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 being Element of NAT st 1 <= j & j + 2 <= width (GoB f) & f /. (k + 1) = (GoB f) * 1,(j + 1) & ( ( f /. k = (GoB f) * 1,j & f /. (k + 2) = (GoB f) * 2,(j + 1) ) or ( f /. (k + 2) = (GoB f) * 1,j & f /. k = (GoB f) * 2,(j + 1) ) ) holds
LSeg (((1 / 2) * (((GoB f) * 1,(j + 1)) + ((GoB f) * 1,(j + 2)))) - |[1,0 ]|),((1 / 2) * (((GoB f) * 1,(j + 1)) + ((GoB f) * 2,(j + 2)))) misses L~ f
let k be Element of NAT ; ( 1 <= k & k + 2 <= len f implies for j being Element of NAT st 1 <= j & j + 2 <= width (GoB f) & f /. (k + 1) = (GoB f) * 1,(j + 1) & ( ( f /. k = (GoB f) * 1,j & f /. (k + 2) = (GoB f) * 2,(j + 1) ) or ( f /. (k + 2) = (GoB f) * 1,j & f /. k = (GoB f) * 2,(j + 1) ) ) holds
LSeg (((1 / 2) * (((GoB f) * 1,(j + 1)) + ((GoB f) * 1,(j + 2)))) - |[1,0 ]|),((1 / 2) * (((GoB f) * 1,(j + 1)) + ((GoB f) * 2,(j + 2)))) misses L~ f )
assume that
A1:
k >= 1
and
A2:
k + 2 <= len f
; for j being Element of NAT st 1 <= j & j + 2 <= width (GoB f) & f /. (k + 1) = (GoB f) * 1,(j + 1) & ( ( f /. k = (GoB f) * 1,j & f /. (k + 2) = (GoB f) * 2,(j + 1) ) or ( f /. (k + 2) = (GoB f) * 1,j & f /. k = (GoB f) * 2,(j + 1) ) ) holds
LSeg (((1 / 2) * (((GoB f) * 1,(j + 1)) + ((GoB f) * 1,(j + 2)))) - |[1,0 ]|),((1 / 2) * (((GoB f) * 1,(j + 1)) + ((GoB f) * 2,(j + 2)))) 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 = LSeg (f /. k),(f /. (k + 1))
by A1, TOPREAL1:def 5;
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;
A6:
1 < len (GoB f)
by GOBOARD7:34;
let j be Element of NAT ; ( 1 <= j & j + 2 <= width (GoB f) & f /. (k + 1) = (GoB f) * 1,(j + 1) & ( ( f /. k = (GoB f) * 1,j & f /. (k + 2) = (GoB f) * 2,(j + 1) ) or ( f /. (k + 2) = (GoB f) * 1,j & f /. k = (GoB f) * 2,(j + 1) ) ) implies LSeg (((1 / 2) * (((GoB f) * 1,(j + 1)) + ((GoB f) * 1,(j + 2)))) - |[1,0 ]|),((1 / 2) * (((GoB f) * 1,(j + 1)) + ((GoB f) * 2,(j + 2)))) misses L~ f )
assume that
A7:
1 <= j
and
A8:
j + 2 <= width (GoB f)
and
A9:
f /. (k + 1) = (GoB f) * 1,(j + 1)
and
A10:
( ( f /. k = (GoB f) * 1,j & f /. (k + 2) = (GoB f) * 2,(j + 1) ) or ( f /. (k + 2) = (GoB f) * 1,j & f /. k = (GoB f) * 2,(j + 1) ) )
; LSeg (((1 / 2) * (((GoB f) * 1,(j + 1)) + ((GoB f) * 1,(j + 2)))) - |[1,0 ]|),((1 / 2) * (((GoB f) * 1,(j + 1)) + ((GoB f) * 2,(j + 2)))) misses L~ f
A11:
(j + 1) + 1 = j + (1 + 1)
;
then A12:
j + 1 < width (GoB f)
by A8, NAT_1:13;
len (GoB f) <> 0
by GOBOARD1:def 5;
then A13:
0 + 1 <= len (GoB f)
by NAT_1:14;
then A14:
L~ f misses Int (cell (GoB f),1,(j + 1))
by A12, GOBOARD7:14;
0 < len (GoB f)
by A13, NAT_1:13;
then
L~ f misses Int (cell (GoB f),0 ,(j + 1))
by A12, GOBOARD7:14;
then A15:
L~ f misses (Int (cell (GoB f),0 ,(j + 1))) \/ (Int (cell (GoB f),1,(j + 1)))
by A14, XBOOLE_1:70;
A16:
1 <= j + 1
by NAT_1:11;
assume
LSeg (((1 / 2) * (((GoB f) * 1,(j + 1)) + ((GoB f) * 1,(j + 2)))) - |[1,0 ]|),((1 / 2) * (((GoB f) * 1,(j + 1)) + ((GoB f) * 2,(j + 2)))) meets L~ f
; contradiction
then
L~ f meets ((Int (cell (GoB f),0 ,(j + 1))) \/ (Int (cell (GoB f),1,(j + 1)))) \/ {((1 / 2) * (((GoB f) * 1,(j + 1)) + ((GoB f) * 1,(j + 2))))}
by A11, A12, A6, A16, GOBOARD6:71, XBOOLE_1:63;
then
L~ f meets {((1 / 2) * (((GoB f) * 1,(j + 1)) + ((GoB f) * 1,(j + 2))))}
by A15, 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) * 1,(j + 2)) = LSeg f,k0
by A8, A9, A13, A11, A16, GOBOARD7:41, ZFMISC_1:56;
A18:
( LSeg f,(k + 1) c= L~ f & 1 + 1 = 2 )
by TOPREAL3:26;
( LSeg f,k0 c= L~ f & LSeg f,k c= L~ f )
by TOPREAL3:26;
hence
contradiction
by A7, A9, A10, A12, A6, A17, A4, A18, A5, GOBOARD7:61; verum