 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 ||.||.x.||.|| = ||.x.||
proof
  set Nx = ||.x.||, NNx=||.Nx.||;
  assume
A1:x is positive;
  o in L_NNx implies o in L_Nx
  proof
    assume
A2: o in L_NNx & not o in L_Nx;
    then reconsider o as Surreal by SURREAL0:def 16;
    o = 0_No or (o in L_Nx & o is positive) by A2,Def9;
    hence thesis by A1,A2,Def9;
  end;
  then
A3: L_NNx c= L_Nx by TARSKI:def 3;
  o in L_Nx implies o in L_NNx
  proof
    assume
A4: o in L_Nx & not o in L_NNx;
    then reconsider o as Surreal by SURREAL0:def 16;
    o = 0_No or (o in L_x & o is positive) by A1,A4,Def9;
    hence thesis by A4,Def9;
  end;
  then L_Nx c= L_NNx by TARSKI:def 3;
  then
A5:L_Nx = L_NNx by A3,XBOOLE_0:def 10;
  o in R_NNx implies o in R_Nx
  proof
    assume
A6: o in R_NNx & not o in R_Nx;
    then reconsider o as Surreal by SURREAL0:def 16;
    o in R_Nx & o is positive by A6,Def9;
    hence thesis by A6;
  end;
  then
A7: R_NNx c= R_Nx by TARSKI:def 3;
  o in R_Nx implies o in R_NNx
  proof
    assume
A8: o in R_Nx & not o in R_NNx;
    then reconsider o as Surreal by SURREAL0:def 16;
    o in R_x & o is positive by A1,A8,Def9;
    hence thesis by A8,Def9;
  end;
  then R_Nx c= R_NNx by TARSKI:def 3;
  then R_Nx = R_NNx by A7,XBOOLE_0:def 10;
  hence thesis by A5,XTUPLE_0:2;
end;
