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

theorem Th9:
  L_p is surreal-membered & R_p is surreal-membered implies
    sqrtL(p,o).n is surreal-membered & sqrtR(p,o).n is surreal-membered
proof
  assume
A1: L_p is surreal-membered & R_p is surreal-membered;
  defpred P[Nat] means
  sqrtL(p,o).$1 is surreal-membered & sqrtR(p,o).$1 is surreal-membered;
A2:P[0] by A1,Th6;
A3:for n st P[n] holds P[n+1]
  proof
    let n such that
A4: P[n];
    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)
    by Th8;
    hence thesis by A4;
  end;
  for n holds P[n] from NAT_1:sch 2(A2,A3);
  hence thesis;
end;
