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<b>let </b><font color="Maroon">G</font> be   <a href="group_1.html#NM1">Group</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>;<br/>
<a NAME="E2:133"/>
( <font color="Maroon">a</font> <a href="group_2.html#K12" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K6" target="_self">(Omega).</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> &amp; <span class="p1">(<span class="default"><a href="group_2.html#K6" target="_self">(Omega).</a> <font color="Maroon">G</font></span>)</span> <a href="group_2.html#K13" target="_self">*</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> )
 
<b>by </b><i><a class="ref" href="group_2.html#T43" target="_self">Th43</a></i>;<br/>
<a NAME="E3:133"/><b>then </b>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K14" target="_self">Left_Cosets</a> <span class="p1">(<span class="default"><a href="group_2.html#K6" target="_self">(Omega).</a> <font color="Maroon">G</font></span>)</span> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K15" target="_self">Right_Cosets</a> <span class="p1">(<span class="default"><a href="group_2.html#K6" target="_self">(Omega).</a> <font color="Maroon">G</font></span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:133"/><b>then </b><i><font color="Green">E35</font></i>: 
( <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K14" target="_self">Left_Cosets</a> <span class="p1">(<span class="default"><a href="group_2.html#K6" target="_self">(Omega).</a> <font color="Maroon">G</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K15" target="_self">Right_Cosets</a> <span class="p1">(<span class="default"><a href="group_2.html#K6" target="_self">(Omega).</a> <font color="Maroon">G</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E5:133"/><i><font color="Green">E36</font></i>: 
 <a href="group_2.html#K14" target="_self">Left_Cosets</a> <span class="p1">(<span class="default"><a href="group_2.html#K6" target="_self">(Omega).</a> <font color="Maroon">G</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font></span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_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="E1:133_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K14" target="_self">Left_Cosets</a> <span class="p1">(<span class="default"><a href="group_2.html#K6" target="_self">(Omega).</a> <font color="Maroon">G</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font> ;<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="E4:133_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_2.html#K12" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K6" target="_self">(Omega).</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i>, </i>;<br/>
<a NAME="E5:133_1"/>
<font color="Maroon">a</font> <a href="group_2.html#K12" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K6" target="_self">(Omega).</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T43" target="_self">Th43</a></i>;<br/>
<b>hence </b><a NAME="E6:133_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:133"/>
 <a href="group_2.html#K15" target="_self">Right_Cosets</a> <span class="p1">(<span class="default"><a href="group_2.html#K6" target="_self">(Omega).</a> <font color="Maroon">G</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font></span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_2"><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="E1:133_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K15" target="_self">Right_Cosets</a> <span class="p1">(<span class="default"><a href="group_2.html#K6" target="_self">(Omega).</a> <font color="Maroon">G</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font> ;<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="E4:133_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K6" target="_self">(Omega).</a> <font color="Maroon">G</font></span>)</span> <a href="group_2.html#K13" target="_self">*</a> <font color="Maroon">a</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E5:133_2"/>
<span class="p1">(<span class="default"><a href="group_2.html#K6" target="_self">(Omega).</a> <font color="Maroon">G</font></span>)</span> <a href="group_2.html#K13" target="_self">*</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T43" target="_self">Th43</a></i>;<br/>
<b>hence </b><a NAME="E6:133_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:133"/>
(  <a href="group_2.html#K14" target="_self">Left_Cosets</a> <span class="p1">(<span class="default"><a href="group_2.html#K6" target="_self">(Omega).</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="group_2.html#K15" target="_self">Right_Cosets</a> <span class="p1">(<span class="default"><a href="group_2.html#K6" target="_self">(Omega).</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
