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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a> ] <b>means </b>( $1 is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>  implies <font color="Maroon">P<sub>1</sub></font>[$1] );<br/>
<a NAME="E1:29_1"/><i><font color="Green" title="E18">A2</font></i>: 
for <font color="Olive" title="b1">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="b2">P</font> being  <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Olive" title="b1">k</font><br/> for <font color="Olive" title="b3">l</font> being   <a href="qc_lang1.html#M1" title="QC_LANG1:mode.1">QC-variable_list</a> of <font color="Olive" title="b1">k</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Olive" title="b3">l</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">k</font> be    <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/><b>let </b><font color="Maroon" title="c2">P</font> be   <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Maroon" title="c1">k</font>;<br/><b>let </b><font color="Maroon" title="c3">l</font> be    <a href="qc_lang1.html#M1" title="QC_LANG1:mode.1">QC-variable_list</a> of <font color="Maroon" title="c1">k</font>;<br/>





<b>assume </b><a NAME="E1:29_1_1"/>
<font color="Maroon" title="c2">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c3">l</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> 
 ;<br/>

<a NAME="E2:29_1_1"/><b>then </b>
( <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">l</font> &amp; <font color="Maroon" title="c3">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K4" title="QC_LANG1:func.4">free_QC-variables</a>  ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">l</font> &amp; <font color="Maroon" title="c3">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K3" title="QC_LANG1:func.3">fixed_QC-variables</a>  ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T17" target="_self" title="CQC_LANG:th.17">Th17</a></i>;<br/>
<a NAME="E3:29_1_1"/><b>then </b>
<font color="Maroon" title="c3">l</font> is   <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of <font color="Maroon" title="c1">k</font>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T15" target="_self" title="CQC_LANG:th.15">Th15</a></i>;<br/>
<b>hence </b><a NAME="E4:29_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c3">l</font>]
 <b>by </b><i><a class="txt" href="cqc_lang.html#E1:29"><i><font color="Green" title="E17">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:29_1"/><i><font color="Green" title="E19">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a> ]
 
<b>by </b><i><a class="txt" href="cqc_lang.html#E1:29"><i><font color="Green" title="E17">A1</font></i></a></i>;<br/>
<a NAME="E3:29_1"/><i><font color="Green" title="E20">A4</font></i>: 
for <font color="Olive" title="b1">p</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <font color="Olive" title="b1">p</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">p</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a> ;<br/>

<b>assume </b><a NAME="E1:29_1_2"/><i><font color="Green" title="E21">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">p</font>]
 ;<br/>

<b>assume </b><a NAME="E2:29_1_2"/>
 <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <font color="Maroon" title="c1">p</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> 
 ;<br/>

<a NAME="E3:29_1_2"/><b>then </b>
<font color="Maroon" title="c1">p</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> 
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T18" target="_self" title="CQC_LANG:th.18">Th18</a></i>;<br/>
<b>hence </b><a NAME="E4:29_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <font color="Maroon" title="c1">p</font>]
 <b>by </b><i><a class="txt" href="cqc_lang.html#E1:29"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="cqc_lang.html#E1:29_1_2"><i><font color="Green" title="E21">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:29_1"/><i><font color="Green" title="E21">A6</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">q</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Olive" title="b2">q</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">q</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a> ;<br/>

<b>assume </b><a NAME="E1:29_1_3"/><i><font color="Green" title="E22">A7</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">p</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">q</font>] )
 ;<br/>

<b>assume </b><a NAME="E2:29_1_3"/>
<font color="Maroon" title="c1">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">q</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> 
 ;<br/>

<a NAME="E3:29_1_3"/><b>then </b>
( <font color="Maroon" title="c1">p</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>  &amp; <font color="Maroon" title="c2">q</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>  )
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T19" target="_self" title="CQC_LANG:th.19">Th19</a></i>;<br/>
<b>hence </b><a NAME="E4:29_1_3"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">q</font>]
 <b>by </b><i><a class="txt" href="cqc_lang.html#E1:29"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="cqc_lang.html#E1:29_1_3"><i><font color="Green" title="E22">A7</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:29_1"/><i><font color="Green" title="E22">A8</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><br/> for <font color="Olive" title="b2">p</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">p</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">p</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">x</font> be   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon" title="c2">p</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a> ;<br/>



<b>assume </b><a NAME="E1:29_1_4"/><i><font color="Green" title="E23">A9</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">p</font>]
 ;<br/>

<b>assume </b><a NAME="E2:29_1_4"/>
 <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">p</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> 
 ;<br/>

<a NAME="E3:29_1_4"/><b>then </b>
<font color="Maroon" title="c2">p</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> 
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T23" target="_self" title="CQC_LANG:th.23">Th23</a></i>;<br/>
<b>hence </b><a NAME="E4:29_1_4"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">p</font>]
 <b>by </b><i><a class="txt" href="cqc_lang.html#E1:29"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="cqc_lang.html#E1:29_1_4"><i><font color="Green" title="E23">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:29_1"/>
for <font color="Olive" title="b1">p</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>]
 
<b>from </b><i><a class="ref" href="qc_lang1.html#S1" title="QC_LANG1:sch.1">QC_LANG1:sch 1</a>(<a class="txt" href="cqc_lang.html#E1:29_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="cqc_lang.html#E2:29_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="cqc_lang.html#E3:29_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="cqc_lang.html#E4:29_1"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="cqc_lang.html#E5:29_1"><i><font color="Green" title="E22">A8</font></i></a>);<br/></i>
<b>hence </b><a NAME="E7:29_1"/>
for <font color="Olive" title="b1">r</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>  holds  <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">r</font>]
 ;<br/>


</div>
