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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>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>









<b>assume </b><a NAME="E1:112"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="group_2.html#K13" target="_self">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_2.html#K4" target="_self">*</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:112"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <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> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_2.html#T38" target="_self">Th38</a></i>;<br/>
<b>then consider </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E4:112"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span>
 <b>and </b><br/><a NAME="E5:112"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">g</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="group_2.html#T29" target="_self">Th29</a></i>;<br/>
<a NAME="E6:112"/>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:112"/><b>then </b><i><font color="Green">E47</font></i>: 
( <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="group_2.html#K13" target="_self">*</a> <font color="Maroon">b</font> &amp; <font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="group_2.html#K13" target="_self">*</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, <a class="ref" href="group_2.html#T29" target="_self">Th29</a></i>;<br/>
<font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="group_2.html#T52" target="_self">Th52</a>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E9:112"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="group_2.html#K13" target="_self">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <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>
 <b>by </b><i/>;<br/>


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