 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 Th40:
  not x == 0_No implies (0_No < x iff 0_No < x")
proof
  assume not x == 0_No;
  then
A1:x * (x") == 1_No by Th33;
A2:0_No < x * (x") by A1,SURREALO:4,Def8;
  thus 0_No < x implies 0_No < x"
  proof
    assume 0_No < x;
    then 0_No <= x;
    hence thesis by A2,SURREALR:72;
  end;
  assume 0_No < x";
  then 0_No <= x";
  hence thesis by A2,SURREALR:72;
end;
