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<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:11"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K3">RAT</a> 
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:11_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1"><b>suppose </b></a><a NAME="E1:11_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:11_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 <a href="real_1.html#K6">/</a> 1
 ;<br/><b>hence </b><a NAME="E3:11_1_1"/>
 ex <font color="Olive">m</font>, <font color="Olive">n</font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Olive">n</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_2"><b>suppose </b></a><a NAME="E1:11_1_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>consider </b><font color="Maroon">m</font> being   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">n</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E3:11_1_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/><b>take </b>
<font color="Maroon">m</font>
;<br/><b>take </b>
<font color="Maroon">n</font>
;<br/><a NAME="E4:11_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K5">"</a> </span>)</span>
 <b>by </b><i/>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:11_1_2_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:11_1_2_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K5">"</a>  <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/><b>hence </b><a NAME="E3:11_1_2_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div>
<b>end;</b></div><b>thus </b><a NAME="E6:11_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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