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

theorem Th6:
  not y <= x implies x <= y
proof
  assume A1:not y <= x & not x <=y;
  then per cases by SURREAL0:43;
  suppose not L_x << {y};
    then consider xl,Y be Surreal such that
    A2: xl in L_x & Y in {y} & Y <= xl;
    L_x <<= {x} & x in {x} by Th5,TARSKI:def 1;
    then A3: xl <= x by A2;
    y <= xl by A2,TARSKI:def 1;
    hence thesis by A1,A3,Th4;
  end;
  suppose not {x} << R_y;
    then consider X,yr be Surreal such that
    A4: X in {x} & yr in R_y & yr <= X;
    {y} <<= R_y & y in {y} by Th5,TARSKI:def 1;
    then A5: y <= yr by A4;
    yr <= x by A4,TARSKI:def 1;
    hence thesis by A1,A5,Th4;
  end;
end;
