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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">CQC-WFF</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:14"/><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_the1.html#K1" title="CQC_THE1:func.1">Cn</a> <font color="Maroon" title="c1">X</font>
 <b>and </b><br/><a NAME="E2:14"/><i><font color="Green" title="E14">A2</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">=&gt;</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_the1.html#K1" title="CQC_THE1:func.1">Cn</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<a NAME="E3:14"/>
for <font color="Olive" title="b1">T</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">CQC-WFF</a>   st <font color="Olive" title="b1">T</font> <a href="cqc_the1.html#NR1" title="CQC_THE1:NR.1">is_a_theory</a>  &amp; <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">T</font> holds <br/><font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">T</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">CQC-WFF</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:14_1"/><i><font color="Green" title="E15">A3</font></i>: 
<font color="Maroon" title="c4">T</font> <a href="cqc_the1.html#NR1" title="CQC_THE1:NR.1">is_a_theory</a> 
 <b>and </b><br/><a NAME="E2:14_1"/><i><font color="Green" title="E16">A4</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">T</font>
 ;<br/>

<a NAME="E3:14_1"/>
( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">T</font> &amp; <font color="Maroon" title="c2">p</font> <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">=&gt;</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">T</font> )
 
<b>by </b><i><a class="txt" href="cqc_the1.html#E1:14"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="cqc_the1.html#E2:14"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="cqc_the1.html#E1:14_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="cqc_the1.html#E2:14_1"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="cqc_the1.html#D2" target="_self" title="CQC_THE1:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E4:14_1"/>
<font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">T</font>
 <b>by </b><i><a class="txt" href="cqc_the1.html#E1:14_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="cqc_the1.html#D1" target="_self" title="CQC_THE1:def.1">Def1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:14"/>
<font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_the1.html#K1" title="CQC_THE1:func.1">Cn</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="cqc_the1.html#D2" target="_self" title="CQC_THE1:def.2">Def2</a></i>;<br/>


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