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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:3"/><i><font color="Green" title="E4">A1</font></i>: 
not <font color="Maroon" title="c1">n</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>and </b><br/><a NAME="E2:3"/><i><font color="Green" title="E5">A2</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K5" title="JORDAN2C:func.5">1.REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>
 ;<br/>

<a NAME="E3:3"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T31" title="EUCLID:th.31">EUCLID:31</a></i>;<br/>
<a NAME="E4:3"/><b>then </b>
 <a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan2c.html#K5" title="JORDAN2C:func.5">1.REAL</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="topreal9.html#E2:3"><i><font color="Green" title="E5">A2</font></i></a>, <a class="ref" href="topreal9.html#T2" target="_self" title="TOPREAL9:th.2">Th2</a></i>;<br/>
<b>hence </b><a NAME="E5:3"/>
contradiction
 <b>by </b><i><a class="txt" href="topreal9.html#E1:3"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="revrot_1.html#T19" title="REVROT_1:th.19">REVROT_1:19</a></i>;<br/>


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