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 Th37:
  f/.1 = E-max L~f implies (E-max L~f)..f < (E-min L~f)..f
proof
  assume f/.1 = E-max L~f;
  then
A1: (E-max L~f)..f = 1 by FINSEQ_6:43;
A2: E-min L~f in rng f by SPRECT_2:45;
  then (E-min L~f)..f in dom f by FINSEQ_4:20;
  then
A3: (E-min L~f)..f >= 1 by FINSEQ_3:25;
  E-max L~f in rng f by SPRECT_2:46;
  then (E-min L~f)..f <> (E-max L~f)..f by A2,FINSEQ_5:9,SPRECT_2:54;
  hence thesis by A3,A1,XXREAL_0:1;
end;
