It's clear that your battery is adequate for what you want to do. Well, maybe. 12 volts times 7.5 Ah is 90 Wh, and your light requirement is 10 W for 4 hours, or 40 Wh, about half of the battery capacity. Coincidentally, you have lucked out, since for regular lead-acid batteries it is a bad idea (in the long run) to discharge more than 50%. It's not that you'll kill the battery immediately, its just that the lifetime drops way off if you do.
Although you say you don't have specs on the solar cell, it is the weak link in your system. But it is possible to get a rough idea of its suitability.
Let's say that you'll get 4 good hours of charging per day. The usual measure is 6 hours for a fixed orientation, but we'll be conservative (unless you live in a very sunny climate with no clouds). Then you'll need to pump 10 watts into the battery to compensate for the night-time loads. Peak sunlight at the earth's surface is roughly 750 watts/square meter. Let's assume that your solar cell/charger/battery charging efficiency is 10%. This may actually be a bit liberal, since the ability of a lead-acid battery to absorb charge decreases as it gets full. In addition, while a good solar cell might have an intrinsic 20% efficiency, there is no guarantee about yours. So let's go with 10% overall.
This means that the power falling on the solar cell must be 100 watts. With an incident power of 750 w/m^2, the solar cell must have an area of 100/750, or .13 square meters. This is an area .36 meters on a side, or about 15 inches square.
Is this the size of your solar cell? If it's bigger, you stand a chance. If not you are almost certainly out of luck.
And one more caution. I wrote of the need to avoid over-discharging your battery. If your solar cell can only provide exactly as much charge as the light uses at night, any cloudy day will cause bad results the next night. After all, assuming a full charge the day before produces a battery which is about 50% discharged each morning. If bad weather only produces half the usual charge, the next night-time discharge will leave the battery at 25% charge, and if you get two days bad weather in a row the battery will be at 0 charge. So a bigger solar cell/bigger battery is never a bad idea.