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