The correct answer is option 3: 4 degree Celsius. When the upper surface of a lake freezes into ice, the water beneath the ice layer remains in a liquid state. This is because water has a unique property called density anomaly. Water reaches its maximum density at around 4 degrees Celsius. As the temperature decreases, water expands and becomes less dense. Hence, the water at the bottom of the lake, which is not affected by external factors such as wind or sunlight, remains at a relatively constant temperature of 4 degrees Celsius.

Option 1: 10 degrees Celsius is not correct because the temperature at the bottom of the lake would not be as high as 10 degrees Celsius.

Option 2: 0 degrees Celsius is incorrect because the surface ice layer indicates that the water has already reached its freezing point. Therefore, the temperature at the bottom of the lake would not be 0 degrees Celsius.

Option 4: 4 degrees Celsius is the correct answer, as explained above.

In summary, the temperature of the water at the bottom of a lake with a frozen surface would be around 4 degrees Celsius.