let k be 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 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_0:32;
A8: i2 <= len G by A5, MATRIX_0:32;
A9: 1 <= i2 by A5, MATRIX_0:32;
A10: j2 <= width G by A5, MATRIX_0:32;
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 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_0:32;
then j1 > 0 ;
then consider j being Nat such that
A15: j + 1 = j1 by NAT_1:6;
A16: |.(i1 - i2).| + |.(j1 - j2).| = 1 by A3, A11, A12, A13, A4, A5, A6, GOBOARD1:def 9;

A17: now :: thesis: ( ( |.(i1 - i2).| = 1 & j1 = j2 & ( i1 = i2 + 1 or i1 + 1 = i2 ) & j1 = j2 ) or ( i1 = i2 & |.(j1 - j2).| = 1 & ( j1 = j2 + 1 or j1 + 1 = j2 ) & i1 = i2 ) )

per cases ( ( |.(i1 - i2).| = 1 & j1 = j2 ) or ( i1 = i2 & |.(j1 - j2).| = 1 ) ) by A16, SEQM_3:42;

case that A18: |.(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;

case that A20: i1 = i2 and
A21: |.(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_0:32;
then A24: j < width G by A15, NAT_1:13;
A25: 0 + 1 <= i1 by A12, MATRIX_0:32;
then i1 > 0 ;
then consider i being Nat such that
A26: i + 1 = i1 by NAT_1:6;
A27: i1 <= len G by A12, MATRIX_0:32;
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 Nat ;

per cases ( ( 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;

suppose A30: ( 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, Th15;
( j1 < width G & left_cell (f,k,G) = cell (G,i,j1) ) by A1, A2, A3, A12, A13, A5, A6, A10, A29, A30, Th14, 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;

suppose A32: ( 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, Th17;
( i1 < len G & left_cell (f,k,G) = cell (G,i1,j1) ) by A1, A2, A3, A12, A13, A5, A6, A8, A32, Th16, 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;

suppose A34: ( 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, Th19;
( i2 < len G & left_cell (f,k,G) = cell (G,i2,j) ) by A1, A2, A3, A12, A13, A5, A6, A27, A22, A34, Th18, 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;

suppose A36: ( 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, Th21;
( j2 < width G & left_cell (f,k,G) = cell (G,i1,j2) ) by A1, A2, A3, A12, A13, A5, A6, A23, A36, Th20, 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;