1.1 Definition. Let $S$ be a set. An order on $S$ is a relation, denoted by $<$ with the following two properties.
(1) For all $x,y\in S$, one and only one of the statements
is true.
(2) $\forall x,y,z\in S (x<y\wedge y<z \Rightarrow x<z)$
1.2 Definition. Suppose $S$ is an ordered set, and $E\subset S$. If there exists a $\beta \in S$ such that $\forall x\in E, x\leq \beta$, we say that $E$ is bounded above, and call $\beta$ an upper bound of $E$.
1.3 Definition. Suppose $S$ is an ordered set, and $E\subset S$, and $E$ is bounded above. Suppose there exists an $\alpha\in S$ with the following properties:
(1) $\alpha$ is an upper bound of $E$.
(2) If $\gamma <\alpha$ then $\gamma$ is not an upper bound of $E$.
Then $\alpha$ is called the supremum of $E$, an we write
Also, the statement
\[\alpha =\inf E\]means that $\alpha$ is a infimum of $E$.
1.4 Definition. An ordered set $S$ is said to have the least-upper-bound property(LUB) if the following is true:
\[\forall E\subset S(E\neq \emptyset\wedge E \text{ is bounded above} \Rightarrow \exists u \in S(u=\sup E))\]1.5 Theorem. Suppose $S$ is an ordered set with the LUB, $B\subset S, B\neq \emptyset, B$ is bounded below. Let $L$ be the set of all lower bounds of $B$. Then,
\[\exists u\in S(u=\sup L),\ \inf B=\sup L.\]1.6 Definition. A field is a set $F$ with two operations, which satisfy the followings:
(A) Addition
(A1) $\forall x,y\in F, x+y\in F$
(A2) $\forall x,y\in F, \,x+y=y+x$
(A3) $\forall x,y,z\in F, \,(x+y)+z=x+(y+z)$
(A4) $\exists y\in F\,(\forall x\in F,\,x+y=x)$ and we write $y=0$
(A5) $\forall x\in F, \exists y\in F\,(x+y=0)$ and we write $y=-x$
(M) Multiplication
(M1) $\forall x,y\in F,\,x\cdot y\in F$
(M2) $\forall x,y\in F,\,x\cdot y=y\cdot x$
(M3) $\forall x,y,z\in F,\,(x\cdot y)\cdot z=x\cdot (y\cdot z)$
(M4) $\exists y\in F\,((\forall x\in F,\,x\cdot y=x)\wedge(y\neq 0))$ and we write $y=1$
(M5) $\forall x\in F\,(x\neq 0),\,\exists y\in F\,(x\cdot y=1)$ and we write $y=1/x$
(D) Distribution