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

theorem Th15:
  L_y << {x} << R_y & z = [L_x \/ L_y, R_x \/ R_y]
    implies x == z
proof
  assume that
  A1: L_y << {x} << R_y and
  A2: z = [L_x \/ L_y, R_x \/ R_y];
  A3: L_x << {x} << R_x by Th11;
  A4: L_z << {z} << R_z by Th11;
  A5:L_z << {x}
  proof
    let a,b be Surreal;
    assume A6:a in L_z & b in {x};
    then a in L_x or a in L_y by A2,XBOOLE_0:def 3;
    hence a < b by A3,A6,A1;
  end;
  A7: {z} << R_x
  proof
    let a,b be Surreal;
    assume A8: a in {z} & b in R_x;
    then b in R_z by A2,XBOOLE_0:def 3;
    hence a < b by A8,A4;
  end;
  A9: L_x << {z}
  proof
    let a,b be Surreal;
    assume A10:a in L_x & b in {z};
    then a in L_z by A2,XBOOLE_0:def 3;
    hence a < b by A10,A4;
  end;
  {x} << R_z
  proof
    let a,b be Surreal;
    assume A11:a in {x} & b in R_z;
    then b in R_x or b in R_y by A2,XBOOLE_0:def 3;
    hence a < b by A3,A11,A1;
  end;
  hence thesis by A7,A5,SURREAL0:43,A9;
end;
