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<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:13"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:13"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E4:13"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E5:13"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:13"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T47">NAT_1:47</a></i>;<br/>
<b>hence </b><a NAME="E7:13"/>
<font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>


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