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<div class="add">

<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">s</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c4">G</font> be  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/><b>let </b><font color="Maroon" title="c5">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">G</font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:8"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>and </b><br/><a NAME="E2:8"/><i><font color="Green" title="E9">A2</font></i>: 
 <a href="group_1.html#K8" title="GROUP_1:func.8">ord</a> <font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>and </b><br/><a NAME="E3:8"/><i><font color="Green" title="E10">A3</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">s</font>
 ;<br/>

<a NAME="E4:8"/><i><font color="Green" title="E11">A4</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c2">p</font> <a href="group_1.html#NR1" title="GROUP_1:NR.1">is_not_of_order_0</a> 
 
<b>by </b><i><a class="ref" href="gr_cy_1.html#T26" title="GR_CY_1:th.26">GR_CY_1:26</a></i>;<br/>
<a NAME="E5:8"/><i><font color="Green" title="E12">A5</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c4">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" title="c5">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c3">s</font> = 
<font color="Maroon" title="c5">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E3:8"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="group_1.html#T50" title="GROUP_1:th.50">GROUP_1:50</a></i>
<br/>.= 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c4">G</font>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E2:8"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="gr_cy_1.html#T29" title="GR_CY_1:th.29">GR_CY_1:29</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E6:8"/><i><font color="Green" title="E13">A6</font></i>: 
<font color="Maroon" title="c3">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E2:8"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="gr_cy_2.html#E3:8"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="group_1.html#T90" title="GROUP_1:th.90">GROUP_1:90</a></i>;<br/>
<a NAME="E7:8"/><i><font color="Green" title="E14">A7</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E2:8"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="gr_cy_2.html#E3:8"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="group_1.html#T90" title="GROUP_1:th.90">GROUP_1:90</a></i>;<br/>
<a NAME="E8:8"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c4">G</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  holds <br/><font color="Maroon" title="c3">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">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" title="c6">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:8_2"/><i><font color="Green" title="E15">A8</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c4">G</font>
 <b>and </b><br/><a NAME="E2:8_2"/><i><font color="Green" title="E16">A9</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 <b>and </b><br/><a NAME="E3:8_2"/><i><font color="Green" title="E17">A10</font></i>: 
<font color="Maroon" title="c3">s</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">k</font>
 ;<br/>

<a NAME="E4:8_2"/><i><font color="Green" title="E18">A11</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">s</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">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" title="c7">p1</font> = <font color="Maroon" title="c2">p</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<a NAME="E2:8_2_1"/>
<font color="Maroon" title="c7">p1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E7:8"><i><font color="Green" title="E14">A7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:8_2_1"/>
<font color="Maroon" title="c2">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c3">s</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E3:8_2"><i><font color="Green" title="E17">A10</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:8_2"/><i><font color="Green" title="E19">A12</font></i>: 
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E1:8"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="gr_cy_2.html#E2:8"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="gr_cy_1.html#T27" title="GR_CY_1:th.27">GR_CY_1:27</a></i>;<br/>
<a NAME="E6:8_2"/><i><font color="Green" title="E20">A13</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E7:8"><i><font color="Green" title="E14">A7</font></i></a>, <a class="txt" href="gr_cy_2.html#E2:8_2"><i><font color="Green" title="E16">A9</font></i></a>, <a class="ref" href="xcmplx_1.html#T6" title="XCMPLX_1:th.6">XCMPLX_1:6</a></i>;<br/>
<a NAME="E7:8_2"/><i><font color="Green" title="E21">A14</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="group_1.html#NR1" title="GROUP_1:NR.1">is_not_of_order_0</a> 
 
<b>by </b><i><a class="ref" href="gr_cy_1.html#T26" title="GR_CY_1:th.26">GR_CY_1:26</a></i>;<br/>
<a NAME="E8:8_2"/>
<font color="Maroon" title="c5">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c4">G</font>
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E1:8_2"><i><font color="Green" title="E15">A8</font></i></a>, <a class="ref" href="group_1.html#T50" title="GROUP_1:th.50">GROUP_1:50</a></i>;<br/>
<b>hence </b><a NAME="E9:8_2"/>
contradiction
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E3:8"><i><font color="Green" title="E10">A3</font></i></a>, <a class="txt" href="gr_cy_2.html#E4:8_2"><i><font color="Green" title="E18">A11</font></i></a>, <a class="txt" href="gr_cy_2.html#E5:8_2"><i><font color="Green" title="E19">A12</font></i></a>, <a class="txt" href="gr_cy_2.html#E6:8_2"><i><font color="Green" title="E20">A13</font></i></a>, <a class="txt" href="gr_cy_2.html#E7:8_2"><i><font color="Green" title="E21">A14</font></i></a>, <a class="ref" href="group_1.html#D12" title="GROUP_1:def.12">GROUP_1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:8"/>
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E4:8"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="gr_cy_2.html#E5:8"><i><font color="Green" title="E12">A5</font></i></a>, <a class="txt" href="gr_cy_2.html#E6:8"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="group_1.html#D12" title="GROUP_1:def.12">GROUP_1:def 12</a></i>;<br/>


</div>
