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



<b>assume </b><a NAME="E1:11"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<a NAME="E2:11"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="int_2.html#K1">abs</a> <font color="Maroon">a</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="int_2.html#K1">abs</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<a NAME="E4:11"/>
<font color="Maroon">a'</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">b'</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/>
<a NAME="E5:11"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="int_2.html#R1">are_relative_prime</a> 
 
<b>by </b><i><a class="ref" href="int_2.html#D4">INT_2:def 4</a></i>;<br/>
<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/>

<a NAME="E6:11"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="int_2.html#K1">abs</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="int_2.html#K1">abs</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E7:11"/>
(  <a href="int_2.html#K1">abs</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">a'</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">m</font></span>)</span> &amp;  <a href="int_2.html#K1">abs</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">b'</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">m</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="int_2.html#K3">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">b'</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="int_2.html#D3">INT_2:def 3</a></i>
<br/>.= 
 <a href="int_2.html#K1">abs</a> <span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i>, <a class="ref" href="int_2.html#T39">INT_2:39</a></i>
<br/>.= 
<font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E9:11"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 ;<br/>


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