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<b>let </b><font color="Maroon">G</font> be   <a href="group_1.html#NM1">Group</a>;<br/>

<b>thus </b><a NAME="E1:52"/>
( <font color="Maroon">G</font> is  <a href="group_1.html#V7">commutative</a> <a href="group_1.html#NM1">Group</a> implies for <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font>  st <font color="Olive">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/><font color="Olive">A</font> <a href="group_3.html#K3" target="_self">|^</a> <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:52_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">G</font> is  <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>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font>;<br/>



<b>assume </b><a NAME="E2:52_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>thus </b><a NAME="E3:52_1"/>
<font color="Maroon">A</font> <a href="group_3.html#K3" target="_self">|^</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
  <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="52_1_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="E1:52_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="group_3.html#K3" target="_self">|^</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E2:52_1_1"/><b>then </b>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Olive">b</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 
;<br/>
<b>hence </b><a NAME="E3:52_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="group_3.html#E2"><i><font color="Green">Lemma41</font></i></a>, </i>;<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="E4:52_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> ;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="group_3.html#T4" target="_self">Th4</a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E8:52_1"/>
<font color="Maroon">a</font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="group_3.html#E2"><i><font color="Green">Lemma41</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E9:52_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="group_3.html#K3" target="_self">|^</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="group_3.html#T4" target="_self">Th4</a>, <a class="ref" href="group_3.html#T5" target="_self">Th5</a></i>;<br/>


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

<b>assume </b><a NAME="E2:52"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font>  st <font color="Olive">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/><font color="Olive">A</font> <a href="group_3.html#K3" target="_self">|^</a> <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><b>now </b></a><div class="add"><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/><span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> = 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="group_3.html#K3" target="_self">|^</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="txt" href="group_3.html#E2"><i><font color="Green">Lemma41</font></i></a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">b</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i/>
;<br/><b>hence </b><a NAME="E2:52_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T6">ZFMISC_1:6</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:52"/>
<font color="Maroon">G</font> is  <a href="group_1.html#V7">commutative</a> <a href="group_1.html#NM1">Group</a>
 <b>by </b><i/>;<br/>


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