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<b>let </b><font color="Maroon">Y</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="cqc_lang.html#K7">CQC-WFF</a> , <font color="Maroon">Z</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:8_1_2"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">t</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  holds <br/> ( <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> iff for <font color="Olive">T</font> being  <a href="subset_1.html#NM2">Subset</a> of <a href="cqc_lang.html#K7">CQC-WFF</a>   st <font color="Olive">T</font> <a href="cqc_the1.html#NR1" target="_self">is_a_theory</a>  &amp; <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">T</font> holds <br/><font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Olive">T</font> )
 <b>and </b><br/><a NAME="E2:8_1_2"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">t</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  holds <br/> ( <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> iff for <font color="Olive">T</font> being  <a href="subset_1.html#NM2">Subset</a> of <a href="cqc_lang.html#K7">CQC-WFF</a>   st <font color="Olive">T</font> <a href="cqc_the1.html#NR1" target="_self">is_a_theory</a>  &amp; <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">T</font> holds <br/><font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Olive">T</font> )
 ;<br/>

<a NAME="E3:8_1_2"/>
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> iff <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E1:8_1_2_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> implies <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_1_2_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">t</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  ;<br/>
<a NAME="E3:8_1_2_1_1"/>
for <font color="Olive">T</font> being  <a href="subset_1.html#NM2">Subset</a> of <a href="cqc_lang.html#K7">CQC-WFF</a>   st <font color="Olive">T</font> <a href="cqc_the1.html#NR1" target="_self">is_a_theory</a>  &amp; <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">T</font> holds <br/><font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Olive">T</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:8_1_2_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E2:8_1_2_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> implies <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_1_2_1_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">t</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  ;<br/>
<a NAME="E3:8_1_2_1_2"/>
for <font color="Olive">T</font> being  <a href="subset_1.html#NM2">Subset</a> of <a href="cqc_lang.html#K7">CQC-WFF</a>   st <font color="Olive">T</font> <a href="cqc_the1.html#NR1" target="_self">is_a_theory</a>  &amp; <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">T</font> holds <br/><font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Olive">T</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:8_1_2_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:8_1_2"/>
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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