Cardinalis means "that upon which a door hinges, pivotal" in Latin, from cardo, "hinge, pivot". From this it acquired its secondary meaning, "important, principal", which it still has in English (e.g. a cardinal sin). Its third meaning is derived from this: a cardinal number is a "principal" number, i.e. one that simply says how many objects there are.
That'll be one pound.
Two billion people might die.
Ordinalis means "in order of succession, of an order". It comes from ordo, "order, rank". An ordinal number is an adjective that denotes what place an object has in a certain order. The names of the ordinal numbers are usually derived from the cardinal numbers by adding -th.
That is my second victory.
This is the tenth time she's dumped me.
While a cardinal number refers to several objects ("three apples"), an ordinal number refers to only one of those ("the third apple"). An ordinal number is hence dependent on the notion of a cardinal number: there can't be a third apple unless there are at least three apples. By contrast, there can be three apples without one being the third, if they are just not arranged in any particular order.
The Romans used these terms the same way. They also had distributive numbers, which indicated "every third apple", or apples "in triads", "three each".
A theorem is a result that can be proven to be true from a set of axioms. The term is used especially in mathematics where the axioms are those of mathematical logic and the systems in question.
A theory is a set of ideas used to explain why something is true, or a set of rules on which a subject is based on.
Best Answer
"At the example of Foobar" strikes me as ungrammatical, except in constructions like "we look at the example of ...", where that "at" belongs to the phrasal verb "look at". I'd use
Although you should note that the "by" belongs to the phrasal verb "illustrated by". Similarly, in the sentence "this can be found in the example of Foobar", the "in" goes with the verb "found".
If you wanted to put the example at the start of a sentence, usually "In the example of Foobar" would be the right preposition, but whether it is or not depends on the rest of the sentence.