let f be non constant standard special_circular_sequence; :: thesis: 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) * (len (GoB f)),(j + 1) & ( ( f /. k = (GoB f) * (len (GoB f)),(j + 2) & f /. (k + 2) = (GoB f) * ((len (GoB f)) -' 1),(j + 1) ) or ( f /. (k + 2) = (GoB f) * (len (GoB f)),(j + 2) & f /. k = (GoB f) * ((len (GoB f)) -' 1),(j + 1) ) ) holds
LSeg ((1 / 2) * (((GoB f) * ((len (GoB f)) -' 1),j) + ((GoB f) * (len (GoB f)),(j + 1)))),(((1 / 2) * (((GoB f) * (len (GoB f)),j) + ((GoB f) * (len (GoB f)),(j + 1)))) + |[1,0 ]|) misses L~ f
let k be Element of NAT ; :: thesis: ( 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) * (len (GoB f)),(j + 1) & ( ( f /. k = (GoB f) * (len (GoB f)),(j + 2) & f /. (k + 2) = (GoB f) * ((len (GoB f)) -' 1),(j + 1) ) or ( f /. (k + 2) = (GoB f) * (len (GoB f)),(j + 2) & f /. k = (GoB f) * ((len (GoB f)) -' 1),(j + 1) ) ) holds
LSeg ((1 / 2) * (((GoB f) * ((len (GoB f)) -' 1),j) + ((GoB f) * (len (GoB f)),(j + 1)))),(((1 / 2) * (((GoB f) * (len (GoB f)),j) + ((GoB f) * (len (GoB f)),(j + 1)))) + |[1,0 ]|) misses L~ f )
assume that
A1: k >= 1 and
A2: k + 2 <= len f ; :: thesis: for j being Element of NAT st 1 <= j & j + 2 <= width (GoB f) & f /. (k + 1) = (GoB f) * (len (GoB f)),(j + 1) & ( ( f /. k = (GoB f) * (len (GoB f)),(j + 2) & f /. (k + 2) = (GoB f) * ((len (GoB f)) -' 1),(j + 1) ) or ( f /. (k + 2) = (GoB f) * (len (GoB f)),(j + 2) & f /. k = (GoB f) * ((len (GoB f)) -' 1),(j + 1) ) ) holds
LSeg ((1 / 2) * (((GoB f) * ((len (GoB f)) -' 1),j) + ((GoB f) * (len (GoB f)),(j + 1)))),(((1 / 2) * (((GoB f) * (len (GoB f)),j) + ((GoB f) * (len (GoB f)),(j + 1)))) + |[1,0 ]|) 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 be Element of NAT ; :: thesis: ( 1 <= j & j + 2 <= width (GoB f) & f /. (k + 1) = (GoB f) * (len (GoB f)),(j + 1) & ( ( f /. k = (GoB f) * (len (GoB f)),(j + 2) & f /. (k + 2) = (GoB f) * ((len (GoB f)) -' 1),(j + 1) ) or ( f /. (k + 2) = (GoB f) * (len (GoB f)),(j + 2) & f /. k = (GoB f) * ((len (GoB f)) -' 1),(j + 1) ) ) implies LSeg ((1 / 2) * (((GoB f) * ((len (GoB f)) -' 1),j) + ((GoB f) * (len (GoB f)),(j + 1)))),(((1 / 2) * (((GoB f) * (len (GoB f)),j) + ((GoB f) * (len (GoB f)),(j + 1)))) + |[1,0 ]|) misses L~ f )
assume that
A6: 1 <= j and
A7: j + 2 <= width (GoB f) and
A8: f /. (k + 1) = (GoB f) * (len (GoB f)),(j + 1) and
A9: ( ( f /. k = (GoB f) * (len (GoB f)),(j + 2) & f /. (k + 2) = (GoB f) * ((len (GoB f)) -' 1),(j + 1) ) or ( f /. (k + 2) = (GoB f) * (len (GoB f)),(j + 2) & f /. k = (GoB f) * ((len (GoB f)) -' 1),(j + 1) ) ) ; :: thesis: LSeg ((1 / 2) * (((GoB f) * ((len (GoB f)) -' 1),j) + ((GoB f) * (len (GoB f)),(j + 1)))),(((1 / 2) * (((GoB f) * (len (GoB f)),j) + ((GoB f) * (len (GoB f)),(j + 1)))) + |[1,0 ]|) misses L~ f
(j + 1) + 1 = j + (1 + 1) ;
then A10: j + 1 < width (GoB f) by A7, NAT_1:13;
then A11: j < width (GoB f) by NAT_1:13;
then A12: L~ f misses Int (cell (GoB f),(len (GoB f)),j) by GOBOARD7:14;
assume A13: LSeg ((1 / 2) * (((GoB f) * ((len (GoB f)) -' 1),j) + ((GoB f) * (len (GoB f)),(j + 1)))),(((1 / 2) * (((GoB f) * (len (GoB f)),j) + ((GoB f) * (len (GoB f)),(j + 1)))) + |[1,0 ]|) meets L~ f ; :: thesis: contradiction
A14: 1 < len (GoB f) by GOBOARD7:34;
then A15: ((len (GoB f)) -' 1) + 1 = len (GoB f) by XREAL_1:237;
then A16: (len (GoB f)) -' 1 < len (GoB f) by NAT_1:13;
then L~ f misses Int (cell (GoB f),((len (GoB f)) -' 1),j) by A11, GOBOARD7:14;
then A17: L~ f misses (Int (cell (GoB f),((len (GoB f)) -' 1),j)) \/ (Int (cell (GoB f),(len (GoB f)),j)) by A12, XBOOLE_1:70;
A18: 1 <= (len (GoB f)) -' 1 by A14, A15, NAT_1:13;
then (1 / 2) * (((GoB f) * ((len (GoB f)) -' 1),j) + ((GoB f) * (len (GoB f)),(j + 1))) = (1 / 2) * (((GoB f) * ((len (GoB f)) -' 1),(j + 1)) + ((GoB f) * (len (GoB f)),j)) by A6, A15, A10, GOBOARD7:11;
then L~ f meets ((Int (cell (GoB f),((len (GoB f)) -' 1),j)) \/ (Int (cell (GoB f),(len (GoB f)),j))) \/ {((1 / 2) * (((GoB f) * (len (GoB f)),j) + ((GoB f) * (len (GoB f)),(j + 1))))} by A6, A14, A11, A13, GOBOARD6:72, XBOOLE_1:63;
then L~ f meets {((1 / 2) * (((GoB f) * (len (GoB f)),j) + ((GoB f) * (len (GoB f)),(j + 1))))} by A17, XBOOLE_1:70;
then consider k0 being Element of NAT such that
1 <= k0 and
k0 + 1 <= len f and
A19: LSeg (f /. (k + 1)),((GoB f) * (len (GoB f)),j) = LSeg f,k0 by A6, A8, A14, A10, 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, A9, A15, A18, A16, A10, A19, A4, A5, GOBOARD7:62; :: thesis: verum