 reserve A,B,O for Ordinal,
      n,m for Nat,
      a,b,o for object,
      x,y,z for Surreal,
      X,Y,Z for set,
      Inv,I1,I2 for Function;

theorem Th10:
  Inv is (L_x\/R_x)\{0_No} -surreal-valued implies
    Union divL(x,Inv) is surreal-membered &
    Union divR(x,Inv) is surreal-membered
proof
  assume
A1: Inv is (L_x\/R_x)\{0_No} -surreal-valued;
  thus Union divL(x,Inv) is surreal-membered
  proof
    let o;
    assume o in Union divL(x,Inv);
    then consider n be object such that
A2: n in dom divL(x,Inv) & o in divL(x,Inv).n by CARD_5:2;
    dom divL(x,Inv)=NAT by Def5;
    then reconsider n as Nat by A2;
    divL(x,Inv).n is surreal-membered by Th9,A1;
    hence thesis by A2;
  end;
  let o;
  assume o in Union divR(x,Inv);
  then consider n be object such that
A3: n in dom divR(x,Inv) & o in divR(x,Inv).n by CARD_5:2;
  dom divR(x,Inv)=NAT by Def6;
  then reconsider n as Nat by A3;
  divR(x,Inv).n is surreal-membered by Th9,A1;
  hence thesis by A3;
end;
