reserve i,j,k,i1,j1 for Nat,
  p for Point of TOP-REAL 2,
  x for set;
reserve f for non constant standard special_circular_sequence;
reserve P for Subset of TOP-REAL 2;

theorem
  LSeg(1/2*((GoB f)*(1,width GoB f -' 1)+(GoB f)*(1,width GoB f))- |[1,0
  ]| , (GoB f)*(1,width GoB f)+|[-1,1]|) misses L~f
proof
A1: 1 <= len GoB f by GOBOARD7:32;
  now
A2: 1 < width GoB f by GOBOARD7:33;
    then
A3: width GoB f -' 1 +1 = width GoB f by XREAL_1:235;
    then
A4: width GoB f -' 1 <= width GoB f by NAT_1:11;
A5: (GoB f)*(1,width GoB f)`1 = (GoB f)*(1,1)`1 by A1,A2,GOBOARD5:2;
    then ((GoB f)*(1,width GoB f)+|[-1,1]|)`1 = (GoB f)*(1,1)`1+|[-1,1]|`1 by
TOPREAL3:2
      .= (GoB f)*(1,1)`1+-1 by EUCLID:52
      .= (GoB f)*(1,1)`1-1;
    then
A6: (GoB f)*(1,width GoB f)+|[-1,1]| = |[(GoB f)*(1,1)`1-1,((GoB f)* (1,
    width GoB f)+|[-1,1]|)`2]| by EUCLID:53;
    1 <= width GoB f -' 1 by A2,A3,NAT_1:13;
    then
A7: (GoB f)*(1,width GoB f -' 1)`1 = (GoB f)*(1,1)`1 by A1,A4,GOBOARD5:2;
    (1/2*((GoB f)*(1,width GoB f -' 1)+(GoB f)*(1,width GoB f))- |[1,0]|)
`1 = (1/2*((GoB f)*(1,width GoB f -' 1) +(GoB f)*(1,width GoB f)))`1- |[1,0]|`1
    by TOPREAL3:3
      .= 1/2*((GoB f)*(1,width GoB f -' 1)+(GoB f)* (1,width GoB f))`1- |[1,
    0]|`1 by TOPREAL3:4
      .= 1/2*((GoB f)*(1,1)`1+(GoB f)*(1,1)`1)- |[1,0]|`1 by A7,A5,TOPREAL3:2
      .= 1*((GoB f)*(1,1))`1-1 by EUCLID:52;
    then
A8: 1/2*((GoB f)*(1,width GoB f -' 1)+(GoB f)*(1,width GoB f))- |[1 ,0 ]|
= |[(GoB f)*(1,1)`1-1,(1/2*((GoB f)*(1,width GoB f -' 1) +(GoB f)*(1,width GoB
    f))- |[1,0]|)`2]| by EUCLID:53;
    let p;
    assume p in LSeg(1/2*((GoB f)*(1,width GoB f -' 1)+(GoB f)* (1,width GoB
    f))- |[1,0]|, (GoB f)*(1,width GoB f)+|[-1,1]|);
    then p`1 = (GoB f)*(1,1)`1 - 1 by A8,A6,TOPREAL3:11;
    hence p`1 < (GoB f)*(1,1)`1 by XREAL_1:44;
  end;
  hence thesis by Th21;
end;
