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 i, j being Element of NAT st 1 <= i & i + 2 <= len (GoB f) & 1 <= j & j + 2 <= width (GoB f) & f /. (k + 1) = (GoB f) * (i + 1),(j + 1) & ( ( f /. k = (GoB f) * i,(j + 1) & f /. (k + 2) = (GoB f) * (i + 1),(j + 2) ) or ( f /. (k + 2) = (GoB f) * i,(j + 1) & f /. k = (GoB f) * (i + 1),(j + 2) ) ) holds
LSeg ((1 / 2) * (((GoB f) * (i + 1),j) + ((GoB f) * (i + 2),(j + 1)))),((1 / 2) * (((GoB f) * (i + 1),(j + 1)) + ((GoB f) * (i + 2),(j + 2)))) misses L~ f
let k be Element of NAT ; :: thesis: ( 1 <= k & k + 2 <= len f implies for i, j being Element of NAT st 1 <= i & i + 2 <= len (GoB f) & 1 <= j & j + 2 <= width (GoB f) & f /. (k + 1) = (GoB f) * (i + 1),(j + 1) & ( ( f /. k = (GoB f) * i,(j + 1) & f /. (k + 2) = (GoB f) * (i + 1),(j + 2) ) or ( f /. (k + 2) = (GoB f) * i,(j + 1) & f /. k = (GoB f) * (i + 1),(j + 2) ) ) holds
LSeg ((1 / 2) * (((GoB f) * (i + 1),j) + ((GoB f) * (i + 2),(j + 1)))),((1 / 2) * (((GoB f) * (i + 1),(j + 1)) + ((GoB f) * (i + 2),(j + 2)))) misses L~ f )
assume that
A1: k >= 1 and
A2: k + 2 <= len f ; :: thesis: for i, j being Element of NAT st 1 <= i & i + 2 <= len (GoB f) & 1 <= j & j + 2 <= width (GoB f) & f /. (k + 1) = (GoB f) * (i + 1),(j + 1) & ( ( f /. k = (GoB f) * i,(j + 1) & f /. (k + 2) = (GoB f) * (i + 1),(j + 2) ) or ( f /. (k + 2) = (GoB f) * i,(j + 1) & f /. k = (GoB f) * (i + 1),(j + 2) ) ) holds
LSeg ((1 / 2) * (((GoB f) * (i + 1),j) + ((GoB f) * (i + 2),(j + 1)))),((1 / 2) * (((GoB f) * (i + 1),(j + 1)) + ((GoB f) * (i + 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 + 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 i, j be Element of NAT ; :: thesis: ( 1 <= i & i + 2 <= len (GoB f) & 1 <= j & j + 2 <= width (GoB f) & f /. (k + 1) = (GoB f) * (i + 1),(j + 1) & ( ( f /. k = (GoB f) * i,(j + 1) & f /. (k + 2) = (GoB f) * (i + 1),(j + 2) ) or ( f /. (k + 2) = (GoB f) * i,(j + 1) & f /. k = (GoB f) * (i + 1),(j + 2) ) ) implies LSeg ((1 / 2) * (((GoB f) * (i + 1),j) + ((GoB f) * (i + 2),(j + 1)))),((1 / 2) * (((GoB f) * (i + 1),(j + 1)) + ((GoB f) * (i + 2),(j + 2)))) misses L~ f )
assume that
A6: 1 <= i and
A7: i + 2 <= len (GoB f) and
A8: 1 <= j and
A9: j + 2 <= width (GoB f) and
A10: f /. (k + 1) = (GoB f) * (i + 1),(j + 1) and
A11: ( ( f /. k = (GoB f) * i,(j + 1) & f /. (k + 2) = (GoB f) * (i + 1),(j + 2) ) or ( f /. (k + 2) = (GoB f) * i,(j + 1) & f /. k = (GoB f) * (i + 1),(j + 2) ) ) ; :: thesis: LSeg ((1 / 2) * (((GoB f) * (i + 1),j) + ((GoB f) * (i + 2),(j + 1)))),((1 / 2) * (((GoB f) * (i + 1),(j + 1)) + ((GoB f) * (i + 2),(j + 2)))) misses L~ f
A12: (i + 1) + 1 = i + (1 + 1) ;
then A13: i + 1 < len (GoB f) by A7, NAT_1:13;
j < j + 2 by XREAL_1:31;
then j < width (GoB f) by A9, XXREAL_0:2;
then A14: L~ f misses Int (cell (GoB f),(i + 1),j) by A13, GOBOARD7:14;
j + 1 <= j + 2 by XREAL_1:8;
then A15: j + 1 <= width (GoB f) by A9, XXREAL_0:2;
then L~ f misses Int (cell (GoB f),(i + 1),(j + 1)) by A13, GOBOARD7:14;
then A16: L~ f misses (Int (cell (GoB f),(i + 1),j)) \/ (Int (cell (GoB f),(i + 1),(j + 1))) by A14, XBOOLE_1:70;
A17: 1 <= j + 1 by NAT_1:11;
A18: (j + 1) + 1 = j + (1 + 1) ;
then A19: j + 1 < width (GoB f) by A9, NAT_1:13;
A20: 1 <= i + 1 by NAT_1:11;
assume LSeg ((1 / 2) * (((GoB f) * (i + 1),j) + ((GoB f) * (i + 2),(j + 1)))),((1 / 2) * (((GoB f) * (i + 1),(j + 1)) + ((GoB f) * (i + 2),(j + 2)))) meets L~ f ; :: thesis: contradiction
then L~ f meets ((Int (cell (GoB f),(i + 1),j)) \/ (Int (cell (GoB f),(i + 1),(j + 1)))) \/ {((1 / 2) * (((GoB f) * (i + 1),(j + 1)) + ((GoB f) * (i + 2),(j + 1))))} by A8, A12, A13, A19, A20, GOBOARD6:67, XBOOLE_1:63;
then L~ f meets {((1 / 2) * (((GoB f) * (i + 1),(j + 1)) + ((GoB f) * (i + 2),(j + 1))))} by A16, XBOOLE_1:70;
then consider k0 being Element of NAT such that
1 <= k0 and
k0 + 1 <= len f and
A21: LSeg (f /. (k + 1)),((GoB f) * (i + 2),(j + 1)) = LSeg f,k0 by A7, A10, A17, A15, A12, A20, GOBOARD7:42, ZFMISC_1:56;
( LSeg f,k0 c= L~ f & LSeg f,k c= L~ f ) by TOPREAL3:26;
hence contradiction by A6, A10, A11, A17, A13, A18, A19, A21, A4, A5, GOBOARD7:63; :: thesis: verum