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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>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:3"/><i><font color="Green">E42</font></i>: 
not <font color="Maroon">n</font> is <a href="xboole_0.html#V1">empty</a>
 <b>and </b><br/><a NAME="E2:3"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K5">1.REAL</a> <font color="Maroon">n</font></span>)</span>
 ;<br/>

<a NAME="E3:3"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>;<br/>
<a NAME="E4:3"/><b>then </b>
 <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="jordan2c.html#K5">1.REAL</a> <font color="Maroon">n</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="revrot_1.html#T19">REVROT_1:19</a></i>;<br/>


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