 reserve n,m for Nat,
      o for object,
      p for pair object,
      x,y,z for Surreal;

theorem Th8:
  sqrtL(p,o).(n+1) = sqrtL(p,o).n \/ sqrt(o,sqrtL(p,o).n,sqrtR(p,o).n)
  & sqrtR(p,o).(n+1) = sqrtR(p,o).n \/
     sqrt(o,sqrtL(p,o).n,sqrtL(p,o).n) \/
     sqrt(o,sqrtR(p,o).n,sqrtR(p,o).n)
proof
  set T=transitions_of(p,o);
A1: sqrtL(p,o).(n+1) = (T.(n+1))`1 & sqrtR(p,o).(n+1) = (T.(n+1))`2
    by Def4,Def5;
  sqrtL(p,o).n = (T.n)`1 & sqrtR(p,o).n = (T.n)`2 by Def4,Def5;
  hence thesis by Def3,A1;
end;
