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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:81"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>reconsider </b><font color="Maroon">z1</font> = <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span>, <font color="Maroon">z2</font> = <font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<a NAME="E3:81"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">z2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">z1</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T47">NAT_1:47</a></i>;<br/>
<b>hence </b><a NAME="E4:81"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</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">n</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
 ;<br/>


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