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<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>;<br/>



<b>assume </b><a NAME="E1:10"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="group_1.html#NR1" title="GROUP_1:NR.1">is_not_of_order_0</a> 
 ;<br/>

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

<a NAME="E2:10"/>
 <a href="group_1.html#K7" title="GROUP_1:func.7">ord</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="group_8.html#E1:10"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="group_1.html#D12" title="GROUP_1:def.12">GROUP_1:def 12</a></i>;<br/>
<a NAME="E3:10"/><b>then </b>
<font color="Maroon" title="c3">m</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">ord</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">ord</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">ord</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="int_1.html#D8" title="INT_1:def.8">INT_1:def 8</a></i>;<br/>
<b>then </b><font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">ord</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p0">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">ord</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">ord</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p5">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">ord</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">ord</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T63" title="GROUP_1:th.63">GROUP_1:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">G</font></span>)</span>
<b>by </b><i><a class="ref" href="group_8.html#T9" target="_self" title="GROUP_8:th.9">Th9</a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c3">m</font>
<b>by </b><i><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E5:10"/>
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">ord</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>
 ;<br/>


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