let f be non constant standard special_circular_sequence; :: thesis: for j being Nat st 1 <= j & j + 2 <= width (GoB f) holds
LSeg ((((1 / 2) * (((GoB f) * (1,j)) + ((GoB f) * (1,(j + 1))))) - |[1,0]|),(((1 / 2) * (((GoB f) * (1,(j + 1))) + ((GoB f) * (1,(j + 2))))) - |[1,0]|)) misses L~ f
let j be Nat; :: thesis: ( 1 <= j & j + 2 <= width (GoB f) implies LSeg ((((1 / 2) * (((GoB f) * (1,j)) + ((GoB f) * (1,(j + 1))))) - |[1,0]|),(((1 / 2) * (((GoB f) * (1,(j + 1))) + ((GoB f) * (1,(j + 2))))) - |[1,0]|)) misses L~ f )
assume that
A1: 1 <= j and
A2: j + 2 <= width (GoB f) ; :: thesis: LSeg ((((1 / 2) * (((GoB f) * (1,j)) + ((GoB f) * (1,(j + 1))))) - |[1,0]|),(((1 / 2) * (((GoB f) * (1,(j + 1))) + ((GoB f) * (1,(j + 2))))) - |[1,0]|)) misses L~ f
A3: 1 <= len (GoB f) by GOBOARD7:32;

now :: thesis: for p being Point of (TOP-REAL 2) st p in LSeg ((((1 / 2) * (((GoB f) * (1,j)) + ((GoB f) * (1,(j + 1))))) - |[1,0]|),(((1 / 2) * (((GoB f) * (1,(j + 1))) + ((GoB f) * (1,(j + 2))))) - |[1,0]|)) holds
p `1 < ((GoB f) * (1,1)) `1

A4: j <= j + 2 by NAT_1:11;
then j <= width (GoB f) by A2, XXREAL_0:2;
then A5: ((GoB f) * (1,j)) `1 = ((GoB f) * (1,1)) `1 by A1, A3, GOBOARD5:2;
j + 1 <= j + 2 by XREAL_1:6;
then ( 1 <= j + 1 & j + 1 <= width (GoB f) ) by A2, NAT_1:11, XXREAL_0:2;
then A6: ((GoB f) * (1,(j + 1))) `1 = ((GoB f) * (1,1)) `1 by A3, GOBOARD5:2;
1 <= j + 2 by A1, A4, XXREAL_0:2;
then A7: ((GoB f) * (1,(j + 2))) `1 = ((GoB f) * (1,1)) `1 by A2, A3, GOBOARD5:2;
(((1 / 2) * (((GoB f) * (1,(j + 1))) + ((GoB f) * (1,(j + 2))))) - |[1,0]|) `1 = (((1 / 2) * (((GoB f) * (1,(j + 1))) + ((GoB f) * (1,(j + 2))))) `1) - (|[1,0]| `1) by TOPREAL3:3
.= ((1 / 2) * ((((GoB f) * (1,(j + 1))) + ((GoB f) * (1,(j + 2)))) `1)) - (|[1,0]| `1) by TOPREAL3:4
.= ((1 / 2) * ((((GoB f) * (1,1)) `1) + (((GoB f) * (1,1)) `1))) - (|[1,0]| `1) by A6, A7, TOPREAL3:2
.= (1 * (((GoB f) * (1,1)) `1)) - 1 by EUCLID:52 ;
then A8: ((1 / 2) * (((GoB f) * (1,(j + 1))) + ((GoB f) * (1,(j + 2))))) - |[1,0]| = |[((((GoB f) * (1,1)) `1) - 1),((((1 / 2) * (((GoB f) * (1,(j + 1))) + ((GoB f) * (1,(j + 2))))) - |[1,0]|) `2)]| by EUCLID:53;
(((1 / 2) * (((GoB f) * (1,j)) + ((GoB f) * (1,(j + 1))))) - |[1,0]|) `1 = (((1 / 2) * (((GoB f) * (1,j)) + ((GoB f) * (1,(j + 1))))) `1) - (|[1,0]| `1) by TOPREAL3:3
.= ((1 / 2) * ((((GoB f) * (1,j)) + ((GoB f) * (1,(j + 1)))) `1)) - (|[1,0]| `1) by TOPREAL3:4
.= ((1 / 2) * ((((GoB f) * (1,1)) `1) + (((GoB f) * (1,1)) `1))) - (|[1,0]| `1) by A5, A6, TOPREAL3:2
.= (1 * (((GoB f) * (1,1)) `1)) - 1 by EUCLID:52 ;
then A9: ((1 / 2) * (((GoB f) * (1,j)) + ((GoB f) * (1,(j + 1))))) - |[1,0]| = |[((((GoB f) * (1,1)) `1) - 1),((((1 / 2) * (((GoB f) * (1,j)) + ((GoB f) * (1,(j + 1))))) - |[1,0]|) `2)]| by EUCLID:53;
let p be Point of (TOP-REAL 2); :: thesis: ( p in LSeg ((((1 / 2) * (((GoB f) * (1,j)) + ((GoB f) * (1,(j + 1))))) - |[1,0]|),(((1 / 2) * (((GoB f) * (1,(j + 1))) + ((GoB f) * (1,(j + 2))))) - |[1,0]|)) implies p `1 < ((GoB f) * (1,1)) `1 )
assume p in LSeg ((((1 / 2) * (((GoB f) * (1,j)) + ((GoB f) * (1,(j + 1))))) - |[1,0]|),(((1 / 2) * (((GoB f) * (1,(j + 1))) + ((GoB f) * (1,(j + 2))))) - |[1,0]|)) ; :: thesis: p `1 < ((GoB f) * (1,1)) `1
then p `1 = (((GoB f) * (1,1)) `1) - 1 by A9, A8, TOPREAL3:11;
hence p `1 < ((GoB f) * (1,1)) `1 by XREAL_1:44; :: thesis: verum

end;

hence LSeg ((((1 / 2) * (((GoB f) * (1,j)) + ((GoB f) * (1,(j + 1))))) - |[1,0]|),(((1 / 2) * (((GoB f) * (1,(j + 1))) + ((GoB f) * (1,(j + 2))))) - |[1,0]|)) misses L~ f by Th21; :: thesis: verum