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









<b>assume </b><b>that </b><br/><a NAME="E1:8"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">G</font> is <a href="group_1.html#V6">finite</a>
 <b>and </b><br/><a NAME="E2:8"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K5">gr</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E3:8"/><i><font color="Green">E50</font></i>: 
 <a href="group_1.html#K9">ord</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E4:8"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">s</font>
 ;<br/>

<a NAME="E5:8"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">p</font> <a href="group_1.html#NR1" target="_self">is_not_of_order_0</a> 
 
<b>by </b><i>, <a class="ref" href="gr_cy_1.html#T26">GR_CY_1:26</a></i>;<br/>
<a NAME="E6:8"/><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">p</font></span>)</span> <a href="group_1.html#K6">|^</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><div class="add">

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">p</font></span>)</span> <a href="group_1.html#K6">|^</a> <font color="Maroon">s</font> = 
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">n</font>
<b>by </b><i>, <a class="ref" href="group_1.html#T50">GROUP_1:50</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font>
<b>by </b><i>, , <a class="ref" href="gr_cy_1.html#T29">GR_CY_1:29</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E7:8"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, , , <a class="ref" href="group_1.html#T90">GROUP_1:90</a></i>;<br/>
<a NAME="E8:8"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, , , <a class="ref" href="group_1.html#T90">GROUP_1:90</a></i>;<br/>
<a NAME="E9:8"/>
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">p</font></span>)</span> <a href="group_1.html#K6">|^</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font> &amp; <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 holds <br/><font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:8_2"/><i><font color="Green">E67</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">p</font></span>)</span> <a href="group_1.html#K6">|^</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E2:8_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>and </b><br/><a NAME="E3:8_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E4:8_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">p</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">p</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<a NAME="E2:8_2_1"/>
<font color="Maroon">p1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:8_2_1"/>
<font color="Maroon">p</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:8_2"/><i><font color="Green">E71</font></i>: 
 <a href="group_1.html#K7">ord</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, , , <a class="ref" href="gr_cy_1.html#T27">GR_CY_1:27</a></i>;<br/>
<a NAME="E6:8_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, , <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/>
<a NAME="E7:8_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">a</font> <a href="group_1.html#NR1" target="_self">is_not_of_order_0</a> 
 
<b>by </b><i>, <a class="ref" href="gr_cy_1.html#T26">GR_CY_1:26</a></i>;<br/>
<a NAME="E8:8_2"/>
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font>
 
<b>by </b><i>, <a class="ref" href="group_1.html#T50">GROUP_1:50</a></i>;<br/>
<a NAME="E9:8_2"/><b>then </b>
 <a href="group_1.html#K7">ord</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font>
 
<b>by </b><i>, , <a class="ref" href="group_1.html#D12">GROUP_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E10:8_2"/>
contradiction
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:8"/>
 <a href="group_1.html#K7">ord</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>by </b><i>, , , <a class="ref" href="group_1.html#D12">GROUP_1:def 12</a></i>;<br/>


</div>
