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<b>let </b><font color="Maroon">G</font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">H</font> be    <a href="group_2.html#M1" target="_self">Subgroup</a> of <font color="Maroon">G</font>;<br/>







<b>thus </b><a NAME="E1:110"/>
( <font color="Maroon">H</font> <a href="group_2.html#K13" target="_self">*</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="group_2.html#K13" target="_self">*</a> <font color="Maroon">b</font> implies <font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:110_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">H</font> <a href="group_2.html#K13" target="_self">*</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="group_2.html#K13" target="_self">*</a> <font color="Maroon">b</font>
 ;<br/>

 <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H</font> = 
<font color="Maroon">H</font> <a href="group_2.html#K13" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">H</font> <a href="group_2.html#K13" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="group_2.html#K13" target="_self">*</a> <font color="Maroon">b</font></span>)</span> <a href="group_2.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T51" target="_self">Th51</a>, <a class="ref" href="group_2.html#T38" target="_self">Th38</a></i>
<br/>.= 
<font color="Maroon">H</font> <a href="group_2.html#K13" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T38" target="_self">Th38</a></i>
;<br/>
<b>hence </b><a NAME="E3:110_1"/>
<font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="group_2.html#T49" target="_self">Th49</a></i>;<br/>


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

<b>assume </b><a NAME="E2:110"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font>
 ;<br/>

<b>thus </b><font color="Maroon">H</font> <a href="group_2.html#K13" target="_self">*</a> <font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="group_2.html#K13" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span></span>)</span> <a href="group_2.html#K4" target="_self">*</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="group_2.html#T51" target="_self">Th51</a>, <a class="ref" href="group_2.html#T49" target="_self">Th49</a></i>
<br/>.= 
<font color="Maroon">H</font> <a href="group_2.html#K13" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T38" target="_self">Th38</a></i>
<br/>.= 
<font color="Maroon">H</font> <a href="group_2.html#K13" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<font color="Maroon">H</font> <a href="group_2.html#K13" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>
<br/>.= 
<font color="Maroon">H</font> <a href="group_2.html#K13" target="_self">*</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
;<br/>


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