If you know in advance that you have to eat in a restaurant, you should try to choose the one that offers the food you can eat. Many restaurants now have websites where they list their menus. Check out the restaurant that offers more than the food you can eat.
You may even be able to call the restaurant and pre-order a special vegetarian or vegan meal just for you. If you do not know in advance where you are going to eat, or if you do not have a choice location, then you should check out the menu carefully before ordering. You can check out vegetarian restaurants online.
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Some restaurants will offer a section for vegetarian dishes. Be on guard though, because many of the sauces on say a pasta dish may not be vegan-friendly; it may include milk, butter, eggs or other animal products.
Another place to be careful when ordering in a restaurant (especially if you're vegan) will be in the dessert section. Remember that most of the cakes made with eggs and milk, and even cake can be made with lard instead of vegetable oil. Stick to fruit cups or maybe even ice pieces (like a sherbet) if you are unsure. Then again, you can also ask your server to confirm with the chef if the pie crust has animal fat in it.