reserve A,B,C,O for Ordinal,
        X for set,
        o for object,
        x,y,z,t,r,l for Surreal;

theorem Th13:
  x < y implies
     (ex xR be Surreal st xR in R_x & x < xR <= y)
  or
     (ex yL be Surreal st yL in L_y & x <= yL < y)
proof
  assume x < y;
  then per cases by SURREAL0:43;
  suppose not L_y << {x};
    then consider l,r such that
    A1: l in L_y & r in {x} & l >= r;
    L_y << {y} & y in {y} by Th11,TARSKI:def 1;
    then x <= l < y by A1,TARSKI:def 1;
    hence thesis by A1;
  end;
  suppose not {y} << R_x;
    then consider l,r such that
    A2: l in {y} & r in R_x & l >= r;
    {x} << R_x & x in {x} by Th11,TARSKI:def 1;
    then x < r <= y by A2,TARSKI:def 1;
    hence thesis by A2;
  end;
end;
