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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><br/>  for <font color="Olive" title="b2">N</font> being   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Olive" title="b1">G</font>  st <font color="Olive" title="b2">N</font> is <font color="Maroon" title="c1">p</font> <a href="group_10.html#V2" title="GROUP_10:attr.2">-group</a>  &amp; <font color="Olive" title="b1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Olive" title="b2">N</font> is <font color="Maroon" title="c1">p</font> <a href="group_10.html#V2" title="GROUP_10:attr.2">-group</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">N</font> being   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G</font>  st <font color="Olive" title="b1">N</font> is <font color="Maroon" title="c1">p</font> <a href="group_10.html#V2" title="GROUP_10:attr.2">-group</a>  &amp; <font color="Maroon" title="c2">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Olive" title="b1">N</font> is <font color="Maroon" title="c1">p</font> <a href="group_10.html#V2" title="GROUP_10:attr.2">-group</a>  holds <br/><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">let </span><font color="Maroon" title="c3">N</font> be   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G</font>; <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="c3">N</font> is <font color="Maroon" title="c1">p</font> <a href="group_10.html#V2" title="GROUP_10:attr.2">-group</a>  &amp; <font color="Maroon" title="c2">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c3">N</font> is <font color="Maroon" title="c1">p</font> <a href="group_10.html#V2" title="GROUP_10:attr.2">-group</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><span class="kw">that </span><br/><a NAME="E1:28"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
<font color="Maroon" title="c3">N</font> is <font color="Maroon" title="c1">p</font> <a href="group_10.html#V2" title="GROUP_10:attr.2">-group</a> 
 <span class="kw">and </span><br/><a NAME="E2:28"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
<font color="Maroon" title="c2">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c3">N</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"> <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">consider </span><font color="Maroon" title="c4">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> <span class="kw"> such that </span><br/><a NAME="E4:28"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c3">N</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="Maroon" title="c4">r1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="group_10.html#D17" title="GROUP_10:def.17">GROUP_10:def 17</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">r2</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="E6:28"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
 <a href="struct_0.html#K7" title="STRUCT_0:func.7">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c3">N</font></span>)</span> <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="Maroon" title="c5">r2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="group_10.html#D17" title="GROUP_10:def.17">GROUP_10:def 17</a></span>;<br/>
<a NAME="E7:28"/>
 <a href="struct_0.html#K7" title="STRUCT_0:func.7">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K18" title="GROUP_2:func.18">index</a> <font color="Maroon" title="c3">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T27" title="GROUP_6:th.27">GROUP_6:27</a></span>;<br/>
<span class="kw">then </span> <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">r1</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">r2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="group_2.html#T147" title="GROUP_2:th.147">GROUP_2:147</a></span>
<br/>.= 
<font color="Maroon" title="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c5">r2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T8" title="NEWTON:th.8">NEWTON:8</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E9:28"/>
<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 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>; <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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