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 Th10:
  p1 in rng f & p2 in rng f & p3 in rng f & p1..f < p2..f & p2..f
  < p3..f implies p3..Rotate(f,p2) < p1..Rotate(f,p2)
proof
  assume that
A1: p1 in rng f and
A2: p2 in rng f and
A3: p3 in rng f and
A4: p1..f < p2..f and
A5: p2..f < p3..f;
A6: p1 in rng Rotate(f,p2) & p3 in rng Rotate(f,p2) by A1,A2,A3,FINSEQ_6:90;
  1 <= p1..f & p3..f <= len f by A1,A3,FINSEQ_4:21;
  then
A7: p3..f + 1 <= len f + p1..f by XREAL_1:7;
A8: p3..Rotate(f,p2) = p3..f - p2..f + 1 by A2,A3,A5,Th4
    .= p3..f + 1 - p2..f;
  p1..Rotate(f,p2) = len f + p1..f - p2..f by A1,A2,A4,Th9;
  then p3..Rotate(f,p2) <= p1..Rotate(f,p2) by A8,A7,XREAL_1:9;
  hence thesis by A4,A5,A6,FINSEQ_5:9,XXREAL_0:1;
end;
