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<b>let </b><font color="Maroon">i</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">j</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:4"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E2:4"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E3:4"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/>
<b>assume </b><a NAME="E4:4"/>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">j</font> is <a href="int_1.html#V1">integer</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">r</font> = <font color="Maroon">i</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">j</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<a NAME="E6:4"/>
( <font color="Maroon">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E7:4"/><b>then </b>
<font color="Maroon">i</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="jordan4.html#T5">JORDAN4:5</a></i>;<br/>
<a NAME="E8:4"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">i</font> <a href="int_1.html#K5">div</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="newton.html#T101">NEWTON:101</a></i>;<br/>
<font color="Maroon">r</font> = 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">r</font></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i><a class="ref" href="int_1.html#T47">INT_1:47</a></i>
<br/>.= 
<font color="Maroon">i</font> <a href="int_1.html#K5">div</a> <font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="int_1.html#D7">INT_1:def 7</a></i>
;<br/>
<b>hence </b><a NAME="E10:4"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="xcmplx_1.html#T50">XCMPLX_1:50</a></i>;<br/>


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