 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 Th6:
  divL(o,Inv).(n+1) = divL(o,Inv).n \/
    divset(divL(o,Inv).n,o,R_o,Inv) \/ divset(divR(o,Inv).n,o,L_o,Inv)
  &
  divR(o,Inv).(n+1) = divR(o,Inv).n \/
    divset(divL(o,Inv).n,o,L_o,Inv) \/ divset(divR(o,Inv).n,o,R_o,Inv)
proof
  set T=transitions_of(o,Inv);
A1: divL(o,Inv).(n+1) = (T.(n+1))`1 & divR(o,Inv).(n+1) = (T.(n+1))`2
  by Def5,Def6;
  divL(o,Inv).n = (T.n)`1 & divR(o,Inv).n = (T.n)`2 by Def5,Def6;
  hence thesis by Def4,A1;
end;
