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<b>let </b><font color="Maroon">a</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">b</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="E1:8"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">a</font>,<font color="Maroon">m</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Maroon">b</font>,<font color="Maroon">m</font> <a href="int_2.html#R2">are_relative_prime</a>  )
 ;<br/>

<a NAME="E2:8"/><b>then </b>
<font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font>,<font color="Maroon">m</font> <a href="int_2.html#R2">are_relative_prime</a> 
 
<b>by </b><i><a class="ref" href="euler_1.html#T15">EULER_1:15</a></i>;<br/>
<a NAME="E3:8"/><b>then </b><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<b>consider </b><font color="Maroon">t</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:8"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</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">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font></span>)</span>
 <b>and </b><br/><a NAME="E6:8"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/>
<a NAME="E7:8"/>
<font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> 0
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/>
<a NAME="E8:8"/><b>then </b><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="int_2.html#K3">gcd</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font></span>)</span> <a href="int_2.html#K3">gcd</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="euler_1.html#T17">EULER_1:17</a></i>;<br/>
<a NAME="E9:8"/><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font></span>)</span> <a href="int_2.html#K3">gcd</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:8"/>
(  <a href="int_2.html#K1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> &amp;  <a href="int_2.html#K1">abs</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> )
 
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E11:8"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, , , <a class="ref" href="int_2.html#D3">INT_2:def 3</a></i>;<br/>
<b>hence </b><a NAME="E12:8"/>
<font color="Maroon">m</font>,<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>by </b><i><a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>


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