<?xml version="1.0"?>
<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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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  <a href="qc_lang3.html#K5" title="QC_LANG3:func.5">Free</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/><font color="Olive" title="b1">p</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font></span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="qc_lang3.html#K5" title="QC_LANG3:func.5">Free</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c2">p</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> )</span><br/>



<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>(  <a href="qc_lang3.html#K5" title="QC_LANG3:func.5">Free</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies $1 <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 );<br/>
<a NAME="E1:52"/><i><font color="Green" title="E32">A1</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="52_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">P</font> being  <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Maroon" title="c3">k</font><br/> for <font color="Olive" title="b2">l</font> being   <a href="qc_lang1.html#M1" title="QC_LANG1:mode.1">QC-variable_list</a> of <font color="Maroon" title="c3">k</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Olive" title="b2">l</font>]</span><br/><b>let </b><font color="Maroon" title="c4">P</font> be   <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Maroon" title="c3">k</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">l</font> being   <a href="qc_lang1.html#M1" title="QC_LANG1:mode.1">QC-variable_list</a> of <font color="Maroon" title="c3">k</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Olive" title="b1">l</font>]</span><br/><b>let </b><font color="Maroon" title="c5">l</font> be    <a href="qc_lang1.html#M1" title="QC_LANG1:mode.1">QC-variable_list</a> of <font color="Maroon" title="c3">k</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c5">l</font>]</span><br/>





<b>assume </b><a NAME="E1:52_1"/><i><font color="Green" title="E33">A2</font></i>: 
 <a href="qc_lang3.html#K5" title="QC_LANG3:func.5">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c5">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c5">l</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c5">l</font></span><br/>

<a NAME="E2:52_1"/><i><font color="Green" title="E34">A3</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2" title="CQC_LANG:func.2">Subst</a> <font color="Maroon" title="c5">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">l</font>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#D3" target="_self" title="CQC_LANG:def.3">Def3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">j</font> &amp; <font color="Maroon" title="c6">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="c5">l</font> implies <span class="p1">(<span class="default"><a href="cqc_lang.html#K2" title="CQC_LANG:func.2">Subst</a> <font color="Maroon" title="c5">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> )</span><br/><a NAME="E1:52_1_1"/><i><font color="Green" title="E35">A4</font></i>: 
<font color="Maroon" title="c6">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>assume </b><a NAME="E2:52_1_1"/><i><font color="Green" title="E36">A5</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">j</font> &amp; <font color="Maroon" title="c6">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="c5">l</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2" title="CQC_LANG:func.2">Subst</a> <font color="Maroon" title="c5">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_1_1_1"/>
<font color="Maroon" title="c5">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:52_1_1_1"/><b>then </b>
 <a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c5">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="c5">l</font> &amp; <font color="Maroon" title="c5">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> 
 <b>by </b><i><a class="txt" href="#E1:52_1_1"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="#E2:52_1_1"><i><font color="Green" title="E36">A5</font></i></a></i>;<br/><b>hence </b><a NAME="E3:52_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:52_1"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="qc_lang3.html#T66" title="QC_LANG3:th.66">QC_LANG3:66</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E4:52_1_1"/>
<span class="p1">(<span class="default"><a href="cqc_lang.html#K2" title="CQC_LANG:func.2">Subst</a> <font color="Maroon" title="c5">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font>
 <b>by </b><i><a class="txt" href="#E1:52_1_1"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="#E2:52_1_1"><i><font color="Green" title="E36">A5</font></i></a>, <a class="ref" href="cqc_lang.html#T11" target="_self" title="CQC_LANG:th.11">Th11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E4:52_1"/><b>then </b>
 <a href="cqc_lang.html#K2" title="CQC_LANG:func.2">Subst</a> <font color="Maroon" title="c5">l</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font>
 
<b>by </b><i><a class="txt" href="#E2:52_1"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>
<b>hence </b><a NAME="E5:52_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c5">l</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c5">l</font>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T30" target="_self" title="CQC_LANG:th.30">Th30</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E2:52"/><i><font color="Green" title="E33">A6</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="ref" href="cqc_lang.html#T28" target="_self" title="CQC_LANG:th.28">Th28</a></i>;<br/>
<a NAME="E3:52"/><i><font color="Green" title="E34">A7</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>]
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T32" target="_self" title="CQC_LANG:th.32">Th32</a>, <a class="ref" href="qc_lang3.html#T67" title="QC_LANG3:th.67">QC_LANG3:67</a></i>;<br/>
<a NAME="E4:52"/><i><font color="Green" title="E35">A8</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="52_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">p</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">q</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c4">q</font>] )</span><br/>



<b>assume </b><a NAME="E1:52_2"/><i><font color="Green" title="E36">A9</font></i>: 
( (  <a href="qc_lang3.html#K5" title="QC_LANG3:func.5">Free</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c3">p</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> ) &amp; (  <a href="qc_lang3.html#K5" title="QC_LANG3:func.5">Free</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c4">q</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c4">q</font>]</span><br/>

<b>assume </b><a NAME="E2:52_2"/>
 <a href="qc_lang3.html#K5" title="QC_LANG3:func.5">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c4">q</font></span><br/>

<a NAME="E3:52_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="qc_lang3.html#K5" title="QC_LANG3:func.5">Free</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="qc_lang3.html#K5" title="QC_LANG3:func.5">Free</a> <font color="Maroon" title="c4">q</font></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="qc_lang3.html#T69" title="QC_LANG3:th.69">QC_LANG3:69</a></i>;<br/>
<b>hence </b><a NAME="E4:52_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c4">q</font>
 <b>by </b><i><a class="txt" href="#E1:52_2"><i><font color="Green" title="E36">A9</font></i></a>, <a class="ref" href="cqc_lang.html#T34" target="_self" title="CQC_LANG:th.34">Th34</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E5:52"/><i><font color="Green" title="E36">A10</font></i>: 
for <font color="Olive" title="b1">y</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">y</font>,<font color="Olive" title="b2">p</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">y</font> be   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c3">y</font>,<font color="Olive" title="b1">p</font>]</span><br/><b>let </b><font color="Maroon" title="c4">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">p</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">p</font>] )</span><br/>



<b>assume </b><a NAME="E1:52_3"/><i><font color="Green" title="E37">A11</font></i>: 
(  <a href="qc_lang3.html#K5" title="QC_LANG3:func.5">Free</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c4">p</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">p</font>]</span><br/>

<b>assume </b><a NAME="E2:52_3"/>
 <a href="qc_lang3.html#K5" title="QC_LANG3:func.5">Free</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">p</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">p</font></span><br/>

<a NAME="E3:52_3"/><b>then </b>
( ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> implies <span class="p1">(<span class="default"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">p</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">p</font> ) &amp; ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y</font> implies <span class="p1">(<span class="default"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">p</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">p</font> ) )
 
<b>by </b><i><a class="txt" href="#E1:52_3"><i><font color="Green" title="E37">A11</font></i></a>, <a class="ref" href="cqc_lang.html#T37" target="_self" title="CQC_LANG:th.37">Th37</a>, <a class="ref" href="cqc_lang.html#T38" target="_self" title="CQC_LANG:th.38">Th38</a>, <a class="ref" href="qc_lang3.html#T70" title="QC_LANG3:th.70">QC_LANG3:70</a></i>;<br/>
<b>hence </b><a NAME="E4:52_3"/>
<span class="p1">(<span class="default"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">p</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E6:52"/>
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="#E1:52"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="#E2:52"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="#E3:52"><i><font color="Green" title="E34">A7</font></i></a>, <a class="txt" href="#E4:52"><i><font color="Green" title="E35">A8</font></i></a>, <a class="txt" href="#E5:52"><i><font color="Green" title="E36">A10</font></i></a>);<br/></i>
<b>hence </b><a NAME="E7:52"/>
(  <a href="qc_lang3.html#K5" title="QC_LANG3:func.5">Free</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c2">p</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
