 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 implies
     [{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"))] is Surreal
proof
  set Nx = ||.x.||;
  assume
A1: x is positive;
  then divset(R_x,x,L_(x")) <=_ divset(R_x,Nx,L_(x")) by Th18,Th35;
  then
A2: {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 A1,Th18,Th35;
  then
A3: {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 A2,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 A1,Th18,Th35;
  then
A4: 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;
  [{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"))] is Surreal
  by A1,Th36;
  hence thesis by TARSKI:1,Th37,A3,A4;
end;
