let k be Nat; :: thesis: for f being standard special_circular_sequence st 1 <= k & k + 1 <= len f holds
(left_cell (f,k)) /\ (right_cell (f,k)) = LSeg (f,k)
let f be standard special_circular_sequence; :: thesis: ( 1 <= k & k + 1 <= len f implies (left_cell (f,k)) /\ (right_cell (f,k)) = LSeg (f,k) )
assume that
A1: 1 <= k and
A2: k + 1 <= len f ; :: thesis: (left_cell (f,k)) /\ (right_cell (f,k)) = LSeg (f,k)
k <= k + 1 by NAT_1:11;
then k <= len f by A2, XXREAL_0:2;
then A3: k in dom f by A1, FINSEQ_3:25;
then consider i1, j1 being Nat such that
A4: [i1,j1] in Indices (GoB f) and
A5: f /. k = (GoB f) * (i1,j1) by Th11;
A6: k + 1 in dom f by A2, FINSEQ_3:25, NAT_1:11;
then consider i2, j2 being Nat such that
A7: [i2,j2] in Indices (GoB f) and
A8: f /. (k + 1) = (GoB f) * (i2,j2) by Th11;
A9: |.(i1 - i2).| + |.(j1 - j2).| = 1 by A3, A4, A5, A6, A7, A8, Th12;

A10: 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 A9, SEQM_3:42;

case that A11: |.(i1 - i2).| = 1 and
A12: j1 = j2 ; :: thesis: ( ( i1 = i2 + 1 or i1 + 1 = i2 ) & j1 = j2 )

( i1 - i2 = 1 or - (i1 - i2) = 1 ) by A11, ABSVALUE:def 1;
hence ( i1 = i2 + 1 or i1 + 1 = i2 ) ; :: thesis: j1 = j2
thus j1 = j2 by A12; :: thesis: verum

end;

case that A13: i1 = i2 and
A14: |.(j1 - j2).| = 1 ; :: thesis: ( ( j1 = j2 + 1 or j1 + 1 = j2 ) & i1 = i2 )

( j1 - j2 = 1 or - (j1 - j2) = 1 ) by A14, ABSVALUE:def 1;
hence ( j1 = j2 + 1 or j1 + 1 = j2 ) ; :: thesis: i1 = i2
thus i1 = i2 by A13; :: thesis: verum

end;

end;

end;

A15: 0 + 1 <= j1 by A4, MATRIX_0:32;
A16: j1 <= width (GoB f) by A4, MATRIX_0:32;
A17: 1 <= j2 by A7, MATRIX_0:32;
A18: j2 <= width (GoB f) by A7, MATRIX_0:32;
A19: 0 + 1 <= i1 by A4, MATRIX_0:32;
A20: i1 <= len (GoB f) by A4, MATRIX_0:32;
A21: 1 <= i2 by A7, MATRIX_0:32;
A22: i2 <= len (GoB f) by A7, MATRIX_0:32;
i1 > 0 by A19;
then consider i being Nat such that
A23: i + 1 = i1 by NAT_1:6;
reconsider i = i as Nat ;
A24: i + 1 = i1 by A23;
A25: i < len (GoB f) by A20, A23, NAT_1:13;
j1 > 0 by A15;
then consider j being Nat such that
A26: j + 1 = j1 by NAT_1:6;
reconsider j = j as Nat ;
A27: j + 1 = j1 by A26;
A28: j < width (GoB f) by A16, A26, NAT_1:13;

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 A10;

suppose A29: ( i1 = i2 & j1 + 1 = j2 ) ; :: thesis: (left_cell (f,k)) /\ (right_cell (f,k)) = LSeg (f,k)

then A30: j1 < width (GoB f) by A18, NAT_1:13;
A31: left_cell (f,k) = cell ((GoB f),i,j1) by A1, A2, A4, A5, A7, A8, A23, A29, Th27;
right_cell (f,k) = cell ((GoB f),i1,j1) by A1, A2, A4, A5, A7, A8, A24, A29, Th27;
hence (left_cell (f,k)) /\ (right_cell (f,k)) = LSeg (((GoB f) * (i1,j1)),((GoB f) * (i1,(j1 + 1)))) by A15, A23, A25, A30, A31, Th25
.= LSeg (f,k) by A1, A2, A5, A8, A29, TOPREAL1:def 3 ;
:: thesis: verum

end;

suppose A32: ( i1 + 1 = i2 & j1 = j2 ) ; :: thesis: (left_cell (f,k)) /\ (right_cell (f,k)) = LSeg (f,k)

then A33: i1 < len (GoB f) by A22, NAT_1:13;
A34: left_cell (f,k) = cell ((GoB f),i1,j1) by A1, A2, A4, A5, A7, A8, A27, A32, Th28;
right_cell (f,k) = cell ((GoB f),i1,j) by A1, A2, A4, A5, A7, A8, A26, A32, Th28;
hence (left_cell (f,k)) /\ (right_cell (f,k)) = LSeg (((GoB f) * (i1,j1)),((GoB f) * ((i1 + 1),j1))) by A19, A26, A28, A33, A34, Th26
.= LSeg (f,k) by A1, A2, A5, A8, A32, TOPREAL1:def 3 ;
:: thesis: verum

end;

suppose A35: ( i1 = i2 + 1 & j1 = j2 ) ; :: thesis: (left_cell (f,k)) /\ (right_cell (f,k)) = LSeg (f,k)

then A36: i2 < len (GoB f) by A20, NAT_1:13;
A37: left_cell (f,k) = cell ((GoB f),i2,j) by A1, A2, A4, A5, A7, A8, A26, A35, Th29;
right_cell (f,k) = cell ((GoB f),i2,j1) by A1, A2, A4, A5, A7, A8, A27, A35, Th29;
hence (left_cell (f,k)) /\ (right_cell (f,k)) = LSeg (((GoB f) * ((i2 + 1),j1)),((GoB f) * (i2,j1))) by A21, A26, A28, A36, A37, Th26
.= LSeg (f,k) by A1, A2, A5, A8, A35, TOPREAL1:def 3 ;
:: thesis: verum

end;

suppose A38: ( i1 = i2 & j1 = j2 + 1 ) ; :: thesis: (left_cell (f,k)) /\ (right_cell (f,k)) = LSeg (f,k)

then A39: j2 < width (GoB f) by A16, NAT_1:13;
A40: left_cell (f,k) = cell ((GoB f),i1,j2) by A1, A2, A4, A5, A7, A8, A24, A38, Th30;
right_cell (f,k) = cell ((GoB f),i,j2) by A1, A2, A4, A5, A7, A8, A23, A38, Th30;
hence (left_cell (f,k)) /\ (right_cell (f,k)) = LSeg (((GoB f) * (i1,(j2 + 1))),((GoB f) * (i1,j2))) by A17, A23, A25, A39, A40, Th25
.= LSeg (f,k) by A1, A2, A5, A8, A38, TOPREAL1:def 3 ;
:: thesis: verum

end;

end;