 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 Th36:
  x is positive
     implies
    x" = [{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"))]
proof
  set Nx = ||.x.||,Inv = No_inverses_on Nx;
  assume
A1: x is positive;
  then not x==0_No;
  then
A2:x"=inv x by A1,Def14;
  set UL=Union divL(Nx,Inv),UR=Union divR(Nx,Inv);
A3: inv x = [ UL,UR] by A1,Th26;
  then UL c= L_(x") \/ R_(x") by A2,XBOOLE_1:7;
  then
A4: divset(UL,Nx,R_Nx,Inv) = divset(R_x,||.x.||,UL) &
  divset(UL,||.x.||,L_Nx,No_inverses_on ||.x.||)=
          divset(L_x,||.x.||,UL) by A1,A2,A3,Th34;
  UR c= L_(x") \/ R_(x") by A2,A3,XBOOLE_1:7;
  then
A5:divset(UR,Nx,L_Nx,Inv)= divset(L_x,||.x.||,UR) &
  divset(UR,Nx,R_Nx,Inv)= divset(R_x,||.x.||,UR) by A1,A2,A3,Th34;
  UL = {0_No}\/divset(UL,Nx,R_Nx,Inv)\/ divset(UR,Nx,L_Nx,Inv) by Th12;
  hence thesis by A4,A5,A3,A2,Th13;
end;
