diff options
Diffstat (limited to 'library/numbers.tex')
| -rw-r--r-- | library/numbers.tex | 18 |
1 files changed, 11 insertions, 7 deletions
diff --git a/library/numbers.tex b/library/numbers.tex index df47d81..a0e2211 100644 --- a/library/numbers.tex +++ b/library/numbers.tex @@ -10,7 +10,7 @@ %\inv{} für inverse benutzen. Per Signatur einfüheren und dann axiomatisch absicher %\cdot für multiklikation verwenden. %< für die relation benutzen. - +% sup und inf einfügen \begin{signature} $\reals$ is a set. @@ -92,7 +92,7 @@ \begin{axiom}\label{reals_axiom_mul_invers} - For all $x \in \reals$ there exist $y \in \reals$ such that $x \neq \zero$ and $x \times y = 1$. + For all $x \in \reals$ such that $x \neq \zero$ there exist $y \in \reals$ such that $x \times y = 1$. \end{axiom} \begin{axiom}\label{reals_axiom_disstro1} @@ -107,7 +107,10 @@ For all $x,y,z \in \reals$ if $x + y = x + z$ then $y = z$. \end{proposition} - +\begin{axiom}\label{reals_axiom_dedekind_complete} + For all $X,Y,x,y$ such that $X,Y \subseteq \reals$ and $x \in X$ and $y \in Y$ and $x < y$ we have there exist $z \in \reals$ + such that $x < z < y$. +\end{axiom} \begin{lemma}\label{order_reals_lemma1} @@ -129,14 +132,15 @@ then $(x \times z) < (x \times y)$. \end{lemma} -\begin{lemma}\label{a} +\begin{lemma}\label{o4rder_reals_lemma} For all $x,y \in \reals$ if $x > y$ then $x \geq y$. \end{lemma} -\begin{lemma}\label{aa} +\begin{lemma}\label{order_reals_lemma5} For all $x,y \in \reals$ if $x < y$ then $x \leq y$. \end{lemma} -\begin{lemma}\label{aaa} +\begin{lemma}\label{order_reals_lemma6} For all $x,y \in \reals$ if $x \leq y \leq x$ then $x=y$. -\end{lemma}
\ No newline at end of file +\end{lemma} + |