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<b>let </b><font color="Maroon">G</font> be  <a href="group_1.html#V7">commutative</a> <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font>, <font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font>;<br/>



<b>thus </b><a NAME="E1:26"/>
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="group_2.html#K2" target="_self">*</a> <font color="Maroon">B</font></span>)</span> <a href="group_2.html#K1" target="_self">"</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="group_2.html#K1" target="_self">"</a> </span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_2.html#K1" target="_self">"</a> </span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<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="E1:26_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="group_2.html#K2" target="_self">*</a> <font color="Maroon">B</font></span>)</span> <a href="group_2.html#K1" target="_self">"</a> 
 ;<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="E3:26_1"/><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#K3">"</a>  &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="group_2.html#K2" target="_self">*</a> <font color="Maroon">B</font> )
 ;<br/>
<b>consider </b><font color="Maroon">g1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <font color="Maroon">g2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E5:26_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">g2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 <b>by </b><i/>;<br/>
<a NAME="E6:26_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="group_1.html#K3">"</a> </span>)</span> &amp; <font color="Maroon">g1</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="group_2.html#K1" target="_self">"</a>  &amp; <font color="Maroon">g2</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="group_2.html#K1" target="_self">"</a>  )
 
<b>by </b><i>, , <a class="ref" href="group_1.html#T94">GROUP_1:94</a></i>;<br/>
<b>hence </b><a NAME="E7:26_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="group_2.html#K1" target="_self">"</a> </span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_2.html#K1" target="_self">"</a> </span>)</span>
 ;<br/>


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

<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="E2:26"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="group_2.html#K1" target="_self">"</a> </span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="group_2.html#K1" target="_self">"</a> </span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">g1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <font color="Maroon">g2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E4:26"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="group_2.html#K1" target="_self">"</a>  &amp; <font color="Maroon">g2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="group_2.html#K1" target="_self">"</a>  )
 ;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E6:26"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K3">"</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E8:26"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="group_1.html#K3">"</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 <b>by </b><i/>;<br/>
<a NAME="E9:26"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K3">"</a>  &amp; <font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="group_2.html#K2" target="_self">*</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, , , <a class="ref" href="group_1.html#T94">GROUP_1:94</a></i>;<br/>
<b>hence </b><a NAME="E10:26"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="group_2.html#K2" target="_self">*</a> <font color="Maroon">B</font></span>)</span> <a href="group_2.html#K1" target="_self">"</a> 
 ;<br/>


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