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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_9_01.]
Pod pretpostavkom da va\v{z}i $p \neq q$ i $A \in p$ i $B \in p$ i $A \in q$ i $B \in q$ pokazati da va\v{z}i $A = B$.
\end{theorem}

{\em Dokaz:}
\setcounter{proofstepnum}{0} 


\proofstep{0} {Va\v{z}i $A = B$ ili $A \neq B$.}  

\proofstep{3}  {Pretpostavimo da va\v{z}i: $A = B$.} 

\proofstep{6} {Zaklju\v{c}ak teoreme sledi iz \v{c}injenice $A = B$.}  

\proofstep{3}  {Pretpostavimo da va\v{z}i: $A \neq B$.} 
\proofstep{6} {Na osnovu \v{c}injenica $A \neq B$ i $A \in p$ i $B \in p$ i $A \in q$ i $B \in q$ va\v{z}i $p = q$ (aksioma $I2$).} 
\proofstep{6} {Na osnovu \v{c}injenica $p \neq q$ i $p = q$ dobijamo kontradikciju.} 
\proofstep{0} {Teorema je dokazana u svim slu\v{c}ajevima.}
QED
\hrule 

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