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<b>let </b><font color="Maroon">p</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">s</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:71_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1"><b>suppose </b></a><a NAME="E1:71_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>then </b><span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K6" target="_self">mod</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K5" target="_self">div</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K6" target="_self">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K6" target="_self">mod</a> <font color="Maroon">n</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K5" target="_self">div</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K6" target="_self">mod</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="nat_1.html#K6" target="_self">mod</a> <font color="Maroon">n</font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K6" target="_self">mod</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K6" target="_self">mod</a> <font color="Maroon">n</font>
<b>by </b><i/>
;<br/><b>hence </b><a NAME="E3:71_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K6" target="_self">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K6" target="_self">mod</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K6" target="_self">mod</a> <font color="Maroon">n</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_2"><b>suppose </b></a><a NAME="E1:71_1_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K6" target="_self">mod</a> <font color="Maroon">n</font> = 
0
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K6" target="_self">mod</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K6" target="_self">mod</a> <font color="Maroon">n</font>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

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