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<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:9"/><i><font color="Green" title="E9">A1</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/>

<b>assume </b><a NAME="E2:9"/>
<font color="Maroon" title="c2">k</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E3:9"/><b>then </b>
<font color="Maroon" title="c2">k</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="newton.html#T101" title="NEWTON:th.101">NEWTON:101</a></i>;<br/>
<a NAME="E4:9"/><b>then </b>
not <font color="Maroon" title="c1">n</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c2">k</font>
 
<b>by </b><i><a class="txt" href="jordan1d.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="int_3.html#T11" title="INT_3:th.11">INT_3:11</a></i>;<br/>
<a NAME="E5:9"/><b>then </b><i><font color="Green" title="E10">A2</font></i>: 
<font color="Maroon" title="c2">k</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">n</font> is  not   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>
 
<b>by </b><i><a class="txt" href="jordan1d.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="wsierp_1.html#T22" title="WSIERP_1:th.22">WSIERP_1:22</a></i>;<br/>
<b>thus </b> <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">n</font></span>)</span> = 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c1">n</font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T101" title="NEWTON:th.101">NEWTON:101</a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">n</font></span>)</span></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#D7" title="INT_1:def.7">INT_1:def 7</a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
<b>by </b><i><a class="txt" href="jordan1d.html#E5:9"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="jordan1d.html#T3" target="_self" title="JORDAN1D:th.3">Th3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
<b>by </b><i><a class="ref" href="int_1.html#D7" title="INT_1:def.7">INT_1:def 7</a></i>
;<br/>


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