
theorem
  for f being non constant standard special_circular_sequence,
  i, j being Nat st 1 <= i & j <= len GoB f & i < j holds
  LSeg((GoB f)*(1,width GoB f), (GoB f)*(i,width GoB f)) /\
  LSeg((GoB f)*(j,width GoB f), (GoB f)*(len GoB f,width GoB f)) = {}
proof
  let f be non constant standard special_circular_sequence,
  i, j be Nat;
  assume that
A1: 1 <= i and
A2: j <= len GoB f and
A3: i < j;
  set A = LSeg((GoB f)*(1,width GoB f), (GoB f)*(i,width GoB f)),
  B = LSeg((GoB f)*(j,width GoB f), (GoB f)*(len GoB f,width GoB f));
  assume A /\ B <> {};
  then A meets B;
  then consider x be object such that
A4: x in A and
A5: x in B by XBOOLE_0:3;
  reconsider x1 = x as Point of TOP-REAL 2 by A4;
A6: 1 <= width GoB f by GOBOARD7:33;
A7: i <= len GoB f by A2,A3,XXREAL_0:2;
  ((GoB f)*(1,width GoB f))`1 <= ((GoB f)*(i,width GoB f))`1
  proof
    per cases by A1,XXREAL_0:1;
    suppose i = 1;
      hence thesis;
    end;
    suppose i > 1;
      hence thesis by A6,A7,GOBOARD5:3;
    end;
  end;
  then
A8: x1`1 <= ((GoB f)*(i,width GoB f) )`1 by A4,TOPREAL1:3;
A9: ((GoB f)*(j,width GoB f))`1 > ((GoB f)*(i,width GoB f))`1
  by A1,A2,A3,A6,GOBOARD5:3;
A10: 1 <= j by A1,A3,XXREAL_0:2;
  ((GoB f)*(j,width GoB f))`1 <= ((GoB f)*(len GoB f,width GoB f))`1
  proof
    per cases by A2,XXREAL_0:1;
    suppose j < len GoB f;
      hence thesis by A6,A10,GOBOARD5:3;
    end;
    suppose j = len GoB f;
      hence thesis;
    end;
  end;
  then x1`1 >= ((GoB f)*(j,width GoB f))`1 by A5,TOPREAL1:3;
  hence thesis by A8,A9,XXREAL_0:2;
end;
