 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 Th17:
  for x be object st (L_x \/ R_x)\{0_No} c= Z & I1|Z = I2|Z holds
    divL(x,I1) = divL(x,I2) & divR(x,I1) = divR(x,I2)
proof
  let x be object such that
A1: (L_x \/ R_x)\{0_No} c= Z & I1|Z = I2|Z;
A2: transitions_of(x,I1) = transitions_of(x,I2) by A1,Th16;
A3: dom divL(x,I1) = NAT = dom divL(x,I2) by Def5;
  o in dom divL(x,I1) implies divL(x,I1).o=divL(x,I2).o
  proof
    assume o in dom divL(x,I1);
    then reconsider n=o as Nat by A3;
    thus divL(x,I1).o = (transitions_of(x,I1).n)`1 by Def5
    .= divL(x,I2).o by A2,Def5;
  end;
  hence divL(x,I1) = divL(x,I2) by A3,FUNCT_1:2;
A4: dom divR(x,I1) = NAT = dom divR(x,I2) by Def6;
  o in dom divR(x,I1) implies divR(x,I1).o=divR(x,I2).o
  proof
    assume o in dom divR(x,I1);
    then reconsider n=o as Nat by A4;
    thus divR(x,I1).o = (transitions_of(x,I1).n)`2 by Def6
    .= divR(x,I2).o by Def6,A2;
  end;
  hence thesis by A4,FUNCT_1:2;
end;
