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<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>let </b><font color="Maroon">G</font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/>





<b>assume </b><a NAME="E1:16"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">G</font> is <a href="group_1.html#V6">finite</a>
 ;<br/>

<b>set </b><font color="Maroon">q</font> =  <a href="group_1.html#K9">ord</a> <font color="Maroon">G</font>;<br/>
<b>set </b><font color="Maroon">p'</font> = <font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span>;<br/>
<a NAME="E2:16"/>
 <a href="group_1.html#K9">ord</a> <font color="Maroon">G</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i>, <a class="ref" href="group_1.html#T90">GROUP_1:90</a></i>;<br/>
<a NAME="E3:16"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="group_1.html#K9">ord</a> <font color="Maroon">G</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E4:16"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="group_1.html#K9">ord</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T46">NAT_1:46</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">q'</font> = <span class="p1">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span></span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">n</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span></span>)</span> = 
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">q'</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T48">GROUP_1:48</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <span class="p5">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">q'</font></span>)</span>
<b>by </b><i>, <a class="ref" href="nat_1.html#T47">NAT_1:47</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <span class="p5">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <span class="p5">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <span class="p5">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <span class="p5">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T64">GROUP_1:64</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <span class="p5">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <span class="p5">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T65">GROUP_1:65</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <span class="p5">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <span class="p5">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <span class="p4">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T71">GROUP_1:71</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <span class="p5">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <span class="p4">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T50">GROUP_1:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font>
<b>by </b><i><a class="ref" href="group_1.html#T42">GROUP_1:42</a></i>
;<br/>
<b>hence </b><a NAME="E7:16"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">n</font></span>)</span> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="group_1.html#T20">GROUP_1:20</a></i>;<br/>


</div>
