let f be non constant standard special_circular_sequence; for k being Nat st 1 <= k & k + 1 <= len f holds
ex i, j being Nat st
( i <= len (GoB f) & j <= width (GoB f) & cell ((GoB f),i,j) = left_cell (f,k) )
let k be Nat; ( 1 <= k & k + 1 <= len f implies ex i, j being Nat st
( i <= len (GoB f) & j <= width (GoB f) & cell ((GoB f),i,j) = left_cell (f,k) ) )
assume that
A1:
1 <= k
and
A2:
k + 1 <= len f
; ex i, j being Nat st
( i <= len (GoB f) & j <= width (GoB f) & cell ((GoB f),i,j) = left_cell (f,k) )
A3:
f is_sequence_on GoB f
by GOBOARD5:def 5;
k <= len f
by A2, NAT_1:13;
then A4:
k in dom f
by A1, FINSEQ_3:25;
then consider i1, j1 being Nat such that
A5:
[i1,j1] in Indices (GoB f)
and
A6:
f /. k = (GoB f) * (i1,j1)
by A3;
1 <= k + 1
by NAT_1:11;
then A7:
k + 1 in dom f
by A2, FINSEQ_3:25;
then consider i2, j2 being Nat such that
A8:
[i2,j2] in Indices (GoB f)
and
A9:
f /. (k + 1) = (GoB f) * (i2,j2)
by A3;
1 <= i1
by A5, MATRIX_0:32;
then A10:
(i1 -' 1) + 1 = i1
by XREAL_1:235;
1 <= j1
by A5, MATRIX_0:32;
then A11:
(j1 -' 1) + 1 = j1
by XREAL_1:235;
reconsider i19 = i1, i29 = i2, j19 = j1, j29 = j2 as Element of REAL by XREAL_0:def 1;
|.(i1 - i2).| + |.(j1 - j2).| = 1
by A3, A4, A5, A6, A7, A8, A9;
then A12:
( ( |.(i19 - i29).| = 1 & j1 = j2 ) or ( |.(j19 - j29).| = 1 & i1 = i2 ) )
by SEQM_3:42;
A13:
i1 <= len (GoB f)
by A5, MATRIX_0:32;
A14:
j1 <= width (GoB f)
by A5, MATRIX_0:32;
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 A12, SEQM_3:41;
suppose A15:
(
i1 = i2 &
j1 + 1
= j2 )
;
ex i, j being Nat st
( i <= len (GoB f) & j <= width (GoB f) & cell ((GoB f),i,j) = left_cell (f,k) )take
i1 -' 1
;
ex j being Nat st
( i1 -' 1 <= len (GoB f) & j <= width (GoB f) & cell ((GoB f),(i1 -' 1),j) = left_cell (f,k) )take
j1
;
( i1 -' 1 <= len (GoB f) & j1 <= width (GoB f) & cell ((GoB f),(i1 -' 1),j1) = left_cell (f,k) )
i1 -' 1
<= i1
by NAT_D:35;
hence
i1 -' 1
<= len (GoB f)
by A13, XXREAL_0:2;
( j1 <= width (GoB f) & cell ((GoB f),(i1 -' 1),j1) = left_cell (f,k) )thus
j1 <= width (GoB f)
by A5, MATRIX_0:32;
cell ((GoB f),(i1 -' 1),j1) = left_cell (f,k)thus
cell (
(GoB f),
(i1 -' 1),
j1)
= left_cell (
f,
k)
by A1, A2, A5, A6, A8, A9, A10, A15, GOBOARD5:27;
verum end; suppose A16:
(
i1 + 1
= i2 &
j1 = j2 )
;
ex i, j being Nat st
( i <= len (GoB f) & j <= width (GoB f) & cell ((GoB f),i,j) = left_cell (f,k) )take
i1
;
ex j being Nat st
( i1 <= len (GoB f) & j <= width (GoB f) & cell ((GoB f),i1,j) = left_cell (f,k) )take
j1
;
( i1 <= len (GoB f) & j1 <= width (GoB f) & cell ((GoB f),i1,j1) = left_cell (f,k) )thus
i1 <= len (GoB f)
by A5, MATRIX_0:32;
( j1 <= width (GoB f) & cell ((GoB f),i1,j1) = left_cell (f,k) )thus
j1 <= width (GoB f)
by A5, MATRIX_0:32;
cell ((GoB f),i1,j1) = left_cell (f,k)thus
cell (
(GoB f),
i1,
j1)
= left_cell (
f,
k)
by A1, A2, A5, A6, A8, A9, A11, A16, GOBOARD5:28;
verum end; suppose A17:
(
i1 = i2 + 1 &
j1 = j2 )
;
ex i, j being Nat st
( i <= len (GoB f) & j <= width (GoB f) & cell ((GoB f),i,j) = left_cell (f,k) )take
i2
;
ex j being Nat st
( i2 <= len (GoB f) & j <= width (GoB f) & cell ((GoB f),i2,j) = left_cell (f,k) )take
j1 -' 1
;
( i2 <= len (GoB f) & j1 -' 1 <= width (GoB f) & cell ((GoB f),i2,(j1 -' 1)) = left_cell (f,k) )thus
i2 <= len (GoB f)
by A8, MATRIX_0:32;
( j1 -' 1 <= width (GoB f) & cell ((GoB f),i2,(j1 -' 1)) = left_cell (f,k) )
j1 -' 1
<= j1
by NAT_D:35;
hence
j1 -' 1
<= width (GoB f)
by A14, XXREAL_0:2;
cell ((GoB f),i2,(j1 -' 1)) = left_cell (f,k)thus
cell (
(GoB f),
i2,
(j1 -' 1))
= left_cell (
f,
k)
by A1, A2, A5, A6, A8, A9, A11, A17, GOBOARD5:29;
verum end; suppose A18:
(
i1 = i2 &
j1 = j2 + 1 )
;
ex i, j being Nat st
( i <= len (GoB f) & j <= width (GoB f) & cell ((GoB f),i,j) = left_cell (f,k) )take
i1
;
ex j being Nat st
( i1 <= len (GoB f) & j <= width (GoB f) & cell ((GoB f),i1,j) = left_cell (f,k) )take
j2
;
( i1 <= len (GoB f) & j2 <= width (GoB f) & cell ((GoB f),i1,j2) = left_cell (f,k) )thus
i1 <= len (GoB f)
by A5, MATRIX_0:32;
( j2 <= width (GoB f) & cell ((GoB f),i1,j2) = left_cell (f,k) )thus
j2 <= width (GoB f)
by A8, MATRIX_0:32;
cell ((GoB f),i1,j2) = left_cell (f,k)thus
cell (
(GoB f),
i1,
j2)
= left_cell (
f,
k)
by A1, A2, A5, A6, A8, A9, A10, A18, GOBOARD5:30;
verum end; end;