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 = W-max L~z implies (S-max L~z)..z < (S-min L~z)..z
proof
  set g = Rotate(z,E-max L~z);
A1: L~z = L~g by REVROT_1:33;
  E-max L~z in rng z by SPRECT_2:46;
  then g/.1 = E-max L~g by A1,FINSEQ_6:92;
  then
A2: (S-max L~g)..g < (S-min L~g)..g & (S-min L~g)..g < (W-max L~g)..g by Lm30
,Th40;
A3: W-max L~g in rng g by SPRECT_2:44;
  assume
A4: z/.1 = W-max L~z;
  for i be Nat st 1 < i & i < len z holds z/.i <> z/.1 by
GOBOARD7:36;
  then
A5: Rotate(g,W-max L~z) = z by A4,FINSEQ_6:181;
  S-max L~g in rng g & S-min L~g in rng g by SPRECT_2:41,42;
  hence thesis by A1,A5,A3,A2,Th11;
end;
