let f be non constant standard special_circular_sequence; for k being Nat st 1 <= k & k + 2 <= len f holds
for i being Nat st 1 <= i & i + 2 <= len (GoB f) & f /. (k + 1) = (GoB f) * ((i + 1),1) & ( ( f /. k = (GoB f) * (i,1) & f /. (k + 2) = (GoB f) * ((i + 1),2) ) or ( f /. (k + 2) = (GoB f) * (i,1) & f /. k = (GoB f) * ((i + 1),2) ) ) holds
LSeg ((((1 / 2) * (((GoB f) * ((i + 1),1)) + ((GoB f) * ((i + 2),1)))) - |[0,1]|),((1 / 2) * (((GoB f) * ((i + 1),1)) + ((GoB f) * ((i + 2),2))))) misses L~ f
let k be Nat; ( 1 <= k & k + 2 <= len f implies for i being Nat st 1 <= i & i + 2 <= len (GoB f) & f /. (k + 1) = (GoB f) * ((i + 1),1) & ( ( f /. k = (GoB f) * (i,1) & f /. (k + 2) = (GoB f) * ((i + 1),2) ) or ( f /. (k + 2) = (GoB f) * (i,1) & f /. k = (GoB f) * ((i + 1),2) ) ) holds
LSeg ((((1 / 2) * (((GoB f) * ((i + 1),1)) + ((GoB f) * ((i + 2),1)))) - |[0,1]|),((1 / 2) * (((GoB f) * ((i + 1),1)) + ((GoB f) * ((i + 2),2))))) misses L~ f )
assume that
A1:
k >= 1
and
A2:
k + 2 <= len f
; for i being Nat st 1 <= i & i + 2 <= len (GoB f) & f /. (k + 1) = (GoB f) * ((i + 1),1) & ( ( f /. k = (GoB f) * (i,1) & f /. (k + 2) = (GoB f) * ((i + 1),2) ) or ( f /. (k + 2) = (GoB f) * (i,1) & f /. k = (GoB f) * ((i + 1),2) ) ) holds
LSeg ((((1 / 2) * (((GoB f) * ((i + 1),1)) + ((GoB f) * ((i + 2),1)))) - |[0,1]|),((1 / 2) * (((GoB f) * ((i + 1),1)) + ((GoB f) * ((i + 2),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 3;
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 3;
A6:
1 < width (GoB f)
by GOBOARD7:33;
let i be Nat; ( 1 <= i & i + 2 <= len (GoB f) & f /. (k + 1) = (GoB f) * ((i + 1),1) & ( ( f /. k = (GoB f) * (i,1) & f /. (k + 2) = (GoB f) * ((i + 1),2) ) or ( f /. (k + 2) = (GoB f) * (i,1) & f /. k = (GoB f) * ((i + 1),2) ) ) implies LSeg ((((1 / 2) * (((GoB f) * ((i + 1),1)) + ((GoB f) * ((i + 2),1)))) - |[0,1]|),((1 / 2) * (((GoB f) * ((i + 1),1)) + ((GoB f) * ((i + 2),2))))) misses L~ f )
assume that
A7:
1 <= i
and
A8:
i + 2 <= len (GoB f)
and
A9:
f /. (k + 1) = (GoB f) * ((i + 1),1)
and
A10:
( ( f /. k = (GoB f) * (i,1) & f /. (k + 2) = (GoB f) * ((i + 1),2) ) or ( f /. (k + 2) = (GoB f) * (i,1) & f /. k = (GoB f) * ((i + 1),2) ) )
; LSeg ((((1 / 2) * (((GoB f) * ((i + 1),1)) + ((GoB f) * ((i + 2),1)))) - |[0,1]|),((1 / 2) * (((GoB f) * ((i + 1),1)) + ((GoB f) * ((i + 2),2))))) misses L~ f
A11:
(i + 1) + 1 = i + (1 + 1)
;
then A12:
i + 1 < len (GoB f)
by A8, NAT_1:13;
width (GoB f) <> 0
by MATRIX_0:def 10;
then A13:
0 + 1 <= width (GoB f)
by NAT_1:14;
then A14:
L~ f misses Int (cell ((GoB f),(i + 1),1))
by A12, GOBOARD7:12;
0 < width (GoB f)
by A13;
then
L~ f misses Int (cell ((GoB f),(i + 1),0))
by A12, GOBOARD7:12;
then A15:
L~ f misses (Int (cell ((GoB f),(i + 1),0))) \/ (Int (cell ((GoB f),(i + 1),1)))
by A14, XBOOLE_1:70;
A16:
1 <= i + 1
by NAT_1:11;
assume
LSeg ((((1 / 2) * (((GoB f) * ((i + 1),1)) + ((GoB f) * ((i + 2),1)))) - |[0,1]|),((1 / 2) * (((GoB f) * ((i + 1),1)) + ((GoB f) * ((i + 2),2))))) meets L~ f
; contradiction
then
L~ f meets ((Int (cell ((GoB f),(i + 1),0))) \/ (Int (cell ((GoB f),(i + 1),1)))) \/ {((1 / 2) * (((GoB f) * ((i + 1),1)) + ((GoB f) * ((i + 2),1))))}
by A11, A12, A6, A16, GOBOARD6:66, XBOOLE_1:63;
then
L~ f meets {((1 / 2) * (((GoB f) * ((i + 1),1)) + ((GoB f) * ((i + 2),1))))}
by A15, XBOOLE_1:70;
then consider k0 being Nat such that
1 <= k0
and
k0 + 1 <= len f
and
A17:
LSeg ((f /. (k + 1)),((GoB f) * ((i + 2),1))) = LSeg (f,k0)
by A8, A9, A13, A11, A16, GOBOARD7:40, ZFMISC_1:50;
A18:
( LSeg (f,(k + 1)) c= L~ f & 1 + 1 = 2 )
by TOPREAL3:19;
( LSeg (f,k0) c= L~ f & LSeg (f,k) c= L~ f )
by TOPREAL3:19;
hence
contradiction
by A7, A9, A10, A12, A6, A17, A4, A18, A5, GOBOARD7:61; verum