 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 Th1:
  divL(o,Inv).0 = {0_No} & divR(o,Inv).0={}
proof
  divL(o,Inv).0 = L_(transitions_of(o,Inv).0) &
    divR(o,Inv).0 = R_(transitions_of(o,Inv).0) by Def5,Def6;
  then divL(o,Inv).0 = L_1_No & divR(o,Inv).0 = R_1_No by Def4;
  hence thesis;
end;
