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

theorem Th10:
  L_p is surreal-membered & R_p is surreal-membered implies
    Union sqrtL(p,o) is surreal-membered & Union sqrtR(p,o) is surreal-membered
proof
  assume
A1:L_p is surreal-membered & R_p is surreal-membered;
  thus Union sqrtL(p,o) is surreal-membered
  proof
    let a be object;
    assume a in Union sqrtL(p,o);
    then consider n be object such that
A2: n in dom sqrtL(p,o) & a in sqrtL(p,o).n by CARD_5:2;
    dom sqrtL(p,o) = NAT by Def4;
    then reconsider n as Nat by A2;
    sqrtL(p,o).n is surreal-membered by A1,Th9;
    hence thesis by A2;
  end;
  let a be object;
  assume a in Union sqrtR(p,o);
  then consider n be object such that
A3: n in dom sqrtR(p,o) & a in sqrtR(p,o).n by CARD_5:2;
  dom sqrtR(p,o)=NAT by Def5;
  then reconsider n as Nat by A3;
  sqrtR(p,o).n is surreal-membered by A1,Th9;
  hence thesis by A3;
end;
