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<div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</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>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">S</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="substut1.html#K38">CQC-Sub-WFF</a> ;<br/><b>let </b><font color="Maroon">xSQ</font> be    <a href="substut1.html#M1">second_Q_comp</a> of <span class="p1"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>;<br/>











<b>thus </b><a NAME="E1:90"/>
( ( for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">J</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <font color="Maroon">v</font>,<font color="Maroon">S</font>,<font color="Maroon">x</font>,<font color="Maroon">xSQ</font></span>)</span></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">a</font></span>)</span> <a href="sublemma.html#R1" target="_self">|=</a> <font color="Maroon">S</font> ) implies for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">J</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <font color="Maroon">v</font>,<font color="Maroon">S</font>,<font color="Maroon">x</font>,<font color="Maroon">xSQ</font></span>)</span></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">a</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:90_1"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">J</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <font color="Maroon">v</font>,<font color="Maroon">S</font>,<font color="Maroon">x</font>,<font color="Maroon">xSQ</font></span>)</span></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">a</font></span>)</span> <a href="sublemma.html#R1" target="_self">|=</a> <font color="Maroon">S</font>
 ;<br/>

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>;<br/>

<a NAME="E2:90_1"/>
( <font color="Maroon">J</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <font color="Maroon">v</font>,<font color="Maroon">S</font>,<font color="Maroon">x</font>,<font color="Maroon">xSQ</font></span>)</span></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sublemma.html#R1" target="_self">|=</a> <font color="Maroon">S</font> iff <font color="Maroon">J</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <font color="Maroon">v</font>,<font color="Maroon">S</font>,<font color="Maroon">x</font>,<font color="Maroon">xSQ</font></span>)</span></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a>  )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:90_1"/>
<font color="Maroon">J</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <font color="Maroon">v</font>,<font color="Maroon">S</font>,<font color="Maroon">x</font>,<font color="Maroon">xSQ</font></span>)</span></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> 
 <b>by </b><i><a class="ref" href="sublemma.html#T3" target="_self">Th3</a></i>;<br/>


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

<b>thus </b><a NAME="E2:90"/>
( ( for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">J</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <font color="Maroon">v</font>,<font color="Maroon">S</font>,<font color="Maroon">x</font>,<font color="Maroon">xSQ</font></span>)</span></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">a</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a>  ) implies for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">J</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <font color="Maroon">v</font>,<font color="Maroon">S</font>,<font color="Maroon">x</font>,<font color="Maroon">xSQ</font></span>)</span></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">a</font></span>)</span> <a href="sublemma.html#R1" target="_self">|=</a> <font color="Maroon">S</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:90_2"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">J</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <font color="Maroon">v</font>,<font color="Maroon">S</font>,<font color="Maroon">x</font>,<font color="Maroon">xSQ</font></span>)</span></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">a</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> 
 ;<br/>

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>;<br/>

<a NAME="E2:90_2"/>
( <font color="Maroon">J</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <font color="Maroon">v</font>,<font color="Maroon">S</font>,<font color="Maroon">x</font>,<font color="Maroon">xSQ</font></span>)</span></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sublemma.html#R1" target="_self">|=</a> <font color="Maroon">S</font> iff <font color="Maroon">J</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <font color="Maroon">v</font>,<font color="Maroon">S</font>,<font color="Maroon">x</font>,<font color="Maroon">xSQ</font></span>)</span></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a>  )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:90_2"/>
<font color="Maroon">J</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <font color="Maroon">v</font>,<font color="Maroon">S</font>,<font color="Maroon">x</font>,<font color="Maroon">xSQ</font></span>)</span></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sublemma.html#R1" target="_self">|=</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="sublemma.html#T3" target="_self">Th3</a></i>;<br/>


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


</div>
