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<b>let </b><font color="Maroon">m</font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>let </b><font color="Maroon">n</font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="rat_1.html#NM1">Rational</a>;<br/>





<b>assume </b><a NAME="E1:49"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1"><b>suppose </b></a><a NAME="E1:49_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:49_1_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><b>then reconsider </b><font color="Maroon">x</font> =  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E4:49_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i/>;<br/><font color="Maroon">p</font> = 
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T189">XCMPLX_1:189</a></i>
;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:49_1_1"/><i><font color="Green">E34</font></i>: 
(  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" target="_self">numerator</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font> )
 <b>by </b><i>, <a class="txt" href="rat_1.html#E4:49_1_1"><i><font color="Green">E29</font></i></a></i>;<br/><b>take </b><font color="Maroon">y</font> =  <a href="real_1.html#K1">-</a> <font color="Maroon">k</font>;<br/><b>thus </b><font color="Maroon">m</font> = 
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K2" target="_self">numerator</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="rat_1.html#K2" target="_self">numerator</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font>

;<br/><b>thus </b><font color="Maroon">n</font> = 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font>

;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_2"><b>suppose </b></a><a NAME="E1:49_1_2"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:49_1_2"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" target="_self">numerator</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font> )
 <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">k</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/><b>take </b>
<font color="Maroon">y</font>
;<br/><b>thus </b><a NAME="E6:49_1_2"/>
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" target="_self">numerator</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" target="_self">denominator</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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