 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
  x is positive & y = [{0_No}\/divset(R_x,x,L_(x"))\/divset(L_x,x,R_(x")),
                       divset(L_x,x,L_(x"))\/divset(R_x,x,R_(x"))]
  implies x" == y
proof
  set Nx = ||.x.||;
  assume
A1: x is positive;
  then
A2: x==Nx by Th18;
  then divset(R_x,x,L_(x")) <==> divset(R_x,Nx,L_(x")) by Th35;
  then
A3:{0_No}\/divset(R_x,x,L_(x")) <==>{0_No}\/divset(R_x,Nx,L_(x"))
  by SURREALO:31;
  divset(L_x,x,R_(x")) <==> divset(L_x,Nx,R_(x")) by A2,Th35;
  then
A4:{0_No}\/divset(R_x,x,L_(x"))\/divset(L_x,x,R_(x"))
  <==> {0_No}\/divset(R_x,Nx,L_(x"))\/divset(L_x,Nx,R_(x"))
  by A3,SURREALO:31;
  divset(L_x,x,L_(x")) <==> divset(L_x,Nx,L_(x")) &
  divset(R_x,x,R_(x")) <==> divset(R_x,Nx,R_(x")) by A2,Th35;
  then
A5:divset(L_x,x,L_(x"))\/divset(R_x,x,R_(x"))
  <==> divset(L_x,Nx,L_(x"))\/divset(R_x,Nx,R_(x"))
  by SURREALO:31;
  assume
A6: y = [{0_No}\/divset(R_x,x,L_(x"))\/divset(L_x,x,R_(x")),
  divset(L_x,x,L_(x"))\/divset(R_x,x,R_(x"))];
  x"=[{0_No}\/divset(R_x,Nx,L_(x"))\/divset(L_x,Nx,R_(x")),
  divset(L_x,Nx,L_(x"))\/divset(R_x,Nx,R_(x"))] by A1,Th36;
  hence thesis by SURREALO:29,A6,A4,A5;
end;
