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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <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>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><br/>  for <font color="Olive" title="b3">p</font> being   <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>   st <font color="Olive" title="b1">a</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Olive" title="b2">b</font> is <font color="Olive" title="b3">p</font> <a href="groupp_1.html#V1" title="GROUPP_1:attr.1">-power</a>  holds <br/><font color="Olive" title="b1">a</font> is <font color="Olive" title="b3">p</font> <a href="groupp_1.html#V1" title="GROUPP_1:attr.1">-power</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>; <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">p</font> being   <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>   st <font color="Maroon" title="c2">a</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c3">b</font> is <font color="Olive" title="b1">p</font> <a href="groupp_1.html#V1" title="GROUPP_1:attr.1">-power</a>  holds <br/><font color="Maroon" title="c2">a</font> is <font color="Olive" title="b1">p</font> <a href="groupp_1.html#V1" title="GROUPP_1:attr.1">-power</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">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"> ( <font color="Maroon" title="c2">a</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c3">b</font> is <font color="Maroon" title="c4">p</font> <a href="groupp_1.html#V1" title="GROUPP_1:attr.1">-power</a>  implies <font color="Maroon" title="c2">a</font> is <font color="Maroon" title="c4">p</font> <a href="groupp_1.html#V1" title="GROUPP_1:attr.1">-power</a>  )</span><br/>







<span class="kw">assume </span><a NAME="E1:17"/>
<font color="Maroon" title="c2">a</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c3">b</font> is <font color="Maroon" title="c4">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">a</font> is <font color="Maroon" title="c4">p</font> <a href="groupp_1.html#V1" title="GROUPP_1:attr.1">-power</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">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> <span class="kw"> such that </span><br/><a NAME="E3:17"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">r</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/>
<a NAME="E4:17"/>
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="groupp_1.html#T6" target="_self" title="GROUPP_1:th.6">Th6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:17"/>
<font color="Maroon" title="c2">a</font> is <font color="Maroon" title="c4">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="ref" href="groupp_1.html#D1" target="_self" title="GROUPP_1:def.1">Def1</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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