The real number system, $\displaystyle \mathbb{R}\times \mathbb{R}$ contains order axioms that show relations and properties of the system that add completeness to the ordered field algebraic laws.

The properties are as follows.

Trichotomy Property:

Given $\displaystyle a$, $\displaystyle b\ \epsilon\ \mathbb{R}$ only one of the following statements holds true $\displaystyle a< b$, $\displaystyle b< a$, or $\displaystyle a=b$.

Transitive Property:

For $\displaystyle a$, $\displaystyle b$, and $\displaystyle c\ \epsilon\ \mathbb{R}$ where $\displaystyle a< b$ and $\displaystyle b< c$ then this implies $\displaystyle a< c$.

Additive Property:

For $\displaystyle a$, $\displaystyle b$, and $\displaystyle c\ \epsilon\ \mathbb{R}$ where $\displaystyle a< b$ and $\displaystyle c\ \epsilon\ \mathbb{R}$ then this implies $\displaystyle a+b< b+c$.

Multiplicative Properties:

For $\displaystyle a$, $\displaystyle b$, and $\displaystyle c\ \epsilon\ \mathbb{R}$ where $\displaystyle a< b$ and $\displaystyle c>0$ then this implies $\displaystyle ac< bc$ and  $\displaystyle a< b$ and $\displaystyle c< 0$ then this implies $\displaystyle bc< ac$.

Therefore looking at the options the Trichotomy Property identifies the property in this particular question.

The real number system, $\displaystyle \mathbb{R}\times \mathbb{R}$ contains order axioms that show relations and properties of the system that add completeness to the ordered field algebraic laws.

The properties are as follows.

Trichotomy Property:

Given $\displaystyle a$, $\displaystyle b\ \epsilon\ \mathbb{R}$ only one of the following statements holds true $\displaystyle a< b$, $\displaystyle b< a$, or $\displaystyle a=b$.

Transitive Property:

For $\displaystyle a$, $\displaystyle b$, and $\displaystyle c\ \epsilon\ \mathbb{R}$ where $\displaystyle a< b$ and $\displaystyle b< c$ then this implies $\displaystyle a< c$.

Additive Property:

For $\displaystyle a$, $\displaystyle b$, and $\displaystyle c\ \epsilon\ \mathbb{R}$ where $\displaystyle a< b$ and $\displaystyle c\ \epsilon\ \mathbb{R}$ then this implies $\displaystyle a+b< b+c$.

Multiplicative Properties:

For $\displaystyle a$, $\displaystyle b$, and $\displaystyle c\ \epsilon\ \mathbb{R}$ where $\displaystyle a< b$ and $\displaystyle c>0$ then this implies $\displaystyle ac< bc$ and  $\displaystyle a< b$ and $\displaystyle c< 0$ then this implies $\displaystyle bc< ac$.

Therefore looking at the options the Transitive Property identifies the property in this particular question.

The real number system, $\displaystyle \mathbb{R}\times \mathbb{R}$ contains order axioms that show relations and properties of the system that add completeness to the ordered field algebraic laws.

The properties are as follows.

Trichotomy Property:

Given $\displaystyle a$, $\displaystyle b\ \epsilon\ \mathbb{R}$ only one of the following statements holds true $\displaystyle a< b$, $\displaystyle b< a$, or $\displaystyle a=b$.

Transitive Property:

For $\displaystyle a$, $\displaystyle b$, and $\displaystyle c\ \epsilon\ \mathbb{R}$ where $\displaystyle a< b$ and $\displaystyle b< c$ then this implies $\displaystyle a< c$.

Additive Property:

For $\displaystyle a$, $\displaystyle b$, and $\displaystyle c\ \epsilon\ \mathbb{R}$ where $\displaystyle a< b$ and $\displaystyle c\ \epsilon\ \mathbb{R}$ then this implies $\displaystyle a+b< b+c$.

Multiplicative Properties:

For $\displaystyle a$, $\displaystyle b$, and $\displaystyle c\ \epsilon\ \mathbb{R}$ where $\displaystyle a< b$ and $\displaystyle c>0$ then this implies $\displaystyle ac< bc$ and  $\displaystyle a< b$ and $\displaystyle c< 0$ then this implies $\displaystyle bc< ac$.

Therefore looking at the options the Additive Property identifies the property in this particular question.

The real number system, $\displaystyle \mathbb{R}\times \mathbb{R}$ contains order axioms that show relations and properties of the system that add completeness to the ordered field algebraic laws.

The properties are as follows.

Trichotomy Property:

Given $\displaystyle a$, $\displaystyle b\ \epsilon\ \mathbb{R}$ only one of the following statements holds true $\displaystyle a< b$, $\displaystyle b< a$, or $\displaystyle a=b$.

Transitive Property:

For $\displaystyle a$, $\displaystyle b$, and $\displaystyle c\ \epsilon\ \mathbb{R}$ where $\displaystyle a< b$ and $\displaystyle b< c$ then this implies $\displaystyle a< c$.

Additive Property:

For $\displaystyle a$, $\displaystyle b$, and $\displaystyle c\ \epsilon\ \mathbb{R}$ where $\displaystyle a< b$ and $\displaystyle c\ \epsilon\ \mathbb{R}$ then this implies $\displaystyle a+b< b+c$.

Multiplicative Properties:

For $\displaystyle a$, $\displaystyle b$, and $\displaystyle c\ \epsilon\ \mathbb{R}$ where $\displaystyle a< b$ and $\displaystyle c>0$ then this implies $\displaystyle ac< bc$ and  $\displaystyle a< b$ and $\displaystyle c< 0$ then this implies $\displaystyle bc< ac$.

Therefore looking at the options the Multiplicative Properties identifies the property in this particular question.

According to the Well-Ordered Principal this statement is true. The following proof illuminate its truth.

Suppose $\displaystyle A\subseteq \mathbb{N}$ is nonempty. From there, it is known that $\displaystyle -A$ is bounded above, by $\displaystyle -1$.

Therefore, by the Completeness Axiom the supremum of $\displaystyle -A$ exists.

Furthermore, if $\displaystyle A\subset \mathbb{Z}$ has a supremum, then $\displaystyle \text{sup}A\ \epsilon\ A$, thus in this particular case $\displaystyle \text{sup}(-A)\ \epsilon\ -A$.

Thus by the Reflection Principal,

$\displaystyle \text{inf}A=-\text{sup}(-A)$ 

exists and 

$\displaystyle \text{inf}A\ \epsilon\ -(-A)=A$.

Therefore proving the statement in question true.