:: GOBRD13 semantic presentation

begin


definition
let E be non empty set ;
let S be non empty FinSequenceSet of the carrier of (TOP-REAL 2);
let F be Function of E,S;
let e be Element of E;
:: original: .
redefine funcF . e ->  FinSequence of (TOP-REAL 2);
coherence
 F . e is FinSequence of (TOP-REAL 2)
proof
thus  F . e is FinSequence of (TOP-REAL 2) by FINSEQ_2:def_3; ::_thesis: verum
end;
end;


theorem :: GOBRD13:1
canceled;

theorem :: GOBRD13:2
canceled;

theorem :: GOBRD13:3
canceled;

theorem :: GOBRD13:4
canceled;

theorem :: GOBRD13:5
canceled;

theorem :: GOBRD13:6
canceled;

theorem :: GOBRD13:7
canceled;

theorem :: GOBRD13:8
for f being FinSequence of (TOP-REAL 2)
for G being Matrix of (TOP-REAL 2) st f is_sequence_on G holds
rng f c= Values G
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: for G being Matrix of (TOP-REAL 2) st f is_sequence_on G holds
rng f c= Values G
let G be Matrix of (TOP-REAL 2); ::_thesis: ( f is_sequence_on G implies rng f c= Values G )
assume A1: f is_sequence_on G ; ::_thesis: rng f c= Values G
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng f or y in Values G )
assume  y in rng f ; ::_thesis: y in Values G
then consider n being Element of NAT  such that
A2: n in dom f and
A3: f /. n = y by PARTFUN2:2;
 ex i, j being Element of NAT st
( [i,j] in Indices G & f /. n = G * (i,j) ) by A1, A2, GOBOARD1:def_9;
then  y in  {  (G * (i,j)) where i, j is Element of NAT : [i,j] in Indices G  }  by A3;
hence  y in Values G by MATRIX_1:45; ::_thesis: verum
end;

theorem Th9: :: GOBRD13:9
for i1, j1, j2 being Element of NAT
for G1, G2 being Go-board st Values G1 c= Values G2 & [i1,j1] in Indices G1 & 1 <= j2 & j2 <= width G2 & G1 * (i1,j1) = G2 * (1,j2) holds
i1 = 1
proof
let i1, j1, j2 be Element of NAT ; ::_thesis: for G1, G2 being Go-board st Values G1 c= Values G2 & [i1,j1] in Indices G1 & 1 <= j2 & j2 <= width G2 & G1 * (i1,j1) = G2 * (1,j2) holds
i1 = 1
let G1, G2 be Go-board; ::_thesis: ( Values G1 c= Values G2 & [i1,j1] in Indices G1 & 1 <= j2 & j2 <= width G2 & G1 * (i1,j1) = G2 * (1,j2) implies i1 = 1 )
assume that
A1: Values G1 c= Values G2 and
A2: [i1,j1] in Indices G1 and
A3: ( 1 <= j2 & j2 <= width G2 ) and
A4: G1 * (i1,j1) = G2 * (1,j2) ; ::_thesis: i1 = 1
set p = G1 * (1,j1);
A5: ( 1 <= j1 & j1 <= width G1 ) by A2, MATRIX_1:38;
assume A6: i1 <> 1 ; ::_thesis: contradiction
 1 <= i1 by A2, MATRIX_1:38;
then A7: 1 < i1 by A6, XXREAL_0:1;
 i1 <= len G1 by A2, MATRIX_1:38;
then A8: (G1 * (1,j1)) `1 < (G1 * (i1,j1)) `1 by A5, A7, GOBOARD5:3;
 0 <> len G1 by GOBOARD1:def_3;
then  1 <= len G1 by NAT_1:14;
then  [1,j1] in Indices G1 by A5, MATRIX_1:36;
then  G1 * (1,j1) in  {  (G1 * (i,j)) where i, j is Element of NAT : [i,j] in Indices G1  }  ;
then  G1 * (1,j1) in Values G1 by MATRIX_1:45;
then  G1 * (1,j1) in Values G2 by A1;
then  G1 * (1,j1) in  {  (G2 * (i,j)) where i, j is Element of NAT : [i,j] in Indices G2  }  by MATRIX_1:45;
then consider i, j being Element of NAT  such that
A9: G1 * (1,j1) = G2 * (i,j) and
A10: [i,j] in Indices G2 ;
A11: ( 1 <= j & j <= width G2 ) by A10, MATRIX_1:38;
 0 <> len G2 by GOBOARD1:def_3;
then A12: 1 <= len G2 by NAT_1:14;
then A13: (G2 * (1,j)) `1 = (G2 * (1,1)) `1 by A11, GOBOARD5:2
.= (G2 * (1,j2)) `1 by A3, A12, GOBOARD5:2 ;
A14: i <= len G2 by A10, MATRIX_1:38;
 1 <= i by A10, MATRIX_1:38;
then  1 < i by A4, A8, A9, A13, XXREAL_0:1;
hence  contradiction by A4, A8, A9, A14, A11, A13, GOBOARD5:3; ::_thesis: verum
end;

theorem Th10: :: GOBRD13:10
for i1, j1, j2 being Element of NAT
for G1, G2 being Go-board st Values G1 c= Values G2 & [i1,j1] in Indices G1 & 1 <= j2 & j2 <= width G2 & G1 * (i1,j1) = G2 * ((len G2),j2) holds
i1 = len G1
proof
let i1, j1, j2 be Element of NAT ; ::_thesis: for G1, G2 being Go-board st Values G1 c= Values G2 & [i1,j1] in Indices G1 & 1 <= j2 & j2 <= width G2 & G1 * (i1,j1) = G2 * ((len G2),j2) holds
i1 = len G1
let G1, G2 be Go-board; ::_thesis: ( Values G1 c= Values G2 & [i1,j1] in Indices G1 & 1 <= j2 & j2 <= width G2 & G1 * (i1,j1) = G2 * ((len G2),j2) implies i1 = len G1 )
assume that
A1: Values G1 c= Values G2 and
A2: [i1,j1] in Indices G1 and
A3: ( 1 <= j2 & j2 <= width G2 ) and
A4: G1 * (i1,j1) = G2 * ((len G2),j2) ; ::_thesis: i1 = len G1
set p = G1 * ((len G1),j1);
A5: ( 1 <= j1 & j1 <= width G1 ) by A2, MATRIX_1:38;
assume A6: i1 <> len G1 ; ::_thesis: contradiction
 i1 <= len G1 by A2, MATRIX_1:38;
then A7: i1 < len G1 by A6, XXREAL_0:1;
 1 <= i1 by A2, MATRIX_1:38;
then A8: (G1 * (i1,j1)) `1 < (G1 * ((len G1),j1)) `1 by A5, A7, GOBOARD5:3;
 0 <> len G1 by GOBOARD1:def_3;
then  1 <= len G1 by NAT_1:14;
then  [(len G1),j1] in Indices G1 by A5, MATRIX_1:36;
then  G1 * ((len G1),j1) in  {  (G1 * (i,j)) where i, j is Element of NAT : [i,j] in Indices G1  }  ;
then  G1 * ((len G1),j1) in Values G1 by MATRIX_1:45;
then  G1 * ((len G1),j1) in Values G2 by A1;
then  G1 * ((len G1),j1) in  {  (G2 * (i,j)) where i, j is Element of NAT : [i,j] in Indices G2  }  by MATRIX_1:45;
then consider i, j being Element of NAT  such that
A9: G1 * ((len G1),j1) = G2 * (i,j) and
A10: [i,j] in Indices G2 ;
A11: ( 1 <= j & j <= width G2 ) by A10, MATRIX_1:38;
 0 <> len G2 by GOBOARD1:def_3;
then A12: 1 <= len G2 by NAT_1:14;
then A13: (G2 * ((len G2),j)) `1 = (G2 * ((len G2),1)) `1 by A11, GOBOARD5:2
.= (G2 * ((len G2),j2)) `1 by A3, A12, GOBOARD5:2 ;
A14: 1 <= i by A10, MATRIX_1:38;
 i <= len G2 by A10, MATRIX_1:38;
then  i < len G2 by A4, A8, A9, A13, XXREAL_0:1;
hence  contradiction by A4, A8, A9, A14, A11, A13, GOBOARD5:3; ::_thesis: verum
end;

theorem Th11: :: GOBRD13:11
for i1, j1, i2 being Element of NAT
for G1, G2 being Go-board st Values G1 c= Values G2 & [i1,j1] in Indices G1 & 1 <= i2 & i2 <= len G2 & G1 * (i1,j1) = G2 * (i2,1) holds
j1 = 1
proof
let i1, j1, i2 be Element of NAT ; ::_thesis: for G1, G2 being Go-board st Values G1 c= Values G2 & [i1,j1] in Indices G1 & 1 <= i2 & i2 <= len G2 & G1 * (i1,j1) = G2 * (i2,1) holds
j1 = 1
let G1, G2 be Go-board; ::_thesis: ( Values G1 c= Values G2 & [i1,j1] in Indices G1 & 1 <= i2 & i2 <= len G2 & G1 * (i1,j1) = G2 * (i2,1) implies j1 = 1 )
assume that
A1: Values G1 c= Values G2 and
A2: [i1,j1] in Indices G1 and
A3: ( 1 <= i2 & i2 <= len G2 ) and
A4: G1 * (i1,j1) = G2 * (i2,1) ; ::_thesis: j1 = 1
set p = G1 * (i1,1);
A5: ( 1 <= i1 & i1 <= len G1 ) by A2, MATRIX_1:38;
assume A6: j1 <> 1 ; ::_thesis: contradiction
 1 <= j1 by A2, MATRIX_1:38;
then A7: 1 < j1 by A6, XXREAL_0:1;
 j1 <= width G1 by A2, MATRIX_1:38;
then A8: (G1 * (i1,1)) `2 < (G1 * (i1,j1)) `2 by A5, A7, GOBOARD5:4;
 0 <> width G1 by GOBOARD1:def_3;
then  1 <= width G1 by NAT_1:14;
then  [i1,1] in Indices G1 by A5, MATRIX_1:36;
then  G1 * (i1,1) in  {  (G1 * (i,j)) where i, j is Element of NAT : [i,j] in Indices G1  }  ;
then  G1 * (i1,1) in Values G1 by MATRIX_1:45;
then  G1 * (i1,1) in Values G2 by A1;
then  G1 * (i1,1) in  {  (G2 * (i,j)) where i, j is Element of NAT : [i,j] in Indices G2  }  by MATRIX_1:45;
then consider i, j being Element of NAT  such that
A9: G1 * (i1,1) = G2 * (i,j) and
A10: [i,j] in Indices G2 ;
A11: ( 1 <= i & i <= len G2 ) by A10, MATRIX_1:38;
 0 <> width G2 by GOBOARD1:def_3;
then A12: 1 <= width G2 by NAT_1:14;
then A13: (G2 * (i,1)) `2 = (G2 * (1,1)) `2 by A11, GOBOARD5:1
.= (G2 * (i2,1)) `2 by A3, A12, GOBOARD5:1 ;
A14: j <= width G2 by A10, MATRIX_1:38;
 1 <= j by A10, MATRIX_1:38;
then  1 < j by A4, A8, A9, A13, XXREAL_0:1;
hence  contradiction by A4, A8, A9, A11, A14, A13, GOBOARD5:4; ::_thesis: verum
end;

theorem Th12: :: GOBRD13:12
for i1, j1, i2 being Element of NAT
for G1, G2 being Go-board st Values G1 c= Values G2 & [i1,j1] in Indices G1 & 1 <= i2 & i2 <= len G2 & G1 * (i1,j1) = G2 * (i2,(width G2)) holds
j1 = width G1
proof
let i1, j1, i2 be Element of NAT ; ::_thesis: for G1, G2 being Go-board st Values G1 c= Values G2 & [i1,j1] in Indices G1 & 1 <= i2 & i2 <= len G2 & G1 * (i1,j1) = G2 * (i2,(width G2)) holds
j1 = width G1
let G1, G2 be Go-board; ::_thesis: ( Values G1 c= Values G2 & [i1,j1] in Indices G1 & 1 <= i2 & i2 <= len G2 & G1 * (i1,j1) = G2 * (i2,(width G2)) implies j1 = width G1 )
assume that
A1: Values G1 c= Values G2 and
A2: [i1,j1] in Indices G1 and
A3: ( 1 <= i2 & i2 <= len G2 ) and
A4: G1 * (i1,j1) = G2 * (i2,(width G2)) ; ::_thesis: j1 = width G1
set p = G1 * (i1,(width G1));
A5: ( 1 <= i1 & i1 <= len G1 ) by A2, MATRIX_1:38;
assume A6: j1 <> width G1 ; ::_thesis: contradiction
 j1 <= width G1 by A2, MATRIX_1:38;
then A7: j1 < width G1 by A6, XXREAL_0:1;
 1 <= j1 by A2, MATRIX_1:38;
then A8: (G1 * (i1,j1)) `2 < (G1 * (i1,(width G1))) `2 by A5, A7, GOBOARD5:4;
 0 <> width G1 by GOBOARD1:def_3;
then  1 <= width G1 by NAT_1:14;
then  [i1,(width G1)] in Indices G1 by A5, MATRIX_1:36;
then  G1 * (i1,(width G1)) in  {  (G1 * (i,j)) where i, j is Element of NAT : [i,j] in Indices G1  }  ;
then  G1 * (i1,(width G1)) in Values G1 by MATRIX_1:45;
then  G1 * (i1,(width G1)) in Values G2 by A1;
then  G1 * (i1,(width G1)) in  {  (G2 * (i,j)) where i, j is Element of NAT : [i,j] in Indices G2  }  by MATRIX_1:45;
then consider i, j being Element of NAT  such that
A9: G1 * (i1,(width G1)) = G2 * (i,j) and
A10: [i,j] in Indices G2 ;
A11: ( 1 <= i & i <= len G2 ) by A10, MATRIX_1:38;
 0 <> width G2 by GOBOARD1:def_3;
then A12: 1 <= width G2 by NAT_1:14;
then A13: (G2 * (i,(width G2))) `2 = (G2 * (1,(width G2))) `2 by A11, GOBOARD5:1
.= (G2 * (i2,(width G2))) `2 by A3, A12, GOBOARD5:1 ;
A14: 1 <= j by A10, MATRIX_1:38;
 j <= width G2 by A10, MATRIX_1:38;
then  j < width G2 by A4, A8, A9, A13, XXREAL_0:1;
hence  contradiction by A4, A8, A9, A11, A14, A13, GOBOARD5:4; ::_thesis: verum
end;

theorem Th13: :: GOBRD13:13
for i1, j1, i2, j2 being Element of NAT
for G1, G2 being Go-board st Values G1 c= Values G2 & 1 <= i1 & i1 < len G1 & 1 <= j1 & j1 <= width G1 & 1 <= i2 & i2 < len G2 & 1 <= j2 & j2 <= width G2 & G1 * (i1,j1) = G2 * (i2,j2) holds
(G2 * ((i2 + 1),j2)) `1 <= (G1 * ((i1 + 1),j1)) `1
proof
let i1, j1, i2, j2 be Element of NAT ; ::_thesis: for G1, G2 being Go-board st Values G1 c= Values G2 & 1 <= i1 & i1 < len G1 & 1 <= j1 & j1 <= width G1 & 1 <= i2 & i2 < len G2 & 1 <= j2 & j2 <= width G2 & G1 * (i1,j1) = G2 * (i2,j2) holds
(G2 * ((i2 + 1),j2)) `1 <= (G1 * ((i1 + 1),j1)) `1
let G1, G2 be Go-board; ::_thesis: ( Values G1 c= Values G2 & 1 <= i1 & i1 < len G1 & 1 <= j1 & j1 <= width G1 & 1 <= i2 & i2 < len G2 & 1 <= j2 & j2 <= width G2 & G1 * (i1,j1) = G2 * (i2,j2) implies (G2 * ((i2 + 1),j2)) `1 <= (G1 * ((i1 + 1),j1)) `1 )
assume that
A1: Values G1 c= Values G2 and
A2: 1 <= i1 and
A3: i1 < len G1 and
A4: ( 1 <= j1 & j1 <= width G1 ) and
A5: 1 <= i2 and
A6: i2 < len G2 and
A7: ( 1 <= j2 & j2 <= width G2 ) and
A8: G1 * (i1,j1) = G2 * (i2,j2) ; ::_thesis: (G2 * ((i2 + 1),j2)) `1 <= (G1 * ((i1 + 1),j1)) `1
set p = G1 * ((i1 + 1),j1);
A9: i1 + 1 <= len G1 by A3, NAT_1:13;
 1 <= i1 + 1 by A2, NAT_1:13;
then  [(i1 + 1),j1] in Indices G1 by A4, A9, MATRIX_1:36;
then  G1 * ((i1 + 1),j1) in  {  (G1 * (i,j)) where i, j is Element of NAT : [i,j] in Indices G1  }  ;
then  G1 * ((i1 + 1),j1) in Values G1 by MATRIX_1:45;
then  G1 * ((i1 + 1),j1) in Values G2 by A1;
then  G1 * ((i1 + 1),j1) in  {  (G2 * (i,j)) where i, j is Element of NAT : [i,j] in Indices G2  }  by MATRIX_1:45;
then consider i, j being Element of NAT  such that
A10: G1 * ((i1 + 1),j1) = G2 * (i,j) and
A11: [i,j] in Indices G2 ;
A12: 1 <= i by A11, MATRIX_1:38;
A13: i <= len G2 by A11, MATRIX_1:38;
 ( 1 <= j & j <= width G2 ) by A11, MATRIX_1:38;
then A14: (G2 * (i,j)) `1 = (G2 * (i,1)) `1 by A12, A13, GOBOARD5:2
.= (G2 * (i,j2)) `1 by A7, A12, A13, GOBOARD5:2 ;
 i1 < i1 + 1 by NAT_1:13;
then A15: (G2 * (i2,j2)) `1 < (G2 * (i,j2)) `1 by A2, A4, A8, A9, A10, A14, GOBOARD5:3;
A16: now__::_thesis:_not_i_<=_i2
assume  i <= i2 ; ::_thesis: contradiction
then  ( i = i2 or i < i2 ) by XXREAL_0:1;
hence  contradiction by A6, A7, A12, A15, GOBOARD5:3; ::_thesis: verum
end;
assume A17: (G1 * ((i1 + 1),j1)) `1 < (G2 * ((i2 + 1),j2)) `1 ; ::_thesis: contradiction
A18: 1 <= i2 + 1 by A5, NAT_1:13;
now__::_thesis:_not_i2_+_1_<=_i
assume  i2 + 1 <= i ; ::_thesis: contradiction
then  ( i2 + 1 = i or i2 + 1 < i ) by XXREAL_0:1;
hence  contradiction by A7, A17, A10, A13, A14, A18, GOBOARD5:3; ::_thesis: verum
end;
hence  contradiction by A16, NAT_1:13; ::_thesis: verum
end;

theorem Th14: :: GOBRD13:14
for i1, j1, i2, j2 being Element of NAT
for G1, G2 being Go-board st G1 * ((i1 -' 1),j1) in Values G2 & 1 < i1 & i1 <= len G1 & 1 <= j1 & j1 <= width G1 & 1 < i2 & i2 <= len G2 & 1 <= j2 & j2 <= width G2 & G1 * (i1,j1) = G2 * (i2,j2) holds
(G1 * ((i1 -' 1),j1)) `1 <= (G2 * ((i2 -' 1),j2)) `1
proof
let i1, j1, i2, j2 be Element of NAT ; ::_thesis: for G1, G2 being Go-board st G1 * ((i1 -' 1),j1) in Values G2 & 1 < i1 & i1 <= len G1 & 1 <= j1 & j1 <= width G1 & 1 < i2 & i2 <= len G2 & 1 <= j2 & j2 <= width G2 & G1 * (i1,j1) = G2 * (i2,j2) holds
(G1 * ((i1 -' 1),j1)) `1 <= (G2 * ((i2 -' 1),j2)) `1
let G1, G2 be Go-board; ::_thesis: ( G1 * ((i1 -' 1),j1) in Values G2 & 1 < i1 & i1 <= len G1 & 1 <= j1 & j1 <= width G1 & 1 < i2 & i2 <= len G2 & 1 <= j2 & j2 <= width G2 & G1 * (i1,j1) = G2 * (i2,j2) implies (G1 * ((i1 -' 1),j1)) `1 <= (G2 * ((i2 -' 1),j2)) `1 )
assume that
A1: G1 * ((i1 -' 1),j1) in Values G2 and
A2: 1 < i1 and
A3: ( i1 <= len G1 & 1 <= j1 & j1 <= width G1 ) and
A4: 1 < i2 and
A5: i2 <= len G2 and
A6: ( 1 <= j2 & j2 <= width G2 ) and
A7: G1 * (i1,j1) = G2 * (i2,j2) ; ::_thesis: (G1 * ((i1 -' 1),j1)) `1 <= (G2 * ((i2 -' 1),j2)) `1
set p = G1 * ((i1 -' 1),j1);
A8: G1 * ((i1 -' 1),j1) in  {  (G2 * (i,j)) where i, j is Element of NAT : [i,j] in Indices G2  }  by A1, MATRIX_1:45;
 1 <= i2 -' 1 by A4, NAT_D:49;
then  i2 -' 1 < i2 by NAT_D:51;
then A9: i2 -' 1 < len G2 by A5, XXREAL_0:2;
consider i, j being Element of NAT  such that
A10: G1 * ((i1 -' 1),j1) = G2 * (i,j) and
A11: [i,j] in Indices G2 by A8;
A12: 1 <= i by A11, MATRIX_1:38;
A13: i <= len G2 by A11, MATRIX_1:38;
 ( 1 <= j & j <= width G2 ) by A11, MATRIX_1:38;
then A14: (G2 * (i,j)) `1 = (G2 * (i,1)) `1 by A12, A13, GOBOARD5:2
.= (G2 * (i,j2)) `1 by A6, A12, A13, GOBOARD5:2 ;
A15: 1 <= i1 -' 1 by A2, NAT_D:49;
then  i1 -' 1 < i1 by NAT_D:51;
then A16: (G2 * (i,j2)) `1 < (G2 * (i2,j2)) `1 by A3, A7, A15, A10, A14, GOBOARD5:3;
A17: now__::_thesis:_not_i2_<=_i
assume  i2 <= i ; ::_thesis: contradiction
then  ( i = i2 or i2 < i ) by XXREAL_0:1;
hence  contradiction by A4, A6, A13, A16, GOBOARD5:3; ::_thesis: verum
end;
assume A18: (G2 * ((i2 -' 1),j2)) `1 < (G1 * ((i1 -' 1),j1)) `1 ; ::_thesis: contradiction
now__::_thesis:_not_i_<=_i2_-'_1
assume  i <= i2 -' 1 ; ::_thesis: contradiction
then  ( i2 -' 1 = i or i < i2 -' 1 ) by XXREAL_0:1;
hence  contradiction by A6, A18, A10, A12, A14, A9, GOBOARD5:3; ::_thesis: verum
end;
hence  contradiction by A17, NAT_D:49; ::_thesis: verum
end;

theorem Th15: :: GOBRD13:15
for i1, j1, i2, j2 being Element of NAT
for G1, G2 being Go-board st G1 * (i1,(j1 + 1)) in Values G2 & 1 <= i1 & i1 <= len G1 & 1 <= j1 & j1 < width G1 & 1 <= i2 & i2 <= len G2 & 1 <= j2 & j2 < width G2 & G1 * (i1,j1) = G2 * (i2,j2) holds
(G2 * (i2,(j2 + 1))) `2 <= (G1 * (i1,(j1 + 1))) `2
proof
let i1, j1, i2, j2 be Element of NAT ; ::_thesis: for G1, G2 being Go-board st G1 * (i1,(j1 + 1)) in Values G2 & 1 <= i1 & i1 <= len G1 & 1 <= j1 & j1 < width G1 & 1 <= i2 & i2 <= len G2 & 1 <= j2 & j2 < width G2 & G1 * (i1,j1) = G2 * (i2,j2) holds
(G2 * (i2,(j2 + 1))) `2 <= (G1 * (i1,(j1 + 1))) `2
let G1, G2 be Go-board; ::_thesis: ( G1 * (i1,(j1 + 1)) in Values G2 & 1 <= i1 & i1 <= len G1 & 1 <= j1 & j1 < width G1 & 1 <= i2 & i2 <= len G2 & 1 <= j2 & j2 < width G2 & G1 * (i1,j1) = G2 * (i2,j2) implies (G2 * (i2,(j2 + 1))) `2 <= (G1 * (i1,(j1 + 1))) `2 )
assume that
A1: G1 * (i1,(j1 + 1)) in Values G2 and
A2: ( 1 <= i1 & i1 <= len G1 & 1 <= j1 ) and
A3: j1 < width G1 and
A4: ( 1 <= i2 & i2 <= len G2 ) and
A5: 1 <= j2 and
A6: j2 < width G2 and
A7: G1 * (i1,j1) = G2 * (i2,j2) ; ::_thesis: (G2 * (i2,(j2 + 1))) `2 <= (G1 * (i1,(j1 + 1))) `2
set p = G1 * (i1,(j1 + 1));
 G1 * (i1,(j1 + 1)) in  {  (G2 * (i,j)) where i, j is Element of NAT : [i,j] in Indices G2  }  by A1, MATRIX_1:45;
then consider i, j being Element of NAT  such that
A8: G1 * (i1,(j1 + 1)) = G2 * (i,j) and
A9: [i,j] in Indices G2 ;
A10: 1 <= j by A9, MATRIX_1:38;
A11: j <= width G2 by A9, MATRIX_1:38;
 ( 1 <= i & i <= len G2 ) by A9, MATRIX_1:38;
then A12: (G2 * (i,j)) `2 = (G2 * (1,j)) `2 by A10, A11, GOBOARD5:1
.= (G2 * (i2,j)) `2 by A4, A10, A11, GOBOARD5:1 ;
 ( j1 < j1 + 1 & j1 + 1 <= width G1 ) by A3, NAT_1:13;
then A13: (G2 * (i2,j2)) `2 < (G2 * (i2,j)) `2 by A2, A7, A8, A12, GOBOARD5:4;
A14: now__::_thesis:_not_j_<=_j2
assume  j <= j2 ; ::_thesis: contradiction
then  ( j = j2 or j < j2 ) by XXREAL_0:1;
hence  contradiction by A4, A6, A10, A13, GOBOARD5:4; ::_thesis: verum
end;
assume A15: (G1 * (i1,(j1 + 1))) `2 < (G2 * (i2,(j2 + 1))) `2 ; ::_thesis: contradiction
A16: 1 <= j2 + 1 by A5, NAT_1:13;
now__::_thesis:_not_j2_+_1_<=_j
assume  j2 + 1 <= j ; ::_thesis: contradiction
then  ( j2 + 1 = j or j2 + 1 < j ) by XXREAL_0:1;
hence  contradiction by A4, A15, A8, A11, A12, A16, GOBOARD5:4; ::_thesis: verum
end;
hence  contradiction by A14, NAT_1:13; ::_thesis: verum
end;

theorem Th16: :: GOBRD13:16
for i1, j1, i2, j2 being Element of NAT
for G1, G2 being Go-board st Values G1 c= Values G2 & 1 <= i1 & i1 <= len G1 & 1 < j1 & j1 <= width G1 & 1 <= i2 & i2 <= len G2 & 1 < j2 & j2 <= width G2 & G1 * (i1,j1) = G2 * (i2,j2) holds
(G1 * (i1,(j1 -' 1))) `2 <= (G2 * (i2,(j2 -' 1))) `2
proof
let i1, j1, i2, j2 be Element of NAT ; ::_thesis: for G1, G2 being Go-board st Values G1 c= Values G2 & 1 <= i1 & i1 <= len G1 & 1 < j1 & j1 <= width G1 & 1 <= i2 & i2 <= len G2 & 1 < j2 & j2 <= width G2 & G1 * (i1,j1) = G2 * (i2,j2) holds
(G1 * (i1,(j1 -' 1))) `2 <= (G2 * (i2,(j2 -' 1))) `2
let G1, G2 be Go-board; ::_thesis: ( Values G1 c= Values G2 & 1 <= i1 & i1 <= len G1 & 1 < j1 & j1 <= width G1 & 1 <= i2 & i2 <= len G2 & 1 < j2 & j2 <= width G2 & G1 * (i1,j1) = G2 * (i2,j2) implies (G1 * (i1,(j1 -' 1))) `2 <= (G2 * (i2,(j2 -' 1))) `2 )
assume that
A1: Values G1 c= Values G2 and
A2: ( 1 <= i1 & i1 <= len G1 ) and
A3: 1 < j1 and
A4: j1 <= width G1 and
A5: ( 1 <= i2 & i2 <= len G2 ) and
A6: 1 < j2 and
A7: j2 <= width G2 and
A8: G1 * (i1,j1) = G2 * (i2,j2) ; ::_thesis: (G1 * (i1,(j1 -' 1))) `2 <= (G2 * (i2,(j2 -' 1))) `2
set p = G1 * (i1,(j1 -' 1));
A9: 1 <= j1 -' 1 by A3, NAT_D:49;
then A10: j1 -' 1 < j1 by NAT_D:51;
then  j1 -' 1 < width G1 by A4, XXREAL_0:2;
then  [i1,(j1 -' 1)] in Indices G1 by A2, A9, MATRIX_1:36;
then  G1 * (i1,(j1 -' 1)) in  {  (G1 * (i,j)) where i, j is Element of NAT : [i,j] in Indices G1  }  ;
then  G1 * (i1,(j1 -' 1)) in Values G1 by MATRIX_1:45;
then  G1 * (i1,(j1 -' 1)) in Values G2 by A1;
then  G1 * (i1,(j1 -' 1)) in  {  (G2 * (i,j)) where i, j is Element of NAT : [i,j] in Indices G2  }  by MATRIX_1:45;
then consider i, j being Element of NAT  such that
A11: G1 * (i1,(j1 -' 1)) = G2 * (i,j) and
A12: [i,j] in Indices G2 ;
A13: 1 <= j by A12, MATRIX_1:38;
A14: j <= width G2 by A12, MATRIX_1:38;
 ( 1 <= i & i <= len G2 ) by A12, MATRIX_1:38;
then A15: (G2 * (i,j)) `2 = (G2 * (1,j)) `2 by A13, A14, GOBOARD5:1
.= (G2 * (i2,j)) `2 by A5, A13, A14, GOBOARD5:1 ;
then A16: (G2 * (i2,j)) `2 < (G2 * (i2,j2)) `2 by A2, A4, A8, A9, A10, A11, GOBOARD5:4;
A17: now__::_thesis:_not_j2_<=_j
assume  j2 <= j ; ::_thesis: contradiction
then  ( j = j2 or j2 < j ) by XXREAL_0:1;
hence  contradiction by A5, A6, A14, A16, GOBOARD5:4; ::_thesis: verum
end;
 1 <= j2 -' 1 by A6, NAT_D:49;
then  j2 -' 1 < j2 by NAT_D:51;
then A18: j2 -' 1 < width G2 by A7, XXREAL_0:2;
assume A19: (G2 * (i2,(j2 -' 1))) `2 < (G1 * (i1,(j1 -' 1))) `2 ; ::_thesis: contradiction
now__::_thesis:_not_j_<=_j2_-'_1
assume  j <= j2 -' 1 ; ::_thesis: contradiction
then  ( j2 -' 1 = j or j < j2 -' 1 ) by XXREAL_0:1;
hence  contradiction by A5, A19, A11, A13, A15, A18, GOBOARD5:4; ::_thesis: verum
end;
hence  contradiction by A17, NAT_D:49; ::_thesis: verum
end;

theorem Th17: :: GOBRD13:17
for i1, j1, i2, j2 being Element of NAT
for G1, G2 being Go-board st Values G1 c= Values G2 & [i1,j1] in Indices G1 & [i2,j2] in Indices G2 & G1 * (i1,j1) = G2 * (i2,j2) holds
cell (G2,i2,j2) c= cell (G1,i1,j1)
proof
let i1, j1, i2, j2 be Element of NAT ; ::_thesis: for G1, G2 being Go-board st Values G1 c= Values G2 & [i1,j1] in Indices G1 & [i2,j2] in Indices G2 & G1 * (i1,j1) = G2 * (i2,j2) holds
cell (G2,i2,j2) c= cell (G1,i1,j1)
let G1, G2 be Go-board; ::_thesis: ( Values G1 c= Values G2 & [i1,j1] in Indices G1 & [i2,j2] in Indices G2 & G1 * (i1,j1) = G2 * (i2,j2) implies cell (G2,i2,j2) c= cell (G1,i1,j1) )
assume that
A1: Values G1 c= Values G2 and
A2: [i1,j1] in Indices G1 and
A3: [i2,j2] in Indices G2 and
A4: G1 * (i1,j1) = G2 * (i2,j2) ; ::_thesis: cell (G2,i2,j2) c= cell (G1,i1,j1)
A5: 1 <= i1 by A2, MATRIX_1:38;
A6: j1 <= width G1 by A2, MATRIX_1:38;
let p be set ; :: according to TARSKI:def_3 ::_thesis: ( not p in cell (G2,i2,j2) or p in cell (G1,i1,j1) )
assume A7: p in cell (G2,i2,j2) ; ::_thesis: p in cell (G1,i1,j1)
A8: 1 <= i2 by A3, MATRIX_1:38;
A9: j2 <= width G2 by A3, MATRIX_1:38;
A10: 1 <= j2 by A3, MATRIX_1:38;
A11: i2 <= len G2 by A3, MATRIX_1:38;
then A12: ( (G2 * (i2,j2)) `1 = (G2 * (i2,1)) `1 & (G2 * (i2,j2)) `2 = (G2 * (1,j2)) `2 ) by A8, A10, A9, GOBOARD5:1, GOBOARD5:2;
A13: 1 <= j1 by A2, MATRIX_1:38;
A14: i1 <= len G1 by A2, MATRIX_1:38;
then A15: ( (G1 * (i1,j1)) `1 = (G1 * (i1,1)) `1 & (G1 * (i1,j1)) `2 = (G1 * (1,j1)) `2 ) by A5, A13, A6, GOBOARD5:1, GOBOARD5:2;
percases ( ( i2 = len G2 & j2 = width G2 ) or ( i2 = len G2 & j2 < width G2 ) or ( i2 < len G2 & j2 = width G2 ) or ( i2 < len G2 & j2 < width G2 ) ) by A11, A9, XXREAL_0:1;
supposeA16: ( i2 = len G2 & j2 = width G2 ) ; ::_thesis: p in cell (G1,i1,j1)
then A17: p in  {  |[r,s]| where r, s is Real : ( (G2 * (i2,j2)) `1 <= r & (G2 * (i2,j2)) `2 <= s )  }  by A7, A12, GOBRD11:28;
 ( i1 = len G1 & j1 = width G1 ) by A1, A2, A4, A8, A10, A16, Th10, Th12;
hence  p in cell (G1,i1,j1) by A4, A15, A17, GOBRD11:28; ::_thesis: verum
end;
supposeA18: ( i2 = len G2 & j2 < width G2 ) ; ::_thesis: p in cell (G1,i1,j1)
then  p in  {  |[r,s]| where r, s is Real : ( (G2 * (i2,j2)) `1 <= r & (G2 * (i2,j2)) `2 <= s & s <= (G2 * (1,(j2 + 1))) `2 )  }  by A7, A10, A12, GOBRD11:29;
then consider r9, s9 being Real such that
A19: ( p = |[r9,s9]| & (G2 * (i2,j2)) `1 <= r9 & (G2 * (i2,j2)) `2 <= s9 ) and
A20: s9 <= (G2 * (1,(j2 + 1))) `2 ;
A21: i1 = len G1 by A1, A2, A4, A10, A18, Th10;
now__::_thesis:_p_in_cell_(G1,i1,j1)
percases ( j1 = width G1 or j1 < width G1 ) by A6, XXREAL_0:1;
supposeA22: j1 = width G1 ; ::_thesis: p in cell (G1,i1,j1)
 p in  {  |[r,s]| where r, s is Real : ( (G1 * (i1,j1)) `1 <= r & (G1 * (i1,j1)) `2 <= s )  }  by A4, A19;
hence  p in cell (G1,i1,j1) by A15, A21, A22, GOBRD11:28; ::_thesis: verum
end;
supposeA23: j1 < width G1 ; ::_thesis: p in cell (G1,i1,j1)
 ( 1 <= j2 + 1 & j2 + 1 <= width G2 ) by A18, NAT_1:12, NAT_1:13;
then A24: (G2 * (i2,(j2 + 1))) `2 = (G2 * (1,(j2 + 1))) `2 by A8, A11, GOBOARD5:1;
 ( 1 <= j1 + 1 & j1 + 1 <= width G1 ) by A23, NAT_1:12, NAT_1:13;
then  ( G1 * (i1,(j1 + 1)) in Values G1 & (G1 * (i1,(j1 + 1))) `2 = (G1 * (1,(j1 + 1))) `2 ) by A5, A14, MATRIX_1:47, GOBOARD5:1;
then  (G2 * (1,(j2 + 1))) `2 <= (G1 * (1,(j1 + 1))) `2 by A1, A4, A5, A14, A13, A8, A10, A18, A23, A24, Th15;
then  s9 <= (G1 * (1,(j1 + 1))) `2 by A20, XXREAL_0:2;
then  p in  {  |[r,s]| where r, s is Real : ( (G1 * (i1,j1)) `1 <= r & (G1 * (i1,j1)) `2 <= s & s <= (G1 * (1,(j1 + 1))) `2 )  }  by A4, A19;
hence  p in cell (G1,i1,j1) by A13, A15, A21, A23, GOBRD11:29; ::_thesis: verum
end;
end;
end;
hence  p in cell (G1,i1,j1) ; ::_thesis: verum
end;
supposeA25: ( i2 < len G2 & j2 = width G2 ) ; ::_thesis: p in cell (G1,i1,j1)
then  p in  {  |[r,s]| where r, s is Real : ( (G2 * (i2,j2)) `1 <= r & r <= (G2 * ((i2 + 1),1)) `1 & (G2 * (i2,j2)) `2 <= s )  }  by A7, A8, A12, GOBRD11:31;
then consider r9, s9 being Real such that
A26: ( p = |[r9,s9]| & (G2 * (i2,j2)) `1 <= r9 ) and
A27: r9 <= (G2 * ((i2 + 1),1)) `1 and
A28: (G2 * (i2,j2)) `2 <= s9 ;
A29: j1 = width G1 by A1, A2, A4, A8, A25, Th12;
now__::_thesis:_p_in_cell_(G1,i1,j1)
percases ( i1 = len G1 or i1 < len G1 ) by A14, XXREAL_0:1;
supposeA30: i1 = len G1 ; ::_thesis: p in cell (G1,i1,j1)
 p in  {  |[r,s]| where r, s is Real : ( (G1 * (i1,j1)) `1 <= r & (G1 * (i1,j1)) `2 <= s )  }  by A4, A26, A28;
hence  p in cell (G1,i1,j1) by A15, A29, A30, GOBRD11:28; ::_thesis: verum
end;
supposeA31: i1 < len G1 ; ::_thesis: p in cell (G1,i1,j1)
 ( 1 <= i2 + 1 & i2 + 1 <= len G2 ) by A25, NAT_1:12, NAT_1:13;
then A32: (G2 * ((i2 + 1),j2)) `1 = (G2 * ((i2 + 1),1)) `1 by A10, A9, GOBOARD5:2;
 ( 1 <= i1 + 1 & i1 + 1 <= len G1 ) by A31, NAT_1:12, NAT_1:13;
then  (G1 * ((i1 + 1),j1)) `1 = (G1 * ((i1 + 1),1)) `1 by A13, A6, GOBOARD5:2;
then  (G2 * ((i2 + 1),1)) `1 <= (G1 * ((i1 + 1),1)) `1 by A1, A4, A5, A13, A6, A8, A10, A25, A31, A32, Th13;
then  r9 <= (G1 * ((i1 + 1),1)) `1 by A27, XXREAL_0:2;
then  p in  {  |[r,s]| where r, s is Real : ( (G1 * (i1,j1)) `1 <= r & r <= (G1 * ((i1 + 1),1)) `1 & (G1 * (i1,j1)) `2 <= s )  }  by A4, A26, A28;
hence  p in cell (G1,i1,j1) by A5, A15, A29, A31, GOBRD11:31; ::_thesis: verum
end;
end;
end;
hence  p in cell (G1,i1,j1) ; ::_thesis: verum
end;
supposeA33: ( i2 < len G2 & j2 < width G2 ) ; ::_thesis: p in cell (G1,i1,j1)
then  ( 1 <= j2 + 1 & j2 + 1 <= width G2 ) by NAT_1:12, NAT_1:13;
then A34: (G2 * (i2,(j2 + 1))) `2 = (G2 * (1,(j2 + 1))) `2 by A8, A11, GOBOARD5:1;
 ( 1 <= i2 + 1 & i2 + 1 <= len G2 ) by A33, NAT_1:12, NAT_1:13;
then  (G2 * ((i2 + 1),j2)) `1 = (G2 * ((i2 + 1),1)) `1 by A10, A9, GOBOARD5:2;
then  p in  {  |[r,s]| where r, s is Real : ( (G2 * (i2,j2)) `1 <= r & r <= (G2 * ((i2 + 1),j2)) `1 & (G2 * (i2,j2)) `2 <= s & s <= (G2 * (i2,(j2 + 1))) `2 )  }  by A7, A8, A10, A12, A33, A34, GOBRD11:32;
then consider r9, s9 being Real such that
A35: ( p = |[r9,s9]| & (G2 * (i2,j2)) `1 <= r9 ) and
A36: r9 <= (G2 * ((i2 + 1),j2)) `1 and
A37: (G2 * (i2,j2)) `2 <= s9 and
A38: s9 <= (G2 * (i2,(j2 + 1))) `2 ;
now__::_thesis:_p_in_cell_(G1,i1,j1)
percases ( ( i1 = len G1 & j1 = width G1 ) or ( i1 = len G1 & j1 < width G1 ) or ( i1 < len G1 & j1 = width G1 ) or ( i1 < len G1 & j1 < width G1 ) ) by A14, A6, XXREAL_0:1;
supposeA39: ( i1 = len G1 & j1 = width G1 ) ; ::_thesis: p in cell (G1,i1,j1)
 p in  {  |[r,s]| where r, s is Real : ( (G1 * (i1,j1)) `1 <= r & (G1 * (i1,j1)) `2 <= s )  }  by A4, A35, A37;
hence  p in cell (G1,i1,j1) by A15, A39, GOBRD11:28; ::_thesis: verum
end;
supposeA40: ( i1 = len G1 & j1 < width G1 ) ; ::_thesis: p in cell (G1,i1,j1)
then  ( 1 <= j1 + 1 & j1 + 1 <= width G1 ) by NAT_1:12, NAT_1:13;
then  ( G1 * (i1,(j1 + 1)) in Values G1 & (G1 * (i1,(j1 + 1))) `2 = (G1 * (1,(j1 + 1))) `2 ) by A5, A14, MATRIX_1:47, GOBOARD5:1;
then  (G2 * (i2,(j2 + 1))) `2 <= (G1 * (1,(j1 + 1))) `2 by A1, A4, A5, A13, A8, A10, A33, A40, Th15;
then  s9 <= (G1 * (1,(j1 + 1))) `2 by A38, XXREAL_0:2;
then  p in  {  |[r,s]| where r, s is Real : ( (G1 * (i1,j1)) `1 <= r & (G1 * (i1,j1)) `2 <= s & s <= (G1 * (1,(j1 + 1))) `2 )  }  by A4, A35, A37;
hence  p in cell (G1,i1,j1) by A13, A15, A40, GOBRD11:29; ::_thesis: verum
end;
supposeA41: ( i1 < len G1 & j1 = width G1 ) ; ::_thesis: p in cell (G1,i1,j1)
then  ( 1 <= i1 + 1 & i1 + 1 <= len G1 ) by NAT_1:12, NAT_1:13;
then  (G1 * ((i1 + 1),j1)) `1 = (G1 * ((i1 + 1),1)) `1 by A13, A6, GOBOARD5:2;
then  (G2 * ((i2 + 1),j2)) `1 <= (G1 * ((i1 + 1),1)) `1 by A1, A4, A5, A13, A8, A10, A33, A41, Th13;
then  r9 <= (G1 * ((i1 + 1),1)) `1 by A36, XXREAL_0:2;
then  p in  {  |[r,s]| where r, s is Real : ( (G1 * (i1,j1)) `1 <= r & r <= (G1 * ((i1 + 1),1)) `1 & (G1 * (i1,j1)) `2 <= s )  }  by A4, A35, A37;
hence  p in cell (G1,i1,j1) by A5, A15, A41, GOBRD11:31; ::_thesis: verum
end;
supposeA42: ( i1 < len G1 & j1 < width G1 ) ; ::_thesis: p in cell (G1,i1,j1)
then  ( 1 <= i1 + 1 & i1 + 1 <= len G1 ) by NAT_1:12, NAT_1:13;
then  (G1 * ((i1 + 1),j1)) `1 = (G1 * ((i1 + 1),1)) `1 by A13, A6, GOBOARD5:2;
then  (G2 * ((i2 + 1),j2)) `1 <= (G1 * ((i1 + 1),1)) `1 by A1, A4, A5, A13, A8, A10, A33, A42, Th13;
then A43: r9 <= (G1 * ((i1 + 1),1)) `1 by A36, XXREAL_0:2;
 ( 1 <= j1 + 1 & j1 + 1 <= width G1 ) by A42, NAT_1:12, NAT_1:13;
then  ( G1 * (i1,(j1 + 1)) in Values G1 & (G1 * (i1,(j1 + 1))) `2 = (G1 * (1,(j1 + 1))) `2 ) by A5, A14, MATRIX_1:47, GOBOARD5:1;
then  (G2 * (i2,(j2 + 1))) `2 <= (G1 * (1,(j1 + 1))) `2 by A1, A4, A5, A13, A8, A10, A33, A42, Th15;
then  s9 <= (G1 * (1,(j1 + 1))) `2 by A38, XXREAL_0:2;
then  p in  {  |[r,s]| where r, s is Real : ( (G1 * (i1,1)) `1 <= r & r <= (G1 * ((i1 + 1),1)) `1 & (G1 * (1,j1)) `2 <= s & s <= (G1 * (1,(j1 + 1))) `2 )  }  by A4, A15, A35, A37, A43;
hence  p in cell (G1,i1,j1) by A5, A13, A42, GOBRD11:32; ::_thesis: verum
end;
end;
end;
hence  p in cell (G1,i1,j1) ; ::_thesis: verum
end;
end;
end;

theorem Th18: :: GOBRD13:18
for i1, j1, i2, j2 being Element of NAT
for G1, G2 being Go-board st Values G1 c= Values G2 & [i1,j1] in Indices G1 & [i2,j2] in Indices G2 & G1 * (i1,j1) = G2 * (i2,j2) holds
cell (G2,(i2 -' 1),j2) c= cell (G1,(i1 -' 1),j1)
proof
let i1, j1, i2, j2 be Element of NAT ; ::_thesis: for G1, G2 being Go-board st Values G1 c= Values G2 & [i1,j1] in Indices G1 & [i2,j2] in Indices G2 & G1 * (i1,j1) = G2 * (i2,j2) holds
cell (G2,(i2 -' 1),j2) c= cell (G1,(i1 -' 1),j1)
let G1, G2 be Go-board; ::_thesis: ( Values G1 c= Values G2 & [i1,j1] in Indices G1 & [i2,j2] in Indices G2 & G1 * (i1,j1) = G2 * (i2,j2) implies cell (G2,(i2 -' 1),j2) c= cell (G1,(i1 -' 1),j1) )
assume that
A1: Values G1 c= Values G2 and
A2: [i1,j1] in Indices G1 and
A3: [i2,j2] in Indices G2 and
A4: G1 * (i1,j1) = G2 * (i2,j2) ; ::_thesis: cell (G2,(i2 -' 1),j2) c= cell (G1,(i1 -' 1),j1)
A5: i2 <= len G2 by A3, MATRIX_1:38;
A6: j1 <= width G1 by A2, MATRIX_1:38;
A7: 1 <= j1 by A2, MATRIX_1:38;
A8: j2 <= width G2 by A3, MATRIX_1:38;
A9: 1 <= j2 by A3, MATRIX_1:38;
A10: 1 <= i2 by A3, MATRIX_1:38;
then A11: (G2 * (i2,j2)) `1 = (G2 * (i2,1)) `1 by A5, A9, A8, GOBOARD5:2;
A12: (G2 * (i2,j2)) `2 = (G2 * (1,j2)) `2 by A10, A5, A9, A8, GOBOARD5:1;
let p be set ; :: according to TARSKI:def_3 ::_thesis: ( not p in cell (G2,(i2 -' 1),j2) or p in cell (G1,(i1 -' 1),j1) )
assume A13: p in cell (G2,(i2 -' 1),j2) ; ::_thesis: p in cell (G1,(i1 -' 1),j1)
A14: 1 <= i1 by A2, MATRIX_1:38;
A15: i1 <= len G1 by A2, MATRIX_1:38;
percases ( ( i1 = 1 & i2 = 1 ) or ( i1 = 1 & 1 < i2 ) or ( 1 < i1 & i2 = 1 ) or ( 1 < i1 & 1 < i2 ) ) by A14, A10, XXREAL_0:1;
supposeA16: ( i1 = 1 & i2 = 1 ) ; ::_thesis: p in cell (G1,(i1 -' 1),j1)
then A17: i1 -' 1 = 0 by XREAL_1:232;
A18: i2 -' 1 = 0 by A16, XREAL_1:232;
now__::_thesis:_p_in_cell_(G1,(i1_-'_1),j1)
percases ( j2 = width G2 or j2 < width G2 ) by A8, XXREAL_0:1;
supposeA19: j2 = width G2 ; ::_thesis: p in cell (G1,(i1 -' 1),j1)
then A20: j1 = width G1 by A1, A2, A4, A10, A5, Th12;
 p in  {  |[r,s]| where r, s is Real : ( r <= (G2 * (1,1)) `1 & (G2 * (1,(width G2))) `2 <= s )  }  by A13, A18, A19, GOBRD11:25;
then consider r9, s9 being Real such that
A21: p = |[r9,s9]| and
A22: r9 <= (G2 * (1,1)) `1 and
A23: (G2 * (1,(width G2))) `2 <= s9 ;
 (G2 * (1,1)) `1 = (G2 * (i1,j2)) `1 by A5, A9, A8, A16, GOBOARD5:2;
then  r9 <= (G1 * (1,1)) `1 by A4, A15, A7, A6, A16, A22, GOBOARD5:2;
then  p in  {  |[r,s]| where r, s is Real : ( r <= (G1 * (1,1)) `1 & (G1 * (1,(width G1))) `2 <= s )  }  by A4, A16, A19, A21, A23, A20;
hence  p in cell (G1,(i1 -' 1),j1) by A17, A20, GOBRD11:25; ::_thesis: verum
end;
supposeA24: j2 < width G2 ; ::_thesis: p in cell (G1,(i1 -' 1),j1)
then  p in  {  |[r,s]| where r, s is Real : ( r <= (G2 * (1,1)) `1 & (G2 * (1,j2)) `2 <= s & s <= (G2 * (1,(j2 + 1))) `2 )  }  by A13, A9, A18, GOBRD11:26;
then consider r9, s9 being Real such that
A25: p = |[r9,s9]| and
A26: r9 <= (G2 * (1,1)) `1 and
A27: (G2 * (1,j2)) `2 <= s9 and
A28: s9 <= (G2 * (1,(j2 + 1))) `2 ;
 (G2 * (1,1)) `1 = (G2 * (i1,j2)) `1 by A5, A9, A8, A16, GOBOARD5:2;
then A29: r9 <= (G1 * (1,1)) `1 by A4, A15, A7, A6, A16, A26, GOBOARD5:2;
now__::_thesis:_p_in_cell_(G1,(i1_-'_1),j1)
percases ( j1 = width G1 or j1 < width G1 ) by A6, XXREAL_0:1;
supposeA30: j1 = width G1 ; ::_thesis: p in cell (G1,(i1 -' 1),j1)
then  p in  {  |[r,s]| where r, s is Real : ( r <= (G1 * (1,1)) `1 & (G1 * (1,(width G1))) `2 <= s )  }  by A4, A16, A25, A27, A29;
hence  p in cell (G1,(i1 -' 1),j1) by A17, A30, GOBRD11:25; ::_thesis: verum
end;
supposeA31: j1 < width G1 ; ::_thesis: p in cell (G1,(i1 -' 1),j1)
then  ( 1 <= j1 + 1 & j1 + 1 <= width G1 ) by NAT_1:11, NAT_1:13;
then  G1 * (i1,(j1 + 1)) in Values G1 by A14, A15, MATRIX_1:47;
then  (G2 * (1,(j2 + 1))) `2 <= (G1 * (1,(j1 + 1))) `2 by A1, A4, A15, A7, A5, A9, A16, A24, A31, Th15;
then  s9 <= (G1 * (1,(j1 + 1))) `2 by A28, XXREAL_0:2;
then  p in  {  |[r,s]| where r, s is Real : ( r <= (G1 * (1,1)) `1 & (G1 * (1,j1)) `2 <= s & s <= (G1 * (1,(j1 + 1))) `2 )  }  by A4, A16, A25, A27, A29;
hence  p in cell (G1,(i1 -' 1),j1) by A7, A17, A31, GOBRD11:26; ::_thesis: verum
end;
end;
end;
hence  p in cell (G1,(i1 -' 1),j1) ; ::_thesis: verum
end;
end;
end;
hence  p in cell (G1,(i1 -' 1),j1) ; ::_thesis: verum
end;
supposethat A32: i1 = 1 and
A33: 1 < i2 ; ::_thesis: p in cell (G1,(i1 -' 1),j1)
A34: i1 -' 1 = 0 by A32, XREAL_1:232;
A35: 1 <= i2 -' 1 by A33, NAT_D:49;
then  i2 -' 1 < i2 by NAT_D:51;
then A36: i2 -' 1 < len G2 by A5, XXREAL_0:2;
A37: (i2 -' 1) + 1 = i2 by A33, XREAL_1:235;
now__::_thesis:_p_in_cell_(G1,(i1_-'_1),j1)
percases ( j2 = width G2 or j2 < width G2 ) by A8, XXREAL_0:1;
supposeA38: j2 = width G2 ; ::_thesis: p in cell (G1,(i1 -' 1),j1)
then  p in  {  |[r,s]| where r, s is Real : ( (G2 * ((i2 -' 1),1)) `1 <= r & r <= (G2 * (i2,1)) `1 & (G2 * (1,j2)) `2 <= s )  }  by A13, A35, A36, A37, GOBRD11:31;
then consider r9, s9 being Real such that
A39: p = |[r9,s9]| and
 (G2 * ((i2 -' 1),1)) `1 <= r9 and
A40: ( r9 <= (G2 * (i2,1)) `1 & (G2 * (1,j2)) `2 <= s9 ) ;
 ( r9 <= (G1 * (1,1)) `1 & (G1 * (1,j1)) `2 <= s9 ) by A4, A15, A7, A6, A11, A12, A32, A40, GOBOARD5:2;
then A41: p in  {  |[r,s]| where r, s is Real : ( r <= (G1 * (1,1)) `1 & (G1 * (1,j1)) `2 <= s )  }  by A39;
 j1 = width G1 by A1, A2, A4, A10, A5, A38, Th12;
hence  p in cell (G1,(i1 -' 1),j1) by A34, A41, GOBRD11:25; ::_thesis: verum
end;
supposeA42: j2 < width G2 ; ::_thesis: p in cell (G1,(i1 -' 1),j1)
then  p in  {  |[r,s]| where r, s is Real : ( (G2 * ((i2 -' 1),1)) `1 <= r & r <= (G2 * (i2,1)) `1 & (G2 * (1,j2)) `2 <= s & s <= (G2 * (1,(j2 + 1))) `2 )  }  by A13, A9, A35, A36, A37, GOBRD11:32;
then consider r9, s9 being Real such that
A43: p = |[r9,s9]| and
 (G2 * ((i2 -' 1),1)) `1 <= r9 and
A44: ( r9 <= (G2 * (i2,1)) `1 & (G2 * (1,j2)) `2 <= s9 ) and
A45: s9 <= (G2 * (1,(j2 + 1))) `2 ;
A46: ( r9 <= (G1 * (1,1)) `1 & (G1 * (1,j1)) `2 <= s9 ) by A4, A15, A7, A6, A11, A12, A32, A44, GOBOARD5:2;
now__::_thesis:_p_in_cell_(G1,(i1_-'_1),j1)
percases ( j1 = width G1 or j1 < width G1 ) by A6, XXREAL_0:1;
supposeA47: j1 = width G1 ; ::_thesis: p in cell (G1,(i1 -' 1),j1)
then  p in  {  |[r,s]| where r, s is Real : ( r <= (G1 * (1,1)) `1 & (G1 * (1,(width G1))) `2 <= s )  }  by A43, A46;
hence  p in cell (G1,(i1 -' 1),j1) by A34, A47, GOBRD11:25; ::_thesis: verum
end;
supposeA48: j1 < width G1 ; ::_thesis: p in cell (G1,(i1 -' 1),j1)
 ( 1 <= j2 + 1 & j2 + 1 <= width G2 ) by A42, NAT_1:12, NAT_1:13;
then A49: (G2 * (i2,(j2 + 1))) `2 = (G2 * (1,(j2 + 1))) `2 by A10, A5, GOBOARD5:1;
 ( 1 <= j1 + 1 & j1 + 1 <= width G1 ) by A48, NAT_1:12, NAT_1:13;
then  ( G1 * (i1,(j1 + 1)) in Values G1 & (G1 * (i1,(j1 + 1))) `2 = (G1 * (1,(j1 + 1))) `2 ) by A14, A15, MATRIX_1:47, GOBOARD5:1;
then  (G2 * (1,(j2 + 1))) `2 <= (G1 * (1,(j1 + 1))) `2 by A1, A4, A14, A15, A7, A10, A5, A9, A42, A48, A49, Th15;
then  s9 <= (G1 * (1,(j1 + 1))) `2 by A45, XXREAL_0:2;
then  p in  {  |[r,s]| where r, s is Real : ( r <= (G1 * (1,1)) `1 & (G1 * (1,j1)) `2 <= s & s <= (G1 * (1,(j1 + 1))) `2 )  }  by A43, A46;
hence  p in cell (G1,(i1 -' 1),j1) by A7, A34, A48, GOBRD11:26; ::_thesis: verum
end;
end;
end;
hence  p in cell (G1,(i1 -' 1),j1) ; ::_thesis: verum
end;
end;
end;
hence  p in cell (G1,(i1 -' 1),j1) ; ::_thesis: verum
end;
suppose ( 1 < i1 & i2 = 1 ) ; ::_thesis: p in cell (G1,(i1 -' 1),j1)
hence  p in cell (G1,(i1 -' 1),j1) by A1, A2, A4, A9, A8, Th9; ::_thesis: verum
end;
supposeA50: ( 1 < i1 & 1 < i2 ) ; ::_thesis: p in cell (G1,(i1 -' 1),j1)
then A51: 1 <= i2 -' 1 by NAT_D:49;
then A52: (i2 -' 1) + 1 = i2 by NAT_D:43;
 i2 -' 1 < i2 by A51, NAT_D:51;
then A53: i2 -' 1 < len G2 by A5, XXREAL_0:2;
then A54: (G2 * ((i2 -' 1),1)) `1 = (G2 * ((i2 -' 1),j2)) `1 by A9, A8, A51, GOBOARD5:2;
A55: 1 <= i1 -' 1 by A50, NAT_D:49;
then A56: (i1 -' 1) + 1 = i1 by NAT_D:43;
 i1 -' 1 < i1 by A55, NAT_D:51;
then A57: i1 -' 1 < len G1 by A15, XXREAL_0:2;
then  ( G1 * ((i1 -' 1),j1) in Values G1 & (G1 * ((i1 -' 1),1)) `1 = (G1 * ((i1 -' 1),j1)) `1 ) by A7, A6, A55, MATRIX_1:47, GOBOARD5:2;
then A58: (G1 * ((i1 -' 1),1)) `1 <= (G2 * ((i2 -' 1),1)) `1 by A1, A4, A15, A7, A6, A5, A9, A8, A50, A54, Th14;
now__::_thesis:_p_in_cell_(G1,(i1_-'_1),j1)
percases ( j2 = width G2 or j2 < width G2 ) by A8, XXREAL_0:1;
supposeA59: j2 = width G2 ; ::_thesis: p in cell (G1,(i1 -' 1),j1)
then  p in  {  |[r,s]| where r, s is Real : ( (G2 * ((i2 -' 1),1)) `1 <= r & r <= (G2 * (i2,1)) `1 & (G2 * (1,j2)) `2 <= s )  }  by A13, A51, A53, A52, GOBRD11:31;
then consider r9, s9 being Real such that
A60: p = |[r9,s9]| and
A61: ( (G2 * ((i2 -' 1),1)) `1 <= r9 & r9 <= (G2 * (i2,1)) `1 ) and
A62: (G2 * (1,j2)) `2 <= s9 ;
A63: (G1 * (1,j1)) `2 <= s9 by A4, A14, A15, A7, A6, A12, A62, GOBOARD5:1;
 ( (G1 * ((i1 -' 1),1)) `1 <= r9 & r9 <= (G1 * (i1,1)) `1 ) by A4, A14, A15, A7, A6, A11, A58, A61, GOBOARD5:2, XXREAL_0:2;
then A64: p in  {  |[r,s]| where r, s is Real : ( (G1 * ((i1 -' 1),1)) `1 <= r & r <= (G1 * (i1,1)) `1 & (G1 * (1,j1)) `2 <= s )  }  by A60, A63;
 j1 = width G1 by A1, A2, A4, A10, A5, A59, Th12;
hence  p in cell (G1,(i1 -' 1),j1) by A55, A57, A56, A64, GOBRD11:31; ::_thesis: verum
end;
supposeA65: j2 < width G2 ; ::_thesis: p in cell (G1,(i1 -' 1),j1)
then  p in  {  |[r,s]| where r, s is Real : ( (G2 * ((i2 -' 1),1)) `1 <= r & r <= (G2 * (i2,1)) `1 & (G2 * (1,j2)) `2 <= s & s <= (G2 * (1,(j2 + 1))) `2 )  }  by A13, A9, A51, A53, A52, GOBRD11:32;
then consider r9, s9 being Real such that
A66: p = |[r9,s9]| and
A67: ( (G2 * ((i2 -' 1),1)) `1 <= r9 & r9 <= (G2 * (i2,1)) `1 ) and
A68: (G2 * (1,j2)) `2 <= s9 and
A69: s9 <= (G2 * (1,(j2 + 1))) `2 ;
A70: (G1 * (1,j1)) `2 <= s9 by A4, A14, A15, A7, A6, A12, A68, GOBOARD5:1;
A71: ( (G1 * ((i1 -' 1),1)) `1 <= r9 & r9 <= (G1 * (i1,1)) `1 ) by A4, A14, A15, A7, A6, A11, A58, A67, GOBOARD5:2, XXREAL_0:2;
now__::_thesis:_p_in_cell_(G1,(i1_-'_1),j1)
percases ( j1 = width G1 or j1 < width G1 ) by A6, XXREAL_0:1;
supposeA72: j1 = width G1 ; ::_thesis: p in cell (G1,(i1 -' 1),j1)
 p in  {  |[r,s]| where r, s is Real : ( (G1 * ((i1 -' 1),1)) `1 <= r & r <= (G1 * (i1,1)) `1 & (G1 * (1,j1)) `2 <= s )  }  by A66, A71, A70;
hence  p in cell (G1,(i1 -' 1),j1) by A55, A57, A56, A72, GOBRD11:31; ::_thesis: verum
end;
supposeA73: j1 < width G1 ; ::_thesis: p in cell (G1,(i1 -' 1),j1)
 ( 1 <= j2 + 1 & j2 + 1 <= width G2 ) by A65, NAT_1:12, NAT_1:13;
then A74: (G2 * (i2,(j2 + 1))) `2 = (G2 * (1,(j2 + 1))) `2 by A10, A5, GOBOARD5:1;
 ( 1 <= j1 + 1 & j1 + 1 <= width G1 ) by A73, NAT_1:12, NAT_1:13;
then  ( G1 * (i1,(j1 + 1)) in Values G1 & (G1 * (i1,(j1 + 1))) `2 = (G1 * (1,(j1 + 1))) `2 ) by A14, A15, MATRIX_1:47, GOBOARD5:1;
then  (G2 * (1,(j2 + 1))) `2 <= (G1 * (1,(j1 + 1))) `2 by A1, A4, A14, A15, A7, A10, A5, A9, A65, A73, A74, Th15;
then  s9 <= (G1 * (1,(j1 + 1))) `2 by A69, XXREAL_0:2;
then  p in  {  |[r,s]| where r, s is Real : ( (G1 * ((i1 -' 1),1)) `1 <= r & r <= (G1 * (i1,1)) `1 & (G1 * (1,j1)) `2 <= s & s <= (G1 * (1,(j1 + 1))) `2 )  }  by A66, A71, A70;
hence  p in cell (G1,(i1 -' 1),j1) by A7, A55, A57, A56, A73, GOBRD11:32; ::_thesis: verum
end;
end;
end;
hence  p in cell (G1,(i1 -' 1),j1) ; ::_thesis: verum
end;
end;
end;
hence  p in cell (G1,(i1 -' 1),j1) ; ::_thesis: verum
end;
end;
end;

theorem Th19: :: GOBRD13:19
for i1, j1, i2, j2 being Element of NAT
for G1, G2 being Go-board st Values G1 c= Values G2 & [i1,j1] in Indices G1 & [i2,j2] in Indices G2 & G1 * (i1,j1) = G2 * (i2,j2) holds
cell (G2,i2,(j2 -' 1)) c= cell (G1,i1,(j1 -' 1))
proof
let i1, j1, i2, j2 be Element of NAT ; ::_thesis: for G1, G2 being Go-board st Values G1 c= Values G2 & [i1,j1] in Indices G1 & [i2,j2] in Indices G2 & G1 * (i1,j1) = G2 * (i2,j2) holds
cell (G2,i2,(j2 -' 1)) c= cell (G1,i1,(j1 -' 1))
let G1, G2 be Go-board; ::_thesis: ( Values G1 c= Values G2 & [i1,j1] in Indices G1 & [i2,j2] in Indices G2 & G1 * (i1,j1) = G2 * (i2,j2) implies cell (G2,i2,(j2 -' 1)) c= cell (G1,i1,(j1 -' 1)) )
assume that
A1: Values G1 c= Values G2 and
A2: [i1,j1] in Indices G1 and
A3: [i2,j2] in Indices G2 and
A4: G1 * (i1,j1) = G2 * (i2,j2) ; ::_thesis: cell (G2,i2,(j2 -' 1)) c= cell (G1,i1,(j1 -' 1))
A5: 1 <= i1 by A2, MATRIX_1:38;
A6: 1 <= j2 by A3, MATRIX_1:38;
A7: 1 <= i2 by A3, MATRIX_1:38;
A8: j1 <= width G1 by A2, MATRIX_1:38;
A9: j2 <= width G2 by A3, MATRIX_1:38;
A10: i2 <= len G2 by A3, MATRIX_1:38;
then A11: (G2 * (i2,j2)) `1 = (G2 * (i2,1)) `1 by A7, A6, A9, GOBOARD5:2;
A12: i1 <= len G1 by A2, MATRIX_1:38;
A13: 1 <= j1 by A2, MATRIX_1:38;
then A14: (G1 * (i1,j1)) `2 = (G1 * (1,j1)) `2 by A5, A12, A8, GOBOARD5:1;
let p be set ; :: according to TARSKI:def_3 ::_thesis: ( not p in cell (G2,i2,(j2 -' 1)) or p in cell (G1,i1,(j1 -' 1)) )
assume A15: p in cell (G2,i2,(j2 -' 1)) ; ::_thesis: p in cell (G1,i1,(j1 -' 1))
A16: (G2 * (i2,j2)) `2 = (G2 * (1,j2)) `2 by A7, A10, A6, A9, GOBOARD5:1;
percases ( ( j1 = 1 & j2 = 1 ) or ( j1 = 1 & 1 < j2 ) or ( 1 < j1 & j2 = 1 ) or ( 1 < j1 & 1 < j2 ) ) by A13, A6, XXREAL_0:1;
supposeA17: ( j1 = 1 & j2 = 1 ) ; ::_thesis: p in cell (G1,i1,(j1 -' 1))
then A18: j1 -' 1 = 0 by XREAL_1:232;
A19: j2 -' 1 = 0 by A17, XREAL_1:232;
now__::_thesis:_p_in_cell_(G1,i1,(j1_-'_1))
percases ( i2 = len G2 or i2 < len G2 ) by A10, XXREAL_0:1;
supposeA20: i2 = len G2 ; ::_thesis: p in cell (G1,i1,(j1 -' 1))
then  p in  {  |[r,s]| where r, s is Real : ( (G2 * ((len G2),1)) `1 <= r & s <= (G2 * (1,1)) `2 )  }  by A15, A19, GOBRD11:27;
then consider r9, s9 being Real such that
A21: ( p = |[r9,s9]| & (G2 * ((len G2),1)) `1 <= r9 ) and
A22: s9 <= (G2 * (1,1)) `2 ;
A23: i1 = len G1 by A1, A2, A4, A6, A9, A20, Th10;
 (G2 * (1,1)) `2 = (G2 * (i2,j2)) `2 by A7, A10, A9, A17, GOBOARD5:1;
then  s9 <= (G1 * (1,1)) `2 by A4, A5, A12, A8, A17, A22, GOBOARD5:1;
then  p in  {  |[r,s]| where r, s is Real : ( (G1 * ((len G1),1)) `1 <= r & s <= (G1 * (1,1)) `2 )  }  by A4, A17, A20, A21, A23;
hence  p in cell (G1,i1,(j1 -' 1)) by A18, A23, GOBRD11:27; ::_thesis: verum
end;
supposeA24: i2 < len G2 ; ::_thesis: p in cell (G1,i1,(j1 -' 1))
then  p in  {  |[r,s]| where r, s is Real : ( (G2 * (i2,1)) `1 <= r & r <= (G2 * ((i2 + 1),1)) `1 & s <= (G2 * (1,1)) `2 )  }  by A15, A7, A19, GOBRD11:30;
then consider r9, s9 being Real such that
A25: ( p = |[r9,s9]| & (G2 * (i2,1)) `1 <= r9 ) and
A26: r9 <= (G2 * ((i2 + 1),1)) `1 and
A27: s9 <= (G2 * (1,1)) `2 ;
 (G2 * (1,1)) `2 = (G2 * (i2,j1)) `2 by A7, A10, A9, A17, GOBOARD5:1;
then A28: s9 <= (G1 * (1,1)) `2 by A4, A5, A12, A8, A17, A27, GOBOARD5:1;
now__::_thesis:_p_in_cell_(G1,i1,(j1_-'_1))
percases ( i1 = len G1 or i1 < len G1 ) by A12, XXREAL_0:1;
supposeA29: i1 = len G1 ; ::_thesis: p in cell (G1,i1,(j1 -' 1))
then  p in  {  |[r,s]| where r, s is Real : ( (G1 * ((len G1),1)) `1 <= r & s <= (G1 * (1,1)) `2 )  }  by A4, A17, A25, A28;
hence  p in cell (G1,i1,(j1 -' 1)) by A18, A29, GOBRD11:27; ::_thesis: verum
end;
supposeA30: i1 < len G1 ; ::_thesis: p in cell (G1,i1,(j1 -' 1))
then  (G2 * ((i2 + 1),1)) `1 <= (G1 * ((i1 + 1),1)) `1 by A1, A4, A5, A8, A7, A9, A17, A24, Th13;
then  r9 <= (G1 * ((i1 + 1),1)) `1 by A26, XXREAL_0:2;
then  p in  {  |[r,s]| where r, s is Real : ( (G1 * (i1,1)) `1 <= r & r <= (G1 * ((i1 + 1),1)) `1 & s <= (G1 * (1,1)) `2 )  }  by A4, A17, A25, A28;
hence  p in cell (G1,i1,(j1 -' 1)) by A5, A18, A30, GOBRD11:30; ::_thesis: verum
end;
end;
end;
hence  p in cell (G1,i1,(j1 -' 1)) ; ::_thesis: verum
end;
end;
end;
hence  p in cell (G1,i1,(j1 -' 1)) ; ::_thesis: verum
end;
supposethat A31: j1 = 1 and
A32: 1 < j2 ; ::_thesis: p in cell (G1,i1,(j1 -' 1))
A33: j1 -' 1 = 0 by A31, XREAL_1:232;
A34: 1 <= j2 -' 1 by A32, NAT_D:49;
then  j2 -' 1 < j2 by NAT_D:51;
then A35: j2 -' 1 < width G2 by A9, XXREAL_0:2;
A36: (j2 -' 1) + 1 = j2 by A32, XREAL_1:235;
now__::_thesis:_p_in_cell_(G1,i1,(j1_-'_1))
percases ( i2 = len G2 or i2 < len G2 ) by A10, XXREAL_0:1;
supposeA37: i2 = len G2 ; ::_thesis: p in cell (G1,i1,(j1 -' 1))
then  p in  {  |[r,s]| where r, s is Real : ( (G2 * (i2,1)) `1 <= r & (G2 * (1,(j2 -' 1))) `2 <= s & s <= (G2 * (1,j2)) `2 )  }  by A15, A34, A35, A36, GOBRD11:29;
then  ex r9, s9 being Real st
( p = |[r9,s9]| & (G2 * (i2,1)) `1 <= r9 & (G2 * (1,(j2 -' 1))) `2 <= s9 & s9 <= (G2 * (1,j2)) `2 ) ;
then A38: p in  {  |[r,s]| where r, s is Real : ( (G1 * (i1,1)) `1 <= r & s <= (G1 * (1,1)) `2 )  }  by A4, A14, A11, A16, A31;
 i1 = len G1 by A1, A2, A4, A6, A9, A37, Th10;
hence  p in cell (G1,i1,(j1 -' 1)) by A33, A38, GOBRD11:27; ::_thesis: verum
end;
supposeA39: i2 < len G2 ; ::_thesis: p in cell (G1,i1,(j1 -' 1))
then  p in  {  |[r,s]| where r, s is Real : ( (G2 * (i2,1)) `1 <= r & r <= (G2 * ((i2 + 1),1)) `1 & (G2 * (1,(j2 -' 1))) `2 <= s & s <= (G2 * (1,j2)) `2 )  }  by A15, A7, A34, A35, A36, GOBRD11:32;
then consider r9, s9 being Real such that
A40: p = |[r9,s9]| and
A41: (G2 * (i2,1)) `1 <= r9 and
A42: r9 <= (G2 * ((i2 + 1),1)) `1 and
 (G2 * (1,(j2 -' 1))) `2 <= s9 and
A43: s9 <= (G2 * (1,j2)) `2 ;
A44: ( s9 <= (G1 * (1,1)) `2 & (G1 * (i1,1)) `1 <= r9 ) by A4, A7, A10, A6, A9, A14, A31, A41, A43, GOBOARD5:1, GOBOARD5:2;
now__::_thesis:_p_in_cell_(G1,i1,(j1_-'_1))
percases ( i1 = len G1 or i1 < len G1 ) by A12, XXREAL_0:1;
supposeA45: i1 = len G1 ; ::_thesis: p in cell (G1,i1,(j1 -' 1))
then  p in  {  |[r,s]| where r, s is Real : ( (G1 * ((len G1),1)) `1 <= r & s <= (G1 * (1,1)) `2 )  }  by A40, A44;
hence  p in cell (G1,i1,(j1 -' 1)) by A33, A45, GOBRD11:27; ::_thesis: verum
end;
supposeA46: i1 < len G1 ; ::_thesis: p in cell (G1,i1,(j1 -' 1))
 ( 1 <= i2 + 1 & i2 + 1 <= len G2 ) by A39, NAT_1:12, NAT_1:13;
then A47: (G2 * ((i2 + 1),j2)) `1 = (G2 * ((i2 + 1),1)) `1 by A6, A9, GOBOARD5:2;
 ( 1 <= i1 + 1 & i1 + 1 <= len G1 ) by A46, NAT_1:12, NAT_1:13;
then  (G1 * ((i1 + 1),j1)) `1 = (G1 * ((i1 + 1),1)) `1 by A13, A8, GOBOARD5:2;
then  (G2 * ((i2 + 1),1)) `1 <= (G1 * ((i1 + 1),1)) `1 by A1, A4, A5, A13, A8, A7, A6, A9, A39, A46, A47, Th13;
then  r9 <= (G1 * ((i1 + 1),1)) `1 by A42, XXREAL_0:2;
then  p in  {  |[r,s]| where r, s is Real : ( (G1 * (i1,1)) `1 <= r & r <= (G1 * ((i1 + 1),1)) `1 & s <= (G1 * (1,1)) `2 )  }  by A40, A44;
hence  p in cell (G1,i1,(j1 -' 1)) by A5, A33, A46, GOBRD11:30; ::_thesis: verum
end;
end;
end;
hence  p in cell (G1,i1,(j1 -' 1)) ; ::_thesis: verum
end;
end;
end;
hence  p in cell (G1,i1,(j1 -' 1)) ; ::_thesis: verum
end;
suppose ( 1 < j1 & j2 = 1 ) ; ::_thesis: p in cell (G1,i1,(j1 -' 1))
hence  p in cell (G1,i1,(j1 -' 1)) by A1, A2, A4, A7, A10, Th11; ::_thesis: verum
end;
supposeA48: ( 1 < j1 & 1 < j2 ) ; ::_thesis: p in cell (G1,i1,(j1 -' 1))
then A49: 1 <= j2 -' 1 by NAT_D:49;
then A50: (j2 -' 1) + 1 = j2 by NAT_D:43;
 j2 -' 1 < j2 by A49, NAT_D:51;
then A51: j2 -' 1 < width G2 by A9, XXREAL_0:2;
then A52: (G2 * (1,(j2 -' 1))) `2 = (G2 * (i2,(j2 -' 1))) `2 by A7, A10, A49, GOBOARD5:1;
A53: 1 <= j1 -' 1 by A48, NAT_D:49;
then A54: (j1 -' 1) + 1 = j1 by NAT_D:43;
 j1 -' 1 < j1 by A53, NAT_D:51;
then A55: j1 -' 1 < width G1 by A8, XXREAL_0:2;
then  (G1 * (1,(j1 -' 1))) `2 = (G1 * (i1,(j1 -' 1))) `2 by A5, A12, A53, GOBOARD5:1;
then A56: (G1 * (1,(j1 -' 1))) `2 <= (G2 * (1,(j2 -' 1))) `2 by A1, A4, A5, A12, A8, A7, A10, A9, A48, A52, Th16;
now__::_thesis:_p_in_cell_(G1,i1,(j1_-'_1))
percases ( i2 = len G2 or i2 < len G2 ) by A10, XXREAL_0:1;
supposeA57: i2 = len G2 ; ::_thesis: p in cell (G1,i1,(j1 -' 1))
then  p in  {  |[r,s]| where r, s is Real : ( (G2 * (i2,1)) `1 <= r & (G2 * (1,(j2 -' 1))) `2 <= s & s <= (G2 * (1,j2)) `2 )  }  by A15, A49, A51, A50, GOBRD11:29;
then consider r9, s9 being Real such that
A58: p = |[r9,s9]| and
A59: (G2 * (i2,1)) `1 <= r9 and
A60: ( (G2 * (1,(j2 -' 1))) `2 <= s9 & s9 <= (G2 * (1,j2)) `2 ) ;
A61: (G1 * (i1,1)) `1 <= r9 by A4, A5, A12, A13, A8, A11, A59, GOBOARD5:2;
 ( (G1 * (1,(j1 -' 1))) `2 <= s9 & s9 <= (G1 * (1,j1)) `2 ) by A4, A5, A12, A13, A8, A16, A56, A60, GOBOARD5:1, XXREAL_0:2;
then A62: p in  {  |[r,s]| where r, s is Real : ( (G1 * (i1,1)) `1 <= r & (G1 * (1,(j1 -' 1))) `2 <= s & s <= (G1 * (1,j1)) `2 )  }  by A58, A61;
 i1 = len G1 by A1, A2, A4, A6, A9, A57, Th10;
hence  p in cell (G1,i1,(j1 -' 1)) by A53, A55, A54, A62, GOBRD11:29; ::_thesis: verum
end;
supposeA63: i2 < len G2 ; ::_thesis: p in cell (G1,i1,(j1 -' 1))
then  p in  {  |[r,s]| where r, s is Real : ( (G2 * (i2,1)) `1 <= r & r <= (G2 * ((i2 + 1),1)) `1 & (G2 * (1,(j2 -' 1))) `2 <= s & s <= (G2 * (1,j2)) `2 )  }  by A15, A7, A49, A51, A50, GOBRD11:32;
then consider r9, s9 being Real such that
A64: p = |[r9,s9]| and
A65: (G2 * (i2,1)) `1 <= r9 and
A66: r9 <= (G2 * ((i2 + 1),1)) `1 and
A67: ( (G2 * (1,(j2 -' 1))) `2 <= s9 & s9 <= (G2 * (1,j2)) `2 ) ;
A68: (G1 * (i1,1)) `1 <= r9 by A4, A5, A12, A13, A8, A11, A65, GOBOARD5:2;
A69: ( (G1 * (1,(j1 -' 1))) `2 <= s9 & s9 <= (G1 * (1,j1)) `2 ) by A4, A5, A12, A13, A8, A16, A56, A67, GOBOARD5:1, XXREAL_0:2;
now__::_thesis:_p_in_cell_(G1,i1,(j1_-'_1))
percases ( i1 = len G1 or i1 < len G1 ) by A12, XXREAL_0:1;
supposeA70: i1 = len G1 ; ::_thesis: p in cell (G1,i1,(j1 -' 1))
 p in  {  |[r,s]| where r, s is Real : ( (G1 * (i1,1)) `1 <= r & (G1 * (1,(j1 -' 1))) `2 <= s & s <= (G1 * (1,j1)) `2 )  }  by A64, A69, A68;
hence  p in cell (G1,i1,(j1 -' 1)) by A53, A55, A54, A70, GOBRD11:29; ::_thesis: verum
end;
supposeA71: i1 < len G1 ; ::_thesis: p in cell (G1,i1,(j1 -' 1))
 ( 1 <= i2 + 1 & i2 + 1 <= len G2 ) by A63, NAT_1:12, NAT_1:13;
then A72: (G2 * ((i2 + 1),j2)) `1 = (G2 * ((i2 + 1),1)) `1 by A6, A9, GOBOARD5:2;
 ( 1 <= i1 + 1 & i1 + 1 <= len G1 ) by A71, NAT_1:12, NAT_1:13;
then  (G1 * ((i1 + 1),j1)) `1 = (G1 * ((i1 + 1),1)) `1 by A13, A8, GOBOARD5:2;
then  (G2 * ((i2 + 1),1)) `1 <= (G1 * ((i1 + 1),1)) `1 by A1, A4, A5, A13, A8, A7, A6, A9, A63, A71, A72, Th13;
then  r9 <= (G1 * ((i1 + 1),1)) `1 by A66, XXREAL_0:2;
then  p in  {  |[r,s]| where r, s is Real : ( (G1 * (i1,1)) `1 <= r & r <= (G1 * ((i1 + 1),1)) `1 & (G1 * (1,(j1 -' 1))) `2 <= s & s <= (G1 * (1,j1)) `2 )  }  by A64, A69, A68;
hence  p in cell (G1,i1,(j1 -' 1)) by A5, A53, A55, A54, A71, GOBRD11:32; ::_thesis: verum
end;
end;
end;
hence  p in cell (G1,i1,(j1 -' 1)) ; ::_thesis: verum
end;
end;
end;
hence  p in cell (G1,i1,(j1 -' 1)) ; ::_thesis: verum
end;
end;
end;

Lm1:  for i, j being Element of NAT
for f being non empty FinSequence of (TOP-REAL 2) st 1 <= i & i <= len (GoB f) & 1 <= j & j <= width (GoB f) holds
ex k being Element of NAT st
( k in dom f & (f /. k) `1 = ((GoB f) * (i,j)) `1 )
proof
let i, j be Element of NAT ; ::_thesis: for f being non empty FinSequence of (TOP-REAL 2) st 1 <= i & i <= len (GoB f) & 1 <= j & j <= width (GoB f) holds
ex k being Element of NAT st
( k in dom f & (f /. k) `1 = ((GoB f) * (i,j)) `1 )
let f be non empty FinSequence of (TOP-REAL 2); ::_thesis: ( 1 <= i & i <= len (GoB f) & 1 <= j & j <= width (GoB f) implies ex k being Element of NAT st
( k in dom f & (f /. k) `1 = ((GoB f) * (i,j)) `1 ) )
assume that
A1: ( 1 <= i & i <= len (GoB f) ) and
A2: ( 1 <= j & j <= width (GoB f) ) ; ::_thesis: ex k being Element of NAT st
( k in dom f & (f /. k) `1 = ((GoB f) * (i,j)) `1 )
A3: GoB f = GoB ((Incr (X_axis f)),(Incr (Y_axis f))) by GOBOARD2:def_2;
then  len (Incr (X_axis f)) = len (GoB f) by GOBOARD2:def_1;
then  i in dom (Incr (X_axis f)) by A1, FINSEQ_3:25;
then  (Incr (X_axis f)) . i in rng (Incr (X_axis f)) by FUNCT_1:def_3;
then  (Incr (X_axis f)) . i in rng (X_axis f) by SEQ_4:def_21;
then consider k being Nat such that
A4: k in dom (X_axis f) and
A5: (X_axis f) . k = (Incr (X_axis f)) . i by FINSEQ_2:10;
 [i,j] in Indices (GoB f) by A1, A2, MATRIX_1:36;
then A6: (GoB f) * (i,j) = |[((Incr (X_axis f)) . i),((Incr (Y_axis f)) . j)]| by A3, GOBOARD2:def_1;
reconsider k = k as Element of NAT by ORDINAL1:def_12;
take  k ; ::_thesis: ( k in dom f & (f /. k) `1 = ((GoB f) * (i,j)) `1 )
 len (X_axis f) = len f by GOBOARD1:def_1;
hence  k in dom f by A4, FINSEQ_3:29; ::_thesis: (f /. k) `1 = ((GoB f) * (i,j)) `1
thus (f /. k) `1 = (Incr (X_axis f)) . i by A4, A5, GOBOARD1:def_1
.= ((GoB f) * (i,j)) `1 by A6, EUCLID:52 ; ::_thesis: verum
end;

Lm2:  for i, j being Element of NAT
for f being non empty FinSequence of (TOP-REAL 2) st 1 <= i & i <= len (GoB f) & 1 <= j & j <= width (GoB f) holds
ex k being Element of NAT st
( k in dom f & (f /. k) `2 = ((GoB f) * (i,j)) `2 )
proof
let i, j be Element of NAT ; ::_thesis: for f being non empty FinSequence of (TOP-REAL 2) st 1 <= i & i <= len (GoB f) & 1 <= j & j <= width (GoB f) holds
ex k being Element of NAT st
( k in dom f & (f /. k) `2 = ((GoB f) * (i,j)) `2 )
let f be non empty FinSequence of (TOP-REAL 2); ::_thesis: ( 1 <= i & i <= len (GoB f) & 1 <= j & j <= width (GoB f) implies ex k being Element of NAT st
( k in dom f & (f /. k) `2 = ((GoB f) * (i,j)) `2 ) )
assume that
A1: ( 1 <= i & i <= len (GoB f) ) and
A2: ( 1 <= j & j <= width (GoB f) ) ; ::_thesis: ex k being Element of NAT st
( k in dom f & (f /. k) `2 = ((GoB f) * (i,j)) `2 )
A3: GoB f = GoB ((Incr (X_axis f)),(Incr (Y_axis f))) by GOBOARD2:def_2;
then  len (Incr (Y_axis f)) = width (GoB f) by GOBOARD2:def_1;
then  j in dom (Incr (Y_axis f)) by A2, FINSEQ_3:25;
then  (Incr (Y_axis f)) . j in rng (Incr (Y_axis f)) by FUNCT_1:def_3;
then  (Incr (Y_axis f)) . j in rng (Y_axis f) by SEQ_4:def_21;
then consider k being Nat such that
A4: k in dom (Y_axis f) and
A5: (Y_axis f) . k = (Incr (Y_axis f)) . j by FINSEQ_2:10;
 [i,j] in Indices (GoB f) by A1, A2, MATRIX_1:36;
then A6: (GoB f) * (i,j) = |[((Incr (X_axis f)) . i),((Incr (Y_axis f)) . j)]| by A3, GOBOARD2:def_1;
reconsider k = k as Element of NAT by ORDINAL1:def_12;
take  k ; ::_thesis: ( k in dom f & (f /. k) `2 = ((GoB f) * (i,j)) `2 )
 len (Y_axis f) = len f by GOBOARD1:def_2;
hence  k in dom f by A4, FINSEQ_3:29; ::_thesis: (f /. k) `2 = ((GoB f) * (i,j)) `2
thus (f /. k) `2 = (Incr (Y_axis f)) . j by A4, A5, GOBOARD1:def_2
.= ((GoB f) * (i,j)) `2 by A6, EUCLID:52 ; ::_thesis: verum
end;

theorem Th20: :: GOBRD13:20
for G being Go-board
for f being standard special_circular_sequence st f is_sequence_on G holds
Values (GoB f) c= Values G
proof
let G be Go-board; ::_thesis: for f being standard special_circular_sequence st f is_sequence_on G holds
Values (GoB f) c= Values G
let f be standard special_circular_sequence; ::_thesis: ( f is_sequence_on G implies Values (GoB f) c= Values G )
assume A1: f is_sequence_on G ; ::_thesis: Values (GoB f) c= Values G
let p be set ; :: according to TARSKI:def_3 ::_thesis: ( not p in Values (GoB f) or p in Values G )
set F = GoB f;
assume  p in Values (GoB f) ; ::_thesis: p in Values G
then  p in  {  ((GoB f) * (i,j)) where i, j is Element of NAT : [i,j] in Indices (GoB f)  }  by MATRIX_1:45;
then consider i, j being Element of NAT  such that
A2: p = (GoB f) * (i,j) and
A3: [i,j] in Indices (GoB f) ;
reconsider p = p as Point of (TOP-REAL 2) by A2;
A4: ( 1 <= j & j <= width (GoB f) ) by A3, MATRIX_1:38;
A5: ( 1 <= i & i <= len (GoB f) ) by A3, MATRIX_1:38;
then consider k1 being Element of NAT  such that
A6: k1 in dom f and
A7: p `1 = (f /. k1) `1 by A2, A4, Lm1;
consider k2 being Element of NAT  such that
A8: k2 in dom f and
A9: p `2 = (f /. k2) `2 by A2, A5, A4, Lm2;
consider i2, j2 being Element of NAT  such that
A10: [i2,j2] in Indices G and
A11: f /. k2 = G * (i2,j2) by A1, A8, GOBOARD1:def_9;
A12: ( 1 <= i2 & i2 <= len G ) by A10, MATRIX_1:38;
consider i1, j1 being Element of NAT  such that
A13: [i1,j1] in Indices G and
A14: f /. k1 = G * (i1,j1) by A1, A6, GOBOARD1:def_9;
A15: ( 1 <= j1 & j1 <= width G ) by A13, MATRIX_1:38;
A16: p = |[(p `1),(p `2)]| by EUCLID:53;
A17: ( 1 <= j2 & j2 <= width G ) by A10, MATRIX_1:38;
A18: ( 1 <= i1 & i1 <= len G ) by A13, MATRIX_1:38;
then A19: [i1,j2] in Indices G by A17, MATRIX_1:36;
A20: (G * (i1,j2)) `2 = (G * (1,j2)) `2 by A18, A17, GOBOARD5:1
.= (G * (i2,j2)) `2 by A12, A17, GOBOARD5:1 ;
(G * (i1,j2)) `1 = (G * (i1,1)) `1 by A18, A17, GOBOARD5:2
.= (G * (i1,j1)) `1 by A18, A15, GOBOARD5:2 ;
then  p = G * (i1,j2) by A7, A9, A14, A11, A20, A16, EUCLID:53;
then  p in  {  (G * (k,l)) where k, l is Element of NAT : [k,l] in Indices G  }  by A19;
hence  p in Values G by MATRIX_1:45; ::_thesis: verum
end;

definition
canceled;
let f be FinSequence of (TOP-REAL 2);
let G be Go-board;
let k be Element of NAT ;
assume  ( 1 <= k & k + 1 <= len f & f is_sequence_on G ) ;
then consider i1, j1, i2, j2 being Element of NAT  such that
A1: ( [i1,j1] in Indices G & f /. k = G * (i1,j1) ) and
A2: ( [i2,j2] in Indices G & f /. (k + 1) = G * (i2,j2) ) and
A3: ( ( 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;
func right_cell (f,k,G) ->  Subset of (TOP-REAL 2) means :Def2: :: GOBRD13:def 2
 for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & it = cell (G,i1,j1) ) & not ( i1 + 1 = i2 & j1 = j2 & it = cell (G,i1,(j1 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & it = cell (G,i2,j2) ) holds
( i1 = i2 & j1 = j2 + 1 & it = cell (G,(i1 -' 1),j2) );
existence
 ex b1 being Subset of (TOP-REAL 2) st
for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & b1 = cell (G,i1,j1) ) & not ( i1 + 1 = i2 & j1 = j2 & b1 = cell (G,i1,(j1 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & b1 = cell (G,i2,j2) ) holds
( i1 = i2 & j1 = j2 + 1 & b1 = cell (G,(i1 -' 1),j2) )
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 A3;
supposeA4: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: ex b1 being Subset of (TOP-REAL 2) st
for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & b1 = cell (G,i1,j1) ) & not ( i1 + 1 = i2 & j1 = j2 & b1 = cell (G,i1,(j1 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & b1 = cell (G,i2,j2) ) holds
( i1 = i2 & j1 = j2 + 1 & b1 = cell (G,(i1 -' 1),j2) )
take  cell (G,i1,j1) ; ::_thesis: for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & cell (G,i1,j1) = cell (G,i1,j1) ) & not ( i1 + 1 = i2 & j1 = j2 & cell (G,i1,j1) = cell (G,i1,(j1 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & cell (G,i1,j1) = cell (G,i2,j2) ) holds
( i1 = i2 & j1 = j2 + 1 & cell (G,i1,j1) = cell (G,(i1 -' 1),j2) )
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices G & [i29,j29] in Indices G & f /. k = G * (i19,j19) & f /. (k + 1) = G * (i29,j29) & not ( i19 = i29 & j19 + 1 = j29 & cell (G,i1,j1) = cell (G,i19,j19) ) & not ( i19 + 1 = i29 & j19 = j29 & cell (G,i1,j1) = cell (G,i19,(j19 -' 1)) ) & not ( i19 = i29 + 1 & j19 = j29 & cell (G,i1,j1) = cell (G,i29,j29) ) implies ( i19 = i29 & j19 = j29 + 1 & cell (G,i1,j1) = cell (G,(i19 -' 1),j29) ) )
assume that
A5: [i19,j19] in Indices G and
A6: [i29,j29] in Indices G and
A7: f /. k = G * (i19,j19) and
A8: f /. (k + 1) = G * (i29,j29) ; ::_thesis: ( ( i19 = i29 & j19 + 1 = j29 & cell (G,i1,j1) = cell (G,i19,j19) ) or ( i19 + 1 = i29 & j19 = j29 & cell (G,i1,j1) = cell (G,i19,(j19 -' 1)) ) or ( i19 = i29 + 1 & j19 = j29 & cell (G,i1,j1) = cell (G,i29,j29) ) or ( i19 = i29 & j19 = j29 + 1 & cell (G,i1,j1) = cell (G,(i19 -' 1),j29) ) )
 ( i1 = i19 & j1 = j19 ) by A1, A5, A7, GOBOARD1:5;
hence  ( ( i19 = i29 & j19 + 1 = j29 & cell (G,i1,j1) = cell (G,i19,j19) ) or ( i19 + 1 = i29 & j19 = j29 & cell (G,i1,j1) = cell (G,i19,(j19 -' 1)) ) or ( i19 = i29 + 1 & j19 = j29 & cell (G,i1,j1) = cell (G,i29,j29) ) or ( i19 = i29 & j19 = j29 + 1 & cell (G,i1,j1) = cell (G,(i19 -' 1),j29) ) ) by A2, A4, A6, A8, GOBOARD1:5; ::_thesis: verum
end;
supposeA9: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: ex b1 being Subset of (TOP-REAL 2) st
for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & b1 = cell (G,i1,j1) ) & not ( i1 + 1 = i2 & j1 = j2 & b1 = cell (G,i1,(j1 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & b1 = cell (G,i2,j2) ) holds
( i1 = i2 & j1 = j2 + 1 & b1 = cell (G,(i1 -' 1),j2) )
take  cell (G,i1,(j1 -' 1)) ; ::_thesis: for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & cell (G,i1,(j1 -' 1)) = cell (G,i1,j1) ) & not ( i1 + 1 = i2 & j1 = j2 & cell (G,i1,(j1 -' 1)) = cell (G,i1,(j1 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & cell (G,i1,(j1 -' 1)) = cell (G,i2,j2) ) holds
( i1 = i2 & j1 = j2 + 1 & cell (G,i1,(j1 -' 1)) = cell (G,(i1 -' 1),j2) )
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices G & [i29,j29] in Indices G & f /. k = G * (i19,j19) & f /. (k + 1) = G * (i29,j29) & not ( i19 = i29 & j19 + 1 = j29 & cell (G,i1,(j1 -' 1)) = cell (G,i19,j19) ) & not ( i19 + 1 = i29 & j19 = j29 & cell (G,i1,(j1 -' 1)) = cell (G,i19,(j19 -' 1)) ) & not ( i19 = i29 + 1 & j19 = j29 & cell (G,i1,(j1 -' 1)) = cell (G,i29,j29) ) implies ( i19 = i29 & j19 = j29 + 1 & cell (G,i1,(j1 -' 1)) = cell (G,(i19 -' 1),j29) ) )
assume that
A10: [i19,j19] in Indices G and
A11: [i29,j29] in Indices G and
A12: f /. k = G * (i19,j19) and
A13: f /. (k + 1) = G * (i29,j29) ; ::_thesis: ( ( i19 = i29 & j19 + 1 = j29 & cell (G,i1,(j1 -' 1)) = cell (G,i19,j19) ) or ( i19 + 1 = i29 & j19 = j29 & cell (G,i1,(j1 -' 1)) = cell (G,i19,(j19 -' 1)) ) or ( i19 = i29 + 1 & j19 = j29 & cell (G,i1,(j1 -' 1)) = cell (G,i29,j29) ) or ( i19 = i29 & j19 = j29 + 1 & cell (G,i1,(j1 -' 1)) = cell (G,(i19 -' 1),j29) ) )
 ( i1 = i19 & j1 = j19 ) by A1, A10, A12, GOBOARD1:5;
hence  ( ( i19 = i29 & j19 + 1 = j29 & cell (G,i1,(j1 -' 1)) = cell (G,i19,j19) ) or ( i19 + 1 = i29 & j19 = j29 & cell (G,i1,(j1 -' 1)) = cell (G,i19,(j19 -' 1)) ) or ( i19 = i29 + 1 & j19 = j29 & cell (G,i1,(j1 -' 1)) = cell (G,i29,j29) ) or ( i19 = i29 & j19 = j29 + 1 & cell (G,i1,(j1 -' 1)) = cell (G,(i19 -' 1),j29) ) ) by A2, A9, A11, A13, GOBOARD1:5; ::_thesis: verum
end;
supposeA14: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: ex b1 being Subset of (TOP-REAL 2) st
for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & b1 = cell (G,i1,j1) ) & not ( i1 + 1 = i2 & j1 = j2 & b1 = cell (G,i1,(j1 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & b1 = cell (G,i2,j2) ) holds
( i1 = i2 & j1 = j2 + 1 & b1 = cell (G,(i1 -' 1),j2) )
take  cell (G,i2,j2) ; ::_thesis: for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & cell (G,i2,j2) = cell (G,i1,j1) ) & not ( i1 + 1 = i2 & j1 = j2 & cell (G,i2,j2) = cell (G,i1,(j1 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & cell (G,i2,j2) = cell (G,i2,j2) ) holds
( i1 = i2 & j1 = j2 + 1 & cell (G,i2,j2) = cell (G,(i1 -' 1),j2) )
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices G & [i29,j29] in Indices G & f /. k = G * (i19,j19) & f /. (k + 1) = G * (i29,j29) & not ( i19 = i29 & j19 + 1 = j29 & cell (G,i2,j2) = cell (G,i19,j19) ) & not ( i19 + 1 = i29 & j19 = j29 & cell (G,i2,j2) = cell (G,i19,(j19 -' 1)) ) & not ( i19 = i29 + 1 & j19 = j29 & cell (G,i2,j2) = cell (G,i29,j29) ) implies ( i19 = i29 & j19 = j29 + 1 & cell (G,i2,j2) = cell (G,(i19 -' 1),j29) ) )
assume A15: ( [i19,j19] in Indices G & [i29,j29] in Indices G & f /. k = G * (i19,j19) & f /. (k + 1) = G * (i29,j29) ) ; ::_thesis: ( ( i19 = i29 & j19 + 1 = j29 & cell (G,i2,j2) = cell (G,i19,j19) ) or ( i19 + 1 = i29 & j19 = j29 & cell (G,i2,j2) = cell (G,i19,(j19 -' 1)) ) or ( i19 = i29 + 1 & j19 = j29 & cell (G,i2,j2) = cell (G,i29,j29) ) or ( i19 = i29 & j19 = j29 + 1 & cell (G,i2,j2) = cell (G,(i19 -' 1),j29) ) )
then  ( i2 = i29 & j1 = j19 ) by A1, A2, GOBOARD1:5;
hence  ( ( i19 = i29 & j19 + 1 = j29 & cell (G,i2,j2) = cell (G,i19,j19) ) or ( i19 + 1 = i29 & j19 = j29 & cell (G,i2,j2) = cell (G,i19,(j19 -' 1)) ) or ( i19 = i29 + 1 & j19 = j29 & cell (G,i2,j2) = cell (G,i29,j29) ) or ( i19 = i29 & j19 = j29 + 1 & cell (G,i2,j2) = cell (G,(i19 -' 1),j29) ) ) by A1, A2, A14, A15, GOBOARD1:5; ::_thesis: verum
end;
supposeA16: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: ex b1 being Subset of (TOP-REAL 2) st
for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & b1 = cell (G,i1,j1) ) & not ( i1 + 1 = i2 & j1 = j2 & b1 = cell (G,i1,(j1 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & b1 = cell (G,i2,j2) ) holds
( i1 = i2 & j1 = j2 + 1 & b1 = cell (G,(i1 -' 1),j2) )
take  cell (G,(i1 -' 1),j2) ; ::_thesis: for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & cell (G,(i1 -' 1),j2) = cell (G,i1,j1) ) & not ( i1 + 1 = i2 & j1 = j2 & cell (G,(i1 -' 1),j2) = cell (G,i1,(j1 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & cell (G,(i1 -' 1),j2) = cell (G,i2,j2) ) holds
( i1 = i2 & j1 = j2 + 1 & cell (G,(i1 -' 1),j2) = cell (G,(i1 -' 1),j2) )
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices G & [i29,j29] in Indices G & f /. k = G * (i19,j19) & f /. (k + 1) = G * (i29,j29) & not ( i19 = i29 & j19 + 1 = j29 & cell (G,(i1 -' 1),j2) = cell (G,i19,j19) ) & not ( i19 + 1 = i29 & j19 = j29 & cell (G,(i1 -' 1),j2) = cell (G,i19,(j19 -' 1)) ) & not ( i19 = i29 + 1 & j19 = j29 & cell (G,(i1 -' 1),j2) = cell (G,i29,j29) ) implies ( i19 = i29 & j19 = j29 + 1 & cell (G,(i1 -' 1),j2) = cell (G,(i19 -' 1),j29) ) )
assume that
A17: [i19,j19] in Indices G and
A18: [i29,j29] in Indices G and
A19: f /. k = G * (i19,j19) and
A20: f /. (k + 1) = G * (i29,j29) ; ::_thesis: ( ( i19 = i29 & j19 + 1 = j29 & cell (G,(i1 -' 1),j2) = cell (G,i19,j19) ) or ( i19 + 1 = i29 & j19 = j29 & cell (G,(i1 -' 1),j2) = cell (G,i19,(j19 -' 1)) ) or ( i19 = i29 + 1 & j19 = j29 & cell (G,(i1 -' 1),j2) = cell (G,i29,j29) ) or ( i19 = i29 & j19 = j29 + 1 & cell (G,(i1 -' 1),j2) = cell (G,(i19 -' 1),j29) ) )
 ( i1 = i19 & j1 = j19 ) by A1, A17, A19, GOBOARD1:5;
hence  ( ( i19 = i29 & j19 + 1 = j29 & cell (G,(i1 -' 1),j2) = cell (G,i19,j19) ) or ( i19 + 1 = i29 & j19 = j29 & cell (G,(i1 -' 1),j2) = cell (G,i19,(j19 -' 1)) ) or ( i19 = i29 + 1 & j19 = j29 & cell (G,(i1 -' 1),j2) = cell (G,i29,j29) ) or ( i19 = i29 & j19 = j29 + 1 & cell (G,(i1 -' 1),j2) = cell (G,(i19 -' 1),j29) ) ) by A2, A16, A18, A20, GOBOARD1:5; ::_thesis: verum
end;
end;
end;
uniqueness
 for b1, b2 being Subset of (TOP-REAL 2) st ( for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & b1 = cell (G,i1,j1) ) & not ( i1 + 1 = i2 & j1 = j2 & b1 = cell (G,i1,(j1 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & b1 = cell (G,i2,j2) ) holds
( i1 = i2 & j1 = j2 + 1 & b1 = cell (G,(i1 -' 1),j2) ) ) & ( for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & b2 = cell (G,i1,j1) ) & not ( i1 + 1 = i2 & j1 = j2 & b2 = cell (G,i1,(j1 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & b2 = cell (G,i2,j2) ) holds
( i1 = i2 & j1 = j2 + 1 & b2 = cell (G,(i1 -' 1),j2) ) ) holds
b1 = b2
proof
let P1, P2 be Subset of (TOP-REAL 2); ::_thesis: ( ( for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & P1 = cell (G,i1,j1) ) & not ( i1 + 1 = i2 & j1 = j2 & P1 = cell (G,i1,(j1 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & P1 = cell (G,i2,j2) ) holds
( i1 = i2 & j1 = j2 + 1 & P1 = cell (G,(i1 -' 1),j2) ) ) & ( for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & P2 = cell (G,i1,j1) ) & not ( i1 + 1 = i2 & j1 = j2 & P2 = cell (G,i1,(j1 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & P2 = cell (G,i2,j2) ) holds
( i1 = i2 & j1 = j2 + 1 & P2 = cell (G,(i1 -' 1),j2) ) ) implies P1 = P2 )
assume that
A21: for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & P1 = cell (G,i1,j1) ) & not ( i1 + 1 = i2 & j1 = j2 & P1 = cell (G,i1,(j1 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & P1 = cell (G,i2,j2) ) holds
( i1 = i2 & j1 = j2 + 1 & P1 = cell (G,(i1 -' 1),j2) ) and
A22: for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & P2 = cell (G,i1,j1) ) & not ( i1 + 1 = i2 & j1 = j2 & P2 = cell (G,i1,(j1 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & P2 = cell (G,i2,j2) ) holds
( i1 = i2 & j1 = j2 + 1 & P2 = cell (G,(i1 -' 1),j2) ) ; ::_thesis: P1 = P2
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 A3;
supposeA23: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: P1 = P2
A24: j2 <= j2 + 1 by NAT_1:11;
A25: j1 < j2 by A23, XREAL_1:29;
hence P1 = cell (G,i1,j1) by A1, A2, A21, A24
.= P2 by A1, A2, A22, A25, A24 ;
::_thesis: verum
end;
supposeA26: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: P1 = P2
A27: i2 <= i2 + 1 by NAT_1:11;
A28: i1 < i2 by A26, XREAL_1:29;
hence P1 = cell (G,i1,(j1 -' 1)) by A1, A2, A21, A27
.= P2 by A1, A2, A22, A28, A27 ;
::_thesis: verum
end;
supposeA29: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: P1 = P2
A30: i1 <= i1 + 1 by NAT_1:11;
A31: i2 < i1 by A29, XREAL_1:29;
hence P1 = cell (G,i2,j2) by A1, A2, A21, A30
.= P2 by A1, A2, A22, A31, A30 ;
::_thesis: verum
end;
supposeA32: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: P1 = P2
A33: j1 <= j1 + 1 by NAT_1:11;
A34: j2 < j1 by A32, XREAL_1:29;
hence P1 = cell (G,(i1 -' 1),j2) by A1, A2, A21, A33
.= P2 by A1, A2, A22, A34, A33 ;
::_thesis: verum
end;
end;
end;
func left_cell (f,k,G) ->  Subset of (TOP-REAL 2) means :Def3: :: GOBRD13:def 3
 for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & it = cell (G,(i1 -' 1),j1) ) & not ( i1 + 1 = i2 & j1 = j2 & it = cell (G,i1,j1) ) & not ( i1 = i2 + 1 & j1 = j2 & it = cell (G,i2,(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & it = cell (G,i1,j2) );
existence
 ex b1 being Subset of (TOP-REAL 2) st
for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & b1 = cell (G,(i1 -' 1),j1) ) & not ( i1 + 1 = i2 & j1 = j2 & b1 = cell (G,i1,j1) ) & not ( i1 = i2 + 1 & j1 = j2 & b1 = cell (G,i2,(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & b1 = cell (G,i1,j2) )
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 A3;
supposeA35: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: ex b1 being Subset of (TOP-REAL 2) st
for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & b1 = cell (G,(i1 -' 1),j1) ) & not ( i1 + 1 = i2 & j1 = j2 & b1 = cell (G,i1,j1) ) & not ( i1 = i2 + 1 & j1 = j2 & b1 = cell (G,i2,(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & b1 = cell (G,i1,j2) )
take  cell (G,(i1 -' 1),j1) ; ::_thesis: for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & cell (G,(i1 -' 1),j1) = cell (G,(i1 -' 1),j1) ) & not ( i1 + 1 = i2 & j1 = j2 & cell (G,(i1 -' 1),j1) = cell (G,i1,j1) ) & not ( i1 = i2 + 1 & j1 = j2 & cell (G,(i1 -' 1),j1) = cell (G,i2,(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & cell (G,(i1 -' 1),j1) = cell (G,i1,j2) )
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices G & [i29,j29] in Indices G & f /. k = G * (i19,j19) & f /. (k + 1) = G * (i29,j29) & not ( i19 = i29 & j19 + 1 = j29 & cell (G,(i1 -' 1),j1) = cell (G,(i19 -' 1),j19) ) & not ( i19 + 1 = i29 & j19 = j29 & cell (G,(i1 -' 1),j1) = cell (G,i19,j19) ) & not ( i19 = i29 + 1 & j19 = j29 & cell (G,(i1 -' 1),j1) = cell (G,i29,(j29 -' 1)) ) implies ( i19 = i29 & j19 = j29 + 1 & cell (G,(i1 -' 1),j1) = cell (G,i19,j29) ) )
assume that
A36: [i19,j19] in Indices G and
A37: [i29,j29] in Indices G and
A38: f /. k = G * (i19,j19) and
A39: f /. (k + 1) = G * (i29,j29) ; ::_thesis: ( ( i19 = i29 & j19 + 1 = j29 & cell (G,(i1 -' 1),j1) = cell (G,(i19 -' 1),j19) ) or ( i19 + 1 = i29 & j19 = j29 & cell (G,(i1 -' 1),j1) = cell (G,i19,j19) ) or ( i19 = i29 + 1 & j19 = j29 & cell (G,(i1 -' 1),j1) = cell (G,i29,(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & cell (G,(i1 -' 1),j1) = cell (G,i19,j29) ) )
 ( i1 = i19 & j1 = j19 ) by A1, A36, A38, GOBOARD1:5;
hence  ( ( i19 = i29 & j19 + 1 = j29 & cell (G,(i1 -' 1),j1) = cell (G,(i19 -' 1),j19) ) or ( i19 + 1 = i29 & j19 = j29 & cell (G,(i1 -' 1),j1) = cell (G,i19,j19) ) or ( i19 = i29 + 1 & j19 = j29 & cell (G,(i1 -' 1),j1) = cell (G,i29,(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & cell (G,(i1 -' 1),j1) = cell (G,i19,j29) ) ) by A2, A35, A37, A39, GOBOARD1:5; ::_thesis: verum
end;
supposeA40: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: ex b1 being Subset of (TOP-REAL 2) st
for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & b1 = cell (G,(i1 -' 1),j1) ) & not ( i1 + 1 = i2 & j1 = j2 & b1 = cell (G,i1,j1) ) & not ( i1 = i2 + 1 & j1 = j2 & b1 = cell (G,i2,(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & b1 = cell (G,i1,j2) )
take  cell (G,i1,j1) ; ::_thesis: for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & cell (G,i1,j1) = cell (G,(i1 -' 1),j1) ) & not ( i1 + 1 = i2 & j1 = j2 & cell (G,i1,j1) = cell (G,i1,j1) ) & not ( i1 = i2 + 1 & j1 = j2 & cell (G,i1,j1) = cell (G,i2,(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & cell (G,i1,j1) = cell (G,i1,j2) )
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices G & [i29,j29] in Indices G & f /. k = G * (i19,j19) & f /. (k + 1) = G * (i29,j29) & not ( i19 = i29 & j19 + 1 = j29 & cell (G,i1,j1) = cell (G,(i19 -' 1),j19) ) & not ( i19 + 1 = i29 & j19 = j29 & cell (G,i1,j1) = cell (G,i19,j19) ) & not ( i19 = i29 + 1 & j19 = j29 & cell (G,i1,j1) = cell (G,i29,(j29 -' 1)) ) implies ( i19 = i29 & j19 = j29 + 1 & cell (G,i1,j1) = cell (G,i19,j29) ) )
assume that
A41: [i19,j19] in Indices G and
A42: [i29,j29] in Indices G and
A43: f /. k = G * (i19,j19) and
A44: f /. (k + 1) = G * (i29,j29) ; ::_thesis: ( ( i19 = i29 & j19 + 1 = j29 & cell (G,i1,j1) = cell (G,(i19 -' 1),j19) ) or ( i19 + 1 = i29 & j19 = j29 & cell (G,i1,j1) = cell (G,i19,j19) ) or ( i19 = i29 + 1 & j19 = j29 & cell (G,i1,j1) = cell (G,i29,(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & cell (G,i1,j1) = cell (G,i19,j29) ) )
 ( i1 = i19 & j1 = j19 ) by A1, A41, A43, GOBOARD1:5;
hence  ( ( i19 = i29 & j19 + 1 = j29 & cell (G,i1,j1) = cell (G,(i19 -' 1),j19) ) or ( i19 + 1 = i29 & j19 = j29 & cell (G,i1,j1) = cell (G,i19,j19) ) or ( i19 = i29 + 1 & j19 = j29 & cell (G,i1,j1) = cell (G,i29,(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & cell (G,i1,j1) = cell (G,i19,j29) ) ) by A2, A40, A42, A44, GOBOARD1:5; ::_thesis: verum
end;
supposeA45: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: ex b1 being Subset of (TOP-REAL 2) st
for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & b1 = cell (G,(i1 -' 1),j1) ) & not ( i1 + 1 = i2 & j1 = j2 & b1 = cell (G,i1,j1) ) & not ( i1 = i2 + 1 & j1 = j2 & b1 = cell (G,i2,(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & b1 = cell (G,i1,j2) )
take  cell (G,i2,(j2 -' 1)) ; ::_thesis: for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & cell (G,i2,(j2 -' 1)) = cell (G,(i1 -' 1),j1) ) & not ( i1 + 1 = i2 & j1 = j2 & cell (G,i2,(j2 -' 1)) = cell (G,i1,j1) ) & not ( i1 = i2 + 1 & j1 = j2 & cell (G,i2,(j2 -' 1)) = cell (G,i2,(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & cell (G,i2,(j2 -' 1)) = cell (G,i1,j2) )
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices G & [i29,j29] in Indices G & f /. k = G * (i19,j19) & f /. (k + 1) = G * (i29,j29) & not ( i19 = i29 & j19 + 1 = j29 & cell (G,i2,(j2 -' 1)) = cell (G,(i19 -' 1),j19) ) & not ( i19 + 1 = i29 & j19 = j29 & cell (G,i2,(j2 -' 1)) = cell (G,i19,j19) ) & not ( i19 = i29 + 1 & j19 = j29 & cell (G,i2,(j2 -' 1)) = cell (G,i29,(j29 -' 1)) ) implies ( i19 = i29 & j19 = j29 + 1 & cell (G,i2,(j2 -' 1)) = cell (G,i19,j29) ) )
assume A46: ( [i19,j19] in Indices G & [i29,j29] in Indices G & f /. k = G * (i19,j19) & f /. (k + 1) = G * (i29,j29) ) ; ::_thesis: ( ( i19 = i29 & j19 + 1 = j29 & cell (G,i2,(j2 -' 1)) = cell (G,(i19 -' 1),j19) ) or ( i19 + 1 = i29 & j19 = j29 & cell (G,i2,(j2 -' 1)) = cell (G,i19,j19) ) or ( i19 = i29 + 1 & j19 = j29 & cell (G,i2,(j2 -' 1)) = cell (G,i29,(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & cell (G,i2,(j2 -' 1)) = cell (G,i19,j29) ) )
then  ( i2 = i29 & j1 = j19 ) by A1, A2, GOBOARD1:5;
hence  ( ( i19 = i29 & j19 + 1 = j29 & cell (G,i2,(j2 -' 1)) = cell (G,(i19 -' 1),j19) ) or ( i19 + 1 = i29 & j19 = j29 & cell (G,i2,(j2 -' 1)) = cell (G,i19,j19) ) or ( i19 = i29 + 1 & j19 = j29 & cell (G,i2,(j2 -' 1)) = cell (G,i29,(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & cell (G,i2,(j2 -' 1)) = cell (G,i19,j29) ) ) by A1, A2, A45, A46, GOBOARD1:5; ::_thesis: verum
end;
supposeA47: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: ex b1 being Subset of (TOP-REAL 2) st
for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & b1 = cell (G,(i1 -' 1),j1) ) & not ( i1 + 1 = i2 & j1 = j2 & b1 = cell (G,i1,j1) ) & not ( i1 = i2 + 1 & j1 = j2 & b1 = cell (G,i2,(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & b1 = cell (G,i1,j2) )
take  cell (G,i1,j2) ; ::_thesis: for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & cell (G,i1,j2) = cell (G,(i1 -' 1),j1) ) & not ( i1 + 1 = i2 & j1 = j2 & cell (G,i1,j2) = cell (G,i1,j1) ) & not ( i1 = i2 + 1 & j1 = j2 & cell (G,i1,j2) = cell (G,i2,(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & cell (G,i1,j2) = cell (G,i1,j2) )
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices G & [i29,j29] in Indices G & f /. k = G * (i19,j19) & f /. (k + 1) = G * (i29,j29) & not ( i19 = i29 & j19 + 1 = j29 & cell (G,i1,j2) = cell (G,(i19 -' 1),j19) ) & not ( i19 + 1 = i29 & j19 = j29 & cell (G,i1,j2) = cell (G,i19,j19) ) & not ( i19 = i29 + 1 & j19 = j29 & cell (G,i1,j2) = cell (G,i29,(j29 -' 1)) ) implies ( i19 = i29 & j19 = j29 + 1 & cell (G,i1,j2) = cell (G,i19,j29) ) )
assume that
A48: [i19,j19] in Indices G and
A49: [i29,j29] in Indices G and
A50: f /. k = G * (i19,j19) and
A51: f /. (k + 1) = G * (i29,j29) ; ::_thesis: ( ( i19 = i29 & j19 + 1 = j29 & cell (G,i1,j2) = cell (G,(i19 -' 1),j19) ) or ( i19 + 1 = i29 & j19 = j29 & cell (G,i1,j2) = cell (G,i19,j19) ) or ( i19 = i29 + 1 & j19 = j29 & cell (G,i1,j2) = cell (G,i29,(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & cell (G,i1,j2) = cell (G,i19,j29) ) )
 ( i1 = i19 & j1 = j19 ) by A1, A48, A50, GOBOARD1:5;
hence  ( ( i19 = i29 & j19 + 1 = j29 & cell (G,i1,j2) = cell (G,(i19 -' 1),j19) ) or ( i19 + 1 = i29 & j19 = j29 & cell (G,i1,j2) = cell (G,i19,j19) ) or ( i19 = i29 + 1 & j19 = j29 & cell (G,i1,j2) = cell (G,i29,(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & cell (G,i1,j2) = cell (G,i19,j29) ) ) by A2, A47, A49, A51, GOBOARD1:5; ::_thesis: verum
end;
end;
end;
uniqueness
 for b1, b2 being Subset of (TOP-REAL 2) st ( for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & b1 = cell (G,(i1 -' 1),j1) ) & not ( i1 + 1 = i2 & j1 = j2 & b1 = cell (G,i1,j1) ) & not ( i1 = i2 + 1 & j1 = j2 & b1 = cell (G,i2,(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & b1 = cell (G,i1,j2) ) ) & ( for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & b2 = cell (G,(i1 -' 1),j1) ) & not ( i1 + 1 = i2 & j1 = j2 & b2 = cell (G,i1,j1) ) & not ( i1 = i2 + 1 & j1 = j2 & b2 = cell (G,i2,(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & b2 = cell (G,i1,j2) ) ) holds
b1 = b2
proof
let P1, P2 be Subset of (TOP-REAL 2); ::_thesis: ( ( for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & P1 = cell (G,(i1 -' 1),j1) ) & not ( i1 + 1 = i2 & j1 = j2 & P1 = cell (G,i1,j1) ) & not ( i1 = i2 + 1 & j1 = j2 & P1 = cell (G,i2,(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & P1 = cell (G,i1,j2) ) ) & ( for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & P2 = cell (G,(i1 -' 1),j1) ) & not ( i1 + 1 = i2 & j1 = j2 & P2 = cell (G,i1,j1) ) & not ( i1 = i2 + 1 & j1 = j2 & P2 = cell (G,i2,(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & P2 = cell (G,i1,j2) ) ) implies P1 = P2 )
assume that
A52: for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & P1 = cell (G,(i1 -' 1),j1) ) & not ( i1 + 1 = i2 & j1 = j2 & P1 = cell (G,i1,j1) ) & not ( i1 = i2 + 1 & j1 = j2 & P1 = cell (G,i2,(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & P1 = cell (G,i1,j2) ) and
A53: for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & P2 = cell (G,(i1 -' 1),j1) ) & not ( i1 + 1 = i2 & j1 = j2 & P2 = cell (G,i1,j1) ) & not ( i1 = i2 + 1 & j1 = j2 & P2 = cell (G,i2,(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & P2 = cell (G,i1,j2) ) ; ::_thesis: P1 = P2
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 A3;
supposeA54: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: P1 = P2
A55: j2 <= j2 + 1 by NAT_1:11;
A56: j1 < j2 by A54, XREAL_1:29;
hence P1 = cell (G,(i1 -' 1),j1) by A1, A2, A52, A55
.= P2 by A1, A2, A53, A56, A55 ;
::_thesis: verum
end;
supposeA57: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: P1 = P2
A58: i2 <= i2 + 1 by NAT_1:11;
A59: i1 < i2 by A57, XREAL_1:29;
hence P1 = cell (G,i1,j1) by A1, A2, A52, A58
.= P2 by A1, A2, A53, A59, A58 ;
::_thesis: verum
end;
supposeA60: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: P1 = P2
A61: i1 <= i1 + 1 by NAT_1:11;
A62: i2 < i1 by A60, XREAL_1:29;
hence P1 = cell (G,i2,(j2 -' 1)) by A1, A2, A52, A61
.= P2 by A1, A2, A53, A62, A61 ;
::_thesis: verum
end;
supposeA63: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: P1 = P2
A64: j1 <= j1 + 1 by NAT_1:11;
A65: j2 < j1 by A63, XREAL_1:29;
hence P1 = cell (G,i1,j2) by A1, A2, A52, A64
.= P2 by A1, A2, A53, A65, A64 ;
::_thesis: verum
end;
end;
end;
end;

:: deftheorem  GOBRD13:def_1_:_
canceled;

:: deftheorem Def2 defines right_cell GOBRD13:def_2_:_
 for f being FinSequence of (TOP-REAL 2)
for G being Go-board
for k being Element of NAT st 1 <= k & k + 1 <= len f & f is_sequence_on G holds
for b4 being Subset of (TOP-REAL 2) holds
( b4 = right_cell (f,k,G) iff for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & b4 = cell (G,i1,j1) ) & not ( i1 + 1 = i2 & j1 = j2 & b4 = cell (G,i1,(j1 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & b4 = cell (G,i2,j2) ) holds
( i1 = i2 & j1 = j2 + 1 & b4 = cell (G,(i1 -' 1),j2) ) );

:: deftheorem Def3 defines left_cell GOBRD13:def_3_:_
 for f being FinSequence of (TOP-REAL 2)
for G being Go-board
for k being Element of NAT st 1 <= k & k + 1 <= len f & f is_sequence_on G holds
for b4 being Subset of (TOP-REAL 2) holds
( b4 = left_cell (f,k,G) iff for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & b4 = cell (G,(i1 -' 1),j1) ) & not ( i1 + 1 = i2 & j1 = j2 & b4 = cell (G,i1,j1) ) & not ( i1 = i2 + 1 & j1 = j2 & b4 = cell (G,i2,(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & b4 = cell (G,i1,j2) ) );

theorem Th21: :: GOBRD13:21
for k, i, j being Element of NAT
for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [i,(j + 1)] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * (i,(j + 1)) holds
left_cell (f,k,G) = cell (G,(i -' 1),j)
proof
let k, i, j be Element of NAT ; ::_thesis: for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [i,(j + 1)] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * (i,(j + 1)) holds
left_cell (f,k,G) = cell (G,(i -' 1),j)
let f be FinSequence of (TOP-REAL 2); ::_thesis: for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [i,(j + 1)] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * (i,(j + 1)) holds
left_cell (f,k,G) = cell (G,(i -' 1),j)
let G be Go-board; ::_thesis: ( 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [i,(j + 1)] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * (i,(j + 1)) implies left_cell (f,k,G) = cell (G,(i -' 1),j) )
A1: ( j < j + 1 & j + 1 <= (j + 1) + 1 ) by XREAL_1:29;
assume  ( 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [i,(j + 1)] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * (i,(j + 1)) ) ; ::_thesis: left_cell (f,k,G) = cell (G,(i -' 1),j)
hence  left_cell (f,k,G) = cell (G,(i -' 1),j) by A1, Def3; ::_thesis: verum
end;

theorem Th22: :: GOBRD13:22
for k, i, j being Element of NAT
for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [i,(j + 1)] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * (i,(j + 1)) holds
right_cell (f,k,G) = cell (G,i,j)
proof
let k, i, j be Element of NAT ; ::_thesis: for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [i,(j + 1)] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * (i,(j + 1)) holds
right_cell (f,k,G) = cell (G,i,j)
let f be FinSequence of (TOP-REAL 2); ::_thesis: for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [i,(j + 1)] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * (i,(j + 1)) holds
right_cell (f,k,G) = cell (G,i,j)
let G be Go-board; ::_thesis: ( 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [i,(j + 1)] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * (i,(j + 1)) implies right_cell (f,k,G) = cell (G,i,j) )
A1: ( j < j + 1 & j + 1 <= (j + 1) + 1 ) by XREAL_1:29;
assume  ( 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [i,(j + 1)] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * (i,(j + 1)) ) ; ::_thesis: right_cell (f,k,G) = cell (G,i,j)
hence  right_cell (f,k,G) = cell (G,i,j) by A1, Def2; ::_thesis: verum
end;

theorem Th23: :: GOBRD13:23
for k, i, j being Element of NAT
for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * ((i + 1),j) holds
left_cell (f,k,G) = cell (G,i,j)
proof
let k, i, j be Element of NAT ; ::_thesis: for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * ((i + 1),j) holds
left_cell (f,k,G) = cell (G,i,j)
let f be FinSequence of (TOP-REAL 2); ::_thesis: for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * ((i + 1),j) holds
left_cell (f,k,G) = cell (G,i,j)
let G be Go-board; ::_thesis: ( 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * ((i + 1),j) implies left_cell (f,k,G) = cell (G,i,j) )
A1: ( i < i + 1 & i + 1 <= (i + 1) + 1 ) by XREAL_1:29;
assume  ( 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * ((i + 1),j) ) ; ::_thesis: left_cell (f,k,G) = cell (G,i,j)
hence  left_cell (f,k,G) = cell (G,i,j) by A1, Def3; ::_thesis: verum
end;

theorem Th24: :: GOBRD13:24
for k, i, j being Element of NAT
for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * ((i + 1),j) holds
right_cell (f,k,G) = cell (G,i,(j -' 1))
proof
let k, i, j be Element of NAT ; ::_thesis: for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * ((i + 1),j) holds
right_cell (f,k,G) = cell (G,i,(j -' 1))
let f be FinSequence of (TOP-REAL 2); ::_thesis: for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * ((i + 1),j) holds
right_cell (f,k,G) = cell (G,i,(j -' 1))
let G be Go-board; ::_thesis: ( 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * ((i + 1),j) implies right_cell (f,k,G) = cell (G,i,(j -' 1)) )
A1: ( i < i + 1 & i + 1 <= (i + 1) + 1 ) by XREAL_1:29;
assume  ( 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * ((i + 1),j) ) ; ::_thesis: right_cell (f,k,G) = cell (G,i,(j -' 1))
hence  right_cell (f,k,G) = cell (G,i,(j -' 1)) by A1, Def2; ::_thesis: verum
end;

theorem Th25: :: GOBRD13:25
for k, i, j being Element of NAT
for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * ((i + 1),j) & f /. (k + 1) = G * (i,j) holds
left_cell (f,k,G) = cell (G,i,(j -' 1))
proof
let k, i, j be Element of NAT ; ::_thesis: for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * ((i + 1),j) & f /. (k + 1) = G * (i,j) holds
left_cell (f,k,G) = cell (G,i,(j -' 1))
let f be FinSequence of (TOP-REAL 2); ::_thesis: for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * ((i + 1),j) & f /. (k + 1) = G * (i,j) holds
left_cell (f,k,G) = cell (G,i,(j -' 1))
let G be Go-board; ::_thesis: ( 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * ((i + 1),j) & f /. (k + 1) = G * (i,j) implies left_cell (f,k,G) = cell (G,i,(j -' 1)) )
A1: ( i < i + 1 & i + 1 <= (i + 1) + 1 ) by XREAL_1:29;
assume  ( 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * ((i + 1),j) & f /. (k + 1) = G * (i,j) ) ; ::_thesis: left_cell (f,k,G) = cell (G,i,(j -' 1))
hence  left_cell (f,k,G) = cell (G,i,(j -' 1)) by A1, Def3; ::_thesis: verum
end;

theorem Th26: :: GOBRD13:26
for k, i, j being Element of NAT
for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * ((i + 1),j) & f /. (k + 1) = G * (i,j) holds
right_cell (f,k,G) = cell (G,i,j)
proof
let k, i, j be Element of NAT ; ::_thesis: for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * ((i + 1),j) & f /. (k + 1) = G * (i,j) holds
right_cell (f,k,G) = cell (G,i,j)
let f be FinSequence of (TOP-REAL 2); ::_thesis: for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * ((i + 1),j) & f /. (k + 1) = G * (i,j) holds
right_cell (f,k,G) = cell (G,i,j)
let G be Go-board; ::_thesis: ( 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * ((i + 1),j) & f /. (k + 1) = G * (i,j) implies right_cell (f,k,G) = cell (G,i,j) )
A1: ( i < i + 1 & i + 1 <= (i + 1) + 1 ) by XREAL_1:29;
assume  ( 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * ((i + 1),j) & f /. (k + 1) = G * (i,j) ) ; ::_thesis: right_cell (f,k,G) = cell (G,i,j)
hence  right_cell (f,k,G) = cell (G,i,j) by A1, Def2; ::_thesis: verum
end;

theorem Th27: :: GOBRD13:27
for k, i, j being Element of NAT
for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,(j + 1)] in Indices G & [i,j] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) holds
left_cell (f,k,G) = cell (G,i,j)
proof
let k, i, j be Element of NAT ; ::_thesis: for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,(j + 1)] in Indices G & [i,j] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) holds
left_cell (f,k,G) = cell (G,i,j)
let f be FinSequence of (TOP-REAL 2); ::_thesis: for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,(j + 1)] in Indices G & [i,j] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) holds
left_cell (f,k,G) = cell (G,i,j)
let G be Go-board; ::_thesis: ( 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,(j + 1)] in Indices G & [i,j] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) implies left_cell (f,k,G) = cell (G,i,j) )
A1: ( j < j + 1 & j + 1 <= (j + 1) + 1 ) by XREAL_1:29;
assume  ( 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,(j + 1)] in Indices G & [i,j] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) ) ; ::_thesis: left_cell (f,k,G) = cell (G,i,j)
hence  left_cell (f,k,G) = cell (G,i,j) by A1, Def3; ::_thesis: verum
end;

theorem Th28: :: GOBRD13:28
for k, i, j being Element of NAT
for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,(j + 1)] in Indices G & [i,j] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) holds
right_cell (f,k,G) = cell (G,(i -' 1),j)
proof
let k, i, j be Element of NAT ; ::_thesis: for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,(j + 1)] in Indices G & [i,j] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) holds
right_cell (f,k,G) = cell (G,(i -' 1),j)
let f be FinSequence of (TOP-REAL 2); ::_thesis: for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,(j + 1)] in Indices G & [i,j] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) holds
right_cell (f,k,G) = cell (G,(i -' 1),j)
let G be Go-board; ::_thesis: ( 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,(j + 1)] in Indices G & [i,j] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) implies right_cell (f,k,G) = cell (G,(i -' 1),j) )
A1: ( j < j + 1 & j + 1 <= (j + 1) + 1 ) by XREAL_1:29;
assume  ( 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,(j + 1)] in Indices G & [i,j] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) ) ; ::_thesis: right_cell (f,k,G) = cell (G,(i -' 1),j)
hence  right_cell (f,k,G) = cell (G,(i -' 1),j) by A1, Def2; ::_thesis: verum
end;

theorem :: GOBRD13:29
for k being Element of NAT
for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G holds
(left_cell (f,k,G)) /\ (right_cell (f,k,G)) = LSeg (f,k)
proof
let k be Element of NAT ; ::_thesis: for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G holds
(left_cell (f,k,G)) /\ (right_cell (f,k,G)) = LSeg (f,k)
let f be FinSequence of (TOP-REAL 2); ::_thesis: for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G holds
(left_cell (f,k,G)) /\ (right_cell (f,k,G)) = LSeg (f,k)
let G be Go-board; ::_thesis: ( 1 <= k & k + 1 <= len f & f is_sequence_on G implies (left_cell (f,k,G)) /\ (right_cell (f,k,G)) = LSeg (f,k) )
assume that
A1: 1 <= k and
A2: k + 1 <= len f and
A3: f is_sequence_on G ; ::_thesis: (left_cell (f,k,G)) /\ (right_cell (f,k,G)) = LSeg (f,k)
 k + 1 >= 1 by NAT_1:11;
then A4: k + 1 in dom f by A2, FINSEQ_3:25;
then consider i2, j2 being Element of NAT  such that
A5: [i2,j2] in Indices G and
A6: f /. (k + 1) = G * (i2,j2) by A3, GOBOARD1:def_9;
A7: 1 <= j2 by A5, MATRIX_1:38;
A8: i2 <= len G by A5, MATRIX_1:38;
A9: 1 <= i2 by A5, MATRIX_1:38;
A10: j2 <= width G by A5, MATRIX_1:38;
 k <= k + 1 by NAT_1:11;
then  k <= len f by A2, XXREAL_0:2;
then A11: k in dom f by A1, FINSEQ_3:25;
then consider i1, j1 being Element of NAT  such that
A12: [i1,j1] in Indices G and
A13: f /. k = G * (i1,j1) by A3, GOBOARD1:def_9;
A14: 0 + 1 <= j1 by A12, MATRIX_1:38;
then  j1 > 0 by NAT_1:13;
then consider j being Nat such that
A15: j + 1 = j1 by NAT_1:6;
A16: (abs (i1 - i2)) + (abs (j1 - j2)) = 1 by A3, A11, A12, A13, A4, A5, A6, GOBOARD1:def_9;
A17: now__::_thesis:_(_(_abs_(i1_-_i2)_=_1_&_j1_=_j2_&_(_i1_=_i2_+_1_or_i1_+_1_=_i2_)_&_j1_=_j2_)_or_(_i1_=_i2_&_abs_(j1_-_j2)_=_1_&_(_j1_=_j2_+_1_or_j1_+_1_=_j2_)_&_i1_=_i2_)_)
percases ( ( abs (i1 - i2) = 1 & j1 = j2 ) or ( i1 = i2 & abs (j1 - j2) = 1 ) ) by A16, SEQM_3:42;
casethat A18: abs (i1 - i2) = 1 and
A19: j1 = j2 ; ::_thesis: ( ( i1 = i2 + 1 or i1 + 1 = i2 ) & j1 = j2 )
 ( i1 - i2 = 1 or - (i1 - i2) = 1 ) by A18, ABSVALUE:def_1;
hence  ( i1 = i2 + 1 or i1 + 1 = i2 ) ; ::_thesis: j1 = j2
thus  j1 = j2 by A19; ::_thesis: verum
end;
casethat A20: i1 = i2 and
A21: abs (j1 - j2) = 1 ; ::_thesis: ( ( j1 = j2 + 1 or j1 + 1 = j2 ) & i1 = i2 )
 ( j1 - j2 = 1 or - (j1 - j2) = 1 ) by A21, ABSVALUE:def_1;
hence  ( j1 = j2 + 1 or j1 + 1 = j2 ) ; ::_thesis: i1 = i2
thus  i1 = i2 by A20; ::_thesis: verum
end;
end;
end;
A22: j1 -' 1 = j by A15, NAT_D:34;
A23: j1 <= width G by A12, MATRIX_1:38;
then A24: j < width G by A15, NAT_1:13;
A25: 0 + 1 <= i1 by A12, MATRIX_1:38;
then  i1 > 0 by NAT_1:13;
then consider i being Nat such that
A26: i + 1 = i1 by NAT_1:6;
A27: i1 <= len G by A12, MATRIX_1:38;
then A28: i < len G by A26, NAT_1:13;
A29: i1 -' 1 = i by A26, NAT_D:34;
reconsider i = i, j = j as Element of NAT by ORDINAL1:def_12;
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 A17;
supposeA30: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: (left_cell (f,k,G)) /\ (right_cell (f,k,G)) = LSeg (f,k)
then A31: right_cell (f,k,G) = cell (G,i1,j1) by A1, A2, A3, A12, A13, A5, A6, Th22;
 ( j1 < width G & left_cell (f,k,G) = cell (G,i,j1) ) by A1, A2, A3, A12, A13, A5, A6, A10, A29, A30, Th21, NAT_1:13;
hence (left_cell (f,k,G)) /\ (right_cell (f,k,G)) = LSeg ((G * (i1,j1)),(G * (i1,(j1 + 1)))) by A14, A26, A28, A31, GOBOARD5:25
.= LSeg (f,k) by A1, A2, A13, A6, A30, TOPREAL1:def_3 ;
::_thesis: verum
end;
supposeA32: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: (left_cell (f,k,G)) /\ (right_cell (f,k,G)) = LSeg (f,k)
then A33: right_cell (f,k,G) = cell (G,i1,j) by A1, A2, A3, A12, A13, A5, A6, A22, Th24;
 ( i1 < len G & left_cell (f,k,G) = cell (G,i1,j1) ) by A1, A2, A3, A12, A13, A5, A6, A8, A32, Th23, NAT_1:13;
hence (left_cell (f,k,G)) /\ (right_cell (f,k,G)) = LSeg ((G * (i1,j1)),(G * ((i1 + 1),j1))) by A25, A15, A24, A33, GOBOARD5:26
.= LSeg (f,k) by A1, A2, A13, A6, A32, TOPREAL1:def_3 ;
::_thesis: verum
end;
supposeA34: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: (left_cell (f,k,G)) /\ (right_cell (f,k,G)) = LSeg (f,k)
then A35: right_cell (f,k,G) = cell (G,i2,j1) by A1, A2, A3, A12, A13, A5, A6, Th26;
 ( i2 < len G & left_cell (f,k,G) = cell (G,i2,j) ) by A1, A2, A3, A12, A13, A5, A6, A27, A22, A34, Th25, NAT_1:13;
hence (left_cell (f,k,G)) /\ (right_cell (f,k,G)) = LSeg ((G * ((i2 + 1),j1)),(G * (i2,j1))) by A9, A15, A24, A35, GOBOARD5:26
.= LSeg (f,k) by A1, A2, A13, A6, A34, TOPREAL1:def_3 ;
::_thesis: verum
end;
supposeA36: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: (left_cell (f,k,G)) /\ (right_cell (f,k,G)) = LSeg (f,k)
then A37: right_cell (f,k,G) = cell (G,i,j2) by A1, A2, A3, A12, A13, A5, A6, A29, Th28;
 ( j2 < width G & left_cell (f,k,G) = cell (G,i1,j2) ) by A1, A2, A3, A12, A13, A5, A6, A23, A36, Th27, NAT_1:13;
hence (left_cell (f,k,G)) /\ (right_cell (f,k,G)) = LSeg ((G * (i1,(j2 + 1))),(G * (i1,j2))) by A7, A26, A28, A37, GOBOARD5:25
.= LSeg (f,k) by A1, A2, A13, A6, A36, TOPREAL1:def_3 ;
::_thesis: verum
end;
end;
end;

theorem :: GOBRD13:30
for k being Element of NAT
for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G holds
right_cell (f,k,G) is closed
proof
let k be Element of NAT ; ::_thesis: for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G holds
right_cell (f,k,G) is closed
let f be FinSequence of (TOP-REAL 2); ::_thesis: for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G holds
right_cell (f,k,G) is closed
let G be Go-board; ::_thesis: ( 1 <= k & k + 1 <= len f & f is_sequence_on G implies right_cell (f,k,G) is closed )
assume A1: ( 1 <= k & k + 1 <= len f & f is_sequence_on G ) ; ::_thesis: right_cell (f,k,G) is closed
then consider i1, j1, i2, j2 being Element of NAT  such that
A2: ( [i1,j1] in Indices G & f /. k = G * (i1,j1) & [i2,j2] in Indices G & f /. (k + 1) = G * (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 JORDAN8:3;
 ( ( i1 = i2 & j1 + 1 = j2 & right_cell (f,k,G) = cell (G,i1,j1) ) or ( i1 + 1 = i2 & j1 = j2 & right_cell (f,k,G) = cell (G,i1,(j1 -' 1)) ) or ( i1 = i2 + 1 & j1 = j2 & right_cell (f,k,G) = cell (G,i2,j2) ) or ( i1 = i2 & j1 = j2 + 1 & right_cell (f,k,G) = cell (G,(i1 -' 1),j2) ) ) by A1, A2, Def2;
hence  right_cell (f,k,G) is closed by GOBRD11:33; ::_thesis: verum
end;

theorem :: GOBRD13:31
for k, n being Element of NAT
for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & k + 1 <= n holds
( left_cell (f,k,G) = left_cell ((f | n),k,G) & right_cell (f,k,G) = right_cell ((f | n),k,G) )
proof
let k, n be Element of NAT ; ::_thesis: for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & k + 1 <= n holds
( left_cell (f,k,G) = left_cell ((f | n),k,G) & right_cell (f,k,G) = right_cell ((f | n),k,G) )
let f be FinSequence of (TOP-REAL 2); ::_thesis: for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & k + 1 <= n holds
( left_cell (f,k,G) = left_cell ((f | n),k,G) & right_cell (f,k,G) = right_cell ((f | n),k,G) )
let G be Go-board; ::_thesis: ( 1 <= k & k + 1 <= len f & f is_sequence_on G & k + 1 <= n implies ( left_cell (f,k,G) = left_cell ((f | n),k,G) & right_cell (f,k,G) = right_cell ((f | n),k,G) ) )
assume that
A1: 1 <= k and
A2: k + 1 <= len f and
A3: f is_sequence_on G and
A4: k + 1 <= n ; ::_thesis: ( left_cell (f,k,G) = left_cell ((f | n),k,G) & right_cell (f,k,G) = right_cell ((f | n),k,G) )
percases ( len f <= n or n < len f ) ;
suppose len f <= n ; ::_thesis: ( left_cell (f,k,G) = left_cell ((f | n),k,G) & right_cell (f,k,G) = right_cell ((f | n),k,G) )
hence  ( left_cell (f,k,G) = left_cell ((f | n),k,G) & right_cell (f,k,G) = right_cell ((f | n),k,G) ) by FINSEQ_1:58; ::_thesis: verum
end;
suppose n < len f ; ::_thesis: ( left_cell (f,k,G) = left_cell ((f | n),k,G) & right_cell (f,k,G) = right_cell ((f | n),k,G) )
then A5: len (f | n) = n by FINSEQ_1:59;
then  k in dom (f | n) by A1, A4, SEQ_4:134;
then A6: (f | n) /. k = f /. k by FINSEQ_4:70;
 k + 1 in dom (f | n) by A1, A4, A5, SEQ_4:134;
then A7: (f | n) /. (k + 1) = f /. (k + 1) by FINSEQ_4:70;
set lf = left_cell (f,k,G);
set lfn = left_cell ((f | n),k,G);
set rf = right_cell (f,k,G);
set rfn = right_cell ((f | n),k,G);
A8: f | n is_sequence_on G by A3, GOBOARD1:22;
consider i1, j1, i2, j2 being Element of NAT  such that
A9: ( [i1,j1] in Indices G & f /. k = G * (i1,j1) & [i2,j2] in Indices G & f /. (k + 1) = G * (i2,j2) ) and
A10: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A1, A2, A3, JORDAN8:3;
A11: ( j1 + 1 > j1 & j2 + 1 > j2 ) by NAT_1:13;
A12: ( i1 + 1 > i1 & i2 + 1 > i2 ) by NAT_1:13;
now__::_thesis:_(_left_cell_(f,k,G)_=_left_cell_((f_|_n),k,G)_&_right_cell_(f,k,G)_=_right_cell_((f_|_n),k,G)_)
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 A10;
supposeA13: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: ( left_cell (f,k,G) = left_cell ((f | n),k,G) & right_cell (f,k,G) = right_cell ((f | n),k,G) )
hence  left_cell (f,k,G) = cell (G,(i1 -' 1),j1) by A1, A2, A3, A9, A11, Def3
.= left_cell ((f | n),k,G) by A1, A4, A9, A11, A8, A5, A6, A7, A13, Def3 ;
::_thesis: right_cell (f,k,G) = right_cell ((f | n),k,G)
thus  right_cell (f,k,G) = cell (G,i1,j1) by A1, A2, A3, A9, A11, A13, Def2
.= right_cell ((f | n),k,G) by A1, A4, A9, A11, A8, A5, A6, A7, A13, Def2 ; ::_thesis: verum
end;
supposeA14: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: ( left_cell (f,k,G) = left_cell ((f | n),k,G) & right_cell (f,k,G) = right_cell ((f | n),k,G) )
hence  left_cell (f,k,G) = cell (G,i1,j1) by A1, A2, A3, A9, A12, Def3
.= left_cell ((f | n),k,G) by A1, A4, A9, A12, A8, A5, A6, A7, A14, Def3 ;
::_thesis: right_cell (f,k,G) = right_cell ((f | n),k,G)
thus  right_cell (f,k,G) = cell (G,i1,(j1 -' 1)) by A1, A2, A3, A9, A12, A14, Def2
.= right_cell ((f | n),k,G) by A1, A4, A9, A12, A8, A5, A6, A7, A14, Def2 ; ::_thesis: verum
end;
supposeA15: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: ( left_cell (f,k,G) = left_cell ((f | n),k,G) & right_cell (f,k,G) = right_cell ((f | n),k,G) )
hence  left_cell (f,k,G) = cell (G,i2,(j2 -' 1)) by A1, A2, A3, A9, A12, Def3
.= left_cell ((f | n),k,G) by A1, A4, A9, A12, A8, A5, A6, A7, A15, Def3 ;
::_thesis: right_cell (f,k,G) = right_cell ((f | n),k,G)
thus  right_cell (f,k,G) = cell (G,i2,j2) by A1, A2, A3, A9, A12, A15, Def2
.= right_cell ((f | n),k,G) by A1, A4, A9, A12, A8, A5, A6, A7, A15, Def2 ; ::_thesis: verum
end;
supposeA16: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: ( left_cell (f,k,G) = left_cell ((f | n),k,G) & right_cell (f,k,G) = right_cell ((f | n),k,G) )
hence  left_cell (f,k,G) = cell (G,i1,j2) by A1, A2, A3, A9, A11, Def3
.= left_cell ((f | n),k,G) by A1, A4, A9, A11, A8, A5, A6, A7, A16, Def3 ;
::_thesis: right_cell (f,k,G) = right_cell ((f | n),k,G)
thus  right_cell (f,k,G) = cell (G,(i1 -' 1),j2) by A1, A2, A3, A9, A11, A16, Def2
.= right_cell ((f | n),k,G) by A1, A4, A9, A11, A8, A5, A6, A7, A16, Def2 ; ::_thesis: verum
end;
end;
end;
hence  ( left_cell (f,k,G) = left_cell ((f | n),k,G) & right_cell (f,k,G) = right_cell ((f | n),k,G) ) ; ::_thesis: verum
end;
end;
end;

theorem :: GOBRD13:32
for k, n being Element of NAT
for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len (f /^ n) & n <= len f & f is_sequence_on G holds
( left_cell (f,(k + n),G) = left_cell ((f /^ n),k,G) & right_cell (f,(k + n),G) = right_cell ((f /^ n),k,G) )
proof
let k, n be Element of NAT ; ::_thesis: for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len (f /^ n) & n <= len f & f is_sequence_on G holds
( left_cell (f,(k + n),G) = left_cell ((f /^ n),k,G) & right_cell (f,(k + n),G) = right_cell ((f /^ n),k,G) )
let f be FinSequence of (TOP-REAL 2); ::_thesis: for G being Go-board st 1 <= k & k + 1 <= len (f /^ n) & n <= len f & f is_sequence_on G holds
( left_cell (f,(k + n),G) = left_cell ((f /^ n),k,G) & right_cell (f,(k + n),G) = right_cell ((f /^ n),k,G) )
let G be Go-board; ::_thesis: ( 1 <= k & k + 1 <= len (f /^ n) & n <= len f & f is_sequence_on G implies ( left_cell (f,(k + n),G) = left_cell ((f /^ n),k,G) & right_cell (f,(k + n),G) = right_cell ((f /^ n),k,G) ) )
set g = f /^ n;
assume that
A1: 1 <= k and
A2: k + 1 <= len (f /^ n) and
A3: n <= len f and
A4: f is_sequence_on G ; ::_thesis: ( left_cell (f,(k + n),G) = left_cell ((f /^ n),k,G) & right_cell (f,(k + n),G) = right_cell ((f /^ n),k,G) )
A5: ( len (f /^ n) = (len f) - n & (k + 1) + n <= (len (f /^ n)) + n ) by A2, A3, RFINSEQ:def_1, XREAL_1:6;
 k in dom (f /^ n) by A1, A2, SEQ_4:134;
then A6: (f /^ n) /. k = f /. (k + n) by FINSEQ_5:27;
set lf = left_cell (f,(k + n),G);
set lfn = left_cell ((f /^ n),k,G);
set rf = right_cell (f,(k + n),G);
set rfn = right_cell ((f /^ n),k,G);
A7: ( (k + 1) + n = (k + n) + 1 & 1 <= k + n ) by A1, NAT_1:12;
 k + 1 in dom (f /^ n) by A1, A2, SEQ_4:134;
then A8: (f /^ n) /. (k + 1) = f /. ((k + 1) + n) by FINSEQ_5:27;
A9: f /^ n is_sequence_on G by A4, JORDAN8:2;
then consider i1, j1, i2, j2 being Element of NAT  such that
A10: ( [i1,j1] in Indices G & (f /^ n) /. k = G * (i1,j1) & [i2,j2] in Indices G & (f /^ n) /. (k + 1) = G * (i2,j2) ) and
A11: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A1, A2, JORDAN8:3;
A12: ( j1 + 1 > j1 & j2 + 1 > j2 ) by NAT_1:13;
A13: ( i1 + 1 > i1 & i2 + 1 > i2 ) by NAT_1:13;
now__::_thesis:_(_left_cell_(f,(k_+_n),G)_=_left_cell_((f_/^_n),k,G)_&_right_cell_(f,(k_+_n),G)_=_right_cell_((f_/^_n),k,G)_)
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 A11;
supposeA14: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: ( left_cell (f,(k + n),G) = left_cell ((f /^ n),k,G) & right_cell (f,(k + n),G) = right_cell ((f /^ n),k,G) )
hence  left_cell (f,(k + n),G) = cell (G,(i1 -' 1),j1) by A4, A10, A12, A6, A8, A5, A7, Def3
.= left_cell ((f /^ n),k,G) by A1, A2, A9, A10, A12, A14, Def3 ;
::_thesis: right_cell (f,(k + n),G) = right_cell ((f /^ n),k,G)
thus  right_cell (f,(k + n),G) = cell (G,i1,j1) by A4, A10, A12, A6, A8, A5, A7, A14, Def2
.= right_cell ((f /^ n),k,G) by A1, A2, A9, A10, A12, A14, Def2 ; ::_thesis: verum
end;
supposeA15: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: ( left_cell (f,(k + n),G) = left_cell ((f /^ n),k,G) & right_cell (f,(k + n),G) = right_cell ((f /^ n),k,G) )
hence  left_cell (f,(k + n),G) = cell (G,i1,j1) by A4, A10, A13, A6, A8, A5, A7, Def3
.= left_cell ((f /^ n),k,G) by A1, A2, A9, A10, A13, A15, Def3 ;
::_thesis: right_cell (f,(k + n),G) = right_cell ((f /^ n),k,G)
thus  right_cell (f,(k + n),G) = cell (G,i1,(j1 -' 1)) by A4, A10, A13, A6, A8, A5, A7, A15, Def2
.= right_cell ((f /^ n),k,G) by A1, A2, A9, A10, A13, A15, Def2 ; ::_thesis: verum
end;
supposeA16: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: ( left_cell (f,(k + n),G) = left_cell ((f /^ n),k,G) & right_cell (f,(k + n),G) = right_cell ((f /^ n),k,G) )
hence  left_cell (f,(k + n),G) = cell (G,i2,(j2 -' 1)) by A4, A10, A13, A6, A8, A5, A7, Def3
.= left_cell ((f /^ n),k,G) by A1, A2, A9, A10, A13, A16, Def3 ;
::_thesis: right_cell (f,(k + n),G) = right_cell ((f /^ n),k,G)
thus  right_cell (f,(k + n),G) = cell (G,i2,j2) by A4, A10, A13, A6, A8, A5, A7, A16, Def2
.= right_cell ((f /^ n),k,G) by A1, A2, A9, A10, A13, A16, Def2 ; ::_thesis: verum
end;
supposeA17: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: ( left_cell (f,(k + n),G) = left_cell ((f /^ n),k,G) & right_cell (f,(k + n),G) = right_cell ((f /^ n),k,G) )
hence  left_cell (f,(k + n),G) = cell (G,i1,j2) by A4, A10, A12, A6, A8, A5, A7, Def3
.= left_cell ((f /^ n),k,G) by A1, A2, A9, A10, A12, A17, Def3 ;
::_thesis: right_cell (f,(k + n),G) = right_cell ((f /^ n),k,G)
thus  right_cell (f,(k + n),G) = cell (G,(i1 -' 1),j2) by A4, A10, A12, A6, A8, A5, A7, A17, Def2
.= right_cell ((f /^ n),k,G) by A1, A2, A9, A10, A12, A17, Def2 ; ::_thesis: verum
end;
end;
end;
hence  ( left_cell (f,(k + n),G) = left_cell ((f /^ n),k,G) & right_cell (f,(k + n),G) = right_cell ((f /^ n),k,G) ) ; ::_thesis: verum
end;

theorem :: GOBRD13:33
for n being Element of NAT
for G being Go-board
for f being standard special_circular_sequence st 1 <= n & n + 1 <= len f & f is_sequence_on G holds
( left_cell (f,n,G) c= left_cell (f,n) & right_cell (f,n,G) c= right_cell (f,n) )
proof
let n be Element of NAT ; ::_thesis: for G being Go-board
for f being standard special_circular_sequence st 1 <= n & n + 1 <= len f & f is_sequence_on G holds
( left_cell (f,n,G) c= left_cell (f,n) & right_cell (f,n,G) c= right_cell (f,n) )
let G be Go-board; ::_thesis: for f being standard special_circular_sequence st 1 <= n & n + 1 <= len f & f is_sequence_on G holds
( left_cell (f,n,G) c= left_cell (f,n) & right_cell (f,n,G) c= right_cell (f,n) )
let f be standard special_circular_sequence; ::_thesis: ( 1 <= n & n + 1 <= len f & f is_sequence_on G implies ( left_cell (f,n,G) c= left_cell (f,n) & right_cell (f,n,G) c= right_cell (f,n) ) )
assume that
A1: ( 1 <= n & n + 1 <= len f ) and
A2: f is_sequence_on G ; ::_thesis: ( left_cell (f,n,G) c= left_cell (f,n) & right_cell (f,n,G) c= right_cell (f,n) )
consider i1, j1, i2, j2 being Element of NAT  such that
A3: [i1,j1] in Indices G and
A4: f /. n = G * (i1,j1) and
A5: [i2,j2] in Indices G and
A6: f /. (n + 1) = G * (i2,j2) and
A7: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A1, A2, JORDAN8:3;
A8: 1 <= j1 by A3, MATRIX_1:38;
A9: ( j1 + 1 > j1 & j2 + 1 > j2 ) by NAT_1:13;
A10: ( i1 + 1 > i1 & i2 + 1 > i2 ) by NAT_1:13;
A11: j2 <= width G by A5, MATRIX_1:38;
A12: j1 <= width G by A3, MATRIX_1:38;
A13: i2 <= len G by A5, MATRIX_1:38;
A14: 1 <= i2 by A5, MATRIX_1:38;
then A15: (G * (i2,j1)) `2 = (G * (1,j1)) `2 by A8, A12, A13, GOBOARD5:1;
A16: 1 <= j2 by A5, MATRIX_1:38;
then A17: (G * (i2,j2)) `1 = (G * (i2,1)) `1 by A14, A13, A11, GOBOARD5:2;
A18: i1 <= len G by A3, MATRIX_1:38;
set F = GoB f;
A19: Values (GoB f) c= Values G by A2, Th20;
 f is_sequence_on GoB f by GOBOARD5:def_5;
then consider m, k, i, j being Element of NAT  such that
A20: [m,k] in Indices (GoB f) and
A21: f /. n = (GoB f) * (m,k) and
A22: [i,j] in Indices (GoB f) and
A23: f /. (n + 1) = (GoB f) * (i,j) and
 ( ( m = i & k + 1 = j ) or ( m + 1 = i & k = j ) or ( m = i + 1 & k = j ) or ( m = i & k = j + 1 ) ) by A1, JORDAN8:3;
A24: 1 <= m by A20, MATRIX_1:38;
A25: 1 <= i1 by A3, MATRIX_1:38;
then A26: (G * (i1,j1)) `1 = (G * (i1,1)) `1 by A18, A8, A12, GOBOARD5:2;
A27: (G * (i1,j1)) `2 = (G * (1,j1)) `2 by A25, A18, A8, A12, GOBOARD5:1;
A28: m <= len (GoB f) by A20, MATRIX_1:38;
A29: j + 1 > j by NAT_1:13;
A30: k + 1 > k by NAT_1:13;
A31: k + 1 >= 1 by NAT_1:12;
A32: j + 1 >= 1 by NAT_1:12;
A33: j <= width (GoB f) by A22, MATRIX_1:38;
A34: i + 1 > i by NAT_1:13;
A35: m + 1 > m by NAT_1:13;
A36: i <= len (GoB f) by A22, MATRIX_1:38;
A37: i + 1 >= 1 by NAT_1:12;
A38: m + 1 >= 1 by NAT_1:12;
A39: k <= width (GoB f) by A20, MATRIX_1:38;
A40: 1 <= j by A22, MATRIX_1:38;
then A41: ((GoB f) * (m,j)) `2 = ((GoB f) * (1,j)) `2 by A24, A28, A33, GOBOARD5:1;
A42: 1 <= i by A22, MATRIX_1:38;
then A43: ((GoB f) * (i,j)) `1 = ((GoB f) * (i,1)) `1 by A36, A40, A33, GOBOARD5:2;
A44: ((GoB f) * (i,j)) `2 = ((GoB f) * (1,j)) `2 by A42, A36, A40, A33, GOBOARD5:1;
A45: 1 <= k by A20, MATRIX_1:38;
then A46: ((GoB f) * (i,k)) `1 = ((GoB f) * (i,1)) `1 by A39, A42, A36, GOBOARD5:2;
percases ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A7;
supposeA47: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: ( left_cell (f,n,G) c= left_cell (f,n) & right_cell (f,n,G) c= right_cell (f,n) )
A48: now__::_thesis:_not_k_+_1_<_j
A49: (G * (i2,j2)) `2 = (G * (1,j2)) `2 by A14, A13, A16, A11, GOBOARD5:1;
assume A50: k + 1 < j ; ::_thesis: contradiction
then A51: k + 1 < width (GoB f) by A33, XXREAL_0:2;
then consider l, i9 being Element of NAT  such that
A52: l in dom f and
A53: [i9,(k + 1)] in Indices (GoB f) and
A54: f /. l = (GoB f) * (i9,(k + 1)) by JORDAN5D:8, NAT_1:12;
A55: ((GoB f) * (m,(k + 1))) `2 = ((GoB f) * (1,(k + 1))) `2 by A24, A28, A31, A51, GOBOARD5:1;
 ( 1 <= i9 & i9 <= len (GoB f) ) by A53, MATRIX_1:38;
then A56: ((GoB f) * (i9,(k + 1))) `2 = ((GoB f) * (1,(k + 1))) `2 by A31, A51, GOBOARD5:1;
consider i19, j19 being Element of NAT  such that
A57: [i19,j19] in Indices G and
A58: f /. l = G * (i19,j19) by A2, A52, GOBOARD1:def_9;
A59: 1 <= j19 by A57, MATRIX_1:38;
A60: ( 1 <= i19 & i19 <= len G ) by A57, MATRIX_1:38;
then A61: (G * (i19,j1)) `2 = (G * (1,j1)) `2 by A8, A12, GOBOARD5:1;
A62: now__::_thesis:_not_j1_>=_j19
assume  j1 >= j19 ; ::_thesis: contradiction
then  (G * (i19,j19)) `2 <= (G * (i1,j1)) `2 by A12, A27, A60, A59, A61, SPRECT_3:12;
hence  contradiction by A21, A24, A28, A45, A4, A30, A51, A54, A56, A55, A58, GOBOARD5:4; ::_thesis: verum
end;
A63: j19 <= width G by A57, MATRIX_1:38;
A64: (G * (i19,j2)) `2 = (G * (1,j2)) `2 by A16, A11, A60, GOBOARD5:1;
now__::_thesis:_not_j2_<=_j19
assume  j2 <= j19 ; ::_thesis: contradiction
then  (G * (i2,j2)) `2 <= (G * (i19,j19)) `2 by A16, A60, A63, A49, A64, SPRECT_3:12;
hence  contradiction by A23, A24, A28, A33, A44, A41, A6, A31, A50, A54, A56, A55, A58, GOBOARD5:4; ::_thesis: verum
end;
hence  contradiction by A47, A62, NAT_1:13; ::_thesis: verum
end;
now__::_thesis:_not_j_<=_k
assume  j <= k ; ::_thesis: contradiction
then A65: ((GoB f) * (i,j)) `2 <= ((GoB f) * (m,k)) `2 by A24, A28, A39, A40, A44, A41, SPRECT_3:12;
 j1 < j2 by A47, NAT_1:13;
hence  contradiction by A21, A23, A4, A6, A8, A14, A13, A11, A27, A15, A65, GOBOARD5:4; ::_thesis: verum
end;
then  k + 1 <= j by NAT_1:13;
then  k + 1 = j by A48, XXREAL_0:1;
then A66: ( right_cell (f,n) = cell ((GoB f),m,k) & left_cell (f,n) = cell ((GoB f),(m -' 1),k) ) by A1, A20, A21, A22, A23, A30, A29, GOBOARD5:def_6, GOBOARD5:def_7;
 ( right_cell (f,n,G) = cell (G,i1,j1) & left_cell (f,n,G) = cell (G,(i1 -' 1),j1) ) by A1, A2, A3, A4, A5, A6, A9, A47, Def2, Def3;
hence  ( left_cell (f,n,G) c= left_cell (f,n) & right_cell (f,n,G) c= right_cell (f,n) ) by A19, A20, A21, A3, A4, A66, Th17, Th18; ::_thesis: verum
end;
supposeA67: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: ( left_cell (f,n,G) c= left_cell (f,n) & right_cell (f,n,G) c= right_cell (f,n) )
A68: now__::_thesis:_not_m_+_1_<_i
assume A69: m + 1 < i ; ::_thesis: contradiction
then A70: m + 1 < len (GoB f) by A36, XXREAL_0:2;
then consider l, j9 being Element of NAT  such that
A71: l in dom f and
A72: [(m + 1),j9] in Indices (GoB f) and
A73: f /. l = (GoB f) * ((m + 1),j9) by JORDAN5D:7, NAT_1:12;
A74: ((GoB f) * ((m + 1),k)) `1 = ((GoB f) * ((m + 1),1)) `1 by A45, A39, A38, A70, GOBOARD5:2;
 ( 1 <= j9 & j9 <= width (GoB f) ) by A72, MATRIX_1:38;
then A75: ((GoB f) * ((m + 1),j9)) `1 = ((GoB f) * ((m + 1),1)) `1 by A38, A70, GOBOARD5:2;
A76: ( 1 <= m + 1 & ((GoB f) * ((m + 1),j)) `1 = ((GoB f) * ((m + 1),1)) `1 ) by A40, A33, A38, A70, GOBOARD5:2;
A77: (G * (i2,j2)) `1 = (G * (i2,1)) `1 by A14, A13, A16, A11, GOBOARD5:2;
consider i19, j19 being Element of NAT  such that
A78: [i19,j19] in Indices G and
A79: f /. l = G * (i19,j19) by A2, A71, GOBOARD1:def_9;
A80: i19 <= len G by A78, MATRIX_1:38;
A81: ( 1 <= j19 & j19 <= width G ) by A78, MATRIX_1:38;
A82: 1 <= i19 by A78, MATRIX_1:38;
then A83: (G * (i19,j19)) `1 = (G * (i19,1)) `1 by A80, A81, GOBOARD5:2;
A84: (G * (i19,j1)) `1 = (G * (i19,1)) `1 by A8, A12, A82, A80, GOBOARD5:2;
A85: now__::_thesis:_not_i1_>=_i19
assume  i1 >= i19 ; ::_thesis: contradiction
then  (G * (i19,j19)) `1 <= (G * (i1,j1)) `1 by A18, A8, A12, A82, A83, A84, SPRECT_3:13;
hence  contradiction by A21, A24, A45, A39, A4, A35, A70, A73, A75, A74, A79, GOBOARD5:3; ::_thesis: verum
end;
A86: (G * (i2,j19)) `1 = (G * (i2,1)) `1 by A14, A13, A81, GOBOARD5:2;
now__::_thesis:_not_i2_<=_i19
assume  i2 <= i19 ; ::_thesis: contradiction
then  (G * (i2,j2)) `1 <= (G * (i19,j19)) `1 by A14, A80, A81, A77, A86, SPRECT_3:13;
hence  contradiction by A23, A36, A40, A33, A6, A69, A73, A75, A76, A79, GOBOARD5:3; ::_thesis: verum
end;
hence  contradiction by A67, A85, NAT_1:13; ::_thesis: verum
end;
now__::_thesis:_not_i_<=_m
assume  i <= m ; ::_thesis: contradiction
then A87: ((GoB f) * (i,j)) `1 <= ((GoB f) * (m,k)) `1 by A28, A45, A39, A42, A43, A46, SPRECT_3:13;
 i1 < i2 by A67, NAT_1:13;
hence  contradiction by A21, A23, A4, A6, A25, A8, A12, A13, A67, A87, GOBOARD5:3; ::_thesis: verum
end;
then  m + 1 <= i by NAT_1:13;
then  m + 1 = i by A68, XXREAL_0:1;
then A88: ( right_cell (f,n) = cell ((GoB f),m,(k -' 1)) & left_cell (f,n) = cell ((GoB f),m,k) ) by A1, A20, A21, A22, A23, A35, A34, GOBOARD5:def_6, GOBOARD5:def_7;
 ( right_cell (f,n,G) = cell (G,i1,(j1 -' 1)) & left_cell (f,n,G) = cell (G,i1,j1) ) by A1, A2, A3, A4, A5, A6, A10, A67, Def2, Def3;
hence  ( left_cell (f,n,G) c= left_cell (f,n) & right_cell (f,n,G) c= right_cell (f,n) ) by A19, A20, A21, A3, A4, A88, Th17, Th19; ::_thesis: verum
end;
supposeA89: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: ( left_cell (f,n,G) c= left_cell (f,n) & right_cell (f,n,G) c= right_cell (f,n) )
A90: now__::_thesis:_not_m_>_i_+_1
assume A91: m > i + 1 ; ::_thesis: contradiction
then A92: i + 1 < len (GoB f) by A28, XXREAL_0:2;
then consider l, j9 being Element of NAT  such that
A93: l in dom f and
A94: [(i + 1),j9] in Indices (GoB f) and
A95: f /. l = (GoB f) * ((i + 1),j9) by JORDAN5D:7, NAT_1:12;
A96: ( 1 <= i + 1 & ((GoB f) * ((i + 1),k)) `1 = ((GoB f) * ((i + 1),1)) `1 ) by A45, A39, A37, A92, GOBOARD5:2;
 ( 1 <= j9 & j9 <= width (GoB f) ) by A94, MATRIX_1:38;
then A97: ((GoB f) * ((i + 1),j9)) `1 = ((GoB f) * ((i + 1),1)) `1 by A37, A92, GOBOARD5:2;
A98: ((GoB f) * ((i + 1),j)) `1 = ((GoB f) * ((i + 1),1)) `1 by A40, A33, A37, A92, GOBOARD5:2;
A99: (G * (i2,j2)) `1 = (G * (i2,1)) `1 by A14, A13, A16, A11, GOBOARD5:2;
consider i19, j19 being Element of NAT  such that
A100: [i19,j19] in Indices G and
A101: f /. l = G * (i19,j19) by A2, A93, GOBOARD1:def_9;
A102: 1 <= i19 by A100, MATRIX_1:38;
A103: ( 1 <= j19 & j19 <= width G ) by A100, MATRIX_1:38;
A104: i19 <= len G by A100, MATRIX_1:38;
then A105: (G * (i19,j19)) `1 = (G * (i19,1)) `1 by A102, A103, GOBOARD5:2;
A106: (G * (i19,j1)) `1 = (G * (i19,1)) `1 by A8, A12, A102, A104, GOBOARD5:2;
A107: now__::_thesis:_not_i1_<=_i19
assume  i1 <= i19 ; ::_thesis: contradiction
then  (G * (i19,j19)) `1 >= (G * (i1,j1)) `1 by A25, A8, A12, A104, A105, A106, SPRECT_3:13;
hence  contradiction by A21, A28, A45, A39, A4, A91, A95, A97, A96, A101, GOBOARD5:3; ::_thesis: verum
end;
A108: (G * (i2,j19)) `1 = (G * (i2,1)) `1 by A14, A13, A103, GOBOARD5:2;
now__::_thesis:_not_i2_>=_i19
assume  i2 >= i19 ; ::_thesis: contradiction
then  (G * (i2,j2)) `1 >= (G * (i19,j19)) `1 by A13, A102, A103, A99, A108, SPRECT_3:13;
hence  contradiction by A23, A42, A40, A33, A6, A34, A92, A95, A97, A98, A101, GOBOARD5:3; ::_thesis: verum
end;
hence  contradiction by A89, A107, NAT_1:13; ::_thesis: verum
end;
now__::_thesis:_not_m_<=_i
assume  m <= i ; ::_thesis: contradiction
then A109: ((GoB f) * (i,j)) `1 >= ((GoB f) * (m,k)) `1 by A24, A45, A39, A36, A43, A46, SPRECT_3:13;
 i1 > i2 by A89, NAT_1:13;
hence  contradiction by A21, A23, A4, A6, A18, A8, A12, A14, A89, A109, GOBOARD5:3; ::_thesis: verum
end;
then  i + 1 <= m by NAT_1:13;
then  i + 1 = m by A90, XXREAL_0:1;
then A110: ( right_cell (f,n) = cell ((GoB f),i,j) & left_cell (f,n) = cell ((GoB f),i,(j -' 1)) ) by A1, A20, A21, A22, A23, A35, A34, GOBOARD5:def_6, GOBOARD5:def_7;
 ( right_cell (f,n,G) = cell (G,i2,j2) & left_cell (f,n,G) = cell (G,i2,(j2 -' 1)) ) by A1, A2, A3, A4, A5, A6, A10, A89, Def2, Def3;
hence  ( left_cell (f,n,G) c= left_cell (f,n) & right_cell (f,n,G) c= right_cell (f,n) ) by A19, A22, A23, A5, A6, A110, Th17, Th19; ::_thesis: verum
end;
supposeA111: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: ( left_cell (f,n,G) c= left_cell (f,n) & right_cell (f,n,G) c= right_cell (f,n) )
A112: now__::_thesis:_not_j_+_1_<_k
A113: (G * (i2,j2)) `2 = (G * (1,j2)) `2 by A14, A13, A16, A11, GOBOARD5:1;
assume A114: j + 1 < k ; ::_thesis: contradiction
then A115: j + 1 < width (GoB f) by A39, XXREAL_0:2;
then consider l, i9 being Element of NAT  such that
A116: l in dom f and
A117: [i9,(j + 1)] in Indices (GoB f) and
A118: f /. l = (GoB f) * (i9,(j + 1)) by JORDAN5D:8, NAT_1:12;
A119: ((GoB f) * (m,(j + 1))) `2 = ((GoB f) * (1,(j + 1))) `2 by A24, A28, A32, A115, GOBOARD5:1;
 ( 1 <= i9 & i9 <= len (GoB f) ) by A117, MATRIX_1:38;
then A120: ((GoB f) * (i9,(j + 1))) `2 = ((GoB f) * (1,(j + 1))) `2 by A32, A115, GOBOARD5:1;
consider i19, j19 being Element of NAT  such that
A121: [i19,j19] in Indices G and
A122: f /. l = G * (i19,j19) by A2, A116, GOBOARD1:def_9;
A123: j19 <= width G by A121, MATRIX_1:38;
A124: ( 1 <= i19 & i19 <= len G ) by A121, MATRIX_1:38;
then A125: (G * (i19,j1)) `2 = (G * (1,j1)) `2 by A8, A12, GOBOARD5:1;
A126: now__::_thesis:_not_j1_<=_j19
assume  j1 <= j19 ; ::_thesis: contradiction
then  (G * (i19,j19)) `2 >= (G * (i1,j1)) `2 by A8, A27, A124, A123, A125, SPRECT_3:12;
hence  contradiction by A21, A24, A28, A39, A4, A32, A114, A118, A120, A119, A122, GOBOARD5:4; ::_thesis: verum
end;
A127: ((GoB f) * (i,(j + 1))) `2 = ((GoB f) * (1,(j + 1))) `2 by A42, A36, A32, A115, GOBOARD5:1;
A128: 1 <= j19 by A121, MATRIX_1:38;
A129: (G * (i19,j2)) `2 = (G * (1,j2)) `2 by A16, A11, A124, GOBOARD5:1;
now__::_thesis:_not_j2_>=_j19
assume  j2 >= j19 ; ::_thesis: contradiction
then  (G * (i2,j2)) `2 >= (G * (i19,j19)) `2 by A11, A124, A128, A113, A129, SPRECT_3:12;
hence  contradiction by A23, A42, A36, A40, A6, A29, A115, A118, A120, A127, A122, GOBOARD5:4; ::_thesis: verum
end;
hence  contradiction by A111, A126, NAT_1:13; ::_thesis: verum
end;
now__::_thesis:_not_j_>=_k
assume  j >= k ; ::_thesis: contradiction
then A130: ((GoB f) * (i,j)) `2 >= ((GoB f) * (m,k)) `2 by A24, A28, A45, A33, A44, A41, SPRECT_3:12;
 j1 > j2 by A111, NAT_1:13;
hence  contradiction by A21, A23, A4, A6, A12, A14, A13, A16, A27, A15, A130, GOBOARD5:4; ::_thesis: verum
end;
then  j + 1 <= k by NAT_1:13;
then  j + 1 = k by A112, XXREAL_0:1;
then A131: ( right_cell (f,n) = cell ((GoB f),(m -' 1),j) & left_cell (f,n) = cell ((GoB f),m,j) ) by A1, A20, A21, A22, A23, A30, A29, GOBOARD5:def_6, GOBOARD5:def_7;
A132: now__::_thesis:_not_m_<>_i
assume A133: m <> i ; ::_thesis: contradiction
percases ( m < i or m > i ) by A133, XXREAL_0:1;
suppose m < i ; ::_thesis: contradiction
hence  contradiction by A21, A23, A24, A45, A39, A36, A43, A46, A4, A6, A26, A17, A111, GOBOARD5:3; ::_thesis: verum
end;
suppose m > i ; ::_thesis: contradiction
hence  contradiction by A21, A23, A28, A45, A39, A42, A43, A46, A4, A6, A26, A17, A111, GOBOARD5:3; ::_thesis: verum
end;
end;
end;
 ( right_cell (f,n,G) = cell (G,(i1 -' 1),j2) & left_cell (f,n,G) = cell (G,i1,j2) ) by A1, A2, A3, A4, A5, A6, A9, A111, Def2, Def3;
hence  ( left_cell (f,n,G) c= left_cell (f,n) & right_cell (f,n,G) c= right_cell (f,n) ) by A19, A22, A23, A5, A6, A111, A132, A131, Th17, Th18; ::_thesis: verum
end;
end;
end;

definition
let f be FinSequence of (TOP-REAL 2);
let G be Go-board;
let k be Element of NAT ;
assume  ( 1 <= k & k + 1 <= len f & f is_sequence_on G ) ;
then consider i1, j1, i2, j2 being Element of NAT  such that
A1: ( [i1,j1] in Indices G & f /. k = G * (i1,j1) & [i2,j2] in Indices G & f /. (k + 1) = G * (i2,j2) ) and
A2: ( ( 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;
func front_right_cell (f,k,G) ->  Subset of (TOP-REAL 2) means :Def4: :: GOBRD13:def 4
 for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & it = cell (G,i2,j2) ) & not ( i1 + 1 = i2 & j1 = j2 & it = cell (G,i2,(j2 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & it = cell (G,(i2 -' 1),j2) ) holds
( i1 = i2 & j1 = j2 + 1 & it = cell (G,(i2 -' 1),(j2 -' 1)) );
existence
 ex b1 being Subset of (TOP-REAL 2) st
for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & b1 = cell (G,i2,j2) ) & not ( i1 + 1 = i2 & j1 = j2 & b1 = cell (G,i2,(j2 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & b1 = cell (G,(i2 -' 1),j2) ) holds
( i1 = i2 & j1 = j2 + 1 & b1 = cell (G,(i2 -' 1),(j2 -' 1)) )
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 A2;
supposeA3: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: ex b1 being Subset of (TOP-REAL 2) st
for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & b1 = cell (G,i2,j2) ) & not ( i1 + 1 = i2 & j1 = j2 & b1 = cell (G,i2,(j2 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & b1 = cell (G,(i2 -' 1),j2) ) holds
( i1 = i2 & j1 = j2 + 1 & b1 = cell (G,(i2 -' 1),(j2 -' 1)) )
take  cell (G,i2,j2) ; ::_thesis: for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & cell (G,i2,j2) = cell (G,i2,j2) ) & not ( i1 + 1 = i2 & j1 = j2 & cell (G,i2,j2) = cell (G,i2,(j2 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & cell (G,i2,j2) = cell (G,(i2 -' 1),j2) ) holds
( i1 = i2 & j1 = j2 + 1 & cell (G,i2,j2) = cell (G,(i2 -' 1),(j2 -' 1)) )
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices G & [i29,j29] in Indices G & f /. k = G * (i19,j19) & f /. (k + 1) = G * (i29,j29) & not ( i19 = i29 & j19 + 1 = j29 & cell (G,i2,j2) = cell (G,i29,j29) ) & not ( i19 + 1 = i29 & j19 = j29 & cell (G,i2,j2) = cell (G,i29,(j29 -' 1)) ) & not ( i19 = i29 + 1 & j19 = j29 & cell (G,i2,j2) = cell (G,(i29 -' 1),j29) ) implies ( i19 = i29 & j19 = j29 + 1 & cell (G,i2,j2) = cell (G,(i29 -' 1),(j29 -' 1)) ) )
assume A4: ( [i19,j19] in Indices G & [i29,j29] in Indices G & f /. k = G * (i19,j19) & f /. (k + 1) = G * (i29,j29) ) ; ::_thesis: ( ( i19 = i29 & j19 + 1 = j29 & cell (G,i2,j2) = cell (G,i29,j29) ) or ( i19 + 1 = i29 & j19 = j29 & cell (G,i2,j2) = cell (G,i29,(j29 -' 1)) ) or ( i19 = i29 + 1 & j19 = j29 & cell (G,i2,j2) = cell (G,(i29 -' 1),j29) ) or ( i19 = i29 & j19 = j29 + 1 & cell (G,i2,j2) = cell (G,(i29 -' 1),(j29 -' 1)) ) )
then  ( i2 = i29 & j1 = j19 ) by A1, GOBOARD1:5;
hence  ( ( i19 = i29 & j19 + 1 = j29 & cell (G,i2,j2) = cell (G,i29,j29) ) or ( i19 + 1 = i29 & j19 = j29 & cell (G,i2,j2) = cell (G,i29,(j29 -' 1)) ) or ( i19 = i29 + 1 & j19 = j29 & cell (G,i2,j2) = cell (G,(i29 -' 1),j29) ) or ( i19 = i29 & j19 = j29 + 1 & cell (G,i2,j2) = cell (G,(i29 -' 1),(j29 -' 1)) ) ) by A1, A3, A4, GOBOARD1:5; ::_thesis: verum
end;
supposeA5: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: ex b1 being Subset of (TOP-REAL 2) st
for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & b1 = cell (G,i2,j2) ) & not ( i1 + 1 = i2 & j1 = j2 & b1 = cell (G,i2,(j2 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & b1 = cell (G,(i2 -' 1),j2) ) holds
( i1 = i2 & j1 = j2 + 1 & b1 = cell (G,(i2 -' 1),(j2 -' 1)) )
take  cell (G,i2,(j2 -' 1)) ; ::_thesis: for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & cell (G,i2,(j2 -' 1)) = cell (G,i2,j2) ) & not ( i1 + 1 = i2 & j1 = j2 & cell (G,i2,(j2 -' 1)) = cell (G,i2,(j2 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & cell (G,i2,(j2 -' 1)) = cell (G,(i2 -' 1),j2) ) holds
( i1 = i2 & j1 = j2 + 1 & cell (G,i2,(j2 -' 1)) = cell (G,(i2 -' 1),(j2 -' 1)) )
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices G & [i29,j29] in Indices G & f /. k = G * (i19,j19) & f /. (k + 1) = G * (i29,j29) & not ( i19 = i29 & j19 + 1 = j29 & cell (G,i2,(j2 -' 1)) = cell (G,i29,j29) ) & not ( i19 + 1 = i29 & j19 = j29 & cell (G,i2,(j2 -' 1)) = cell (G,i29,(j29 -' 1)) ) & not ( i19 = i29 + 1 & j19 = j29 & cell (G,i2,(j2 -' 1)) = cell (G,(i29 -' 1),j29) ) implies ( i19 = i29 & j19 = j29 + 1 & cell (G,i2,(j2 -' 1)) = cell (G,(i29 -' 1),(j29 -' 1)) ) )
assume A6: ( [i19,j19] in Indices G & [i29,j29] in Indices G & f /. k = G * (i19,j19) & f /. (k + 1) = G * (i29,j29) ) ; ::_thesis: ( ( i19 = i29 & j19 + 1 = j29 & cell (G,i2,(j2 -' 1)) = cell (G,i29,j29) ) or ( i19 + 1 = i29 & j19 = j29 & cell (G,i2,(j2 -' 1)) = cell (G,i29,(j29 -' 1)) ) or ( i19 = i29 + 1 & j19 = j29 & cell (G,i2,(j2 -' 1)) = cell (G,(i29 -' 1),j29) ) or ( i19 = i29 & j19 = j29 + 1 & cell (G,i2,(j2 -' 1)) = cell (G,(i29 -' 1),(j29 -' 1)) ) )
then  ( i2 = i29 & j1 = j19 ) by A1, GOBOARD1:5;
hence  ( ( i19 = i29 & j19 + 1 = j29 & cell (G,i2,(j2 -' 1)) = cell (G,i29,j29) ) or ( i19 + 1 = i29 & j19 = j29 & cell (G,i2,(j2 -' 1)) = cell (G,i29,(j29 -' 1)) ) or ( i19 = i29 + 1 & j19 = j29 & cell (G,i2,(j2 -' 1)) = cell (G,(i29 -' 1),j29) ) or ( i19 = i29 & j19 = j29 + 1 & cell (G,i2,(j2 -' 1)) = cell (G,(i29 -' 1),(j29 -' 1)) ) ) by A1, A5, A6, GOBOARD1:5; ::_thesis: verum
end;
supposeA7: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: ex b1 being Subset of (TOP-REAL 2) st
for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & b1 = cell (G,i2,j2) ) & not ( i1 + 1 = i2 & j1 = j2 & b1 = cell (G,i2,(j2 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & b1 = cell (G,(i2 -' 1),j2) ) holds
( i1 = i2 & j1 = j2 + 1 & b1 = cell (G,(i2 -' 1),(j2 -' 1)) )
take  cell (G,(i2 -' 1),j2) ; ::_thesis: for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & cell (G,(i2 -' 1),j2) = cell (G,i2,j2) ) & not ( i1 + 1 = i2 & j1 = j2 & cell (G,(i2 -' 1),j2) = cell (G,i2,(j2 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & cell (G,(i2 -' 1),j2) = cell (G,(i2 -' 1),j2) ) holds
( i1 = i2 & j1 = j2 + 1 & cell (G,(i2 -' 1),j2) = cell (G,(i2 -' 1),(j2 -' 1)) )
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices G & [i29,j29] in Indices G & f /. k = G * (i19,j19) & f /. (k + 1) = G * (i29,j29) & not ( i19 = i29 & j19 + 1 = j29 & cell (G,(i2 -' 1),j2) = cell (G,i29,j29) ) & not ( i19 + 1 = i29 & j19 = j29 & cell (G,(i2 -' 1),j2) = cell (G,i29,(j29 -' 1)) ) & not ( i19 = i29 + 1 & j19 = j29 & cell (G,(i2 -' 1),j2) = cell (G,(i29 -' 1),j29) ) implies ( i19 = i29 & j19 = j29 + 1 & cell (G,(i2 -' 1),j2) = cell (G,(i29 -' 1),(j29 -' 1)) ) )
assume A8: ( [i19,j19] in Indices G & [i29,j29] in Indices G & f /. k = G * (i19,j19) & f /. (k + 1) = G * (i29,j29) ) ; ::_thesis: ( ( i19 = i29 & j19 + 1 = j29 & cell (G,(i2 -' 1),j2) = cell (G,i29,j29) ) or ( i19 + 1 = i29 & j19 = j29 & cell (G,(i2 -' 1),j2) = cell (G,i29,(j29 -' 1)) ) or ( i19 = i29 + 1 & j19 = j29 & cell (G,(i2 -' 1),j2) = cell (G,(i29 -' 1),j29) ) or ( i19 = i29 & j19 = j29 + 1 & cell (G,(i2 -' 1),j2) = cell (G,(i29 -' 1),(j29 -' 1)) ) )
then  ( i2 = i29 & j1 = j19 ) by A1, GOBOARD1:5;
hence  ( ( i19 = i29 & j19 + 1 = j29 & cell (G,(i2 -' 1),j2) = cell (G,i29,j29) ) or ( i19 + 1 = i29 & j19 = j29 & cell (G,(i2 -' 1),j2) = cell (G,i29,(j29 -' 1)) ) or ( i19 = i29 + 1 & j19 = j29 & cell (G,(i2 -' 1),j2) = cell (G,(i29 -' 1),j29) ) or ( i19 = i29 & j19 = j29 + 1 & cell (G,(i2 -' 1),j2) = cell (G,(i29 -' 1),(j29 -' 1)) ) ) by A1, A7, A8, GOBOARD1:5; ::_thesis: verum
end;
supposeA9: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: ex b1 being Subset of (TOP-REAL 2) st
for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & b1 = cell (G,i2,j2) ) & not ( i1 + 1 = i2 & j1 = j2 & b1 = cell (G,i2,(j2 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & b1 = cell (G,(i2 -' 1),j2) ) holds
( i1 = i2 & j1 = j2 + 1 & b1 = cell (G,(i2 -' 1),(j2 -' 1)) )
take  cell (G,(i2 -' 1),(j2 -' 1)) ; ::_thesis: for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & cell (G,(i2 -' 1),(j2 -' 1)) = cell (G,i2,j2) ) & not ( i1 + 1 = i2 & j1 = j2 & cell (G,(i2 -' 1),(j2 -' 1)) = cell (G,i2,(j2 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & cell (G,(i2 -' 1),(j2 -' 1)) = cell (G,(i2 -' 1),j2) ) holds
( i1 = i2 & j1 = j2 + 1 & cell (G,(i2 -' 1),(j2 -' 1)) = cell (G,(i2 -' 1),(j2 -' 1)) )
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices G & [i29,j29] in Indices G & f /. k = G * (i19,j19) & f /. (k + 1) = G * (i29,j29) & not ( i19 = i29 & j19 + 1 = j29 & cell (G,(i2 -' 1),(j2 -' 1)) = cell (G,i29,j29) ) & not ( i19 + 1 = i29 & j19 = j29 & cell (G,(i2 -' 1),(j2 -' 1)) = cell (G,i29,(j29 -' 1)) ) & not ( i19 = i29 + 1 & j19 = j29 & cell (G,(i2 -' 1),(j2 -' 1)) = cell (G,(i29 -' 1),j29) ) implies ( i19 = i29 & j19 = j29 + 1 & cell (G,(i2 -' 1),(j2 -' 1)) = cell (G,(i29 -' 1),(j29 -' 1)) ) )
assume A10: ( [i19,j19] in Indices G & [i29,j29] in Indices G & f /. k = G * (i19,j19) & f /. (k + 1) = G * (i29,j29) ) ; ::_thesis: ( ( i19 = i29 & j19 + 1 = j29 & cell (G,(i2 -' 1),(j2 -' 1)) = cell (G,i29,j29) ) or ( i19 + 1 = i29 & j19 = j29 & cell (G,(i2 -' 1),(j2 -' 1)) = cell (G,i29,(j29 -' 1)) ) or ( i19 = i29 + 1 & j19 = j29 & cell (G,(i2 -' 1),(j2 -' 1)) = cell (G,(i29 -' 1),j29) ) or ( i19 = i29 & j19 = j29 + 1 & cell (G,(i2 -' 1),(j2 -' 1)) = cell (G,(i29 -' 1),(j29 -' 1)) ) )
then  ( i2 = i29 & j1 = j19 ) by A1, GOBOARD1:5;
hence  ( ( i19 = i29 & j19 + 1 = j29 & cell (G,(i2 -' 1),(j2 -' 1)) = cell (G,i29,j29) ) or ( i19 + 1 = i29 & j19 = j29 & cell (G,(i2 -' 1),(j2 -' 1)) = cell (G,i29,(j29 -' 1)) ) or ( i19 = i29 + 1 & j19 = j29 & cell (G,(i2 -' 1),(j2 -' 1)) = cell (G,(i29 -' 1),j29) ) or ( i19 = i29 & j19 = j29 + 1 & cell (G,(i2 -' 1),(j2 -' 1)) = cell (G,(i29 -' 1),(j29 -' 1)) ) ) by A1, A9, A10, GOBOARD1:5; ::_thesis: verum
end;
end;
end;
uniqueness
 for b1, b2 being Subset of (TOP-REAL 2) st ( for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & b1 = cell (G,i2,j2) ) & not ( i1 + 1 = i2 & j1 = j2 & b1 = cell (G,i2,(j2 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & b1 = cell (G,(i2 -' 1),j2) ) holds
( i1 = i2 & j1 = j2 + 1 & b1 = cell (G,(i2 -' 1),(j2 -' 1)) ) ) & ( for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & b2 = cell (G,i2,j2) ) & not ( i1 + 1 = i2 & j1 = j2 & b2 = cell (G,i2,(j2 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & b2 = cell (G,(i2 -' 1),j2) ) holds
( i1 = i2 & j1 = j2 + 1 & b2 = cell (G,(i2 -' 1),(j2 -' 1)) ) ) holds
b1 = b2
proof
let P1, P2 be Subset of (TOP-REAL 2); ::_thesis: ( ( for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & P1 = cell (G,i2,j2) ) & not ( i1 + 1 = i2 & j1 = j2 & P1 = cell (G,i2,(j2 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & P1 = cell (G,(i2 -' 1),j2) ) holds
( i1 = i2 & j1 = j2 + 1 & P1 = cell (G,(i2 -' 1),(j2 -' 1)) ) ) & ( for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & P2 = cell (G,i2,j2) ) & not ( i1 + 1 = i2 & j1 = j2 & P2 = cell (G,i2,(j2 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & P2 = cell (G,(i2 -' 1),j2) ) holds
( i1 = i2 & j1 = j2 + 1 & P2 = cell (G,(i2 -' 1),(j2 -' 1)) ) ) implies P1 = P2 )
assume that
A11: for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & P1 = cell (G,i2,j2) ) & not ( i1 + 1 = i2 & j1 = j2 & P1 = cell (G,i2,(j2 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & P1 = cell (G,(i2 -' 1),j2) ) holds
( i1 = i2 & j1 = j2 + 1 & P1 = cell (G,(i2 -' 1),(j2 -' 1)) ) and
A12: for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & P2 = cell (G,i2,j2) ) & not ( i1 + 1 = i2 & j1 = j2 & P2 = cell (G,i2,(j2 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & P2 = cell (G,(i2 -' 1),j2) ) holds
( i1 = i2 & j1 = j2 + 1 & P2 = cell (G,(i2 -' 1),(j2 -' 1)) ) ; ::_thesis: P1 = P2
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 A2;
supposeA13: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: P1 = P2
A14: j2 <= j2 + 1 by NAT_1:11;
A15: j1 < j2 by A13, XREAL_1:29;
hence P1 = cell (G,i2,j2) by A1, A11, A14
.= P2 by A1, A12, A15, A14 ;
::_thesis: verum
end;
supposeA16: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: P1 = P2
A17: i2 <= i2 + 1 by NAT_1:11;
A18: i1 < i2 by A16, XREAL_1:29;
hence P1 = cell (G,i2,(j2 -' 1)) by A1, A11, A17
.= P2 by A1, A12, A18, A17 ;
::_thesis: verum
end;
supposeA19: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: P1 = P2
A20: i1 <= i1 + 1 by NAT_1:11;
A21: i2 < i1 by A19, XREAL_1:29;
hence P1 = cell (G,(i2 -' 1),j2) by A1, A11, A20
.= P2 by A1, A12, A21, A20 ;
::_thesis: verum
end;
supposeA22: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: P1 = P2
A23: j1 <= j1 + 1 by NAT_1:11;
A24: j2 < j1 by A22, XREAL_1:29;
hence P1 = cell (G,(i2 -' 1),(j2 -' 1)) by A1, A11, A23
.= P2 by A1, A12, A24, A23 ;
::_thesis: verum
end;
end;
end;
func front_left_cell (f,k,G) ->  Subset of (TOP-REAL 2) means :Def5: :: GOBRD13:def 5
 for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & it = cell (G,(i2 -' 1),j2) ) & not ( i1 + 1 = i2 & j1 = j2 & it = cell (G,i2,j2) ) & not ( i1 = i2 + 1 & j1 = j2 & it = cell (G,(i2 -' 1),(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & it = cell (G,i2,(j2 -' 1)) );
existence
 ex b1 being Subset of (TOP-REAL 2) st
for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & b1 = cell (G,(i2 -' 1),j2) ) & not ( i1 + 1 = i2 & j1 = j2 & b1 = cell (G,i2,j2) ) & not ( i1 = i2 + 1 & j1 = j2 & b1 = cell (G,(i2 -' 1),(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & b1 = cell (G,i2,(j2 -' 1)) )
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 A2;
supposeA25: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: ex b1 being Subset of (TOP-REAL 2) st
for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & b1 = cell (G,(i2 -' 1),j2) ) & not ( i1 + 1 = i2 & j1 = j2 & b1 = cell (G,i2,j2) ) & not ( i1 = i2 + 1 & j1 = j2 & b1 = cell (G,(i2 -' 1),(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & b1 = cell (G,i2,(j2 -' 1)) )
take  cell (G,(i2 -' 1),j2) ; ::_thesis: for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & cell (G,(i2 -' 1),j2) = cell (G,(i2 -' 1),j2) ) & not ( i1 + 1 = i2 & j1 = j2 & cell (G,(i2 -' 1),j2) = cell (G,i2,j2) ) & not ( i1 = i2 + 1 & j1 = j2 & cell (G,(i2 -' 1),j2) = cell (G,(i2 -' 1),(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & cell (G,(i2 -' 1),j2) = cell (G,i2,(j2 -' 1)) )
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices G & [i29,j29] in Indices G & f /. k = G * (i19,j19) & f /. (k + 1) = G * (i29,j29) & not ( i19 = i29 & j19 + 1 = j29 & cell (G,(i2 -' 1),j2) = cell (G,(i29 -' 1),j29) ) & not ( i19 + 1 = i29 & j19 = j29 & cell (G,(i2 -' 1),j2) = cell (G,i29,j29) ) & not ( i19 = i29 + 1 & j19 = j29 & cell (G,(i2 -' 1),j2) = cell (G,(i29 -' 1),(j29 -' 1)) ) implies ( i19 = i29 & j19 = j29 + 1 & cell (G,(i2 -' 1),j2) = cell (G,i29,(j29 -' 1)) ) )
assume A26: ( [i19,j19] in Indices G & [i29,j29] in Indices G & f /. k = G * (i19,j19) & f /. (k + 1) = G * (i29,j29) ) ; ::_thesis: ( ( i19 = i29 & j19 + 1 = j29 & cell (G,(i2 -' 1),j2) = cell (G,(i29 -' 1),j29) ) or ( i19 + 1 = i29 & j19 = j29 & cell (G,(i2 -' 1),j2) = cell (G,i29,j29) ) or ( i19 = i29 + 1 & j19 = j29 & cell (G,(i2 -' 1),j2) = cell (G,(i29 -' 1),(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & cell (G,(i2 -' 1),j2) = cell (G,i29,(j29 -' 1)) ) )
then  ( i2 = i29 & j1 = j19 ) by A1, GOBOARD1:5;
hence  ( ( i19 = i29 & j19 + 1 = j29 & cell (G,(i2 -' 1),j2) = cell (G,(i29 -' 1),j29) ) or ( i19 + 1 = i29 & j19 = j29 & cell (G,(i2 -' 1),j2) = cell (G,i29,j29) ) or ( i19 = i29 + 1 & j19 = j29 & cell (G,(i2 -' 1),j2) = cell (G,(i29 -' 1),(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & cell (G,(i2 -' 1),j2) = cell (G,i29,(j29 -' 1)) ) ) by A1, A25, A26, GOBOARD1:5; ::_thesis: verum
end;
supposeA27: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: ex b1 being Subset of (TOP-REAL 2) st
for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & b1 = cell (G,(i2 -' 1),j2) ) & not ( i1 + 1 = i2 & j1 = j2 & b1 = cell (G,i2,j2) ) & not ( i1 = i2 + 1 & j1 = j2 & b1 = cell (G,(i2 -' 1),(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & b1 = cell (G,i2,(j2 -' 1)) )
take  cell (G,i2,j2) ; ::_thesis: for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & cell (G,i2,j2) = cell (G,(i2 -' 1),j2) ) & not ( i1 + 1 = i2 & j1 = j2 & cell (G,i2,j2) = cell (G,i2,j2) ) & not ( i1 = i2 + 1 & j1 = j2 & cell (G,i2,j2) = cell (G,(i2 -' 1),(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & cell (G,i2,j2) = cell (G,i2,(j2 -' 1)) )
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices G & [i29,j29] in Indices G & f /. k = G * (i19,j19) & f /. (k + 1) = G * (i29,j29) & not ( i19 = i29 & j19 + 1 = j29 & cell (G,i2,j2) = cell (G,(i29 -' 1),j29) ) & not ( i19 + 1 = i29 & j19 = j29 & cell (G,i2,j2) = cell (G,i29,j29) ) & not ( i19 = i29 + 1 & j19 = j29 & cell (G,i2,j2) = cell (G,(i29 -' 1),(j29 -' 1)) ) implies ( i19 = i29 & j19 = j29 + 1 & cell (G,i2,j2) = cell (G,i29,(j29 -' 1)) ) )
assume A28: ( [i19,j19] in Indices G & [i29,j29] in Indices G & f /. k = G * (i19,j19) & f /. (k + 1) = G * (i29,j29) ) ; ::_thesis: ( ( i19 = i29 & j19 + 1 = j29 & cell (G,i2,j2) = cell (G,(i29 -' 1),j29) ) or ( i19 + 1 = i29 & j19 = j29 & cell (G,i2,j2) = cell (G,i29,j29) ) or ( i19 = i29 + 1 & j19 = j29 & cell (G,i2,j2) = cell (G,(i29 -' 1),(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & cell (G,i2,j2) = cell (G,i29,(j29 -' 1)) ) )
then  ( i2 = i29 & j1 = j19 ) by A1, GOBOARD1:5;
hence  ( ( i19 = i29 & j19 + 1 = j29 & cell (G,i2,j2) = cell (G,(i29 -' 1),j29) ) or ( i19 + 1 = i29 & j19 = j29 & cell (G,i2,j2) = cell (G,i29,j29) ) or ( i19 = i29 + 1 & j19 = j29 & cell (G,i2,j2) = cell (G,(i29 -' 1),(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & cell (G,i2,j2) = cell (G,i29,(j29 -' 1)) ) ) by A1, A27, A28, GOBOARD1:5; ::_thesis: verum
end;
supposeA29: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: ex b1 being Subset of (TOP-REAL 2) st
for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & b1 = cell (G,(i2 -' 1),j2) ) & not ( i1 + 1 = i2 & j1 = j2 & b1 = cell (G,i2,j2) ) & not ( i1 = i2 + 1 & j1 = j2 & b1 = cell (G,(i2 -' 1),(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & b1 = cell (G,i2,(j2 -' 1)) )
take  cell (G,(i2 -' 1),(j2 -' 1)) ; ::_thesis: for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & cell (G,(i2 -' 1),(j2 -' 1)) = cell (G,(i2 -' 1),j2) ) & not ( i1 + 1 = i2 & j1 = j2 & cell (G,(i2 -' 1),(j2 -' 1)) = cell (G,i2,j2) ) & not ( i1 = i2 + 1 & j1 = j2 & cell (G,(i2 -' 1),(j2 -' 1)) = cell (G,(i2 -' 1),(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & cell (G,(i2 -' 1),(j2 -' 1)) = cell (G,i2,(j2 -' 1)) )
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices G & [i29,j29] in Indices G & f /. k = G * (i19,j19) & f /. (k + 1) = G * (i29,j29) & not ( i19 = i29 & j19 + 1 = j29 & cell (G,(i2 -' 1),(j2 -' 1)) = cell (G,(i29 -' 1),j29) ) & not ( i19 + 1 = i29 & j19 = j29 & cell (G,(i2 -' 1),(j2 -' 1)) = cell (G,i29,j29) ) & not ( i19 = i29 + 1 & j19 = j29 & cell (G,(i2 -' 1),(j2 -' 1)) = cell (G,(i29 -' 1),(j29 -' 1)) ) implies ( i19 = i29 & j19 = j29 + 1 & cell (G,(i2 -' 1),(j2 -' 1)) = cell (G,i29,(j29 -' 1)) ) )
assume A30: ( [i19,j19] in Indices G & [i29,j29] in Indices G & f /. k = G * (i19,j19) & f /. (k + 1) = G * (i29,j29) ) ; ::_thesis: ( ( i19 = i29 & j19 + 1 = j29 & cell (G,(i2 -' 1),(j2 -' 1)) = cell (G,(i29 -' 1),j29) ) or ( i19 + 1 = i29 & j19 = j29 & cell (G,(i2 -' 1),(j2 -' 1)) = cell (G,i29,j29) ) or ( i19 = i29 + 1 & j19 = j29 & cell (G,(i2 -' 1),(j2 -' 1)) = cell (G,(i29 -' 1),(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & cell (G,(i2 -' 1),(j2 -' 1)) = cell (G,i29,(j29 -' 1)) ) )
then  ( i2 = i29 & j1 = j19 ) by A1, GOBOARD1:5;
hence  ( ( i19 = i29 & j19 + 1 = j29 & cell (G,(i2 -' 1),(j2 -' 1)) = cell (G,(i29 -' 1),j29) ) or ( i19 + 1 = i29 & j19 = j29 & cell (G,(i2 -' 1),(j2 -' 1)) = cell (G,i29,j29) ) or ( i19 = i29 + 1 & j19 = j29 & cell (G,(i2 -' 1),(j2 -' 1)) = cell (G,(i29 -' 1),(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & cell (G,(i2 -' 1),(j2 -' 1)) = cell (G,i29,(j29 -' 1)) ) ) by A1, A29, A30, GOBOARD1:5; ::_thesis: verum
end;
supposeA31: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: ex b1 being Subset of (TOP-REAL 2) st
for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & b1 = cell (G,(i2 -' 1),j2) ) & not ( i1 + 1 = i2 & j1 = j2 & b1 = cell (G,i2,j2) ) & not ( i1 = i2 + 1 & j1 = j2 & b1 = cell (G,(i2 -' 1),(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & b1 = cell (G,i2,(j2 -' 1)) )
take  cell (G,i2,(j2 -' 1)) ; ::_thesis: for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & cell (G,i2,(j2 -' 1)) = cell (G,(i2 -' 1),j2) ) & not ( i1 + 1 = i2 & j1 = j2 & cell (G,i2,(j2 -' 1)) = cell (G,i2,j2) ) & not ( i1 = i2 + 1 & j1 = j2 & cell (G,i2,(j2 -' 1)) = cell (G,(i2 -' 1),(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & cell (G,i2,(j2 -' 1)) = cell (G,i2,(j2 -' 1)) )
let i19, j19, i29, j29 be Element of NAT ; ::_thesis: ( [i19,j19] in Indices G & [i29,j29] in Indices G & f /. k = G * (i19,j19) & f /. (k + 1) = G * (i29,j29) & not ( i19 = i29 & j19 + 1 = j29 & cell (G,i2,(j2 -' 1)) = cell (G,(i29 -' 1),j29) ) & not ( i19 + 1 = i29 & j19 = j29 & cell (G,i2,(j2 -' 1)) = cell (G,i29,j29) ) & not ( i19 = i29 + 1 & j19 = j29 & cell (G,i2,(j2 -' 1)) = cell (G,(i29 -' 1),(j29 -' 1)) ) implies ( i19 = i29 & j19 = j29 + 1 & cell (G,i2,(j2 -' 1)) = cell (G,i29,(j29 -' 1)) ) )
assume A32: ( [i19,j19] in Indices G & [i29,j29] in Indices G & f /. k = G * (i19,j19) & f /. (k + 1) = G * (i29,j29) ) ; ::_thesis: ( ( i19 = i29 & j19 + 1 = j29 & cell (G,i2,(j2 -' 1)) = cell (G,(i29 -' 1),j29) ) or ( i19 + 1 = i29 & j19 = j29 & cell (G,i2,(j2 -' 1)) = cell (G,i29,j29) ) or ( i19 = i29 + 1 & j19 = j29 & cell (G,i2,(j2 -' 1)) = cell (G,(i29 -' 1),(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & cell (G,i2,(j2 -' 1)) = cell (G,i29,(j29 -' 1)) ) )
then  ( i2 = i29 & j1 = j19 ) by A1, GOBOARD1:5;
hence  ( ( i19 = i29 & j19 + 1 = j29 & cell (G,i2,(j2 -' 1)) = cell (G,(i29 -' 1),j29) ) or ( i19 + 1 = i29 & j19 = j29 & cell (G,i2,(j2 -' 1)) = cell (G,i29,j29) ) or ( i19 = i29 + 1 & j19 = j29 & cell (G,i2,(j2 -' 1)) = cell (G,(i29 -' 1),(j29 -' 1)) ) or ( i19 = i29 & j19 = j29 + 1 & cell (G,i2,(j2 -' 1)) = cell (G,i29,(j29 -' 1)) ) ) by A1, A31, A32, GOBOARD1:5; ::_thesis: verum
end;
end;
end;
uniqueness
 for b1, b2 being Subset of (TOP-REAL 2) st ( for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & b1 = cell (G,(i2 -' 1),j2) ) & not ( i1 + 1 = i2 & j1 = j2 & b1 = cell (G,i2,j2) ) & not ( i1 = i2 + 1 & j1 = j2 & b1 = cell (G,(i2 -' 1),(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & b1 = cell (G,i2,(j2 -' 1)) ) ) & ( for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & b2 = cell (G,(i2 -' 1),j2) ) & not ( i1 + 1 = i2 & j1 = j2 & b2 = cell (G,i2,j2) ) & not ( i1 = i2 + 1 & j1 = j2 & b2 = cell (G,(i2 -' 1),(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & b2 = cell (G,i2,(j2 -' 1)) ) ) holds
b1 = b2
proof
let P1, P2 be Subset of (TOP-REAL 2); ::_thesis: ( ( for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & P1 = cell (G,(i2 -' 1),j2) ) & not ( i1 + 1 = i2 & j1 = j2 & P1 = cell (G,i2,j2) ) & not ( i1 = i2 + 1 & j1 = j2 & P1 = cell (G,(i2 -' 1),(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & P1 = cell (G,i2,(j2 -' 1)) ) ) & ( for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & P2 = cell (G,(i2 -' 1),j2) ) & not ( i1 + 1 = i2 & j1 = j2 & P2 = cell (G,i2,j2) ) & not ( i1 = i2 + 1 & j1 = j2 & P2 = cell (G,(i2 -' 1),(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & P2 = cell (G,i2,(j2 -' 1)) ) ) implies P1 = P2 )
assume that
A33: for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & P1 = cell (G,(i2 -' 1),j2) ) & not ( i1 + 1 = i2 & j1 = j2 & P1 = cell (G,i2,j2) ) & not ( i1 = i2 + 1 & j1 = j2 & P1 = cell (G,(i2 -' 1),(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & P1 = cell (G,i2,(j2 -' 1)) ) and
A34: for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & P2 = cell (G,(i2 -' 1),j2) ) & not ( i1 + 1 = i2 & j1 = j2 & P2 = cell (G,i2,j2) ) & not ( i1 = i2 + 1 & j1 = j2 & P2 = cell (G,(i2 -' 1),(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & P2 = cell (G,i2,(j2 -' 1)) ) ; ::_thesis: P1 = P2
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 A2;
supposeA35: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: P1 = P2
A36: j2 <= j2 + 1 by NAT_1:11;
A37: j1 < j2 by A35, XREAL_1:29;
hence P1 = cell (G,(i2 -' 1),j2) by A1, A33, A36
.= P2 by A1, A34, A37, A36 ;
::_thesis: verum
end;
supposeA38: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: P1 = P2
A39: i2 <= i2 + 1 by NAT_1:11;
A40: i1 < i2 by A38, XREAL_1:29;
hence P1 = cell (G,i2,j2) by A1, A33, A39
.= P2 by A1, A34, A40, A39 ;
::_thesis: verum
end;
supposeA41: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: P1 = P2
A42: i1 <= i1 + 1 by NAT_1:11;
A43: i2 < i1 by A41, XREAL_1:29;
hence P1 = cell (G,(i2 -' 1),(j2 -' 1)) by A1, A33, A42
.= P2 by A1, A34, A43, A42 ;
::_thesis: verum
end;
supposeA44: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: P1 = P2
A45: j1 <= j1 + 1 by NAT_1:11;
A46: j2 < j1 by A44, XREAL_1:29;
hence P1 = cell (G,i2,(j2 -' 1)) by A1, A33, A45
.= P2 by A1, A34, A46, A45 ;
::_thesis: verum
end;
end;
end;
end;

:: deftheorem Def4 defines front_right_cell GOBRD13:def_4_:_
 for f being FinSequence of (TOP-REAL 2)
for G being Go-board
for k being Element of NAT st 1 <= k & k + 1 <= len f & f is_sequence_on G holds
for b4 being Subset of (TOP-REAL 2) holds
( b4 = front_right_cell (f,k,G) iff for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & b4 = cell (G,i2,j2) ) & not ( i1 + 1 = i2 & j1 = j2 & b4 = cell (G,i2,(j2 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & b4 = cell (G,(i2 -' 1),j2) ) holds
( i1 = i2 & j1 = j2 + 1 & b4 = cell (G,(i2 -' 1),(j2 -' 1)) ) );

:: deftheorem Def5 defines front_left_cell GOBRD13:def_5_:_
 for f being FinSequence of (TOP-REAL 2)
for G being Go-board
for k being Element of NAT st 1 <= k & k + 1 <= len f & f is_sequence_on G holds
for b4 being Subset of (TOP-REAL 2) holds
( b4 = front_left_cell (f,k,G) iff for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & b4 = cell (G,(i2 -' 1),j2) ) & not ( i1 + 1 = i2 & j1 = j2 & b4 = cell (G,i2,j2) ) & not ( i1 = i2 + 1 & j1 = j2 & b4 = cell (G,(i2 -' 1),(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & b4 = cell (G,i2,(j2 -' 1)) ) );

theorem :: GOBRD13:34
for k, i, j being Element of NAT
for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [i,(j + 1)] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * (i,(j + 1)) holds
front_left_cell (f,k,G) = cell (G,(i -' 1),(j + 1))
proof
let k, i, j be Element of NAT ; ::_thesis: for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [i,(j + 1)] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * (i,(j + 1)) holds
front_left_cell (f,k,G) = cell (G,(i -' 1),(j + 1))
let f be FinSequence of (TOP-REAL 2); ::_thesis: for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [i,(j + 1)] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * (i,(j + 1)) holds
front_left_cell (f,k,G) = cell (G,(i -' 1),(j + 1))
let G be Go-board; ::_thesis: ( 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [i,(j + 1)] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * (i,(j + 1)) implies front_left_cell (f,k,G) = cell (G,(i -' 1),(j + 1)) )
A1: ( j < j + 1 & j + 1 <= (j + 1) + 1 ) by XREAL_1:29;
assume  ( 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [i,(j + 1)] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * (i,(j + 1)) ) ; ::_thesis: front_left_cell (f,k,G) = cell (G,(i -' 1),(j + 1))
hence  front_left_cell (f,k,G) = cell (G,(i -' 1),(j + 1)) by A1, Def5; ::_thesis: verum
end;

theorem :: GOBRD13:35
for k, i, j being Element of NAT
for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [i,(j + 1)] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * (i,(j + 1)) holds
front_right_cell (f,k,G) = cell (G,i,(j + 1))
proof
let k, i, j be Element of NAT ; ::_thesis: for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [i,(j + 1)] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * (i,(j + 1)) holds
front_right_cell (f,k,G) = cell (G,i,(j + 1))
let f be FinSequence of (TOP-REAL 2); ::_thesis: for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [i,(j + 1)] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * (i,(j + 1)) holds
front_right_cell (f,k,G) = cell (G,i,(j + 1))
let G be Go-board; ::_thesis: ( 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [i,(j + 1)] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * (i,(j + 1)) implies front_right_cell (f,k,G) = cell (G,i,(j + 1)) )
A1: ( j < j + 1 & j + 1 <= (j + 1) + 1 ) by XREAL_1:29;
assume  ( 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [i,(j + 1)] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * (i,(j + 1)) ) ; ::_thesis: front_right_cell (f,k,G) = cell (G,i,(j + 1))
hence  front_right_cell (f,k,G) = cell (G,i,(j + 1)) by A1, Def4; ::_thesis: verum
end;

theorem :: GOBRD13:36
for k, i, j being Element of NAT
for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * ((i + 1),j) holds
front_left_cell (f,k,G) = cell (G,(i + 1),j)
proof
let k, i, j be Element of NAT ; ::_thesis: for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * ((i + 1),j) holds
front_left_cell (f,k,G) = cell (G,(i + 1),j)
let f be FinSequence of (TOP-REAL 2); ::_thesis: for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * ((i + 1),j) holds
front_left_cell (f,k,G) = cell (G,(i + 1),j)
let G be Go-board; ::_thesis: ( 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * ((i + 1),j) implies front_left_cell (f,k,G) = cell (G,(i + 1),j) )
A1: ( i < i + 1 & i + 1 <= (i + 1) + 1 ) by XREAL_1:29;
assume  ( 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * ((i + 1),j) ) ; ::_thesis: front_left_cell (f,k,G) = cell (G,(i + 1),j)
hence  front_left_cell (f,k,G) = cell (G,(i + 1),j) by A1, Def5; ::_thesis: verum
end;

theorem :: GOBRD13:37
for k, i, j being Element of NAT
for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * ((i + 1),j) holds
front_right_cell (f,k,G) = cell (G,(i + 1),(j -' 1))
proof
let k, i, j be Element of NAT ; ::_thesis: for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * ((i + 1),j) holds
front_right_cell (f,k,G) = cell (G,(i + 1),(j -' 1))
let f be FinSequence of (TOP-REAL 2); ::_thesis: for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * ((i + 1),j) holds
front_right_cell (f,k,G) = cell (G,(i + 1),(j -' 1))
let G be Go-board; ::_thesis: ( 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * ((i + 1),j) implies front_right_cell (f,k,G) = cell (G,(i + 1),(j -' 1)) )
A1: ( i < i + 1 & i + 1 <= (i + 1) + 1 ) by XREAL_1:29;
assume  ( 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * (i,j) & f /. (k + 1) = G * ((i + 1),j) ) ; ::_thesis: front_right_cell (f,k,G) = cell (G,(i + 1),(j -' 1))
hence  front_right_cell (f,k,G) = cell (G,(i + 1),(j -' 1)) by A1, Def4; ::_thesis: verum
end;

theorem :: GOBRD13:38
for k, i, j being Element of NAT
for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * ((i + 1),j) & f /. (k + 1) = G * (i,j) holds
front_left_cell (f,k,G) = cell (G,(i -' 1),(j -' 1))
proof
let k, i, j be Element of NAT ; ::_thesis: for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * ((i + 1),j) & f /. (k + 1) = G * (i,j) holds
front_left_cell (f,k,G) = cell (G,(i -' 1),(j -' 1))
let f be FinSequence of (TOP-REAL 2); ::_thesis: for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * ((i + 1),j) & f /. (k + 1) = G * (i,j) holds
front_left_cell (f,k,G) = cell (G,(i -' 1),(j -' 1))
let G be Go-board; ::_thesis: ( 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * ((i + 1),j) & f /. (k + 1) = G * (i,j) implies front_left_cell (f,k,G) = cell (G,(i -' 1),(j -' 1)) )
A1: ( i < i + 1 & i + 1 <= (i + 1) + 1 ) by XREAL_1:29;
assume  ( 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * ((i + 1),j) & f /. (k + 1) = G * (i,j) ) ; ::_thesis: front_left_cell (f,k,G) = cell (G,(i -' 1),(j -' 1))
hence  front_left_cell (f,k,G) = cell (G,(i -' 1),(j -' 1)) by A1, Def5; ::_thesis: verum
end;

theorem :: GOBRD13:39
for k, i, j being Element of NAT
for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * ((i + 1),j) & f /. (k + 1) = G * (i,j) holds
front_right_cell (f,k,G) = cell (G,(i -' 1),j)
proof
let k, i, j be Element of NAT ; ::_thesis: for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * ((i + 1),j) & f /. (k + 1) = G * (i,j) holds
front_right_cell (f,k,G) = cell (G,(i -' 1),j)
let f be FinSequence of (TOP-REAL 2); ::_thesis: for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * ((i + 1),j) & f /. (k + 1) = G * (i,j) holds
front_right_cell (f,k,G) = cell (G,(i -' 1),j)
let G be Go-board; ::_thesis: ( 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * ((i + 1),j) & f /. (k + 1) = G * (i,j) implies front_right_cell (f,k,G) = cell (G,(i -' 1),j) )
A1: ( i < i + 1 & i + 1 <= (i + 1) + 1 ) by XREAL_1:29;
assume  ( 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,j] in Indices G & [(i + 1),j] in Indices G & f /. k = G * ((i + 1),j) & f /. (k + 1) = G * (i,j) ) ; ::_thesis: front_right_cell (f,k,G) = cell (G,(i -' 1),j)
hence  front_right_cell (f,k,G) = cell (G,(i -' 1),j) by A1, Def4; ::_thesis: verum
end;

theorem :: GOBRD13:40
for k, i, j being Element of NAT
for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,(j + 1)] in Indices G & [i,j] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) holds
front_left_cell (f,k,G) = cell (G,i,(j -' 1))
proof
let k, i, j be Element of NAT ; ::_thesis: for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,(j + 1)] in Indices G & [i,j] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) holds
front_left_cell (f,k,G) = cell (G,i,(j -' 1))
let f be FinSequence of (TOP-REAL 2); ::_thesis: for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,(j + 1)] in Indices G & [i,j] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) holds
front_left_cell (f,k,G) = cell (G,i,(j -' 1))
let G be Go-board; ::_thesis: ( 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,(j + 1)] in Indices G & [i,j] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) implies front_left_cell (f,k,G) = cell (G,i,(j -' 1)) )
A1: ( j < j + 1 & j + 1 <= (j + 1) + 1 ) by XREAL_1:29;
assume  ( 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,(j + 1)] in Indices G & [i,j] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) ) ; ::_thesis: front_left_cell (f,k,G) = cell (G,i,(j -' 1))
hence  front_left_cell (f,k,G) = cell (G,i,(j -' 1)) by A1, Def5; ::_thesis: verum
end;

theorem :: GOBRD13:41
for k, i, j being Element of NAT
for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,(j + 1)] in Indices G & [i,j] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) holds
front_right_cell (f,k,G) = cell (G,(i -' 1),(j -' 1))
proof
let k, i, j be Element of NAT ; ::_thesis: for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,(j + 1)] in Indices G & [i,j] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) holds
front_right_cell (f,k,G) = cell (G,(i -' 1),(j -' 1))
let f be FinSequence of (TOP-REAL 2); ::_thesis: for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,(j + 1)] in Indices G & [i,j] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) holds
front_right_cell (f,k,G) = cell (G,(i -' 1),(j -' 1))
let G be Go-board; ::_thesis: ( 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,(j + 1)] in Indices G & [i,j] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) implies front_right_cell (f,k,G) = cell (G,(i -' 1),(j -' 1)) )
A1: ( j < j + 1 & j + 1 <= (j + 1) + 1 ) by XREAL_1:29;
assume  ( 1 <= k & k + 1 <= len f & f is_sequence_on G & [i,(j + 1)] in Indices G & [i,j] in Indices G & f /. k = G * (i,(j + 1)) & f /. (k + 1) = G * (i,j) ) ; ::_thesis: front_right_cell (f,k,G) = cell (G,(i -' 1),(j -' 1))
hence  front_right_cell (f,k,G) = cell (G,(i -' 1),(j -' 1)) by A1, Def4; ::_thesis: verum
end;

theorem :: GOBRD13:42
for k, n being Element of NAT
for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & k + 1 <= n holds
( front_left_cell (f,k,G) = front_left_cell ((f | n),k,G) & front_right_cell (f,k,G) = front_right_cell ((f | n),k,G) )
proof
let k, n be Element of NAT ; ::_thesis: for f being FinSequence of (TOP-REAL 2)
for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & k + 1 <= n holds
( front_left_cell (f,k,G) = front_left_cell ((f | n),k,G) & front_right_cell (f,k,G) = front_right_cell ((f | n),k,G) )
let f be FinSequence of (TOP-REAL 2); ::_thesis: for G being Go-board st 1 <= k & k + 1 <= len f & f is_sequence_on G & k + 1 <= n holds
( front_left_cell (f,k,G) = front_left_cell ((f | n),k,G) & front_right_cell (f,k,G) = front_right_cell ((f | n),k,G) )
let G be Go-board; ::_thesis: ( 1 <= k & k + 1 <= len f & f is_sequence_on G & k + 1 <= n implies ( front_left_cell (f,k,G) = front_left_cell ((f | n),k,G) & front_right_cell (f,k,G) = front_right_cell ((f | n),k,G) ) )
assume that
A1: 1 <= k and
A2: k + 1 <= len f and
A3: f is_sequence_on G and
A4: k + 1 <= n ; ::_thesis: ( front_left_cell (f,k,G) = front_left_cell ((f | n),k,G) & front_right_cell (f,k,G) = front_right_cell ((f | n),k,G) )
percases ( len f <= n or n < len f ) ;
suppose len f <= n ; ::_thesis: ( front_left_cell (f,k,G) = front_left_cell ((f | n),k,G) & front_right_cell (f,k,G) = front_right_cell ((f | n),k,G) )
hence  ( front_left_cell (f,k,G) = front_left_cell ((f | n),k,G) & front_right_cell (f,k,G) = front_right_cell ((f | n),k,G) ) by FINSEQ_1:58; ::_thesis: verum
end;
suppose n < len f ; ::_thesis: ( front_left_cell (f,k,G) = front_left_cell ((f | n),k,G) & front_right_cell (f,k,G) = front_right_cell ((f | n),k,G) )
then A5: len (f | n) = n by FINSEQ_1:59;
then  k in dom (f | n) by A1, A4, SEQ_4:134;
then A6: (f | n) /. k = f /. k by FINSEQ_4:70;
 k + 1 in dom (f | n) by A1, A4, A5, SEQ_4:134;
then A7: (f | n) /. (k + 1) = f /. (k + 1) by FINSEQ_4:70;
set lf = front_left_cell (f,k,G);
set lfn = front_left_cell ((f | n),k,G);
set rf = front_right_cell (f,k,G);
set rfn = front_right_cell ((f | n),k,G);
A8: f | n is_sequence_on G by A3, GOBOARD1:22;
consider i1, j1, i2, j2 being Element of NAT  such that
A9: ( [i1,j1] in Indices G & f /. k = G * (i1,j1) & [i2,j2] in Indices G & f /. (k + 1) = G * (i2,j2) ) and
A10: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A1, A2, A3, JORDAN8:3;
A11: ( j1 + 1 > j1 & j2 + 1 > j2 ) by NAT_1:13;
A12: ( i1 + 1 > i1 & i2 + 1 > i2 ) by NAT_1:13;
now__::_thesis:_(_front_left_cell_(f,k,G)_=_front_left_cell_((f_|_n),k,G)_&_front_right_cell_(f,k,G)_=_front_right_cell_((f_|_n),k,G)_)
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 A10;
supposeA13: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: ( front_left_cell (f,k,G) = front_left_cell ((f | n),k,G) & front_right_cell (f,k,G) = front_right_cell ((f | n),k,G) )
hence  front_left_cell (f,k,G) = cell (G,(i2 -' 1),j2) by A1, A2, A3, A9, A11, Def5
.= front_left_cell ((f | n),k,G) by A1, A4, A9, A11, A8, A5, A6, A7, A13, Def5 ;
::_thesis: front_right_cell (f,k,G) = front_right_cell ((f | n),k,G)
thus  front_right_cell (f,k,G) = cell (G,i2,j2) by A1, A2, A3, A9, A11, A13, Def4
.= front_right_cell ((f | n),k,G) by A1, A4, A9, A11, A8, A5, A6, A7, A13, Def4 ; ::_thesis: verum
end;
supposeA14: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: ( front_left_cell (f,k,G) = front_left_cell ((f | n),k,G) & front_right_cell (f,k,G) = front_right_cell ((f | n),k,G) )
hence  front_left_cell (f,k,G) = cell (G,i2,j2) by A1, A2, A3, A9, A12, Def5
.= front_left_cell ((f | n),k,G) by A1, A4, A9, A12, A8, A5, A6, A7, A14, Def5 ;
::_thesis: front_right_cell (f,k,G) = front_right_cell ((f | n),k,G)
thus  front_right_cell (f,k,G) = cell (G,i2,(j2 -' 1)) by A1, A2, A3, A9, A12, A14, Def4
.= front_right_cell ((f | n),k,G) by A1, A4, A9, A12, A8, A5, A6, A7, A14, Def4 ; ::_thesis: verum
end;
supposeA15: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: ( front_left_cell (f,k,G) = front_left_cell ((f | n),k,G) & front_right_cell (f,k,G) = front_right_cell ((f | n),k,G) )
hence  front_left_cell (f,k,G) = cell (G,(i2 -' 1),(j2 -' 1)) by A1, A2, A3, A9, A12, Def5
.= front_left_cell ((f | n),k,G) by A1, A4, A9, A12, A8, A5, A6, A7, A15, Def5 ;
::_thesis: front_right_cell (f,k,G) = front_right_cell ((f | n),k,G)
thus  front_right_cell (f,k,G) = cell (G,(i2 -' 1),j2) by A1, A2, A3, A9, A12, A15, Def4
.= front_right_cell ((f | n),k,G) by A1, A4, A9, A12, A8, A5, A6, A7, A15, Def4 ; ::_thesis: verum
end;
supposeA16: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: ( front_left_cell (f,k,G) = front_left_cell ((f | n),k,G) & front_right_cell (f,k,G) = front_right_cell ((f | n),k,G) )
hence  front_left_cell (f,k,G) = cell (G,i2,(j2 -' 1)) by A1, A2, A3, A9, A11, Def5
.= front_left_cell ((f | n),k,G) by A1, A4, A9, A11, A8, A5, A6, A7, A16, Def5 ;
::_thesis: front_right_cell (f,k,G) = front_right_cell ((f | n),k,G)
thus  front_right_cell (f,k,G) = cell (G,(i2 -' 1),(j2 -' 1)) by A1, A2, A3, A9, A11, A16, Def4
.= front_right_cell ((f | n),k,G) by A1, A4, A9, A11, A8, A5, A6, A7, A16, Def4 ; ::_thesis: verum
end;
end;
end;
hence  ( front_left_cell (f,k,G) = front_left_cell ((f | n),k,G) & front_right_cell (f,k,G) = front_right_cell ((f | n),k,G) ) ; ::_thesis: verum
end;
end;
end;

definition
let D be set ;
let f be FinSequence of D;
let G be Matrix of D;
let k be Element of NAT ;
predf turns_right k,G means :Def6: :: GOBRD13:def 6
 for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & [(i2 + 1),j2] in Indices G & f /. (k + 2) = G * ((i2 + 1),j2) ) & not ( i1 + 1 = i2 & j1 = j2 & [i2,(j2 -' 1)] in Indices G & f /. (k + 2) = G * (i2,(j2 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & [i2,(j2 + 1)] in Indices G & f /. (k + 2) = G * (i2,(j2 + 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & [(i2 -' 1),j2] in Indices G & f /. (k + 2) = G * ((i2 -' 1),j2) );
predf turns_left k,G means :Def7: :: GOBRD13:def 7
 for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & [(i2 -' 1),j2] in Indices G & f /. (k + 2) = G * ((i2 -' 1),j2) ) & not ( i1 + 1 = i2 & j1 = j2 & [i2,(j2 + 1)] in Indices G & f /. (k + 2) = G * (i2,(j2 + 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & [i2,(j2 -' 1)] in Indices G & f /. (k + 2) = G * (i2,(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & [(i2 + 1),j2] in Indices G & f /. (k + 2) = G * ((i2 + 1),j2) );
predf goes_straight k,G means :Def8: :: GOBRD13:def 8
 for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & [i2,(j2 + 1)] in Indices G & f /. (k + 2) = G * (i2,(j2 + 1)) ) & not ( i1 + 1 = i2 & j1 = j2 & [(i2 + 1),j2] in Indices G & f /. (k + 2) = G * ((i2 + 1),j2) ) & not ( i1 = i2 + 1 & j1 = j2 & [(i2 -' 1),j2] in Indices G & f /. (k + 2) = G * ((i2 -' 1),j2) ) holds
( i1 = i2 & j1 = j2 + 1 & [i2,(j2 -' 1)] in Indices G & f /. (k + 2) = G * (i2,(j2 -' 1)) );
end;

:: deftheorem Def6 defines turns_right GOBRD13:def_6_:_
 for D being set
for f being FinSequence of D
for G being Matrix of D
for k being Element of NAT holds
( f turns_right k,G iff for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & [(i2 + 1),j2] in Indices G & f /. (k + 2) = G * ((i2 + 1),j2) ) & not ( i1 + 1 = i2 & j1 = j2 & [i2,(j2 -' 1)] in Indices G & f /. (k + 2) = G * (i2,(j2 -' 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & [i2,(j2 + 1)] in Indices G & f /. (k + 2) = G * (i2,(j2 + 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & [(i2 -' 1),j2] in Indices G & f /. (k + 2) = G * ((i2 -' 1),j2) ) );

:: deftheorem Def7 defines turns_left GOBRD13:def_7_:_
 for D being set
for f being FinSequence of D
for G being Matrix of D
for k being Element of NAT holds
( f turns_left k,G iff for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & [(i2 -' 1),j2] in Indices G & f /. (k + 2) = G * ((i2 -' 1),j2) ) & not ( i1 + 1 = i2 & j1 = j2 & [i2,(j2 + 1)] in Indices G & f /. (k + 2) = G * (i2,(j2 + 1)) ) & not ( i1 = i2 + 1 & j1 = j2 & [i2,(j2 -' 1)] in Indices G & f /. (k + 2) = G * (i2,(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & [(i2 + 1),j2] in Indices G & f /. (k + 2) = G * ((i2 + 1),j2) ) );

:: deftheorem Def8 defines goes_straight GOBRD13:def_8_:_
 for D being set
for f being FinSequence of D
for G being Matrix of D
for k being Element of NAT holds
( f goes_straight k,G iff for i1, j1, i2, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & f /. k = G * (i1,j1) & f /. (k + 1) = G * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & [i2,(j2 + 1)] in Indices G & f /. (k + 2) = G * (i2,(j2 + 1)) ) & not ( i1 + 1 = i2 & j1 = j2 & [(i2 + 1),j2] in Indices G & f /. (k + 2) = G * ((i2 + 1),j2) ) & not ( i1 = i2 + 1 & j1 = j2 & [(i2 -' 1),j2] in Indices G & f /. (k + 2) = G * ((i2 -' 1),j2) ) holds
( i1 = i2 & j1 = j2 + 1 & [i2,(j2 -' 1)] in Indices G & f /. (k + 2) = G * (i2,(j2 -' 1)) ) );


theorem :: GOBRD13:43
for k, n being Element of NAT
for D being set
for f being FinSequence of D
for G being Matrix of D st 1 <= k & k + 2 <= n & f | n turns_right k,G holds
f turns_right k,G
proof
let k, n be Element of NAT ; ::_thesis: for D being set
for f being FinSequence of D
for G being Matrix of D st 1 <= k & k + 2 <= n & f | n turns_right k,G holds
f turns_right k,G
let D be set ; ::_thesis: for f being FinSequence of D
for G being Matrix of D st 1 <= k & k + 2 <= n & f | n turns_right k,G holds
f turns_right k,G
let f be FinSequence of D; ::_thesis: for G being Matrix of D st 1 <= k & k + 2 <= n & f | n turns_right k,G holds
f turns_right k,G
let G be Matrix of D; ::_thesis: ( 1 <= k & k + 2 <= n & f | n turns_right k,G implies f turns_right k,G )
assume that
A1: ( 1 <= k & k + 2 <= n ) and
A2: f | n turns_right k,G ; ::_thesis: f turns_right k,G
percases ( len f <= n or n < len f ) ;
suppose len f <= n ; ::_thesis: f turns_right k,G
hence  f turns_right k,G by A2, FINSEQ_1:58; ::_thesis: verum
end;
supposeA3: n < len f ; ::_thesis: f turns_right k,G
let i19, j19, i29, j29 be Element of NAT ; :: according to GOBRD13:def_6 ::_thesis: ( [i19,j19] in Indices G & [i29,j29] in Indices G & f /. k = G * (i19,j19) & f /. (k + 1) = G * (i29,j29) & not ( i19 = i29 & j19 + 1 = j29 & [(i29 + 1),j29] in Indices G & f /. (k + 2) = G * ((i29 + 1),j29) ) & not ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 -' 1)] in Indices G & f /. (k + 2) = G * (i29,(j29 -' 1)) ) & not ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 + 1)] in Indices G & f /. (k + 2) = G * (i29,(j29 + 1)) ) implies ( i19 = i29 & j19 = j29 + 1 & [(i29 -' 1),j29] in Indices G & f /. (k + 2) = G * ((i29 -' 1),j29) ) )
A4: len (f | n) = n by A3, FINSEQ_1:59;
then  k + 1 in dom (f | n) by A1, SEQ_4:135;
then A5: (f | n) /. (k + 1) = f /. (k + 1) by FINSEQ_4:70;
 k + 2 in dom (f | n) by A1, A4, SEQ_4:135;
then A6: (f | n) /. (k + 2) = f /. (k + 2) by FINSEQ_4:70;
 k in dom (f | n) by A1, A4, SEQ_4:135;
then  (f | n) /. k = f /. k by FINSEQ_4:70;
hence  ( [i19,j19] in Indices G & [i29,j29] in Indices G & f /. k = G * (i19,j19) & f /. (k + 1) = G * (i29,j29) & not ( i19 = i29 & j19 + 1 = j29 & [(i29 + 1),j29] in Indices G & f /. (k + 2) = G * ((i29 + 1),j29) ) & not ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 -' 1)] in Indices G & f /. (k + 2) = G * (i29,(j29 -' 1)) ) & not ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 + 1)] in Indices G & f /. (k + 2) = G * (i29,(j29 + 1)) ) implies ( i19 = i29 & j19 = j29 + 1 & [(i29 -' 1),j29] in Indices G & f /. (k + 2) = G * ((i29 -' 1),j29) ) ) by A2, A5, A6, Def6; ::_thesis: verum
end;
end;
end;

theorem :: GOBRD13:44
for k, n being Element of NAT
for D being set
for f being FinSequence of D
for G being Matrix of D st 1 <= k & k + 2 <= n & f | n turns_left k,G holds
f turns_left k,G
proof
let k, n be Element of NAT ; ::_thesis: for D being set
for f being FinSequence of D
for G being Matrix of D st 1 <= k & k + 2 <= n & f | n turns_left k,G holds
f turns_left k,G
let D be set ; ::_thesis: for f being FinSequence of D
for G being Matrix of D st 1 <= k & k + 2 <= n & f | n turns_left k,G holds
f turns_left k,G
let f be FinSequence of D; ::_thesis: for G being Matrix of D st 1 <= k & k + 2 <= n & f | n turns_left k,G holds
f turns_left k,G
let G be Matrix of D; ::_thesis: ( 1 <= k & k + 2 <= n & f | n turns_left k,G implies f turns_left k,G )
assume that
A1: ( 1 <= k & k + 2 <= n ) and
A2: f | n turns_left k,G ; ::_thesis: f turns_left k,G
percases ( len f <= n or n < len f ) ;
suppose len f <= n ; ::_thesis: f turns_left k,G
hence  f turns_left k,G by A2, FINSEQ_1:58; ::_thesis: verum
end;
supposeA3: n < len f ; ::_thesis: f turns_left k,G
let i19, j19, i29, j29 be Element of NAT ; :: according to GOBRD13:def_7 ::_thesis: ( [i19,j19] in Indices G & [i29,j29] in Indices G & f /. k = G * (i19,j19) & f /. (k + 1) = G * (i29,j29) & not ( i19 = i29 & j19 + 1 = j29 & [(i29 -' 1),j29] in Indices G & f /. (k + 2) = G * ((i29 -' 1),j29) ) & not ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 + 1)] in Indices G & f /. (k + 2) = G * (i29,(j29 + 1)) ) & not ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 -' 1)] in Indices G & f /. (k + 2) = G * (i29,(j29 -' 1)) ) implies ( i19 = i29 & j19 = j29 + 1 & [(i29 + 1),j29] in Indices G & f /. (k + 2) = G * ((i29 + 1),j29) ) )
A4: len (f | n) = n by A3, FINSEQ_1:59;
then  k + 1 in dom (f | n) by A1, SEQ_4:135;
then A5: (f | n) /. (k + 1) = f /. (k + 1) by FINSEQ_4:70;
 k + 2 in dom (f | n) by A1, A4, SEQ_4:135;
then A6: (f | n) /. (k + 2) = f /. (k + 2) by FINSEQ_4:70;
 k in dom (f | n) by A1, A4, SEQ_4:135;
then  (f | n) /. k = f /. k by FINSEQ_4:70;
hence  ( [i19,j19] in Indices G & [i29,j29] in Indices G & f /. k = G * (i19,j19) & f /. (k + 1) = G * (i29,j29) & not ( i19 = i29 & j19 + 1 = j29 & [(i29 -' 1),j29] in Indices G & f /. (k + 2) = G * ((i29 -' 1),j29) ) & not ( i19 + 1 = i29 & j19 = j29 & [i29,(j29 + 1)] in Indices G & f /. (k + 2) = G * (i29,(j29 + 1)) ) & not ( i19 = i29 + 1 & j19 = j29 & [i29,(j29 -' 1)] in Indices G & f /. (k + 2) = G * (i29,(j29 -' 1)) ) implies ( i19 = i29 & j19 = j29 + 1 & [(i29 + 1),j29] in Indices G & f /. (k + 2) = G * ((i29 + 1),j29) ) ) by A2, A5, A6, Def7; ::_thesis: verum
end;
end;
end;

theorem :: GOBRD13:45
for k, n being Element of NAT
for D being set
for f being FinSequence of D
for G being Matrix of D st 1 <= k & k + 2 <= n & f | n goes_straight k,G holds
f goes_straight k,G
proof
let k, n be Element of NAT ; ::_thesis: for D being set
for f being FinSequence of D
for G being Matrix of D st 1 <= k & k + 2 <= n & f | n goes_straight k,G holds
f goes_straight k,G
let D be set ; ::_thesis: for f being FinSequence of D
for G being Matrix of D st 1 <= k & k + 2 <= n & f | n goes_straight k,G holds
f goes_straight k,G
let f be FinSequence of D; ::_thesis: for G being Matrix of D st 1 <= k & k + 2 <= n & f | n goes_straight k,G holds
f goes_straight k,G
let G be Matrix of D; ::_thesis: ( 1 <= k & k + 2 <= n & f | n goes_straight k,G implies f goes_straight k,G )
assume that
A1: ( 1 <= k & k + 2 <= n ) and
A2: f | n goes_straight k,G ; ::_thesis: f goes_straight k,G
percases ( len f <= n or n < len f ) ;
suppose len f <= n ; ::_thesis: f goes_straight k,G
hence  f goes_straight k,G by A2, FINSEQ_1:58; ::_thesis: verum
end;
supposeA3: n < len f ; ::_thesis: f goes_straight k,G
let i19, j19, i29, j29 be Element of NAT ; :: according to GOBRD13:def_8 ::_thesis: ( [i19,j19] in Indices G & [i29,j29] in Indices G & f /. k = G * (i19,j19) & f /. (k + 1) = G * (i29,j29) & not ( i19 = i29 & j19 + 1 = j29 & [i29,(j29 + 1)] in Indices G & f /. (k + 2) = G * (i29,(j29 + 1)) ) & not ( i19 + 1 = i29 & j19 = j29 & [(i29 + 1),j29] in Indices G & f /. (k + 2) = G * ((i29 + 1),j29) ) & not ( i19 = i29 + 1 & j19 = j29 & [(i29 -' 1),j29] in Indices G & f /. (k + 2) = G * ((i29 -' 1),j29) ) implies ( i19 = i29 & j19 = j29 + 1 & [i29,(j29 -' 1)] in Indices G & f /. (k + 2) = G * (i29,(j29 -' 1)) ) )
A4: len (f | n) = n by A3, FINSEQ_1:59;
then  k + 1 in dom (f | n) by A1, SEQ_4:135;
then A5: (f | n) /. (k + 1) = f /. (k + 1) by FINSEQ_4:70;
 k + 2 in dom (f | n) by A1, A4, SEQ_4:135;
then A6: (f | n) /. (k + 2) = f /. (k + 2) by FINSEQ_4:70;
 k in dom (f | n) by A1, A4, SEQ_4:135;
then  (f | n) /. k = f /. k by FINSEQ_4:70;
hence  ( [i19,j19] in Indices G & [i29,j29] in Indices G & f /. k = G * (i19,j19) & f /. (k + 1) = G * (i29,j29) & not ( i19 = i29 & j19 + 1 = j29 & [i29,(j29 + 1)] in Indices G & f /. (k + 2) = G * (i29,(j29 + 1)) ) & not ( i19 + 1 = i29 & j19 = j29 & [(i29 + 1),j29] in Indices G & f /. (k + 2) = G * ((i29 + 1),j29) ) & not ( i19 = i29 + 1 & j19 = j29 & [(i29 -' 1),j29] in Indices G & f /. (k + 2) = G * ((i29 -' 1),j29) ) implies ( i19 = i29 & j19 = j29 + 1 & [i29,(j29 -' 1)] in Indices G & f /. (k + 2) = G * (i29,(j29 -' 1)) ) ) by A2, A5, A6, Def8; ::_thesis: verum
end;
end;
end;

theorem :: GOBRD13:46
for k being Element of NAT
for D being set
for f1, f2 being FinSequence of D
for G being Matrix of D st 1 < k & k + 1 <= len f1 & k + 1 <= len f2 & f1 is_sequence_on G & f1 | k = f2 | k & f1 turns_right k -' 1,G & f2 turns_right k -' 1,G holds
f1 | (k + 1) = f2 | (k + 1)
proof
let k be Element of NAT ; ::_thesis: for D being set
for f1, f2 being FinSequence of D
for G being Matrix of D st 1 < k & k + 1 <= len f1 & k + 1 <= len f2 & f1 is_sequence_on G & f1 | k = f2 | k & f1 turns_right k -' 1,G & f2 turns_right k -' 1,G holds
f1 | (k + 1) = f2 | (k + 1)
let D be set ; ::_thesis: for f1, f2 being FinSequence of D
for G being Matrix of D st 1 < k & k + 1 <= len f1 & k + 1 <= len f2 & f1 is_sequence_on G & f1 | k = f2 | k & f1 turns_right k -' 1,G & f2 turns_right k -' 1,G holds
f1 | (k + 1) = f2 | (k + 1)
let f1, f2 be FinSequence of D; ::_thesis: for G being Matrix of D st 1 < k & k + 1 <= len f1 & k + 1 <= len f2 & f1 is_sequence_on G & f1 | k = f2 | k & f1 turns_right k -' 1,G & f2 turns_right k -' 1,G holds
f1 | (k + 1) = f2 | (k + 1)
let G be Matrix of D; ::_thesis: ( 1 < k & k + 1 <= len f1 & k + 1 <= len f2 & f1 is_sequence_on G & f1 | k = f2 | k & f1 turns_right k -' 1,G & f2 turns_right k -' 1,G implies f1 | (k + 1) = f2 | (k + 1) )
assume that
A1: 1 < k and
A2: k + 1 <= len f1 and
A3: k + 1 <= len f2 and
A4: f1 is_sequence_on G and
A5: f1 | k = f2 | k and
A6: f1 turns_right k -' 1,G and
A7: f2 turns_right k -' 1,G ; ::_thesis: f1 | (k + 1) = f2 | (k + 1)
A8: 1 <= k -' 1 by A1, NAT_D:49;
A9: k <= k + 1 by NAT_1:12;
then  k <= len (f1 | k) by A2, FINSEQ_1:59, XXREAL_0:2;
then A10: k in dom (f1 | k) by A1, FINSEQ_3:25;
then A11: f2 /. k = (f2 | k) /. k by A5, FINSEQ_4:70;
 k -' 1 <= k by NAT_D:35;
then  k -' 1 <= len (f1 | k) by A2, A9, FINSEQ_1:59, XXREAL_0:2;
then A12: k -' 1 in dom (f1 | k) by A8, FINSEQ_3:25;
then A13: f2 /. (k -' 1) = (f2 | k) /. (k -' 1) by A5, FINSEQ_4:70;
A14: f1 /. k = (f1 | k) /. k by A10, FINSEQ_4:70;
A15: f1 /. (k -' 1) = (f1 | k) /. (k -' 1) by A12, FINSEQ_4:70;
A16: k = (k -' 1) + 1 by A1, XREAL_1:235;
then A17: k + 1 = (k -' 1) + (1 + 1) ;
 k <= len f1 by A2, A9, XXREAL_0:2;
then consider i1, j1, i2, j2 being Element of NAT  such that
A18: ( [i1,j1] in Indices G & f1 /. (k -' 1) = G * (i1,j1) & [i2,j2] in Indices G & f1 /. k = G * (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 A4, A8, A16, JORDAN8:3;
A20: ( j1 + 1 > j1 & j2 + 1 > j2 ) by NAT_1:13;
A21: ( i1 + 1 > i1 & i2 + 1 > i2 ) by NAT_1:13;
now__::_thesis:_f1_/._(k_+_1)_=_f2_/._(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 A19;
supposeA22: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: f1 /. (k + 1) = f2 /. (k + 1)
hence f1 /. (k + 1) = G * ((i2 + 1),j2) by A6, A16, A17, A18, A20, Def6
.= f2 /. (k + 1) by A5, A7, A16, A15, A14, A13, A11, A17, A18, A20, A22, Def6 ;
::_thesis: verum
end;
supposeA23: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: f1 /. (k + 1) = f2 /. (k + 1)
hence f1 /. (k + 1) = G * (i2,(j2 -' 1)) by A6, A16, A17, A18, A21, Def6
.= f2 /. (k + 1) by A5, A7, A16, A15, A14, A13, A11, A17, A18, A21, A23, Def6 ;
::_thesis: verum
end;
supposeA24: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: f1 /. (k + 1) = f2 /. (k + 1)
hence f1 /. (k + 1) = G * (i2,(j2 + 1)) by A6, A16, A17, A18, A21, Def6
.= f2 /. (k + 1) by A5, A7, A16, A15, A14, A13, A11, A17, A18, A21, A24, Def6 ;
::_thesis: verum
end;
supposeA25: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: f1 /. (k + 1) = f2 /. (k + 1)
hence f1 /. (k + 1) = G * ((i2 -' 1),j2) by A6, A16, A17, A18, A20, Def6
.= f2 /. (k + 1) by A5, A7, A16, A15, A14, A13, A11, A17, A18, A20, A25, Def6 ;
::_thesis: verum
end;
end;
end;
hence f1 | (k + 1) = (f2 | k) ^ <*(f2 /. (k + 1))*> by A2, A5, FINSEQ_5:82
.= f2 | (k + 1) by A3, FINSEQ_5:82 ;
::_thesis: verum
end;

theorem :: GOBRD13:47
for k being Element of NAT
for D being set
for f1, f2 being FinSequence of D
for G being Matrix of D st 1 < k & k + 1 <= len f1 & k + 1 <= len f2 & f1 is_sequence_on G & f1 | k = f2 | k & f1 turns_left k -' 1,G & f2 turns_left k -' 1,G holds
f1 | (k + 1) = f2 | (k + 1)
proof
let k be Element of NAT ; ::_thesis: for D being set
for f1, f2 being FinSequence of D
for G being Matrix of D st 1 < k & k + 1 <= len f1 & k + 1 <= len f2 & f1 is_sequence_on G & f1 | k = f2 | k & f1 turns_left k -' 1,G & f2 turns_left k -' 1,G holds
f1 | (k + 1) = f2 | (k + 1)
let D be set ; ::_thesis: for f1, f2 being FinSequence of D
for G being Matrix of D st 1 < k & k + 1 <= len f1 & k + 1 <= len f2 & f1 is_sequence_on G & f1 | k = f2 | k & f1 turns_left k -' 1,G & f2 turns_left k -' 1,G holds
f1 | (k + 1) = f2 | (k + 1)
let f1, f2 be FinSequence of D; ::_thesis: for G being Matrix of D st 1 < k & k + 1 <= len f1 & k + 1 <= len f2 & f1 is_sequence_on G & f1 | k = f2 | k & f1 turns_left k -' 1,G & f2 turns_left k -' 1,G holds
f1 | (k + 1) = f2 | (k + 1)
let G be Matrix of D; ::_thesis: ( 1 < k & k + 1 <= len f1 & k + 1 <= len f2 & f1 is_sequence_on G & f1 | k = f2 | k & f1 turns_left k -' 1,G & f2 turns_left k -' 1,G implies f1 | (k + 1) = f2 | (k + 1) )
assume that
A1: 1 < k and
A2: k + 1 <= len f1 and
A3: k + 1 <= len f2 and
A4: f1 is_sequence_on G and
A5: f1 | k = f2 | k and
A6: f1 turns_left k -' 1,G and
A7: f2 turns_left k -' 1,G ; ::_thesis: f1 | (k + 1) = f2 | (k + 1)
A8: 1 <= k -' 1 by A1, NAT_D:49;
A9: k <= k + 1 by NAT_1:12;
then  k <= len (f1 | k) by A2, FINSEQ_1:59, XXREAL_0:2;
then A10: k in dom (f1 | k) by A1, FINSEQ_3:25;
then A11: f2 /. k = (f2 | k) /. k by A5, FINSEQ_4:70;
 k -' 1 <= k by NAT_D:35;
then  k -' 1 <= len (f1 | k) by A2, A9, FINSEQ_1:59, XXREAL_0:2;
then A12: k -' 1 in dom (f1 | k) by A8, FINSEQ_3:25;
then A13: f2 /. (k -' 1) = (f2 | k) /. (k -' 1) by A5, FINSEQ_4:70;
A14: f1 /. k = (f1 | k) /. k by A10, FINSEQ_4:70;
A15: f1 /. (k -' 1) = (f1 | k) /. (k -' 1) by A12, FINSEQ_4:70;
A16: k = (k -' 1) + 1 by A1, XREAL_1:235;
then A17: k + 1 = (k -' 1) + (1 + 1) ;
 k <= len f1 by A2, A9, XXREAL_0:2;
then consider i1, j1, i2, j2 being Element of NAT  such that
A18: ( [i1,j1] in Indices G & f1 /. (k -' 1) = G * (i1,j1) & [i2,j2] in Indices G & f1 /. k = G * (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 A4, A8, A16, JORDAN8:3;
A20: ( j1 + 1 > j1 & j2 + 1 > j2 ) by NAT_1:13;
A21: ( i1 + 1 > i1 & i2 + 1 > i2 ) by NAT_1:13;
now__::_thesis:_f1_/._(k_+_1)_=_f2_/._(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 A19;
supposeA22: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: f1 /. (k + 1) = f2 /. (k + 1)
hence f1 /. (k + 1) = G * ((i2 -' 1),j2) by A6, A16, A17, A18, A20, Def7
.= f2 /. (k + 1) by A5, A7, A16, A15, A14, A13, A11, A17, A18, A20, A22, Def7 ;
::_thesis: verum
end;
supposeA23: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: f1 /. (k + 1) = f2 /. (k + 1)
hence f1 /. (k + 1) = G * (i2,(j2 + 1)) by A6, A16, A17, A18, A21, Def7
.= f2 /. (k + 1) by A5, A7, A16, A15, A14, A13, A11, A17, A18, A21, A23, Def7 ;
::_thesis: verum
end;
supposeA24: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: f1 /. (k + 1) = f2 /. (k + 1)
hence f1 /. (k + 1) = G * (i2,(j2 -' 1)) by A6, A16, A17, A18, A21, Def7
.= f2 /. (k + 1) by A5, A7, A16, A15, A14, A13, A11, A17, A18, A21, A24, Def7 ;
::_thesis: verum
end;
supposeA25: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: f1 /. (k + 1) = f2 /. (k + 1)
hence f1 /. (k + 1) = G * ((i2 + 1),j2) by A6, A16, A17, A18, A20, Def7
.= f2 /. (k + 1) by A5, A7, A16, A15, A14, A13, A11, A17, A18, A20, A25, Def7 ;
::_thesis: verum
end;
end;
end;
hence f1 | (k + 1) = (f2 | k) ^ <*(f2 /. (k + 1))*> by A2, A5, FINSEQ_5:82
.= f2 | (k + 1) by A3, FINSEQ_5:82 ;
::_thesis: verum
end;

theorem :: GOBRD13:48
for k being Element of NAT
for D being set
for f1, f2 being FinSequence of D
for G being Matrix of D st 1 < k & k + 1 <= len f1 & k + 1 <= len f2 & f1 is_sequence_on G & f1 | k = f2 | k & f1 goes_straight k -' 1,G & f2 goes_straight k -' 1,G holds
f1 | (k + 1) = f2 | (k + 1)
proof
let k be Element of NAT ; ::_thesis: for D being set
for f1, f2 being FinSequence of D
for G being Matrix of D st 1 < k & k + 1 <= len f1 & k + 1 <= len f2 & f1 is_sequence_on G & f1 | k = f2 | k & f1 goes_straight k -' 1,G & f2 goes_straight k -' 1,G holds
f1 | (k + 1) = f2 | (k + 1)
let D be set ; ::_thesis: for f1, f2 being FinSequence of D
for G being Matrix of D st 1 < k & k + 1 <= len f1 & k + 1 <= len f2 & f1 is_sequence_on G & f1 | k = f2 | k & f1 goes_straight k -' 1,G & f2 goes_straight k -' 1,G holds
f1 | (k + 1) = f2 | (k + 1)
let f1, f2 be FinSequence of D; ::_thesis: for G being Matrix of D st 1 < k & k + 1 <= len f1 & k + 1 <= len f2 & f1 is_sequence_on G & f1 | k = f2 | k & f1 goes_straight k -' 1,G & f2 goes_straight k -' 1,G holds
f1 | (k + 1) = f2 | (k + 1)
let G be Matrix of D; ::_thesis: ( 1 < k & k + 1 <= len f1 & k + 1 <= len f2 & f1 is_sequence_on G & f1 | k = f2 | k & f1 goes_straight k -' 1,G & f2 goes_straight k -' 1,G implies f1 | (k + 1) = f2 | (k + 1) )
assume that
A1: 1 < k and
A2: k + 1 <= len f1 and
A3: k + 1 <= len f2 and
A4: f1 is_sequence_on G and
A5: f1 | k = f2 | k and
A6: f1 goes_straight k -' 1,G and
A7: f2 goes_straight k -' 1,G ; ::_thesis: f1 | (k + 1) = f2 | (k + 1)
A8: 1 <= k -' 1 by A1, NAT_D:49;
A9: k <= k + 1 by NAT_1:12;
then  k <= len (f1 | k) by A2, FINSEQ_1:59, XXREAL_0:2;
then A10: k in dom (f1 | k) by A1, FINSEQ_3:25;
then A11: f2 /. k = (f2 | k) /. k by A5, FINSEQ_4:70;
 k -' 1 <= k by NAT_D:35;
then  k -' 1 <= len (f1 | k) by A2, A9, FINSEQ_1:59, XXREAL_0:2;
then A12: k -' 1 in dom (f1 | k) by A8, FINSEQ_3:25;
then A13: f2 /. (k -' 1) = (f2 | k) /. (k -' 1) by A5, FINSEQ_4:70;
A14: f1 /. k = (f1 | k) /. k by A10, FINSEQ_4:70;
A15: f1 /. (k -' 1) = (f1 | k) /. (k -' 1) by A12, FINSEQ_4:70;
A16: k = (k -' 1) + 1 by A1, XREAL_1:235;
then A17: k + 1 = (k -' 1) + (1 + 1) ;
 k <= len f1 by A2, A9, XXREAL_0:2;
then consider i1, j1, i2, j2 being Element of NAT  such that
A18: ( [i1,j1] in Indices G & f1 /. (k -' 1) = G * (i1,j1) & [i2,j2] in Indices G & f1 /. k = G * (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 A4, A8, A16, JORDAN8:3;
A20: ( j1 + 1 > j1 & j2 + 1 > j2 ) by NAT_1:13;
A21: ( i1 + 1 > i1 & i2 + 1 > i2 ) by NAT_1:13;
now__::_thesis:_f1_/._(k_+_1)_=_f2_/._(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 A19;
supposeA22: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: f1 /. (k + 1) = f2 /. (k + 1)
hence f1 /. (k + 1) = G * (i2,(j2 + 1)) by A6, A16, A17, A18, A20, Def8
.= f2 /. (k + 1) by A5, A7, A16, A15, A14, A13, A11, A17, A18, A20, A22, Def8 ;
::_thesis: verum
end;
supposeA23: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: f1 /. (k + 1) = f2 /. (k + 1)
hence f1 /. (k + 1) = G * ((i2 + 1),j2) by A6, A16, A17, A18, A21, Def8
.= f2 /. (k + 1) by A5, A7, A16, A15, A14, A13, A11, A17, A18, A21, A23, Def8 ;
::_thesis: verum
end;
supposeA24: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: f1 /. (k + 1) = f2 /. (k + 1)
hence f1 /. (k + 1) = G * ((i2 -' 1),j2) by A6, A16, A17, A18, A21, Def8
.= f2 /. (k + 1) by A5, A7, A16, A15, A14, A13, A11, A17, A18, A21, A24, Def8 ;
::_thesis: verum
end;
supposeA25: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: f1 /. (k + 1) = f2 /. (k + 1)
hence f1 /. (k + 1) = G * (i2,(j2 -' 1)) by A6, A16, A17, A18, A20, Def8
.= f2 /. (k + 1) by A5, A7, A16, A15, A14, A13, A11, A17, A18, A20, A25, Def8 ;
::_thesis: verum
end;
end;
end;
hence f1 | (k + 1) = (f2 | k) ^ <*(f2 /. (k + 1))*> by A2, A5, FINSEQ_5:82
.= f2 | (k + 1) by A3, FINSEQ_5:82 ;
::_thesis: verum
end;