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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ;<br/>



<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> ] means (  <a href="qc_lang3.html#K5">Free</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">a<sub>1</sub></font> <a href="cqc_lang.html#K17" target="_self">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E1:57"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Olive">b<sub>1</sub></font><br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="qc_lang1.html#M1">QC-variable_list</a> of <font color="Olive">b<sub>1</sub></font> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font> <a href="qc_lang1.html#K9">!</a> <font color="Olive">b<sub>3</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="qc_lang1.html#M1">QC-variable_list</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>





<b>assume </b><a NAME="E1:57_1"/><i><font color="Green">E36</font></i>: 
 <a href="qc_lang3.html#K5">Free</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:57_1"/><i><font color="Green">E37</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K5" target="_self">Subst</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6" target="_self">.--&gt;</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#D3" target="_self">Def3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:57_1_1"/><i><font color="Green">E38</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:57_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang3.html#K3">a.</a> 0
 ;<br/><a NAME="E2:57_1_1_1"/><b>then </b>
 <a href="qc_lang3.html#K3">a.</a> 0 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K4">free_QC-variables</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="cqc_lang.html#E1:57_1_1"><i><font color="Green">E38</font></i></a></i>;<br/><b>hence </b><a NAME="E3:57_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="cqc_lang.html#E1:57_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="qc_lang3.html#T66">QC_LANG3:66</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:57_1_1"/>
<span class="p1">(<span class="default"><a href="cqc_lang.html#K5" target="_self">Subst</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6" target="_self">.--&gt;</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="cqc_lang.html#E1:57_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="cqc_lang.html#T11" target="_self">Th11</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:57_1"/><b>then </b>
 <a href="cqc_lang.html#K5" target="_self">Subst</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6" target="_self">.--&gt;</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="cqc_lang.html#E2:57_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>
<b>hence </b><a NAME="E5:57_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="cqc_lang.html#K17" target="_self">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T30" target="_self">Th30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:57"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K9" target="_self">VERUM</a> ]
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T28" target="_self">Th28</a></i>;<br/>
<a NAME="E3:57"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K12">'not'</a> <font color="Olive">b<sub>1</sub></font>]
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T32" target="_self">Th32</a>, <a class="ref" href="qc_lang3.html#T67">QC_LANG3:67</a></i>;<br/>
<a NAME="E4:57"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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">b<sub>1</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></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="E1:57_2"/><i><font color="Green">E39</font></i>: 
( (  <a href="qc_lang3.html#K5">Free</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">c<sub>3</sub></font> <a href="cqc_lang.html#K17" target="_self">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> ) &amp; (  <a href="qc_lang3.html#K5">Free</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">c<sub>4</sub></font> <a href="cqc_lang.html#K17" target="_self">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> ) )
 ;<br/>

<b>assume </b><a NAME="E2:57_2"/>
 <a href="qc_lang3.html#K5">Free</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E3:57_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="qc_lang3.html#K5">Free</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="qc_lang3.html#K5">Free</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="qc_lang3.html#T69">QC_LANG3:69</a></i>;<br/>
<b>hence </b><a NAME="E4:57_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="cqc_lang.html#K17" target="_self">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="cqc_lang.html#E1:57_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="cqc_lang.html#T34" target="_self">Th34</a>, <a class="ref" href="xboole_1.html#T15">XBOOLE_1:15</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:57"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a><br/>for <font color="Olive">b<sub>2</sub></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">b<sub>2</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K14">All</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></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="E1:57_3"/><i><font color="Green">E40</font></i>: 
(  <a href="qc_lang3.html#K5">Free</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">c<sub>4</sub></font> <a href="cqc_lang.html#K17" target="_self">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<b>assume </b><a NAME="E2:57_3"/>
 <a href="qc_lang3.html#K5">Free</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K14">All</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E3:57_3"/><b>then </b>
( ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> implies <span class="p1">(<span class="default"><a href="qc_lang1.html#K14">All</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="cqc_lang.html#K17" target="_self">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K14">All</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> ) &amp; ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> implies <span class="p1">(<span class="default"><a href="qc_lang1.html#K14">All</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="cqc_lang.html#K17" target="_self">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K14">All</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> ) )
 
<b>by </b><i><a class="txt" href="cqc_lang.html#E1:57_3"><i><font color="Green">E40</font></i></a>, <a class="ref" href="cqc_lang.html#T37" target="_self">Th37</a>, <a class="ref" href="cqc_lang.html#T38" target="_self">Th38</a>, <a class="ref" href="qc_lang3.html#T70">QC_LANG3:70</a></i>;<br/>
<b>hence </b><a NAME="E4:57_3"/>
<span class="p1">(<span class="default"><a href="qc_lang1.html#K14">All</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="cqc_lang.html#K17" target="_self">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K14">All</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:57"/>
for <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="qc_lang1.html#S1">QC_LANG1:sch 1</a>(<a class="txt" href="cqc_lang.html#E1:57"><i><font color="Green">E35</font></i></a>, <a class="txt" href="cqc_lang.html#E2:57"><i><font color="Green">E36</font></i></a>, <a class="txt" href="cqc_lang.html#E3:57"><i><font color="Green">E37</font></i></a>, <a class="txt" href="cqc_lang.html#E4:57"><i><font color="Green">E38</font></i></a>, <a class="txt" href="cqc_lang.html#E5:57"><i><font color="Green">E39</font></i></a>);<br/></i>
<b>hence </b><a NAME="E7:57"/>
(  <a href="qc_lang3.html#K5">Free</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K17" target="_self">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>


</div>
