reserve i,j,k,m,n for Nat,
  D for non empty set,
  p for Element of D,
  f for FinSequence of D;
reserve D for non empty set,
  p for Element of D,
  f for FinSequence of D;
reserve f for circular FinSequence of D;
reserve f,g for FinSequence of TOP-REAL 2;
reserve p for Point of TOP-REAL 2,
  f for FinSequence of TOP-REAL 2;

theorem Th27:
  for f being circular FinSequence of TOP-REAL 2 holds Incr Y_axis
  f = Incr Y_axis Rotate(f,p)
proof
  let f be circular FinSequence of TOP-REAL 2;
  per cases;
  suppose
    not p in rng f;
    hence thesis by FINSEQ_6:def 2;
  end;
  suppose
    p in rng f;
    then rng Rotate(f,p) = rng f by FINSEQ_6:90;
    then rng Y_axis Rotate(f,p) = rng Y_axis f by Th23;
    then
    rng Incr Y_axis Rotate(f,p) = rng Y_axis f & len Incr Y_axis Rotate(f,
    p) = card rng Y_axis f by SEQ_4:def 21;
    hence thesis by SEQ_4:def 21;
  end;
end;
