\begin{definition}\label{collinear}
    $a$ is collinear with $b$ and $c$ iff $\Betw{a}{b}{c}$ or $\Betw{b}{c}{a}$ or $\Betw{c}{a}{b}$.
\end{definition}

\begin{axiom}[Reflexivity of congruence]\label{cong_refl_swap} % SST A1
    Let $a, b$ be points.
    Then $\Cong{a}{b}{b}{a}$.
\end{axiom}

\begin{axiom}[Pseudotransitivity of congruence]\label{cong_pseudotransitive} % SST A2
    Let $a, b, c, d$ be points.
    If $\Cong{c}{d}{a}{b}$ and $\Cong{c}{d}{e}{f}$ then $\Cong{a}{b}{e}{f}$.
\end{axiom}

\begin{axiom}[Identity of congruence]\label{cong_id} % SST A3
    Let $a, b, c$ be points.
    If $\Cong{a}{b}{c}{c}$ then $a = b$.
\end{axiom}


\begin{axiom}[Segment construction]\label{segment_construction} % SST A4
    There exists a point $c$ such that $\Betw{a}{b}{c}$ and $\Cong{b}{c}{d}{e}$.
\end{axiom}


\begin{definition}\label{ofs}
    Let $x,y,z,r,u,v,w,p$ be points.
    $\OFS{x}{y}{z}{r}{u}{v}{w}{p}$ if and only if
    $\Betw{x}{y}{z}
      \land \Betw{u}{v}{w}$
    and
    $\Cong{x}{y}{u}{v}
      \land \Cong{y}{z}{v}{w}
      \land \Cong{x}{r}{u}{p}
      \land \Cong{y}{r}{v}{p}$.
\end{definition}


\begin{axiom}[Five segment axiom]\label{five_segment} % SST A5
    Let $a,b,c,d,a',b',c',d'$ be points.
    If $\OFS{a}{b}{c}{d}{a'}{b'}{c'}{d'}$ and
    $a \neq b$ then $\Cong{c}{d}{c'}{d'}$.
\end{axiom}


\begin{axiom}[Identity of betweenness]\label{betw_id}
    Let $a,b$ be points.
    If $\Betw{a}{b}{a}$ then $a = b$.
\end{axiom}


\begin{axiom}[Inner Pasch]\label{innerpasch} % SST A7
    Let $x,y,z,u,v,w$ be points.
    If $\Betw{x}{u}{z}$ and $\Betw{y}{v}{z}$ then there exists a point $w$
    such that $\Betw{u}{w}{y}$ and $\Betw{v}{w}{x}$.
\end{axiom}


% \begin{axiom}[Lower dimension] % SST A8
%     %There exist points $a, b, c$ such that
%     There exists $a, b, c$ such that $a,b,c$ are points and
%     $a$ is not collinear with $b$ and $c$.
% \end{axiom}


\begin{axiom}[Upper dimension]\label{upperdim} % SST A9
    Let $x,y,z,u,v$ be points.
    If $\Cong{x}{u}{x}{v}$ and $\Cong{y}{u}{y}{v}$
    and $\Cong{z}{u}{z}{v}$ and $u \neq v$
    then $x$ is collinear with $y$ and $z$.
\end{axiom}


% \begin{axiom}[Euclid]
%     Assume $x \neq r$.
%     If $\Betw{x}{r}{v}$ and $\Betw{y}{r}{z}$
%     then there exist points $s,t$ such that
%     $\Betw{x}{y}{s}$ and $\Betw{x}{z}{t}$ and $\Betw{s}{v}{t}$.
% \end{axiom}


\begin{lemma}[Reflexivity of congruence]\label{cong_refl} % Satz 2.1
    Let $a, b$ be points.
    Then $\Cong{a}{b}{a}{b}$.
\end{lemma}


\begin{lemma}[Symmetry of congruence]\label{cong_sym} % Satz 2.2
    Let $a, b, c, d$ be points.
    Suppose $\Cong{a}{b}{c}{d}$.
    Then $\Cong{c}{d}{a}{b}$.
\end{lemma}


\begin{lemma}[Transitivity of congruence]\label{cong_transitive} % Satz 2.3
    Let $a, b, c, d, e, f$ be points.
    If $\Cong{a}{b}{c}{d}$ and $\Cong{c}{d}{e}{f}$
    then $\Cong{a}{b}{e}{f}$.
\end{lemma}


\begin{lemma}\label{cong_shuffle_left} % Satz 2.4
    Let $a, b, c, d$ be points.
    If $\Cong{a}{b}{c}{d}$
    then $\Cong{b}{a}{c}{d}$.
\end{lemma}


\begin{lemma}\label{cong_shuffle_right} % Satz 2.5
    Let $a, b, c, d$ be points.
    If $\Cong{a}{b}{c}{d}$
    then $\Cong{b}{a}{d}{c}$.
\end{lemma}


\begin{lemma}[Zero segments are congruent]\label{cong_zero} % Satz 2.8
    Let $a,b$ be points.
    Then $\Cong{a}{a}{b}{b}$.
\end{lemma}

% Somewhat slow for now.
%\begin{lemma}[Concatenation of segments]\label{segment_concat} % Satz 2.11
%    Assume $\Betw{x}{y}{z}$ and $\Betw{r}{v}{w}$.
%    Assume $\Cong{x}{y}{r}{v}$ and $\Cong{y}{z}{v}{w}$.
%    Then $\Cong{x}{z}{r}{w}$.
%\end{lemma}
%\begin{proof}
%    % We have
%    $\OFS{x}{y}{z}{x}{r}{v}{w}{r}$. % By previous results and assumption.
%    If $x = y$ then $r = v$.                % Axiom A3 gives this implication.
%    If $x \neq y$ then $\Cong{x}{z}{r}{w}$. % Axiom A5 completes the proof.
%\end{proof}
