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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7" target="_self">CQC-WFF</a>  implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>] );<br/>
<a NAME="E1:34_1"/><i><font color="Green">E43</font></i>: 
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">P</font> being  <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Olive">k</font><br/> for <font color="Olive">l</font> being   <a href="qc_lang1.html#M1">QC-variable_list</a> of <font color="Olive">k</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">P</font> <a href="qc_lang1.html#K9">!</a> <font color="Olive">l</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1"><b>proof </b></a><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">P</font> be   <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Maroon">k</font>;<br/><b>let </b><font color="Maroon">l</font> be    <a href="qc_lang1.html#M1">QC-variable_list</a> of <font color="Maroon">k</font>;<br/>





<b>assume </b><a NAME="E1:34_1_1"/>
<font color="Maroon">P</font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7" target="_self">CQC-WFF</a> 
 ;<br/>

<a NAME="E2:34_1_1"/><b>then </b>
( <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K4">free_QC-variables</a>  ) </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K3">fixed_QC-variables</a>  ) </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E3:34_1_1"/><b>then </b>
<font color="Maroon">l</font> is   <a href="cqc_lang.html#NM2">CQC-variable_list</a> of <font color="Maroon">k</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:34_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">P</font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">l</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:34_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K9" target="_self">VERUM</a> ]
 
<b>by </b><i/>;<br/>
<a NAME="E3:34_1"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">p</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K12">'not'</a> <font color="Olive">p</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ;<br/>

<b>assume </b><a NAME="E1:34_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">p</font>]
 ;<br/>

<b>assume </b><a NAME="E2:34_1_2"/>
 <a href="qc_lang1.html#K12">'not'</a> <font color="Maroon">p</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7" target="_self">CQC-WFF</a> 
 ;<br/>

<a NAME="E3:34_1_2"/><b>then </b>
<font color="Maroon">p</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7" target="_self">CQC-WFF</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:34_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="qc_lang1.html#K12">'not'</a> <font color="Maroon">p</font>]
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:34_1"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">q</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Olive">q</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> , <font color="Maroon">q</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ;<br/>

<b>assume </b><a NAME="E1:34_1_3"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">p</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">q</font>] )
 ;<br/>

<b>assume </b><a NAME="E2:34_1_3"/>
<font color="Maroon">p</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7" target="_self">CQC-WFF</a> 
 ;<br/>

<a NAME="E3:34_1_3"/><b>then </b>
( <font color="Maroon">p</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7" target="_self">CQC-WFF</a>  &amp; <font color="Maroon">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7" target="_self">CQC-WFF</a>  )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:34_1_3"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">p</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">q</font>]
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:34_1"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">x</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a><br/> for <font color="Olive">p</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K14">All</a> <font color="Olive">x</font>,<font color="Olive">p</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_4"><b>proof </b></a><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">p</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ;<br/>



<b>assume </b><a NAME="E1:34_1_4"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">p</font>]
 ;<br/>

<b>assume </b><a NAME="E2:34_1_4"/>
 <a href="qc_lang1.html#K14">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7" target="_self">CQC-WFF</a> 
 ;<br/>

<a NAME="E3:34_1_4"/><b>then </b>
<font color="Maroon">p</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7" target="_self">CQC-WFF</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:34_1_4"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="qc_lang1.html#K14">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font>]
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:34_1"/>
for <font color="Olive">p</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>]
 
<b>from </b><i><a class="ref" href="qc_lang1.html#S1">QC_LANG1:sch 1</a>(, , , , );<br/></i>
<b>hence </b><a NAME="E7:34_1"/>
for <font color="Olive">r</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7" target="_self">CQC-WFF</a>  holds  <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">r</font>]
 ;<br/>


</div>
