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<span class="kw">let </span><font color="Maroon" title="c1">p</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">G</font> being   <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>  st (  for <font color="Olive" title="b2">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">G</font> holds  <font color="Olive" title="b2">a</font> is <font color="Maroon" title="c1">p</font> <a href="groupp_1.html#V1" title="GROUPP_1:attr.1">-power</a>  ) holds <br/><font color="Olive" title="b1">G</font> is <font color="Maroon" title="c1">p</font> <a href="group_10.html#V2" title="GROUP_10:attr.2">-group</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> holds  <font color="Olive" title="b1">a</font> is <font color="Maroon" title="c1">p</font> <a href="groupp_1.html#V1" title="GROUPP_1:attr.1">-power</a>  ) implies <font color="Maroon" title="c2">G</font> is <font color="Maroon" title="c1">p</font> <a href="group_10.html#V2" title="GROUP_10:attr.2">-group</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:26"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
 for <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> holds  <font color="Olive" title="b1">a</font> is <font color="Maroon" title="c1">p</font> <a href="groupp_1.html#V1" title="GROUPP_1:attr.1">-power</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">G</font> is <font color="Maroon" title="c1">p</font> <a href="group_10.html#V2" title="GROUP_10:attr.2">-group</a> </span><br/>

<span class="kw">assume </span><a NAME="E2:26"/>
 not <font color="Maroon" title="c2">G</font> is <font color="Maroon" title="c1">p</font> <a href="group_10.html#V2" title="GROUP_10:attr.2">-group</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:26"/><span class="kw">then </span>
 for <font color="Olive" title="b1">r</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   holds  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b1">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_10.html#D17" title="GROUP_10:def.17">GROUP_10:def 17</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">s</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E5:26"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
( <font color="Maroon" title="c3">s</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  &amp; <font color="Maroon" title="c3">s</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font> )
 <span class="kw">and </span><br/><a NAME="E6:26"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
<font color="Maroon" title="c3">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="groupp_1.html#T1" target="_self" title="GROUPP_1:th.1">Th1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font><span class="kw"> such that </span><br/><a NAME="E8:26"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="group_10.html#T11" title="GROUP_10:th.11">GROUP_10:11</a></span>;<br/>
<a NAME="E9:26"/>
<font color="Maroon" title="c4">b</font> is <font color="Maroon" title="c1">p</font> <a href="groupp_1.html#V1" title="GROUPP_1:attr.1">-power</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E16">A1</font></span></a></span>;<br/>
<a NAME="E10:26"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">r1</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b1">r1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="groupp_1.html#D1" target="_self" title="GROUPP_1:def.1">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:26"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="nat_3.html#T6" title="NAT_3:th.6">NAT_3:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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