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\begin{document}

\title{High school geometry theorems}
\author{Hilbert's axiomatic system.\\
Formalizovano od strane: Sana Stojanovic Djurdjevic \\
Dokaz generisan uz pomo\'{c}: ArgoGeoChecker.}
\date{30.03.2018.}
\maketitle

\begin{theorem}[th_18_01.]
Pod pretpostavkom da va\v{z}i $bet(A, B, C)$ i $bet(A, D, C)$ pokazati da postoji prava $p$ tako da va\v{z}i $A \in p$ i $B \in p$ i $C \in p$ i $D \in p$.
\end{theorem}

{\em Dokaz:}
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\proofstep{0} {Na osnovu \v{c}injenice $bet(A, B, C)$ va\v{z}i $A \neq B$ i $A \neq C$ i $B \neq C$ i $col(A, B, C)$ i $bet(C, B, A)$ (aksioma $II1$).} 
\proofstep{0} {Na osnovu \v{c}injenice $bet(A, D, C)$ va\v{z}i $A \neq D$ i $A \neq C$ i $D \neq C$ i $col(A, D, C)$ i $bet(C, D, A)$ (aksioma $II1$).} 
\proofstep{0} {Na osnovu \v{c}injenice $col(A, D, C)$ va\v{z}i $col(A, C, D)$ i $col(D, A, C)$ i $col(D, C, A)$ i $col(C, A, D)$ i $col(C, D, A)$ (aksioma $sym_col$).} 
\proofstep{0} {Na osnovu \v{c}injenice $col(A, B, C)$ postoji prava $p$  tako da va\v{z}i $A \in p$ i $B \in p$ i $C \in p$ (aksioma $D2$).} 
\proofstep{0} {Na osnovu \v{c}injenice $col(A, C, D)$ postoji prava $q$  tako da va\v{z}i $A \in q$ i $C \in q$ i $D \in q$ (aksioma $D2$).} 
\proofstep{0} {Na osnovu \v{c}injenica $A \neq C$ i $A \in p$ i $C \in p$ i $A \in q$ i $C \in q$ va\v{z}i $p = q$ (aksioma $I2$).} 
\proofstep{0} {Na osnovu \v{c}injenica $D \in q$ i $p = q$ va\v{z}i $D \in p$.} 

\proofstep{0} {Zaklju\v{c}ak teoreme sledi iz \v{c}injenica $A \in p$ i $B \in p$ i $C \in p$ i $D \in p$.} 
QED
\hrule 
\begin{theorem}[th_18_02.]
Pod pretpostavkom da va\v{z}i $bet(A, B, C)$ i $bet(A, D, C)$ i $A \in t$ i $B \in t$ i $C \in t$ i $D \in t$ pokazati da va\v{z}i $bet(A, B, D)$.
\end{theorem}

\hrule 
\begin{theorem}[th_18_03.]
Pod pretpostavkom da va\v{z}i $bet(A, B, C)$ i $bet(A, D, C)$ i $A \in t$ i $B \in t$ i $C \in t$ i $D \in t$ i $bet(A, B, D)$ pokazati da va\v{z}i $bet(B, C, D)$.
\end{theorem}

\hrule 

\end{document}
