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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:23"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="cqc_the1.html#K1" target="_self">Cn</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:23"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="cqc_the1.html#K1" target="_self">Cn</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E3:23"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <a href="cqc_lang.html#K7">CQC-WFF</a>  st <font color="Olive">b<sub>1</sub></font> <a href="cqc_the1.html#NR1" target="_self">is_a_theory</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></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="E1:23_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="cqc_the1.html#NR1" target="_self">is_a_theory</a> 
 <b>and </b><br/><a NAME="E2:23_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E3:23_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="cqc_the1.html#E1:23"><i><font color="Green">E13</font></i></a>, <a class="txt" href="cqc_the1.html#E2:23"><i><font color="Green">E14</font></i></a>, <a class="txt" href="cqc_the1.html#E1:23_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="cqc_the1.html#E2:23_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="cqc_the1.html#D2" target="_self">Def2</a></i>;<br/>
<b>hence </b><a NAME="E4:23_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="cqc_the1.html#E1:23_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="cqc_the1.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:23"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="cqc_the1.html#K1" target="_self">Cn</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="cqc_the1.html#D2" target="_self">Def2</a></i>;<br/>


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