let f be non constant standard special_circular_sequence; :: thesis: 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; :: thesis: ( 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 ; :: thesis: 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; :: thesis: ( 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) ) ) ; :: thesis: 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 ; :: thesis: 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; :: thesis: verum