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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">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="subset_1.html#M1">Element</a> of  <a href="valuat_1.html#K2">Valuations_in</a> <font color="Maroon">A</font>;<br/>









<a NAME="E1:10"/><i><font color="Green">E54</font></i>: 
( not <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="valuat_1.html#R1">|=</a>  <a href="cqc_lang.html#K16">Ex</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span> iff for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>  holds not <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="sublemma.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <font color="Olive">a</font></span>)</span> <a href="valuat_1.html#R1">|=</a>  <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E2:10"/>
( ( for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>  holds not <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="sublemma.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <font color="Olive">a</font></span>)</span> <a href="valuat_1.html#R1">|=</a>  <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font> ) iff 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>,<font color="Maroon">v</font> <a href="sublemma.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <font color="Olive">a</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:10_1"/>
( ( for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>  holds not <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="sublemma.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <font color="Olive">a</font></span>)</span> <a href="valuat_1.html#R1">|=</a>  <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</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>,<font color="Maroon">v</font> <a href="sublemma.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <font color="Olive">a</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_1_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 not <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="sublemma.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <font color="Olive">a</font></span>)</span> <a href="valuat_1.html#R1">|=</a>  <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</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="E2:10_1_1"/>
not <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="sublemma.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <font color="Maroon">a</font></span>)</span> <a href="valuat_1.html#R1">|=</a>  <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:10_1_1"/>
<font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="sublemma.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <font color="Maroon">a</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="valuat_1.html#T28">VALUAT_1:28</a></i>;<br/>


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

<b>thus </b><a NAME="E2:10_1"/>
( ( 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>,<font color="Maroon">v</font> <a href="sublemma.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <font color="Olive">a</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> ) implies for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>  holds not <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="sublemma.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <font color="Olive">a</font></span>)</span> <a href="valuat_1.html#R1">|=</a>  <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_1_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>,<font color="Maroon">v</font> <a href="sublemma.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <font color="Olive">a</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</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="E2:10_1_2"/>
<font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="sublemma.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <font color="Maroon">a</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:10_1_2"/>
not <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="sublemma.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <font color="Maroon">a</font></span>)</span> <a href="valuat_1.html#R1">|=</a>  <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="valuat_1.html#T28">VALUAT_1:28</a></i>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:10"/>
( <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="valuat_1.html#R1">|=</a>  <a href="cqc_lang.html#K10">'not'</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K16">Ex</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span>)</span> iff <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="valuat_1.html#R1">|=</a>  <a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font> )
 <b>by </b><i>, <a class="ref" href="sublemma.html#T51">SUBLEMMA:51</a>, <a class="ref" href="valuat_1.html#T28">VALUAT_1:28</a></i>;<br/>


</div>
