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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">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" target="_self">interpretation</a> of <font color="Maroon">A</font>;<br/>





<b>set </b><font color="Maroon">D</font> =  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font></span>)</span>,<a href="margrel1.html#K6">BOOLEAN</a> ;<br/>
<b>set </b><font color="Maroon">V</font> = <span class="p1">(<span class="default"><a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font></span>)</span> <a href="margrel1.html#K1">--&gt;</a> <a href="margrel1.html#K8">TRUE</a> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,  <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Maroon">a<sub>1</sub></font>,  <a href="cqc_lang.html#NM2">CQC-variable_list</a> of <font color="Maroon">a<sub>1</sub></font>) <b>-&gt; </b>   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font></span>)</span>,<a href="margrel1.html#K6">BOOLEAN</a>  = <font color="Maroon">a<sub>3</sub></font> <a href="valuat_1.html#K9" target="_self">'in'</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="valuat_1.html#K11" target="_self">.</a> <font color="Maroon">a<sub>2</sub></font></span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font></span>)</span>,<a href="margrel1.html#K6">BOOLEAN</a> ) <b>-&gt; </b>   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font></span>)</span>,<a href="margrel1.html#K6">BOOLEAN</a>  =  <a href="valuat_1.html#K5" target="_self">'not'</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font></span>)</span>,<a href="margrel1.html#K6">BOOLEAN</a> ,   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font></span>)</span>,<a href="margrel1.html#K6">BOOLEAN</a> ) <b>-&gt; </b>   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font></span>)</span>,<a href="margrel1.html#K6">BOOLEAN</a>  = <font color="Maroon">a<sub>1</sub></font> <a href="valuat_1.html#K6" target="_self">'&amp;'</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(  <a href="qc_lang1.html#NM3">bound_QC-variable</a>,   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font></span>)</span>,<a href="margrel1.html#K6">BOOLEAN</a> ) <b>-&gt; </b>   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font></span>)</span>,<a href="margrel1.html#K6">BOOLEAN</a>  =  <a href="valuat_1.html#K7" target="_self">FOR_ALL</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>5</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ) <b>-&gt; </b>   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font></span>)</span>,<a href="margrel1.html#K6">BOOLEAN</a>  =  <a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">J</font>;<br/>
<a NAME="E1:21"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">p</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> <br/> for <font color="Olive">d</font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font></span>)</span>,<a href="margrel1.html#K6">BOOLEAN</a>  holds <br/> ( <font color="Olive">d</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">p</font>) iff  ex <font color="Olive">F</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> , <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font></span>)</span>,<a href="margrel1.html#K6">BOOLEAN</a>  st <br/>( <font color="Olive">d</font> <a href="funct_2.html#R2">=</a> <font color="Olive">F</font> <a href="valuat_1.html#K10" target="_self">.</a> <font color="Olive">p</font> &amp; <font color="Olive">F</font> <a href="valuat_1.html#K10" target="_self">.</a> <a href="cqc_lang.html#K9">VERUM</a>  <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font></span>)</span> <a href="margrel1.html#K1">--&gt;</a> <a href="margrel1.html#K8">TRUE</a>  &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> <br/> for <font color="Olive">x</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a><br/> for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">ll</font> being  <a href="cqc_lang.html#NM2">CQC-variable_list</a> of <font color="Olive">k</font><br/> for <font color="Olive">P</font> being  <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Olive">k</font> holds <br/> ( <font color="Olive">F</font> <a href="valuat_1.html#K10" target="_self">.</a> <span class="p1">(<span class="default"><font color="Olive">P</font> <a href="cqc_lang.html#K8">!</a> <font color="Olive">ll</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>,<font color="Olive">P</font>,<font color="Olive">ll</font>) &amp; <font color="Olive">F</font> <a href="valuat_1.html#K10" target="_self">.</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Olive">p</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">F</font> <a href="valuat_1.html#K10" target="_self">.</a> <font color="Olive">p</font>) &amp; <font color="Olive">F</font> <a href="valuat_1.html#K10" target="_self">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Olive">q</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">F</font> <a href="valuat_1.html#K10" target="_self">.</a> <font color="Olive">p</font>,<font color="Olive">F</font> <a href="valuat_1.html#K10" target="_self">.</a> <font color="Olive">q</font>) &amp; <font color="Olive">F</font> <a href="valuat_1.html#K10" target="_self">.</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Olive">x</font>,<font color="Olive">p</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">x</font>,<font color="Olive">F</font> <a href="valuat_1.html#K10" target="_self">.</a> <font color="Olive">p</font>) ) ) ) )
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E2:21"/>
<font color="Maroon">H<sub>5</sub></font>( <a href="cqc_lang.html#K9">VERUM</a> ) <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font></span>)</span> <a href="margrel1.html#K1">--&gt;</a> <a href="margrel1.html#K8">TRUE</a> 
 <b>from </b><i><a class="ref" href="cqc_lang.html#S5">CQC_LANG:sch 5</a>();</i><br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">ll</font> be   <a href="cqc_lang.html#NM2">CQC-variable_list</a> of <font color="Maroon">k</font>;<br/><b>let </b><font color="Maroon">P</font> be   <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Maroon">k</font>;<br/><b>thus </b><a NAME="E1:21_1"/>
<font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">P</font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">ll</font>) <a href="funct_2.html#R2">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">k</font>,<font color="Maroon">P</font>,<font color="Maroon">ll</font>)
 <b>from </b><i><a class="ref" href="cqc_lang.html#S6">CQC_LANG:sch 6</a>();</i><br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><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>thus </b><a NAME="E1:21_2"/>
<font color="Maroon">H<sub>5</sub></font>( <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font>) <a href="funct_2.html#R2">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">p</font>))
 <b>from </b><i><a class="ref" href="cqc_lang.html#S7">CQC_LANG:sch 7</a>();</i><br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><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>thus </b><a NAME="E1:21_3"/>
<font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">p</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">q</font>) <a href="funct_2.html#R2">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">p</font>),<font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">q</font>))
 <b>from </b><i><a class="ref" href="cqc_lang.html#S8">CQC_LANG:sch 8</a>();</i><br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/>

<b>thus </b><a NAME="E6:21"/>
<font color="Maroon">H<sub>5</sub></font>( <a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font>) <a href="funct_2.html#R2">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">x</font>,<font color="Maroon">H<sub>5</sub></font>(<font color="Maroon">p</font>))
 <b>from </b><i><a class="ref" href="cqc_lang.html#S9">CQC_LANG:sch 9</a>();</i><br/>


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