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

<b>let </b><font color="Maroon">F</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> ;<br/><b>let </b><font color="Maroon">G</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> ;<br/><b>let </b><font color="Maroon">s</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">F</font>, <font color="Olive">G</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> <br/> for <font color="Olive">s</font> being  <a href="finseq_1.html#NM1">FinSequence</a>  st  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Olive">F</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp;  <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Olive">G</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Olive">s</font> holds <br/> <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Olive">G</font>;<br/>
<a NAME="E1:31"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st ( for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">n</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:31_1"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> holds <br/>for <font color="Olive">F</font>, <font color="Olive">G</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> <br/> for <font color="Olive">s</font> being  <a href="finseq_1.html#NM1">FinSequence</a>  st  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Olive">F</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> &amp;  <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Olive">G</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Olive">s</font> holds <br/> <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Olive">G</font>
 ;<br/>

<b>let </b><font color="Maroon">F</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> , <font color="Maroon">G</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> ;<br/><b>let </b><font color="Maroon">s</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E2:31_1"/><i><font color="Green">E34</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E3:31_1"/><i><font color="Green">E36</font></i>: 
 <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">G</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s</font>
 ;<br/>

<a NAME="E4:31_1"/><i><font color="Green">E42</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">G</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E5:31_1"/><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">G</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:31_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">G</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i/>;<br/>
<a NAME="E7:31_1"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="qc_lang1.html#T10" target="_self">Th10</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E8:31_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> 1
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_1_1_1"/>
( <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K11" target="_self">VERUM</a>  or <font color="Maroon">F</font> is <a href="qc_lang1.html#V2" target="_self">atomic</a> or <font color="Maroon">F</font> is <a href="qc_lang1.html#V3" target="_self">negative</a> or <font color="Maroon">F</font> is <a href="qc_lang1.html#V4" target="_self">conjunctive</a> or <font color="Maroon">F</font> is <a href="qc_lang1.html#V5" target="_self">universal</a> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_1"><b>suppose </b></a><a NAME="E1:31_1_1_1_1"/>
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K11" target="_self">VERUM</a> 
 ;<br/><div class="add"><a NAME="E2:31_1_1_1_1"/><b>then </b>
 <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K11" target="_self">VERUM</a> 
 ;<br/><a NAME="E3:31_1_1_1_1"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/><a NAME="E4:31_1_1_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">G</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">G</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i>, <a class="ref" href="qc_lang1.html#T4" target="_self">Th4</a>, , <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E5:31_1_1_1_1"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 0 <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="qc_lang1.html#T4" target="_self">Th4</a>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E6:31_1_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E7:31_1_1_1_1"/><b>then </b>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><b>hence </b><a NAME="E8:31_1_1_1_1"/>
 <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_2"><b>suppose </b></a><a NAME="E1:31_1_1_1_2"/>
<font color="Maroon">F</font> is <a href="qc_lang1.html#V2" target="_self">atomic</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">P</font> being   <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Maroon">k</font>, <font color="Maroon">ll</font> being    <a href="qc_lang1.html#M1" target="_self">QC-variable_list</a> of <font color="Maroon">k</font><b> such that </b><br/><a NAME="E3:31_1_1_1_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="qc_lang1.html#K9" target="_self">!</a> <font color="Maroon">ll</font>
 <b>by </b><i/>;<br/><a NAME="E4:31_1_1_1_2"/><i><font color="Green">E61</font></i>: 
 <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">P</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">ll</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:31_1_1_1_2"/><b>then </b><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T58">FINSEQ_1:58</a></i>;<br/><a NAME="E6:31_1_1_1_2"/><b>then </b>
<font color="Maroon">G</font> is <a href="qc_lang1.html#V2" target="_self">atomic</a>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">k'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">P'</font> being   <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Maroon">k'</font>, <font color="Maroon">ll'</font> being    <a href="qc_lang1.html#M1" target="_self">QC-variable_list</a> of <font color="Maroon">k'</font><b> such that </b><br/><a NAME="E8:31_1_1_1_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P'</font> <a href="qc_lang1.html#K9" target="_self">!</a> <font color="Maroon">ll'</font>
 <b>by </b><i/>;<br/><a NAME="E9:31_1_1_1_2"/><i><font color="Green">E64</font></i>: 
 <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">P'</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">ll'</font>
 <b>by </b><i><a class="txt" href="qc_lang1.html#E1:31_1"><i><font color="Green">E33</font></i></a>, </i>;<br/><a NAME="E10:31_1_1_1_2"/><b>then </b><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">P'</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T58">FINSEQ_1:58</a></i>;<br/><i><font color="Green">E66</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">ll'</font> = 
<font color="Maroon">k'</font>
<b>by </b><i/>
<br/>.= 
 <a href="qc_lang1.html#K6" target="_self">the_arity_of</a> <font color="Maroon">P</font>
<b>by </b><i><a class="txt" href="qc_lang1.html#E1:31"><i><font color="Green">E32</font></i></a>, , <a class="ref" href="qc_lang1.html#T2" target="_self">Th2</a></i>
<br/>.= 
<font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="qc_lang1.html#T2" target="_self">Th2</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">ll</font>
<b>by </b><i/>
;<br/><a NAME="E12:31_1_1_1_2"/>
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">P</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">ll</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">P</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><font color="Maroon">ll'</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="qc_lang1.html#D6" target="_self">Def6</a>, <a class="txt" href="qc_lang1.html#E1:31"><i><font color="Green">E32</font></i></a>, <a class="txt" href="qc_lang1.html#E2:31_1"><i><font color="Green">E34</font></i></a>, , <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/><a NAME="E13:31_1_1_1_2"/><b>then </b>
<font color="Maroon">ll</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ll'</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/><a NAME="E14:31_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">ll</font></span>)</span> <a href="nat_1.html#K1">+</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">ll'</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E15:31_1_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="qc_lang1.html#E3:31_1"><i><font color="Green">E36</font></i></a></i>;<br/><a NAME="E16:31_1_1_1_2"/><b>then </b>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><b>hence </b><a NAME="E17:31_1_1_1_2"/>
 <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_3"><b>suppose </b></a><a NAME="E1:31_1_1_1_3"/>
<font color="Maroon">F</font> is <a href="qc_lang1.html#V3" target="_self">negative</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">p</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> <b> such that </b><br/><a NAME="E3:31_1_1_1_3"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K12" target="_self">'not'</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/><a NAME="E4:31_1_1_1_3"/><i><font color="Green">E93</font></i>: 
 <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E5:31_1_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T58">FINSEQ_1:58</a></i>;<br/><a NAME="E6:31_1_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/><a NAME="E7:31_1_1_1_3"/><b>then </b>
<font color="Maroon">G</font> is <a href="qc_lang1.html#V3" target="_self">negative</a>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">p'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> <b> such that </b><br/><a NAME="E9:31_1_1_1_3"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K12" target="_self">'not'</a> <font color="Maroon">p'</font>
 <b>by </b><i/>;<br/><a NAME="E10:31_1_1_1_3"/>
 <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p'</font></span>)</span>
 <b>by </b><i><a class="txt" href="qc_lang1.html#E1:31"><i><font color="Green">E32</font></i></a></i>;<br/><a NAME="E11:31_1_1_1_3"/><b>then </b>
<span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p'</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="qc_lang1.html#D6" target="_self">Def6</a>, <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/><a NAME="E12:31_1_1_1_3"/><b>then </b><i><font color="Green">E95</font></i>: 
 <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p'</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/> <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="qc_lang1.html#D6" target="_self">Def6</a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
;<br/><a NAME="E14:31_1_1_1_3"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E15:31_1_1_1_3"/><b>then </b>
 <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p'</font>
 <b>by </b><i>, , <a class="txt" href="qc_lang1.html#E1:31_1"><i><font color="Green">E33</font></i></a></i>;<br/><a NAME="E16:31_1_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p'</font></span>)</span> <a href="finseq_1.html#K7">^</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p'</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="txt" href="qc_lang1.html#E1:31_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/><a NAME="E17:31_1_1_1_3"/><b>then </b>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/><b>hence </b><a NAME="E18:31_1_1_1_3"/>
 <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_4"><b>suppose </b></a><a NAME="E1:31_1_1_1_4"/>
<font color="Maroon">F</font> is <a href="qc_lang1.html#V4" target="_self">conjunctive</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">p</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> , <font color="Maroon">q</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> <b> such that </b><br/><a NAME="E3:31_1_1_1_4"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="qc_lang1.html#K13" target="_self">'&amp;'</a> <font color="Maroon">q</font>
 <b>by </b><i/>;<br/><a NAME="E4:31_1_1_1_4"/><i><font color="Green">E97</font></i>: 
 <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,0</span><a href="tarski.html#K4">]</a></span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">q</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/><a NAME="E5:31_1_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,0</span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T58">FINSEQ_1:58</a></i>;<br/><a NAME="E6:31_1_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 2
 <b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/><a NAME="E7:31_1_1_1_4"/><b>then </b>
<font color="Maroon">G</font> is <a href="qc_lang1.html#V4" target="_self">conjunctive</a>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">p'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> , <font color="Maroon">q'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> <b> such that </b><br/><a NAME="E9:31_1_1_1_4"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> <a href="qc_lang1.html#K13" target="_self">'&amp;'</a> <font color="Maroon">q'</font>
 <b>by </b><i/>;<br/><a NAME="E10:31_1_1_1_4"/><i><font color="Green">E99</font></i>: 
 <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,0</span><a href="tarski.html#K4">]</a></span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p'</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">q'</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="qc_lang1.html#E1:31"><i><font color="Green">E32</font></i></a>, <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/><a NAME="E11:31_1_1_1_4"/><b>then </b>
<span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,0</span><a href="tarski.html#K4">]</a></span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,0</span><a href="tarski.html#K4">]</a></span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p'</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">q'</font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="qc_lang1.html#D6" target="_self">Def6</a>, <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/><b>then </b><i><font color="Green">E100</font></i>: <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">q</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p'</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">q'</font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p'</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">q'</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>
;<br/><a NAME="E13:31_1_1_1_4"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p'</font></span>)</span> or  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p'</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p</font></span>)</span> )
 ;<br/><b>then consider </b><font color="Maroon">a</font> being   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">b</font> being   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">c</font> being   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">d</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E15:31_1_1_1_4"/><i><font color="Green">E105</font></i>: 
( ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p'</font> ) or ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p'</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p</font> ) )
 <b>and </b><br/><a NAME="E16:31_1_1_1_4"/><i><font color="Green">E106</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E17:31_1_1_1_4"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">a</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="txt" href="qc_lang1.html#E2:31_1"><i><font color="Green">E34</font></i></a></i>;<br/><a NAME="E18:31_1_1_1_4"/><i><font color="Green">E108</font></i>: 
 ex <font color="Olive">t</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <font color="Maroon">a</font> <a href="finseq_1.html#K7">^</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="qc_lang1.html#E3:31_1"><i><font color="Green">E36</font></i></a>, , <a class="ref" href="finseq_1.html#T64">FINSEQ_1:64</a></i>;<br/><i><font color="Green">E110</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default">2,0</span><a href="tarski.html#K4">]</a></span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="qc_lang1.html#D6" target="_self">Def6</a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
;<br/><a NAME="E20:31_1_1_1_4"/><b>then </b><i><font color="Green">E111</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E21:31_1_1_1_4"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">q</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E22:31_1_1_1_4"/><b>then </b><i><font color="Green">E112</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E23:31_1_1_1_4"/><i><font color="Green">E113</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p'</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">q'</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="qc_lang1.html#E2:31_1"><i><font color="Green">E34</font></i></a>, , <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E24:31_1_1_1_4"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p'</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p'</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">q'</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E25:31_1_1_1_4"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p'</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="qc_lang1.html#E4:31_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E26:31_1_1_1_4"/><b>then </b><i><font color="Green">E114</font></i>: 
 <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p'</font>
 <b>by </b><i>, , , , </i>;<br/><a NAME="E27:31_1_1_1_4"/><b>then </b><i><font color="Green">E115</font></i>: 
 <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">q'</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="txt" href="qc_lang1.html#E2:31_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/><a NAME="E28:31_1_1_1_4"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">q</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">q</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E29:31_1_1_1_4"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">q</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E30:31_1_1_1_4"/>
 <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">G</font>
 <b>by </b><i>, , <a class="ref" href="qc_lang1.html#D6" target="_self">Def6</a>, <a class="txt" href="qc_lang1.html#E1:31_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="qc_lang1.html#E5:31_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="qc_lang1.html#D8" target="_self">Def8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_5"><b>suppose </b></a><a NAME="E1:31_1_1_1_5"/>
<font color="Maroon">F</font> is <a href="qc_lang1.html#V5" target="_self">universal</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being   <a href="qc_lang1.html#NM3">bound_QC-variable</a>, <font color="Maroon">p</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> <b> such that </b><br/><a NAME="E3:31_1_1_1_5"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K14" target="_self">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font>
 <b>by </b><i/>;<br/><a NAME="E4:31_1_1_1_5"/><i><font color="Green">E117</font></i>: 
 <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default">3,0</span><a href="tarski.html#K4">]</a></span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E5:31_1_1_1_5"/><b>then </b><i><font color="Green">E118</font></i>: 
 <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,0</span><a href="tarski.html#K4">]</a></span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/><a NAME="E6:31_1_1_1_5"/><b>then </b>
<span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,0</span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T58">FINSEQ_1:58</a></i>;<br/><a NAME="E7:31_1_1_1_5"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 3
 <b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/><a NAME="E8:31_1_1_1_5"/><b>then </b>
<font color="Maroon">G</font> is <a href="qc_lang1.html#V5" target="_self">universal</a>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">x'</font> being   <a href="qc_lang1.html#NM3">bound_QC-variable</a>, <font color="Maroon">p'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> <b> such that </b><br/><a NAME="E10:31_1_1_1_5"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K14" target="_self">All</a> <font color="Maroon">x'</font>,<font color="Maroon">p'</font>
 <b>by </b><i/>;<br/><a NAME="E11:31_1_1_1_5"/><i><font color="Green">E121</font></i>: 
 <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default">3,0</span><a href="tarski.html#K4">]</a></span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x'</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p'</font></span>)</span>
 <b>by </b><i><a class="txt" href="qc_lang1.html#E1:31_1"><i><font color="Green">E33</font></i></a></i>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default">3,0</span><a href="tarski.html#K4">]</a></span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default">3,0</span><a href="tarski.html#K4">]</a></span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x'</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p'</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s</font>
<b>by </b><i>, <a class="ref" href="qc_lang1.html#D6" target="_self">Def6</a>, <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,0</span><a href="tarski.html#K4">]</a></span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x'</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p'</font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>
;<br/><b>then </b><i><font color="Green">E122</font></i>: <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x'</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p'</font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s</font>
<b>by </b><i><a class="ref" href="qc_lang1.html#D6" target="_self">Def6</a>, <a class="txt" href="qc_lang1.html#E1:31"><i><font color="Green">E32</font></i></a>, <a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x'</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p'</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>
;<br/><b>then </b><i><font color="Green">E123</font></i>: <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x'</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p'</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T58">FINSEQ_1:58</a></i>
<br/>.= 
<font color="Maroon">x'</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T58">FINSEQ_1:58</a></i>
;<br/><a NAME="E15:31_1_1_1_5"/><b>then </b><i><font color="Green">E124</font></i>: 
 <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p'</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/> <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default">3,0</span><a href="tarski.html#K4">]</a></span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="qc_lang1.html#E1:31"><i><font color="Green">E32</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
;<br/><a NAME="E17:31_1_1_1_5"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E18:31_1_1_1_5"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E19:31_1_1_1_5"/>
 <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">G</font>
 <b>by </b><i>, , <a class="ref" href="qc_lang1.html#D6" target="_self">Def6</a>, <a class="txt" href="qc_lang1.html#E2:31_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="qc_lang1.html#E3:31_1"><i><font color="Green">E36</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E10:31_1"/>
 <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">G</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:31"/><i><font color="Green">E125</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S4">NAT_1:sch 4</a>();<br/></i>
<a NAME="E3:31"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E4:31"/>
(  <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">G</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s</font> implies  <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">G</font> )
 <b>by </b><i/>;<br/>


</div>
