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

theorem
  card (L_x) (+) card (R_x) = 1 iff ex y be Surreal st
    x= [{},{y}] or x = [{y},{}]
proof
  thus card (L_x) (+) card (R_x) = 1 implies ex y be Surreal st
  x= [{},{y}] or x = [{y},{}]
  proof
    assume A1:card (L_x) (+) card (R_x) = 1;
    then card (L_x) c= 1 & card (R_x) c= 1 by ORDINAL7:86;
    then reconsider c1=card (L_x),c2=card (R_x) as Nat;
    A2:c1+c2 = 1 by A1,ORDINAL7:76;
    then per cases by NAT_1:11,25;
    suppose A3: c1=0;
      then consider y be object such that
      A4:{y} = R_x by A2,CARD_2:42;
      y in {y} by TARSKI:def 1;
      then reconsider y as Surreal by A4,SURREAL0:def 16;
      take y;
      L_x = {} by A3;
      hence thesis by A4;
    end;
    suppose A5:c1=1;
      then consider y be object such that
      A6:{y} = L_x by CARD_2:42;
      y in {y} by TARSKI:def 1;
      then reconsider y as Surreal by A6,SURREAL0:def 16;
      take y;
      R_x = {} by A5,A2;
      hence thesis by A6;
    end;
  end;
  given y be Surreal such that
  A7: x= [{},{y}] or x = [{y},{}];
  (card (L_x) =0 & card (R_x) =1) or (card (R_x) =0 & card (L_x) =1)
  by A7,CARD_1:30;
  then card (L_x) (+) card (R_x) = 1+0 by ORDINAL7:76;
  hence thesis;
end;
