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<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>;<br/>



<b>thus </b><a NAME="E1:53"/>
( <font color="Maroon" title="c2">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_4.html#K6" title="GROUP_4:func.6">Phi</a> <font color="Maroon" title="c1">G</font> implies  not <font color="Maroon" title="c2">a</font> is <a href="group_4.html#V1" title="GROUP_4:attr.1">generating</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:53_1"/>
<font color="Maroon" title="c2">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_4.html#K6" title="GROUP_4:func.6">Phi</a> <font color="Maroon" title="c1">G</font>
 ;<br/>

<a NAME="E2:53_1"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K6" title="GROUP_4:func.6">Phi</a> <font color="Maroon" title="c1">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E3:53_1"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> : not <font color="Olive" title="b1">b</font> is <a href="group_4.html#V1" title="GROUP_4:attr.1">generating</a> </span>}</span> 
 
<b>by </b><i><a class="ref" href="group_4.html#T55" target="_self" title="GROUP_4:th.55">Th55</a></i>;<br/>
<a NAME="E4:53_1"/><b>then </b>
 ex <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; not <font color="Olive" title="b1">b</font> is <a href="group_4.html#V1" title="GROUP_4:attr.1">generating</a> )
 
;<br/>
<b>hence </b><a NAME="E5:53_1"/>
not <font color="Maroon" title="c2">a</font> is <a href="group_4.html#V1" title="GROUP_4:attr.1">generating</a>
 ;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:53"/>
not <font color="Maroon" title="c2">a</font> is <a href="group_4.html#V1" title="GROUP_4:attr.1">generating</a>
 ;<br/>

<a NAME="E3:53"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> : not <font color="Olive" title="b1">b</font> is <a href="group_4.html#V1" title="GROUP_4:attr.1">generating</a> </span>}</span> 
 
;<br/>
<a NAME="E4:53"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K6" title="GROUP_4:func.6">Phi</a> <font color="Maroon" title="c1">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_4.html#T55" target="_self" title="GROUP_4:th.55">Th55</a></i>;<br/>
<b>hence </b><a NAME="E5:53"/>
<font color="Maroon" title="c2">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_4.html#K6" title="GROUP_4:func.6">Phi</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>


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