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<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c1">G</font> holds <br/> ( <font color="Olive" title="b1">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> iff not <font color="Olive" title="b1">a</font> is  <a href="group_4.html#V1" title="GROUP_4:attr.1">generating</a>  )</span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> iff not <font color="Maroon" title="c2">a</font> is  <a href="group_4.html#V1" title="GROUP_4:attr.1">generating</a>  )</span><br/>



<span class="kw">thus </span><a NAME="E1:66"/>
( <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>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not <font color="Maroon" title="c2">a</font> is  <a href="group_4.html#V1" title="GROUP_4:attr.1">generating</a>  implies <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> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:66_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c2">a</font> is  <a href="group_4.html#V1" title="GROUP_4:attr.1">generating</a> </span><br/>

<a NAME="E2:66_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E3:66_1"/><span class="kw">then </span>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_4.html#T55" target="_self" title="GROUP_4:th.55">Th55</a></span>;<br/>
<a NAME="E4:66_1"/><span class="kw">then </span>
 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/>
<span class="kw">hence </span><a NAME="E5:66_1"/>
not <font color="Maroon" title="c2">a</font> is  <a href="group_4.html#V1" title="GROUP_4:attr.1">generating</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:66"/>
not <font color="Maroon" title="c2">a</font> is  <a href="group_4.html#V1" title="GROUP_4:attr.1">generating</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<a NAME="E3:66"/><span class="kw">then </span>
<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="E4:66"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_4.html#T55" target="_self" title="GROUP_4:th.55">Th55</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:66"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</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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