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<b>let </b><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">j</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:4"/><i><font color="Green" title="E5">A1</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">j</font>
 <b>and </b><br/><a NAME="E2:4"/><i><font color="Green" title="E6">A2</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E3:4"/><i><font color="Green" title="E7">A3</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></i>;<br/>
<b>assume </b><a NAME="E4:4"/>
<font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">j</font> is <a href="int_1.html#V1" title="INT_1:attr.1">integer</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c3">r</font> = <font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">j</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<a NAME="E6:4"/>
<font color="Maroon" title="c1">i</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="scmpds_9.html#E1:4"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="nat_d.html#T27" title="NAT_D:th.27">NAT_D:27</a></i>;<br/>
<a NAME="E7:4"/><b>then </b><i><font color="Green" title="E8">A4</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="ref" href="newton.html#T101" title="NEWTON:th.101">NEWTON:101</a></i>;<br/>
<font color="Maroon" title="c3">r</font> = 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
<b>by </b><i><a class="ref" href="int_1.html#T47" title="INT_1:th.47">INT_1:47</a></i>
<br/>.= 
<font color="Maroon" title="c1">i</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c2">j</font>
<b>by </b><i><a class="ref" href="int_1.html#D7" title="INT_1:def.7">INT_1:def 7</a></i>
;<br/>
<b>hence </b><a NAME="E9:4"/>
contradiction
 <b>by </b><i><a class="txt" href="scmpds_9.html#E1:4"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="scmpds_9.html#E2:4"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="scmpds_9.html#E3:4"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="scmpds_9.html#E7:4"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="xcmplx_1.html#T50" title="XCMPLX_1:th.50">XCMPLX_1:50</a></i>;<br/>


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