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<span class="kw">let </span><font color="Maroon" title="c1">Gc</font> be   <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="gr_cy_1.html#V1" title="GR_CY_1:attr.1">cyclic</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="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">k</font> being   <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>  st  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">k</font> implies  ex <font color="Olive" title="b1">H</font> being   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">Gc</font> st <br/>(  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Olive" title="b1">H</font> is   <a href="gr_cy_1.html#V1" title="GR_CY_1:attr.1">cyclic</a>  <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a> ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">n</font> =  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font>;<br/>
<span class="kw">assume </span><a NAME="E1:30"/>
 ex <font color="Olive" title="b1">k</font> being   <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>  st  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">Gc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">H</font> being   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">Gc</font> st <br/>(  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Olive" title="b1">H</font> is   <a href="gr_cy_1.html#V1" title="GR_CY_1:attr.1">cyclic</a>  <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">g1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Gc</font><span class="kw"> such that </span><br/><a NAME="E3:30"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c3">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">and </span><br/><a NAME="E4:30"/>
for <font color="Olive" title="b1">g2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Gc</font>  st  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Olive" title="b1">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 holds <br/><font color="Maroon" title="c3">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gr_cy_2.html#T30" target="_self" title="GR_CY_2:th.30">Th30</a></span>;<br/>
<span class="kw">take </span>
 <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="c3">g1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">g1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp;  <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="c3">g1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is   <a href="gr_cy_1.html#V1" title="GR_CY_1:attr.1">cyclic</a>  <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a> )</span><br/>

<span class="kw">thus </span><a NAME="E5:30"/>
(  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">g1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp;  <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="c3">g1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is   <a href="gr_cy_1.html#V1" title="GR_CY_1:attr.1">cyclic</a>  <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="gr_cy_2.html#T10" target="_self" title="GR_CY_2:th.10">Th10</a>, <a class="ref" href="gr_cy_1.html#T27" title="GR_CY_1:th.27">GR_CY_1:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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