let f be non constant standard special_circular_sequence; for k being Element of NAT st 1 <= k & k + 2 <= len f holds
for j, i being Element of NAT st 1 <= j & j + 1 <= width (GoB f) & 1 <= i & i + 2 <= len (GoB f) & f /. (k + 1) = (GoB f) * (i + 1),j & ( ( f /. k = (GoB f) * i,j & f /. (k + 2) = (GoB f) * (i + 2),j ) or ( f /. (k + 2) = (GoB f) * i,j & f /. k = (GoB f) * (i + 2),j ) ) holds
LSeg ((1 / 2) * (((GoB f) * i,j) + ((GoB f) * (i + 1),(j + 1)))),((1 / 2) * (((GoB f) * (i + 1),j) + ((GoB f) * (i + 2),(j + 1)))) misses L~ f
let k be Element of NAT ; ( 1 <= k & k + 2 <= len f implies for j, i being Element of NAT st 1 <= j & j + 1 <= width (GoB f) & 1 <= i & i + 2 <= len (GoB f) & f /. (k + 1) = (GoB f) * (i + 1),j & ( ( f /. k = (GoB f) * i,j & f /. (k + 2) = (GoB f) * (i + 2),j ) or ( f /. (k + 2) = (GoB f) * i,j & f /. k = (GoB f) * (i + 2),j ) ) holds
LSeg ((1 / 2) * (((GoB f) * i,j) + ((GoB f) * (i + 1),(j + 1)))),((1 / 2) * (((GoB f) * (i + 1),j) + ((GoB f) * (i + 2),(j + 1)))) misses L~ f )
assume that
A1:
k >= 1
and
A2:
k + 2 <= len f
; for j, i being Element of NAT st 1 <= j & j + 1 <= width (GoB f) & 1 <= i & i + 2 <= len (GoB f) & f /. (k + 1) = (GoB f) * (i + 1),j & ( ( f /. k = (GoB f) * i,j & f /. (k + 2) = (GoB f) * (i + 2),j ) or ( f /. (k + 2) = (GoB f) * i,j & f /. k = (GoB f) * (i + 2),j ) ) holds
LSeg ((1 / 2) * (((GoB f) * i,j) + ((GoB f) * (i + 1),(j + 1)))),((1 / 2) * (((GoB f) * (i + 1),j) + ((GoB f) * (i + 2),(j + 1)))) misses L~ f
A3:
(k + 1) + 1 = k + (1 + 1)
;
then
k + 1 < len f
by A2, NAT_1:13;
then A4:
( LSeg f,(k + 1) c= L~ f & LSeg f,k = LSeg (f /. k),(f /. (k + 1)) )
by A1, TOPREAL1:def 5, TOPREAL3:26;
1 <= k + 1
by NAT_1:11;
then A5:
LSeg f,(k + 1) = LSeg (f /. (k + 1)),(f /. (k + 2))
by A2, A3, TOPREAL1:def 5;
let j, i be Element of NAT ; ( 1 <= j & j + 1 <= width (GoB f) & 1 <= i & i + 2 <= len (GoB f) & f /. (k + 1) = (GoB f) * (i + 1),j & ( ( f /. k = (GoB f) * i,j & f /. (k + 2) = (GoB f) * (i + 2),j ) or ( f /. (k + 2) = (GoB f) * i,j & f /. k = (GoB f) * (i + 2),j ) ) implies LSeg ((1 / 2) * (((GoB f) * i,j) + ((GoB f) * (i + 1),(j + 1)))),((1 / 2) * (((GoB f) * (i + 1),j) + ((GoB f) * (i + 2),(j + 1)))) misses L~ f )
assume that
A6:
1 <= j
and
A7:
j + 1 <= width (GoB f)
and
A8:
1 <= i
and
A9:
i + 2 <= len (GoB f)
and
A10:
f /. (k + 1) = (GoB f) * (i + 1),j
and
A11:
( ( f /. k = (GoB f) * i,j & f /. (k + 2) = (GoB f) * (i + 2),j ) or ( f /. (k + 2) = (GoB f) * i,j & f /. k = (GoB f) * (i + 2),j ) )
; LSeg ((1 / 2) * (((GoB f) * i,j) + ((GoB f) * (i + 1),(j + 1)))),((1 / 2) * (((GoB f) * (i + 1),j) + ((GoB f) * (i + 2),(j + 1)))) misses L~ f
A12:
j < width (GoB f)
by A7, NAT_1:13;
i < i + 2
by XREAL_1:31;
then
i <= len (GoB f)
by A9, XXREAL_0:2;
then A13:
L~ f misses Int (cell (GoB f),i,j)
by A12, GOBOARD7:14;
i + 1 <= i + 2
by XREAL_1:8;
then A14:
i + 1 <= len (GoB f)
by A9, XXREAL_0:2;
then
L~ f misses Int (cell (GoB f),(i + 1),j)
by A12, GOBOARD7:14;
then A15:
L~ f misses (Int (cell (GoB f),i,j)) \/ (Int (cell (GoB f),(i + 1),j))
by A13, XBOOLE_1:70;
(i + 1) + 1 = i + (1 + 1)
;
then A16:
i + 1 < len (GoB f)
by A9, NAT_1:13;
assume
LSeg ((1 / 2) * (((GoB f) * i,j) + ((GoB f) * (i + 1),(j + 1)))),((1 / 2) * (((GoB f) * (i + 1),j) + ((GoB f) * (i + 2),(j + 1)))) meets L~ f
; contradiction
then
L~ f meets ((Int (cell (GoB f),i,j)) \/ (Int (cell (GoB f),(i + 1),j))) \/ {((1 / 2) * (((GoB f) * (i + 1),j) + ((GoB f) * (i + 1),(j + 1))))}
by A6, A8, A16, A12, GOBOARD6:68, XBOOLE_1:63;
then
( 1 <= i + 1 & L~ f meets {((1 / 2) * (((GoB f) * (i + 1),j) + ((GoB f) * (i + 1),(j + 1))))} )
by A15, NAT_1:11, XBOOLE_1:70;
then consider k0 being Element of NAT such that
1 <= k0
and
k0 + 1 <= len f
and
A17:
LSeg (f /. (k + 1)),((GoB f) * (i + 1),(j + 1)) = LSeg f,k0
by A6, A7, A10, A14, GOBOARD7:41, ZFMISC_1:56;
( LSeg f,k0 c= L~ f & LSeg f,k c= L~ f )
by TOPREAL3:26;
hence
contradiction
by A6, A8, A10, A11, A16, A12, A17, A4, A5, GOBOARD7:63; verum