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<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,  <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of $1,  <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of $1) <span class="kw">-&gt; </span>   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font></span>)</span>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>) = $3 <a href="valuat_1.html#K5" title="VALUAT_1:func.5">'in'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">J</font> <a href="valuat_1.html#K7" title="VALUAT_1:func.7">.</a> $2</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">V</font> = <span class="p1">(<span class="default"><a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">D</font> =  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font></span>)</span>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>);<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font></span>)</span>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>)) <span class="kw">-&gt; </span>   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font></span>)</span>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>) =  <a href="margrel1.html#K15" title="MARGREL1:func.15">'not'</a> $1;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>3</sub></font>(   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font></span>)</span>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>),   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font></span>)</span>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>)) <span class="kw">-&gt; </span>   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font></span>)</span>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>) = $1 <a href="margrel1.html#K16" title="MARGREL1:func.16">'&amp;'</a> $2;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>4</sub></font>(  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>,   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font></span>)</span>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>)) <span class="kw">-&gt; </span>   <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font></span>)</span>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>) =  <a href="valuat_1.html#K3" title="VALUAT_1:func.3">FOR_ALL</a> ($1,$2);<br/>
<span class="kw">thus </span><a NAME="E1:14_1_1"/>
(  ex <font color="Olive" title="b1">d</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font></span>)</span>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>) ex <font color="Olive" title="b2">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a>,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p2">(<span class="default"><a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font></span>)</span>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>)</span>)</span> st <br/>( <font color="Olive" title="b1">d</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">F</font> <a href="valuat_1.html#K6" title="VALUAT_1:func.6">.</a> <font color="Maroon" title="c3">p</font> &amp; <font color="Olive" title="b2">F</font> <a href="valuat_1.html#K6" title="VALUAT_1:func.6">.</a> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">VERUM</a> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> &amp; (  for <font color="Olive" title="b3">r</font>, <font color="Olive" title="b4">s</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> <br/>  for <font color="Olive" title="b5">x</font> being   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><br/>  for <font color="Olive" title="b6">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/>  for <font color="Olive" title="b7">l</font> being   <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of <font color="Olive" title="b6">k</font><br/>  for <font color="Olive" title="b8">P</font> being   <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Olive" title="b6">k</font> holds <br/> ( <font color="Olive" title="b2">F</font> <a href="valuat_1.html#K6" title="VALUAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b8">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Olive" title="b7">l</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b6">k</font>,<font color="Olive" title="b8">P</font>,<font color="Olive" title="b7">l</font>) &amp; <font color="Olive" title="b2">F</font> <a href="valuat_1.html#K6" title="VALUAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Olive" title="b3">r</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">F</font> <a href="valuat_1.html#K6" title="VALUAT_1:func.6">.</a> <font color="Olive" title="b3">r</font>) &amp; <font color="Olive" title="b2">F</font> <a href="valuat_1.html#K6" title="VALUAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">r</font> <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'&amp;'</a> <font color="Olive" title="b4">s</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b2">F</font> <a href="valuat_1.html#K6" title="VALUAT_1:func.6">.</a> <font color="Olive" title="b3">r</font>,<font color="Olive" title="b2">F</font> <a href="valuat_1.html#K6" title="VALUAT_1:func.6">.</a> <font color="Olive" title="b4">s</font>) &amp; <font color="Olive" title="b2">F</font> <a href="valuat_1.html#K6" title="VALUAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Olive" title="b5">x</font>,<font color="Olive" title="b3">r</font>)</span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b5">x</font>,<font color="Olive" title="b2">F</font> <a href="valuat_1.html#K6" title="VALUAT_1:func.6">.</a> <font color="Olive" title="b3">r</font>) ) ) ) &amp; (  for <font color="Olive" title="b1">d1</font>, <font color="Olive" title="b2">d2</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font></span>)</span>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>)  st  ex <font color="Olive" title="b3">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a>,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p2">(<span class="default"><a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font></span>)</span>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>)</span>)</span> st <br/>( <font color="Olive" title="b1">d1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b3">F</font> <a href="valuat_1.html#K6" title="VALUAT_1:func.6">.</a> <font color="Maroon" title="c3">p</font> &amp; <font color="Olive" title="b3">F</font> <a href="valuat_1.html#K6" title="VALUAT_1:func.6">.</a> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">VERUM</a> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> &amp; (  for <font color="Olive" title="b4">r</font>, <font color="Olive" title="b5">s</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> <br/>  for <font color="Olive" title="b6">x</font> being   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><br/>  for <font color="Olive" title="b7">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/>  for <font color="Olive" title="b8">l</font> being   <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of <font color="Olive" title="b7">k</font><br/>  for <font color="Olive" title="b9">P</font> being   <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Olive" title="b7">k</font> holds <br/> ( <font color="Olive" title="b3">F</font> <a href="valuat_1.html#K6" title="VALUAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b9">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Olive" title="b8">l</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b7">k</font>,<font color="Olive" title="b9">P</font>,<font color="Olive" title="b8">l</font>) &amp; <font color="Olive" title="b3">F</font> <a href="valuat_1.html#K6" title="VALUAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Olive" title="b4">r</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b3">F</font> <a href="valuat_1.html#K6" title="VALUAT_1:func.6">.</a> <font color="Olive" title="b4">r</font>) &amp; <font color="Olive" title="b3">F</font> <a href="valuat_1.html#K6" title="VALUAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">r</font> <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'&amp;'</a> <font color="Olive" title="b5">s</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b3">F</font> <a href="valuat_1.html#K6" title="VALUAT_1:func.6">.</a> <font color="Olive" title="b4">r</font>,<font color="Olive" title="b3">F</font> <a href="valuat_1.html#K6" title="VALUAT_1:func.6">.</a> <font color="Olive" title="b5">s</font>) &amp; <font color="Olive" title="b3">F</font> <a href="valuat_1.html#K6" title="VALUAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Olive" title="b6">x</font>,<font color="Olive" title="b4">r</font>)</span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b6">x</font>,<font color="Olive" title="b3">F</font> <a href="valuat_1.html#K6" title="VALUAT_1:func.6">.</a> <font color="Olive" title="b4">r</font>) ) ) ) &amp;  ex <font color="Olive" title="b3">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a>,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p2">(<span class="default"><a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font></span>)</span>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>)</span>)</span> st <br/>( <font color="Olive" title="b2">d2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b3">F</font> <a href="valuat_1.html#K6" title="VALUAT_1:func.6">.</a> <font color="Maroon" title="c3">p</font> &amp; <font color="Olive" title="b3">F</font> <a href="valuat_1.html#K6" title="VALUAT_1:func.6">.</a> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">VERUM</a> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> &amp; (  for <font color="Olive" title="b4">r</font>, <font color="Olive" title="b5">s</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> <br/>  for <font color="Olive" title="b6">x</font> being   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><br/>  for <font color="Olive" title="b7">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/>  for <font color="Olive" title="b8">l</font> being   <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of <font color="Olive" title="b7">k</font><br/>  for <font color="Olive" title="b9">P</font> being   <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Olive" title="b7">k</font> holds <br/> ( <font color="Olive" title="b3">F</font> <a href="valuat_1.html#K6" title="VALUAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b9">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Olive" title="b8">l</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b7">k</font>,<font color="Olive" title="b9">P</font>,<font color="Olive" title="b8">l</font>) &amp; <font color="Olive" title="b3">F</font> <a href="valuat_1.html#K6" title="VALUAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Olive" title="b4">r</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b3">F</font> <a href="valuat_1.html#K6" title="VALUAT_1:func.6">.</a> <font color="Olive" title="b4">r</font>) &amp; <font color="Olive" title="b3">F</font> <a href="valuat_1.html#K6" title="VALUAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">r</font> <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'&amp;'</a> <font color="Olive" title="b5">s</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b3">F</font> <a href="valuat_1.html#K6" title="VALUAT_1:func.6">.</a> <font color="Olive" title="b4">r</font>,<font color="Olive" title="b3">F</font> <a href="valuat_1.html#K6" title="VALUAT_1:func.6">.</a> <font color="Olive" title="b5">s</font>) &amp; <font color="Olive" title="b3">F</font> <a href="valuat_1.html#K6" title="VALUAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Olive" title="b6">x</font>,<font color="Olive" title="b4">r</font>)</span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b6">x</font>,<font color="Olive" title="b3">F</font> <a href="valuat_1.html#K6" title="VALUAT_1:func.6">.</a> <font color="Olive" title="b4">r</font>) ) ) ) holds <br/><font color="Olive" title="b1">d1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">d2</font> ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="cqc_lang.html#S4" title="CQC_LANG:sch.4">CQC_LANG:sch 4</a>();</span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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