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<b>let </b><font color="Maroon">HX</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>, <font color="Maroon">GX</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:11_1_2"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">Z</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> holds <br/> ( <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">HX</font> iff  ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">Z</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> ) )
 <b>and </b><br/><a NAME="E2:11_1_2"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> holds <br/> ( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font> iff  ex <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">X</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> ) )
 ;<br/>

<a NAME="E3:11_1_2"/>
<font color="Maroon">HX</font> <a href="hidden.html#R1">=</a> <font color="Maroon">GX</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:11_1_2_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">HX</font>
 ;<br/><b>then reconsider </b><font color="Maroon">Z'</font> = <font color="Maroon">Z</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/><a NAME="E3:11_1_2_1_1"/>
 ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">Z'</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:11_1_2_1_1"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E2:11_1_2_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">HX</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">GX</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_2_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:11_1_2_1_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GX</font>
 ;<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">T</font> ;<br/><a NAME="E3:11_1_2_1_2"/>
 ex <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">X'</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:11_1_2_1_2"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">HX</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E4:11_1_2_1"/><b>then </b>
<font color="Maroon">GX</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">HX</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:11_1_2_1"/>
<font color="Maroon">HX</font> <a href="hidden.html#R1">=</a> <font color="Maroon">GX</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:11_1_2"/>
<font color="Maroon">HX</font> <a href="hidden.html#R1">=</a> <font color="Maroon">GX</font>
 ;<br/>


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