reserve A,B for Ordinal,
        o for object,
        x,y,z for Surreal,
        n for Nat,
        r,r1,r2 for Real;

theorem
  for r be Sequence of REAL
    for y,s be Sequence
      for alpha, beta be Ordinal st beta c= alpha &
        s,y,r simplest_up_to alpha
  holds s,y,r simplest_up_to beta;
