The following is quoted from
Edmund Landau, *Foundations of Analysis*, Chelsea, 1951,
pp. 1-18.

**Theorem 1:**- If then .
**Theorem 2:**- .
**Theorem 3:**- If there exists one (and hence,
by Axiom 4, exactly one)
*u*such that*x*=*u*'. **Theorem 4,**- and at the same time
**Definition 1:**To every pair of numbers*x*,*y*, we may assign in exactly one way a natural number, called*x*+*y*(+ to be read ``plus''), such that**1.**-
*x*+1=*x*' for every*x*. **2.**-
*x*+*y*'=(*x*+*y*)' for every*x*and every*y*.

*x*+*y*is called the sum of*x*and*y*, or the number obtained by addition of*y*to*x*. **Theorem 5 (Associative Law of Addition):**- (
*x*+*y*)+*z*=*x*+(*y*+*z*). **Theorem 6 (Commutative Law of Addition):**-
*x*+*y*=*y*+*x*. **Theorem 7:**- .
**Theorem 8:**- If then .
**Theorem 9:**- For given
*x*and*y*, exactly one of the following must be the case:**1.**-
*x*=*y*. **2.**- There exists a
*u*(exactly one, by Theorem 8) such that*x*=*y*+*u*. **3.**- There exists a
*v*(exactly one, by Theorem 8) such that*y*=*x*+*v*.

**Definition 2:**- If
*x*=*y*+*u*then*x*>*y*. (> to be read ``is greater than.'') **Definition 3:**- If
*y*=*x*+*v*then*x*<*y*. (< to be read ``is less than.'') **Theorem 10:**- For any given
*x*,*y*, we have exactly one of the cases*x*=*y*,*x*>*y*,*x*<*y*. **Theorem 11:**- If
*x*>*y*then*y*<*x*. **Theorem 12:**- If
*x*<*y*then*y*>*x*. **Definition 4:**- means
*x*>*y*or*x*=*y*. ( to be read ``is greater than or equal to.'') **Definition 5:**- means
*x*<*y*or*x*=*y*. ( to be read ``is less than or equal to.'') **Theorem 13:**- If then .
**Theorem 14:**- If then .
**Theorem 15 (Transitivity of Ordering:)**- If
*x*<*y*,*y*<*z*, then*x*<*z*. **Theorem 16:**- If ,
*y*<*z*or*x*<*y*, , then*x*<*z*. **Theorem 17:**- If , , then .
**Theorem 18:**-
*x*+*y*>*x*. **Theorem 19:**- If
*x*>*y*, or*x*=*y*, or*x*<*y*, then*x*+*z*>*y*+*z*, or*x*+*z*=*y*+*z*, or*x*+*z*<*y*+*z*, respectively. **Theorem 20:**- If
*x*+*z*>*y*+*z*, or*x*+*z*=*y*+*z*, or*x*+*z*<*y*+*z*, then*x*>*y*, or*x*=*y*, or*x*<*y*, respectively. **Theorem 21:**- If
*x*>*y*,*z*>*u*, then*x*+*z*>*y*+*u*. **Theorem 22:**- If ,
*z*>*u*or*x*>*y*, , then*x*+*z*>*y*+*u*. **Theorem 23:**- If , , then .
**Theorem 24:**- .
**Theorem 25:**- If
*y*>*x*then . **Theorem 26:**- If
*y*<*x*+1 then . **Theorem 27:**- In every non-empty set of natural numbers
there is a least one (i.e., one which is less than any other number of
the set).
**Theorem 28**- and at the same time
**Definition 6:**To every pair of numbers*x*,*y*, we may assign in exactly one way a natural number, called ( to be read ``times''; however, the dot is usually omitted), such that**1.**- for every
*x*, **2.**- for every
*x*and every*y*.

*x*and*y*, or the number obtained from multiplication of*x*by*y*. **Theorem 29 (Commutative Law of Multiplication):**-
*xy*=*yx*. **Theorem 30 (Distributive Law):**-
*x*(*y*+*z*)=*xy*+*xz*. **Theorem 31 (Associative Law of Multiplication):**-
(
*xy*)*z*=*x*(*yz*). **Theorem 32:**- If
*x*>*y*, or*x*=*y*, or*x*<*y*, then*xz*>*yz*,*xz*=*yz*, or*xz*<*yz*, respectively. **Theorem 33:**- If
*xz*>*yz*, or*xz*=*yz*, or*xz*<*yz*, then*x*>*y*, or*x*=*y*, or*x*<*y*, respectively. **Theorem 34:**- If
*x*>*y*,*z*>*u*, then*xz*>*yu*. **Theorem 35:**- If ,
*z*>*u*or*x*>*y*, , then*xz*>*yu*. **Theorem 36:**- If , , then .

Wed Sep 16 09:26:16 EDT 1998