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;

theorem
  p1 in rng f & p2 in rng f & p3 in rng f & p1..f <= p2..f & p2..f < p3
  ..f implies p1..Rotate(f,p3) <= p2..Rotate(f,p3)
proof
  assume that
A1: p1 in rng f & p2 in rng f and
A2: p3 in rng f and
A3: p1..f <= p2..f and
A4: p2..f < p3..f;
  per cases by A3,XXREAL_0:1;
  suppose
    p1..f < p2..f;
    hence thesis by A1,A2,A4,Th11;
  end;
  suppose
    p1..f = p2..f;
    hence thesis by A1,FINSEQ_5:9;
  end;
end;
