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<b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon">q</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">J</font> be    <a href="valuat_1.html#M1">interpretation</a> of <font color="Maroon">A</font>;<br/>







<b>assume </b><a NAME="E1:79"/><i><font color="Green">E55</font></i>: 
( ( for <font color="Olive">v</font>, <font color="Olive">w</font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2">Valuations_in</a> <font color="Maroon">A</font>  st <font color="Olive">v</font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">w</font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font></span>)</span> holds <br/>( <font color="Maroon">J</font>,<font color="Olive">v</font> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> iff <font color="Maroon">J</font>,<font color="Olive">w</font> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> ) ) &amp; ( for <font color="Olive">v</font>, <font color="Olive">w</font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2">Valuations_in</a> <font color="Maroon">A</font>  st <font color="Olive">v</font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">w</font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">q</font></span>)</span> holds <br/>( <font color="Maroon">J</font>,<font color="Olive">v</font> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">q</font> iff <font color="Maroon">J</font>,<font color="Olive">w</font> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">q</font> ) ) )
 ;<br/>

<b>let </b><font color="Maroon">v</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2">Valuations_in</a> <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">w</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2">Valuations_in</a> <font color="Maroon">A</font>;<br/>



<b>assume </b><a NAME="E2:79"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">v</font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">q</font></span>)</span></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">X</font> = <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">q</font></span>)</span>;<br/>
<a NAME="E3:79"/><i><font color="Green">E57</font></i>: 
 <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="qc_lang3.html#T14">QC_LANG3:14</a></i>;<br/>
<a NAME="E4:79"/><i><font color="Green">E58</font></i>: 
 <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E5:79"/>
 <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E6:79"/><b>then </b><i><font color="Green">E59</font></i>: 
( <font color="Maroon">v</font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font></span>)</span> &amp; <font color="Maroon">v</font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">q</font></span>)</span> )
 
<b>by </b><i>, , , <a class="ref" href="relat_1.html#T188">RELAT_1:188</a></i>;<br/>
<a NAME="E7:79"/>
( <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">q</font> iff ( <font color="Maroon">J</font>,<font color="Maroon">w</font> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">J</font>,<font color="Maroon">w</font> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">q</font> ) )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E8:79"/>
( <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">q</font> iff <font color="Maroon">J</font>,<font color="Maroon">w</font> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="ref" href="valuat_1.html#T29">VALUAT_1:29</a></i>;<br/>


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