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

theorem Th16:
  for x,y st L_y << {x} << R_y &
    for z st L_y << {z} << R_y holds born x c= born z
  holds x == y
proof
  let x,y be Surreal such that
  A1: L_y << {x} << R_y  and
  A2:for x1 be Surreal st L_y << {x1} << R_y holds born x c= born x1;
  set X=L_x \/ L_y, Y = R_x \/ R_y;
  A3:x in {x} by TARSKI:def 1;
  reconsider z=[X,Y] as Surreal by Th14,A1;
  A4: L_x << {x} << R_x by Th11;
  A5: L_y << {y} << R_y by Th11;
  A6: L_z << {z} << R_z by Th11;
  A7: L_x << {z}
  proof
    let a,b be Surreal;
    assume A8:a in L_x & b in {z};
    then a in L_z by XBOOLE_0:def 3;
    hence a < b by A8,A6;
  end;
  {x} << R_z
  proof
    let a,b be Surreal;
    assume A9:a in {x} & b in R_z;
    then b in R_x or b in R_y by XBOOLE_0:def 3;
    hence a < b by A4,A9,A1;
  end;
  then A10:x <= z by A7,SURREAL0:43;
  A11: L_y << {z}
  proof
    let a,b be Surreal;
    assume A12:a in L_y & b in {z};
    then a < z by A1,A3,A10,Th4;
    hence thesis by A12,TARSKI:def 1;
  end;
  A13: x==z by Th15,A1;
  A14: {y} << R_z
  proof
    let a,b be Surreal;
    assume A15:a in {y} & b in R_z;
    then per cases by XBOOLE_0: def 3;
    suppose b in R_y;
      hence thesis by A5,A15;
    end;
    suppose A16: b in R_x;
      assume not a < b;
      then b <= y by A15,TARSKI:def 1;
      then
      A17:{b} << R_y by SURREAL0:43;
      L_y << {b}
      proof
        let c,d be Surreal;
        assume A18:c in L_y & d in {b};
        then c <= x by A3,A1;
        then c < b by A4,A3,A16,Th4;
        hence c < d by A18,TARSKI:def 1;
      end;
      then A19:born x c= born b by A17,A2;
      A20: b in L_x \/ R_x by A16,XBOOLE_0:def 3;
      then
      A21: born b in born x by Th1;
      born b c= born x by A20,Th1,ORDINAL1:def 2;
      then born b = born x by A19,XBOOLE_0:def 10;
      hence thesis by A21;
    end;
  end;
  A22: {z} << R_y
  proof
    let a,b be Surreal;
    assume A23:a in {z} & b in R_y;
    then z < b by A1,A3,A13,Th4;
    hence thesis by A23,TARSKI:def 1;
  end;
  L_z << {y}
  proof
    let b,a be Surreal;
    assume A24:b in L_z & a in {y};
    then per cases by XBOOLE_0: def 3;
    suppose b in L_y;
      hence thesis by A5,A24;
    end;
    suppose A25: b in L_x;
      assume not b < a;
      then y <= b by A24,TARSKI:def 1;
      then A26: L_y << {b} by SURREAL0:43;
      {b} << R_y
      proof
        let d,c be Surreal;
        assume A27:d in {b} & c in R_y;
        then x <= c by A3,A1;
        then b < c by A4,A3,A25,Th4;
        hence d < c by A27,TARSKI:def 1;
      end;
      then A28:born x c= born b by A26,A2;
      A29: b in L_x \/ R_x by A25,XBOOLE_0:def 3;
      then
      A30: born b in born x by Th1;
      born b c= born x by A29,Th1,ORDINAL1:def 2;
      then born b = born x by A28,XBOOLE_0:def 10;
      hence thesis by A30;
    end;
  end;
  then z == y by A22,SURREAL0:43,A14,A11;
  hence thesis by A13,Th4;
end;
