let f be non constant standard special_circular_sequence; :: thesis: for k being Nat st 1 <= k & k + 2 <= len f holds
for j, i being Nat st 1 <= j & j + 2 <= width (GoB f) & 1 <= i & i + 2 <= len (GoB f) & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & ( ( f /. k = (GoB f) * ((i + 1),(j + 2)) & f /. (k + 2) = (GoB f) * ((i + 2),(j + 1)) ) or ( f /. (k + 2) = (GoB f) * ((i + 1),(j + 2)) & f /. k = (GoB f) * ((i + 2),(j + 1)) ) ) 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 Nat; :: thesis: ( 1 <= k & k + 2 <= len f implies for j, i being Nat st 1 <= j & j + 2 <= width (GoB f) & 1 <= i & i + 2 <= len (GoB f) & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & ( ( f /. k = (GoB f) * ((i + 1),(j + 2)) & f /. (k + 2) = (GoB f) * ((i + 2),(j + 1)) ) or ( f /. (k + 2) = (GoB f) * ((i + 1),(j + 2)) & f /. k = (GoB f) * ((i + 2),(j + 1)) ) ) 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 ; :: thesis: for j, i being Nat st 1 <= j & j + 2 <= width (GoB f) & 1 <= i & i + 2 <= len (GoB f) & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & ( ( f /. k = (GoB f) * ((i + 1),(j + 2)) & f /. (k + 2) = (GoB f) * ((i + 2),(j + 1)) ) or ( f /. (k + 2) = (GoB f) * ((i + 1),(j + 2)) & f /. k = (GoB f) * ((i + 2),(j + 1)) ) ) 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 3, TOPREAL3:19;
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;
let j, i be Nat; :: thesis: ( 1 <= j & j + 2 <= width (GoB f) & 1 <= i & i + 2 <= len (GoB f) & f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) & ( ( f /. k = (GoB f) * ((i + 1),(j + 2)) & f /. (k + 2) = (GoB f) * ((i + 2),(j + 1)) ) or ( f /. (k + 2) = (GoB f) * ((i + 1),(j + 2)) & f /. k = (GoB f) * ((i + 2),(j + 1)) ) ) 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 + 2 <= width (GoB f) and
A8: 1 <= i and
A9: i + 2 <= len (GoB f) and
A10: f /. (k + 1) = (GoB f) * ((i + 1),(j + 1)) and
A11: ( ( f /. k = (GoB f) * ((i + 1),(j + 2)) & f /. (k + 2) = (GoB f) * ((i + 2),(j + 1)) ) or ( f /. (k + 2) = (GoB f) * ((i + 1),(j + 2)) & f /. k = (GoB f) * ((i + 2),(j + 1)) ) ) ; :: thesis: 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
(j + 1) + 1 = j + (1 + 1) ;
then A12: j + 1 < width (GoB f) by A7, NAT_1:13;
then A13: j < width (GoB f) by NAT_1:13;
i < i + 2 by XREAL_1:29;
then i < len (GoB f) by A9, XXREAL_0:2;
then A14: L~ f misses Int (cell ((GoB f),i,j)) by A13, GOBOARD7:12;
i + 1 <= i + 2 by XREAL_1:6;
then A15: i + 1 <= len (GoB f) by A9, XXREAL_0:2;
then L~ f misses Int (cell ((GoB f),(i + 1),j)) by A13, GOBOARD7:12;
then A16: L~ f misses (Int (cell ((GoB f),i,j))) \/ (Int (cell ((GoB f),(i + 1),j))) by A14, XBOOLE_1:70;
A17: 1 <= i + 1 by NAT_1:11;
A18: (i + 1) + 1 = i + (1 + 1) ;
then A19: 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 ; :: thesis: 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, A19, A13, GOBOARD6:65, XBOOLE_1:63;
then L~ f meets {((1 / 2) * (((GoB f) * ((i + 1),j)) + ((GoB f) * ((i + 1),(j + 1)))))} by A16, XBOOLE_1:70;
then consider k0 being Nat such that
1 <= k0 and
k0 + 1 <= len f and
A20: LSeg ((f /. (k + 1)),((GoB f) * ((i + 1),j))) = LSeg (f,k0) by A6, A10, A17, A15, A12, GOBOARD7:39, ZFMISC_1:50;
( LSeg (f,k0) c= L~ f & LSeg (f,k) c= L~ f ) by TOPREAL3:19;
hence contradiction by A6, A10, A11, A17, A12, A18, A19, A20, A4, A5, GOBOARD7:59; :: thesis: verum