBOARD1:5;
case ( i19 = i29 & j19 + 1 = j29 ) ; ::_thesis: ( [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) )
now__::_thesis:_not_i2_-'_1_<_1
assume  i2 -' 1 < 1 ; ::_thesis: contradiction
then  i2 <= 1 by NAT_1:14, NAT_D:36;
then  i2 = 1 by A22, XXREAL_0:1;
then  cell ((Gauge (C,n)),(1 -' 1),j2) meets C by A10, A14, A11, A15, A16, A17, A18, A121, A122, GOBRD13:34;
then  cell ((Gauge (C,n)),0,j2) meets C by XREAL_1:232;
hence  contradiction by A2, A27, JORDAN8:18; ::_thesis: verum
end;
hence  [(i29 -' 1),j29] in Indices (Gauge (C,n)) by A25, A27, A30, A131, A132, MATRIX_1:36; ::_thesis: f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29)
thus  f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) by A12, A131, A132, FINSEQ_4:67; ::_thesis: verum
end;
case ( i19 + 1 = i29 & j19 = j29 ) ; ::_thesis: ( [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) )
hence  ( [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) by A122, A128, A131; ::_thesis: verum
end;
case ( i19 = i29 + 1 & j19 = j29 ) ; ::_thesis: ( [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) )
hence  ( [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) by A122, A128, A131; ::_thesis: verum
end;
case ( i19 = i29 & j19 = j29 + 1 ) ; ::_thesis: ( [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) )
hence  ( [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) by A122, A129, A132; ::_thesis: verum
end;
end;
end;
end;
hence  S1[k,x,f1] by A7, A9, A10, A121; ::_thesis: verum
end;
supposeA133: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: ex y being set st S1[k,x,y]
take f1 = f ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*>; ::_thesis: S1[k,x,f1]
now__::_thesis:_ex_i_being_Element_of_NAT_ex_j_being_Element_of_NAT_st_f1_turns_left_(len_f)_-'_1,_Gauge_(C,n)
take i = i2; ::_thesis: ex j being Element of NAT st f1 turns_left (len f) -' 1, Gauge (C,n)
take j = j2 + 1; ::_thesis: f1 turns_left (len f) -' 1, Gauge (C,n)
thus  f1 turns_left (len f) -' 1, Gauge (C,n) ::_thesis: verum
proof
let i19, j19, i29, j29 be Element of NAT ; :: according to GOBRD13:def_7 ::_thesis: ( not [i19,j19] in Indices (Gauge (C,n)) or not [i29,j29] in Indices (Gauge (C,n)) or not f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) or not f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) or ( i19 = i29 & j19 + 1 = j29 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) or ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) )
assume that
A134: [i19,j19] in Indices (Gauge (C,n)) and
A135: [i29,j29] in Indices (Gauge (C,n)) and
A136: f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) and
A137: f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) ; ::_thesis: ( ( i19 = i29 & j19 + 1 = j29 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) or ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) )
A138: f /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) by A13, A137, FINSEQ_4:68;
then A139: i2 = i29 by A17, A18, A135, GOBOARD1:5;
A140: f /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) by A26, A136, FINSEQ_4:68;
then A141: i1 = i19 by A15, A16, A134, GOBOARD1:5;
A142: j2 = j29 by A17, A18, A135, A138, GOBOARD1:5;
percases ( ( i19 = i29 & j19 + 1 = j29 ) or ( i19 + 1 = i29 & j19 = j29 ) or ( i19 = i29 + 1 & j19 = j29 ) or ( i19 = i29 & j19 = j29 + 1 ) ) by A15, A16, A19, A134, A140, A139, A142, GOBOARD1:5;
case ( i19 = i29 & j19 + 1 = j29 ) ; ::_thesis: ( [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) )
hence  ( [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) by A133, A141, A139; ::_thesis: verum
end;
case ( i19 + 1 = i29 & j19 = j29 ) ; ::_thesis: ( [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) )
now__::_thesis:_not_j2_+_1_>_len_(Gauge_(C,n))
assume  j2 + 1 > len (Gauge (C,n)) ; ::_thesis: contradiction
then A143: (len (Gauge (C,n))) + 1 <= j2 + 1 by NAT_1:13;
 j2 + 1 <= (len (Gauge (C,n))) + 1 by A2, A27, XREAL_1:6;
then  j2 + 1 = (len (Gauge (C,n))) + 1 by A143, XXREAL_0:1;
then  cell ((Gauge (C,n)),i2,(len (Gauge (C,n)))) meets C by A10, A14, A11, A15, A16, A17, A18, A121, A133, GOBRD13:36;
hence  contradiction by A29, JORDAN8:15; ::_thesis: verum
end;
hence  [i29,(j29 + 1)] in Indices (Gauge (C,n)) by A2, A22, A29, A21, A139, A142, MATRIX_1:36; ::_thesis: f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1))
thus  f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) by A12, A139, A142, FINSEQ_4:67; ::_thesis: verum
end;
case ( i19 = i29 + 1 & j19 = j29 ) ; ::_thesis: ( [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) )
hence  ( [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) by A133, A141, A139; ::_thesis: verum
end;
case ( i19 = i29 & j19 = j29 + 1 ) ; ::_thesis: ( [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) )
hence  ( [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) by A133, A141, A139; ::_thesis: verum
end;
end;
end;
end;
hence  S1[k,x,f1] by A7, A9, A10, A121; ::_thesis: verum
end;
supposeA144: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: ex y being set st S1[k,x,y]
take f1 = f ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*>; ::_thesis: S1[k,x,f1]
now__::_thesis:_ex_i_being_Element_of_NAT_ex_j_being_Element_of_NAT_st_f1_turns_left_(len_f)_-'_1,_Gauge_(C,n)
take i = i2; ::_thesis: ex j being Element of NAT st f1 turns_left (len f) -' 1, Gauge (C,n)
take j = j2 -' 1; ::_thesis: f1 turns_left (len f) -' 1, Gauge (C,n)
thus  f1 turns_left (len f) -' 1, Gauge (C,n) ::_thesis: verum
proof
let i19, j19, i29, j29 be Element of NAT ; :: according to GOBRD13:def_7 ::_thesis: ( not [i19,j19] in Indices (Gauge (C,n)) or not [i29,j29] in Indices (Gauge (C,n)) or not f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) or not f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) or ( i19 = i29 & j19 + 1 = j29 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) or ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) )
assume that
A145: [i19,j19] in Indices (Gauge (C,n)) and
A146: [i29,j29] in Indices (Gauge (C,n)) and
A147: f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) and
A148: f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) ; ::_thesis: ( ( i19 = i29 & j19 + 1 = j29 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) or ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) )
A149: f /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) by A13, A148, FINSEQ_4:68;
then A150: i2 = i29 by A17, A18, A146, GOBOARD1:5;
A151: f /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) by A26, A147, FINSEQ_4:68;
then A152: i1 = i19 by A15, A16, A145, GOBOARD1:5;
A153: j2 = j29 by A17, A18, A146, A149, GOBOARD1:5;
percases ( ( i19 = i29 & j19 + 1 = j29 ) or ( i19 + 1 = i29 & j19 = j29 ) or ( i19 = i29 + 1 & j19 = j29 ) or ( i19 = i29 & j19 = j29 + 1 ) ) by A15, A16, A19, A145, A151, A150, A153, GOBOARD1:5;
case ( i19 = i29 & j19 + 1 = j29 ) ; ::_thesis: ( [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) )
hence  ( [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) by A144, A152, A150; ::_thesis: verum
end;
case ( i19 + 1 = i29 & j19 = j29 ) ; ::_thesis: ( [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) )
hence  ( [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) by A144, A152, A150; ::_thesis: verum
end;
case ( i19 = i29 + 1 & j19 = j29 ) ; ::_thesis: ( [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) )
now__::_thesis:_not_j2_-'_1_<_1
assume  j2 -' 1 < 1 ; ::_thesis: contradiction
then  j2 <= 1 by NAT_1:14, NAT_D:36;
then  j2 = 1 by A25, XXREAL_0:1;
then  cell ((Gauge (C,n)),(i2 -' 1),(1 -' 1)) meets C by A10, A14, A11, A15, A16, A17, A18, A121, A144, GOBRD13:38;
then  cell ((Gauge (C,n)),(i2 -' 1),0) meets C by XREAL_1:232;
hence  contradiction by A29, JORDAN8:17, NAT_D:44; ::_thesis: verum
end;
hence  [i29,(j29 -' 1)] in Indices (Gauge (C,n)) by A22, A29, A28, A150, A153, MATRIX_1:36; ::_thesis: f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1))
thus  f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) by A12, A150, A153, FINSEQ_4:67; ::_thesis: verum
end;
case ( i19 = i29 & j19 = j29 + 1 ) ; ::_thesis: ( [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) )
hence  ( [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) by A144, A152, A150; ::_thesis: verum
end;
end;
end;
end;
hence  S1[k,x,f1] by A7, A9, A10, A121; ::_thesis: verum
end;
supposeA154: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: ex y being set st S1[k,x,y]
take f1 = f ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*>; ::_thesis: S1[k,x,f1]
now__::_thesis:_ex_i_being_Element_of_NAT_ex_j_being_Element_of_NAT_st_f1_turns_left_(len_f)_-'_1,_Gauge_(C,n)
take i = i2 + 1; ::_thesis: ex j being Element of NAT st f1 turns_left (len f) -' 1, Gauge (C,n)
take j = j2; ::_thesis: f1 turns_left (len f) -' 1, Gauge (C,n)
thus  f1 turns_left (len f) -' 1, Gauge (C,n) ::_thesis: verum
proof
let i19, j19, i29, j29 be Element of NAT ; :: according to GOBRD13:def_7 ::_thesis: ( not [i19,j19] in Indices (Gauge (C,n)) or not [i29,j29] in Indices (Gauge (C,n)) or not f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) or not f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) or ( i19 = i29 & j19 + 1 = j29 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) or ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) )
assume that
A155: [i19,j19] in Indices (Gauge (C,n)) and
A156: [i29,j29] in Indices (Gauge (C,n)) and
A157: f1 /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) and
A158: f1 /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) ; ::_thesis: ( ( i19 = i29 & j19 + 1 = j29 & [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) or ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) or ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) ) )
A159: f /. ((len f) -' 1) = (Gauge (C,n)) * (i19,j19) by A26, A157, FINSEQ_4:68;
then A160: i1 = i19 by A15, A16, A155, GOBOARD1:5;
A161: j1 = j19 by A15, A16, A155, A159, GOBOARD1:5;
A162: f /. (((len f) -' 1) + 1) = (Gauge (C,n)) * (i29,j29) by A13, A158, FINSEQ_4:68;
then A163: i2 = i29 by A17, A18, A156, GOBOARD1:5;
A164: j2 = j29 by A17, A18, A156, A162, GOBOARD1:5;
percases ( ( i19 = i29 & j19 + 1 = j29 ) or ( i19 + 1 = i29 & j19 = j29 ) or ( i19 = i29 + 1 & j19 = j29 ) or ( i19 = i29 & j19 = j29 + 1 ) ) by A15, A16, A19, A155, A159, A163, A164, GOBOARD1:5;
case ( i19 = i29 & j19 + 1 = j29 ) ; ::_thesis: ( [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) )
hence  ( [(i29 -' 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 -' 1),j29) ) by A154, A161, A164; ::_thesis: verum
end;
case ( i19 + 1 = i29 & j19 = j29 ) ; ::_thesis: ( [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) )
hence  ( [i29,(j29 + 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 + 1)) ) by A154, A160, A163; ::_thesis: verum
end;
case ( i19 = i29 + 1 & j19 = j29 ) ; ::_thesis: ( [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) )
hence  ( [i29,(j29 -' 1)] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * (i29,(j29 -' 1)) ) by A154, A160, A163; ::_thesis: verum
end;
case ( i19 = i29 & j19 = j29 + 1 ) ; ::_thesis: ( [(i29 + 1),j29] in Indices (Gauge (C,n)) & f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) )
now__::_thesis:_not_i2_+_1_>_len_(Gauge_(C,n))
assume  i2 + 1 > len (Gauge (C,n)) ; ::_thesis: contradiction
then A165: (len (Gauge (C,n))) + 1 <= i2 + 1 by NAT_1:13;
 i2 + 1 <= (len (Gauge (C,n))) + 1 by A29, XREAL_1:6;
then  i2 + 1 = (len (Gauge (C,n))) + 1 by A165, XXREAL_0:1;
then  cell ((Gauge (C,n)),(len (Gauge (C,n))),(j2 -' 1)) meets C by A10, A14, A11, A15, A16, A17, A18, A121, A154, GOBRD13:40;
hence  contradiction by A2, A27, JORDAN8:16, NAT_D:44; ::_thesis: verum
end;
hence  [(i29 + 1),j29] in Indices (Gauge (C,n)) by A25, A27, A24, A163, A164, MATRIX_1:36; ::_thesis: f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29)
thus  f1 /. (((len f) -' 1) + 2) = (Gauge (C,n)) * ((i29 + 1),j29) by A12, A163, A164, FINSEQ_4:67; ::_thesis: verum
end;
end;
end;
end;
hence  S1[k,x,f1] by A7, A9, A10, A121; ::_thesis: verum
end;
end;
end;
end;
end;
end;
end;
supposeA166: ( not f is_sequence_on Gauge (C,n) or right_cell (f,((len f) -' 1),(Gauge (C,n))) misses C ) ; ::_thesis: ex y being set st S1[k,x,y]
take  f ^ <*((Gauge (C,n)) * (1,1))*> ; ::_thesis: S1[k,x,f ^ <*((Gauge (C,n)) * (1,1))*>]
thus  S1[k,x,f ^ <*((Gauge (C,n)) * (1,1))*>] by A7, A9, A166; ::_thesis: verum
end;
end;
end;
end;
supposeA167: len f <> k ; ::_thesis: ex y being set st S1[k,x,y]
take  {} ; ::_thesis: S1[k,x, {} ]
thus  S1[k,x, {} ] by A7, A167; ::_thesis: verum
end;
end;
end;
end;
supposeA168: ( k > 1 & x is not FinSequence of (TOP-REAL 2) ) ; ::_thesis: ex y being set st S1[k,x,y]
take  {} ; ::_thesis: S1[k,x, {} ]
thus  S1[k,x, {} ] by A168; ::_thesis: verum
end;
end;
end;
consider F being Function such that
A169: dom F = NAT and
A170: F . 0 = {} and
A171: for k being Element of NAT holds S1[k,F . k,F . (k + 1)] from RECDEF_1:sch_1(A3);
defpred S2[ Element of NAT ] means F . $1 is FinSequence of (TOP-REAL 2);
A172: {} = <*> the carrier of (TOP-REAL 2) ;
A173: for k being Element of NAT st S2[k] holds
S2[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S2[k] implies S2[k + 1] )
assume A174: F . k is FinSequence of (TOP-REAL 2) ; ::_thesis: S2[k + 1]
A175: S1[k,F . k,F . (k + 1)] by A171;
percases ( k = 0 or k = 1 or k > 1 ) by NAT_1:25;
suppose k = 0 ; ::_thesis: S2[k + 1]
hence  S2[k + 1] by A175; ::_thesis: verum
end;
suppose k = 1 ; ::_thesis: S2[k + 1]
hence  S2[k + 1] by A175; ::_thesis: verum
end;
supposeA176: k > 1 ; ::_thesis: S2[k + 1]
thus  S2[k + 1] ::_thesis: verum
proof
reconsider f = F . k as FinSequence of (TOP-REAL 2) by A174;
percases ( len f = k or len f <> k ) ;
supposeA177: len f = k ; ::_thesis: S2[k + 1]
thus  S2[k + 1] ::_thesis: verum
proof
percases ( ( f is_sequence_on Gauge (C,n) & right_cell (f,((len f) -' 1),(Gauge (C,n))) meets C ) or not f is_sequence_on Gauge (C,n) or right_cell (f,((len f) -' 1),(Gauge (C,n))) misses C ) ;
supposeA178: ( f is_sequence_on Gauge (C,n) & right_cell (f,((len f) -' 1),(Gauge (C,n))) meets C ) ; ::_thesis: S2[k + 1]
then A179: ( front_left_cell (f,((len f) -' 1),(Gauge (C,n))) meets C implies ex i, j being Element of NAT st
( f ^ <*((Gauge (C,n)) * (i,j))*> turns_left (len f) -' 1, Gauge (C,n) & F . (k + 1) = f ^ <*((Gauge (C,n)) * (i,j))*> ) ) by A171, A176, A177;
A180: ( front_left_cell (f,((len f) -' 1),(Gauge (C,n))) misses C & front_right_cell (f,((len f) -' 1),(Gauge (C,n))) meets C implies ex i, j being Element of NAT st
( f ^ <*((Gauge (C,n)) * (i,j))*> goes_straight (len f) -' 1, Gauge (C,n) & F . (k + 1) = f ^ <*((Gauge (C,n)) * (i,j))*> ) ) by A171, A176, A177, A178;
 ( front_left_cell (f,((len f) -' 1),(Gauge (C,n))) misses C & front_right_cell (f,((len f) -' 1),(Gauge (C,n))) misses C implies ex i, j being Element of NAT st
( f ^ <*((Gauge (C,n)) * (i,j))*> turns_right (len f) -' 1, Gauge (C,n) & F . (k + 1) = f ^ <*((Gauge (C,n)) * (i,j))*> ) ) by A171, A176, A177, A178;
hence  S2[k + 1] by A180, A179; ::_thesis: verum
end;
supposeA181: ( not f is_sequence_on Gauge (C,n) or right_cell (f,((len f) -' 1),(Gauge (C,n))) misses C ) ; ::_thesis: S2[k + 1]
 f ^ <*((Gauge (C,n)) * (1,1))*> is FinSequence of (TOP-REAL 2) ;
hence  S2[k + 1] by A171, A176, A177, A181; ::_thesis: verum
end;
end;
end;
end;
suppose len f <> k ; ::_thesis: S2[k + 1]
hence  S2[k + 1] by A171, A172, A176; ::_thesis: verum
end;
end;
end;
end;
end;
end;
A182: S2[ 0 ] by A170, A172;
A183: for k being Element of NAT holds S2[k] from NAT_1:sch_1(A182, A173);
 rng F c= the carrier of (TOP-REAL 2) *
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng F or y in the carrier of (TOP-REAL 2) * )
assume  y in rng F ; ::_thesis: y in the carrier of (TOP-REAL 2) *
then  ex x being set st
( x in dom F & F . x = y ) by FUNCT_1:def_3;
then  y is FinSequence of (TOP-REAL 2) by A169, A183;
hence  y in the carrier of (TOP-REAL 2) * by FINSEQ_1:def_11; ::_thesis: verum
end;
then reconsider F = F as Function of NAT,( the carrier of (TOP-REAL 2) *) by A169, FUNCT_2:def_1, RELSET_1:4;
defpred S3[ Element of NAT ] means  len (F . $1) = $1;
A184: for k being Element of NAT st S3[k] holds
S3[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S3[k] implies S3[k + 1] )
assume A185: len (F . k) = k ; ::_thesis: S3[k + 1]
A186: S1[k,F . k,F . (k + 1)] by A171;
percases ( k = 0 or k = 1 or k > 1 ) by NAT_1:25;
suppose k = 0 ; ::_thesis: S3[k + 1]
hence  S3[k + 1] by A186, FINSEQ_1:39; ::_thesis: verum
end;
suppose k = 1 ; ::_thesis: S3[k + 1]
hence  S3[k + 1] by A186, FINSEQ_1:44; ::_thesis: verum
end;
supposeA187: k > 1 ; ::_thesis: S3[k + 1]
thus  S3[k + 1] ::_thesis: verum
proof
percases ( ( F . k is_sequence_on Gauge (C,n) & right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ) or not F . k is_sequence_on Gauge (C,n) or right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C ) ;
supposeA188: ( F . k is_sequence_on Gauge (C,n) & right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ) ; ::_thesis: S3[k + 1]
then A189: ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C implies ex i, j being Element of NAT st
( (F . k) ^ <*((Gauge (C,n)) * (i,j))*> turns_left (len (F . k)) -' 1, Gauge (C,n) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) ) by A171, A185, A187;
A190: ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C implies ex i, j being Element of NAT st
( (F . k) ^ <*((Gauge (C,n)) * (i,j))*> goes_straight (len (F . k)) -' 1, Gauge (C,n) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) ) by A171, A185, A187, A188;
 ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C implies ex i, j being Element of NAT st
( (F . k) ^ <*((Gauge (C,n)) * (i,j))*> turns_right (len (F . k)) -' 1, Gauge (C,n) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) ) by A171, A185, A187, A188;
hence  S3[k + 1] by A185, A190, A189, FINSEQ_2:16; ::_thesis: verum
end;
suppose ( not F . k is_sequence_on Gauge (C,n) or right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C ) ; ::_thesis: S3[k + 1]
then  F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (1,1))*> by A171, A185, A187;
hence  S3[k + 1] by A185, FINSEQ_2:16; ::_thesis: verum
end;
end;
end;
end;
end;
end;
A191: S3[ 0 ] by A170, CARD_1:27;
A192: for k being Element of NAT holds S3[k] from NAT_1:sch_1(A191, A184);
A193: now__::_thesis:_for_k_being_Element_of_NAT_st_F_._k_is_sequence_on_Gauge_(C,n)_&_(_for_m_being_Element_of_NAT_st_1_<=_m_&_m_+_1_<=_len_(F_._k)_holds_
(_left_cell_((F_._k),m,(Gauge_(C,n)))_misses_C_&_right_cell_((F_._k),m,(Gauge_(C,n)))_meets_C_)_)_&_k_>_1_holds_
for_i1,_j1,_i2,_j2_being_Element_of_NAT_st_[i1,j1]_in_Indices_(Gauge_(C,n))_&_(F_._k)_/._((len_(F_._k))_-'_1)_=_(Gauge_(C,n))_*_(i1,j1)_&_[i2,j2]_in_Indices_(Gauge_(C,n))_&_(F_._k)_/._(len_(F_._k))_=_(Gauge_(C,n))_*_(i2,j2)_holds_
(_(_i1_=_i2_&_j1_+_1_=_j2_implies_[(i2_+_1),j2]_in_Indices_(Gauge_(C,n))_)_&_(_i1_+_1_=_i2_&_j1_=_j2_implies_[i2,(j2_-'_1)]_in_Indices_(Gauge_(C,n))_)_&_(_i1_=_i2_+_1_&_j1_=_j2_implies_[i2,(j2_+_1)]_in_Indices_(Gauge_(C,n))_)_&_(_i1_=_i2_&_j1_=_j2_+_1_implies_[(i2_-'_1),j2]_in_Indices_(Gauge_(C,n))_)_&_(_front_right_cell_((F_._k),((len_(F_._k))_-'_1),(Gauge_(C,n)))_meets_C_&_i1_=_i2_&_j1_+_1_=_j2_implies_[i2,(j2_+_1)]_in_Indices_(Gauge_(C,n))_)_&_(_front_right_cell_((F_._k),((len_(F_._k))_-'_1),(Gauge_(C,n)))_meets_C_&_i1_+_1_=_i2_&_j1_=_j2_implies_[(i2_+_1),j2]_in_Indices_(Gauge_(C,n))_)_&_(_front_right_cell_((F_._k),((len_(F_._k))_-'_1),(Gauge_(C,n)))_meets_C_&_i1_=_i2_+_1_&_j1_=_j2_implies_[(i2_-'_1),j2]_in_Indices_(Gauge_(C,n))_)_&_(_front_right_cell_((F_._k),((len_(F_._k))_-'_1),(Gauge_(C,n)))_meets_C_&_i1_=_i2_&_j1_=_j2_+_1_implies_[i2,(j2_-'_1)]_in_Indices_(Gauge_(C,n))_)_&_(_front_left_cell_((F_._k),((len_(F_._k))_-'_1),(Gauge_(C,n)))_meets_C_&_i1_=_i2_&_j1_+_1_=_j2_implies_[(i2_-'_1),j2]_in_Indices_(Gauge_(C,n))_)_&_(_front_left_cell_((F_._k),((len_(F_._k))_-'_1),(Gauge_(C,n)))_meets_C_&_i1_+_1_=_i2_&_j1_=_j2_implies_[i2,(j2_+_1)]_in_Indices_(Gauge_(C,n))_)_&_(_front_left_cell_((F_._k),((len_(F_._k))_-'_1),(Gauge_(C,n)))_meets_C_&_i1_=_i2_+_1_&_j1_=_j2_implies_[i2,(j2_-'_1)]_in_Indices_(Gauge_(C,n))_)_&_(_front_left_cell_((F_._k),((len_(F_._k))_-'_1),(Gauge_(C,n)))_meets_C_&_i1_=_i2_&_j1_=_j2_+_1_implies_[(i2_+_1),j2]_in_Indices_(Gauge_(C,n))_)_)
let k be Element of NAT ; ::_thesis: ( F . k is_sequence_on Gauge (C,n) & ( for m being Element of NAT st 1 <= m & m + 1 <= len (F . k) holds
( left_cell ((F . k),m,(Gauge (C,n))) misses C & right_cell ((F . k),m,(Gauge (C,n))) meets C ) ) & k > 1 implies for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices (Gauge (C,n)) & (F . k) /. ((len (F . k)) -' 1) = (Gauge (C,n)) * (i1,j1) & [i2,j2] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) holds
( ( i1 = i2 & j1 + 1 = j2 implies [(i2 + 1),j2] in Indices (Gauge (C,n)) ) & ( i1 + 1 = i2 & j1 = j2 implies [i2,(j2 -' 1)] in Indices (Gauge (C,n)) ) & ( i1 = i2 + 1 & j1 = j2 implies [i2,(j2 + 1)] in Indices (Gauge (C,n)) ) & ( i1 = i2 & j1 = j2 + 1 implies [(i2 -' 1),j2] in Indices (Gauge (C,n)) ) & ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 + 1 = j2 implies [i2,(j2 + 1)] in Indices (Gauge (C,n)) ) & ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 + 1 = i2 & j1 = j2 implies [(i2 + 1),j2] in Indices (Gauge (C,n)) ) & ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 + 1 & j1 = j2 implies [(i2 -' 1),j2] in Indices (Gauge (C,n)) ) & ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 = j2 + 1 implies [i2,(j2 -' 1)] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 + 1 = j2 implies [(i2 -' 1),j2] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 + 1 = i2 & j1 = j2 implies [i2,(j2 + 1)] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 + 1 & j1 = j2 implies [i2,(j2 -' 1)] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 = j2 + 1 implies [(i2 + 1),j2] in Indices (Gauge (C,n)) ) ) )
assume that
A194: F . k is_sequence_on Gauge (C,n) and
A195: for m being Element of NAT st 1 <= m & m + 1 <= len (F . k) holds
( left_cell ((F . k),m,(Gauge (C,n))) misses C & right_cell ((F . k),m,(Gauge (C,n))) meets C ) and
A196: k > 1 ; ::_thesis: for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices (Gauge (C,n)) & (F . k) /. ((len (F . k)) -' 1) = (Gauge (C,n)) * (i1,j1) & [i2,j2] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) holds
( ( i1 = i2 & j1 + 1 = j2 implies [(i2 + 1),j2] in Indices (Gauge (C,n)) ) & ( i1 + 1 = i2 & j1 = j2 implies [i2,(j2 -' 1)] in Indices (Gauge (C,n)) ) & ( i1 = i2 + 1 & j1 = j2 implies [i2,(j2 + 1)] in Indices (Gauge (C,n)) ) & ( i1 = i2 & j1 = j2 + 1 implies [(i2 -' 1),j2] in Indices (Gauge (C,n)) ) & ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 + 1 = j2 implies [i2,(j2 + 1)] in Indices (Gauge (C,n)) ) & ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 + 1 = i2 & j1 = j2 implies [(i2 + 1),j2] in Indices (Gauge (C,n)) ) & ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 + 1 & j1 = j2 implies [(i2 -' 1),j2] in Indices (Gauge (C,n)) ) & ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 = j2 + 1 implies [i2,(j2 -' 1)] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 + 1 = j2 implies [(i2 -' 1),j2] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 + 1 = i2 & j1 = j2 implies [i2,(j2 + 1)] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 + 1 & j1 = j2 implies [i2,(j2 -' 1)] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 = j2 + 1 implies [(i2 + 1),j2] in Indices (Gauge (C,n)) ) )
 len (F . k) = k by A192;
then A197: ( 1 <= (len (F . k)) -' 1 & ((len (F . k)) -' 1) + 1 = len (F . k) ) by A196, NAT_D:49, XREAL_1:235;
then A198: right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C by A195;
let i1, j1, i2, j2 be Element of NAT ; ::_thesis: ( [i1,j1] in Indices (Gauge (C,n)) & (F . k) /. ((len (F . k)) -' 1) = (Gauge (C,n)) * (i1,j1) & [i2,j2] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) implies ( ( i1 = i2 & j1 + 1 = j2 implies [(i2 + 1),j2] in Indices (Gauge (C,n)) ) & ( i1 + 1 = i2 & j1 = j2 implies [i2,(j2 -' 1)] in Indices (Gauge (C,n)) ) & ( i1 = i2 + 1 & j1 = j2 implies [i2,(j2 + 1)] in Indices (Gauge (C,n)) ) & ( i1 = i2 & j1 = j2 + 1 implies [(i2 -' 1),j2] in Indices (Gauge (C,n)) ) & ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 + 1 = j2 implies [i2,(j2 + 1)] in Indices (Gauge (C,n)) ) & ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 + 1 = i2 & j1 = j2 implies [(i2 + 1),j2] in Indices (Gauge (C,n)) ) & ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 + 1 & j1 = j2 implies [(i2 -' 1),j2] in Indices (Gauge (C,n)) ) & ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 = j2 + 1 implies [i2,(j2 -' 1)] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 + 1 = j2 implies [(i2 -' 1),j2] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 + 1 = i2 & j1 = j2 implies [i2,(j2 + 1)] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 + 1 & j1 = j2 implies [i2,(j2 -' 1)] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 = j2 + 1 implies [(i2 + 1),j2] in Indices (Gauge (C,n)) ) ) )
assume that
A199: [i1,j1] in Indices (Gauge (C,n)) and
A200: (F . k) /. ((len (F . k)) -' 1) = (Gauge (C,n)) * (i1,j1) and
A201: [i2,j2] in Indices (Gauge (C,n)) and
A202: (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) ; ::_thesis: ( ( i1 = i2 & j1 + 1 = j2 implies [(i2 + 1),j2] in Indices (Gauge (C,n)) ) & ( i1 + 1 = i2 & j1 = j2 implies [i2,(j2 -' 1)] in Indices (Gauge (C,n)) ) & ( i1 = i2 + 1 & j1 = j2 implies [i2,(j2 + 1)] in Indices (Gauge (C,n)) ) & ( i1 = i2 & j1 = j2 + 1 implies [(i2 -' 1),j2] in Indices (Gauge (C,n)) ) & ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 + 1 = j2 implies [i2,(j2 + 1)] in Indices (Gauge (C,n)) ) & ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 + 1 = i2 & j1 = j2 implies [(i2 + 1),j2] in Indices (Gauge (C,n)) ) & ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 + 1 & j1 = j2 implies [(i2 -' 1),j2] in Indices (Gauge (C,n)) ) & ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 = j2 + 1 implies [i2,(j2 -' 1)] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 + 1 = j2 implies [(i2 -' 1),j2] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 + 1 = i2 & j1 = j2 implies [i2,(j2 + 1)] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 + 1 & j1 = j2 implies [i2,(j2 -' 1)] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 = j2 + 1 implies [(i2 + 1),j2] in Indices (Gauge (C,n)) ) )
A203: i2 <= len (Gauge (C,n)) by A201, MATRIX_1:38;
A204: 1 <= i2 + 1 by NAT_1:12;
A205: 1 <= j2 by A201, MATRIX_1:38;
A206: j2 <= width (Gauge (C,n)) by A201, MATRIX_1:38;
A207: j1 <= width (Gauge (C,n)) by A199, MATRIX_1:38;
hereby  ::_thesis: ( ( i1 + 1 = i2 & j1 = j2 implies [i2,(j2 -' 1)] in Indices (Gauge (C,n)) ) & ( i1 = i2 + 1 & j1 = j2 implies [i2,(j2 + 1)] in Indices (Gauge (C,n)) ) & ( i1 = i2 & j1 = j2 + 1 implies [(i2 -' 1),j2] in Indices (Gauge (C,n)) ) & ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 + 1 = j2 implies [i2,(j2 + 1)] in Indices (Gauge (C,n)) ) & ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 + 1 = i2 & j1 = j2 implies [(i2 + 1),j2] in Indices (Gauge (C,n)) ) & ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 + 1 & j1 = j2 implies [(i2 -' 1),j2] in Indices (Gauge (C,n)) ) & ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 = j2 + 1 implies [i2,(j2 -' 1)] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 + 1 = j2 implies [(i2 -' 1),j2] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 + 1 = i2 & j1 = j2 implies [i2,(j2 + 1)] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 + 1 & j1 = j2 implies [i2,(j2 -' 1)] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 = j2 + 1 implies [(i2 + 1),j2] in Indices (Gauge (C,n)) ) )
assume A208: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: [(i2 + 1),j2] in Indices (Gauge (C,n))
now__::_thesis:_not_i2_+_1_>_len_(Gauge_(C,n))
assume  i2 + 1 > len (Gauge (C,n)) ; ::_thesis: contradiction
then A209: (len (Gauge (C,n))) + 1 <= i2 + 1 by NAT_1:13;
 i2 + 1 <= (len (Gauge (C,n))) + 1 by A203, XREAL_1:6;
then  i2 + 1 = (len (Gauge (C,n))) + 1 by A209, XXREAL_0:1;
then  cell ((Gauge (C,n)),(len (Gauge (C,n))),j1) meets C by A194, A199, A200, A201, A202, A197, A198, A208, GOBRD13:22;
hence  contradiction by A2, A207, JORDAN8:16; ::_thesis: verum
end;
hence  [(i2 + 1),j2] in Indices (Gauge (C,n)) by A205, A206, A204, MATRIX_1:36; ::_thesis: verum
end;
A210: i1 <= len (Gauge (C,n)) by A199, MATRIX_1:38;
A211: 1 <= i2 by A201, MATRIX_1:38;
A212: j2 -' 1 <= width (Gauge (C,n)) by A206, NAT_D:44;
hereby  ::_thesis: ( ( i1 = i2 + 1 & j1 = j2 implies [i2,(j2 + 1)] in Indices (Gauge (C,n)) ) & ( i1 = i2 & j1 = j2 + 1 implies [(i2 -' 1),j2] in Indices (Gauge (C,n)) ) & ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 + 1 = j2 implies [i2,(j2 + 1)] in Indices (Gauge (C,n)) ) & ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 + 1 = i2 & j1 = j2 implies [(i2 + 1),j2] in Indices (Gauge (C,n)) ) & ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 + 1 & j1 = j2 implies [(i2 -' 1),j2] in Indices (Gauge (C,n)) ) & ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 = j2 + 1 implies [i2,(j2 -' 1)] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 + 1 = j2 implies [(i2 -' 1),j2] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 + 1 = i2 & j1 = j2 implies [i2,(j2 + 1)] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 + 1 & j1 = j2 implies [i2,(j2 -' 1)] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 = j2 + 1 implies [(i2 + 1),j2] in Indices (Gauge (C,n)) ) )
assume A213: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: [i2,(j2 -' 1)] in Indices (Gauge (C,n))
now__::_thesis:_not_j2_-'_1_<_1
assume  j2 -' 1 < 1 ; ::_thesis: contradiction
then  j2 <= 1 by NAT_1:14, NAT_D:36;
then  j2 = 1 by A205, XXREAL_0:1;
then  cell ((Gauge (C,n)),i1,(1 -' 1)) meets C by A194, A199, A200, A201, A202, A197, A198, A213, GOBRD13:24;
then  cell ((Gauge (C,n)),i1,0) meets C by XREAL_1:232;
hence  contradiction by A210, JORDAN8:17; ::_thesis: verum
end;
hence  [i2,(j2 -' 1)] in Indices (Gauge (C,n)) by A211, A203, A212, MATRIX_1:36; ::_thesis: verum
end;
A214: 1 <= j2 + 1 by NAT_1:12;
hereby  ::_thesis: ( ( i1 = i2 & j1 = j2 + 1 implies [(i2 -' 1),j2] in Indices (Gauge (C,n)) ) & ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 + 1 = j2 implies [i2,(j2 + 1)] in Indices (Gauge (C,n)) ) & ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 + 1 = i2 & j1 = j2 implies [(i2 + 1),j2] in Indices (Gauge (C,n)) ) & ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 + 1 & j1 = j2 implies [(i2 -' 1),j2] in Indices (Gauge (C,n)) ) & ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 = j2 + 1 implies [i2,(j2 -' 1)] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 + 1 = j2 implies [(i2 -' 1),j2] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 + 1 = i2 & j1 = j2 implies [i2,(j2 + 1)] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 + 1 & j1 = j2 implies [i2,(j2 -' 1)] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 = j2 + 1 implies [(i2 + 1),j2] in Indices (Gauge (C,n)) ) )
assume A215: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: [i2,(j2 + 1)] in Indices (Gauge (C,n))
now__::_thesis:_not_j2_+_1_>_len_(Gauge_(C,n))
assume  j2 + 1 > len (Gauge (C,n)) ; ::_thesis: contradiction
then A216: (len (Gauge (C,n))) + 1 <= j2 + 1 by NAT_1:13;
 j2 + 1 <= (len (Gauge (C,n))) + 1 by A2, A206, XREAL_1:6;
then  j2 + 1 = (len (Gauge (C,n))) + 1 by A216, XXREAL_0:1;
then  cell ((Gauge (C,n)),i2,(len (Gauge (C,n)))) meets C by A194, A199, A200, A201, A202, A197, A198, A215, GOBRD13:26;
hence  contradiction by A203, JORDAN8:15; ::_thesis: verum
end;
hence  [i2,(j2 + 1)] in Indices (Gauge (C,n)) by A2, A211, A203, A214, MATRIX_1:36; ::_thesis: verum
end;
A217: i2 -' 1 <= len (Gauge (C,n)) by A203, NAT_D:44;
hereby  ::_thesis: ( ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 + 1 = j2 implies [i2,(j2 + 1)] in Indices (Gauge (C,n)) ) & ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 + 1 = i2 & j1 = j2 implies [(i2 + 1),j2] in Indices (Gauge (C,n)) ) & ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 + 1 & j1 = j2 implies [(i2 -' 1),j2] in Indices (Gauge (C,n)) ) & ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 = j2 + 1 implies [i2,(j2 -' 1)] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 + 1 = j2 implies [(i2 -' 1),j2] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 + 1 = i2 & j1 = j2 implies [i2,(j2 + 1)] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 + 1 & j1 = j2 implies [i2,(j2 -' 1)] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 = j2 + 1 implies [(i2 + 1),j2] in Indices (Gauge (C,n)) ) )
assume A218: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: [(i2 -' 1),j2] in Indices (Gauge (C,n))
now__::_thesis:_not_i2_-'_1_<_1
assume  i2 -' 1 < 1 ; ::_thesis: contradiction
then  i2 <= 1 by NAT_1:14, NAT_D:36;
then  i2 = 1 by A211, XXREAL_0:1;
then  cell ((Gauge (C,n)),(1 -' 1),j2) meets C by A194, A199, A200, A201, A202, A197, A198, A218, GOBRD13:28;
then  cell ((Gauge (C,n)),0,j2) meets C by XREAL_1:232;
hence  contradiction by A2, A206, JORDAN8:18; ::_thesis: verum
end;
hence  [(i2 -' 1),j2] in Indices (Gauge (C,n)) by A205, A206, A217, MATRIX_1:36; ::_thesis: verum
end;
hereby  ::_thesis: ( ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 + 1 = i2 & j1 = j2 implies [(i2 + 1),j2] in Indices (Gauge (C,n)) ) & ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 + 1 & j1 = j2 implies [(i2 -' 1),j2] in Indices (Gauge (C,n)) ) & ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 = j2 + 1 implies [i2,(j2 -' 1)] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 + 1 = j2 implies [(i2 -' 1),j2] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 + 1 = i2 & j1 = j2 implies [i2,(j2 + 1)] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 + 1 & j1 = j2 implies [i2,(j2 -' 1)] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 = j2 + 1 implies [(i2 + 1),j2] in Indices (Gauge (C,n)) ) )
assume A219: ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: [i2,(j2 + 1)] in Indices (Gauge (C,n))
now__::_thesis:_not_j2_+_1_>_len_(Gauge_(C,n))
assume  j2 + 1 > len (Gauge (C,n)) ; ::_thesis: contradiction
then A220: (len (Gauge (C,n))) + 1 <= j2 + 1 by NAT_1:13;
 j2 + 1 <= (len (Gauge (C,n))) + 1 by A2, A206, XREAL_1:6;
then  j2 + 1 = (len (Gauge (C,n))) + 1 by A220, XXREAL_0:1;
then  cell ((Gauge (C,n)),i1,(len (Gauge (C,n)))) meets C by A194, A199, A200, A201, A202, A197, A219, GOBRD13:35;
hence  contradiction by A210, JORDAN8:15; ::_thesis: verum
end;
hence  [i2,(j2 + 1)] in Indices (Gauge (C,n)) by A2, A211, A203, A214, MATRIX_1:36; ::_thesis: verum
end;
hereby  ::_thesis: ( ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 + 1 & j1 = j2 implies [(i2 -' 1),j2] in Indices (Gauge (C,n)) ) & ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 = j2 + 1 implies [i2,(j2 -' 1)] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 + 1 = j2 implies [(i2 -' 1),j2] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 + 1 = i2 & j1 = j2 implies [i2,(j2 + 1)] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 + 1 & j1 = j2 implies [i2,(j2 -' 1)] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 = j2 + 1 implies [(i2 + 1),j2] in Indices (Gauge (C,n)) ) )
assume A221: ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: [(i2 + 1),j2] in Indices (Gauge (C,n))
now__::_thesis:_not_i2_+_1_>_len_(Gauge_(C,n))
assume  i2 + 1 > len (Gauge (C,n)) ; ::_thesis: contradiction
then A222: (len (Gauge (C,n))) + 1 <= i2 + 1 by NAT_1:13;
 i2 + 1 <= (len (Gauge (C,n))) + 1 by A203, XREAL_1:6;
then  i2 + 1 = (len (Gauge (C,n))) + 1 by A222, XXREAL_0:1;
then  cell ((Gauge (C,n)),(len (Gauge (C,n))),(j1 -' 1)) meets C by A194, A199, A200, A201, A202, A197, A221, GOBRD13:37;
hence  contradiction by A2, A207, JORDAN8:16, NAT_D:44; ::_thesis: verum
end;
hence  [(i2 + 1),j2] in Indices (Gauge (C,n)) by A205, A206, A204, MATRIX_1:36; ::_thesis: verum
end;
hereby  ::_thesis: ( ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 = j2 + 1 implies [i2,(j2 -' 1)] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 + 1 = j2 implies [(i2 -' 1),j2] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 + 1 = i2 & j1 = j2 implies [i2,(j2 + 1)] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 + 1 & j1 = j2 implies [i2,(j2 -' 1)] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 = j2 + 1 implies [(i2 + 1),j2] in Indices (Gauge (C,n)) ) )
assume A223: ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: [(i2 -' 1),j2] in Indices (Gauge (C,n))
now__::_thesis:_not_i2_-'_1_<_1
assume  i2 -' 1 < 1 ; ::_thesis: contradiction
then  i2 <= 1 by NAT_1:14, NAT_D:36;
then  i2 = 1 by A211, XXREAL_0:1;
then  cell ((Gauge (C,n)),(1 -' 1),j1) meets C by A194, A199, A200, A201, A202, A197, A223, GOBRD13:39;
then  cell ((Gauge (C,n)),0,j1) meets C by XREAL_1:232;
hence  contradiction by A2, A207, JORDAN8:18; ::_thesis: verum
end;
hence  [(i2 -' 1),j2] in Indices (Gauge (C,n)) by A205, A206, A217, MATRIX_1:36; ::_thesis: verum
end;
hereby  ::_thesis: ( ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 + 1 = j2 implies [(i2 -' 1),j2] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 + 1 = i2 & j1 = j2 implies [i2,(j2 + 1)] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 + 1 & j1 = j2 implies [i2,(j2 -' 1)] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 = j2 + 1 implies [(i2 + 1),j2] in Indices (Gauge (C,n)) ) )
assume A224: ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: [i2,(j2 -' 1)] in Indices (Gauge (C,n))
now__::_thesis:_not_j2_-'_1_<_1
assume  j2 -' 1 < 1 ; ::_thesis: contradiction
then  j2 <= 1 by NAT_1:14, NAT_D:36;
then  j2 = 1 by A205, XXREAL_0:1;
then  cell ((Gauge (C,n)),(i1 -' 1),(1 -' 1)) meets C by A194, A199, A200, A201, A202, A197, A224, GOBRD13:41;
then  cell ((Gauge (C,n)),(i1 -' 1),0) meets C by XREAL_1:232;
hence  contradiction by A210, JORDAN8:17, NAT_D:44; ::_thesis: verum
end;
hence  [i2,(j2 -' 1)] in Indices (Gauge (C,n)) by A211, A203, A212, MATRIX_1:36; ::_thesis: verum
end;
hereby  ::_thesis: ( ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 + 1 = i2 & j1 = j2 implies [i2,(j2 + 1)] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 + 1 & j1 = j2 implies [i2,(j2 -' 1)] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 = j2 + 1 implies [(i2 + 1),j2] in Indices (Gauge (C,n)) ) )
assume A225: ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: [(i2 -' 1),j2] in Indices (Gauge (C,n))
now__::_thesis:_not_i2_-'_1_<_1
assume  i2 -' 1 < 1 ; ::_thesis: contradiction
then  i2 <= 1 by NAT_1:14, NAT_D:36;
then  i2 = 1 by A211, XXREAL_0:1;
then  cell ((Gauge (C,n)),(1 -' 1),j2) meets C by A194, A199, A200, A201, A202, A197, A225, GOBRD13:34;
then  cell ((Gauge (C,n)),0,j2) meets C by XREAL_1:232;
hence  contradiction by A2, A206, JORDAN8:18; ::_thesis: verum
end;
hence  [(i2 -' 1),j2] in Indices (Gauge (C,n)) by A205, A206, A217, MATRIX_1:36; ::_thesis: verum
end;
hereby  ::_thesis: ( ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 + 1 & j1 = j2 implies [i2,(j2 -' 1)] in Indices (Gauge (C,n)) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 = j2 + 1 implies [(i2 + 1),j2] in Indices (Gauge (C,n)) ) )
assume A226: ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: [i2,(j2 + 1)] in Indices (Gauge (C,n))
now__::_thesis:_not_j2_+_1_>_len_(Gauge_(C,n))
assume  j2 + 1 > len (Gauge (C,n)) ; ::_thesis: contradiction
then A227: (len (Gauge (C,n))) + 1 <= j2 + 1 by NAT_1:13;
 j2 + 1 <= (len (Gauge (C,n))) + 1 by A2, A206, XREAL_1:6;
then  j2 + 1 = (len (Gauge (C,n))) + 1 by A227, XXREAL_0:1;
then  cell ((Gauge (C,n)),i2,(len (Gauge (C,n)))) meets C by A194, A199, A200, A201, A202, A197, A226, GOBRD13:36;
hence  contradiction by A203, JORDAN8:15; ::_thesis: verum
end;
hence  [i2,(j2 + 1)] in Indices (Gauge (C,n)) by A2, A211, A203, A214, MATRIX_1:36; ::_thesis: verum
end;
hereby  ::_thesis: ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 = j2 + 1 implies [(i2 + 1),j2] in Indices (Gauge (C,n)) )
assume A228: ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: [i2,(j2 -' 1)] in Indices (Gauge (C,n))
now__::_thesis:_not_j2_-'_1_<_1
assume  j2 -' 1 < 1 ; ::_thesis: contradiction
then  j2 <= 1 by NAT_1:14, NAT_D:36;
then  j2 = 1 by A205, XXREAL_0:1;
then  cell ((Gauge (C,n)),(i2 -' 1),(1 -' 1)) meets C by A194, A199, A200, A201, A202, A197, A228, GOBRD13:38;
then  cell ((Gauge (C,n)),(i2 -' 1),0) meets C by XREAL_1:232;
hence  contradiction by A203, JORDAN8:17, NAT_D:44; ::_thesis: verum
end;
hence  [i2,(j2 -' 1)] in Indices (Gauge (C,n)) by A211, A203, A212, MATRIX_1:36; ::_thesis: verum
end;
hereby  ::_thesis: verum
assume A229: ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: [(i2 + 1),j2] in Indices (Gauge (C,n))
now__::_thesis:_not_i2_+_1_>_len_(Gauge_(C,n))
assume  i2 + 1 > len (Gauge (C,n)) ; ::_thesis: contradiction
then A230: (len (Gauge (C,n))) + 1 <= i2 + 1 by NAT_1:13;
 i2 + 1 <= (len (Gauge (C,n))) + 1 by A203, XREAL_1:6;
then  i2 + 1 = (len (Gauge (C,n))) + 1 by A230, XXREAL_0:1;
then  cell ((Gauge (C,n)),(len (Gauge (C,n))),(j2 -' 1)) meets C by A194, A199, A200, A201, A202, A197, A229, GOBRD13:40;
hence  contradiction by A2, A206, JORDAN8:16, NAT_D:44; ::_thesis: verum
end;
hence  [(i2 + 1),j2] in Indices (Gauge (C,n)) by A205, A206, A204, MATRIX_1:36; ::_thesis: verum
end;
end;
defpred S4[ Element of NAT ] means ( F . $1 is_sequence_on Gauge (C,n) & ( for m being Element of NAT st 1 <= m & m + 1 <= len (F . $1) holds
( left_cell ((F . $1),m,(Gauge (C,n))) misses C & right_cell ((F . $1),m,(Gauge (C,n))) meets C ) ) );
A231: len (Gauge (C,n)) = (2 |^ n) + 3 by JORDAN8:def_1;
A232: for k being Element of NAT st S4[k] holds
S4[k + 1]
proof
A233: 2 |^ n > 0 by NEWTON:83;
A234: 1 <= len (Gauge (C,n)) by A231, NAT_1:12;
let k be Element of NAT ; ::_thesis: ( S4[k] implies S4[k + 1] )
assume that
A235: F . k is_sequence_on Gauge (C,n) and
A236: for m being Element of NAT st 1 <= m & m + 1 <= len (F . k) holds
( left_cell ((F . k),m,(Gauge (C,n))) misses C & right_cell ((F . k),m,(Gauge (C,n))) meets C ) ; ::_thesis: S4[k + 1]
A237: len (F . k) = k by A192;
A238: len (F . (k + 1)) = k + 1 by A192;
percases ( k = 0 or k = 1 or k > 1 ) by NAT_1:25;
supposeA239: k = 0 ; ::_thesis: S4[k + 1]
then consider i being Element of NAT  such that
A240: 1 <= i and
A241: i + 1 <= len (Gauge (C,n)) and
 N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) and
 N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1)) and
A242: F . (k + 1) = <*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> by A171;
 i < len (Gauge (C,n)) by A241, NAT_1:13;
then A243: [i,(len (Gauge (C,n)))] in Indices (Gauge (C,n)) by A2, A234, A240, MATRIX_1:36;
A244: now__::_thesis:_for_l_being_Element_of_NAT_st_l_in_dom_(F_._(k_+_1))_holds_
ex_i,_j_being_Element_of_NAT_st_
(_[i,j]_in_Indices_(Gauge_(C,n))_&_(F_._(k_+_1))_/._l_=_(Gauge_(C,n))_*_(i,j)_)
let l be Element of NAT ; ::_thesis: ( l in dom (F . (k + 1)) implies ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & (F . (k + 1)) /. l = (Gauge (C,n)) * (i,j) ) )
assume  l in dom (F . (k + 1)) ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & (F . (k + 1)) /. l = (Gauge (C,n)) * (i,j) )
then  ( 1 <= l & l <= 1 ) by A238, A239, FINSEQ_3:25;
then  l = 1 by XXREAL_0:1;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & (F . (k + 1)) /. l = (Gauge (C,n)) * (i,j) ) by A2, A242, A243, FINSEQ_4:16; ::_thesis: verum
end;
now__::_thesis:_for_l_being_Element_of_NAT_st_l_in_dom_(F_._(k_+_1))_&_l_+_1_in_dom_(F_._(k_+_1))_holds_
for_i1,_j1,_i2,_j2_being_Element_of_NAT_st_[i1,j1]_in_Indices_(Gauge_(C,n))_&_[i2,j2]_in_Indices_(Gauge_(C,n))_&_(F_._(k_+_1))_/._l_=_(Gauge_(C,n))_*_(i1,j1)_&_(F_._(k_+_1))_/._(l_+_1)_=_(Gauge_(C,n))_*_(i2,j2)_holds_
(abs_(i1_-_i2))_+_(abs_(j1_-_j2))_=_1
let l be Element of NAT ; ::_thesis: ( l in dom (F . (k + 1)) & l + 1 in dom (F . (k + 1)) implies for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices (Gauge (C,n)) & [i2,j2] in Indices (Gauge (C,n)) & (F . (k + 1)) /. l = (Gauge (C,n)) * (i1,j1) & (F . (k + 1)) /. (l + 1) = (Gauge (C,n)) * (i2,j2) holds
(abs (i1 - i2)) + (abs (j1 - j2)) = 1 )
assume that
A245: l in dom (F . (k + 1)) and
A246: l + 1 in dom (F . (k + 1)) ; ::_thesis: for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices (Gauge (C,n)) & [i2,j2] in Indices (Gauge (C,n)) & (F . (k + 1)) /. l = (Gauge (C,n)) * (i1,j1) & (F . (k + 1)) /. (l + 1) = (Gauge (C,n)) * (i2,j2) holds
(abs (i1 - i2)) + (abs (j1 - j2)) = 1
 ( 1 <= l & l <= 1 ) by A238, A239, A245, FINSEQ_3:25;
then  l = 1 by XXREAL_0:1;
hence  for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices (Gauge (C,n)) & [i2,j2] in Indices (Gauge (C,n)) & (F . (k + 1)) /. l = (Gauge (C,n)) * (i1,j1) & (F . (k + 1)) /. (l + 1) = (Gauge (C,n)) * (i2,j2) holds
(abs (i1 - i2)) + (abs (j1 - j2)) = 1 by A238, A239, A246, FINSEQ_3:25; ::_thesis: verum
end;
hence  F . (k + 1) is_sequence_on Gauge (C,n) by A244, GOBOARD1:def_9; ::_thesis: for m being Element of NAT st 1 <= m & m + 1 <= len (F . (k + 1)) holds
( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
let m be Element of NAT ; ::_thesis: ( 1 <= m & m + 1 <= len (F . (k + 1)) implies ( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C ) )
assume that
A247: 1 <= m and
A248: m + 1 <= len (F . (k + 1)) ; ::_thesis: ( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
 1 <= m + 1 by NAT_1:12;
then  m + 1 = 0 + 1 by A238, A239, A248, XXREAL_0:1;
hence  ( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C ) by A247; ::_thesis: verum
end;
supposeA249: k = 1 ; ::_thesis: S4[k + 1]
then consider i being Element of NAT  such that
A250: 1 <= i and
A251: i + 1 <= len (Gauge (C,n)) and
A252: N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) and
 N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1)) and
A253: F . (k + 1) = <*((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * ((i + 1),(width (Gauge (C,n)))))*> by A171;
A254: i < len (Gauge (C,n)) by A251, NAT_1:13;
then A255: [i,(len (Gauge (C,n)))] in Indices (Gauge (C,n)) by A2, A234, A250, MATRIX_1:36;
 1 <= i + 1 by A250, NAT_1:13;
then A256: [(i + 1),(len (Gauge (C,n)))] in Indices (Gauge (C,n)) by A2, A234, A251, MATRIX_1:36;
A257: ( (F . (k + 1)) /. 1 = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) & (F . (k + 1)) /. 2 = (Gauge (C,n)) * ((i + 1),(width (Gauge (C,n)))) ) by A253, FINSEQ_4:17;
A258: now__::_thesis:_for_l_being_Element_of_NAT_st_l_in_dom_(F_._(k_+_1))_&_l_+_1_in_dom_(F_._(k_+_1))_holds_
for_i1,_j1,_i2,_j2_being_Element_of_NAT_st_[i1,j1]_in_Indices_(Gauge_(C,n))_&_[i2,j2]_in_Indices_(Gauge_(C,n))_&_(F_._(k_+_1))_/._l_=_(Gauge_(C,n))_*_(i1,j1)_&_(F_._(k_+_1))_/._(l_+_1)_=_(Gauge_(C,n))_*_(i2,j2)_holds_
(abs_(i1_-_i2))_+_(abs_(j1_-_j2))_=_1
let l be Element of NAT ; ::_thesis: ( l in dom (F . (k + 1)) & l + 1 in dom (F . (k + 1)) implies for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices (Gauge (C,n)) & [i2,j2] in Indices (Gauge (C,n)) & (F . (k + 1)) /. l = (Gauge (C,n)) * (i1,j1) & (F . (k + 1)) /. (l + 1) = (Gauge (C,n)) * (i2,j2) holds
(abs (i1 - i2)) + (abs (j1 - j2)) = 1 )
assume that
A259: l in dom (F . (k + 1)) and
A260: l + 1 in dom (F . (k + 1)) ; ::_thesis: for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices (Gauge (C,n)) & [i2,j2] in Indices (Gauge (C,n)) & (F . (k + 1)) /. l = (Gauge (C,n)) * (i1,j1) & (F . (k + 1)) /. (l + 1) = (Gauge (C,n)) * (i2,j2) holds
(abs (i1 - i2)) + (abs (j1 - j2)) = 1
 l <= 2 by A238, A249, A259, FINSEQ_3:25;
then A261: ( l = 0 or l = 1 or l = 2 ) by NAT_1:26;
let i1, j1, i2, j2 be Element of NAT ; ::_thesis: ( [i1,j1] in Indices (Gauge (C,n)) & [i2,j2] in Indices (Gauge (C,n)) & (F . (k + 1)) /. l = (Gauge (C,n)) * (i1,j1) & (F . (k + 1)) /. (l + 1) = (Gauge (C,n)) * (i2,j2) implies (abs (i1 - i2)) + (abs (j1 - j2)) = 1 )
assume A262: ( [i1,j1] in Indices (Gauge (C,n)) & [i2,j2] in Indices (Gauge (C,n)) & (F . (k + 1)) /. l = (Gauge (C,n)) * (i1,j1) & (F . (k + 1)) /. (l + 1) = (Gauge (C,n)) * (i2,j2) ) ; ::_thesis: (abs (i1 - i2)) + (abs (j1 - j2)) = 1
 ( j1 = len (Gauge (C,n)) & j2 = len (Gauge (C,n)) ) by A2, A238, A249, A257, A255, A256, A259, A260, A261, A262, FINSEQ_3:25, GOBOARD1:5;
then A263: abs (j1 - j2) = 0 by ABSVALUE:def_1;
 ( i1 = i & i2 = i + 1 ) by A2, A238, A249, A257, A255, A256, A259, A260, A261, A262, FINSEQ_3:25, GOBOARD1:5;
then  abs (i2 - i1) = 1 by ABSVALUE:def_1;
hence  (abs (i1 - i2)) + (abs (j1 - j2)) = 1 by A263, UNIFORM1:11; ::_thesis: verum
end;
now__::_thesis:_for_l_being_Element_of_NAT_st_l_in_dom_(F_._(k_+_1))_holds_
ex_i,_j_being_Element_of_NAT_st_
(_[i,j]_in_Indices_(Gauge_(C,n))_&_(F_._(k_+_1))_/._l_=_(Gauge_(C,n))_*_(i,j)_)
let l be Element of NAT ; ::_thesis: ( l in dom (F . (k + 1)) implies ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & (F . (k + 1)) /. l = (Gauge (C,n)) * (i,j) ) )
assume A264: l in dom (F . (k + 1)) ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & (F . (k + 1)) /. l = (Gauge (C,n)) * (i,j) )
then  l <= 2 by A238, A249, FINSEQ_3:25;
then  ( l = 0 or l = 1 or l = 2 ) by NAT_1:26;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & (F . (k + 1)) /. l = (Gauge (C,n)) * (i,j) ) by A2, A257, A255, A256, A264, FINSEQ_3:25; ::_thesis: verum
end;
hence A265: F . (k + 1) is_sequence_on Gauge (C,n) by A258, GOBOARD1:def_9; ::_thesis: for m being Element of NAT st 1 <= m & m + 1 <= len (F . (k + 1)) holds
( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
A266: ( i < i + 1 & i + 1 < (i + 1) + 1 ) by NAT_1:13;
let m be Element of NAT ; ::_thesis: ( 1 <= m & m + 1 <= len (F . (k + 1)) implies ( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C ) )
assume that
A267: 1 <= m and
A268: m + 1 <= len (F . (k + 1)) ; ::_thesis: ( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
 1 + 1 <= m + 1 by A267, XREAL_1:6;
then A269: m + 1 = 1 + 1 by A238, A249, A268, XXREAL_0:1;
then A270: left_cell ((F . (k + 1)),m,(Gauge (C,n))) = cell ((Gauge (C,n)),i,(len (Gauge (C,n)))) by A2, A257, A255, A256, A265, A268, A266, GOBRD13:def_3;
now__::_thesis:_not_left_cell_((F_._(k_+_1)),m,(Gauge_(C,n)))_meets_C
 N-bound C > S-bound C by JORDAN8:9;
then  (N-bound C) - (S-bound C) > (S-bound C) - (S-bound C) by XREAL_1:9;
then  ((N-bound C) - (S-bound C)) / (2 |^ n) > 0 by A233, XREAL_1:139;
then A271: (N-bound C) + 0 < (N-bound C) + (((N-bound C) - (S-bound C)) / (2 |^ n)) by XREAL_1:6;
 [1,(len (Gauge (C,n)))] in Indices (Gauge (C,n)) by A2, A234, MATRIX_1:36;
then  (Gauge (C,n)) * (1,(len (Gauge (C,n)))) = |[((W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * (1 - 2))),((S-bound C) + ((((N-bound C) - (S-bound C)) / (2 |^ n)) * ((len (Gauge (C,n))) - 2)))]| by JORDAN8:def_1;
then A272: ((Gauge (C,n)) * (1,(len (Gauge (C,n))))) `2 = (S-bound C) + ((((N-bound C) - (S-bound C)) / (2 |^ n)) * ((len (Gauge (C,n))) - 2)) by EUCLID:52;
A273: cell ((Gauge (C,n)),i,(len (Gauge (C,n)))) =  {  |[r,s]| where r, s is Real : ( ((Gauge (C,n)) * (i,1)) `1 <= r & r <= ((Gauge (C,n)) * ((i + 1),1)) `1 & ((Gauge (C,n)) * (1,(len (Gauge (C,n))))) `2 <= s )  }  by A2, A250, A254, GOBRD11:31;
assume  left_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C ; ::_thesis: contradiction
then consider p being set  such that
A274: p in cell ((Gauge (C,n)),i,(len (Gauge (C,n)))) and
A275: p in C by A270, XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A274;
reconsider p = p as Element of (TOP-REAL 2) ;
A276: p `2 <= N-bound C by A275, PSCOMP_1:24;
consider r, s being Real such that
A277: p = |[r,s]| and
 ((Gauge (C,n)) * (i,1)) `1 <= r and
 r <= ((Gauge (C,n)) * ((i + 1),1)) `1 and
A278: ((Gauge (C,n)) * (1,(len (Gauge (C,n))))) `2 <= s by A274, A273;
(((N-bound C) - (S-bound C)) / (2 |^ n)) * ((len (Gauge (C,n))) - 2) = ((((N-bound C) - (S-bound C)) / (2 |^ n)) * (2 |^ n)) + ((((N-bound C) - (S-bound C)) / (2 |^ n)) * 1) by A231
.= ((N-bound C) - (S-bound C)) + (((N-bound C) - (S-bound C)) / (2 |^ n)) by A233, XCMPLX_1:87 ;
then  N-bound C < s by A278, A272, A271, XXREAL_0:2;
hence  contradiction by A277, A276, EUCLID:52; ::_thesis: verum
end;
hence  left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C ; ::_thesis: right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C
 ( N-min C in C & N-min C in right_cell ((F . (k + 1)),m,(Gauge (C,n))) ) by A2, A252, A257, A255, A256, A265, A268, A269, A266, GOBRD13:def_2, SPRECT_1:11;
hence  right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C by XBOOLE_0:3; ::_thesis: verum
end;
supposeA279: k > 1 ; ::_thesis: S4[k + 1]
then A280: len (F . k) in dom (F . k) by A237, FINSEQ_3:25;
A281: ((len (F . k)) -' 1) + 1 = len (F . k) by A237, A279, XREAL_1:235;
then A282: ((len (F . k)) -' 1) + (1 + 1) = (len (F . k)) + 1 ;
A283: 1 <= (len (F . k)) -' 1 by A237, A279, NAT_D:49;
then consider i1, j1, i2, j2 being Element of NAT  such that
A284: [i1,j1] in Indices (Gauge (C,n)) and
A285: (F . k) /. ((len (F . k)) -' 1) = (Gauge (C,n)) * (i1,j1) and
A286: [i2,j2] in Indices (Gauge (C,n)) and
A287: (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) and
 ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A235, A281, JORDAN8:3;
A288: ( i1 = i2 & j1 + 1 = j2 implies [(i2 + 1),j2] in Indices (Gauge (C,n)) ) by A193, A235, A236, A279, A284, A285, A286, A287;
A289: ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 + 1 & j1 = j2 implies [i2,(j2 -' 1)] in Indices (Gauge (C,n)) ) by A193, A235, A236, A279, A284, A285, A286, A287;
A290: ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 + 1 = i2 & j1 = j2 implies [i2,(j2 + 1)] in Indices (Gauge (C,n)) ) by A193, A235, A236, A279, A284, A285, A286, A287;
A291: ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 + 1 = j2 implies [(i2 -' 1),j2] in Indices (Gauge (C,n)) ) by A193, A235, A236, A279, A284, A285, A286, A287;
A292: ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 = j2 + 1 implies [i2,(j2 -' 1)] in Indices (Gauge (C,n)) ) by A193, A235, A236, A279, A284, A285, A286, A287;
A293: ( i1 = i2 + 1 & j1 = j2 implies [i2,(j2 + 1)] in Indices (Gauge (C,n)) ) by A193, A235, A236, A279, A284, A285, A286, A287;
A294: ( i1 + 1 = i2 & j1 = j2 implies [i2,(j2 -' 1)] in Indices (Gauge (C,n)) ) by A193, A235, A236, A279, A284, A285, A286, A287;
A295: 1 <= j2 by A286, MATRIX_1:38;
A296: right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C by A236, A283, A281;
A297: ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 + 1 & j1 = j2 implies [(i2 -' 1),j2] in Indices (Gauge (C,n)) ) by A193, A235, A236, A279, A284, A285, A286, A287;
A298: ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 + 1 = i2 & j1 = j2 implies [(i2 + 1),j2] in Indices (Gauge (C,n)) ) by A193, A235, A236, A279, A284, A285, A286, A287;
A299: ( front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 + 1 = j2 implies [i2,(j2 + 1)] in Indices (Gauge (C,n)) ) by A193, A235, A236, A279, A284, A285, A286, A287;
A300: ( i1 = i2 & j1 = j2 + 1 implies [(i2 -' 1),j2] in Indices (Gauge (C,n)) ) by A193, A235, A236, A279, A284, A285, A286, A287;
A301: ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C & i1 = i2 & j1 = j2 + 1 implies [(i2 + 1),j2] in Indices (Gauge (C,n)) ) by A193, A235, A236, A279, A284, A285, A286, A287;
 (len (F . k)) -' 1 <= len (F . k) by NAT_D:35;
then A302: (len (F . k)) -' 1 in dom (F . k) by A283, FINSEQ_3:25;
A303: 1 <= i2 by A286, MATRIX_1:38;
thus A304: F . (k + 1) is_sequence_on Gauge (C,n) ::_thesis: for m being Element of NAT st 1 <= m & m + 1 <= len (F . (k + 1)) holds
( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
proof
percases ( ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C ) or ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ) or front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ) ;
suppose ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C ) ; ::_thesis: F . (k + 1) is_sequence_on Gauge (C,n)
then consider i, j being Element of NAT  such that
A305: (F . k) ^ <*((Gauge (C,n)) * (i,j))*> turns_right (len (F . k)) -' 1, Gauge (C,n) and
A306: F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> by A171, A235, A237, A279, A296;
thus  F . (k + 1) is_sequence_on Gauge (C,n) ::_thesis: verum
proof
set f = (F . k) ^ <*((Gauge (C,n)) * (i,j))*>;
A307: ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i,j) by FINSEQ_4:67;
A308: ( ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) -' 1) = (Gauge (C,n)) * (i1,j1) & ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) ) by A285, A287, A302, A280, FINSEQ_4:68;
percases ( ( i1 = i2 & j1 + 1 = j2 & ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 + 1),j2) ) or ( i1 + 1 = i2 & j1 = j2 & ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 -' 1)) ) or ( i1 = i2 + 1 & j1 = j2 & ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 + 1)) ) or ( i1 = i2 & j1 = j2 + 1 & ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 -' 1),j2) ) ) by A281, A284, A286, A282, A305, A308, GOBRD13:def_6;
supposethat A309: ( i1 = i2 & j1 + 1 = j2 ) and
A310: ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 + 1),j2) ; ::_thesis: F . (k + 1) is_sequence_on Gauge (C,n)
now__::_thesis:_for_i19,_j19,_i29,_j29_being_Element_of_NAT_st_[i19,j19]_in_Indices_(Gauge_(C,n))_&_[i29,j29]_in_Indices_(Gauge_(C,n))_&_(F_._k)_/._(len_(F_._k))_=_(Gauge_(C,n))_*_(i19,j19)_&_(Gauge_(C,n))_*_((i2_+_1),j2)_=_(Gauge_(C,n))_*_(i29,j29)_holds_
(abs_(i29_-_i19))_+_(abs_(j29_-_j19))_=_1
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) & (Gauge (C,n)) * ((i2 + 1),j2) = (Gauge (C,n)) * (i29,j29) implies (abs (i29 - i19)) + (abs (j29 - j19)) = 1 )
assume A311: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) & (Gauge (C,n)) * ((i2 + 1),j2) = (Gauge (C,n)) * (i29,j29) ) ; ::_thesis: (abs (i29 - i19)) + (abs (j29 - j19)) = 1
then  ( j2 = j19 & j2 = j29 ) by A286, A287, A288, A309, GOBOARD1:5;
then A312: abs (j29 - j19) = 0 by ABSVALUE:def_1;
 ( i2 = i19 & i2 + 1 = i29 ) by A286, A287, A288, A309, A311, GOBOARD1:5;
hence  (abs (i29 - i19)) + (abs (j29 - j19)) = 1 by A312, ABSVALUE:def_1; ::_thesis: verum
end;
hence  F . (k + 1) is_sequence_on Gauge (C,n) by A235, A237, A279, A288, A306, A307, A309, A310, CARD_1:27, JORDAN8:6; ::_thesis: verum
end;
supposethat A313: ( i1 + 1 = i2 & j1 = j2 ) and
A314: ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 -' 1)) ; ::_thesis: F . (k + 1) is_sequence_on Gauge (C,n)
now__::_thesis:_for_i19,_j19,_i29,_j29_being_Element_of_NAT_st_[i19,j19]_in_Indices_(Gauge_(C,n))_&_[i29,j29]_in_Indices_(Gauge_(C,n))_&_(F_._k)_/._(len_(F_._k))_=_(Gauge_(C,n))_*_(i19,j19)_&_(Gauge_(C,n))_*_(i2,(j2_-'_1))_=_(Gauge_(C,n))_*_(i29,j29)_holds_
(abs_(i29_-_i19))_+_(abs_(j29_-_j19))_=_1
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) & (Gauge (C,n)) * (i2,(j2 -' 1)) = (Gauge (C,n)) * (i29,j29) implies (abs (i29 - i19)) + (abs (j29 - j19)) = 1 )
assume A315: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) & (Gauge (C,n)) * (i2,(j2 -' 1)) = (Gauge (C,n)) * (i29,j29) ) ; ::_thesis: (abs (i29 - i19)) + (abs (j29 - j19)) = 1
then  ( j2 = j19 & j2 -' 1 = j29 ) by A286, A287, A294, A313, GOBOARD1:5;
then  j19 - j29 = j2 - (j2 - 1) by A295, XREAL_1:233;
then A316: abs (j19 - j29) = 1 by ABSVALUE:def_1;
 ( i2 = i19 & i2 = i29 ) by A286, A287, A294, A313, A315, GOBOARD1:5;
then  abs (i29 - i19) = 0 by ABSVALUE:def_1;
hence  (abs (i29 - i19)) + (abs (j29 - j19)) = 1 by A316, UNIFORM1:11; ::_thesis: verum
end;
hence  F . (k + 1) is_sequence_on Gauge (C,n) by A235, A237, A279, A294, A306, A307, A313, A314, CARD_1:27, JORDAN8:6; ::_thesis: verum
end;
supposethat A317: ( i1 = i2 + 1 & j1 = j2 ) and
A318: ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 + 1)) ; ::_thesis: F . (k + 1) is_sequence_on Gauge (C,n)
now__::_thesis:_for_i19,_j19,_i29,_j29_being_Element_of_NAT_st_[i19,j19]_in_Indices_(Gauge_(C,n))_&_[i29,j29]_in_Indices_(Gauge_(C,n))_&_(F_._k)_/._(len_(F_._k))_=_(Gauge_(C,n))_*_(i19,j19)_&_(Gauge_(C,n))_*_(i2,(j2_+_1))_=_(Gauge_(C,n))_*_(i29,j29)_holds_
(abs_(i29_-_i19))_+_(abs_(j29_-_j19))_=_1
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) & (Gauge (C,n)) * (i2,(j2 + 1)) = (Gauge (C,n)) * (i29,j29) implies (abs (i29 - i19)) + (abs (j29 - j19)) = 1 )
assume A319: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) & (Gauge (C,n)) * (i2,(j2 + 1)) = (Gauge (C,n)) * (i29,j29) ) ; ::_thesis: (abs (i29 - i19)) + (abs (j29 - j19)) = 1
then  ( i2 = i19 & i2 = i29 ) by A286, A287, A293, A317, GOBOARD1:5;
then A320: abs (i29 - i19) = 0 by ABSVALUE:def_1;
 ( j2 = j19 & j2 + 1 = j29 ) by A286, A287, A293, A317, A319, GOBOARD1:5;
hence  (abs (i29 - i19)) + (abs (j29 - j19)) = 1 by A320, ABSVALUE:def_1; ::_thesis: verum
end;
hence  F . (k + 1) is_sequence_on Gauge (C,n) by A235, A237, A279, A293, A306, A307, A317, A318, CARD_1:27, JORDAN8:6; ::_thesis: verum
end;
supposethat A321: ( i1 = i2 & j1 = j2 + 1 ) and
A322: ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 -' 1),j2) ; ::_thesis: F . (k + 1) is_sequence_on Gauge (C,n)
now__::_thesis:_for_i19,_j19,_i29,_j29_being_Element_of_NAT_st_[i19,j19]_in_Indices_(Gauge_(C,n))_&_[i29,j29]_in_Indices_(Gauge_(C,n))_&_(F_._k)_/._(len_(F_._k))_=_(Gauge_(C,n))_*_(i19,j19)_&_(Gauge_(C,n))_*_((i2_-'_1),j2)_=_(Gauge_(C,n))_*_(i29,j29)_holds_
(abs_(i29_-_i19))_+_(abs_(j29_-_j19))_=_1
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) & (Gauge (C,n)) * ((i2 -' 1),j2) = (Gauge (C,n)) * (i29,j29) implies (abs (i29 - i19)) + (abs (j29 - j19)) = 1 )
assume A323: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) & (Gauge (C,n)) * ((i2 -' 1),j2) = (Gauge (C,n)) * (i29,j29) ) ; ::_thesis: (abs (i29 - i19)) + (abs (j29 - j19)) = 1
then  ( i2 = i19 & i2 -' 1 = i29 ) by A286, A287, A300, A321, GOBOARD1:5;
then  i19 - i29 = i2 - (i2 - 1) by A303, XREAL_1:233;
then A324: abs (i19 - i29) = 1 by ABSVALUE:def_1;
 ( j2 = j19 & j2 = j29 ) by A286, A287, A300, A321, A323, GOBOARD1:5;
then  abs (j29 - j19) = 0 by ABSVALUE:def_1;
hence  (abs (i29 - i19)) + (abs (j29 - j19)) = 1 by A324, UNIFORM1:11; ::_thesis: verum
end;
hence  F . (k + 1) is_sequence_on Gauge (C,n) by A235, A237, A279, A300, A306, A307, A321, A322, CARD_1:27, JORDAN8:6; ::_thesis: verum
end;
end;
end;
end;
supposeA325: ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ) ; ::_thesis: F . (k + 1) is_sequence_on Gauge (C,n)
then consider i, j being Element of NAT  such that
A326: (F . k) ^ <*((Gauge (C,n)) * (i,j))*> goes_straight (len (F . k)) -' 1, Gauge (C,n) and
A327: F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> by A171, A235, A237, A279, A296;
thus  F . (k + 1) is_sequence_on Gauge (C,n) ::_thesis: verum
proof
set f = (F . k) ^ <*((Gauge (C,n)) * (i,j))*>;
A328: ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i,j) by FINSEQ_4:67;
A329: ( ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) -' 1) = (Gauge (C,n)) * (i1,j1) & ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) ) by A285, A287, A302, A280, FINSEQ_4:68;
percases ( ( i1 = i2 & j1 + 1 = j2 & ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 + 1)) ) or ( i1 + 1 = i2 & j1 = j2 & ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 + 1),j2) ) or ( i1 = i2 + 1 & j1 = j2 & ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 -' 1),j2) ) or ( i1 = i2 & j1 = j2 + 1 & ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 -' 1)) ) ) by A281, A284, A286, A282, A326, A329, GOBRD13:def_8;
supposethat A330: ( i1 = i2 & j1 + 1 = j2 ) and
A331: ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 + 1)) ; ::_thesis: F . (k + 1) is_sequence_on Gauge (C,n)
now__::_thesis:_for_i19,_j19,_i29,_j29_being_Element_of_NAT_st_[i19,j19]_in_Indices_(Gauge_(C,n))_&_[i29,j29]_in_Indices_(Gauge_(C,n))_&_(F_._k)_/._(len_(F_._k))_=_(Gauge_(C,n))_*_(i19,j19)_&_(Gauge_(C,n))_*_(i2,(j2_+_1))_=_(Gauge_(C,n))_*_(i29,j29)_holds_
(abs_(i29_-_i19))_+_(abs_(j29_-_j19))_=_1
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) & (Gauge (C,n)) * (i2,(j2 + 1)) = (Gauge (C,n)) * (i29,j29) implies (abs (i29 - i19)) + (abs (j29 - j19)) = 1 )
assume A332: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) & (Gauge (C,n)) * (i2,(j2 + 1)) = (Gauge (C,n)) * (i29,j29) ) ; ::_thesis: (abs (i29 - i19)) + (abs (j29 - j19)) = 1
then  ( i2 = i19 & i2 = i29 ) by A286, A287, A299, A325, A330, GOBOARD1:5;
then A333: abs (i29 - i19) = 0 by ABSVALUE:def_1;
 ( j2 = j19 & j2 + 1 = j29 ) by A286, A287, A299, A325, A330, A332, GOBOARD1:5;
hence  (abs (i29 - i19)) + (abs (j29 - j19)) = 1 by A333, ABSVALUE:def_1; ::_thesis: verum
end;
hence  F . (k + 1) is_sequence_on Gauge (C,n) by A235, A237, A279, A299, A325, A327, A328, A330, A331, CARD_1:27, JORDAN8:6; ::_thesis: verum
end;
supposethat A334: ( i1 + 1 = i2 & j1 = j2 ) and
A335: ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 + 1),j2) ; ::_thesis: F . (k + 1) is_sequence_on Gauge (C,n)
now__::_thesis:_for_i19,_j19,_i29,_j29_being_Element_of_NAT_st_[i19,j19]_in_Indices_(Gauge_(C,n))_&_[i29,j29]_in_Indices_(Gauge_(C,n))_&_(F_._k)_/._(len_(F_._k))_=_(Gauge_(C,n))_*_(i19,j19)_&_(Gauge_(C,n))_*_((i2_+_1),j2)_=_(Gauge_(C,n))_*_(i29,j29)_holds_
(abs_(i29_-_i19))_+_(abs_(j29_-_j19))_=_1
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) & (Gauge (C,n)) * ((i2 + 1),j2) = (Gauge (C,n)) * (i29,j29) implies (abs (i29 - i19)) + (abs (j29 - j19)) = 1 )
assume A336: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) & (Gauge (C,n)) * ((i2 + 1),j2) = (Gauge (C,n)) * (i29,j29) ) ; ::_thesis: (abs (i29 - i19)) + (abs (j29 - j19)) = 1
then  ( j2 = j19 & j2 = j29 ) by A286, A287, A298, A325, A334, GOBOARD1:5;
then A337: abs (j29 - j19) = 0 by ABSVALUE:def_1;
 ( i2 = i19 & i2 + 1 = i29 ) by A286, A287, A298, A325, A334, A336, GOBOARD1:5;
hence  (abs (i29 - i19)) + (abs (j29 - j19)) = 1 by A337, ABSVALUE:def_1; ::_thesis: verum
end;
hence  F . (k + 1) is_sequence_on Gauge (C,n) by A235, A237, A279, A298, A325, A327, A328, A334, A335, CARD_1:27, JORDAN8:6; ::_thesis: verum
end;
supposethat A338: ( i1 = i2 + 1 & j1 = j2 ) and
A339: ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 -' 1),j2) ; ::_thesis: F . (k + 1) is_sequence_on Gauge (C,n)
now__::_thesis:_for_i19,_j19,_i29,_j29_being_Element_of_NAT_st_[i19,j19]_in_Indices_(Gauge_(C,n))_&_[i29,j29]_in_Indices_(Gauge_(C,n))_&_(F_._k)_/._(len_(F_._k))_=_(Gauge_(C,n))_*_(i19,j19)_&_(Gauge_(C,n))_*_((i2_-'_1),j2)_=_(Gauge_(C,n))_*_(i29,j29)_holds_
(abs_(i29_-_i19))_+_(abs_(j29_-_j19))_=_1
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) & (Gauge (C,n)) * ((i2 -' 1),j2) = (Gauge (C,n)) * (i29,j29) implies (abs (i29 - i19)) + (abs (j29 - j19)) = 1 )
assume A340: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) & (Gauge (C,n)) * ((i2 -' 1),j2) = (Gauge (C,n)) * (i29,j29) ) ; ::_thesis: (abs (i29 - i19)) + (abs (j29 - j19)) = 1
then  ( i2 = i19 & i2 -' 1 = i29 ) by A286, A287, A297, A325, A338, GOBOARD1:5;
then  i19 - i29 = i2 - (i2 - 1) by A303, XREAL_1:233;
then A341: abs (i19 - i29) = 1 by ABSVALUE:def_1;
 ( j2 = j19 & j2 = j29 ) by A286, A287, A297, A325, A338, A340, GOBOARD1:5;
then  abs (j29 - j19) = 0 by ABSVALUE:def_1;
hence  (abs (i29 - i19)) + (abs (j29 - j19)) = 1 by A341, UNIFORM1:11; ::_thesis: verum
end;
hence  F . (k + 1) is_sequence_on Gauge (C,n) by A235, A237, A279, A297, A325, A327, A328, A338, A339, CARD_1:27, JORDAN8:6; ::_thesis: verum
end;
supposethat A342: ( i1 = i2 & j1 = j2 + 1 ) and
A343: ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 -' 1)) ; ::_thesis: F . (k + 1) is_sequence_on Gauge (C,n)
now__::_thesis:_for_i19,_j19,_i29,_j29_being_Element_of_NAT_st_[i19,j19]_in_Indices_(Gauge_(C,n))_&_[i29,j29]_in_Indices_(Gauge_(C,n))_&_(F_._k)_/._(len_(F_._k))_=_(Gauge_(C,n))_*_(i19,j19)_&_(Gauge_(C,n))_*_(i2,(j2_-'_1))_=_(Gauge_(C,n))_*_(i29,j29)_holds_
(abs_(i29_-_i19))_+_(abs_(j29_-_j19))_=_1
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) & (Gauge (C,n)) * (i2,(j2 -' 1)) = (Gauge (C,n)) * (i29,j29) implies (abs (i29 - i19)) + (abs (j29 - j19)) = 1 )
assume A344: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) & (Gauge (C,n)) * (i2,(j2 -' 1)) = (Gauge (C,n)) * (i29,j29) ) ; ::_thesis: (abs (i29 - i19)) + (abs (j29 - j19)) = 1
then  ( j2 = j19 & j2 -' 1 = j29 ) by A286, A287, A292, A325, A342, GOBOARD1:5;
then  j19 - j29 = j2 - (j2 - 1) by A295, XREAL_1:233;
then A345: abs (j19 - j29) = 1 by ABSVALUE:def_1;
 ( i2 = i19 & i2 = i29 ) by A286, A287, A292, A325, A342, A344, GOBOARD1:5;
then  abs (i29 - i19) = 0 by ABSVALUE:def_1;
hence  (abs (i29 - i19)) + (abs (j29 - j19)) = 1 by A345, UNIFORM1:11; ::_thesis: verum
end;
hence  F . (k + 1) is_sequence_on Gauge (C,n) by A235, A237, A279, A292, A325, A327, A328, A342, A343, CARD_1:27, JORDAN8:6; ::_thesis: verum
end;
end;
end;
end;
supposeA346: front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ; ::_thesis: F . (k + 1) is_sequence_on Gauge (C,n)
then consider i, j being Element of NAT  such that
A347: (F . k) ^ <*((Gauge (C,n)) * (i,j))*> turns_left (len (F . k)) -' 1, Gauge (C,n) and
A348: F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> by A171, A235, A237, A279, A296;
thus  F . (k + 1) is_sequence_on Gauge (C,n) ::_thesis: verum
proof
set f = (F . k) ^ <*((Gauge (C,n)) * (i,j))*>;
A349: ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i,j) by FINSEQ_4:67;
A350: ( ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) -' 1) = (Gauge (C,n)) * (i1,j1) & ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) ) by A285, A287, A302, A280, FINSEQ_4:68;
percases ( ( i1 = i2 & j1 + 1 = j2 & ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 -' 1),j2) ) or ( i1 + 1 = i2 & j1 = j2 & ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 + 1)) ) or ( i1 = i2 + 1 & j1 = j2 & ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 -' 1)) ) or ( i1 = i2 & j1 = j2 + 1 & ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 + 1),j2) ) ) by A281, A284, A286, A282, A347, A350, GOBRD13:def_7;
supposethat A351: ( i1 = i2 & j1 + 1 = j2 ) and
A352: ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 -' 1),j2) ; ::_thesis: F . (k + 1) is_sequence_on Gauge (C,n)
now__::_thesis:_for_i19,_j19,_i29,_j29_being_Element_of_NAT_st_[i19,j19]_in_Indices_(Gauge_(C,n))_&_[i29,j29]_in_Indices_(Gauge_(C,n))_&_(F_._k)_/._(len_(F_._k))_=_(Gauge_(C,n))_*_(i19,j19)_&_(Gauge_(C,n))_*_((i2_-'_1),j2)_=_(Gauge_(C,n))_*_(i29,j29)_holds_
(abs_(i29_-_i19))_+_(abs_(j29_-_j19))_=_1
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) & (Gauge (C,n)) * ((i2 -' 1),j2) = (Gauge (C,n)) * (i29,j29) implies (abs (i29 - i19)) + (abs (j29 - j19)) = 1 )
assume A353: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) & (Gauge (C,n)) * ((i2 -' 1),j2) = (Gauge (C,n)) * (i29,j29) ) ; ::_thesis: (abs (i29 - i19)) + (abs (j29 - j19)) = 1
then  ( i2 = i19 & i2 -' 1 = i29 ) by A286, A287, A291, A346, A351, GOBOARD1:5;
then  i19 - i29 = i2 - (i2 - 1) by A303, XREAL_1:233;
then A354: abs (i19 - i29) = 1 by ABSVALUE:def_1;
 ( j2 = j19 & j2 = j29 ) by A286, A287, A291, A346, A351, A353, GOBOARD1:5;
then  abs (j29 - j19) = 0 by ABSVALUE:def_1;
hence  (abs (i29 - i19)) + (abs (j29 - j19)) = 1 by A354, UNIFORM1:11; ::_thesis: verum
end;
hence  F . (k + 1) is_sequence_on Gauge (C,n) by A235, A237, A279, A291, A346, A348, A349, A351, A352, CARD_1:27, JORDAN8:6; ::_thesis: verum
end;
supposethat A355: ( i1 + 1 = i2 & j1 = j2 ) and
A356: ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 + 1)) ; ::_thesis: F . (k + 1) is_sequence_on Gauge (C,n)
now__::_thesis:_for_i19,_j19,_i29,_j29_being_Element_of_NAT_st_[i19,j19]_in_Indices_(Gauge_(C,n))_&_[i29,j29]_in_Indices_(Gauge_(C,n))_&_(F_._k)_/._(len_(F_._k))_=_(Gauge_(C,n))_*_(i19,j19)_&_(Gauge_(C,n))_*_(i2,(j2_+_1))_=_(Gauge_(C,n))_*_(i29,j29)_holds_
(abs_(i29_-_i19))_+_(abs_(j29_-_j19))_=_1
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) & (Gauge (C,n)) * (i2,(j2 + 1)) = (Gauge (C,n)) * (i29,j29) implies (abs (i29 - i19)) + (abs (j29 - j19)) = 1 )
assume A357: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) & (Gauge (C,n)) * (i2,(j2 + 1)) = (Gauge (C,n)) * (i29,j29) ) ; ::_thesis: (abs (i29 - i19)) + (abs (j29 - j19)) = 1
then  ( i2 = i19 & i2 = i29 ) by A286, A287, A290, A346, A355, GOBOARD1:5;
then A358: abs (i29 - i19) = 0 by ABSVALUE:def_1;
 ( j2 = j19 & j2 + 1 = j29 ) by A286, A287, A290, A346, A355, A357, GOBOARD1:5;
hence  (abs (i29 - i19)) + (abs (j29 - j19)) = 1 by A358, ABSVALUE:def_1; ::_thesis: verum
end;
hence  F . (k + 1) is_sequence_on Gauge (C,n) by A235, A237, A279, A290, A346, A348, A349, A355, A356, CARD_1:27, JORDAN8:6; ::_thesis: verum
end;
supposethat A359: ( i1 = i2 + 1 & j1 = j2 ) and
A360: ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 -' 1)) ; ::_thesis: F . (k + 1) is_sequence_on Gauge (C,n)
now__::_thesis:_for_i19,_j19,_i29,_j29_being_Element_of_NAT_st_[i19,j19]_in_Indices_(Gauge_(C,n))_&_[i29,j29]_in_Indices_(Gauge_(C,n))_&_(F_._k)_/._(len_(F_._k))_=_(Gauge_(C,n))_*_(i19,j19)_&_(Gauge_(C,n))_*_(i2,(j2_-'_1))_=_(Gauge_(C,n))_*_(i29,j29)_holds_
(abs_(i29_-_i19))_+_(abs_(j29_-_j19))_=_1
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) & (Gauge (C,n)) * (i2,(j2 -' 1)) = (Gauge (C,n)) * (i29,j29) implies (abs (i29 - i19)) + (abs (j29 - j19)) = 1 )
assume A361: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) & (Gauge (C,n)) * (i2,(j2 -' 1)) = (Gauge (C,n)) * (i29,j29) ) ; ::_thesis: (abs (i29 - i19)) + (abs (j29 - j19)) = 1
then  ( j2 = j19 & j2 -' 1 = j29 ) by A286, A287, A289, A346, A359, GOBOARD1:5;
then  j19 - j29 = j2 - (j2 - 1) by A295, XREAL_1:233;
then A362: abs (j19 - j29) = 1 by ABSVALUE:def_1;
 ( i2 = i19 & i2 = i29 ) by A286, A287, A289, A346, A359, A361, GOBOARD1:5;
then  abs (i29 - i19) = 0 by ABSVALUE:def_1;
hence  (abs (i29 - i19)) + (abs (j29 - j19)) = 1 by A362, UNIFORM1:11; ::_thesis: verum
end;
hence  F . (k + 1) is_sequence_on Gauge (C,n) by A235, A237, A279, A289, A346, A348, A349, A359, A360, CARD_1:27, JORDAN8:6; ::_thesis: verum
end;
supposethat A363: ( i1 = i2 & j1 = j2 + 1 ) and
A364: ((F . k) ^ <*((Gauge (C,n)) * (i,j))*>) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 + 1),j2) ; ::_thesis: F . (k + 1) is_sequence_on Gauge (C,n)
now__::_thesis:_for_i19,_j19,_i29,_j29_being_Element_of_NAT_st_[i19,j19]_in_Indices_(Gauge_(C,n))_&_[i29,j29]_in_Indices_(Gauge_(C,n))_&_(F_._k)_/._(len_(F_._k))_=_(Gauge_(C,n))_*_(i19,j19)_&_(Gauge_(C,n))_*_((i2_+_1),j2)_=_(Gauge_(C,n))_*_(i29,j29)_holds_
(abs_(i29_-_i19))_+_(abs_(j29_-_j19))_=_1
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) & (Gauge (C,n)) * ((i2 + 1),j2) = (Gauge (C,n)) * (i29,j29) implies (abs (i29 - i19)) + (abs (j29 - j19)) = 1 )
assume A365: ( [i19,j19] in Indices (Gauge (C,n)) & [i29,j29] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i19,j19) & (Gauge (C,n)) * ((i2 + 1),j2) = (Gauge (C,n)) * (i29,j29) ) ; ::_thesis: (abs (i29 - i19)) + (abs (j29 - j19)) = 1
then  ( j2 = j19 & j2 = j29 ) by A286, A287, A301, A346, A363, GOBOARD1:5;
then A366: abs (j29 - j19) = 0 by ABSVALUE:def_1;
 ( i2 = i19 & i2 + 1 = i29 ) by A286, A287, A301, A346, A363, A365, GOBOARD1:5;
hence  (abs (i29 - i19)) + (abs (j29 - j19)) = 1 by A366, ABSVALUE:def_1; ::_thesis: verum
end;
hence  F . (k + 1) is_sequence_on Gauge (C,n) by A235, A237, A279, A301, A346, A348, A349, A363, A364, CARD_1:27, JORDAN8:6; ::_thesis: verum
end;
end;
end;
end;
end;
end;
let m be Element of NAT ; ::_thesis: ( 1 <= m & m + 1 <= len (F . (k + 1)) implies ( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C ) )
assume that
A367: 1 <= m and
A368: m + 1 <= len (F . (k + 1)) ; ::_thesis: ( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
A369: left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C by A236, A283, A281;
now__::_thesis:_(_left_cell_((F_._(k_+_1)),m,(Gauge_(C,n)))_misses_C_&_right_cell_((F_._(k_+_1)),m,(Gauge_(C,n)))_meets_C_)
percases ( m + 1 = len (F . (k + 1)) or m + 1 <> len (F . (k + 1)) ) ;
supposeA370: m + 1 = len (F . (k + 1)) ; ::_thesis: ( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
A371: (j2 -' 1) + 1 = j2 by A295, XREAL_1:235;
A372: (i2 -' 1) + 1 = i2 by A303, XREAL_1:235;
thus  ( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C ) ::_thesis: verum
proof
percases ( ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C ) or ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ) or front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ) ;
supposeA373: ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C ) ; ::_thesis: ( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
then A374: ex i, j being Element of NAT st
( (F . k) ^ <*((Gauge (C,n)) * (i,j))*> turns_right (len (F . k)) -' 1, Gauge (C,n) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) by A171, A235, A237, A279, A296;
then A375: (F . (k + 1)) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) by A287, A280, FINSEQ_4:68;
A376: (F . (k + 1)) /. ((len (F . k)) -' 1) = (Gauge (C,n)) * (i1,j1) by A285, A302, A374, FINSEQ_4:68;
now__::_thesis:_(_left_cell_((F_._(k_+_1)),m,(Gauge_(C,n)))_misses_C_&_right_cell_((F_._(k_+_1)),m,(Gauge_(C,n)))_meets_C_)
percases ( ( i1 = i2 & j1 + 1 = j2 & (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 + 1),j2) ) or ( i1 + 1 = i2 & j1 = j2 & (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 -' 1)) ) or ( i1 = i2 + 1 & j1 = j2 & (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 + 1)) ) or ( i1 = i2 & j1 = j2 + 1 & (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 -' 1),j2) ) ) by A281, A284, A286, A282, A374, A376, A375, GOBRD13:def_6;
supposethat A377: ( i1 = i2 & j1 + 1 = j2 ) and
A378: (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 + 1),j2) ; ::_thesis: ( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
 front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),i1,j2) by A235, A283, A281, A284, A285, A286, A287, A377, GOBRD13:35;
hence  left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C by A237, A238, A286, A288, A304, A367, A370, A373, A375, A377, A378, GOBRD13:23; ::_thesis: right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C
 ( j2 -' 1 = j1 & cell ((Gauge (C,n)),i1,j1) meets C ) by A235, A283, A281, A284, A285, A286, A287, A296, A377, GOBRD13:22, NAT_D:34;
hence  right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C by A237, A238, A286, A288, A304, A367, A370, A375, A377, A378, GOBRD13:24; ::_thesis: verum
end;
supposethat A379: ( i1 + 1 = i2 & j1 = j2 ) and
A380: (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 -' 1)) ; ::_thesis: ( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
 front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),i2,(j2 -' 1)) by A235, A283, A281, A284, A285, A286, A287, A379, GOBRD13:37;
hence  left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C by A237, A238, A286, A294, A304, A367, A370, A371, A373, A375, A379, A380, GOBRD13:27; ::_thesis: right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C
 ( i2 -' 1 = i1 & cell ((Gauge (C,n)),i1,(j1 -' 1)) meets C ) by A235, A283, A281, A284, A285, A286, A287, A296, A379, GOBRD13:24, NAT_D:34;
hence  right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C by A237, A238, A286, A294, A304, A367, A370, A371, A375, A379, A380, GOBRD13:28; ::_thesis: verum
end;
supposethat A381: ( i1 = i2 + 1 & j1 = j2 ) and
A382: (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 + 1)) ; ::_thesis: ( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
 front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),(i2 -' 1),j2) by A235, A283, A281, A284, A285, A286, A287, A381, GOBRD13:39;
hence  left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C by A237, A238, A286, A293, A304, A367, A370, A373, A375, A381, A382, GOBRD13:21; ::_thesis: right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C
 cell ((Gauge (C,n)),i2,j2) meets C by A235, A283, A281, A284, A285, A286, A287, A296, A381, GOBRD13:26;
hence  right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C by A237, A238, A286, A293, A304, A367, A370, A375, A381, A382, GOBRD13:22; ::_thesis: verum
end;
supposethat A383: ( i1 = i2 & j1 = j2 + 1 ) and
A384: (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 -' 1),j2) ; ::_thesis: ( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
 front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),(i2 -' 1),(j2 -' 1)) by A235, A283, A281, A284, A285, A286, A287, A383, GOBRD13:41;
hence  left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C by A237, A238, A286, A300, A304, A367, A370, A372, A373, A375, A383, A384, GOBRD13:25; ::_thesis: right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C
 cell ((Gauge (C,n)),(i2 -' 1),j2) meets C by A235, A283, A281, A284, A285, A286, A287, A296, A383, GOBRD13:28;
hence  right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C by A237, A238, A286, A300, A304, A367, A370, A372, A375, A383, A384, GOBRD13:26; ::_thesis: verum
end;
end;
end;
hence  ( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C ) ; ::_thesis: verum
end;
supposeA385: ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ) ; ::_thesis: ( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
then A386: ex i, j being Element of NAT st
( (F . k) ^ <*((Gauge (C,n)) * (i,j))*> goes_straight (len (F . k)) -' 1, Gauge (C,n) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) by A171, A235, A237, A279, A296;
then A387: (F . (k + 1)) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) by A287, A280, FINSEQ_4:68;
A388: (F . (k + 1)) /. ((len (F . k)) -' 1) = (Gauge (C,n)) * (i1,j1) by A285, A302, A386, FINSEQ_4:68;
now__::_thesis:_(_left_cell_((F_._(k_+_1)),m,(Gauge_(C,n)))_misses_C_&_right_cell_((F_._(k_+_1)),m,(Gauge_(C,n)))_meets_C_)
percases ( ( i1 = i2 & j1 + 1 = j2 & (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 + 1)) ) or ( i1 + 1 = i2 & j1 = j2 & (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 + 1),j2) ) or ( i1 = i2 + 1 & j1 = j2 & (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 -' 1),j2) ) or ( i1 = i2 & j1 = j2 + 1 & (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 -' 1)) ) ) by A281, A284, A286, A282, A386, A388, A387, GOBRD13:def_8;
supposethat A389: ( i1 = i2 & j1 + 1 = j2 ) and
A390: (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 + 1)) ; ::_thesis: ( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
 front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),(i1 -' 1),j2) by A235, A283, A281, A284, A285, A286, A287, A389, GOBRD13:34;
hence  left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C by A237, A238, A286, A299, A304, A367, A370, A385, A387, A389, A390, GOBRD13:21; ::_thesis: right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C
 front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),i1,j2) by A235, A283, A281, A284, A285, A286, A287, A389, GOBRD13:35;
hence  right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C by A237, A238, A286, A299, A304, A367, A370, A385, A387, A389, A390, GOBRD13:22; ::_thesis: verum
end;
supposethat A391: ( i1 + 1 = i2 & j1 = j2 ) and
A392: (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 + 1),j2) ; ::_thesis: ( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
 front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),i2,j2) by A235, A283, A281, A284, A285, A286, A287, A391, GOBRD13:36;
hence  left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C by A237, A238, A286, A298, A304, A367, A370, A385, A387, A391, A392, GOBRD13:23; ::_thesis: right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C
 front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),i2,(j2 -' 1)) by A235, A283, A281, A284, A285, A286, A287, A391, GOBRD13:37;
hence  right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C by A237, A238, A286, A298, A304, A367, A370, A385, A387, A391, A392, GOBRD13:24; ::_thesis: verum
end;
supposethat A393: ( i1 = i2 + 1 & j1 = j2 ) and
A394: (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 -' 1),j2) ; ::_thesis: ( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
 front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),(i2 -' 1),(j2 -' 1)) by A235, A283, A281, A284, A285, A286, A287, A393, GOBRD13:38;
hence  left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C by A237, A238, A286, A297, A304, A367, A370, A372, A385, A387, A393, A394, GOBRD13:25; ::_thesis: right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C
 front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),(i2 -' 1),j2) by A235, A283, A281, A284, A285, A286, A287, A393, GOBRD13:39;
hence  right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C by A237, A238, A286, A297, A304, A367, A370, A372, A385, A387, A393, A394, GOBRD13:26; ::_thesis: verum
end;
supposethat A395: ( i1 = i2 & j1 = j2 + 1 ) and
A396: (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 -' 1)) ; ::_thesis: ( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
 front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),i2,(j2 -' 1)) by A235, A283, A281, A284, A285, A286, A287, A395, GOBRD13:40;
hence  left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C by A237, A238, A286, A292, A304, A367, A370, A371, A385, A387, A395, A396, GOBRD13:27; ::_thesis: right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C
 front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),(i2 -' 1),(j2 -' 1)) by A235, A283, A281, A284, A285, A286, A287, A395, GOBRD13:41;
hence  right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C by A237, A238, A286, A292, A304, A367, A370, A371, A385, A387, A395, A396, GOBRD13:28; ::_thesis: verum
end;
end;
end;
hence  ( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C ) ; ::_thesis: verum
end;
supposeA397: front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ; ::_thesis: ( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
then A398: ex i, j being Element of NAT st
( (F . k) ^ <*((Gauge (C,n)) * (i,j))*> turns_left (len (F . k)) -' 1, Gauge (C,n) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) by A171, A235, A237, A279, A296;
then A399: (F . (k + 1)) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) by A287, A280, FINSEQ_4:68;
A400: (F . (k + 1)) /. ((len (F . k)) -' 1) = (Gauge (C,n)) * (i1,j1) by A285, A302, A398, FINSEQ_4:68;
now__::_thesis:_(_left_cell_((F_._(k_+_1)),m,(Gauge_(C,n)))_misses_C_&_right_cell_((F_._(k_+_1)),m,(Gauge_(C,n)))_meets_C_)
percases ( ( i1 = i2 & j1 + 1 = j2 & (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 -' 1),j2) ) or ( i1 + 1 = i2 & j1 = j2 & (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 + 1)) ) or ( i1 = i2 + 1 & j1 = j2 & (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 -' 1)) ) or ( i1 = i2 & j1 = j2 + 1 & (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 + 1),j2) ) ) by A281, A284, A286, A282, A398, A400, A399, GOBRD13:def_7;
supposethat A401: ( i1 = i2 & j1 + 1 = j2 ) and
A402: (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 -' 1),j2) ; ::_thesis: ( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
 ( j2 -' 1 = j1 & cell ((Gauge (C,n)),(i1 -' 1),j1) misses C ) by A235, A283, A281, A284, A285, A286, A287, A369, A401, GOBRD13:21, NAT_D:34;
hence  left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C by A237, A238, A286, A291, A304, A367, A370, A372, A397, A399, A401, A402, GOBRD13:25; ::_thesis: right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C
 front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),(i1 -' 1),j2) by A235, A283, A281, A284, A285, A286, A287, A401, GOBRD13:34;
hence  right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C by A237, A238, A286, A291, A304, A367, A370, A372, A397, A399, A401, A402, GOBRD13:26; ::_thesis: verum
end;
supposethat A403: ( i1 + 1 = i2 & j1 = j2 ) and
A404: (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 + 1)) ; ::_thesis: ( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
 ( i2 -' 1 = i1 & cell ((Gauge (C,n)),i1,j1) misses C ) by A235, A283, A281, A284, A285, A286, A287, A369, A403, GOBRD13:23, NAT_D:34;
hence  left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C by A237, A238, A286, A290, A304, A367, A370, A397, A399, A403, A404, GOBRD13:21; ::_thesis: right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C
 front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),i2,j2) by A235, A283, A281, A284, A285, A286, A287, A403, GOBRD13:36;
hence  right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C by A237, A238, A286, A290, A304, A367, A370, A397, A399, A403, A404, GOBRD13:22; ::_thesis: verum
end;
supposethat A405: ( i1 = i2 + 1 & j1 = j2 ) and
A406: (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i2,(j2 -' 1)) ; ::_thesis: ( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
 cell ((Gauge (C,n)),i2,(j2 -' 1)) misses C by A235, A283, A281, A284, A285, A286, A287, A369, A405, GOBRD13:25;
hence  left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C by A237, A238, A286, A289, A304, A367, A370, A371, A397, A399, A405, A406, GOBRD13:27; ::_thesis: right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C
 front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),(i2 -' 1),(j2 -' 1)) by A235, A283, A281, A284, A285, A286, A287, A405, GOBRD13:38;
hence  right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C by A237, A238, A286, A289, A304, A367, A370, A371, A397, A399, A405, A406, GOBRD13:28; ::_thesis: verum
end;
supposethat A407: ( i1 = i2 & j1 = j2 + 1 ) and
A408: (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * ((i2 + 1),j2) ; ::_thesis: ( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
 cell ((Gauge (C,n)),i2,j2) misses C by A235, A283, A281, A284, A285, A286, A287, A369, A407, GOBRD13:27;
hence  left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C by A237, A238, A286, A301, A304, A367, A370, A397, A399, A407, A408, GOBRD13:23; ::_thesis: right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C
 front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),i2,(j2 -' 1)) by A235, A283, A281, A284, A285, A286, A287, A407, GOBRD13:40;
hence  right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C by A237, A238, A286, A301, A304, A367, A370, A397, A399, A407, A408, GOBRD13:24; ::_thesis: verum
end;
end;
end;
hence  ( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C ) ; ::_thesis: verum
end;
end;
end;
end;
suppose m + 1 <> len (F . (k + 1)) ; ::_thesis: ( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
then  m + 1 < len (F . (k + 1)) by A368, XXREAL_0:1;
then A409: m + 1 <= len (F . k) by A237, A238, NAT_1:13;
then consider i1, j1, i2, j2 being Element of NAT  such that
A410: [i1,j1] in Indices (Gauge (C,n)) and
A411: (F . k) /. m = (Gauge (C,n)) * (i1,j1) and
A412: [i2,j2] in Indices (Gauge (C,n)) and
A413: (F . k) /. (m + 1) = (Gauge (C,n)) * (i2,j2) and
A414: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A235, A367, JORDAN8:3;
A415: ( left_cell ((F . k),m,(Gauge (C,n))) misses C & right_cell ((F . k),m,(Gauge (C,n))) meets C ) by A236, A367, A409;
A416: now__::_thesis:_ex_i,_j_being_Element_of_NAT_st_F_._(k_+_1)_=_(F_._k)_^_<*((Gauge_(C,n))_*_(i,j))*>
percases ( ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C ) or ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ) or front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ) ;
suppose ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C ) ; ::_thesis: ex i, j being Element of NAT st F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*>
then consider i, j being Element of NAT  such that
 (F . k) ^ <*((Gauge (C,n)) * (i,j))*> turns_right (len (F . k)) -' 1, Gauge (C,n) and
A417: F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> by A171, A235, A237, A279, A296;
take i = i; ::_thesis: ex j being Element of NAT st F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*>
take j = j; ::_thesis: F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*>
thus  F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> by A417; ::_thesis: verum
end;
suppose ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ) ; ::_thesis: ex i, j being Element of NAT st F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*>
then consider i, j being Element of NAT  such that
 (F . k) ^ <*((Gauge (C,n)) * (i,j))*> goes_straight (len (F . k)) -' 1, Gauge (C,n) and
A418: F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> by A171, A235, A237, A279, A296;
take i = i; ::_thesis: ex j being Element of NAT st F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*>
take j = j; ::_thesis: F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*>
thus  F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> by A418; ::_thesis: verum
end;
suppose front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ; ::_thesis: ex i, j being Element of NAT st F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*>
then consider i, j being Element of NAT  such that
 (F . k) ^ <*((Gauge (C,n)) * (i,j))*> turns_left (len (F . k)) -' 1, Gauge (C,n) and
A419: F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> by A171, A235, A237, A279, A296;
take i = i; ::_thesis: ex j being Element of NAT st F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*>
take j = j; ::_thesis: F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*>
thus  F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> by A419; ::_thesis: verum
end;
end;
end;
 1 <= m + 1 by NAT_1:12;
then  m + 1 in dom (F . k) by A409, FINSEQ_3:25;
then A420: (F . (k + 1)) /. (m + 1) = (Gauge (C,n)) * (i2,j2) by A413, A416, FINSEQ_4:68;
 m <= len (F . k) by A409, NAT_1:13;
then  m in dom (F . k) by A367, FINSEQ_3:25;
then A421: (F . (k + 1)) /. m = (Gauge (C,n)) * (i1,j1) by A411, A416, FINSEQ_4:68;
now__::_thesis:_(_left_cell_((F_._(k_+_1)),m,(Gauge_(C,n)))_misses_C_&_right_cell_((F_._(k_+_1)),m,(Gauge_(C,n)))_meets_C_)
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 A414;
supposeA422: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: ( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
then  ( left_cell ((F . k),m,(Gauge (C,n))) = cell ((Gauge (C,n)),(i1 -' 1),j1) & right_cell ((F . k),m,(Gauge (C,n))) = cell ((Gauge (C,n)),i1,j1) ) by A235, A367, A409, A410, A411, A412, A413, GOBRD13:21, GOBRD13:22;
hence  ( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C ) by A304, A367, A368, A410, A412, A415, A421, A420, A422, GOBRD13:21, GOBRD13:22; ::_thesis: verum
end;
supposeA423: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: ( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
then  ( left_cell ((F . k),m,(Gauge (C,n))) = cell ((Gauge (C,n)),i1,j1) & right_cell ((F . k),m,(Gauge (C,n))) = cell ((Gauge (C,n)),i1,(j1 -' 1)) ) by A235, A367, A409, A410, A411, A412, A413, GOBRD13:23, GOBRD13:24;
hence  ( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C ) by A304, A367, A368, A410, A412, A415, A421, A420, A423, GOBRD13:23, GOBRD13:24; ::_thesis: verum
end;
supposeA424: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: ( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
then  ( left_cell ((F . k),m,(Gauge (C,n))) = cell ((Gauge (C,n)),i2,(j2 -' 1)) & right_cell ((F . k),m,(Gauge (C,n))) = cell ((Gauge (C,n)),i2,j2) ) by A235, A367, A409, A410, A411, A412, A413, GOBRD13:25, GOBRD13:26;
hence  ( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C ) by A304, A367, A368, A410, A412, A415, A421, A420, A424, GOBRD13:25, GOBRD13:26; ::_thesis: verum
end;
supposeA425: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: ( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C )
then  ( left_cell ((F . k),m,(Gauge (C,n))) = cell ((Gauge (C,n)),i2,j2) & right_cell ((F . k),m,(Gauge (C,n))) = cell ((Gauge (C,n)),(i1 -' 1),j2) ) by A235, A367, A409, A410, A411, A412, A413, GOBRD13:27, GOBRD13:28;
hence  ( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C ) by A304, A367, A368, A410, A412, A415, A421, A420, A425, GOBRD13:27, GOBRD13:28; ::_thesis: verum
end;
end;
end;
hence  ( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C ) ; ::_thesis: verum
end;
end;
end;
hence  ( left_cell ((F . (k + 1)),m,(Gauge (C,n))) misses C & right_cell ((F . (k + 1)),m,(Gauge (C,n))) meets C ) ; ::_thesis: verum
end;
end;
end;
defpred S5[ Nat] means  ex w being Element of NAT st
( w = $1 & $1 >= 1 & ex m being Element of NAT st
( m in dom (F . w) & m <> len (F . w) & (F . w) /. m = (F . w) /. (len (F . w)) ) );
A426: S4[ 0 ]
proof
 ( ( for n being Element of NAT st n in dom (F . 0) holds
ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & (F . 0) /. n = (Gauge (C,n)) * (i,j) ) ) & ( for n being Element of NAT st n in dom (F . 0) & n + 1 in dom (F . 0) holds
for m, k, i, j being Element of NAT st [m,k] in Indices (Gauge (C,n)) & [i,j] in Indices (Gauge (C,n)) & (F . 0) /. n = (Gauge (C,n)) * (m,k) & (F . 0) /. (n + 1) = (Gauge (C,n)) * (i,j) holds
(abs (m - i)) + (abs (k - j)) = 1 ) ) by A170;
hence  F . 0 is_sequence_on Gauge (C,n) by GOBOARD1:def_9; ::_thesis: for m being Element of NAT st 1 <= m & m + 1 <= len (F . 0) holds
( left_cell ((F . 0),m,(Gauge (C,n))) misses C & right_cell ((F . 0),m,(Gauge (C,n))) meets C )
let m be Element of NAT ; ::_thesis: ( 1 <= m & m + 1 <= len (F . 0) implies ( left_cell ((F . 0),m,(Gauge (C,n))) misses C & right_cell ((F . 0),m,(Gauge (C,n))) meets C ) )
assume that
 1 <= m and
A427: m + 1 <= len (F . 0) ; ::_thesis: ( left_cell ((F . 0),m,(Gauge (C,n))) misses C & right_cell ((F . 0),m,(Gauge (C,n))) meets C )
thus  ( left_cell ((F . 0),m,(Gauge (C,n))) misses C & right_cell ((F . 0),m,(Gauge (C,n))) meets C ) by A170, A427, CARD_1:27; ::_thesis: verum
end;
A428: for k being Element of NAT holds S4[k] from NAT_1:sch_1(A426, A232);
A429: for k, i1, i2, j1, j2 being Element of NAT st k > 1 & [i1,j1] in Indices (Gauge (C,n)) & (F . k) /. ((len (F . k)) -' 1) = (Gauge (C,n)) * (i1,j1) & [i2,j2] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) holds
( ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C implies ( F . (k + 1) turns_right (len (F . k)) -' 1, Gauge (C,n) & ( i1 = i2 & j1 + 1 = j2 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) & ( i1 + 1 = i2 & j1 = j2 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) & ( i1 = i2 + 1 & j1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) ) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C implies ( F . (k + 1) goes_straight (len (F . k)) -' 1, Gauge (C,n) & ( i1 = i2 & j1 + 1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) & ( i1 + 1 = i2 & j1 = j2 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) & ( i1 = i2 + 1 & j1 = j2 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) ) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C implies ( F . (k + 1) turns_left (len (F . k)) -' 1, Gauge (C,n) & ( i1 = i2 & j1 + 1 = j2 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) & ( i1 + 1 = i2 & j1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) & ( i1 = i2 + 1 & j1 = j2 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) ) ) )
proof
let k, i1, i2, j1, j2 be Element of NAT ; ::_thesis: ( k > 1 & [i1,j1] in Indices (Gauge (C,n)) & (F . k) /. ((len (F . k)) -' 1) = (Gauge (C,n)) * (i1,j1) & [i2,j2] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) implies ( ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C implies ( F . (k + 1) turns_right (len (F . k)) -' 1, Gauge (C,n) & ( i1 = i2 & j1 + 1 = j2 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) & ( i1 + 1 = i2 & j1 = j2 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) & ( i1 = i2 + 1 & j1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) ) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C implies ( F . (k + 1) goes_straight (len (F . k)) -' 1, Gauge (C,n) & ( i1 = i2 & j1 + 1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) & ( i1 + 1 = i2 & j1 = j2 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) & ( i1 = i2 + 1 & j1 = j2 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) ) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C implies ( F . (k + 1) turns_left (len (F . k)) -' 1, Gauge (C,n) & ( i1 = i2 & j1 + 1 = j2 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) & ( i1 + 1 = i2 & j1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) & ( i1 = i2 + 1 & j1 = j2 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) ) ) ) )
assume that
A430: k > 1 and
A431: [i1,j1] in Indices (Gauge (C,n)) and
A432: (F . k) /. ((len (F . k)) -' 1) = (Gauge (C,n)) * (i1,j1) and
A433: [i2,j2] in Indices (Gauge (C,n)) and
A434: (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) ; ::_thesis: ( ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C implies ( F . (k + 1) turns_right (len (F . k)) -' 1, Gauge (C,n) & ( i1 = i2 & j1 + 1 = j2 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) & ( i1 + 1 = i2 & j1 = j2 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) & ( i1 = i2 + 1 & j1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) ) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C implies ( F . (k + 1) goes_straight (len (F . k)) -' 1, Gauge (C,n) & ( i1 = i2 & j1 + 1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) & ( i1 + 1 = i2 & j1 = j2 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) & ( i1 = i2 + 1 & j1 = j2 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) ) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C implies ( F . (k + 1) turns_left (len (F . k)) -' 1, Gauge (C,n) & ( i1 = i2 & j1 + 1 = j2 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) & ( i1 + 1 = i2 & j1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) & ( i1 = i2 + 1 & j1 = j2 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) ) ) )
A435: len (F . k) = k by A192;
then A436: ((len (F . k)) -' 1) + 1 = len (F . k) by A430, XREAL_1:235;
then A437: ((len (F . k)) -' 1) + (1 + 1) = (len (F . k)) + 1 ;
A438: 1 <= (len (F . k)) -' 1 by A430, A435, NAT_D:49;
then A439: right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C by A428, A436;
 (len (F . k)) -' 1 <= len (F . k) by NAT_D:35;
then A440: (len (F . k)) -' 1 in dom (F . k) by A438, FINSEQ_3:25;
A441: ( j1 + 1 > j1 & j2 + 1 > j2 ) by NAT_1:13;
A442: F . k is_sequence_on Gauge (C,n) by A428;
A443: ( i1 + 1 > i1 & i2 + 1 > i2 ) by NAT_1:13;
A444: len (F . k) in dom (F . k) by A430, A435, FINSEQ_3:25;
hereby  ::_thesis: ( ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C implies ( F . (k + 1) goes_straight (len (F . k)) -' 1, Gauge (C,n) & ( i1 = i2 & j1 + 1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) & ( i1 + 1 = i2 & j1 = j2 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) & ( i1 = i2 + 1 & j1 = j2 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) ) ) & ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C implies ( F . (k + 1) turns_left (len (F . k)) -' 1, Gauge (C,n) & ( i1 = i2 & j1 + 1 = j2 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) & ( i1 + 1 = i2 & j1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) & ( i1 = i2 + 1 & j1 = j2 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) ) ) )
assume  ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C ) ; ::_thesis: ( F . (k + 1) turns_right (len (F . k)) -' 1, Gauge (C,n) & ( i1 = i2 & j1 + 1 = j2 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) & ( i1 + 1 = i2 & j1 = j2 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) & ( i1 = i2 + 1 & j1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) )
then consider i, j being Element of NAT  such that
A445: (F . k) ^ <*((Gauge (C,n)) * (i,j))*> turns_right (len (F . k)) -' 1, Gauge (C,n) and
A446: F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> by A171, A430, A442, A435, A439;
thus  F . (k + 1) turns_right (len (F . k)) -' 1, Gauge (C,n) by A445, A446; ::_thesis: ( ( i1 = i2 & j1 + 1 = j2 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) & ( i1 + 1 = i2 & j1 = j2 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) & ( i1 = i2 + 1 & j1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) )
A447: (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i,j) by A446, FINSEQ_4:67;
A448: ( (F . (k + 1)) /. ((len (F . k)) -' 1) = (Gauge (C,n)) * (i1,j1) & (F . (k + 1)) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) ) by A432, A434, A440, A444, A446, FINSEQ_4:68;
hence  ( i1 = i2 & j1 + 1 = j2 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) by A431, A433, A436, A437, A441, A445, A446, A447, GOBRD13:def_6; ::_thesis: ( ( i1 + 1 = i2 & j1 = j2 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) & ( i1 = i2 + 1 & j1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) )
thus  ( i1 + 1 = i2 & j1 = j2 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) by A431, A433, A436, A437, A443, A445, A446, A448, A447, GOBRD13:def_6; ::_thesis: ( ( i1 = i2 + 1 & j1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) )
thus  ( i1 = i2 + 1 & j1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) by A431, A433, A436, A437, A443, A445, A446, A448, A447, GOBRD13:def_6; ::_thesis: ( i1 = i2 & j1 = j2 + 1 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) )
thus  ( i1 = i2 & j1 = j2 + 1 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) by A431, A433, A436, A437, A441, A445, A446, A448, A447, GOBRD13:def_6; ::_thesis: verum
end;
hereby  ::_thesis: ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C implies ( F . (k + 1) turns_left (len (F . k)) -' 1, Gauge (C,n) & ( i1 = i2 & j1 + 1 = j2 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) & ( i1 + 1 = i2 & j1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) & ( i1 = i2 + 1 & j1 = j2 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) ) )
assume  ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ) ; ::_thesis: ( F . (k + 1) goes_straight (len (F . k)) -' 1, Gauge (C,n) & ( i1 = i2 & j1 + 1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) & ( i1 + 1 = i2 & j1 = j2 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) & ( i1 = i2 + 1 & j1 = j2 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) )
then consider i, j being Element of NAT  such that
A449: (F . k) ^ <*((Gauge (C,n)) * (i,j))*> goes_straight (len (F . k)) -' 1, Gauge (C,n) and
A450: F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> by A171, A430, A442, A435, A439;
thus  F . (k + 1) goes_straight (len (F . k)) -' 1, Gauge (C,n) by A449, A450; ::_thesis: ( ( i1 = i2 & j1 + 1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) & ( i1 + 1 = i2 & j1 = j2 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) & ( i1 = i2 + 1 & j1 = j2 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) )
A451: (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i,j) by A450, FINSEQ_4:67;
A452: ( (F . (k + 1)) /. ((len (F . k)) -' 1) = (Gauge (C,n)) * (i1,j1) & (F . (k + 1)) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) ) by A432, A434, A440, A444, A450, FINSEQ_4:68;
hence  ( i1 = i2 & j1 + 1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) by A431, A433, A436, A437, A441, A449, A450, A451, GOBRD13:def_8; ::_thesis: ( ( i1 + 1 = i2 & j1 = j2 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) & ( i1 = i2 + 1 & j1 = j2 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) )
thus  ( i1 + 1 = i2 & j1 = j2 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) by A431, A433, A436, A437, A443, A449, A450, A452, A451, GOBRD13:def_8; ::_thesis: ( ( i1 = i2 + 1 & j1 = j2 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) )
thus  ( i1 = i2 + 1 & j1 = j2 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) by A431, A433, A436, A437, A443, A449, A450, A452, A451, GOBRD13:def_8; ::_thesis: ( i1 = i2 & j1 = j2 + 1 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) )
thus  ( i1 = i2 & j1 = j2 + 1 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) by A431, A433, A436, A437, A441, A449, A450, A452, A451, GOBRD13:def_8; ::_thesis: verum
end;
assume  front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ; ::_thesis: ( F . (k + 1) turns_left (len (F . k)) -' 1, Gauge (C,n) & ( i1 = i2 & j1 + 1 = j2 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) & ( i1 + 1 = i2 & j1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) & ( i1 = i2 + 1 & j1 = j2 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) )
then consider i, j being Element of NAT  such that
A453: (F . k) ^ <*((Gauge (C,n)) * (i,j))*> turns_left (len (F . k)) -' 1, Gauge (C,n) and
A454: F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> by A171, A430, A442, A435, A439;
A455: (F . (k + 1)) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) by A434, A444, A454, FINSEQ_4:68;
thus  F . (k + 1) turns_left (len (F . k)) -' 1, Gauge (C,n) by A453, A454; ::_thesis: ( ( i1 = i2 & j1 + 1 = j2 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) & ( i1 + 1 = i2 & j1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) & ( i1 = i2 + 1 & j1 = j2 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) )
A456: (F . (k + 1)) /. ((len (F . k)) + 1) = (Gauge (C,n)) * (i,j) by A454, FINSEQ_4:67;
A457: (F . (k + 1)) /. ((len (F . k)) -' 1) = (Gauge (C,n)) * (i1,j1) by A432, A440, A454, FINSEQ_4:68;
hence  ( i1 = i2 & j1 + 1 = j2 implies ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) ) by A431, A433, A436, A437, A441, A453, A454, A455, A456, GOBRD13:def_7; ::_thesis: ( ( i1 + 1 = i2 & j1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) & ( i1 = i2 + 1 & j1 = j2 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) )
thus  ( i1 + 1 = i2 & j1 = j2 implies ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) ) by A431, A433, A436, A437, A443, A453, A454, A457, A455, A456, GOBRD13:def_7; ::_thesis: ( ( i1 = i2 + 1 & j1 = j2 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) & ( i1 = i2 & j1 = j2 + 1 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) )
thus  ( i1 = i2 + 1 & j1 = j2 implies ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) ) by A431, A433, A436, A437, A443, A453, A454, A457, A455, A456, GOBRD13:def_7; ::_thesis: ( i1 = i2 & j1 = j2 + 1 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) )
thus  ( i1 = i2 & j1 = j2 + 1 implies ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) ) by A431, A433, A436, A437, A441, A453, A454, A457, A455, A456, GOBRD13:def_7; ::_thesis: verum
end;
A458: for k being Element of NAT st k > 1 holds
( ( front_left_cell ((F . k),(k -' 1),(Gauge (C,n))) misses C & front_right_cell ((F . k),(k -' 1),(Gauge (C,n))) misses C implies F . (k + 1) turns_right k -' 1, Gauge (C,n) ) & ( front_left_cell ((F . k),(k -' 1),(Gauge (C,n))) misses C & front_right_cell ((F . k),(k -' 1),(Gauge (C,n))) meets C implies F . (k + 1) goes_straight k -' 1, Gauge (C,n) ) & ( front_left_cell ((F . k),(k -' 1),(Gauge (C,n))) meets C implies F . (k + 1) turns_left k -' 1, Gauge (C,n) ) )
proof
let k be Element of NAT ; ::_thesis: ( k > 1 implies ( ( front_left_cell ((F . k),(k -' 1),(Gauge (C,n))) misses C & front_right_cell ((F . k),(k -' 1),(Gauge (C,n))) misses C implies F . (k + 1) turns_right k -' 1, Gauge (C,n) ) & ( front_left_cell ((F . k),(k -' 1),(Gauge (C,n))) misses C & front_right_cell ((F . k),(k -' 1),(Gauge (C,n))) meets C implies F . (k + 1) goes_straight k -' 1, Gauge (C,n) ) & ( front_left_cell ((F . k),(k -' 1),(Gauge (C,n))) meets C implies F . (k + 1) turns_left k -' 1, Gauge (C,n) ) ) )
assume A459: k > 1 ; ::_thesis: ( ( front_left_cell ((F . k),(k -' 1),(Gauge (C,n))) misses C & front_right_cell ((F . k),(k -' 1),(Gauge (C,n))) misses C implies F . (k + 1) turns_right k -' 1, Gauge (C,n) ) & ( front_left_cell ((F . k),(k -' 1),(Gauge (C,n))) misses C & front_right_cell ((F . k),(k -' 1),(Gauge (C,n))) meets C implies F . (k + 1) goes_straight k -' 1, Gauge (C,n) ) & ( front_left_cell ((F . k),(k -' 1),(Gauge (C,n))) meets C implies F . (k + 1) turns_left k -' 1, Gauge (C,n) ) )
A460: F . k is_sequence_on Gauge (C,n) by A428;
A461: len (F . k) = k by A192;
then  ( 1 <= (len (F . k)) -' 1 & ((len (F . k)) -' 1) + 1 = len (F . k) ) by A459, NAT_D:49, XREAL_1:235;
then  ex i1, j1, i2, j2 being Element of NAT st
( [i1,j1] in Indices (Gauge (C,n)) & (F . k) /. ((len (F . k)) -' 1) = (Gauge (C,n)) * (i1,j1) & [i2,j2] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) & ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) ) by A460, JORDAN8:3;
hence  ( ( front_left_cell ((F . k),(k -' 1),(Gauge (C,n))) misses C & front_right_cell ((F . k),(k -' 1),(Gauge (C,n))) misses C implies F . (k + 1) turns_right k -' 1, Gauge (C,n) ) & ( front_left_cell ((F . k),(k -' 1),(Gauge (C,n))) misses C & front_right_cell ((F . k),(k -' 1),(Gauge (C,n))) meets C implies F . (k + 1) goes_straight k -' 1, Gauge (C,n) ) & ( front_left_cell ((F . k),(k -' 1),(Gauge (C,n))) meets C implies F . (k + 1) turns_left k -' 1, Gauge (C,n) ) ) by A429, A459, A461; ::_thesis: verum
end;
defpred S6[ Element of NAT ] means  for m being Element of NAT st m <= $1 holds
(F . $1) | m = F . m;
A462: S6[ 0 ]
proof
let m be Element of NAT ; ::_thesis: ( m <= 0 implies (F . 0) | m = F . m )
assume A463: m <= 0 ; ::_thesis: (F . 0) | m = F . m
then  (F . 0) | 0 = (F . 0) | m ;
hence  (F . 0) | m = F . m by A170, A463; ::_thesis: verum
end;
A464: for k being Element of NAT ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
proof
let k be Element of NAT ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
A465: F . k is_sequence_on Gauge (C,n) by A428;
A466: len (F . k) = k by A192;
 len (Gauge (C,n)) >= 4 by JORDAN8:10;
then A467: ( len (Gauge (C,n)) = width (Gauge (C,n)) & 1 < len (Gauge (C,n)) ) by JORDAN8:def_1, XXREAL_0:2;
percases ( k < 1 or k = 1 or k > 1 ) by XXREAL_0:1;
supposeA468: k < 1 ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
consider i being Element of NAT  such that
A469: 1 <= i and
A470: i + 1 <= len (Gauge (C,n)) and
 N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) and
 N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1)) and
A471: F . (0 + 1) = <*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> by A171;
take  i ; ::_thesis: ex j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
take j = width (Gauge (C,n)); ::_thesis: ( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
 i < len (Gauge (C,n)) by A470, NAT_1:13;
hence  [i,j] in Indices (Gauge (C,n)) by A467, A469, MATRIX_1:36; ::_thesis: F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*>
 k = 0 by A468, NAT_1:14;
hence  F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> by A170, A471, FINSEQ_1:34; ::_thesis: verum
end;
supposeA472: k = 1 ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
consider i being Element of NAT  such that
A473: 1 <= i and
A474: i + 1 <= len (Gauge (C,n)) and
A475: ( N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) & N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1)) ) and
A476: F . (0 + 1) = <*((Gauge (C,n)) * (i,(width (Gauge (C,n)))))*> by A171;
take  i + 1 ; ::_thesis: ex j being Element of NAT st
( [(i + 1),j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i + 1),j))*> )
take j = width (Gauge (C,n)); ::_thesis: ( [(i + 1),j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i + 1),j))*> )
 1 <= i + 1 by A473, NAT_1:13;
hence  [(i + 1),j] in Indices (Gauge (C,n)) by A467, A474, MATRIX_1:36; ::_thesis: F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i + 1),j))*>
consider i9 being Element of NAT  such that
A477: ( 1 <= i9 & i9 + 1 <= len (Gauge (C,n)) & N-min C in cell ((Gauge (C,n)),i9,((width (Gauge (C,n))) -' 1)) & N-min C <> (Gauge (C,n)) * (i9,((width (Gauge (C,n))) -' 1)) ) and
A478: F . (1 + 1) = <*((Gauge (C,n)) * (i9,(width (Gauge (C,n))))),((Gauge (C,n)) * ((i9 + 1),(width (Gauge (C,n)))))*> by A171;
 i = i9 by A473, A474, A475, A477, Th29;
hence  F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i + 1),j))*> by A472, A476, A478, FINSEQ_1:def_9; ::_thesis: verum
end;
supposeA479: k > 1 ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
then  ( 1 <= (len (F . k)) -' 1 & ((len (F . k)) -' 1) + 1 = len (F . k) ) by A466, NAT_D:49, XREAL_1:235;
then consider i1, j1, i2, j2 being Element of NAT  such that
A480: ( [i1,j1] in Indices (Gauge (C,n)) & (F . k) /. ((len (F . k)) -' 1) = (Gauge (C,n)) * (i1,j1) & [i2,j2] in Indices (Gauge (C,n)) & (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) ) and
A481: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A465, JORDAN8:3;
now__::_thesis:_ex_i,_j_being_Element_of_NAT_st_
(_[i,j]_in_Indices_(Gauge_(C,n))_&_F_._(k_+_1)_=_(F_._k)_^_<*((Gauge_(C,n))_*_(i,j))*>_)
percases ( ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C ) or ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ) or front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ) ;
supposeA482: ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C ) ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
now__::_thesis:_ex_i,_j_being_Element_of_NAT_st_
(_[i,j]_in_Indices_(Gauge_(C,n))_&_F_._(k_+_1)_=_(F_._k)_^_<*((Gauge_(C,n))_*_(i,j))*>_)
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 A481;
suppose ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
then  ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) by A429, A479, A480, A482;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) ; ::_thesis: verum
end;
suppose ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
then  ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) by A429, A479, A480, A482;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) ; ::_thesis: verum
end;
suppose ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
then  ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) by A429, A479, A480, A482;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) ; ::_thesis: verum
end;
suppose ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
then  ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) by A429, A479, A480, A482;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) ; ::_thesis: verum
end;
end;
end;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) ; ::_thesis: verum
end;
supposeA483: ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ) ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
now__::_thesis:_ex_i,_j_being_Element_of_NAT_st_
(_[i,j]_in_Indices_(Gauge_(C,n))_&_F_._(k_+_1)_=_(F_._k)_^_<*((Gauge_(C,n))_*_(i,j))*>_)
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 A481;
suppose ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
then  ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) by A429, A479, A480, A483;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) ; ::_thesis: verum
end;
suppose ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
then  ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) by A429, A479, A480, A483;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) ; ::_thesis: verum
end;
suppose ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
then  ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) by A429, A479, A480, A483;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) ; ::_thesis: verum
end;
suppose ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
then  ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) by A429, A479, A480, A483;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) ; ::_thesis: verum
end;
end;
end;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) ; ::_thesis: verum
end;
supposeA484: front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
now__::_thesis:_ex_i,_j_being_Element_of_NAT_st_
(_[i,j]_in_Indices_(Gauge_(C,n))_&_F_._(k_+_1)_=_(F_._k)_^_<*((Gauge_(C,n))_*_(i,j))*>_)
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 A481;
suppose ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
then  ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> ) by A429, A479, A480, A484;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) ; ::_thesis: verum
end;
suppose ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
then  ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> ) by A429, A479, A480, A484;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) ; ::_thesis: verum
end;
suppose ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
then  ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> ) by A429, A479, A480, A484;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) ; ::_thesis: verum
end;
suppose ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> )
then  ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> ) by A429, A479, A480, A484;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) ; ::_thesis: verum
end;
end;
end;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) ; ::_thesis: verum
end;
end;
end;
hence  ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) ; ::_thesis: verum
end;
end;
end;
A485: for k being Element of NAT st S6[k] holds
S6[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S6[k] implies S6[k + 1] )
assume A486: for m being Element of NAT st m <= k holds
(F . k) | m = F . m ; ::_thesis: S6[k + 1]
let m be Element of NAT ; ::_thesis: ( m <= k + 1 implies (F . (k + 1)) | m = F . m )
assume A487: m <= k + 1 ; ::_thesis: (F . (k + 1)) | m = F . m
percases ( m < k + 1 or m = k + 1 ) by A487, XXREAL_0:1;
suppose m < k + 1 ; ::_thesis: (F . (k + 1)) | m = F . m
then A488: m <= k by NAT_1:13;
 ( len (F . k) = k & ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) ) by A192, A464;
then  (F . (k + 1)) | m = (F . k) | m by A488, FINSEQ_5:22;
hence  (F . (k + 1)) | m = F . m by A486, A488; ::_thesis: verum
end;
supposeA489: m = k + 1 ; ::_thesis: (F . (k + 1)) | m = F . m
 len (F . (k + 1)) = k + 1 by A192;
hence  (F . (k + 1)) | m = F . m by A489, FINSEQ_1:58; ::_thesis: verum
end;
end;
end;
A490: for k being Element of NAT holds S6[k] from NAT_1:sch_1(A462, A485);
defpred S7[ Element of NAT ] means F . $1 is unfolded ;
A491: for k being Element of NAT st S7[k] holds
S7[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S7[k] implies S7[k + 1] )
assume A492: F . k is unfolded ; ::_thesis: S7[k + 1]
A493: F . k is_sequence_on Gauge (C,n) by A428;
percases ( k <= 1 or k > 1 ) ;
suppose k <= 1 ; ::_thesis: S7[k + 1]
then  k + 1 <= 1 + 1 by XREAL_1:6;
then  len (F . (k + 1)) <= 2 by A192;
hence  S7[k + 1] by SPPOL_2:26; ::_thesis: verum
end;
supposeA494: k > 1 ; ::_thesis: S7[k + 1]
set m = k -' 1;
A495: (k -' 1) + 1 = k by A494, XREAL_1:235;
A496: len (F . k) = k by A192;
A497: 1 <= k -' 1 by A494, NAT_D:49;
then consider i1, j1, i2, j2 being Element of NAT  such that
A498: [i1,j1] in Indices (Gauge (C,n)) and
A499: (F . k) /. (k -' 1) = (Gauge (C,n)) * (i1,j1) and
A500: [i2,j2] in Indices (Gauge (C,n)) and
A501: (F . k) /. (len (F . k)) = (Gauge (C,n)) * (i2,j2) and
A502: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A493, A495, A496, JORDAN8:3;
A503: LSeg ((F . k),(k -' 1)) = LSeg (((Gauge (C,n)) * (i1,j1)),((Gauge (C,n)) * (i2,j2))) by A497, A495, A496, A499, A501, TOPREAL1:def_3;
A504: 1 <= j2 by A500, MATRIX_1:38;
then A505: (j2 -' 1) + 1 = j2 by XREAL_1:235;
A506: 1 <= j1 by A498, MATRIX_1:38;
A507: 1 <= i2 by A500, MATRIX_1:38;
then A508: (i2 -' 1) + 1 = i2 by XREAL_1:235;
A509: i1 <= len (Gauge (C,n)) by A498, MATRIX_1:38;
A510: j2 <= width (Gauge (C,n)) by A500, MATRIX_1:38;
A511: 1 <= i1 by A498, MATRIX_1:38;
A512: j1 <= width (Gauge (C,n)) by A498, MATRIX_1:38;
A513: i2 <= len (Gauge (C,n)) by A500, MATRIX_1:38;
now__::_thesis:_S7[k_+_1]
percases ( ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C ) or ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ) or front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ) ;
supposeA514: ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C ) ; ::_thesis: S7[k + 1]
now__::_thesis:_S7[k_+_1]
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 A502;
supposeA515: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: S7[k + 1]
then  [(i2 + 1),j2] in Indices (Gauge (C,n)) by A429, A494, A496, A498, A499, A500, A501, A514;
then  i2 + 1 <= len (Gauge (C,n)) by MATRIX_1:38;
then A516: (LSeg ((F . k),(k -' 1))) /\ (LSeg (((F . k) /. (len (F . k))),((Gauge (C,n)) * ((i2 + 1),j2)))) = {((F . k) /. (len (F . k)))} by A501, A511, A506, A510, A503, A515, GOBOARD7:15;
 F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> by A429, A494, A496, A498, A499, A500, A501, A514, A515;
hence  S7[k + 1] by A492, A495, A496, A516, SPPOL_2:30; ::_thesis: verum
end;
supposeA517: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: S7[k + 1]
then  [i2,(j2 -' 1)] in Indices (Gauge (C,n)) by A429, A494, A496, A498, A499, A500, A501, A514;
then  1 <= j2 -' 1 by MATRIX_1:38;
then A518: (LSeg ((F . k),(k -' 1))) /\ (LSeg (((F . k) /. (len (F . k))),((Gauge (C,n)) * (i2,(j2 -' 1))))) = {((F . k) /. (len (F . k)))} by A501, A511, A512, A513, A505, A503, A517, GOBOARD7:16;
 F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> by A429, A494, A496, A498, A499, A500, A501, A514, A517;
hence  S7[k + 1] by A492, A495, A496, A518, SPPOL_2:30; ::_thesis: verum
end;
supposeA519: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: S7[k + 1]
then  [i2,(j2 + 1)] in Indices (Gauge (C,n)) by A429, A494, A496, A498, A499, A500, A501, A514;
then  j2 + 1 <= width (Gauge (C,n)) by MATRIX_1:38;
then A520: (LSeg ((F . k),(k -' 1))) /\ (LSeg (((F . k) /. (len (F . k))),((Gauge (C,n)) * (i2,(j2 + 1))))) = {((F . k) /. (len (F . k)))} by A501, A509, A506, A507, A503, A519, GOBOARD7:17;
 F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> by A429, A494, A496, A498, A499, A500, A501, A514, A519;
hence  S7[k + 1] by A492, A495, A496, A520, SPPOL_2:30; ::_thesis: verum
end;
supposeA521: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: S7[k + 1]
then  [(i2 -' 1),j2] in Indices (Gauge (C,n)) by A429, A494, A496, A498, A499, A500, A501, A514;
then  1 <= i2 -' 1 by MATRIX_1:38;
then A522: (LSeg ((F . k),(k -' 1))) /\ (LSeg (((F . k) /. (len (F . k))),((Gauge (C,n)) * ((i2 -' 1),j2)))) = {((F . k) /. (len (F . k)))} by A501, A509, A512, A504, A508, A503, A521, GOBOARD7:18;
 F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> by A429, A494, A496, A498, A499, A500, A501, A514, A521;
hence  S7[k + 1] by A492, A495, A496, A522, SPPOL_2:30; ::_thesis: verum
end;
end;
end;
hence  S7[k + 1] ; ::_thesis: verum
end;
supposeA523: ( front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) misses C & front_right_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ) ; ::_thesis: S7[k + 1]
now__::_thesis:_S7[k_+_1]
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 A502;
supposeA524: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: S7[k + 1]
then  [i2,(j2 + 1)] in Indices (Gauge (C,n)) by A429, A494, A496, A498, A499, A500, A501, A523;
then A525: j2 + 1 <= width (Gauge (C,n)) by MATRIX_1:38;
 j2 + 1 = j1 + (1 + 1) by A524;
then A526: (LSeg ((F . k),(k -' 1))) /\ (LSeg (((F . k) /. (len (F . k))),((Gauge (C,n)) * (i2,(j2 + 1))))) = {((F . k) /. (len (F . k)))} by A501, A511, A509, A506, A503, A524, A525, GOBOARD7:13;
 F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> by A429, A494, A496, A498, A499, A500, A501, A523, A524;
hence  S7[k + 1] by A492, A495, A496, A526, SPPOL_2:30; ::_thesis: verum
end;
supposeA527: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: S7[k + 1]
then  [(i2 + 1),j2] in Indices (Gauge (C,n)) by A429, A494, A496, A498, A499, A500, A501, A523;
then A528: i2 + 1 <= len (Gauge (C,n)) by MATRIX_1:38;
 i2 + 1 = i1 + (1 + 1) by A527;
then A529: (LSeg ((F . k),(k -' 1))) /\ (LSeg (((F . k) /. (len (F . k))),((Gauge (C,n)) * ((i2 + 1),j2)))) = {((F . k) /. (len (F . k)))} by A501, A511, A506, A512, A503, A527, A528, GOBOARD7:14;
 F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> by A429, A494, A496, A498, A499, A500, A501, A523, A527;
hence  S7[k + 1] by A492, A495, A496, A529, SPPOL_2:30; ::_thesis: verum
end;
supposeA530: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: S7[k + 1]
then  [(i2 -' 1),j2] in Indices (Gauge (C,n)) by A429, A494, A496, A498, A499, A500, A501, A523;
then A531: 1 <= i2 -' 1 by MATRIX_1:38;
 ((i2 -' 1) + 1) + 1 = (i2 -' 1) + (1 + 1) ;
then A532: (LSeg ((F . k),(k -' 1))) /\ (LSeg (((F . k) /. (len (F . k))),((Gauge (C,n)) * ((i2 -' 1),j2)))) = {((F . k) /. (len (F . k)))} by A501, A509, A506, A512, A508, A503, A530, A531, GOBOARD7:14;
 F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> by A429, A494, A496, A498, A499, A500, A501, A523, A530;
hence  S7[k + 1] by A492, A495, A496, A532, SPPOL_2:30; ::_thesis: verum
end;
supposeA533: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: S7[k + 1]
then  [i2,(j2 -' 1)] in Indices (Gauge (C,n)) by A429, A494, A496, A498, A499, A500, A501, A523;
then A534: 1 <= j2 -' 1 by MATRIX_1:38;
 ((j2 -' 1) + 1) + 1 = (j2 -' 1) + (1 + 1) ;
then A535: (LSeg ((F . k),(k -' 1))) /\ (LSeg (((F . k) /. (len (F . k))),((Gauge (C,n)) * (i2,(j2 -' 1))))) = {((F . k) /. (len (F . k)))} by A501, A511, A509, A512, A505, A503, A533, A534, GOBOARD7:13;
 F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> by A429, A494, A496, A498, A499, A500, A501, A523, A533;
hence  S7[k + 1] by A492, A495, A496, A535, SPPOL_2:30; ::_thesis: verum
end;
end;
end;
hence  S7[k + 1] ; ::_thesis: verum
end;
supposeA536: front_left_cell ((F . k),((len (F . k)) -' 1),(Gauge (C,n))) meets C ; ::_thesis: S7[k + 1]
now__::_thesis:_S7[k_+_1]
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 A502;
supposeA537: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: S7[k + 1]
then  [(i2 -' 1),j2] in Indices (Gauge (C,n)) by A429, A494, A496, A498, A499, A500, A501, A536;
then  1 <= i2 -' 1 by MATRIX_1:38;
then A538: (LSeg ((F . k),(k -' 1))) /\ (LSeg (((F . k) /. (len (F . k))),((Gauge (C,n)) * ((i2 -' 1),j2)))) = {((F . k) /. (len (F . k)))} by A501, A509, A506, A510, A508, A503, A537, GOBOARD7:16;
 F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 -' 1),j2))*> by A429, A494, A496, A498, A499, A500, A501, A536, A537;
hence  S7[k + 1] by A492, A495, A496, A538, SPPOL_2:30; ::_thesis: verum
end;
supposeA539: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: S7[k + 1]
then  [i2,(j2 + 1)] in Indices (Gauge (C,n)) by A429, A494, A496, A498, A499, A500, A501, A536;
then  j2 + 1 <= width (Gauge (C,n)) by MATRIX_1:38;
then A540: (LSeg ((F . k),(k -' 1))) /\ (LSeg (((F . k) /. (len (F . k))),((Gauge (C,n)) * (i2,(j2 + 1))))) = {((F . k) /. (len (F . k)))} by A501, A511, A506, A513, A503, A539, GOBOARD7:18;
 F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 + 1)))*> by A429, A494, A496, A498, A499, A500, A501, A536, A539;
hence  S7[k + 1] by A492, A495, A496, A540, SPPOL_2:30; ::_thesis: verum
end;
supposeA541: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: S7[k + 1]
then  [i2,(j2 -' 1)] in Indices (Gauge (C,n)) by A429, A494, A496, A498, A499, A500, A501, A536;
then  1 <= j2 -' 1 by MATRIX_1:38;
then A542: (LSeg ((F . k),(k -' 1))) /\ (LSeg (((F . k) /. (len (F . k))),((Gauge (C,n)) * (i2,(j2 -' 1))))) = {((F . k) /. (len (F . k)))} by A501, A509, A512, A507, A505, A503, A541, GOBOARD7:15;
 F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i2,(j2 -' 1)))*> by A429, A494, A496, A498, A499, A500, A501, A536, A541;
hence  S7[k + 1] by A492, A495, A496, A542, SPPOL_2:30; ::_thesis: verum
end;
supposeA543: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: S7[k + 1]
then  [(i2 + 1),j2] in Indices (Gauge (C,n)) by A429, A494, A496, A498, A499, A500, A501, A536;
then  i2 + 1 <= len (Gauge (C,n)) by MATRIX_1:38;
then A544: (LSeg ((F . k),(k -' 1))) /\ (LSeg (((F . k) /. (len (F . k))),((Gauge (C,n)) * ((i2 + 1),j2)))) = {((F . k) /. (len (F . k)))} by A501, A511, A512, A504, A503, A543, GOBOARD7:17;
 F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * ((i2 + 1),j2))*> by A429, A494, A496, A498, A499, A500, A501, A536, A543;
hence  S7[k + 1] by A492, A495, A496, A544, SPPOL_2:30; ::_thesis: verum
end;
end;
end;
hence  S7[k + 1] ; ::_thesis: verum
end;
end;
end;
hence  S7[k + 1] ; ::_thesis: verum
end;
end;
end;
now__::_thesis:_ex_k_being_Element_of_NAT_st_
(_k_>=_1_&_ex_m_being_Element_of_NAT_st_
(_m_in_dom_(F_._k)_&_m_<>_len_(F_._k)_&_not_(F_._k)_/._m_<>_(F_._k)_/._(len_(F_._k))_)_)
defpred S8[ Element of NAT ] means F . $1 is one-to-one ;
assume A545: for k being Element of NAT st k >= 1 holds
for m being Element of NAT st m in dom (F . k) & m <> len (F . k) holds
(F . k) /. m <> (F . k) /. (len (F . k)) ; ::_thesis: contradiction
A546: for k being Element of NAT st S8[k] holds
S8[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S8[k] implies S8[k + 1] )
assume A547: F . k is one-to-one ; ::_thesis: S8[k + 1]
now__::_thesis:_for_n,_m_being_Element_of_NAT_st_n_in_dom_(F_._(k_+_1))_&_m_in_dom_(F_._(k_+_1))_&_(F_._(k_+_1))_/._n_=_(F_._(k_+_1))_/._m_holds_
n_=_m
let n, m be Element of NAT ; ::_thesis: ( n in dom (F . (k + 1)) & m in dom (F . (k + 1)) & (F . (k + 1)) /. n = (F . (k + 1)) /. m implies b1 = b2 )
assume that
A548: n in dom (F . (k + 1)) and
A549: m in dom (F . (k + 1)) and
A550: (F . (k + 1)) /. n = (F . (k + 1)) /. m ; ::_thesis: b1 = b2
A551: ( n <= len (F . (k + 1)) & m <= len (F . (k + 1)) ) by A548, A549, FINSEQ_3:25;
A552: 1 <= m by A549, FINSEQ_3:25;
A553: 1 <= n by A548, FINSEQ_3:25;
A554: ex i, j being Element of NAT st
( [i,j] in Indices (Gauge (C,n)) & F . (k + 1) = (F . k) ^ <*((Gauge (C,n)) * (i,j))*> ) by A464;
A555: len (F . k) = k by A192;
A556: len (F . (k + 1)) = k + 1 by A192;
percases ( ( n <= k & m <= k ) or ( n = k + 1 & m <= k ) or ( n <= k & m = k + 1 ) or ( n = k + 1 & m = k + 1 ) ) by A551, A556, NAT_1:8;
supposeA557: ( n <= k & m <= k ) ; ::_thesis: b1 = b2
then A558: m in dom (F . k) by A552, A555, FINSEQ_3:25;
then A559: (F . (k + 1)) /. m = (F . k) /. m by A554, FINSEQ_4:68;
A560: n in dom (F . k) by A553, A555, A557, FINSEQ_3:25;
then  (F . (k + 1)) /. n = (F . k) /. n by A554, FINSEQ_4:68;
hence  n = m by A547, A550, A560, A558, A559, PARTFUN2:10; ::_thesis: verum
end;
suppose ( n = k + 1 & m <= k ) ; ::_thesis: b1 = b2
hence  n = m by A545, A549, A550, A556, NAT_1:12; ::_thesis: verum
end;
suppose ( n <= k & m = k + 1 ) ; ::_thesis: b1 = b2
hence  n = m by A545, A548, A550, A556, NAT_1:12; ::_thesis: verum
end;
suppose ( n = k + 1 & m = k + 1 ) ; ::_thesis: b1 = b2
hence  n = m ; ::_thesis: verum
end;
end;
end;
hence  S8[k + 1] by PARTFUN2:9; ::_thesis: verum
end;
A561: S8[ 0 ] by A170;
A562: for k being Element of NAT holds S8[k] from NAT_1:sch_1(A561, A546);
A563: for k being Element of NAT holds card (rng (F . k)) = k
proof
let k be Element of NAT ; ::_thesis: card (rng (F . k)) = k
 F . k is one-to-one by A562;
hence  card (rng (F . k)) = len (F . k) by FINSEQ_4:62
.= k by A192 ;
::_thesis: verum
end;
set k = ((len (Gauge (C,n))) * (width (Gauge (C,n)))) + 1;
 F . (((len (Gauge (C,n))) * (width (Gauge (C,n)))) + 1) is_sequence_on Gauge (C,n) by A428;
then  ( card (Values (Gauge (C,n))) <= (len (Gauge (C,n))) * (width (Gauge (C,n))) & card (rng (F . (((len (Gauge (C,n))) * (width (Gauge (C,n)))) + 1))) <= card (Values (Gauge (C,n))) ) by GOBRD13:8, MATRIX_1:46, NAT_1:43;
then  ( ((len (Gauge (C,n))) * (width (Gauge (C,n)))) + 1 > (len (Gauge (C,n))) * (width (Gauge (C,n))) & card (rng (F . (((len (Gauge (C,n))) * (width (Gauge (C,n)))) + 1))) <= (len (Gauge (C,n))) * (width (Gauge (C,n))) ) by NAT_1:13, XXREAL_0:2;
hence  contradiction by A563; ::_thesis: verum
end;
then A564: ex k being Nat st S5[k] ;
consider k being Nat such that
A565: S5[k] and
A566: for l being Nat st S5[l] holds
k <= l from NAT_1:sch_5(A564);
reconsider k = k as Element of NAT by ORDINAL1:def_12;
consider m being Element of NAT  such that
A567: m in dom (F . k) and
A568: m <> len (F . k) and
A569: (F . k) /. m = (F . k) /. (len (F . k)) by A565;
A570: 1 <= m by A567, FINSEQ_3:25;
reconsider f = F . k as non empty FinSequence of (TOP-REAL 2) by A565;
A571: f is_sequence_on Gauge (C,n) by A428;
A572: m <= len f by A567, FINSEQ_3:25;
then A573: m < len f by A568, XXREAL_0:1;
then  1 < len f by A570, XXREAL_0:2;
then A574: len f >= 1 + 1 by NAT_1:13;
A575: S7[ 0 ] by A170, CARD_1:27, SPPOL_2:26;
 for k being Element of NAT holds S7[k] from NAT_1:sch_1(A575, A491);
then reconsider f = f as non empty non constant special unfolded FinSequence of (TOP-REAL 2) by A571, A574, JORDAN8:4, JORDAN8:5;
A576: m + 1 <= len f by A573, NAT_1:13;
set g = f /^ (m -' 1);
 m -' 1 <= m by NAT_D:44;
then  m -' 1 < m + 1 by NAT_1:13;
then A577: m -' 1 < len f by A576, XXREAL_0:2;
then A578: len (f /^ (m -' 1)) = (len f) - (m -' 1) by RFINSEQ:def_1;
then  (m -' 1) - (m -' 1) < len (f /^ (m -' 1)) by A577, XREAL_1:9;
then reconsider g = f /^ (m -' 1) as non empty FinSequence of (TOP-REAL 2) by CARD_1:27;
 len g in dom g by FINSEQ_5:6;
then A579: g /. (len g) = f /. ((m -' 1) + (len g)) by FINSEQ_5:27
.= f /. (len f) by A578 ;
A580: len (F . k) = k by A192;
A581: for j, i being Element of NAT st 1 <= i & i < len g & 1 <= j & j < len g & g /. i = g /. j holds
i = j
proof
let j, i be Element of NAT ; ::_thesis: ( 1 <= i & i < len g & 1 <= j & j < len g & g /. i = g /. j implies i = j )
assume that
A582: 1 <= i and
A583: i < len g and
A584: 1 <= j and
A585: j < len g and
A586: g /. i = g /. j and
A587: i <> j ; ::_thesis: contradiction
A588: i in dom g by A582, A583, FINSEQ_3:25;
then A589: g /. i = f /. ((m -' 1) + i) by FINSEQ_5:27;
A590: j in dom g by A584, A585, FINSEQ_3:25;
then A591: g /. j = f /. ((m -' 1) + j) by FINSEQ_5:27;
percases ( i < j or j < i ) by A587, XXREAL_0:1;
supposeA592: i < j ; ::_thesis: contradiction
set l = (m -' 1) + j;
set m9 = (m -' 1) + i;
A593: (m -' 1) + i < (m -' 1) + j by A592, XREAL_1:6;
A594: len (F . ((m -' 1) + j)) = (m -' 1) + j by A192;
A595: (m -' 1) + j < k by A580, A578, A585, XREAL_1:20;
then A596: f | ((m -' 1) + j) = F . ((m -' 1) + j) by A490;
 0 + j <= (m -' 1) + j by XREAL_1:6;
then A597: 1 <= (m -' 1) + j by A584, XXREAL_0:2;
then  (m -' 1) + j in dom (F . ((m -' 1) + j)) by A594, FINSEQ_3:25;
then A598: (F . ((m -' 1) + j)) /. ((m -' 1) + j) = f /. ((m -' 1) + j) by A596, FINSEQ_4:70;
 0 + i <= (m -' 1) + i by XREAL_1:6;
then  1 <= (m -' 1) + i by A582, XXREAL_0:2;
then A599: (m -' 1) + i in dom (F . ((m -' 1) + j)) by A593, A594, FINSEQ_3:25;
then  (F . ((m -' 1) + j)) /. ((m -' 1) + i) = f /. ((m -' 1) + i) by A596, FINSEQ_4:70;
hence  contradiction by A566, A586, A589, A590, A593, A595, A597, A594, A599, A598, FINSEQ_5:27; ::_thesis: verum
end;
supposeA600: j < i ; ::_thesis: contradiction
set l = (m -' 1) + i;
set m9 = (m -' 1) + j;
A601: (m -' 1) + j < (m -' 1) + i by A600, XREAL_1:6;
A602: len (F . ((m -' 1) + i)) = (m -' 1) + i by A192;
A603: (m -' 1) + i < k by A580, A578, A583, XREAL_1:20;
then A604: f | ((m -' 1) + i) = F . ((m -' 1) + i) by A490;
 0 + i <= (m -' 1) + i by XREAL_1:6;
then A605: 1 <= (m -' 1) + i by A582, XXREAL_0:2;
then  (m -' 1) + i in dom (F . ((m -' 1) + i)) by A602, FINSEQ_3:25;
then A606: (F . ((m -' 1) + i)) /. ((m -' 1) + i) = f /. ((m -' 1) + i) by A604, FINSEQ_4:70;
 0 + j <= (m -' 1) + j by XREAL_1:6;
then  1 <= (m -' 1) + j by A584, XXREAL_0:2;
then A607: (m -' 1) + j in dom (F . ((m -' 1) + i)) by A601, A602, FINSEQ_3:25;
then  (F . ((m -' 1) + i)) /. ((m -' 1) + j) = f /. ((m -' 1) + j) by A604, FINSEQ_4:70;
hence  contradiction by A566, A586, A588, A591, A601, A603, A605, A602, A607, A606, FINSEQ_5:27; ::_thesis: verum
end;
end;
end;
A608: now__::_thesis:_ex_i_being_Element_of_NAT_st_
(_1_<=_i_&_i_+_1_<=_len_(Gauge_(C,n))_&_f_/._1_=_(Gauge_(C,n))_*_(i,(width_(Gauge_(C,n))))_&_f_/._2_=_(Gauge_(C,n))_*_((i_+_1),(width_(Gauge_(C,n))))_&_N-min_C_in_cell_((Gauge_(C,n)),i,((width_(Gauge_(C,n)))_-'_1))_&_N-min_C_<>_(Gauge_(C,n))_*_(i,((width_(Gauge_(C,n)))_-'_1))_)
consider i being Element of NAT  such that
A609: ( 1 <= i & i + 1 <= len (Gauge (C,n)) ) and
A610: ( N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) & N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1)) ) and
A611: F . (1 + 1) = <*((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * ((i + 1),(width (Gauge (C,n)))))*> by A171;
take i = i; ::_thesis: ( 1 <= i & i + 1 <= len (Gauge (C,n)) & f /. 1 = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) & f /. 2 = (Gauge (C,n)) * ((i + 1),(width (Gauge (C,n)))) & N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) & N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1)) )
thus  ( 1 <= i & i + 1 <= len (Gauge (C,n)) ) by A609; ::_thesis: ( f /. 1 = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) & f /. 2 = (Gauge (C,n)) * ((i + 1),(width (Gauge (C,n)))) & N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) & N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1)) )
A612: f | 2 = F . 2 by A490, A580, A574;
A613: len (f | 2) = 2 by A574, FINSEQ_1:59;
then  1 in dom (f | 2) by FINSEQ_3:25;
hence f /. 1 = (f | 2) /. 1 by FINSEQ_4:70
.= (Gauge (C,n)) * (i,(width (Gauge (C,n)))) by A611, A612, FINSEQ_4:17 ;
::_thesis: ( f /. 2 = (Gauge (C,n)) * ((i + 1),(width (Gauge (C,n)))) & N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) & N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1)) )
 2 in dom (f | 2) by A613, FINSEQ_3:25;
hence f /. 2 = (f | 2) /. 2 by FINSEQ_4:70
.= (Gauge (C,n)) * ((i + 1),(width (Gauge (C,n)))) by A611, A612, FINSEQ_4:17 ;
::_thesis: ( N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) & N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1)) )
thus  ( N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) & N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1)) ) by A610; ::_thesis: verum
end;
 1 in dom g by FINSEQ_5:6;
then A614: g /. 1 = f /. ((m -' 1) + 1) by FINSEQ_5:27
.= f /. m by A570, XREAL_1:235 ;
A615: for j, i being Element of NAT st 1 < i & i < j & j <= len g holds
g /. i <> g /. j
proof
let j, i be Element of NAT ; ::_thesis: ( 1 < i & i < j & j <= len g implies g /. i <> g /. j )
assume that
A616: 1 < i and
A617: i < j and
A618: j <= len g and
A619: g /. i = g /. j ; ::_thesis: contradiction
A620: 1 < j by A616, A617, XXREAL_0:2;
A621: i < len g by A617, A618, XXREAL_0:2;
then A622: 1 < len g by A616, XXREAL_0:2;
percases ( j <> len g or j = len g ) ;
suppose j <> len g ; ::_thesis: contradiction
then  j < len g by A618, XXREAL_0:1;
hence  contradiction by A581, A616, A617, A619, A620, A621; ::_thesis: verum
end;
suppose j = len g ; ::_thesis: contradiction
hence  contradiction by A569, A614, A579, A581, A616, A617, A619, A622; ::_thesis: verum
end;
end;
end;
 (m + 1) - (m -' 1) <= len g by A576, A578, XREAL_1:9;
then A623: (m + 1) - (m - 1) <= len g by A570, XREAL_1:233;
then A624: ((1 + m) - m) + 1 <= len g ;
A625: g is_sequence_on Gauge (C,n) by A428, JORDAN8:2;
then A626: g is standard by JORDAN8:4;
A627: not g is constant
proof
take  1 ; :: according to SEQM_3:def_10 ::_thesis: ex b1 being Element of NAT st
( 1 in dom g & b1 in dom g & not g . 1 = g . b1 )
take  2 ; ::_thesis: ( 1 in dom g & 2 in dom g & not g . 1 = g . 2 )
thus A628: 1 in dom g by FINSEQ_5:6; ::_thesis: ( 2 in dom g & not g . 1 = g . 2 )
thus A629: 2 in dom g by A623, FINSEQ_3:25; ::_thesis: not g . 1 = g . 2
then  g /. 1 <> g /. (1 + 1) by A626, FINSEQ_5:6, GOBOARD7:29;
then  g . 1 <> g /. (1 + 1) by A628, PARTFUN1:def_6;
hence  not g . 1 = g . 2 by A629, PARTFUN1:def_6; ::_thesis: verum
end;
A630: for j, i being Element of NAT st 1 <= i & i < j & j < len g holds
g /. i <> g /. j
proof
let j, i be Element of NAT ; ::_thesis: ( 1 <= i & i < j & j < len g implies g /. i <> g /. j )
assume that
A631: ( 1 <= i & i < j & j < len g ) and
A632: g /. i = g /. j ; ::_thesis: contradiction
 ( 1 < j & i < len g ) by A631, XXREAL_0:2;
hence  contradiction by A581, A631, A632; ::_thesis: verum
end;
 g is s.c.c.
proof
let i be Element of NAT ; :: according to GOBOARD5:def_4 ::_thesis: for b1 being Element of NAT holds
( b1 <= i + 1 or ( ( i <= 1 or len g <= b1 ) & len g <= b1 + 1 ) or LSeg (g,i) misses LSeg (g,b1) )
let j be Element of NAT ; ::_thesis: ( j <= i + 1 or ( ( i <= 1 or len g <= j ) & len g <= j + 1 ) or LSeg (g,i) misses LSeg (g,j) )
assume that
A633: i + 1 < j and
A634: ( ( i > 1 & j < len g ) or j + 1 < len g ) ; ::_thesis: LSeg (g,i) misses LSeg (g,j)
A635: 1 < j by A633, NAT_1:12;
A636: 1 <= i + 1 by NAT_1:12;
A637: j <= j + 1 by NAT_1:12;
then A638: i + 1 < j + 1 by A633, XXREAL_0:2;
 i < j by A633, NAT_1:13;
then A639: i < j + 1 by A637, XXREAL_0:2;
percases ( ( i > 1 & j < len g ) or ( i = 0 & j + 1 < len g ) or ( 1 <= i & j + 1 < len g ) ) by A634, NAT_1:14;
supposeA640: ( i > 1 & j < len g ) ; ::_thesis: LSeg (g,i) misses LSeg (g,j)
then A641: j + 1 <= len g by NAT_1:13;
then A642: LSeg (g,j) = LSeg ((g /. j),(g /. (j + 1))) by A635, TOPREAL1:def_3;
consider i19, j19, i29, j29 being Element of NAT  such that
A643: [i19,j19] in Indices (Gauge (C,n)) and
A644: g /. j = (Gauge (C,n)) * (i19,j19) and
A645: [i29,j29] in Indices (Gauge (C,n)) and
A646: g /. (j + 1) = (Gauge (C,n)) * (i29,j29) and
A647: ( ( i19 = i29 & j19 + 1 = j29 ) or ( i19 + 1 = i29 & j19 = j29 ) or ( i19 = i29 + 1 & j19 = j29 ) or ( i19 = i29 & j19 = j29 + 1 ) ) by A625, A635, A641, JORDAN8:3;
A648: 1 <= i19 by A643, MATRIX_1:38;
A649: j29 <= width (Gauge (C,n)) by A645, MATRIX_1:38;
A650: 1 <= i29 by A645, MATRIX_1:38;
A651: i19 <= len (Gauge (C,n)) by A643, MATRIX_1:38;
A652: 1 <= j29 by A645, MATRIX_1:38;
A653: j19 <= width (Gauge (C,n)) by A643, MATRIX_1:38;
A654: i29 <= len (Gauge (C,n)) by A645, MATRIX_1:38;
A655: 1 <= j19 by A643, MATRIX_1:38;
A656: i + 1 < len g by A633, A640, XXREAL_0:2;
then A657: LSeg (g,i) = LSeg ((g /. i),(g /. (i + 1))) by A640, TOPREAL1:def_3;
A658: i < len g by A656, NAT_1:13;
consider i1, j1, i2, j2 being Element of NAT  such that
A659: [i1,j1] in Indices (Gauge (C,n)) and
A660: g /. i = (Gauge (C,n)) * (i1,j1) and
A661: [i2,j2] in Indices (Gauge (C,n)) and
A662: g /. (i + 1) = (Gauge (C,n)) * (i2,j2) and
A663: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A625, A640, A656, JORDAN8:3;
A664: 1 <= i1 by A659, MATRIX_1:38;
A665: j2 <= width (Gauge (C,n)) by A661, MATRIX_1:38;
A666: j1 <= width (Gauge (C,n)) by A659, MATRIX_1:38;
A667: 1 <= j2 by A661, MATRIX_1:38;
A668: 1 <= j1 by A659, MATRIX_1:38;
A669: i2 <= len (Gauge (C,n)) by A661, MATRIX_1:38;
A670: i1 <= len (Gauge (C,n)) by A659, MATRIX_1:38;
A671: 1 < i + 1 by A640, NAT_1:13;
assume A672: LSeg (g,i) meets LSeg (g,j) ; ::_thesis: contradiction
A673: 1 <= i2 by A661, MATRIX_1:38;
now__::_thesis:_contradiction
percases ( ( i1 = i2 & j1 + 1 = j2 & i19 = i29 & j19 + 1 = j29 ) or ( i1 = i2 & j1 + 1 = j2 & i19 + 1 = i29 & j19 = j29 ) or ( i1 = i2 & j1 + 1 = j2 & i19 = i29 + 1 & j19 = j29 ) or ( i1 = i2 & j1 + 1 = j2 & i19 = i29 & j19 = j29 + 1 ) or ( i1 + 1 = i2 & j1 = j2 & i19 = i29 & j19 + 1 = j29 ) or ( i1 + 1 = i2 & j1 = j2 & i19 + 1 = i29 & j19 = j29 ) or ( i1 + 1 = i2 & j1 = j2 & i19 = i29 + 1 & j19 = j29 ) or ( i1 + 1 = i2 & j1 = j2 & i19 = i29 & j19 = j29 + 1 ) or ( i1 = i2 + 1 & j1 = j2 & i19 = i29 & j19 + 1 = j29 ) or ( i1 = i2 + 1 & j1 = j2 & i19 + 1 = i29 & j19 = j29 ) or ( i1 = i2 + 1 & j1 = j2 & i19 = i29 + 1 & j19 = j29 ) or ( i1 = i2 + 1 & j1 = j2 & i19 = i29 & j19 = j29 + 1 ) or ( i1 = i2 & j1 = j2 + 1 & i19 = i29 & j19 + 1 = j29 ) or ( i1 = i2 & j1 = j2 + 1 & i19 + 1 = i29 & j19 = j29 ) or ( i1 = i2 & j1 = j2 + 1 & i19 = i29 + 1 & j19 = j29 ) or ( i1 = i2 & j1 = j2 + 1 & i19 = i29 & j19 = j29 + 1 ) ) by A663, A647;
supposeA674: ( i1 = i2 & j1 + 1 = j2 & i19 = i29 & j19 + 1 = j29 ) ; ::_thesis: contradiction
then A675: ( j1 = j19 or j1 = j19 + 1 or j1 + 1 = j19 ) by A657, A660, A662, A664, A670, A668, A665, A642, A644, A646, A648, A651, A655, A649, A672, GOBOARD7:22;
 i1 = i19 by A657, A660, A662, A664, A670, A668, A665, A642, A644, A646, A648, A651, A655, A649, A672, A674, GOBOARD7:19;
hence  contradiction by A581, A615, A633, A637, A636, A635, A639, A640, A656, A658, A660, A662, A641, A644, A646, A674, A675; ::_thesis: verum
end;
supposeA676: ( i1 = i2 & j1 + 1 = j2 & i19 + 1 = i29 & j19 = j29 ) ; ::_thesis: contradiction
then  ( ( i1 = i19 & j1 = j19 ) or ( i1 = i19 & j1 + 1 = j19 ) or ( i1 = i19 + 1 & j1 = j19 ) or ( i1 = i19 + 1 & j1 + 1 = j19 ) ) by A657, A660, A662, A664, A670, A668, A665, A642, A644, A646, A648, A655, A653, A654, A672, GOBOARD7:21;
hence  contradiction by A581, A615, A633, A638, A635, A639, A640, A671, A658, A660, A662, A641, A644, A646, A676; ::_thesis: verum
end;
supposeA677: ( i1 = i2 & j1 + 1 = j2 & i19 = i29 + 1 & j19 = j29 ) ; ::_thesis: contradiction
then  ( ( i1 = i29 & j19 = j1 ) or ( i1 = i29 & j1 + 1 = j19 ) or ( i1 = i29 + 1 & j19 = j1 ) or ( i1 = i29 + 1 & j1 + 1 = j19 ) ) by A657, A660, A662, A664, A670, A668, A665, A642, A644, A646, A651, A655, A653, A650, A672, GOBOARD7:21;
hence  contradiction by A581, A615, A633, A638, A635, A639, A640, A671, A658, A660, A662, A641, A644, A646, A677; ::_thesis: verum
end;
supposeA678: ( i1 = i2 & j1 + 1 = j2 & i19 = i29 & j19 = j29 + 1 ) ; ::_thesis: contradiction
then A679: ( j1 = j29 or j1 = j29 + 1 or j1 + 1 = j29 ) by A657, A660, A662, A664, A670, A668, A665, A642, A644, A646, A648, A651, A653, A652, A672, GOBOARD7:22;
 i1 = i19 by A657, A660, A662, A664, A670, A668, A665, A642, A644, A646, A648, A651, A653, A652, A672, A678, GOBOARD7:19;
hence  contradiction by A581, A615, A633, A638, A635, A640, A671, A658, A660, A662, A641, A644, A646, A678, A679; ::_thesis: verum
end;
supposeA680: ( i1 + 1 = i2 & j1 = j2 & i19 = i29 & j19 + 1 = j29 ) ; ::_thesis: contradiction
then  ( ( i19 = i1 & j1 = j19 ) or ( i19 = i1 & j19 + 1 = j1 ) or ( i19 = i1 + 1 & j1 = j19 ) or ( i19 = i1 + 1 & j19 + 1 = j1 ) ) by A657, A660, A662, A664, A668, A666, A669, A642, A644, A646, A648, A651, A655, A649, A672, GOBOARD7:21;
hence  contradiction by A581, A615, A633, A638, A635, A639, A640, A671, A658, A660, A662, A641, A644, A646, A680; ::_thesis: verum
end;
supposeA681: ( i1 + 1 = i2 & j1 = j2 & i19 + 1 = i29 & j19 = j29 ) ; ::_thesis: contradiction
then A682: ( i1 = i19 or i1 = i19 + 1 or i1 + 1 = i19 ) by A657, A660, A662, A664, A668, A666, A669, A642, A644, A646, A648, A655, A653, A654, A672, GOBOARD7:23;
 j1 = j19 by A657, A660, A662, A664, A668, A666, A669, A642, A644, A646, A648, A655, A653, A654, A672, A681, GOBOARD7:20;
hence  contradiction by A581, A615, A633, A637, A636, A635, A639, A640, A656, A658, A660, A662, A641, A644, A646, A681, A682; ::_thesis: verum
end;
supposeA683: ( i1 + 1 = i2 & j1 = j2 & i19 = i29 + 1 & j19 = j29 ) ; ::_thesis: contradiction
then A684: ( i1 = i29 or i1 = i29 + 1 or i1 + 1 = i29 ) by A657, A660, A662, A664, A668, A666, A669, A642, A644, A646, A651, A655, A653, A650, A672, GOBOARD7:23;
 j1 = j19 by A657, A660, A662, A664, A668, A666, A669, A642, A644, A646, A651, A655, A653, A650, A672, A683, GOBOARD7:20;
hence  contradiction by A581, A615, A638, A635, A639, A640, A671, A658, A660, A662, A641, A644, A646, A683, A684; ::_thesis: verum
end;
supposeA685: ( i1 + 1 = i2 & j1 = j2 & i19 = i29 & j19 = j29 + 1 ) ; ::_thesis: contradiction
then  ( ( i19 = i1 & j1 = j29 ) or ( i19 = i1 & j29 + 1 = j1 ) or ( i19 = i1 + 1 & j1 = j29 ) or ( i19 = i1 + 1 & j29 + 1 = j1 ) ) by A657, A660, A662, A664, A668, A666, A669, A642, A644, A646, A648, A651, A653, A652, A672, GOBOARD7:21;
hence  contradiction by A581, A615, A633, A638, A635, A639, A640, A671, A658, A660, A662, A641, A644, A646, A685; ::_thesis: verum
end;
supposeA686: ( i1 = i2 + 1 & j1 = j2 & i19 = i29 & j19 + 1 = j29 ) ; ::_thesis: contradiction
then  ( ( i19 = i2 & j19 = j1 ) or ( i19 = i2 & j19 + 1 = j1 ) or ( i19 = i2 + 1 & j19 = j1 ) or ( i19 = i2 + 1 & j19 + 1 = j1 ) ) by A657, A660, A662, A670, A668, A666, A673, A642, A644, A646, A648, A651, A655, A649, A672, GOBOARD7:21;
hence  contradiction by A581, A615, A633, A638, A635, A639, A640, A671, A658, A660, A662, A641, A644, A646, A686; ::_thesis: verum
end;
supposeA687: ( i1 = i2 + 1 & j1 = j2 & i19 + 1 = i29 & j19 = j29 ) ; ::_thesis: contradiction
then A688: ( i2 = i19 or i2 = i19 + 1 or i2 + 1 = i19 ) by A657, A660, A662, A670, A668, A666, A673, A642, A644, A646, A648, A655, A653, A654, A672, GOBOARD7:23;
 j1 = j19 by A657, A660, A662, A670, A668, A666, A673, A642, A644, A646, A648, A655, A653, A654, A672, A687, GOBOARD7:20;
hence  contradiction by A581, A615, A633, A638, A635, A640, A671, A658, A660, A662, A641, A644, A646, A687, A688; ::_thesis: verum
end;
supposeA689: ( i1 = i2 + 1 & j1 = j2 & i19 = i29 + 1 & j19 = j29 ) ; ::_thesis: contradiction
then A690: ( i2 = i29 or i2 = i29 + 1 or i2 + 1 = i29 ) by A657, A660, A662, A670, A668, A666, A673, A642, A644, A646, A651, A655, A653, A650, A672, GOBOARD7:23;
 j1 = j19 by A657, A660, A662, A670, A668, A666, A673, A642, A644, A646, A651, A655, A653, A650, A672, A689, GOBOARD7:20;
hence  contradiction by A615, A633, A638, A639, A640, A671, A660, A662, A641, A644, A646, A689, A690; ::_thesis: verum
end;
supposeA691: ( i1 = i2 + 1 & j1 = j2 & i19 = i29 & j19 = j29 + 1 ) ; ::_thesis: contradiction
then  ( ( i19 = i2 & j29 = j1 ) or ( i19 = i2 & j29 + 1 = j1 ) or ( i19 = i2 + 1 & j29 = j1 ) or ( i19 = i2 + 1 & j29 + 1 = j1 ) ) by A657, A660, A662, A670, A668, A666, A673, A642, A644, A646, A648, A651, A653, A652, A672, GOBOARD7:21;
hence  contradiction by A581, A615, A633, A638, A635, A639, A640, A671, A658, A660, A662, A641, A644, A646, A691; ::_thesis: verum
end;
supposeA692: ( i1 = i2 & j1 = j2 + 1 & i19 = i29 & j19 + 1 = j29 ) ; ::_thesis: contradiction
then A693: ( j2 = j19 or j2 = j19 + 1 or j2 + 1 = j19 ) by A657, A660, A662, A664, A670, A666, A667, A642, A644, A646, A648, A651, A655, A649, A672, GOBOARD7:22;
 i1 = i19 by A657, A660, A662, A664, A670, A666, A667, A642, A644, A646, A648, A651, A655, A649, A672, A692, GOBOARD7:19;
hence  contradiction by A581, A615, A633, A638, A635, A640, A671, A658, A660, A662, A641, A644, A646, A692, A693; ::_thesis: verum
end;
supposeA694: ( i1 = i2 & j1 = j2 + 1 & i19 + 1 = i29 & j19 = j29 ) ; ::_thesis: contradiction
then  ( ( i1 = i19 & j2 = j19 ) or ( i1 = i19 & j2 + 1 = j19 ) or ( i1 = i19 + 1 & j2 = j19 ) or ( i1 = i19 + 1 & j2 + 1 = j19 ) ) by A657, A660, A662, A664, A670, A666, A667, A642, A644, A646, A648, A655, A653, A654, A672, GOBOARD7:21;
hence  contradiction by A581, A615, A633, A638, A635, A639, A640, A671, A658, A660, A662, A641, A644, A646, A694; ::_thesis: verum
end;
supposeA695: ( i1 = i2 & j1 = j2 + 1 & i19 = i29 + 1 & j19 = j29 ) ; ::_thesis: contradiction
then  ( ( i1 = i29 & j2 = j19 ) or ( i1 = i29 & j2 + 1 = j19 ) or ( i1 = i29 + 1 & j2 = j19 ) or ( i1 = i29 + 1 & j2 + 1 = j19 ) ) by A657, A660, A662, A664, A670, A666, A667, A642, A644, A646, A651, A655, A653, A650, A672, GOBOARD7:21;
hence  contradiction by A581, A615, A633, A638, A635, A639, A640, A671, A658, A660, A662, A641, A644, A646, A695; ::_thesis: verum
end;
supposeA696: ( i1 = i2 & j1 = j2 + 1 & i19 = i29 & j19 = j29 + 1 ) ; ::_thesis: contradiction
then A697: ( j2 = j29 or j2 = j29 + 1 or j2 + 1 = j29 ) by A657, A660, A662, A664, A670, A666, A667, A642, A644, A646, A648, A651, A653, A652, A672, GOBOARD7:22;
 i1 = i19 by A657, A660, A662, A664, A670, A666, A667, A642, A644, A646, A648, A651, A653, A652, A672, A696, GOBOARD7:19;
hence  contradiction by A615, A633, A638, A639, A640, A671, A660, A662, A641, A644, A646, A696, A697; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
suppose ( i = 0 & j + 1 < len g ) ; ::_thesis: LSeg (g,i) misses LSeg (g,j)
then  LSeg (g,i) = {} by TOPREAL1:def_3;
hence  LSeg (g,i) misses LSeg (g,j) by XBOOLE_1:65; ::_thesis: verum
end;
supposeA698: ( 1 <= i & j + 1 < len g ) ; ::_thesis: LSeg (g,i) misses LSeg (g,j)
then A699: i + 1 < len g by A638, XXREAL_0:2;
then A700: LSeg (g,i) = LSeg ((g /. i),(g /. (i + 1))) by A698, TOPREAL1:def_3;
A701: i < len g by A699, NAT_1:13;
consider i1, j1, i2, j2 being Element of NAT  such that
A702: [i1,j1] in Indices (Gauge (C,n)) and
A703: g /. i = (Gauge (C,n)) * (i1,j1) and
A704: [i2,j2] in Indices (Gauge (C,n)) and
A705: g /. (i + 1) = (Gauge (C,n)) * (i2,j2) and
A706: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A625, A698, A699, JORDAN8:3;
A707: 1 <= i1 by A702, MATRIX_1:38;
A708: j2 <= width (Gauge (C,n)) by A704, MATRIX_1:38;
A709: j1 <= width (Gauge (C,n)) by A702, MATRIX_1:38;
A710: 1 <= j2 by A704, MATRIX_1:38;
A711: 1 <= j1 by A702, MATRIX_1:38;
A712: i2 <= len (Gauge (C,n)) by A704, MATRIX_1:38;
A713: i1 <= len (Gauge (C,n)) by A702, MATRIX_1:38;
A714: 1 < i + 1 by A698, NAT_1:13;
assume A715: LSeg (g,i) meets LSeg (g,j) ; ::_thesis: contradiction
consider i19, j19, i29, j29 being Element of NAT  such that
A716: [i19,j19] in Indices (Gauge (C,n)) and
A717: g /. j = (Gauge (C,n)) * (i19,j19) and
A718: [i29,j29] in Indices (Gauge (C,n)) and
A719: g /. (j + 1) = (Gauge (C,n)) * (i29,j29) and
A720: ( ( i19 = i29 & j19 + 1 = j29 ) or ( i19 + 1 = i29 & j19 = j29 ) or ( i19 = i29 + 1 & j19 = j29 ) or ( i19 = i29 & j19 = j29 + 1 ) ) by A625, A635, A698, JORDAN8:3;
A721: 1 <= i19 by A716, MATRIX_1:38;
A722: j29 <= width (Gauge (C,n)) by A718, MATRIX_1:38;
A723: 1 <= i29 by A718, MATRIX_1:38;
A724: i19 <= len (Gauge (C,n)) by A716, MATRIX_1:38;
A725: 1 <= j29 by A718, MATRIX_1:38;
A726: j19 <= width (Gauge (C,n)) by A716, MATRIX_1:38;
A727: i29 <= len (Gauge (C,n)) by A718, MATRIX_1:38;
A728: 1 <= j19 by A716, MATRIX_1:38;
A729: j < len g by A698, NAT_1:12;
A730: LSeg (g,j) = LSeg ((g /. j),(g /. (j + 1))) by A635, A698, TOPREAL1:def_3;
A731: 1 <= i2 by A704, MATRIX_1:38;
now__::_thesis:_contradiction
percases ( ( i1 = i2 & j1 + 1 = j2 & i19 = i29 & j19 + 1 = j29 ) or ( i1 = i2 & j1 + 1 = j2 & i19 + 1 = i29 & j19 = j29 ) or ( i1 = i2 & j1 + 1 = j2 & i19 = i29 + 1 & j19 = j29 ) or ( i1 = i2 & j1 + 1 = j2 & i19 = i29 & j19 = j29 + 1 ) or ( i1 + 1 = i2 & j1 = j2 & i19 = i29 & j19 + 1 = j29 ) or ( i1 + 1 = i2 & j1 = j2 & i19 + 1 = i29 & j19 = j29 ) or ( i1 + 1 = i2 & j1 = j2 & i19 = i29 + 1 & j19 = j29 ) or ( i1 + 1 = i2 & j1 = j2 & i19 = i29 & j19 = j29 + 1 ) or ( i1 = i2 + 1 & j1 = j2 & i19 = i29 & j19 + 1 = j29 ) or ( i1 = i2 + 1 & j1 = j2 & i19 + 1 = i29 & j19 = j29 ) or ( i1 = i2 + 1 & j1 = j2 & i19 = i29 + 1 & j19 = j29 ) or ( i1 = i2 + 1 & j1 = j2 & i19 = i29 & j19 = j29 + 1 ) or ( i1 = i2 & j1 = j2 + 1 & i19 = i29 & j19 + 1 = j29 ) or ( i1 = i2 & j1 = j2 + 1 & i19 + 1 = i29 & j19 = j29 ) or ( i1 = i2 & j1 = j2 + 1 & i19 = i29 + 1 & j19 = j29 ) or ( i1 = i2 & j1 = j2 + 1 & i19 = i29 & j19 = j29 + 1 ) ) by A706, A720;
supposeA732: ( i1 = i2 & j1 + 1 = j2 & i19 = i29 & j19 + 1 = j29 ) ; ::_thesis: contradiction
then A733: ( j1 = j19 or j1 = j19 + 1 or j1 + 1 = j19 ) by A700, A703, A705, A707, A713, A711, A708, A730, A717, A719, A721, A724, A728, A722, A715, GOBOARD7:22;
 i1 = i19 by A700, A703, A705, A707, A713, A711, A708, A730, A717, A719, A721, A724, A728, A722, A715, A732, GOBOARD7:19;
hence  contradiction by A630, A633, A638, A636, A639, A698, A729, A703, A705, A717, A719, A732, A733; ::_thesis: verum
end;
supposeA734: ( i1 = i2 & j1 + 1 = j2 & i19 + 1 = i29 & j19 = j29 ) ; ::_thesis: contradiction
then  ( ( i1 = i19 & j1 = j19 ) or ( i1 = i19 & j1 + 1 = j19 ) or ( i1 = i19 + 1 & j1 = j19 ) or ( i1 = i19 + 1 & j1 + 1 = j19 ) ) by A700, A703, A705, A707, A713, A711, A708, A730, A717, A719, A721, A728, A726, A727, A715, GOBOARD7:21;
hence  contradiction by A581, A630, A633, A638, A635, A639, A698, A714, A701, A729, A703, A705, A717, A719, A734; ::_thesis: verum
end;
supposeA735: ( i1 = i2 & j1 + 1 = j2 & i19 = i29 + 1 & j19 = j29 ) ; ::_thesis: contradiction
then  ( ( i1 = i29 & j19 = j1 ) or ( i1 = i29 & j1 + 1 = j19 ) or ( i1 = i29 + 1 & j19 = j1 ) or ( i1 = i29 + 1 & j1 + 1 = j19 ) ) by A700, A703, A705, A707, A713, A711, A708, A730, A717, A719, A724, A728, A726, A723, A715, GOBOARD7:21;
hence  contradiction by A581, A630, A633, A638, A635, A639, A698, A714, A701, A729, A703, A705, A717, A719, A735; ::_thesis: verum
end;
supposeA736: ( i1 = i2 & j1 + 1 = j2 & i19 = i29 & j19 = j29 + 1 ) ; ::_thesis: contradiction
then A737: ( j1 = j29 or j1 = j29 + 1 or j1 + 1 = j29 ) by A700, A703, A705, A707, A713, A711, A708, A730, A717, A719, A721, A724, A726, A725, A715, GOBOARD7:22;
 i1 = i19 by A700, A703, A705, A707, A713, A711, A708, A730, A717, A719, A721, A724, A726, A725, A715, A736, GOBOARD7:19;
hence  contradiction by A581, A615, A633, A638, A635, A698, A714, A701, A729, A703, A705, A717, A719, A736, A737; ::_thesis: verum
end;
supposeA738: ( i1 + 1 = i2 & j1 = j2 & i19 = i29 & j19 + 1 = j29 ) ; ::_thesis: contradiction
then  ( ( i19 = i1 & j1 = j19 ) or ( i19 = i1 & j19 + 1 = j1 ) or ( i19 = i1 + 1 & j1 = j19 ) or ( i19 = i1 + 1 & j19 + 1 = j1 ) ) by A700, A703, A705, A707, A711, A709, A712, A730, A717, A719, A721, A724, A728, A722, A715, GOBOARD7:21;
hence  contradiction by A581, A630, A633, A638, A636, A635, A639, A698, A701, A729, A703, A705, A717, A719, A738; ::_thesis: verum
end;
supposeA739: ( i1 + 1 = i2 & j1 = j2 & i19 + 1 = i29 & j19 = j29 ) ; ::_thesis: contradiction
then A740: ( i1 = i19 or i1 = i19 + 1 or i1 + 1 = i19 ) by A700, A703, A705, A707, A711, A709, A712, A730, A717, A719, A721, A728, A726, A727, A715, GOBOARD7:23;
 j1 = j19 by A700, A703, A705, A707, A711, A709, A712, A730, A717, A719, A721, A728, A726, A727, A715, A739, GOBOARD7:20;
hence  contradiction by A630, A633, A638, A636, A639, A698, A729, A703, A705, A717, A719, A739, A740; ::_thesis: verum
end;
supposeA741: ( i1 + 1 = i2 & j1 = j2 & i19 = i29 + 1 & j19 = j29 ) ; ::_thesis: contradiction
then A742: ( i1 = i29 or i1 = i29 + 1 or i1 + 1 = i29 ) by A700, A703, A705, A707, A711, A709, A712, A730, A717, A719, A724, A728, A726, A723, A715, GOBOARD7:23;
 j1 = j19 by A700, A703, A705, A707, A711, A709, A712, A730, A717, A719, A724, A728, A726, A723, A715, A741, GOBOARD7:20;
hence  contradiction by A581, A630, A638, A635, A639, A698, A714, A701, A729, A703, A705, A717, A719, A741, A742; ::_thesis: verum
end;
supposeA743: ( i1 + 1 = i2 & j1 = j2 & i19 = i29 & j19 = j29 + 1 ) ; ::_thesis: contradiction
then  ( ( i19 = i1 & j1 = j29 ) or ( i19 = i1 & j29 + 1 = j1 ) or ( i19 = i1 + 1 & j1 = j29 ) or ( i19 = i1 + 1 & j29 + 1 = j1 ) ) by A700, A703, A705, A707, A711, A709, A712, A730, A717, A719, A721, A724, A726, A725, A715, GOBOARD7:21;
hence  contradiction by A581, A630, A633, A638, A635, A639, A698, A714, A701, A729, A703, A705, A717, A719, A743; ::_thesis: verum
end;
supposeA744: ( i1 = i2 + 1 & j1 = j2 & i19 = i29 & j19 + 1 = j29 ) ; ::_thesis: contradiction
then  ( ( i19 = i2 & j19 = j1 ) or ( i19 = i2 & j19 + 1 = j1 ) or ( i19 = i2 + 1 & j19 = j1 ) or ( i19 = i2 + 1 & j19 + 1 = j1 ) ) by A700, A703, A705, A713, A711, A709, A731, A730, A717, A719, A721, A724, A728, A722, A715, GOBOARD7:21;
hence  contradiction by A581, A630, A633, A638, A635, A639, A698, A714, A701, A729, A703, A705, A717, A719, A744; ::_thesis: verum
end;
supposeA745: ( i1 = i2 + 1 & j1 = j2 & i19 + 1 = i29 & j19 = j29 ) ; ::_thesis: contradiction
then A746: ( i2 = i19 or i2 = i19 + 1 or i2 + 1 = i19 ) by A700, A703, A705, A713, A711, A709, A731, A730, A717, A719, A721, A728, A726, A727, A715, GOBOARD7:23;
 j1 = j19 by A700, A703, A705, A713, A711, A709, A731, A730, A717, A719, A721, A728, A726, A727, A715, A745, GOBOARD7:20;
hence  contradiction by A581, A615, A633, A638, A635, A698, A714, A701, A729, A703, A705, A717, A719, A745, A746; ::_thesis: verum
end;
supposeA747: ( i1 = i2 + 1 & j1 = j2 & i19 = i29 + 1 & j19 = j29 ) ; ::_thesis: contradiction
then A748: ( i2 = i29 or i2 = i29 + 1 or i2 + 1 = i29 ) by A700, A703, A705, A713, A711, A709, A731, A730, A717, A719, A724, A728, A726, A723, A715, GOBOARD7:23;
 j1 = j19 by A700, A703, A705, A713, A711, A709, A731, A730, A717, A719, A724, A728, A726, A723, A715, A747, GOBOARD7:20;
hence  contradiction by A630, A633, A638, A639, A698, A714, A729, A703, A705, A717, A719, A747, A748; ::_thesis: verum
end;
supposeA749: ( i1 = i2 + 1 & j1 = j2 & i19 = i29 & j19 = j29 + 1 ) ; ::_thesis: contradiction
then  ( ( i19 = i2 & j29 = j1 ) or ( i19 = i2 & j29 + 1 = j1 ) or ( i19 = i2 + 1 & j29 = j1 ) or ( i19 = i2 + 1 & j29 + 1 = j1 ) ) by A700, A703, A705, A713, A711, A709, A731, A730, A717, A719, A721, A724, A726, A725, A715, GOBOARD7:21;
hence  contradiction by A581, A630, A633, A638, A635, A639, A698, A714, A701, A729, A703, A705, A717, A719, A749; ::_thesis: verum
end;
supposeA750: ( i1 = i2 & j1 = j2 + 1 & i19 = i29 & j19 + 1 = j29 ) ; ::_thesis: contradiction
then A751: ( j2 = j19 or j2 = j19 + 1 or j2 + 1 = j19 ) by A700, A703, A705, A707, A713, A709, A710, A730, A717, A719, A721, A724, A728, A722, A715, GOBOARD7:22;
 i1 = i19 by A700, A703, A705, A707, A713, A709, A710, A730, A717, A719, A721, A724, A728, A722, A715, A750, GOBOARD7:19;
hence  contradiction by A581, A615, A633, A638, A635, A698, A714, A701, A729, A703, A705, A717, A719, A750, A751; ::_thesis: verum
end;
supposeA752: ( i1 = i2 & j1 = j2 + 1 & i19 + 1 = i29 & j19 = j29 ) ; ::_thesis: contradiction
then  ( ( i1 = i19 & j2 = j19 ) or ( i1 = i19 & j2 + 1 = j19 ) or ( i1 = i19 + 1 & j2 = j19 ) or ( i1 = i19 + 1 & j2 + 1 = j19 ) ) by A700, A703, A705, A707, A713, A709, A710, A730, A717, A719, A721, A728, A726, A727, A715, GOBOARD7:21;
hence  contradiction by A581, A630, A633, A638, A635, A639, A698, A714, A701, A729, A703, A705, A717, A719, A752; ::_thesis: verum
end;
supposeA753: ( i1 = i2 & j1 = j2 + 1 & i19 = i29 + 1 & j19 = j29 ) ; ::_thesis: contradiction
then  ( ( i1 = i29 & j2 = j19 ) or ( i1 = i29 & j2 + 1 = j19 ) or ( i1 = i29 + 1 & j2 = j19 ) or ( i1 = i29 + 1 & j2 + 1 = j19 ) ) by A700, A703, A705, A707, A713, A709, A710, A730, A717, A719, A724, A728, A726, A723, A715, GOBOARD7:21;
hence  contradiction by A581, A630, A633, A638, A635, A639, A698, A714, A701, A729, A703, A705, A717, A719, A753; ::_thesis: verum
end;
supposeA754: ( i1 = i2 & j1 = j2 + 1 & i19 = i29 & j19 = j29 + 1 ) ; ::_thesis: contradiction
then A755: ( j2 = j29 or j2 = j29 + 1 or j2 + 1 = j29 ) by A700, A703, A705, A707, A713, A709, A710, A730, A717, A719, A721, A724, A726, A725, A715, GOBOARD7:22;
 i1 = i19 by A700, A703, A705, A707, A713, A709, A710, A730, A717, A719, A721, A724, A726, A725, A715, A754, GOBOARD7:19;
hence  contradiction by A630, A633, A638, A639, A698, A714, A729, A703, A705, A717, A719, A754, A755; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
end;
end;
then reconsider g = g as non constant standard special_circular_sequence by A569, A614, A579, A625, A627, FINSEQ_6:def_1, JORDAN8:4;
A756: for i being Element of NAT st 1 <= i & i + 1 <= len f holds
right_cell (f,i,(Gauge (C,n))) = Cl (Int (right_cell (f,i,(Gauge (C,n)))))
proof
let i be Element of NAT ; ::_thesis: ( 1 <= i & i + 1 <= len f implies right_cell (f,i,(Gauge (C,n))) = Cl (Int (right_cell (f,i,(Gauge (C,n))))) )
assume A757: ( 1 <= i & i + 1 <= len f ) ; ::_thesis: right_cell (f,i,(Gauge (C,n))) = Cl (Int (right_cell (f,i,(Gauge (C,n)))))
consider i1, j1, i2, j2 being Element of NAT  such that
A758: [i1,j1] in Indices (Gauge (C,n)) and
A759: f /. i = (Gauge (C,n)) * (i1,j1) and
A760: [i2,j2] in Indices (Gauge (C,n)) and
A761: f /. (i + 1) = (Gauge (C,n)) * (i2,j2) and
A762: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A571, A757, JORDAN8:3;
A763: i1 <= len (Gauge (C,n)) by A758, MATRIX_1:38;
A764: j2 <= width (Gauge (C,n)) by A760, MATRIX_1:38;
A765: j1 <= width (Gauge (C,n)) by A758, MATRIX_1:38;
A766: ( j1 + 1 > j1 & j2 + 1 > j2 ) by NAT_1:13;
A767: i2 <= len (Gauge (C,n)) by A760, MATRIX_1:38;
A768: ( i1 + 1 > i1 & i2 + 1 > i2 ) by NAT_1:13;
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 A762;
suppose ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: right_cell (f,i,(Gauge (C,n))) = Cl (Int (right_cell (f,i,(Gauge (C,n)))))
then  right_cell (f,i,(Gauge (C,n))) = cell ((Gauge (C,n)),i1,j1) by A571, A757, A758, A759, A760, A761, A766, GOBRD13:def_2;
hence  right_cell (f,i,(Gauge (C,n))) = Cl (Int (right_cell (f,i,(Gauge (C,n))))) by A763, A765, GOBRD11:35; ::_thesis: verum
end;
supposeA769: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: right_cell (f,i,(Gauge (C,n))) = Cl (Int (right_cell (f,i,(Gauge (C,n)))))
A770: j1 -' 1 <= width (Gauge (C,n)) by A765, NAT_D:44;
 right_cell (f,i,(Gauge (C,n))) = cell ((Gauge (C,n)),i1,(j1 -' 1)) by A571, A757, A758, A759, A760, A761, A768, A769, GOBRD13:def_2;
hence  right_cell (f,i,(Gauge (C,n))) = Cl (Int (right_cell (f,i,(Gauge (C,n))))) by A763, A770, GOBRD11:35; ::_thesis: verum
end;
suppose ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: right_cell (f,i,(Gauge (C,n))) = Cl (Int (right_cell (f,i,(Gauge (C,n)))))
then  right_cell (f,i,(Gauge (C,n))) = cell ((Gauge (C,n)),i2,j2) by A571, A757, A758, A759, A760, A761, A768, GOBRD13:def_2;
hence  right_cell (f,i,(Gauge (C,n))) = Cl (Int (right_cell (f,i,(Gauge (C,n))))) by A767, A764, GOBRD11:35; ::_thesis: verum
end;
supposeA771: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: right_cell (f,i,(Gauge (C,n))) = Cl (Int (right_cell (f,i,(Gauge (C,n)))))
A772: i1 -' 1 <= len (Gauge (C,n)) by A763, NAT_D:44;
 right_cell (f,i,(Gauge (C,n))) = cell ((Gauge (C,n)),(i1 -' 1),j2) by A571, A757, A758, A759, A760, A761, A766, A771, GOBRD13:def_2;
hence  right_cell (f,i,(Gauge (C,n))) = Cl (Int (right_cell (f,i,(Gauge (C,n))))) by A764, A772, GOBRD11:35; ::_thesis: verum
end;
end;
end;
now__::_thesis:_not_m_<>_1
A773: for h being non constant standard special_circular_sequence st L~ h c= L~ f holds
for Comp being Subset of (TOP-REAL 2) st Comp is_a_component_of (L~ h) ` holds
for n being Element of NAT st 1 <= n & n + 1 <= len f & f /. n in Comp & not f /. n in L~ h holds
C meets Comp
proof
let h be non constant standard special_circular_sequence; ::_thesis: ( L~ h c= L~ f implies for Comp being Subset of (TOP-REAL 2) st Comp is_a_component_of (L~ h) ` holds
for n being Element of NAT st 1 <= n & n + 1 <= len f & f /. n in Comp & not f /. n in L~ h holds
C meets Comp )
assume A774: L~ h c= L~ f ; ::_thesis: for Comp being Subset of (TOP-REAL 2) st Comp is_a_component_of (L~ h) ` holds
for n being Element of NAT st 1 <= n & n + 1 <= len f & f /. n in Comp & not f /. n in L~ h holds
C meets Comp
let Comp be Subset of (TOP-REAL 2); ::_thesis: ( Comp is_a_component_of (L~ h) ` implies for n being Element of NAT st 1 <= n & n + 1 <= len f & f /. n in Comp & not f /. n in L~ h holds
C meets Comp )
assume A775: Comp is_a_component_of (L~ h) ` ; ::_thesis: for n being Element of NAT st 1 <= n & n + 1 <= len f & f /. n in Comp & not f /. n in L~ h holds
C meets Comp
let n be Element of NAT ; ::_thesis: ( 1 <= n & n + 1 <= len f & f /. n in Comp & not f /. n in L~ h implies C meets Comp )
assume that
A776: ( 1 <= n & n + 1 <= len f ) and
A777: f /. n in Comp and
A778: not f /. n in L~ h ; ::_thesis: C meets Comp
reconsider rc = (right_cell (f,n,(Gauge (C,n)))) \ (L~ h) as Subset of (TOP-REAL 2) ;
 f /. n in right_cell (f,n,(Gauge (C,n))) by A571, A776, Th8;
then  f /. n in rc by A778, XBOOLE_0:def_5;
then A779: rc meets Comp by A777, XBOOLE_0:3;
A780: rc meets C
proof
 right_cell (f,n,(Gauge (C,n))) meets C by A428, A776;
then consider p being set  such that
A781: p in right_cell (f,n,(Gauge (C,n))) and
A782: p in C by XBOOLE_0:3;
reconsider p = p as Element of (TOP-REAL 2) by A781;
now__::_thesis:_ex_a_being_Element_of_(TOP-REAL_2)_st_
(_a_in_rc_&_a_in_C_)
take a = p; ::_thesis: ( a in rc & a in C )
now__::_thesis:_not_p_in_L~_h
assume  p in L~ h ; ::_thesis: contradiction
then consider j being Element of NAT  such that
A783: ( 1 <= j & j + 1 <= len f ) and
A784: p in LSeg (f,j) by A774, SPPOL_2:13;
 p in (left_cell (f,j,(Gauge (C,n)))) /\ (right_cell (f,j,(Gauge (C,n)))) by A428, A783, A784, GOBRD13:29;
then A785: p in left_cell (f,j,(Gauge (C,n))) by XBOOLE_0:def_4;
 left_cell (f,j,(Gauge (C,n))) misses C by A428, A783;
hence  contradiction by A782, A785, XBOOLE_0:3; ::_thesis: verum
end;
hence  a in rc by A781, XBOOLE_0:def_5; ::_thesis: a in C
thus  a in C by A782; ::_thesis: verum
end;
hence  rc meets C by XBOOLE_0:3; ::_thesis: verum
end;
 Int (right_cell (f,n,(Gauge (C,n)))) misses L~ f by A571, A776, Th15;
then  Int (right_cell (f,n,(Gauge (C,n)))) misses L~ h by A774, XBOOLE_1:63;
then A786: Int (right_cell (f,n,(Gauge (C,n)))) c= (L~ h) ` by SUBSET_1:23;
A787: rc = (right_cell (f,n,(Gauge (C,n)))) /\ ((L~ h) `) by SUBSET_1:13;
then A788: rc c= (L~ h) ` by XBOOLE_1:17;
 rc c= right_cell (f,n,(Gauge (C,n))) by XBOOLE_1:36;
then A789: ( Int (right_cell (f,n,(Gauge (C,n)))) c= right_cell (f,n,(Gauge (C,n))) & rc c= Cl (Int (right_cell (f,n,(Gauge (C,n))))) ) by A571, A776, Th11, TOPS_1:16;
 Int (right_cell (f,n,(Gauge (C,n)))) is convex by A571, A776, Th10;
then  rc is connected by A787, A786, A789, CONNSP_1:18, XBOOLE_1:19;
then  rc c= Comp by A775, A779, A788, GOBOARD9:4;
hence  C meets Comp by A780, XBOOLE_1:63; ::_thesis: verum
end;
 (L~ g) ` is open by TOPS_1:3;
then A790: (L~ g) ` = Int ((L~ g) `) by TOPS_1:23;
A791: L~ g c= L~ f by JORDAN3:40;
A792: for j, k being Element of NAT st 1 <= j & j <= k holds
(F . k) /. j = (F . j) /. j
proof
let j, k be Element of NAT ; ::_thesis: ( 1 <= j & j <= k implies (F . k) /. j = (F . j) /. j )
assume that
A793: 1 <= j and
A794: j <= k ; ::_thesis: (F . k) /. j = (F . j) /. j
 j <= len (F . k) by A192, A794;
then  len ((F . k) | j) = j by FINSEQ_1:59;
then A795: j in dom ((F . k) | j) by A793, FINSEQ_3:25;
 (F . k) | j = F . j by A490, A794;
hence  (F . k) /. j = (F . j) /. j by A795, FINSEQ_4:70; ::_thesis: verum
end;
assume  m <> 1 ; ::_thesis: contradiction
then A796: 1 < m by A570, XXREAL_0:1;
A797: for n being Element of NAT st 1 <= n & n <= m -' 1 holds
not f /. n in L~ g
proof
A798: 2 <= len (Gauge (C,n)) by A231, NAT_1:12;
let n be Element of NAT ; ::_thesis: ( 1 <= n & n <= m -' 1 implies not f /. n in L~ g )
assume that
A799: 1 <= n and
A800: n <= m -' 1 ; ::_thesis: not f /. n in L~ g
set p = f /. n;
A801: n <= len f by A577, A800, XXREAL_0:2;
then A802: f /. n in Values (Gauge (C,n)) by A428, A799, Th6;
assume  f /. n in L~ g ; ::_thesis: contradiction
then consider j being Element of NAT  such that
A803: (m -' 1) + 1 <= j and
A804: j + 1 <= len f and
A805: f /. n in LSeg (f,j) by A577, Th7;
A806: j < k by A580, A804, NAT_1:13;
A807: n < (m -' 1) + 1 by A800, NAT_1:13;
then A808: n < j by A803, XXREAL_0:2;
A809: (m -' 1) + 1 = m by A570, XREAL_1:235;
then A810: 1 < j by A796, A803, XXREAL_0:2;
percases ( f /. n = f /. j or f /. n = f /. (j + 1) ) by A2, A428, A804, A805, A810, A798, A802, Th23;
supposeA811: f /. n = f /. j ; ::_thesis: contradiction
A812: n <> len (F . j) by A192, A803, A807;
 n <= len (F . j) by A192, A808;
then A813: n in dom (F . j) by A799, FINSEQ_3:25;
(F . j) /. n = (F . n) /. n by A792, A799, A808
.= f /. n by A580, A792, A799, A801
.= (F . j) /. j by A792, A810, A806, A811
.= (F . j) /. (len (F . j)) by A192 ;
hence  contradiction by A566, A810, A806, A813, A812; ::_thesis: verum
end;
supposeA814: f /. n = f /. (j + 1) ; ::_thesis: contradiction
now__::_thesis:_contradiction
percases ( j + 1 = k or j + 1 < k ) by A580, A804, XXREAL_0:1;
supposeA815: j + 1 = k ; ::_thesis: contradiction
A816: n <> len (F . m) by A192, A807, A809;
 n <= len (F . m) by A192, A807, A809;
then A817: n in dom (F . m) by A799, FINSEQ_3:25;
(F . m) /. n = (F . n) /. n by A792, A799, A807, A809
.= (F . k) /. k by A580, A792, A799, A801, A814, A815
.= (F . m) /. m by A569, A580, A570, A572, A792
.= (F . m) /. (len (F . m)) by A192 ;
hence  contradiction by A566, A580, A570, A573, A817, A816; ::_thesis: verum
end;
supposeA818: j + 1 < k ; ::_thesis: contradiction
set l = j + 1;
A819: 1 <= j + 1 by NAT_1:11;
A820: ( n < n + 1 & n + 1 < j + 1 ) by A808, XREAL_1:6, XREAL_1:29;
then A821: n <> len (F . (j + 1)) by A192;
A822: n < j + 1 by A820, XXREAL_0:2;
then  n <= len (F . (j + 1)) by A192;
then A823: n in dom (F . (j + 1)) by A799, FINSEQ_3:25;
(F . (j + 1)) /. n = (F . n) /. n by A792, A799, A822
.= f /. n by A580, A792, A799, A801
.= (F . (j + 1)) /. (j + 1) by A792, A814, A818, A819
.= (F . (j + 1)) /. (len (F . (j + 1))) by A192 ;
hence  contradiction by A566, A818, A823, A821, NAT_1:11; ::_thesis: verum
end;
end;
end;
hence  contradiction ; ::_thesis: verum
end;
end;
end;
 C meets RightComp (Rev g)
proof
 1 <= len g by A624, XREAL_1:145;
then A824: ((len g) -' 1) + 2 = (len g) + 1 by Lm1;
A825: 1 - 1 < m - 1 by A796, XREAL_1:9;
A826: (m -' 1) + 2 = m + 1 by A570, Lm1;
set l = (m -' 1) + ((len g) -' 1);
set a = f /. (m -' 1);
set rg = Rev g;
set p = (Rev g) /. 1;
set q = (Rev g) /. 2;
A827: (1 + 1) - 1 <= (len g) - 1 by A623, XREAL_1:9;
 (1 + 1) -' 1 <= (len g) -' 1 by A623, NAT_D:42;
then A828: 1 <= (len g) -' 1 by NAT_D:34;
then  (m -' 1) + 1 <= (m -' 1) + ((len g) -' 1) by XREAL_1:6;
then  m -' 1 < (m -' 1) + ((len g) -' 1) by NAT_1:13;
then A829: m -' 1 <> len (F . ((m -' 1) + ((len g) -' 1))) by A192;
A830: 1 + 1 <= len (Rev g) by A623, FINSEQ_5:def_3;
then  (1 + 1) -' 1 <= (len (Rev g)) -' 1 by NAT_D:42;
then A831: 1 <= (len (Rev g)) -' 1 by NAT_D:34;
 1 < len (Rev g) by A830, NAT_1:13;
then A832: ((len (Rev g)) -' 1) + 1 = len (Rev g) by XREAL_1:235;
A833: Rev g is_sequence_on Gauge (C,n) by A625, Th5;
then consider p1, p2, q1, q2 being Element of NAT  such that
A834: [p1,p2] in Indices (Gauge (C,n)) and
A835: (Rev g) /. 1 = (Gauge (C,n)) * (p1,p2) and
A836: [q1,q2] in Indices (Gauge (C,n)) and
A837: (Rev g) /. 2 = (Gauge (C,n)) * (q1,q2) and
A838: ( ( p1 = q1 & p2 + 1 = q2 ) or ( p1 + 1 = q1 & p2 = q2 ) or ( p1 = q1 + 1 & p2 = q2 ) or ( p1 = q1 & p2 = q2 + 1 ) ) by A830, JORDAN8:3;
A839: 1 <= p1 by A834, MATRIX_1:38;
A840: p2 <= width (Gauge (C,n)) by A834, MATRIX_1:38;
A841: p1 <= len (Gauge (C,n)) by A834, MATRIX_1:38;
A842: 1 <= p2 by A834, MATRIX_1:38;
A843: (Rev g) /. 1 = f /. m by A569, A579, FINSEQ_5:65;
 (len g) -' 1 <= len g by NAT_D:44;
then A844: (len g) -' 1 in dom g by A828, FINSEQ_3:25;
then A845: (Rev g) /. 2 = g /. ((len g) -' 1) by A824, FINSEQ_5:66
.= f /. ((m -' 1) + ((len g) -' 1)) by A844, FINSEQ_5:27 ;
(m -' 1) + ((len g) -' 1) = (m + ((len g) -' 1)) -' 1 by A570, NAT_D:38
.= (((len g) -' 1) + m) - 1 by A828, NAT_D:37
.= (((len g) - 1) + m) - 1 by A827, XREAL_0:def_2
.= (((k - (m - 1)) - 1) + m) - 1 by A580, A578, A825, XREAL_0:def_2
.= k - 1 ;
then A846: k = ((m -' 1) + ((len g) -' 1)) + 1 ;
then A847: (m -' 1) + ((len g) -' 1) < k by XREAL_1:29;
 (len g) -' 1 <= (m -' 1) + ((len g) -' 1) by NAT_1:11;
then A848: 1 <= (m -' 1) + ((len g) -' 1) by A828, XXREAL_0:2;
then A849: right_cell (f,((m -' 1) + ((len g) -' 1)),(Gauge (C,n))) meets C by A428, A580, A846;
A850: (m -' 1) + 1 = m by A570, XREAL_1:235;
then A851: 1 <= m -' 1 by A796, NAT_1:13;
then A852: right_cell (f,(m -' 1),(Gauge (C,n))) meets C by A428, A572, A850;
 m -' 1 <= (m -' 1) + ((len g) -' 1) by NAT_1:11;
then  m -' 1 <= len (F . ((m -' 1) + ((len g) -' 1))) by A192;
then A853: m -' 1 in dom (F . ((m -' 1) + ((len g) -' 1))) by A851, FINSEQ_3:25;
 not f /. (m -' 1) in L~ g by A797, A851;
then A854: not f /. (m -' 1) in L~ (Rev g) by SPPOL_2:22;
percases ( ( p1 = q1 & p2 + 1 = q2 ) or ( p1 + 1 = q1 & p2 = q2 ) or ( p1 = q1 + 1 & p2 = q2 ) or ( p1 = q1 & p2 = q2 + 1 ) ) by A838;
supposeA855: ( p1 = q1 & p2 + 1 = q2 ) ; ::_thesis: C meets RightComp (Rev g)
consider a1, a2, p91, p92 being Element of NAT  such that
A856: [a1,a2] in Indices (Gauge (C,n)) and
A857: f /. (m -' 1) = (Gauge (C,n)) * (a1,a2) and
A858: ( [p91,p92] in Indices (Gauge (C,n)) & (Rev g) /. 1 = (Gauge (C,n)) * (p91,p92) ) and
A859: ( ( a1 = p91 & a2 + 1 = p92 ) or ( a1 + 1 = p91 & a2 = p92 ) or ( a1 = p91 + 1 & a2 = p92 ) or ( a1 = p91 & a2 = p92 + 1 ) ) by A571, A572, A843, A850, A851, JORDAN8:3;
A860: 1 <= a2 by A856, MATRIX_1:38;
thus  C meets RightComp (Rev g) ::_thesis: verum
proof
percases ( ( a1 = p91 & a2 + 1 = p92 ) or ( a1 + 1 = p91 & a2 = p92 ) or ( a1 = p91 + 1 & a2 = p92 ) or ( a1 = p91 & a2 = p92 + 1 ) ) by A859;
supposeA861: ( a1 = p91 & a2 + 1 = p92 ) ; ::_thesis: C meets RightComp (Rev g)
A862: ( (m -' 1) + 1 <= len (F . m) & f /. ((m -' 1) + 1) = (F . m) /. m ) by A192, A580, A570, A572, A792, A850;
A863: (F . k) | (m + 1) = F . (m + 1) by A490, A580, A576;
A864: a1 = p1 by A834, A835, A858, A861, GOBOARD1:5;
A865: m -' 1 <= m by A850, NAT_1:11;
A866: f /. (m -' 1) = (F . (m -' 1)) /. (m -' 1) by A580, A577, A792, A851
.= (F . m) /. (m -' 1) by A792, A851, A865 ;
A867: 2 in dom g by A623, FINSEQ_3:25;
A868: a2 + 1 = p2 by A834, A835, A858, A861, GOBOARD1:5;
then A869: p2 -' 1 = a2 by NAT_D:34;
right_cell (f,((m -' 1) + ((len g) -' 1)),(Gauge (C,n))) = cell ((Gauge (C,n)),(p1 -' 1),p2) by A428, A569, A580, A843, A846, A848, A845, A834, A835, A836, A837, A855, GOBRD13:28
.= front_left_cell ((F . m),(m -' 1),(Gauge (C,n))) by A428, A843, A850, A851, A834, A835, A856, A857, A864, A868, A866, A862, GOBRD13:34 ;
then  F . (m + 1) turns_left m -' 1, Gauge (C,n) by A458, A796, A849;
then A870: ( a2 + 1 > a2 & f turns_left m -' 1, Gauge (C,n) ) by A851, A826, A863, GOBRD13:44, NAT_1:13;
 ((len (Rev g)) -' 1) + 2 = (len g) + 1 by A824, FINSEQ_5:def_3;
then A871: (Rev g) /. ((len (Rev g)) -' 1) = g /. 2 by A867, FINSEQ_5:66
.= f /. (m + 1) by A826, A867, FINSEQ_5:27 ;
A872: (Rev g) /. 1 = g /. 1 by A569, A614, A579, FINSEQ_5:65
.= (Rev g) /. (len g) by FINSEQ_5:65
.= (Rev g) /. (len (Rev g)) by FINSEQ_5:def_3 ;
set rc = (right_cell ((Rev g),((len (Rev g)) -' 1),(Gauge (C,n)))) \ (L~ (Rev g));
A873: RightComp (Rev g) is_a_component_of (L~ (Rev g)) ` by GOBOARD9:def_2;
A874: (p1 -' 1) + 1 = p1 by A839, XREAL_1:235;
A875: p2 + 1 > a2 + 1 by A868, NAT_1:13;
then A876: [(p1 -' 1),p2] in Indices (Gauge (C,n)) by A843, A850, A834, A835, A856, A857, A870, GOBRD13:def_7;
then A877: 1 <= p1 -' 1 by MATRIX_1:38;
 f /. (m + 1) = (Gauge (C,n)) * ((p1 -' 1),p2) by A843, A850, A826, A834, A835, A856, A857, A875, A870, GOBRD13:def_7;
then  right_cell ((Rev g),((len (Rev g)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),(p1 -' 1),a2) by A833, A831, A832, A834, A835, A876, A869, A874, A871, A872, GOBRD13:24;
then  f /. (m -' 1) in right_cell ((Rev g),((len (Rev g)) -' 1),(Gauge (C,n))) by A841, A840, A857, A860, A864, A868, A877, A874, Th20;
then A878: f /. (m -' 1) in (right_cell ((Rev g),((len (Rev g)) -' 1),(Gauge (C,n)))) \ (L~ (Rev g)) by A854, XBOOLE_0:def_5;
 ( (right_cell ((Rev g),((len (Rev g)) -' 1),(Gauge (C,n)))) \ (L~ (Rev g)) c= RightComp (Rev g) & L~ (Rev g) c= L~ f ) by A791, A833, A831, A832, Th27, SPPOL_2:22;
hence  C meets RightComp (Rev g) by A572, A773, A850, A851, A854, A878, A873; ::_thesis: verum
end;
supposeA879: ( a1 + 1 = p91 & a2 = p92 ) ; ::_thesis: C meets RightComp (Rev g)
then  a1 + 1 = p1 by A834, A835, A858, GOBOARD1:5;
then A880: q1 -' 1 = a1 by A855, NAT_D:34;
 a2 = p2 by A834, A835, A858, A879, GOBOARD1:5;
then  right_cell (f,((m -' 1) + ((len g) -' 1)),(Gauge (C,n))) = cell ((Gauge (C,n)),a1,a2) by A428, A569, A580, A843, A846, A848, A845, A834, A835, A836, A837, A855, A880, GOBRD13:28
.= left_cell (f,(m -' 1),(Gauge (C,n))) by A428, A572, A843, A850, A851, A856, A857, A858, A879, GOBRD13:23 ;
hence  C meets RightComp (Rev g) by A428, A572, A850, A851, A849; ::_thesis: verum
end;
suppose ( a1 = p91 + 1 & a2 = p92 ) ; ::_thesis: C meets RightComp (Rev g)
then  ( a1 = p1 + 1 & a2 = p2 ) by A834, A835, A858, GOBOARD1:5;
then  right_cell (f,(m -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),p1,p2) by A428, A572, A843, A850, A851, A834, A835, A856, A857, GOBRD13:26
.= left_cell (f,((m -' 1) + ((len g) -' 1)),(Gauge (C,n))) by A428, A569, A580, A843, A846, A848, A845, A834, A835, A836, A837, A855, GOBRD13:27 ;
hence  C meets RightComp (Rev g) by A428, A580, A846, A848, A852; ::_thesis: verum
end;
suppose ( a1 = p91 & a2 = p92 + 1 ) ; ::_thesis: C meets RightComp (Rev g)
then A881: ( a1 = q1 & a2 = q2 ) by A834, A835, A855, A858, GOBOARD1:5;
(F . ((m -' 1) + ((len g) -' 1))) /. (m -' 1) = (F . (m -' 1)) /. (m -' 1) by A792, A851, NAT_1:11
.= (Rev g) /. 2 by A580, A577, A792, A851, A837, A857, A881
.= (F . ((m -' 1) + ((len g) -' 1))) /. ((m -' 1) + ((len g) -' 1)) by A792, A847, A848, A845
.= (F . ((m -' 1) + ((len g) -' 1))) /. (len (F . ((m -' 1) + ((len g) -' 1)))) by A192 ;
hence  C meets RightComp (Rev g) by A566, A847, A848, A853, A829; ::_thesis: verum
end;
end;
end;
end;
supposeA882: ( p1 + 1 = q1 & p2 = q2 ) ; ::_thesis: C meets RightComp (Rev g)
consider a1, a2, p91, p92 being Element of NAT  such that
A883: [a1,a2] in Indices (Gauge (C,n)) and
A884: f /. (m -' 1) = (Gauge (C,n)) * (a1,a2) and
A885: ( [p91,p92] in Indices (Gauge (C,n)) & (Rev g) /. 1 = (Gauge (C,n)) * (p91,p92) ) and
A886: ( ( a1 = p91 & a2 + 1 = p92 ) or ( a1 + 1 = p91 & a2 = p92 ) or ( a1 = p91 + 1 & a2 = p92 ) or ( a1 = p91 & a2 = p92 + 1 ) ) by A571, A572, A843, A850, A851, JORDAN8:3;
A887: 1 <= a1 by A883, MATRIX_1:38;
thus  C meets RightComp (Rev g) ::_thesis: verum
proof
percases ( ( a1 = p91 & a2 + 1 = p92 ) or ( a1 + 1 = p91 & a2 = p92 ) or ( a1 = p91 + 1 & a2 = p92 ) or ( a1 = p91 & a2 = p92 + 1 ) ) by A886;
supposeA888: ( a1 = p91 & a2 + 1 = p92 ) ; ::_thesis: C meets RightComp (Rev g)
then  a2 + 1 = p2 by A834, A835, A885, GOBOARD1:5;
then A889: q2 -' 1 = a2 by A882, NAT_D:34;
A890: a1 = p1 by A834, A835, A885, A888, GOBOARD1:5;
right_cell (f,(m -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),a1,a2) by A428, A572, A843, A850, A851, A883, A884, A885, A888, GOBRD13:22
.= left_cell (f,((m -' 1) + ((len g) -' 1)),(Gauge (C,n))) by A428, A569, A580, A843, A846, A848, A845, A834, A835, A836, A837, A882, A890, A889, GOBRD13:25 ;
hence  C meets RightComp (Rev g) by A428, A580, A846, A848, A852; ::_thesis: verum
end;
supposeA891: ( a1 + 1 = p91 & a2 = p92 ) ; ::_thesis: C meets RightComp (Rev g)
A892: m -' 1 <= m by A850, NAT_1:11;
A893: f /. (m -' 1) = (F . (m -' 1)) /. (m -' 1) by A580, A577, A792, A851
.= (F . m) /. (m -' 1) by A792, A851, A892 ;
A894: 2 in dom g by A623, FINSEQ_3:25;
 ((len (Rev g)) -' 1) + 2 = (len g) + 1 by A824, FINSEQ_5:def_3;
then A895: (Rev g) /. ((len (Rev g)) -' 1) = g /. 2 by A894, FINSEQ_5:66
.= f /. (m + 1) by A826, A894, FINSEQ_5:27 ;
A896: (Rev g) /. 1 = g /. 1 by A569, A614, A579, FINSEQ_5:65
.= (Rev g) /. (len g) by FINSEQ_5:65
.= (Rev g) /. (len (Rev g)) by FINSEQ_5:def_3 ;
A897: a2 = p2 by A834, A835, A885, A891, GOBOARD1:5;
A898: ( (m -' 1) + 1 <= len (F . m) & f /. ((m -' 1) + 1) = (F . m) /. m ) by A192, A580, A570, A572, A792, A850;
A899: (F . k) | (m + 1) = F . (m + 1) by A490, A580, A576;
set rc = (right_cell ((Rev g),((len (Rev g)) -' 1),(Gauge (C,n)))) \ (L~ (Rev g));
A900: ( a1 < a1 + 1 & p1 < p1 + 1 ) by XREAL_1:29;
A901: a1 + 1 = p1 by A834, A835, A885, A891, GOBOARD1:5;
then A902: a1 = p1 -' 1 by NAT_D:34;
right_cell (f,((m -' 1) + ((len g) -' 1)),(Gauge (C,n))) = cell ((Gauge (C,n)),p1,p2) by A428, A569, A580, A843, A846, A848, A845, A834, A835, A836, A837, A882, GOBRD13:26
.= front_left_cell ((F . m),(m -' 1),(Gauge (C,n))) by A428, A843, A850, A851, A834, A835, A883, A884, A901, A897, A893, A898, GOBRD13:36 ;
then  F . (m + 1) turns_left m -' 1, Gauge (C,n) by A458, A796, A849;
then A903: f turns_left m -' 1, Gauge (C,n) by A851, A826, A899, GOBRD13:44;
then A904: [p1,(p2 + 1)] in Indices (Gauge (C,n)) by A843, A850, A834, A835, A883, A884, A901, A900, GOBRD13:def_7;
then A905: p2 + 1 <= width (Gauge (C,n)) by MATRIX_1:38;
 f /. (m + 1) = (Gauge (C,n)) * (p1,(p2 + 1)) by A843, A850, A826, A834, A835, A883, A884, A901, A900, A903, GOBRD13:def_7;
then  right_cell ((Rev g),((len (Rev g)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),(p1 -' 1),a2) by A833, A831, A832, A834, A835, A897, A904, A895, A896, GOBRD13:28;
then  f /. (m -' 1) in right_cell ((Rev g),((len (Rev g)) -' 1),(Gauge (C,n))) by A841, A842, A884, A887, A901, A897, A905, A902, Th20;
then A906: f /. (m -' 1) in (right_cell ((Rev g),((len (Rev g)) -' 1),(Gauge (C,n)))) \ (L~ (Rev g)) by A854, XBOOLE_0:def_5;
A907: RightComp (Rev g) is_a_component_of (L~ (Rev g)) ` by GOBOARD9:def_2;
 ( (right_cell ((Rev g),((len (Rev g)) -' 1),(Gauge (C,n)))) \ (L~ (Rev g)) c= RightComp (Rev g) & L~ (Rev g) c= L~ f ) by A791, A833, A831, A832, Th27, SPPOL_2:22;
hence  C meets RightComp (Rev g) by A572, A773, A850, A851, A854, A906, A907; ::_thesis: verum
end;
suppose ( a1 = p91 + 1 & a2 = p92 ) ; ::_thesis: C meets RightComp (Rev g)
then A908: ( a1 = q1 & a2 = q2 ) by A834, A835, A882, A885, GOBOARD1:5;
(F . ((m -' 1) + ((len g) -' 1))) /. (m -' 1) = (F . (m -' 1)) /. (m -' 1) by A792, A851, NAT_1:11
.= (Rev g) /. 2 by A580, A577, A792, A851, A837, A884, A908
.= (F . ((m -' 1) + ((len g) -' 1))) /. ((m -' 1) + ((len g) -' 1)) by A792, A847, A848, A845
.= (F . ((m -' 1) + ((len g) -' 1))) /. (len (F . ((m -' 1) + ((len g) -' 1)))) by A192 ;
hence  C meets RightComp (Rev g) by A566, A847, A848, A853, A829; ::_thesis: verum
end;
suppose ( a1 = p91 & a2 = p92 + 1 ) ; ::_thesis: C meets RightComp (Rev g)
then A909: ( a1 = p1 & a2 = p2 + 1 ) by A834, A835, A885, GOBOARD1:5;
right_cell (f,((m -' 1) + ((len g) -' 1)),(Gauge (C,n))) = cell ((Gauge (C,n)),p1,p2) by A428, A569, A580, A843, A846, A848, A845, A834, A835, A836, A837, A882, GOBRD13:26
.= left_cell (f,(m -' 1),(Gauge (C,n))) by A428, A572, A843, A850, A851, A834, A835, A883, A884, A909, GOBRD13:27 ;
hence  C meets RightComp (Rev g) by A428, A572, A850, A851, A849; ::_thesis: verum
end;
end;
end;
end;
supposeA910: ( p1 = q1 + 1 & p2 = q2 ) ; ::_thesis: C meets RightComp (Rev g)
consider a1, a2, p91, p92 being Element of NAT  such that
A911: [a1,a2] in Indices (Gauge (C,n)) and
A912: f /. (m -' 1) = (Gauge (C,n)) * (a1,a2) and
A913: ( [p91,p92] in Indices (Gauge (C,n)) & (Rev g) /. 1 = (Gauge (C,n)) * (p91,p92) ) and
A914: ( ( a1 = p91 & a2 + 1 = p92 ) or ( a1 + 1 = p91 & a2 = p92 ) or ( a1 = p91 + 1 & a2 = p92 ) or ( a1 = p91 & a2 = p92 + 1 ) ) by A571, A572, A843, A850, A851, JORDAN8:3;
A915: a1 <= len (Gauge (C,n)) by A911, MATRIX_1:38;
thus  C meets RightComp (Rev g) ::_thesis: verum
proof
percases ( ( a1 = p91 & a2 + 1 = p92 ) or ( a1 + 1 = p91 & a2 = p92 ) or ( a1 = p91 + 1 & a2 = p92 ) or ( a1 = p91 & a2 = p92 + 1 ) ) by A914;
supposeA916: ( a1 = p91 & a2 + 1 = p92 ) ; ::_thesis: C meets RightComp (Rev g)
then  a2 + 1 = p2 by A834, A835, A913, GOBOARD1:5;
then A917: q2 -' 1 = a2 by A910, NAT_D:34;
 a1 = p1 by A834, A835, A913, A916, GOBOARD1:5;
then A918: q1 = a1 -' 1 by A910, NAT_D:34;
right_cell (f,((m -' 1) + ((len g) -' 1)),(Gauge (C,n))) = cell ((Gauge (C,n)),q1,(q2 -' 1)) by A428, A569, A580, A843, A846, A848, A845, A834, A835, A836, A837, A910, GOBRD13:24
.= left_cell (f,(m -' 1),(Gauge (C,n))) by A428, A572, A843, A850, A851, A911, A912, A913, A916, A918, A917, GOBRD13:21 ;
hence  C meets RightComp (Rev g) by A428, A572, A850, A851, A849; ::_thesis: verum
end;
suppose ( a1 + 1 = p91 & a2 = p92 ) ; ::_thesis: C meets RightComp (Rev g)
then A919: ( a1 + 1 = p1 & a2 = p2 ) by A834, A835, A913, GOBOARD1:5;
(F . ((m -' 1) + ((len g) -' 1))) /. (m -' 1) = (F . (m -' 1)) /. (m -' 1) by A792, A851, NAT_1:11
.= (Rev g) /. 2 by A580, A577, A792, A851, A837, A910, A912, A919
.= (F . ((m -' 1) + ((len g) -' 1))) /. ((m -' 1) + ((len g) -' 1)) by A792, A847, A848, A845
.= (F . ((m -' 1) + ((len g) -' 1))) /. (len (F . ((m -' 1) + ((len g) -' 1)))) by A192 ;
hence  C meets RightComp (Rev g) by A566, A847, A848, A853, A829; ::_thesis: verum
end;
supposeA920: ( a1 = p91 + 1 & a2 = p92 ) ; ::_thesis: C meets RightComp (Rev g)
A921: (Rev g) /. 1 = g /. 1 by A569, A614, A579, FINSEQ_5:65
.= (Rev g) /. (len g) by FINSEQ_5:65
.= (Rev g) /. (len (Rev g)) by FINSEQ_5:def_3 ;
A922: a1 = p1 + 1 by A834, A835, A913, A920, GOBOARD1:5;
 p1 + 1 > p1 by XREAL_1:29;
then A923: a1 + 1 > p1 by A922, NAT_1:13;
A924: m -' 1 <= m by A850, NAT_1:11;
A925: f /. (m -' 1) = (F . (m -' 1)) /. (m -' 1) by A580, A577, A792, A851
.= (F . m) /. (m -' 1) by A792, A851, A924 ;
A926: 2 in dom g by A623, FINSEQ_3:25;
 ((len (Rev g)) -' 1) + 2 = (len g) + 1 by A824, FINSEQ_5:def_3;
then A927: (Rev g) /. ((len (Rev g)) -' 1) = g /. 2 by A926, FINSEQ_5:66
.= f /. (m + 1) by A826, A926, FINSEQ_5:27 ;
A928: (F . k) | (m + 1) = F . (m + 1) by A490, A580, A576;
set rc = (right_cell ((Rev g),((len (Rev g)) -' 1),(Gauge (C,n)))) \ (L~ (Rev g));
A929: RightComp (Rev g) is_a_component_of (L~ (Rev g)) ` by GOBOARD9:def_2;
A930: (p2 -' 1) + 1 = p2 by A842, XREAL_1:235;
A931: f /. ((m -' 1) + 1) = (F . m) /. m by A580, A570, A572, A792, A850;
A932: ( p1 -' 1 = q1 & (m -' 1) + 1 <= len (F . m) ) by A192, A850, A910, NAT_D:34;
A933: a2 = p2 by A834, A835, A913, A920, GOBOARD1:5;
right_cell (f,((m -' 1) + ((len g) -' 1)),(Gauge (C,n))) = cell ((Gauge (C,n)),q1,(q2 -' 1)) by A428, A569, A580, A843, A846, A848, A845, A834, A835, A836, A837, A910, GOBRD13:24
.= front_left_cell ((F . m),(m -' 1),(Gauge (C,n))) by A428, A843, A850, A851, A834, A835, A910, A911, A912, A922, A933, A932, A925, A931, GOBRD13:38 ;
then  F . (m + 1) turns_left m -' 1, Gauge (C,n) by A458, A796, A849;
then A934: f turns_left m -' 1, Gauge (C,n) by A851, A826, A928, GOBRD13:44;
A935: a2 + 1 > p2 by A933, NAT_1:13;
then A936: [p1,(p2 -' 1)] in Indices (Gauge (C,n)) by A843, A850, A834, A835, A911, A912, A933, A923, A934, GOBRD13:def_7;
then A937: 1 <= p2 -' 1 by MATRIX_1:38;
 f /. (m + 1) = (Gauge (C,n)) * (p1,(p2 -' 1)) by A843, A850, A826, A834, A835, A911, A912, A933, A935, A923, A934, GOBRD13:def_7;
then  right_cell ((Rev g),((len (Rev g)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),p1,(p2 -' 1)) by A833, A831, A832, A834, A835, A936, A930, A927, A921, GOBRD13:22;
then  f /. (m -' 1) in right_cell ((Rev g),((len (Rev g)) -' 1),(Gauge (C,n))) by A839, A840, A912, A915, A922, A933, A937, A930, Th20;
then A938: f /. (m -' 1) in (right_cell ((Rev g),((len (Rev g)) -' 1),(Gauge (C,n)))) \ (L~ (Rev g)) by A854, XBOOLE_0:def_5;
 ( (right_cell ((Rev g),((len (Rev g)) -' 1),(Gauge (C,n)))) \ (L~ (Rev g)) c= RightComp (Rev g) & L~ (Rev g) c= L~ f ) by A791, A833, A831, A832, Th27, SPPOL_2:22;
hence  C meets RightComp (Rev g) by A572, A773, A850, A851, A854, A938, A929; ::_thesis: verum
end;
supposeA939: ( a1 = p91 & a2 = p92 + 1 ) ; ::_thesis: C meets RightComp (Rev g)
then  a1 = p1 by A834, A835, A913, GOBOARD1:5;
then A940: q1 = a1 -' 1 by A910, NAT_D:34;
 a2 = p2 + 1 by A834, A835, A913, A939, GOBOARD1:5;
then  right_cell (f,(m -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),q1,q2) by A428, A572, A843, A850, A851, A910, A911, A912, A913, A939, A940, GOBRD13:28
.= left_cell (f,((m -' 1) + ((len g) -' 1)),(Gauge (C,n))) by A428, A569, A580, A843, A846, A848, A845, A834, A835, A836, A837, A910, GOBRD13:23 ;
hence  C meets RightComp (Rev g) by A428, A580, A846, A848, A852; ::_thesis: verum
end;
end;
end;
end;
supposeA941: ( p1 = q1 & p2 = q2 + 1 ) ; ::_thesis: C meets RightComp (Rev g)
consider a1, a2, p91, p92 being Element of NAT  such that
A942: [a1,a2] in Indices (Gauge (C,n)) and
A943: f /. (m -' 1) = (Gauge (C,n)) * (a1,a2) and
A944: ( [p91,p92] in Indices (Gauge (C,n)) & (Rev g) /. 1 = (Gauge (C,n)) * (p91,p92) ) and
A945: ( ( a1 = p91 & a2 + 1 = p92 ) or ( a1 + 1 = p91 & a2 = p92 ) or ( a1 = p91 + 1 & a2 = p92 ) or ( a1 = p91 & a2 = p92 + 1 ) ) by A571, A572, A843, A850, A851, JORDAN8:3;
A946: a2 <= width (Gauge (C,n)) by A942, MATRIX_1:38;
thus  C meets RightComp (Rev g) ::_thesis: verum
proof
percases ( ( a1 = p91 & a2 + 1 = p92 ) or ( a1 + 1 = p91 & a2 = p92 ) or ( a1 = p91 + 1 & a2 = p92 ) or ( a1 = p91 & a2 = p92 + 1 ) ) by A945;
suppose ( a1 = p91 & a2 + 1 = p92 ) ; ::_thesis: C meets RightComp (Rev g)
then A947: ( a1 = p1 & a2 + 1 = p2 ) by A834, A835, A944, GOBOARD1:5;
(F . ((m -' 1) + ((len g) -' 1))) /. (m -' 1) = (F . (m -' 1)) /. (m -' 1) by A792, A851, NAT_1:11
.= (Rev g) /. 2 by A580, A577, A792, A851, A837, A941, A943, A947
.= (F . ((m -' 1) + ((len g) -' 1))) /. ((m -' 1) + ((len g) -' 1)) by A792, A847, A848, A845
.= (F . ((m -' 1) + ((len g) -' 1))) /. (len (F . ((m -' 1) + ((len g) -' 1)))) by A192 ;
hence  C meets RightComp (Rev g) by A566, A847, A848, A853, A829; ::_thesis: verum
end;
supposeA948: ( a1 + 1 = p91 & a2 = p92 ) ; ::_thesis: C meets RightComp (Rev g)
then  a2 = p2 by A834, A835, A944, GOBOARD1:5;
then A949: a2 -' 1 = q2 by A941, NAT_D:34;
 a1 + 1 = p1 by A834, A835, A944, A948, GOBOARD1:5;
then A950: a1 = q1 -' 1 by A941, NAT_D:34;
right_cell (f,(m -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),a1,(a2 -' 1)) by A428, A572, A843, A850, A851, A942, A943, A944, A948, GOBRD13:24
.= left_cell (f,((m -' 1) + ((len g) -' 1)),(Gauge (C,n))) by A428, A569, A580, A843, A846, A848, A845, A834, A835, A836, A837, A941, A950, A949, GOBRD13:21 ;
hence  C meets RightComp (Rev g) by A428, A580, A846, A848, A852; ::_thesis: verum
end;
supposeA951: ( a1 = p91 + 1 & a2 = p92 ) ; ::_thesis: C meets RightComp (Rev g)
then  a2 = p2 by A834, A835, A944, GOBOARD1:5;
then A952: a2 -' 1 = q2 by A941, NAT_D:34;
A953: a1 = p1 + 1 by A834, A835, A944, A951, GOBOARD1:5;
right_cell (f,((m -' 1) + ((len g) -' 1)),(Gauge (C,n))) = cell ((Gauge (C,n)),q1,q2) by A428, A569, A580, A843, A846, A848, A845, A834, A835, A836, A837, A941, GOBRD13:22
.= left_cell (f,(m -' 1),(Gauge (C,n))) by A428, A572, A843, A850, A851, A941, A942, A943, A944, A951, A953, A952, GOBRD13:25 ;
hence  C meets RightComp (Rev g) by A428, A572, A850, A851, A849; ::_thesis: verum
end;
supposeA954: ( a1 = p91 & a2 = p92 + 1 ) ; ::_thesis: C meets RightComp (Rev g)
set rc = (right_cell ((Rev g),((len (Rev g)) -' 1),(Gauge (C,n)))) \ (L~ (Rev g));
A955: RightComp (Rev g) is_a_component_of (L~ (Rev g)) ` by GOBOARD9:def_2;
A956: 2 in dom g by A623, FINSEQ_3:25;
 ((len (Rev g)) -' 1) + 2 = (len g) + 1 by A824, FINSEQ_5:def_3;
then A957: (Rev g) /. ((len (Rev g)) -' 1) = g /. 2 by A956, FINSEQ_5:66
.= f /. (m + 1) by A826, A956, FINSEQ_5:27 ;
A958: (Rev g) /. 1 = g /. 1 by A569, A614, A579, FINSEQ_5:65
.= (Rev g) /. (len g) by FINSEQ_5:65
.= (Rev g) /. (len (Rev g)) by FINSEQ_5:def_3 ;
A959: a1 = p1 by A834, A835, A944, A954, GOBOARD1:5;
A960: m -' 1 <= m by A850, NAT_1:11;
A961: f /. (m -' 1) = (F . (m -' 1)) /. (m -' 1) by A580, A577, A792, A851
.= (F . m) /. (m -' 1) by A792, A851, A960 ;
A962: ( p2 -' 1 = q2 & (m -' 1) + 1 <= len (F . m) ) by A192, A850, A941, NAT_D:34;
A963: f /. ((m -' 1) + 1) = (F . m) /. m by A580, A570, A572, A792, A850;
A964: (F . k) | (m + 1) = F . (m + 1) by A490, A580, A576;
A965: a2 = p2 + 1 by A834, A835, A944, A954, GOBOARD1:5;
right_cell (f,((m -' 1) + ((len g) -' 1)),(Gauge (C,n))) = cell ((Gauge (C,n)),q1,q2) by A428, A569, A580, A843, A846, A848, A845, A834, A835, A836, A837, A941, GOBRD13:22
.= front_left_cell ((F . m),(m -' 1),(Gauge (C,n))) by A428, A843, A850, A851, A834, A835, A941, A942, A943, A959, A965, A962, A961, A963, GOBRD13:40 ;
then  F . (m + 1) turns_left m -' 1, Gauge (C,n) by A458, A796, A849;
then A966: ( p2 + 1 > p2 & f turns_left m -' 1, Gauge (C,n) ) by A851, A826, A964, GOBRD13:44, NAT_1:13;
A967: a2 + 1 > p2 + 1 by A965, NAT_1:13;
then A968: [(p1 + 1),p2] in Indices (Gauge (C,n)) by A843, A850, A834, A835, A942, A943, A966, GOBRD13:def_7;
then A969: p1 + 1 <= len (Gauge (C,n)) by MATRIX_1:38;
 f /. (m + 1) = (Gauge (C,n)) * ((p1 + 1),p2) by A843, A850, A826, A834, A835, A942, A943, A967, A966, GOBRD13:def_7;
then  right_cell ((Rev g),((len (Rev g)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),p1,p2) by A833, A831, A832, A834, A835, A968, A957, A958, GOBRD13:26;
then  f /. (m -' 1) in right_cell ((Rev g),((len (Rev g)) -' 1),(Gauge (C,n))) by A839, A842, A943, A946, A959, A965, A969, Th20;
then A970: f /. (m -' 1) in (right_cell ((Rev g),((len (Rev g)) -' 1),(Gauge (C,n)))) \ (L~ (Rev g)) by A854, XBOOLE_0:def_5;
 ( (right_cell ((Rev g),((len (Rev g)) -' 1),(Gauge (C,n)))) \ (L~ (Rev g)) c= RightComp (Rev g) & L~ (Rev g) c= L~ f ) by A791, A833, A831, A832, Th27, SPPOL_2:22;
hence  C meets RightComp (Rev g) by A572, A773, A850, A851, A854, A970, A955; ::_thesis: verum
end;
end;
end;
end;
end;
end;
then A971: ( LeftComp g is_a_component_of (L~ g) ` & C meets LeftComp g ) by GOBOARD9:24, GOBOARD9:def_1;
reconsider Lg9 = (L~ g) ` as Subset of (TOP-REAL 2) ;
A972: RightComp g is_a_component_of (L~ g) ` by GOBOARD9:def_2;
A973: C c= Lg9
proof
let c be set ; :: according to TARSKI:def_3 ::_thesis: ( not c in C or c in Lg9 )
assume that
A974: c in C and
A975: not c in Lg9 ; ::_thesis: contradiction
reconsider c = c as Point of (TOP-REAL 2) by A974;
consider i being Element of NAT  such that
A976: 1 <= i and
A977: i + 1 <= len g and
A978: c in LSeg ((g /. i),(g /. (i + 1))) by A975, SPPOL_2:14, SUBSET_1:29;
A979: 1 <= i + (m -' 1) by A976, NAT_1:12;
 i + 1 in dom g by A976, A977, SEQ_4:134;
then A980: g /. (i + 1) = f /. ((i + 1) + (m -' 1)) by FINSEQ_5:27;
 (i + 1) + (m -' 1) = (i + (m -' 1)) + 1 ;
then A981: (i + (m -' 1)) + 1 <= (len g) + (m -' 1) by A977, XREAL_1:6;
 i in dom g by A976, A977, SEQ_4:134;
then  g /. i = f /. (i + (m -' 1)) by FINSEQ_5:27;
then  c in LSeg (f,(i + (m -' 1))) by A578, A978, A980, A979, A981, TOPREAL1:def_3;
then  c in (left_cell (f,(i + (m -' 1)),(Gauge (C,n)))) /\ (right_cell (f,(i + (m -' 1)),(Gauge (C,n)))) by A428, A578, A979, A981, GOBRD13:29;
then  c in left_cell (f,(i + (m -' 1)),(Gauge (C,n))) by XBOOLE_0:def_4;
then  left_cell (f,(i + (m -' 1)),(Gauge (C,n))) meets C by A974, XBOOLE_0:3;
hence  contradiction by A428, A578, A979, A981; ::_thesis: verum
end;
A982: TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) = TopSpaceMetr (Euclid 2) by EUCLID:def_8;
 C meets RightComp g
proof
 right_cell (f,m,(Gauge (C,n))) meets C by A428, A570, A576;
then consider p being set  such that
A983: p in right_cell (f,m,(Gauge (C,n))) and
A984: p in C by XBOOLE_0:3;
reconsider p = p as Element of (TOP-REAL 2) by A983;
now__::_thesis:_ex_a_being_Element_of_(TOP-REAL_2)_st_
(_a_in_C_&_a_in_RightComp_g_)
reconsider u = p as Element of (Euclid 2) by TOPREAL3:8;
take a = p; ::_thesis: ( a in C & a in RightComp g )
thus  a in C by A984; ::_thesis: a in RightComp g
consider r being real number  such that
A985: r > 0 and
A986: Ball (u,r) c= (L~ g) ` by A973, A790, A984, GOBOARD6:5;
reconsider r = r as Real by XREAL_0:def_1;
A987: p in Ball (u,r) by A985, GOBOARD6:1;
reconsider B = Ball (u,r) as non empty Subset of (TOP-REAL 2) by A982, A985, GOBOARD6:1, TOPMETR:12;
A988: p in B by A985, GOBOARD6:1;
 ( right_cell (f,m,(Gauge (C,n))) = Cl (Int (right_cell (f,m,(Gauge (C,n))))) & B is open ) by A570, A576, A756, GOBOARD6:3;
then A989: Int (right_cell (f,m,(Gauge (C,n)))) meets B by A983, A987, TOPS_1:12;
 ( Int (right_cell (g,1,(Gauge (C,n)))) c= Int (right_cell (g,1)) & Int (right_cell (g,1)) c= RightComp g ) by A625, A624, GOBOARD9:25, GOBRD13:33, TOPS_1:19;
then  Int (right_cell (g,1,(Gauge (C,n)))) c= RightComp g by XBOOLE_1:1;
then  Int (right_cell (f,((m -' 1) + 1),(Gauge (C,n)))) c= RightComp g by A571, A577, A624, GOBRD13:32;
then  ( B is connected & Int (right_cell (f,m,(Gauge (C,n)))) c= RightComp g ) by A570, SPRECT_3:7, XREAL_1:235;
then  B c= RightComp g by A972, A986, A989, GOBOARD9:4;
hence  a in RightComp g by A988; ::_thesis: verum
end;
hence  C meets RightComp g by XBOOLE_0:3; ::_thesis: verum
end;
hence  contradiction by A1, A972, A973, A971, Th1, SPRECT_4:6; ::_thesis: verum
end;
then g = f /^ 0 by XREAL_1:232
.= f by FINSEQ_5:28 ;
then reconsider f = f as non constant standard special_circular_sequence ;
 f is clockwise_oriented
proof
 f /. 2 in LSeg ((f /. 1),(f /. (1 + 1))) by RLTOPSP1:68;
then A990: f /. 2 in L~ f by A574, SPPOL_2:15;
 (NW-corner (L~ f)) `1 = W-bound (L~ f) by EUCLID:52;
then A991: (NW-corner (L~ f)) `1 <= (f /. 2) `1 by A990, PSCOMP_1:24;
 len (Gauge (C,n)) >= 3 by A231, NAT_1:12;
then A992: 1 < len (Gauge (C,n)) by XXREAL_0:2;
 (NE-corner (L~ f)) `1 = E-bound (L~ f) by EUCLID:52;
then A993: (f /. 2) `1 <= (NE-corner (L~ f)) `1 by A990, PSCOMP_1:24;
 for k being Element of NAT st 1 <= k & k + 1 <= len f holds
( left_cell (f,k,(Gauge (C,n))) misses C & right_cell (f,k,(Gauge (C,n))) meets C ) by A428;
then A994: N-min (L~ f) = f /. 1 by A571, A608, Th30;
consider i being Element of NAT  such that
A995: 1 <= i and
A996: i + 1 <= len (Gauge (C,n)) and
A997: ( f /. 1 = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) & f /. 2 = (Gauge (C,n)) * ((i + 1),(width (Gauge (C,n)))) ) and
 N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) and
 N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1)) by A608;
 i < len (Gauge (C,n)) by A996, NAT_1:13;
then A998: ( (N-min (L~ f)) `2 = N-bound (L~ f) & ((Gauge (C,n)) * (i,(width (Gauge (C,n))))) `2 = ((Gauge (C,n)) * (1,(width (Gauge (C,n))))) `2 ) by A2, A992, A995, EUCLID:52, GOBOARD5:1;
 1 <= i + 1 by NAT_1:12;
then A999: ((Gauge (C,n)) * ((i + 1),(width (Gauge (C,n))))) `2 = ((Gauge (C,n)) * (1,(width (Gauge (C,n))))) `2 by A2, A992, A996, GOBOARD5:1;
 ( (NW-corner (L~ f)) `2 = (NE-corner (L~ f)) `2 & (NE-corner (L~ f)) `2 = N-bound (L~ f) ) by EUCLID:52, PSCOMP_1:27;
then  f /. 2 in LSeg ((NW-corner (L~ f)),(NE-corner (L~ f))) by A994, A997, A998, A999, A991, A993, GOBOARD7:8;
then  f /. 2 in (LSeg ((NW-corner (L~ f)),(NE-corner (L~ f)))) /\ (L~ f) by A990, XBOOLE_0:def_4;
hence  f is clockwise_oriented by A994, SPRECT_2:30; ::_thesis: verum
end;
then reconsider f = f as non constant standard clockwise_oriented special_circular_sequence ;
take  f ; ::_thesis: ( f is_sequence_on Gauge (C,n) & ex i being Element of NAT st
( 1 <= i & i + 1 <= len (Gauge (C,n)) & f /. 1 = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) & f /. 2 = (Gauge (C,n)) * ((i + 1),(width (Gauge (C,n)))) & N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) & N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1)) ) & ( for k being Element of NAT st 1 <= k & k + 2 <= len f holds
( ( front_left_cell (f,k,(Gauge (C,n))) misses C & front_right_cell (f,k,(Gauge (C,n))) misses C implies f turns_right k, Gauge (C,n) ) & ( front_left_cell (f,k,(Gauge (C,n))) misses C & front_right_cell (f,k,(Gauge (C,n))) meets C implies f goes_straight k, Gauge (C,n) ) & ( front_left_cell (f,k,(Gauge (C,n))) meets C implies f turns_left k, Gauge (C,n) ) ) ) )
thus  f is_sequence_on Gauge (C,n) by A428; ::_thesis: ( ex i being Element of NAT st
( 1 <= i & i + 1 <= len (Gauge (C,n)) & f /. 1 = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) & f /. 2 = (Gauge (C,n)) * ((i + 1),(width (Gauge (C,n)))) & N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) & N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1)) ) & ( for k being Element of NAT st 1 <= k & k + 2 <= len f holds
( ( front_left_cell (f,k,(Gauge (C,n))) misses C & front_right_cell (f,k,(Gauge (C,n))) misses C implies f turns_right k, Gauge (C,n) ) & ( front_left_cell (f,k,(Gauge (C,n))) misses C & front_right_cell (f,k,(Gauge (C,n))) meets C implies f goes_straight k, Gauge (C,n) ) & ( front_left_cell (f,k,(Gauge (C,n))) meets C implies f turns_left k, Gauge (C,n) ) ) ) )
thus  ex i being Element of NAT st
( 1 <= i & i + 1 <= len (Gauge (C,n)) & f /. 1 = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) & f /. 2 = (Gauge (C,n)) * ((i + 1),(width (Gauge (C,n)))) & N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) & N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1)) ) by A608; ::_thesis: for k being Element of NAT st 1 <= k & k + 2 <= len f holds
( ( front_left_cell (f,k,(Gauge (C,n))) misses C & front_right_cell (f,k,(Gauge (C,n))) misses C implies f turns_right k, Gauge (C,n) ) & ( front_left_cell (f,k,(Gauge (C,n))) misses C & front_right_cell (f,k,(Gauge (C,n))) meets C implies f goes_straight k, Gauge (C,n) ) & ( front_left_cell (f,k,(Gauge (C,n))) meets C implies f turns_left k, Gauge (C,n) ) )
let m be Element of NAT ; ::_thesis: ( 1 <= m & m + 2 <= len f implies ( ( front_left_cell (f,m,(Gauge (C,n))) misses C & front_right_cell (f,m,(Gauge (C,n))) misses C implies f turns_right m, Gauge (C,n) ) & ( front_left_cell (f,m,(Gauge (C,n))) misses C & front_right_cell (f,m,(Gauge (C,n))) meets C implies f goes_straight m, Gauge (C,n) ) & ( front_left_cell (f,m,(Gauge (C,n))) meets C implies f turns_left m, Gauge (C,n) ) ) )
assume that
A1000: 1 <= m and
A1001: m + 2 <= len f ; ::_thesis: ( ( front_left_cell (f,m,(Gauge (C,n))) misses C & front_right_cell (f,m,(Gauge (C,n))) misses C implies f turns_right m, Gauge (C,n) ) & ( front_left_cell (f,m,(Gauge (C,n))) misses C & front_right_cell (f,m,(Gauge (C,n))) meets C implies f goes_straight m, Gauge (C,n) ) & ( front_left_cell (f,m,(Gauge (C,n))) meets C implies f turns_left m, Gauge (C,n) ) )
A1002: F . ((m + 1) + 1) = f | ((m + 1) + 1) by A490, A580, A1001;
A1003: m + 1 < m + 2 by XREAL_1:6;
then A1004: f | (m + 1) = F . (m + 1) by A490, A580, A1001, XXREAL_0:2;
A1005: m + 1 <= len f by A1001, A1003, XXREAL_0:2;
then A1006: front_left_cell ((F . (m + 1)),m,(Gauge (C,n))) = front_left_cell (f,m,(Gauge (C,n))) by A571, A1000, A1004, GOBRD13:42;
A1007: ( m = (m + 1) -' 1 & m + 1 > 1 ) by A1000, NAT_1:13, NAT_D:34;
A1008: front_right_cell ((F . (m + 1)),m,(Gauge (C,n))) = front_right_cell (f,m,(Gauge (C,n))) by A571, A1000, A1005, A1004, GOBRD13:42;
hereby  ::_thesis: ( ( front_left_cell (f,m,(Gauge (C,n))) misses C & front_right_cell (f,m,(Gauge (C,n))) meets C implies f goes_straight m, Gauge (C,n) ) & ( front_left_cell (f,m,(Gauge (C,n))) meets C implies f turns_left m, Gauge (C,n) ) )
assume  ( front_left_cell (f,m,(Gauge (C,n))) misses C & front_right_cell (f,m,(Gauge (C,n))) misses C ) ; ::_thesis: f turns_right m, Gauge (C,n)
then  F . ((m + 1) + 1) turns_right m, Gauge (C,n) by A458, A1007, A1006, A1008;
hence  f turns_right m, Gauge (C,n) by A1000, A1001, A1002, GOBRD13:43; ::_thesis: verum
end;
hereby  ::_thesis: ( front_left_cell (f,m,(Gauge (C,n))) meets C implies f turns_left m, Gauge (C,n) )
assume  ( front_left_cell (f,m,(Gauge (C,n))) misses C & front_right_cell (f,m,(Gauge (C,n))) meets C ) ; ::_thesis: f goes_straight m, Gauge (C,n)
then  F . ((m + 1) + 1) goes_straight m, Gauge (C,n) by A458, A1007, A1006, A1008;
hence  f goes_straight m, Gauge (C,n) by A1000, A1001, A1002, GOBRD13:45; ::_thesis: verum
end;
assume  front_left_cell (f,m,(Gauge (C,n))) meets C ; ::_thesis: f turns_left m, Gauge (C,n)
then  F . ((m + 1) + 1) turns_left m, Gauge (C,n) by A458, A1007, A1006;
hence  f turns_left m, Gauge (C,n) by A1000, A1001, A1002, GOBRD13:44; ::_thesis: verum
end;
uniqueness
 for b1, b2 being non constant standard clockwise_oriented special_circular_sequence st b1 is_sequence_on Gauge (C,n) & ex i being Element of NAT st
( 1 <= i & i + 1 <= len (Gauge (C,n)) & b1 /. 1 = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) & b1 /. 2 = (Gauge (C,n)) * ((i + 1),(width (Gauge (C,n)))) & N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) & N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1)) ) & ( for k being Element of NAT st 1 <= k & k + 2 <= len b1 holds
( ( front_left_cell (b1,k,(Gauge (C,n))) misses C & front_right_cell (b1,k,(Gauge (C,n))) misses C implies b1 turns_right k, Gauge (C,n) ) & ( front_left_cell (b1,k,(Gauge (C,n))) misses C & front_right_cell (b1,k,(Gauge (C,n))) meets C implies b1 goes_straight k, Gauge (C,n) ) & ( front_left_cell (b1,k,(Gauge (C,n))) meets C implies b1 turns_left k, Gauge (C,n) ) ) ) & b2 is_sequence_on Gauge (C,n) & ex i being Element of NAT st
( 1 <= i & i + 1 <= len (Gauge (C,n)) & b2 /. 1 = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) & b2 /. 2 = (Gauge (C,n)) * ((i + 1),(width (Gauge (C,n)))) & N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) & N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1)) ) & ( for k being Element of NAT st 1 <= k & k + 2 <= len b2 holds
( ( front_left_cell (b2,k,(Gauge (C,n))) misses C & front_right_cell (b2,k,(Gauge (C,n))) misses C implies b2 turns_right k, Gauge (C,n) ) & ( front_left_cell (b2,k,(Gauge (C,n))) misses C & front_right_cell (b2,k,(Gauge (C,n))) meets C implies b2 goes_straight k, Gauge (C,n) ) & ( front_left_cell (b2,k,(Gauge (C,n))) meets C implies b2 turns_left k, Gauge (C,n) ) ) ) holds
b1 = b2
proof
let f1, f2 be non constant standard clockwise_oriented special_circular_sequence; ::_thesis: ( f1 is_sequence_on Gauge (C,n) & ex i being Element of NAT st
( 1 <= i & i + 1 <= len (Gauge (C,n)) & f1 /. 1 = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) & f1 /. 2 = (Gauge (C,n)) * ((i + 1),(width (Gauge (C,n)))) & N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) & N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1)) ) & ( for k being Element of NAT st 1 <= k & k + 2 <= len f1 holds
( ( front_left_cell (f1,k,(Gauge (C,n))) misses C & front_right_cell (f1,k,(Gauge (C,n))) misses C implies f1 turns_right k, Gauge (C,n) ) & ( front_left_cell (f1,k,(Gauge (C,n))) misses C & front_right_cell (f1,k,(Gauge (C,n))) meets C implies f1 goes_straight k, Gauge (C,n) ) & ( front_left_cell (f1,k,(Gauge (C,n))) meets C implies f1 turns_left k, Gauge (C,n) ) ) ) & f2 is_sequence_on Gauge (C,n) & ex i being Element of NAT st
( 1 <= i & i + 1 <= len (Gauge (C,n)) & f2 /. 1 = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) & f2 /. 2 = (Gauge (C,n)) * ((i + 1),(width (Gauge (C,n)))) & N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) & N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1)) ) & ( for k being Element of NAT st 1 <= k & k + 2 <= len f2 holds
( ( front_left_cell (f2,k,(Gauge (C,n))) misses C & front_right_cell (f2,k,(Gauge (C,n))) misses C implies f2 turns_right k, Gauge (C,n) ) & ( front_left_cell (f2,k,(Gauge (C,n))) misses C & front_right_cell (f2,k,(Gauge (C,n))) meets C implies f2 goes_straight k, Gauge (C,n) ) & ( front_left_cell (f2,k,(Gauge (C,n))) meets C implies f2 turns_left k, Gauge (C,n) ) ) ) implies f1 = f2 )
assume A1009: f1 is_sequence_on Gauge (C,n) ; ::_thesis: ( for i being Element of NAT holds
( not 1 <= i or not i + 1 <= len (Gauge (C,n)) or not f1 /. 1 = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) or not f1 /. 2 = (Gauge (C,n)) * ((i + 1),(width (Gauge (C,n)))) or not N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) or not N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1)) ) or ex k being Element of NAT st
( 1 <= k & k + 2 <= len f1 & not ( ( front_left_cell (f1,k,(Gauge (C,n))) misses C & front_right_cell (f1,k,(Gauge (C,n))) misses C implies f1 turns_right k, Gauge (C,n) ) & ( front_left_cell (f1,k,(Gauge (C,n))) misses C & front_right_cell (f1,k,(Gauge (C,n))) meets C implies f1 goes_straight k, Gauge (C,n) ) & ( front_left_cell (f1,k,(Gauge (C,n))) meets C implies f1 turns_left k, Gauge (C,n) ) ) ) or not f2 is_sequence_on Gauge (C,n) or for i being Element of NAT holds
( not 1 <= i or not i + 1 <= len (Gauge (C,n)) or not f2 /. 1 = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) or not f2 /. 2 = (Gauge (C,n)) * ((i + 1),(width (Gauge (C,n)))) or not N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) or not N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1)) ) or ex k being Element of NAT st
( 1 <= k & k + 2 <= len f2 & not ( ( front_left_cell (f2,k,(Gauge (C,n))) misses C & front_right_cell (f2,k,(Gauge (C,n))) misses C implies f2 turns_right k, Gauge (C,n) ) & ( front_left_cell (f2,k,(Gauge (C,n))) misses C & front_right_cell (f2,k,(Gauge (C,n))) meets C implies f2 goes_straight k, Gauge (C,n) ) & ( front_left_cell (f2,k,(Gauge (C,n))) meets C implies f2 turns_left k, Gauge (C,n) ) ) ) or f1 = f2 )
defpred S1[ Element of NAT ] means f1 | $1 = f2 | $1;
given i1 being Element of NAT  such that A1010: ( 1 <= i1 & i1 + 1 <= len (Gauge (C,n)) ) and
A1011: f1 /. 1 = (Gauge (C,n)) * (i1,(width (Gauge (C,n)))) and
A1012: f1 /. 2 = (Gauge (C,n)) * ((i1 + 1),(width (Gauge (C,n)))) and
A1013: ( N-min C in cell ((Gauge (C,n)),i1,((width (Gauge (C,n))) -' 1)) & N-min C <> (Gauge (C,n)) * (i1,((width (Gauge (C,n))) -' 1)) ) ; ::_thesis: ( ex k being Element of NAT st
( 1 <= k & k + 2 <= len f1 & not ( ( front_left_cell (f1,k,(Gauge (C,n))) misses C & front_right_cell (f1,k,(Gauge (C,n))) misses C implies f1 turns_right k, Gauge (C,n) ) & ( front_left_cell (f1,k,(Gauge (C,n))) misses C & front_right_cell (f1,k,(Gauge (C,n))) meets C implies f1 goes_straight k, Gauge (C,n) ) & ( front_left_cell (f1,k,(Gauge (C,n))) meets C implies f1 turns_left k, Gauge (C,n) ) ) ) or not f2 is_sequence_on Gauge (C,n) or for i being Element of NAT holds
( not 1 <= i or not i + 1 <= len (Gauge (C,n)) or not f2 /. 1 = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) or not f2 /. 2 = (Gauge (C,n)) * ((i + 1),(width (Gauge (C,n)))) or not N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) or not N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1)) ) or ex k being Element of NAT st
( 1 <= k & k + 2 <= len f2 & not ( ( front_left_cell (f2,k,(Gauge (C,n))) misses C & front_right_cell (f2,k,(Gauge (C,n))) misses C implies f2 turns_right k, Gauge (C,n) ) & ( front_left_cell (f2,k,(Gauge (C,n))) misses C & front_right_cell (f2,k,(Gauge (C,n))) meets C implies f2 goes_straight k, Gauge (C,n) ) & ( front_left_cell (f2,k,(Gauge (C,n))) meets C implies f2 turns_left k, Gauge (C,n) ) ) ) or f1 = f2 )
assume that
A1014: for k being Element of NAT st 1 <= k & k + 2 <= len f1 holds
( ( front_left_cell (f1,k,(Gauge (C,n))) misses C & front_right_cell (f1,k,(Gauge (C,n))) misses C implies f1 turns_right k, Gauge (C,n) ) & ( front_left_cell (f1,k,(Gauge (C,n))) misses C & front_right_cell (f1,k,(Gauge (C,n))) meets C implies f1 goes_straight k, Gauge (C,n) ) & ( front_left_cell (f1,k,(Gauge (C,n))) meets C implies f1 turns_left k, Gauge (C,n) ) ) and
A1015: f2 is_sequence_on Gauge (C,n) ; ::_thesis: ( for i being Element of NAT holds
( not 1 <= i or not i + 1 <= len (Gauge (C,n)) or not f2 /. 1 = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) or not f2 /. 2 = (Gauge (C,n)) * ((i + 1),(width (Gauge (C,n)))) or not N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) or not N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1)) ) or ex k being Element of NAT st
( 1 <= k & k + 2 <= len f2 & not ( ( front_left_cell (f2,k,(Gauge (C,n))) misses C & front_right_cell (f2,k,(Gauge (C,n))) misses C implies f2 turns_right k, Gauge (C,n) ) & ( front_left_cell (f2,k,(Gauge (C,n))) misses C & front_right_cell (f2,k,(Gauge (C,n))) meets C implies f2 goes_straight k, Gauge (C,n) ) & ( front_left_cell (f2,k,(Gauge (C,n))) meets C implies f2 turns_left k, Gauge (C,n) ) ) ) or f1 = f2 )
given i2 being Element of NAT  such that A1016: ( 1 <= i2 & i2 + 1 <= len (Gauge (C,n)) ) and
A1017: f2 /. 1 = (Gauge (C,n)) * (i2,(width (Gauge (C,n)))) and
A1018: f2 /. 2 = (Gauge (C,n)) * ((i2 + 1),(width (Gauge (C,n)))) and
A1019: ( N-min C in cell ((Gauge (C,n)),i2,((width (Gauge (C,n))) -' 1)) & N-min C <> (Gauge (C,n)) * (i2,((width (Gauge (C,n))) -' 1)) ) ; ::_thesis: ( ex k being Element of NAT st
( 1 <= k & k + 2 <= len f2 & not ( ( front_left_cell (f2,k,(Gauge (C,n))) misses C & front_right_cell (f2,k,(Gauge (C,n))) misses C implies f2 turns_right k, Gauge (C,n) ) & ( front_left_cell (f2,k,(Gauge (C,n))) misses C & front_right_cell (f2,k,(Gauge (C,n))) meets C implies f2 goes_straight k, Gauge (C,n) ) & ( front_left_cell (f2,k,(Gauge (C,n))) meets C implies f2 turns_left k, Gauge (C,n) ) ) ) or f1 = f2 )
assume A1020: for k being Element of NAT st 1 <= k & k + 2 <= len f2 holds
( ( front_left_cell (f2,k,(Gauge (C,n))) misses C & front_right_cell (f2,k,(Gauge (C,n))) misses C implies f2 turns_right k, Gauge (C,n) ) & ( front_left_cell (f2,k,(Gauge (C,n))) misses C & front_right_cell (f2,k,(Gauge (C,n))) meets C implies f2 goes_straight k, Gauge (C,n) ) & ( front_left_cell (f2,k,(Gauge (C,n))) meets C implies f2 turns_left k, Gauge (C,n) ) ) ; ::_thesis: f1 = f2
A1021: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
A1022: len f1 > 4 by GOBOARD7:34;
A1023: ( f1 | 1 = <*(f1 /. 1)*> & f2 | 1 = <*(f2 /. 1)*> ) by FINSEQ_5:20;
A1024: ( i1 = i2 & len f2 > 4 ) by A1010, A1013, A1016, A1019, Th29, GOBOARD7:34;
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A1025: f1 | k = f2 | k ; ::_thesis: S1[k + 1]
percases ( k = 0 or k = 1 or k > 1 ) by NAT_1:25;
suppose k = 0 ; ::_thesis: S1[k + 1]
hence  S1[k + 1] by A1010, A1011, A1013, A1016, A1017, A1019, A1023, Th29; ::_thesis: verum
end;
supposeA1026: k = 1 ; ::_thesis: S1[k + 1]
 f1 | 2 = <*(f1 /. 1),(f1 /. 2)*> by A1022, FINSEQ_5:81, XXREAL_0:2;
hence  S1[k + 1] by A1011, A1012, A1017, A1018, A1024, A1026, FINSEQ_5:81, XXREAL_0:2; ::_thesis: verum
end;
supposeA1027: k > 1 ; ::_thesis: S1[k + 1]
A1028: ( f1 /. 1 = f1 /. (len f1) & f2 /. 1 = f2 /. (len f2) ) by FINSEQ_6:def_1;
now__::_thesis:_S1[k_+_1]
percases ( len f1 > k or k >= len f1 ) ;
supposeA1029: len f1 > k ; ::_thesis: S1[k + 1]
set m = k -' 1;
A1030: 1 <= k -' 1 by A1027, NAT_D:49;
then A1031: (k -' 1) + 1 = k by NAT_D:43;
then A1032: front_left_cell (f1,(k -' 1),(Gauge (C,n))) = front_left_cell ((f1 | k),(k -' 1),(Gauge (C,n))) by A1009, A1029, A1030, GOBRD13:42;
A1033: (k -' 1) + (1 + 1) = k + 1 by A1031;
A1034: k + 1 <= len f1 by A1029, NAT_1:13;
A1035: now__::_thesis:_not_len_f2_<=_k
A1036: 1 < len f2 by GOBOARD7:34, XXREAL_0:2;
assume A1037: len f2 <= k ; ::_thesis: contradiction
then A1038: f2 = f2 | k by FINSEQ_1:58;
then  len f2 in dom (f2 | k) by FINSEQ_5:6;
then A1039: (f1 | k) /. (len f2) = f1 /. (len f2) by A1025, FINSEQ_4:70;
 ( 1 in dom (f2 | k) & len f2 <= len f1 ) by A1025, A1038, FINSEQ_5:6, FINSEQ_5:16;
hence  contradiction by A1025, A1028, A1029, A1037, A1038, A1039, A1036, FINSEQ_4:70, GOBOARD7:38; ::_thesis: verum
end;
then A1040: k + 1 <= len f2 by NAT_1:13;
A1041: front_right_cell (f2,(k -' 1),(Gauge (C,n))) = front_right_cell ((f2 | k),(k -' 1),(Gauge (C,n))) by A1015, A1030, A1031, A1035, GOBRD13:42;
A1042: front_left_cell (f2,(k -' 1),(Gauge (C,n))) = front_left_cell ((f2 | k),(k -' 1),(Gauge (C,n))) by A1015, A1030, A1031, A1035, GOBRD13:42;
A1043: front_right_cell (f1,(k -' 1),(Gauge (C,n))) = front_right_cell ((f1 | k),(k -' 1),(Gauge (C,n))) by A1009, A1029, A1030, A1031, GOBRD13:42;
now__::_thesis:_S1[k_+_1]
percases ( ( front_left_cell (f1,(k -' 1),(Gauge (C,n))) misses C & front_right_cell (f1,(k -' 1),(Gauge (C,n))) misses C ) or ( front_left_cell (f1,(k -' 1),(Gauge (C,n))) misses C & front_right_cell (f1,(k -' 1),(Gauge (C,n))) meets C ) or front_left_cell (f1,(k -' 1),(Gauge (C,n))) meets C ) ;
suppose ( front_left_cell (f1,(k -' 1),(Gauge (C,n))) misses C & front_right_cell (f1,(k -' 1),(Gauge (C,n))) misses C ) ; ::_thesis: S1[k + 1]
then  ( f1 turns_right k -' 1, Gauge (C,n) & f2 turns_right k -' 1, Gauge (C,n) ) by A1014, A1020, A1025, A1030, A1040, A1034, A1032, A1043, A1042, A1041, A1033;
hence  S1[k + 1] by A1015, A1025, A1027, A1040, A1034, GOBRD13:46; ::_thesis: verum
end;
suppose ( front_left_cell (f1,(k -' 1),(Gauge (C,n))) misses C & front_right_cell (f1,(k -' 1),(Gauge (C,n))) meets C ) ; ::_thesis: S1[k + 1]
then  ( f1 goes_straight k -' 1, Gauge (C,n) & f2 goes_straight k -' 1, Gauge (C,n) ) by A1014, A1020, A1025, A1030, A1040, A1034, A1032, A1043, A1042, A1041, A1033;
hence  S1[k + 1] by A1015, A1025, A1027, A1040, A1034, GOBRD13:48; ::_thesis: verum
end;
suppose front_left_cell (f1,(k -' 1),(Gauge (C,n))) meets C ; ::_thesis: S1[k + 1]
then  ( f1 turns_left k -' 1, Gauge (C,n) & f2 turns_left k -' 1, Gauge (C,n) ) by A1014, A1020, A1025, A1030, A1040, A1034, A1032, A1042, A1033;
hence  S1[k + 1] by A1015, A1025, A1027, A1040, A1034, GOBRD13:47; ::_thesis: verum
end;
end;
end;
hence  S1[k + 1] ; ::_thesis: verum
end;
supposeA1044: k >= len f1 ; ::_thesis: f1 | (k + 1) = f2 | (k + 1)
A1045: 1 < len f1 by GOBOARD7:34, XXREAL_0:2;
A1046: f1 = f1 | k by A1044, FINSEQ_1:58;
then  len f1 in dom (f1 | k) by FINSEQ_5:6;
then A1047: (f2 | k) /. (len f1) = f2 /. (len f1) by A1025, FINSEQ_4:70;
 ( 1 in dom (f1 | k) & len f1 <= len f2 ) by A1025, A1046, FINSEQ_5:6, FINSEQ_5:16;
then A1048: len f2 = len f1 by A1025, A1028, A1046, A1047, A1045, FINSEQ_4:70, GOBOARD7:38;
A1049: k + 1 > len f1 by A1044, NAT_1:13;
hence f1 | (k + 1) = f1 by FINSEQ_1:58
.= f2 by A1025, A1044, A1046, A1048, FINSEQ_1:58
.= f2 | (k + 1) by A1048, A1049, FINSEQ_1:58 ;
::_thesis: verum
end;
end;
end;
hence  S1[k + 1] ; ::_thesis: verum
end;
end;
end;
A1050: S1[ 0 ] ;
 for k being Element of NAT holds S1[k] from NAT_1:sch_1(A1050, A1021);
hence  f1 = f2 by Th2; ::_thesis: verum
end;
end;

:: deftheorem Def1 defines Cage JORDAN9:def_1_:_
 for C being non empty compact non horizontal non vertical Subset of (TOP-REAL 2)
for n being Nat st C is connected holds
for b3 being non constant standard clockwise_oriented special_circular_sequence holds
( b3 = Cage (C,n) iff ( b3 is_sequence_on Gauge (C,n) & ex i being Element of NAT st
( 1 <= i & i + 1 <= len (Gauge (C,n)) & b3 /. 1 = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) & b3 /. 2 = (Gauge (C,n)) * ((i + 1),(width (Gauge (C,n)))) & N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) & N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1)) ) & ( for k being Element of NAT st 1 <= k & k + 2 <= len b3 holds
( ( front_left_cell (b3,k,(Gauge (C,n))) misses C & front_right_cell (b3,k,(Gauge (C,n))) misses C implies b3 turns_right k, Gauge (C,n) ) & ( front_left_cell (b3,k,(Gauge (C,n))) misses C & front_right_cell (b3,k,(Gauge (C,n))) meets C implies b3 goes_straight k, Gauge (C,n) ) & ( front_left_cell (b3,k,(Gauge (C,n))) meets C implies b3 turns_left k, Gauge (C,n) ) ) ) ) );

theorem Th31: :: JORDAN9:31
for k, n being Element of NAT
for C being non empty compact non horizontal non vertical Subset of (TOP-REAL 2) st C is connected & 1 <= k & k + 1 <= len (Cage (C,n)) holds
( left_cell ((Cage (C,n)),k,(Gauge (C,n))) misses C & right_cell ((Cage (C,n)),k,(Gauge (C,n))) meets C )
proof
let k, n be Element of NAT ; ::_thesis: for C being non empty compact non horizontal non vertical Subset of (TOP-REAL 2) st C is connected & 1 <= k & k + 1 <= len (Cage (C,n)) holds
( left_cell ((Cage (C,n)),k,(Gauge (C,n))) misses C & right_cell ((Cage (C,n)),k,(Gauge (C,n))) meets C )
let C be non empty compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( C is connected & 1 <= k & k + 1 <= len (Cage (C,n)) implies ( left_cell ((Cage (C,n)),k,(Gauge (C,n))) misses C & right_cell ((Cage (C,n)),k,(Gauge (C,n))) meets C ) )
set G = Gauge (C,n);
set f = Cage (C,n);
set W = W-bound C;
set E = E-bound C;
set S = S-bound C;
set N = N-bound C;
defpred S1[ Element of NAT ] means  for m being Element of NAT st 1 <= m & m + 1 <= len ((Cage (C,n)) | $1) holds
( left_cell (((Cage (C,n)) | $1),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | $1),m,(Gauge (C,n))) meets C );
A1: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
assume A2: C is connected ; ::_thesis: ( not 1 <= k or not k + 1 <= len (Cage (C,n)) or ( left_cell ((Cage (C,n)),k,(Gauge (C,n))) misses C & right_cell ((Cage (C,n)),k,(Gauge (C,n))) meets C ) )
then A3: Cage (C,n) is_sequence_on Gauge (C,n) by Def1;
A4: len (Gauge (C,n)) = (2 |^ n) + 3 by JORDAN8:def_1;
A5: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A6: for m being Element of NAT st 1 <= m & m + 1 <= len ((Cage (C,n)) | k) holds
( left_cell (((Cage (C,n)) | k),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | k),m,(Gauge (C,n))) meets C ) ; ::_thesis: S1[k + 1]
percases ( k >= len (Cage (C,n)) or k < len (Cage (C,n)) ) ;
suppose k >= len (Cage (C,n)) ; ::_thesis: S1[k + 1]
then  ( (Cage (C,n)) | k = Cage (C,n) & (Cage (C,n)) | (k + 1) = Cage (C,n) ) by FINSEQ_1:58, NAT_1:12;
hence  S1[k + 1] by A6; ::_thesis: verum
end;
supposeA7: k < len (Cage (C,n)) ; ::_thesis: S1[k + 1]
then A8: len ((Cage (C,n)) | k) = k by FINSEQ_1:59;
A9: 1 <= len (Gauge (C,n)) by A4, NAT_1:12;
A10: (Cage (C,n)) | k is_sequence_on Gauge (C,n) by A3, GOBOARD1:22;
A11: (Cage (C,n)) | (k + 1) is_sequence_on Gauge (C,n) by A3, GOBOARD1:22;
consider i being Element of NAT  such that
A12: 1 <= i and
A13: i + 1 <= len (Gauge (C,n)) and
A14: ( (Cage (C,n)) /. 1 = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) & (Cage (C,n)) /. 2 = (Gauge (C,n)) * ((i + 1),(width (Gauge (C,n)))) ) and
A15: N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) and
 N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1)) by A2, Def1;
let m be Element of NAT ; ::_thesis: ( 1 <= m & m + 1 <= len ((Cage (C,n)) | (k + 1)) implies ( left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C ) )
assume that
A16: 1 <= m and
A17: m + 1 <= len ((Cage (C,n)) | (k + 1)) ; ::_thesis: ( left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C )
A18: k + 1 <= len (Cage (C,n)) by A7, NAT_1:13;
then A19: len ((Cage (C,n)) | (k + 1)) = k + 1 by FINSEQ_1:59;
A20: 2 |^ n >= n + 1 by NEWTON:85;
now__::_thesis:_(_left_cell_(((Cage_(C,n))_|_(k_+_1)),m,(Gauge_(C,n)))_misses_C_&_right_cell_(((Cage_(C,n))_|_(k_+_1)),m,(Gauge_(C,n)))_meets_C_)
percases ( k = 0 or k = 1 or k > 1 ) by NAT_1:25;
supposeA21: k = 0 ; ::_thesis: ( left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C )
 1 <= m + 1 by NAT_1:12;
then  m + 1 = 0 + 1 by A19, A17, A21, XXREAL_0:1;
hence  ( left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C ) by A16; ::_thesis: verum
end;
supposeA22: k = 1 ; ::_thesis: ( left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C )
 1 + 1 <= m + 1 by A16, XREAL_1:6;
then A23: m + 1 = 1 + 1 by A19, A17, A22, XXREAL_0:1;
 (Cage (C,n)) | (k + 1) = <*((Gauge (C,n)) * (i,(width (Gauge (C,n))))),((Gauge (C,n)) * ((i + 1),(width (Gauge (C,n)))))*> by A18, A14, A22, FINSEQ_5:81;
then A24: ( ((Cage (C,n)) | (k + 1)) /. 1 = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) & ((Cage (C,n)) | (k + 1)) /. 2 = (Gauge (C,n)) * ((i + 1),(width (Gauge (C,n)))) ) by FINSEQ_4:17;
 1 <= i + 1 by A12, NAT_1:13;
then A25: [(i + 1),(len (Gauge (C,n)))] in Indices (Gauge (C,n)) by A1, A13, A9, MATRIX_1:36;
A26: i < len (Gauge (C,n)) by A13, NAT_1:13;
then A27: [i,(len (Gauge (C,n)))] in Indices (Gauge (C,n)) by A1, A12, A9, MATRIX_1:36;
A28: ( i < i + 1 & i + 1 < (i + 1) + 1 ) by NAT_1:13;
then A29: left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) = cell ((Gauge (C,n)),i,(len (Gauge (C,n)))) by A1, A11, A17, A24, A27, A25, A23, GOBRD13:def_3;
now__::_thesis:_not_left_cell_(((Cage_(C,n))_|_(k_+_1)),m,(Gauge_(C,n)))_meets_C
 N-bound C > S-bound C by JORDAN8:9;
then  (N-bound C) - (S-bound C) > (S-bound C) - (S-bound C) by XREAL_1:9;
then  ((N-bound C) - (S-bound C)) / (2 |^ n) > 0 by A20, XREAL_1:139;
then A30: (N-bound C) + 0 < (N-bound C) + (((N-bound C) - (S-bound C)) / (2 |^ n)) by XREAL_1:6;
assume  left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C ; ::_thesis: contradiction
then consider p being set  such that
A31: p in cell ((Gauge (C,n)),i,(len (Gauge (C,n)))) and
A32: p in C by A29, XBOOLE_0:3;
reconsider p = p as Element of (TOP-REAL 2) by A31;
A33: p `2 <= N-bound C by A32, PSCOMP_1:24;
 [1,(len (Gauge (C,n)))] in Indices (Gauge (C,n)) by A1, A9, MATRIX_1:36;
then  (Gauge (C,n)) * (1,(len (Gauge (C,n)))) = |[((W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * (1 - 2))),((S-bound C) + ((((N-bound C) - (S-bound C)) / (2 |^ n)) * ((len (Gauge (C,n))) - 2)))]| by JORDAN8:def_1;
then A34: ((Gauge (C,n)) * (1,(len (Gauge (C,n))))) `2 = (S-bound C) + ((((N-bound C) - (S-bound C)) / (2 |^ n)) * ((len (Gauge (C,n))) - 2)) by EUCLID:52;
 cell ((Gauge (C,n)),i,(len (Gauge (C,n)))) =  {  |[r,s]| where r, s is Real : ( ((Gauge (C,n)) * (i,1)) `1 <= r & r <= ((Gauge (C,n)) * ((i + 1),1)) `1 & ((Gauge (C,n)) * (1,(len (Gauge (C,n))))) `2 <= s )  }  by A1, A12, A26, GOBRD11:31;
then consider r, s being Real such that
A35: p = |[r,s]| and
 ((Gauge (C,n)) * (i,1)) `1 <= r and
 r <= ((Gauge (C,n)) * ((i + 1),1)) `1 and
A36: ((Gauge (C,n)) * (1,(len (Gauge (C,n))))) `2 <= s by A31;
(((N-bound C) - (S-bound C)) / (2 |^ n)) * ((len (Gauge (C,n))) - 2) = ((((N-bound C) - (S-bound C)) / (2 |^ n)) * (2 |^ n)) + ((((N-bound C) - (S-bound C)) / (2 |^ n)) * 1) by A4
.= ((N-bound C) - (S-bound C)) + (((N-bound C) - (S-bound C)) / (2 |^ n)) by A20, XCMPLX_1:87 ;
then  N-bound C < s by A36, A34, A30, XXREAL_0:2;
hence  contradiction by A35, A33, EUCLID:52; ::_thesis: verum
end;
hence  left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C ; ::_thesis: right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C
 ( N-min C in C & N-min C in right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) ) by A1, A11, A15, A17, A24, A27, A25, A23, A28, GOBRD13:def_2, SPRECT_1:11;
hence  right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C by XBOOLE_0:3; ::_thesis: verum
end;
supposeA37: k > 1 ; ::_thesis: ( left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C )
then A38: ((len ((Cage (C,n)) | k)) -' 1) + 1 = len ((Cage (C,n)) | k) by A8, XREAL_1:235;
A39: 1 <= (len ((Cage (C,n)) | k)) -' 1 by A8, A37, NAT_D:49;
now__::_thesis:_(_left_cell_(((Cage_(C,n))_|_(k_+_1)),m,(Gauge_(C,n)))_misses_C_&_right_cell_(((Cage_(C,n))_|_(k_+_1)),m,(Gauge_(C,n)))_meets_C_)
percases ( m + 1 = len ((Cage (C,n)) | (k + 1)) or m + 1 <> len ((Cage (C,n)) | (k + 1)) ) ;
supposeA40: m + 1 = len ((Cage (C,n)) | (k + 1)) ; ::_thesis: ( left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C )
A41: len ((Cage (C,n)) | k) <= len (Cage (C,n)) by FINSEQ_5:16;
now__::_thesis:_(_left_cell_(((Cage_(C,n))_|_(k_+_1)),m,(Gauge_(C,n)))_misses_C_&_right_cell_(((Cage_(C,n))_|_(k_+_1)),m,(Gauge_(C,n)))_meets_C_)
 left_cell (((Cage (C,n)) | k),((len ((Cage (C,n)) | k)) -' 1),(Gauge (C,n))) misses C by A6, A39, A38;
then A42: left_cell ((Cage (C,n)),((len ((Cage (C,n)) | k)) -' 1),(Gauge (C,n))) misses C by A3, A8, A39, A38, A41, GOBRD13:31;
A43: ((len ((Cage (C,n)) | k)) -' 1) + (1 + 1) = (len ((Cage (C,n)) | k)) + 1 by A38;
 right_cell (((Cage (C,n)) | k),((len ((Cage (C,n)) | k)) -' 1),(Gauge (C,n))) meets C by A6, A39, A38;
then A44: right_cell ((Cage (C,n)),((len ((Cage (C,n)) | k)) -' 1),(Gauge (C,n))) meets C by A3, A8, A39, A38, A41, GOBRD13:31;
consider i1, j1, i2, j2 being Element of NAT  such that
A45: ( [i1,j1] in Indices (Gauge (C,n)) & (Cage (C,n)) /. ((len ((Cage (C,n)) | k)) -' 1) = (Gauge (C,n)) * (i1,j1) ) and
A46: [i2,j2] in Indices (Gauge (C,n)) and
A47: (Cage (C,n)) /. (len ((Cage (C,n)) | k)) = (Gauge (C,n)) * (i2,j2) and
 ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A3, A7, A8, A39, A38, JORDAN8:3;
 1 <= i2 by A46, MATRIX_1:38;
then A48: (i2 -' 1) + 1 = i2 by XREAL_1:235;
 1 <= j2 by A46, MATRIX_1:38;
then A49: (j2 -' 1) + 1 = j2 by XREAL_1:235;
percases ( ( front_left_cell ((Cage (C,n)),((len ((Cage (C,n)) | k)) -' 1),(Gauge (C,n))) misses C & front_right_cell ((Cage (C,n)),((len ((Cage (C,n)) | k)) -' 1),(Gauge (C,n))) misses C ) or ( front_left_cell ((Cage (C,n)),((len ((Cage (C,n)) | k)) -' 1),(Gauge (C,n))) misses C & front_right_cell ((Cage (C,n)),((len ((Cage (C,n)) | k)) -' 1),(Gauge (C,n))) meets C ) or front_left_cell ((Cage (C,n)),((len ((Cage (C,n)) | k)) -' 1),(Gauge (C,n))) meets C ) ;
supposeA50: ( front_left_cell ((Cage (C,n)),((len ((Cage (C,n)) | k)) -' 1),(Gauge (C,n))) misses C & front_right_cell ((Cage (C,n)),((len ((Cage (C,n)) | k)) -' 1),(Gauge (C,n))) misses C ) ; ::_thesis: ( left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C )
then A51: Cage (C,n) turns_right (len ((Cage (C,n)) | k)) -' 1, Gauge (C,n) by A2, A18, A8, A39, A43, Def1;
now__::_thesis:_(_left_cell_(((Cage_(C,n))_|_(k_+_1)),m,(Gauge_(C,n)))_misses_C_&_right_cell_(((Cage_(C,n))_|_(k_+_1)),m,(Gauge_(C,n)))_meets_C_)
percases ( ( i1 = i2 & j1 + 1 = j2 & [(i2 + 1),j2] in Indices (Gauge (C,n)) & (Cage (C,n)) /. ((len ((Cage (C,n)) | k)) + 1) = (Gauge (C,n)) * ((i2 + 1),j2) ) or ( i1 + 1 = i2 & j1 = j2 & [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & (Cage (C,n)) /. ((len ((Cage (C,n)) | k)) + 1) = (Gauge (C,n)) * (i2,(j2 -' 1)) ) or ( i1 = i2 + 1 & j1 = j2 & [i2,(j2 + 1)] in Indices (Gauge (C,n)) & (Cage (C,n)) /. ((len ((Cage (C,n)) | k)) + 1) = (Gauge (C,n)) * (i2,(j2 + 1)) ) or ( i1 = i2 & j1 = j2 + 1 & [(i2 -' 1),j2] in Indices (Gauge (C,n)) & (Cage (C,n)) /. ((len ((Cage (C,n)) | k)) + 1) = (Gauge (C,n)) * ((i2 -' 1),j2) ) ) by A38, A45, A46, A47, A43, A51, GOBRD13:def_6;
supposethat A52: ( i1 = i2 & j1 + 1 = j2 ) and
A53: ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & (Cage (C,n)) /. ((len ((Cage (C,n)) | k)) + 1) = (Gauge (C,n)) * ((i2 + 1),j2) ) ; ::_thesis: ( left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C )
 front_right_cell ((Cage (C,n)),((len ((Cage (C,n)) | k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),i1,j2) by A3, A39, A38, A41, A45, A46, A47, A52, GOBRD13:35;
then  left_cell ((Cage (C,n)),m,(Gauge (C,n))) misses C by A3, A18, A8, A19, A16, A40, A46, A47, A50, A52, A53, GOBRD13:23;
hence  left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C by A3, A18, A19, A16, A40, GOBRD13:31; ::_thesis: right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C
 ( j2 -' 1 = j1 & cell ((Gauge (C,n)),i1,j1) meets C ) by A3, A39, A38, A41, A45, A46, A47, A44, A52, GOBRD13:22, NAT_D:34;
then  right_cell ((Cage (C,n)),m,(Gauge (C,n))) meets C by A3, A18, A8, A19, A16, A40, A46, A47, A52, A53, GOBRD13:24;
hence  right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C by A3, A18, A19, A16, A40, GOBRD13:31; ::_thesis: verum
end;
supposethat A54: ( i1 + 1 = i2 & j1 = j2 ) and
A55: ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & (Cage (C,n)) /. ((len ((Cage (C,n)) | k)) + 1) = (Gauge (C,n)) * (i2,(j2 -' 1)) ) ; ::_thesis: ( left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C )
 front_right_cell ((Cage (C,n)),((len ((Cage (C,n)) | k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),i2,(j2 -' 1)) by A3, A39, A38, A41, A45, A46, A47, A54, GOBRD13:37;
then  left_cell ((Cage (C,n)),m,(Gauge (C,n))) misses C by A3, A18, A8, A19, A16, A40, A46, A47, A49, A50, A55, GOBRD13:27;
hence  left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C by A3, A18, A19, A16, A40, GOBRD13:31; ::_thesis: right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C
 ( i2 -' 1 = i1 & cell ((Gauge (C,n)),i1,(j1 -' 1)) meets C ) by A3, A39, A38, A41, A45, A46, A47, A44, A54, GOBRD13:24, NAT_D:34;
then  right_cell ((Cage (C,n)),m,(Gauge (C,n))) meets C by A3, A18, A8, A19, A16, A40, A46, A47, A49, A54, A55, GOBRD13:28;
hence  right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C by A3, A18, A19, A16, A40, GOBRD13:31; ::_thesis: verum
end;
supposethat A56: ( i1 = i2 + 1 & j1 = j2 ) and
A57: ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & (Cage (C,n)) /. ((len ((Cage (C,n)) | k)) + 1) = (Gauge (C,n)) * (i2,(j2 + 1)) ) ; ::_thesis: ( left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C )
 front_right_cell ((Cage (C,n)),((len ((Cage (C,n)) | k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),(i2 -' 1),j2) by A3, A39, A38, A41, A45, A46, A47, A56, GOBRD13:39;
then  left_cell ((Cage (C,n)),m,(Gauge (C,n))) misses C by A3, A18, A8, A19, A16, A40, A46, A47, A50, A57, GOBRD13:21;
hence  left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C by A3, A18, A19, A16, A40, GOBRD13:31; ::_thesis: right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C
 cell ((Gauge (C,n)),i2,j2) meets C by A3, A39, A38, A41, A45, A46, A47, A44, A56, GOBRD13:26;
then  right_cell ((Cage (C,n)),m,(Gauge (C,n))) meets C by A3, A18, A8, A19, A16, A40, A46, A47, A57, GOBRD13:22;
hence  right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C by A3, A18, A19, A16, A40, GOBRD13:31; ::_thesis: verum
end;
supposethat A58: ( i1 = i2 & j1 = j2 + 1 ) and
A59: ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & (Cage (C,n)) /. ((len ((Cage (C,n)) | k)) + 1) = (Gauge (C,n)) * ((i2 -' 1),j2) ) ; ::_thesis: ( left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C )
 front_right_cell ((Cage (C,n)),((len ((Cage (C,n)) | k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),(i2 -' 1),(j2 -' 1)) by A3, A39, A38, A41, A45, A46, A47, A58, GOBRD13:41;
then  left_cell ((Cage (C,n)),m,(Gauge (C,n))) misses C by A3, A18, A8, A19, A16, A40, A46, A47, A48, A50, A59, GOBRD13:25;
hence  left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C by A3, A18, A19, A16, A40, GOBRD13:31; ::_thesis: right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C
 cell ((Gauge (C,n)),(i2 -' 1),j2) meets C by A3, A39, A38, A41, A45, A46, A47, A44, A58, GOBRD13:28;
then  right_cell ((Cage (C,n)),m,(Gauge (C,n))) meets C by A3, A18, A8, A19, A16, A40, A46, A47, A48, A59, GOBRD13:26;
hence  right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C by A3, A18, A19, A16, A40, GOBRD13:31; ::_thesis: verum
end;
end;
end;
hence  ( left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C ) ; ::_thesis: verum
end;
supposeA60: ( front_left_cell ((Cage (C,n)),((len ((Cage (C,n)) | k)) -' 1),(Gauge (C,n))) misses C & front_right_cell ((Cage (C,n)),((len ((Cage (C,n)) | k)) -' 1),(Gauge (C,n))) meets C ) ; ::_thesis: ( left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C )
then A61: Cage (C,n) goes_straight (len ((Cage (C,n)) | k)) -' 1, Gauge (C,n) by A2, A18, A8, A39, A43, Def1;
now__::_thesis:_(_left_cell_(((Cage_(C,n))_|_(k_+_1)),m,(Gauge_(C,n)))_misses_C_&_right_cell_(((Cage_(C,n))_|_(k_+_1)),m,(Gauge_(C,n)))_meets_C_)
percases ( ( i1 = i2 & j1 + 1 = j2 & [i2,(j2 + 1)] in Indices (Gauge (C,n)) & (Cage (C,n)) /. ((len ((Cage (C,n)) | k)) + 1) = (Gauge (C,n)) * (i2,(j2 + 1)) ) or ( i1 + 1 = i2 & j1 = j2 & [(i2 + 1),j2] in Indices (Gauge (C,n)) & (Cage (C,n)) /. ((len ((Cage (C,n)) | k)) + 1) = (Gauge (C,n)) * ((i2 + 1),j2) ) or ( i1 = i2 + 1 & j1 = j2 & [(i2 -' 1),j2] in Indices (Gauge (C,n)) & (Cage (C,n)) /. ((len ((Cage (C,n)) | k)) + 1) = (Gauge (C,n)) * ((i2 -' 1),j2) ) or ( i1 = i2 & j1 = j2 + 1 & [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & (Cage (C,n)) /. ((len ((Cage (C,n)) | k)) + 1) = (Gauge (C,n)) * (i2,(j2 -' 1)) ) ) by A38, A45, A46, A47, A43, A61, GOBRD13:def_8;
supposethat A62: ( i1 = i2 & j1 + 1 = j2 ) and
A63: ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & (Cage (C,n)) /. ((len ((Cage (C,n)) | k)) + 1) = (Gauge (C,n)) * (i2,(j2 + 1)) ) ; ::_thesis: ( left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C )
 front_left_cell ((Cage (C,n)),((len ((Cage (C,n)) | k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),(i1 -' 1),j2) by A3, A39, A38, A41, A45, A46, A47, A62, GOBRD13:34;
then  left_cell ((Cage (C,n)),m,(Gauge (C,n))) misses C by A3, A18, A8, A19, A16, A40, A46, A47, A60, A62, A63, GOBRD13:21;
hence  left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C by A3, A18, A19, A16, A40, GOBRD13:31; ::_thesis: right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C
 front_right_cell ((Cage (C,n)),((len ((Cage (C,n)) | k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),i1,j2) by A3, A39, A38, A41, A45, A46, A47, A62, GOBRD13:35;
then  right_cell ((Cage (C,n)),m,(Gauge (C,n))) meets C by A3, A18, A8, A19, A16, A40, A46, A47, A60, A62, A63, GOBRD13:22;
hence  right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C by A3, A18, A19, A16, A40, GOBRD13:31; ::_thesis: verum
end;
supposethat A64: ( i1 + 1 = i2 & j1 = j2 ) and
A65: ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & (Cage (C,n)) /. ((len ((Cage (C,n)) | k)) + 1) = (Gauge (C,n)) * ((i2 + 1),j2) ) ; ::_thesis: ( left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C )
 front_left_cell ((Cage (C,n)),((len ((Cage (C,n)) | k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),i2,j2) by A3, A39, A38, A41, A45, A46, A47, A64, GOBRD13:36;
then  left_cell ((Cage (C,n)),m,(Gauge (C,n))) misses C by A3, A18, A8, A19, A16, A40, A46, A47, A60, A65, GOBRD13:23;
hence  left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C by A3, A18, A19, A16, A40, GOBRD13:31; ::_thesis: right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C
 front_right_cell ((Cage (C,n)),((len ((Cage (C,n)) | k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),i2,(j2 -' 1)) by A3, A39, A38, A41, A45, A46, A47, A64, GOBRD13:37;
then  right_cell ((Cage (C,n)),m,(Gauge (C,n))) meets C by A3, A18, A8, A19, A16, A40, A46, A47, A60, A65, GOBRD13:24;
hence  right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C by A3, A18, A19, A16, A40, GOBRD13:31; ::_thesis: verum
end;
supposethat A66: ( i1 = i2 + 1 & j1 = j2 ) and
A67: ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & (Cage (C,n)) /. ((len ((Cage (C,n)) | k)) + 1) = (Gauge (C,n)) * ((i2 -' 1),j2) ) ; ::_thesis: ( left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C )
 front_left_cell ((Cage (C,n)),((len ((Cage (C,n)) | k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),(i2 -' 1),(j2 -' 1)) by A3, A39, A38, A41, A45, A46, A47, A66, GOBRD13:38;
then  left_cell ((Cage (C,n)),m,(Gauge (C,n))) misses C by A3, A18, A8, A19, A16, A40, A46, A47, A48, A60, A67, GOBRD13:25;
hence  left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C by A3, A18, A19, A16, A40, GOBRD13:31; ::_thesis: right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C
 front_right_cell ((Cage (C,n)),((len ((Cage (C,n)) | k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),(i2 -' 1),j2) by A3, A39, A38, A41, A45, A46, A47, A66, GOBRD13:39;
then  right_cell ((Cage (C,n)),m,(Gauge (C,n))) meets C by A3, A18, A8, A19, A16, A40, A46, A47, A48, A60, A67, GOBRD13:26;
hence  right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C by A3, A18, A19, A16, A40, GOBRD13:31; ::_thesis: verum
end;
supposethat A68: ( i1 = i2 & j1 = j2 + 1 ) and
A69: ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & (Cage (C,n)) /. ((len ((Cage (C,n)) | k)) + 1) = (Gauge (C,n)) * (i2,(j2 -' 1)) ) ; ::_thesis: ( left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C )
 front_left_cell ((Cage (C,n)),((len ((Cage (C,n)) | k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),i2,(j2 -' 1)) by A3, A39, A38, A41, A45, A46, A47, A68, GOBRD13:40;
then  left_cell ((Cage (C,n)),m,(Gauge (C,n))) misses C by A3, A18, A8, A19, A16, A40, A46, A47, A49, A60, A69, GOBRD13:27;
hence  left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C by A3, A18, A19, A16, A40, GOBRD13:31; ::_thesis: right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C
 front_right_cell ((Cage (C,n)),((len ((Cage (C,n)) | k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),(i2 -' 1),(j2 -' 1)) by A3, A39, A38, A41, A45, A46, A47, A68, GOBRD13:41;
then  right_cell ((Cage (C,n)),m,(Gauge (C,n))) meets C by A3, A18, A8, A19, A16, A40, A46, A47, A49, A60, A69, GOBRD13:28;
hence  right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C by A3, A18, A19, A16, A40, GOBRD13:31; ::_thesis: verum
end;
end;
end;
hence  ( left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C ) ; ::_thesis: verum
end;
supposeA70: front_left_cell ((Cage (C,n)),((len ((Cage (C,n)) | k)) -' 1),(Gauge (C,n))) meets C ; ::_thesis: ( left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C )
then A71: Cage (C,n) turns_left (len ((Cage (C,n)) | k)) -' 1, Gauge (C,n) by A2, A18, A8, A39, A43, Def1;
now__::_thesis:_(_left_cell_(((Cage_(C,n))_|_(k_+_1)),m,(Gauge_(C,n)))_misses_C_&_right_cell_(((Cage_(C,n))_|_(k_+_1)),m,(Gauge_(C,n)))_meets_C_)
percases ( ( i1 = i2 & j1 + 1 = j2 & [(i2 -' 1),j2] in Indices (Gauge (C,n)) & (Cage (C,n)) /. ((len ((Cage (C,n)) | k)) + 1) = (Gauge (C,n)) * ((i2 -' 1),j2) ) or ( i1 + 1 = i2 & j1 = j2 & [i2,(j2 + 1)] in Indices (Gauge (C,n)) & (Cage (C,n)) /. ((len ((Cage (C,n)) | k)) + 1) = (Gauge (C,n)) * (i2,(j2 + 1)) ) or ( i1 = i2 + 1 & j1 = j2 & [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & (Cage (C,n)) /. ((len ((Cage (C,n)) | k)) + 1) = (Gauge (C,n)) * (i2,(j2 -' 1)) ) or ( i1 = i2 & j1 = j2 + 1 & [(i2 + 1),j2] in Indices (Gauge (C,n)) & (Cage (C,n)) /. ((len ((Cage (C,n)) | k)) + 1) = (Gauge (C,n)) * ((i2 + 1),j2) ) ) by A38, A45, A46, A47, A43, A71, GOBRD13:def_7;
supposethat A72: ( i1 = i2 & j1 + 1 = j2 ) and
A73: ( [(i2 -' 1),j2] in Indices (Gauge (C,n)) & (Cage (C,n)) /. ((len ((Cage (C,n)) | k)) + 1) = (Gauge (C,n)) * ((i2 -' 1),j2) ) ; ::_thesis: ( left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C )
 ( j2 -' 1 = j1 & cell ((Gauge (C,n)),(i1 -' 1),j1) misses C ) by A3, A39, A38, A41, A45, A46, A47, A42, A72, GOBRD13:21, NAT_D:34;
then  left_cell ((Cage (C,n)),m,(Gauge (C,n))) misses C by A3, A18, A8, A19, A16, A40, A46, A47, A48, A72, A73, GOBRD13:25;
hence  left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C by A3, A18, A19, A16, A40, GOBRD13:31; ::_thesis: right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C
 front_left_cell ((Cage (C,n)),((len ((Cage (C,n)) | k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),(i1 -' 1),j2) by A3, A39, A38, A41, A45, A46, A47, A72, GOBRD13:34;
then  right_cell ((Cage (C,n)),m,(Gauge (C,n))) meets C by A3, A18, A8, A19, A16, A40, A46, A47, A48, A70, A72, A73, GOBRD13:26;
hence  right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C by A3, A18, A19, A16, A40, GOBRD13:31; ::_thesis: verum
end;
supposethat A74: ( i1 + 1 = i2 & j1 = j2 ) and
A75: ( [i2,(j2 + 1)] in Indices (Gauge (C,n)) & (Cage (C,n)) /. ((len ((Cage (C,n)) | k)) + 1) = (Gauge (C,n)) * (i2,(j2 + 1)) ) ; ::_thesis: ( left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C )
 ( i2 -' 1 = i1 & cell ((Gauge (C,n)),i1,j1) misses C ) by A3, A39, A38, A41, A45, A46, A47, A42, A74, GOBRD13:23, NAT_D:34;
then  left_cell ((Cage (C,n)),m,(Gauge (C,n))) misses C by A3, A18, A8, A19, A16, A40, A46, A47, A74, A75, GOBRD13:21;
hence  left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C by A3, A18, A19, A16, A40, GOBRD13:31; ::_thesis: right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C
 front_left_cell ((Cage (C,n)),((len ((Cage (C,n)) | k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),i2,j2) by A3, A39, A38, A41, A45, A46, A47, A74, GOBRD13:36;
then  right_cell ((Cage (C,n)),m,(Gauge (C,n))) meets C by A3, A18, A8, A19, A16, A40, A46, A47, A70, A75, GOBRD13:22;
hence  right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C by A3, A18, A19, A16, A40, GOBRD13:31; ::_thesis: verum
end;
supposethat A76: ( i1 = i2 + 1 & j1 = j2 ) and
A77: ( [i2,(j2 -' 1)] in Indices (Gauge (C,n)) & (Cage (C,n)) /. ((len ((Cage (C,n)) | k)) + 1) = (Gauge (C,n)) * (i2,(j2 -' 1)) ) ; ::_thesis: ( left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C )
 cell ((Gauge (C,n)),i2,(j2 -' 1)) misses C by A3, A39, A38, A41, A45, A46, A47, A42, A76, GOBRD13:25;
then  left_cell ((Cage (C,n)),m,(Gauge (C,n))) misses C by A3, A18, A8, A19, A16, A40, A46, A47, A49, A77, GOBRD13:27;
hence  left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C by A3, A18, A19, A16, A40, GOBRD13:31; ::_thesis: right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C
 front_left_cell ((Cage (C,n)),((len ((Cage (C,n)) | k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),(i2 -' 1),(j2 -' 1)) by A3, A39, A38, A41, A45, A46, A47, A76, GOBRD13:38;
then  right_cell ((Cage (C,n)),m,(Gauge (C,n))) meets C by A3, A18, A8, A19, A16, A40, A46, A47, A49, A70, A77, GOBRD13:28;
hence  right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C by A3, A18, A19, A16, A40, GOBRD13:31; ::_thesis: verum
end;
supposethat A78: ( i1 = i2 & j1 = j2 + 1 ) and
A79: ( [(i2 + 1),j2] in Indices (Gauge (C,n)) & (Cage (C,n)) /. ((len ((Cage (C,n)) | k)) + 1) = (Gauge (C,n)) * ((i2 + 1),j2) ) ; ::_thesis: ( left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C )
 cell ((Gauge (C,n)),i2,j2) misses C by A3, A39, A38, A41, A45, A46, A47, A42, A78, GOBRD13:27;
then  left_cell ((Cage (C,n)),m,(Gauge (C,n))) misses C by A3, A18, A8, A19, A16, A40, A46, A47, A79, GOBRD13:23;
hence  left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C by A3, A18, A19, A16, A40, GOBRD13:31; ::_thesis: right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C
 front_left_cell ((Cage (C,n)),((len ((Cage (C,n)) | k)) -' 1),(Gauge (C,n))) = cell ((Gauge (C,n)),i2,(j2 -' 1)) by A3, A39, A38, A41, A45, A46, A47, A78, GOBRD13:40;
then  right_cell ((Cage (C,n)),m,(Gauge (C,n))) meets C by A3, A18, A8, A19, A16, A40, A46, A47, A70, A79, GOBRD13:24;
hence  right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C by A3, A18, A19, A16, A40, GOBRD13:31; ::_thesis: verum
end;
end;
end;
hence  ( left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C ) ; ::_thesis: verum
end;
end;
end;
hence  ( left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C ) ; ::_thesis: verum
end;
suppose m + 1 <> len ((Cage (C,n)) | (k + 1)) ; ::_thesis: ( left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C )
then  m + 1 < len ((Cage (C,n)) | (k + 1)) by A17, XXREAL_0:1;
then A80: m + 1 <= len ((Cage (C,n)) | k) by A8, A19, NAT_1:13;
then consider i1, j1, i2, j2 being Element of NAT  such that
A81: [i1,j1] in Indices (Gauge (C,n)) and
A82: ((Cage (C,n)) | k) /. m = (Gauge (C,n)) * (i1,j1) and
A83: [i2,j2] in Indices (Gauge (C,n)) and
A84: ((Cage (C,n)) | k) /. (m + 1) = (Gauge (C,n)) * (i2,j2) and
A85: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A10, A16, JORDAN8:3;
A86: ( left_cell (((Cage (C,n)) | k),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | k),m,(Gauge (C,n))) meets C ) by A6, A16, A80;
A87: (Cage (C,n)) | (k + 1) = ((Cage (C,n)) | k) ^ <*((Cage (C,n)) /. (k + 1))*> by A18, FINSEQ_5:82;
 1 <= m + 1 by NAT_1:12;
then  m + 1 in dom ((Cage (C,n)) | k) by A80, FINSEQ_3:25;
then A88: ((Cage (C,n)) | (k + 1)) /. (m + 1) = (Gauge (C,n)) * (i2,j2) by A84, A87, FINSEQ_4:68;
 m <= len ((Cage (C,n)) | k) by A80, NAT_1:13;
then  m in dom ((Cage (C,n)) | k) by A16, FINSEQ_3:25;
then A89: ((Cage (C,n)) | (k + 1)) /. m = (Gauge (C,n)) * (i1,j1) by A82, A87, FINSEQ_4:68;
now__::_thesis:_(_left_cell_(((Cage_(C,n))_|_(k_+_1)),m,(Gauge_(C,n)))_misses_C_&_right_cell_(((Cage_(C,n))_|_(k_+_1)),m,(Gauge_(C,n)))_meets_C_)
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 A85;
supposeA90: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: ( left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C )
then  ( left_cell (((Cage (C,n)) | k),m,(Gauge (C,n))) = cell ((Gauge (C,n)),(i1 -' 1),j1) & right_cell (((Cage (C,n)) | k),m,(Gauge (C,n))) = cell ((Gauge (C,n)),i1,j1) ) by A10, A16, A80, A81, A82, A83, A84, GOBRD13:21, GOBRD13:22;
hence  ( left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C ) by A11, A16, A17, A81, A83, A86, A89, A88, A90, GOBRD13:21, GOBRD13:22; ::_thesis: verum
end;
supposeA91: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: ( left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C )
then  ( left_cell (((Cage (C,n)) | k),m,(Gauge (C,n))) = cell ((Gauge (C,n)),i1,j1) & right_cell (((Cage (C,n)) | k),m,(Gauge (C,n))) = cell ((Gauge (C,n)),i1,(j1 -' 1)) ) by A10, A16, A80, A81, A82, A83, A84, GOBRD13:23, GOBRD13:24;
hence  ( left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C ) by A11, A16, A17, A81, A83, A86, A89, A88, A91, GOBRD13:23, GOBRD13:24; ::_thesis: verum
end;
supposeA92: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: ( left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C )
then  ( left_cell (((Cage (C,n)) | k),m,(Gauge (C,n))) = cell ((Gauge (C,n)),i2,(j2 -' 1)) & right_cell (((Cage (C,n)) | k),m,(Gauge (C,n))) = cell ((Gauge (C,n)),i2,j2) ) by A10, A16, A80, A81, A82, A83, A84, GOBRD13:25, GOBRD13:26;
hence  ( left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C ) by A11, A16, A17, A81, A83, A86, A89, A88, A92, GOBRD13:25, GOBRD13:26; ::_thesis: verum
end;
supposeA93: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: ( left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C )
then  ( left_cell (((Cage (C,n)) | k),m,(Gauge (C,n))) = cell ((Gauge (C,n)),i2,j2) & right_cell (((Cage (C,n)) | k),m,(Gauge (C,n))) = cell ((Gauge (C,n)),(i1 -' 1),j2) ) by A10, A16, A80, A81, A82, A83, A84, GOBRD13:27, GOBRD13:28;
hence  ( left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C ) by A11, A16, A17, A81, A83, A86, A89, A88, A93, GOBRD13:27, GOBRD13:28; ::_thesis: verum
end;
end;
end;
hence  ( left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C ) ; ::_thesis: verum
end;
end;
end;
hence  ( left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C ) ; ::_thesis: verum
end;
end;
end;
hence  ( left_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) misses C & right_cell (((Cage (C,n)) | (k + 1)),m,(Gauge (C,n))) meets C ) ; ::_thesis: verum
end;
end;
end;
A94: (Cage (C,n)) | (len (Cage (C,n))) = Cage (C,n) by FINSEQ_1:58;
A95: S1[ 0 ] by CARD_1:27;
 for k being Element of NAT holds S1[k] from NAT_1:sch_1(A95, A5);
hence  ( not 1 <= k or not k + 1 <= len (Cage (C,n)) or ( left_cell ((Cage (C,n)),k,(Gauge (C,n))) misses C & right_cell ((Cage (C,n)),k,(Gauge (C,n))) meets C ) ) by A94; ::_thesis: verum
end;

theorem :: JORDAN9:32
for n being Element of NAT
for C being non empty compact non horizontal non vertical Subset of (TOP-REAL 2) st C is connected holds
N-min (L~ (Cage (C,n))) = (Cage (C,n)) /. 1
proof
let n be Element of NAT ; ::_thesis: for C being non empty compact non horizontal non vertical Subset of (TOP-REAL 2) st C is connected holds
N-min (L~ (Cage (C,n))) = (Cage (C,n)) /. 1
let C be non empty compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ( C is connected implies N-min (L~ (Cage (C,n))) = (Cage (C,n)) /. 1 )
set f = Cage (C,n);
assume A1: C is connected ; ::_thesis: N-min (L~ (Cage (C,n))) = (Cage (C,n)) /. 1
then A2: for k being Element of NAT st 1 <= k & k + 1 <= len (Cage (C,n)) holds
( left_cell ((Cage (C,n)),k,(Gauge (C,n))) misses C & right_cell ((Cage (C,n)),k,(Gauge (C,n))) meets C ) by Th31;
 ( Cage (C,n) is_sequence_on Gauge (C,n) & ex i being Element of NAT st
( 1 <= i & i + 1 <= len (Gauge (C,n)) & (Cage (C,n)) /. 1 = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) & (Cage (C,n)) /. 2 = (Gauge (C,n)) * ((i + 1),(width (Gauge (C,n)))) & N-min C in cell ((Gauge (C,n)),i,((width (Gauge (C,n))) -' 1)) & N-min C <> (Gauge (C,n)) * (i,((width (Gauge (C,n))) -' 1)) ) ) by A1, Def1;
hence  N-min (L~ (Cage (C,n))) = (Cage (C,n)) /. 1 by A2, Th30; ::_thesis: verum
end;