reserve D for non empty set,
  f for FinSequence of D,
  g for circular FinSequence of D,
  p,p1,p2,p3,q for Element of D;
reserve f for non constant standard special_circular_sequence,
  p,p1,p2,p3,q for Point of TOP-REAL 2;
reserve z for clockwise_oriented non constant standard
  special_circular_sequence;

theorem Th35:
  z/.1 = S-max L~z & N-max L~z <> E-max L~z implies (N-max L~z)..z
  < (E-max L~z)..z
proof
  set g = Rotate(z,W-min L~z);
A1: L~z = L~g by REVROT_1:33;
  W-min L~z in rng z by SPRECT_2:43;
  then
A2: g/.1 = W-min L~g by A1,FINSEQ_6:92;
  then
A3: (E-max L~g)..g < (S-max L~g)..g by Lm11;
  assume that
A4: z/.1 = S-max L~z and
A5: N-max L~z <> E-max L~z;
  for i be Nat st 1 < i & i < len z holds z/.i <> z/.1 by
GOBOARD7:36;
  then
A6: Rotate(g,S-max L~z) = z by A4,FINSEQ_6:181;
A7: N-max L~g in rng g & E-max L~g in rng g by SPRECT_2:40,46;
  S-max L~g in rng g & (N-max L~g)..g < (E-max L~g)..g by A1,A5,A2,Th25,
SPRECT_2:42;
  hence thesis by A1,A6,A7,A3,Th11;
end;
