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<div class="add">

<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> : <font color="Maroon">A</font> <a href="group_3.html#K4" target="_self">|^</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> </span>}</span> ;<br/>
<a NAME="E1:139_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> : <font color="Maroon">A</font> <a href="group_3.html#K4" target="_self">|^</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="139_1_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="E1:139_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> : <font color="Maroon">A</font> <a href="group_3.html#K4" target="_self">|^</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> </span>}</span> 
 ;<br/>

<a NAME="E2:139_1_1_1"/><b>then </b>
 ex <font color="Olive">h</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">h</font> &amp; <font color="Maroon">A</font> <a href="group_3.html#K4" target="_self">|^</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 
;<br/>
<b>hence </b><a NAME="E3:139_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> : <font color="Maroon">A</font> <a href="group_3.html#K4" target="_self">|^</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font> ;<br/>
<a NAME="E3:139_1_1"/>
<font color="Maroon">A</font> <a href="group_3.html#K4" target="_self">|^</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="group_3.html#E4:139_1_1"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E4:139_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
;<br/>
<div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="139_1_1_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/><b>assume </b><b>that </b><br/><a NAME="E1:139_1_1_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E2:139_1_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E3:139_1_1_2"/><i><font color="Green">E50</font></i>: 
 ex <font color="Olive">g</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> &amp; <font color="Maroon">A</font> <a href="group_3.html#K4" target="_self">|^</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="group_3.html#T65" target="_self">Th65</a></i>;<br/><a NAME="E4:139_1_1_2"/>
 ex <font color="Olive">h</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> st <br/>( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> &amp; <font color="Maroon">A</font> <a href="group_3.html#K4" target="_self">|^</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="group_3.html#D6" target="_self">Def6</a></i>;<br/><a NAME="E5:139_1_1_2"/><b>then </b>
<font color="Maroon">A</font> <a href="group_3.html#K4" 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> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="group_3.html#T70" target="_self">Th70</a>, </i>;<br/><b>hence </b><a NAME="E6:139_1_1_2"/>
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="139_1_1_3"><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>assume </b><a NAME="E1:139_1_1_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:139_1_1_3"/><b>then </b>
 ex <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">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; <font color="Maroon">A</font> <a href="group_3.html#K4" target="_self">|^</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 ;<br/><a NAME="E3:139_1_1_3"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="group_3.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="txt" href="group_3.html#E5:139_1_1"><i><font color="Green">E39</font></i></a></i>;<br/><b>hence </b><a NAME="E4:139_1_1_3"/>
<font color="Maroon">a</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:139_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> : <font color="Maroon">A</font> <a href="group_3.html#K4" target="_self">|^</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="group_3.html#T61" target="_self">Th61</a>, <a class="ref" href="group_3.html#T63" target="_self">Th63</a>, <a class="ref" href="group_2.html#T61">GROUP_2:61</a></i>;<br/>


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