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
  f/.1 = S-min L~f & S-min L~f <> W-min L~f implies (S-min L~f)..f < (
  W-min L~f)..f
proof
  assume that
A1: f/.1 = S-min L~f and
A2: S-min L~f <> W-min L~f;
A3: W-min L~f in rng f by SPRECT_2:43;
  then (W-min L~f)..f in dom f by FINSEQ_4:20;
  then
A4: (W-min L~f)..f >= 1 by FINSEQ_3:25;
  S-min L~f in rng f & (S-min L~f)..f = 1 by A1,FINSEQ_6:43,SPRECT_2:41;
  hence thesis by A3,A2,A4,FINSEQ_5:9,XXREAL_0:1;
end;
