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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>given </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:5_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="facirc_1.html#D1" target="_self">FACIRC_1:def 1</a><br/>

<a NAME="E2:5_1_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span> &amp; 0 <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E3:5_1_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> </span>}</span>  )
 
<b>by </b><i>, <a class="ref" href="axioms.html#T30">AXIOMS:30</a>, <a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<a NAME="E4:5_1_1"/><b>then </b>
0 <a href="hidden.html#R2">in</a> <font color="Maroon">n</font>
 
;<br/>
<b>hence </b><a NAME="E5:5_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


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