<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><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>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="valuat_1.html#M1" target="_self">interpretation</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>







<i><font color="Green">E20</font></i>: <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="valuat_1.html#T22" target="_self">Th22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K11">'not'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="valuat_1.html#T20" target="_self">Th20</a></i>
;<br/>
<div><i><font color="Green">E21</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_2"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/><a NAME="E2:28_2"/><b>then </b>
 <a href="margrel1.html#K11">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="ref" href="margrel1.html#T41">MARGREL1:41</a></i>;<br/><b>hence </b><a NAME="E3:28_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K11">'not'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="ref" href="margrel1.html#T45">MARGREL1:45</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:28"/>
( <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K11">'not'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 
<b>by </b><i><a class="ref" href="margrel1.html#T45">MARGREL1:45</a></i>;<br/>
<b>hence </b><a NAME="E4:28"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="valuat_1.html#E2:28"><i><font color="Green">E21</font></i></a>, <a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>;<br/>


</div>
