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<b>let </b><font color="Maroon">a</font> be   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">b</font> be   <a href="int_1.html#NM1">Integer</a>;<br/><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>assume </b><a NAME="E1:3"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>then </b><span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="int_1.html#D8">INT_1:def 8</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 0

;<br/>
<a NAME="E3:3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font>
 
;<br/>
<a NAME="E4:3"/><b>then </b>
<font color="Maroon">n</font> <a href="int_1.html#R2">divides</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="int_1.html#D9">INT_1:def 9</a></i>;<br/>
<a NAME="E5:3"/><b>then </b>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span> <a href="int_1.html#R1">are_congruent_mod</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="int_2.html#T19">INT_2:19</a></i>;<br/>
<b>hence </b><a NAME="E6:3"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="int_3.html#T12">INT_3:12</a></i>;<br/>


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