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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">b</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/>





<b>assume </b><a NAME="E1:23"/>
<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">m</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 ;<br/>

<a NAME="E2:23"/><b>then </b>
<font color="Maroon" title="c1">a</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">s</font>, <font color="Maroon" title="c5">t</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E4:23"/><i><font color="Green" title="E22">A2</font></i>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="int_3.html#T16" title="INT_3:th.16">INT_3:16</a></i>;<br/>
<b>take </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c3">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">s</font>;<br/>

<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">m</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">m</font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">t</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">m</font>
<b>by </b><i><a class="txt" href="int_4.html#E4:23"><i><font color="Green" title="E22">A2</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c5">t</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">m</font>

<br/>.= 
0 <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">m</font>
<b>by </b><i><a class="ref" href="int_3.html#T8" title="INT_3:th.8">INT_3:8</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="int_4.html#T12" target="_self" title="INT_4:th.12">Th8</a></i>
;<br/>
<b>hence </b><a NAME="E6:23"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/>


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