reserve p, q for Point of TOP-REAL 2,
  r for Real,
  h for non constant standard special_circular_sequence,
  g for FinSequence of TOP-REAL 2,
  f for non empty FinSequence of TOP-REAL 2,
  I, i1, i, j, k for Nat;

theorem
  n_e_n h <> n_e_s h
proof
  set i1 = n_e_n h, i2 = n_e_s h;
A1: i1 <= i1 + 1 by NAT_1:11;
A2: 1 <= i1 by Def16;
  i1+1 <= len h by Def16;
  then i1 <= len h by A1,XXREAL_0:2;
  then i1 in dom h by A2,FINSEQ_3:25;
  then
A3: h.i1 = h/.i1 by PARTFUN1:def 6;
A4: i2 <= i2 + 1 by NAT_1:11;
A5: h.i2 = S-max L~h by Def14;
A6: 1 <= i2 by Def14;
  i2+1 <= len h by Def14;
  then i2 <= len h by A4,XXREAL_0:2;
  then i2 in dom h by A6,FINSEQ_3:25;
  then
A7: h.i2 = h/.i2 by PARTFUN1:def 6;
A8: h.i1 = N-max L~h by Def16;
  thus i1 <> i2
  proof
    assume i1 = i2;
    then
A9: N-bound(L~h) = (h/.i2)`2 by A8,A3,EUCLID:52
      .= S-bound(L~h) by A5,A7,EUCLID:52;
A10: 1 <= len h by GOBOARD7:34,XXREAL_0:2;
    then
A11: (h/.1)`2 >= S-bound (L~h) by Th11;
    consider ii be Nat such that
A12: ii in dom h and
A13: (h/.ii)`2 <> (h/.1)`2 by GOBOARD7:31;
A14: ii <= len h by A12,FINSEQ_3:25;
A15: 1 <= ii by A12,FINSEQ_3:25;
    then
A16: (h/.ii)`2 <= N-bound (L~h) by A14,Th11;
A17: (h/.ii)`2 >= S-bound L~h by A15,A14,Th11;
    (h/.1)`2 <= N-bound (L~h) by A10,Th11;
    then (h/.1)`2 = N-bound (L~h) by A9,A11,XXREAL_0:1;
    hence thesis by A9,A13,A16,A17,XXREAL_0:1;
  end;
end;
