 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 Th9:
   Inv is (L_x\/R_x)\{0_No} -surreal-valued implies
     divL(x,Inv).n is  surreal-membered &
     divR(x,Inv).n is  surreal-membered
proof
  assume
A1:Inv is (L_x\/R_x)\{0_No} -surreal-valued;
  defpred P[Nat] means
  divL(x,Inv).$1 is  surreal-membered &
  divR(x,Inv).$1 is  surreal-membered;
  {} is surreal-membered;
  then
A2: P[0] by Th1;
A3: P[m] implies P[m+1]
  proof
    assume
A4: P[m];
    L_x\{0_No} c= (L_x\/R_x)\{0_No} & R_x\{0_No} c= (L_x\/R_x)\{0_No}
    by XBOOLE_1:7,33;
    then
A5: Inv is L_x\{0_No} -surreal-valued & Inv is R_x\{0_No} -surreal-valued
    by A1;
    divset(divL(x,Inv).m,x,L_x\{0_No},Inv) is surreal-membered &
    divset(divL(x,Inv).m,x,R_x\{0_No},Inv) is surreal-membered &
    divset(divR(x,Inv).m,x,L_x\{0_No},Inv) is surreal-membered &
    divset(divR(x,Inv).m,x,R_x\{0_No},Inv) is surreal-membered
    by A5,Th5,A4;
    then
A6: divset(divL(x,Inv).m,x,L_x,Inv) is surreal-membered &
    divset(divL(x,Inv).m,x,R_x,Inv) is surreal-membered &
    divset(divR(x,Inv).m,x,L_x,Inv) is surreal-membered &
    divset(divR(x,Inv).m,x,R_x,Inv) is surreal-membered by Th8;
    divL(x,Inv).(m+1) = divL(x,Inv).m \/
       divset(divL(x,Inv).m,x,R_x,Inv) \/ divset(divR(x,Inv).m,x,L_x,Inv)
     & divR(x,Inv).(m+1) = divR(x,Inv).m \/
       divset(divL(x,Inv).m,x,L_x,Inv) \/ divset(divR(x,Inv).m,x,R_x,Inv)
        by Th6;
    hence thesis by A4,A6;
  end;
  P[m] from NAT_1:sch 2(A2,A3);
  hence thesis;
end;
