名詞解釋
Theorem：就是定理，比較重要的，簡稱是 Thm。
Lemma：小小的定理，通常是為了證明后面的定理，如果證明的篇幅很長時，可能會把證明拆成幾個部分來論述，雖然篇幅可能變多，但派絡卻很清楚。
Corollary：推論。由定理立即可推知的結果。
Property：性質，結果雖然值得一記，卻沒定理來的深刻。
Proposition：有人翻譯為命題， 有些作者喜歡用，大概也可以算是比較簡單的定理的一種稱呼。
Claim：證明時先論述一個結果，再作證明。看的人比較輕松。
Note：通常只是一個注解。
Remark：涉及一些結論，比較起來 "Note" 比較像說明， "remark" 則常是非正式的定理。
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首先、定義和公理是任何理論的基礎，定義解決了概念的范疇，公理使得理論能夠被人的理性所接受。
其次、定理和命題就是在定義和公理的基礎上通過理性的加工使得理論的再延伸，我認為它們的區別主要在於，定理的理論高度比命題高些，定理主要是描述各定義（范疇）間的邏輯關系，命題一般描述的是某種對應關系（非范疇性的）。而推論就是某一定理的附屬品，是該定理的簡單應用。
最后、引理就是在證明某一定理時所必須用到的其它定理。而在一般情況下，就像前面所提到的定理的證明是依賴於定義和公理的。
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1.引理和定理應該是根據文章目的不同而區分的，同樣的論點在這篇文章可以是引理，在那篇文章可以是定理。
2.如果為了說明一個問題進行論證，但是在論證前需要證明若干個小問題，那么這些若干個小問題的結論就是引理，而這個問題的論證將會需要引用到前面的引理，該問題的結論就是定理。
3.引理是為定理作准備的。文章中的定理才是需要說明的主要問題或者目的。
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就如doppler 說的，
"Theorem" 本身是一個大 result
"Lemma" 是 prove “Theorem“ 之前用的一個 result
"Corollary" 是可以從 "Theorem" 里直接 deduce/prove 出來的 result
" Proposition" 是一個還無法大到變成 "Theorem" 的一個 result (當作小 theorem )
(1) Definition（定義）------a precise and unambiguous description of the meaning of a mathematical term. It characterizes the meaning of a word by giving all the properties and only those properties that must be true.
(2) Theorem（定理）----a mathematical statement that is proved using rigorous mathemat-ical reasoning. In a mathematical paper, the term theorem is often reserved for the most important results.
(3) Lemma（引理）----a minor result whose sole purpose is to help in proving a theorem. It is a stepping stone on the path to proving a theorem. Very occasionally lemmas can take on a life of their own (Zorn's lemma, Urysohn's lemma, Burnside's lemma,Sperner's lemma).
(4) Corollary（推論）-----a result in which the (usually short) proof relies heavily on a given theorem (we often say that \this is a corollary of Theorem A").
(5) Proposition（命題）-----a proved and often interesting result, but generally less important than a theorem.
(6) Conjecture（推測，猜想）----a statement that is unproved, but is believed to be true (Collatz conjecture, Goldbach conjecture, twin prime conjecture).
(7) Claim（斷言）-----an assertion that is then proved. It is often used like an informal lemma.
(8) Axiom/Postulate------（公理/假定）a statement that is assumed to be true without proof. These are the basic building blocks from which all theorems are proved (Eu-clid's ve postulates, Zermelo-Frankel axioms, Peano axioms).
(9) Identity（恆等式）-----a mathematical expression giving the equality of two (often variable) quantities (trigonometric identities, Euler's identity).
(10) Paradox（悖論）----a statement that can be shown, using a given set of axioms and de nitions, to be both true and false. Paradoxes are often used to show the inconsistencies in a awed theory (Russell's paradox). The term paradox is often used informally to describe a surprising or counterintuitive result that follows from a given set of rules (Banach-Tarski paradox, Alabama paradox, Gabriel's horn).
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