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
  z/.1 = E-min L~z & S-min L~z <> W-min L~z implies (S-min L~z)..z < (
  W-min L~z)..z
proof
  set g = Rotate(z,N-min L~z);
A1: L~z = L~g by REVROT_1:33;
  N-min L~z in rng z by SPRECT_2:39;
  then
A2: g/.1 = N-min L~g by A1,FINSEQ_6:92;
  then
A3: W-min L~g in rng g & (E-min L~g)..g < (S-min L~g)..g by Lm4,SPRECT_2:43;
  assume that
A4: z/.1 = E-min L~z and
A5: S-min L~z <> W-min L~z;
  for i be Nat st 1 < i & i < len z holds z/.i <> z/.1 by
GOBOARD7:36;
  then
A6: Rotate(g,E-min L~z) = z by A4,FINSEQ_6:181;
A7: S-min L~g in rng g & E-min L~g in rng g by SPRECT_2:41,45;
  (S-min L~g)..g < (W-min L~g)..g by A1,A5,A2,SPRECT_2:74;
  hence thesis by A1,A6,A7,A3,Th5;
end;
