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

<span class="kw">let </span><font color="Maroon" title="c1">A</font> be    <a href="qc_lang1.html#M1" title="QC_LANG1:mode.1">QC-alphabet</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F</font>, <font color="Olive" title="b2">G</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon" title="c1">A</font><br/>  for <font color="Olive" title="b3">s</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Olive" title="b2">G</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b3">s</font> holds <br/> <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Olive" title="b2">G</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">F</font>, <font color="Maroon" title="c3">G</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">s</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c3">G</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b1">s</font> holds <br/> <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c3">G</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">s</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c3">G</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c4">s</font> implies  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c3">G</font> )</span><br/>







<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span> for <font color="Olive" title="b1">F</font>, <font color="Olive" title="b2">G</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon" title="c1">A</font><br/>  for <font color="Olive" title="b3">s</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Olive" title="b1">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp;  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Olive" title="b2">G</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b3">s</font> holds <br/> <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Olive" title="b2">G</font>;<br/>
<a NAME="E1:43"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st (  for <font color="Olive" title="b2">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">k</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">n</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:43_1"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">n</font> holds <br/> for <font color="Olive" title="b2">F</font>, <font color="Olive" title="b3">G</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon" title="c1">A</font><br/>  for <font color="Olive" title="b4">s</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Olive" title="b2">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> &amp;  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Olive" title="b3">G</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b4">s</font> holds <br/> <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Olive" title="b3">G</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font>]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">F</font>, <font color="Maroon" title="c7">G</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">s</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font> &amp;  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b1">s</font> holds <br/> <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c7">G</font></span><br/><span class="kw">let </span><font color="Maroon" title="c8">s</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font> &amp;  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c8">s</font> implies  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c7">G</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:43_1"/><span class="lab"><font color="Green" title="E26">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n</font>
 <span class="kw">and </span><br/><a NAME="E3:43_1"/><span class="lab"><font color="Green" title="E27">A4</font></span>: 
 <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c8">s</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c7">G</font></span><br/>

<a NAME="E4:43_1"/>
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c7">G</font></span>)</span></span>)</span> &amp; 1 <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> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c7">G</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang1.html#T10" target="_self" onmouseover="rs('qc_lang1/T10')" onmouseout="rh()">Th10</a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E5:43_1"/><span class="kw">then </span>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c7">G</font></span>)</span>
 
;<br/>
<a NAME="E6:43_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A5</font></span>: 
<span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="ref" href="finseq_1.html#D7" onmouseover="rs('finseq_1/D7')" onmouseout="rh()">FINSEQ_1:def 7</a></span>;<br/>
<a NAME="E7:43_1"/><span class="lab"><font color="Green" title="E29">A6</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c7">G</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">s</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c7">G</font></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:43_1_1_1"/>
( <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">VERUM</a> <font color="Maroon" title="c1">A</font> or <font color="Maroon" title="c6">F</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  or <font color="Maroon" title="c6">F</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  or <font color="Maroon" title="c6">F</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  or <font color="Maroon" title="c6">F</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang1.html#T9" target="_self" onmouseover="rs('qc_lang1/T9')" onmouseout="rh()">Th9</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:43_1_1_1_1"/><span class="lab"><font color="Green" title="E30">A7</font></span>: 
<font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">VERUM</a> <font color="Maroon" title="c1">A</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c7">G</font></span><br/><div class="add"><a NAME="E2:43_1_1_1_1"/><span class="lab"><font color="Green" title="E31">A8</font></span>: 
1 <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> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c7">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang1.html#T10" target="_self" onmouseover="rs('qc_lang1/T10')" onmouseout="rh()">Th10</a></span>;<br/><a NAME="E3:43_1_1_1_1"/><span class="lab"><font color="Green" title="E32">A9</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1_1_1_1"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="ref" href="finseq_1.html#T39" onmouseover="rs('finseq_1/T39')" onmouseout="rh()">FINSEQ_1:39</a></span>;<br/><a NAME="E4:43_1_1_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E7:43_1"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/><a NAME="E5:43_1_1_1_1"/><span class="kw">then </span>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E7:43_1"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="txt" href="#E3:43_1_1_1_1"><span class="lab"><font color="Green" title="E32">A9</font></span></a>, <a class="txt" href="#E2:43_1_1_1_1"><span class="lab"><font color="Green" title="E31">A8</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E6:43_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><span class="kw">hence </span><a NAME="E7:43_1_1_1_1"/>
 <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c7">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="ref" href="finseq_1.html#T34" onmouseover="rs('finseq_1/T34')" onmouseout="rh()">FINSEQ_1:34</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:43_1_1_1_2"/>
<font color="Maroon" title="c6">F</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c7">G</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>, <font color="Maroon" title="c10">P</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-pred_symbol</a> of <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c11">ll</font> being   <a href="qc_lang1.html#NM9" title="QC_LANG1:NM.9">QC-variable_list</a> of <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c1">A</font><span class="kw"> such that </span><br/><a NAME="E3:43_1_1_1_2"/><span class="lab"><font color="Green" title="E30">A10</font></span>: 
<font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">P</font> <a href="qc_lang1.html#K10" title="QC_LANG1:func.10">!</a> <font color="Maroon" title="c11">ll</font>
 ;<br/><a NAME="E4:43_1_1_1_2"/><span class="lab"><font color="Green" title="E31">A11</font></span>: 
 <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c10">P</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c11">ll</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1_1_1_2"><span class="lab"><font color="Green" title="E30">A10</font></span></a>, <a class="ref" href="qc_lang1.html#T8" target="_self" onmouseover="rs('qc_lang1/T8')" onmouseout="rh()">Th8</a></span>;<br/><a NAME="E5:43_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A12</font></span>: 
<span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43_1"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="ref" href="finseq_1.html#T41" onmouseover="rs('finseq_1/T41')" onmouseout="rh()">FINSEQ_1:41</a></span>;<br/><a NAME="E6:43_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">G</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang1.html#T12" target="_self" onmouseover="rs('qc_lang1/T12')" onmouseout="rh()">Th12</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">k9</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>, <font color="Maroon" title="c13">P9</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-pred_symbol</a> of <font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c14">ll9</font> being   <a href="qc_lang1.html#NM9" title="QC_LANG1:NM.9">QC-variable_list</a> of <font color="Maroon" title="c12">k9</font>,<font color="Maroon" title="c1">A</font><span class="kw"> such that </span><br/><a NAME="E8:43_1_1_1_2"/><span class="lab"><font color="Green" title="E33">A13</font></span>: 
<font color="Maroon" title="c7">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">P9</font> <a href="qc_lang1.html#K10" title="QC_LANG1:func.10">!</a> <font color="Maroon" title="c14">ll9</font>
 ;<br/><a NAME="E9:43_1_1_1_2"/><span class="lab"><font color="Green" title="E34">A14</font></span>: 
 <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c7">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c13">P9</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c14">ll9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:43_1_1_1_2"><span class="lab"><font color="Green" title="E33">A13</font></span></a>, <a class="ref" href="qc_lang1.html#T8" target="_self" onmouseover="rs('qc_lang1/T8')" onmouseout="rh()">Th8</a></span>;<br/><a NAME="E10:43_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A15</font></span>: 
<span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">P9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T41" onmouseover="rs('finseq_1/T41')" onmouseout="rh()">FINSEQ_1:41</a></span>;<br/><a NAME="E11:43_1_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c10">P</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c11">ll</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c10">P</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">ll9</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c8">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E4:43_1_1_1_2"><span class="lab"><font color="Green" title="E31">A11</font></span></a>, <a class="txt" href="#E5:43_1_1_1_2"><span class="lab"><font color="Green" title="E32">A12</font></span></a>, <a class="txt" href="#E9:43_1_1_1_2"><span class="lab"><font color="Green" title="E34">A14</font></span></a>, <a class="ref" href="finseq_1.html#T32" onmouseover="rs('finseq_1/T32')" onmouseout="rh()">FINSEQ_1:32</a></span>;<br/><a NAME="E12:43_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">ll</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">ll9</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c8">s</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T33" onmouseover="rs('finseq_1/T33')" onmouseout="rh()">FINSEQ_1:33</a></span>;<br/><a NAME="E13:43_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A16</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">ll</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">ll9</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>;<br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">ll9</font> = 
<font color="Maroon" title="c12">k9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a></span>
<br/>.= 
 <a href="qc_lang1.html#K7" title="QC_LANG1:func.7">the_arity_of</a> <font color="Maroon" title="c10">P</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43_1_1_1_2"><span class="lab"><font color="Green" title="E32">A12</font></span></a>, <a class="txt" href="#E10:43_1_1_1_2"><span class="lab"><font color="Green" title="E35">A15</font></span></a>, <a class="ref" href="qc_lang1.html#T11" target="_self" onmouseover="rs('qc_lang1/T11')" onmouseout="rh()">Th11</a></span>
<br/>.= 
<font color="Maroon" title="c9">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang1.html#T11" target="_self" onmouseover="rs('qc_lang1/T11')" onmouseout="rh()">Th11</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">ll</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a></span>
;<br/><a NAME="E15:43_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43_1_1_1_2"><span class="lab"><font color="Green" title="E36">A16</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E16:43_1_1_1_2"/>
 <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c7">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="ref" href="finseq_1.html#T34" onmouseover="rs('finseq_1/T34')" onmouseout="rh()">FINSEQ_1:34</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:43_1_1_1_3"/>
<font color="Maroon" title="c6">F</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c7">G</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon" title="c1">A</font><span class="kw"> such that </span><br/><a NAME="E3:43_1_1_1_3"/><span class="lab"><font color="Green" title="E30">A17</font></span>: 
<font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'not'</a> <font color="Maroon" title="c9">p</font>
 ;<br/><a NAME="E4:43_1_1_1_3"/>
<span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1_1_1_3"><span class="lab"><font color="Green" title="E30">A17</font></span></a>, <a class="ref" href="finseq_1.html#T41" onmouseover="rs('finseq_1/T41')" onmouseout="rh()">FINSEQ_1:41</a></span>;<br/><a NAME="E5:43_1_1_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43_1"><span class="lab"><font color="Green" title="E28">A5</font></span></a></span>;<br/><a NAME="E6:43_1_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c7">G</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang1.html#T12" target="_self" onmouseover="rs('qc_lang1/T12')" onmouseout="rh()">Th12</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">p9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon" title="c1">A</font><span class="kw"> such that </span><br/><a NAME="E8:43_1_1_1_3"/><span class="lab"><font color="Green" title="E31">A18</font></span>: 
<font color="Maroon" title="c7">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'not'</a> <font color="Maroon" title="c10">p9</font>
 ;<br/><a NAME="E9:43_1_1_1_3"/>
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c9">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c10">p9</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c8">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E3:43_1_1_1_3"><span class="lab"><font color="Green" title="E30">A17</font></span></a>, <a class="txt" href="#E8:43_1_1_1_3"><span class="lab"><font color="Green" title="E31">A18</font></span></a>, <a class="ref" href="finseq_1.html#T32" onmouseover="rs('finseq_1/T32')" onmouseout="rh()">FINSEQ_1:32</a></span>;<br/><a NAME="E10:43_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A19</font></span>: 
 <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c10">p9</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c8">s</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T33" onmouseover="rs('finseq_1/T33')" onmouseout="rh()">FINSEQ_1:33</a></span>;<br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c6">F</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c9">p</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1_1_1_3"><span class="lab"><font color="Green" title="E30">A17</font></span></a>, <a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c9">p</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T40" onmouseover="rs('finseq_1/T40')" onmouseout="rh()">FINSEQ_1:40</a></span>
;<br/><a NAME="E12:43_1_1_1_3"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c9">p</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c6">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E13:43_1_1_1_3"/><span class="kw">then </span>
 <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c10">p9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="txt" href="#E2:43_1"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="txt" href="#E10:43_1_1_1_3"><span class="lab"><font color="Green" title="E32">A19</font></span></a></span>;<br/><a NAME="E14:43_1_1_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c10">p9</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c10">p9</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c8">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43_1_1_1_3"><span class="lab"><font color="Green" title="E32">A19</font></span></a>, <a class="ref" href="finseq_1.html#T34" onmouseover="rs('finseq_1/T34')" onmouseout="rh()">FINSEQ_1:34</a></span>;<br/><a NAME="E15:43_1_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T33" onmouseover="rs('finseq_1/T33')" onmouseout="rh()">FINSEQ_1:33</a></span>;<br/><span class="kw">hence </span><a NAME="E16:43_1_1_1_3"/>
 <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c7">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="ref" href="finseq_1.html#T34" onmouseover="rs('finseq_1/T34')" onmouseout="rh()">FINSEQ_1:34</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:43_1_1_1_4"/>
<font color="Maroon" title="c6">F</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c7">G</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">p</font>, <font color="Maroon" title="c10">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon" title="c1">A</font><span class="kw"> such that </span><br/><a NAME="E3:43_1_1_1_4"/><span class="lab"><font color="Green" title="E30">A20</font></span>: 
<font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">'&amp;'</a> <font color="Maroon" title="c10">q</font>
 ;<br/><a NAME="E4:43_1_1_1_4"/><span class="lab"><font color="Green" title="E31">A21</font></span>: 
 <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c9">p</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c10">q</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1_1_1_4"><span class="lab"><font color="Green" title="E30">A20</font></span></a>, <a class="ref" href="finseq_1.html#T32" onmouseover="rs('finseq_1/T32')" onmouseout="rh()">FINSEQ_1:32</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A22</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c6">F</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c9">p</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c10">q</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c9">p</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c10">q</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T40" onmouseover="rs('finseq_1/T40')" onmouseout="rh()">FINSEQ_1:40</a></span>
;<br/><a NAME="E6:43_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A23</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c9">p</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c10">q</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>;<br/><a NAME="E7:43_1_1_1_4"/>
<span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_1_1_1_4"><span class="lab"><font color="Green" title="E31">A21</font></span></a>, <a class="ref" href="finseq_1.html#T41" onmouseover="rs('finseq_1/T41')" onmouseout="rh()">FINSEQ_1:41</a></span>;<br/><a NAME="E8:43_1_1_1_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43_1"><span class="lab"><font color="Green" title="E28">A5</font></span></a></span>;<br/><a NAME="E9:43_1_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c7">G</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang1.html#T12" target="_self" onmouseover="rs('qc_lang1/T12')" onmouseout="rh()">Th12</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">p9</font>, <font color="Maroon" title="c12">q9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon" title="c1">A</font><span class="kw"> such that </span><br/><a NAME="E11:43_1_1_1_4"/><span class="lab"><font color="Green" title="E34">A24</font></span>: 
<font color="Maroon" title="c7">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p9</font> <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">'&amp;'</a> <font color="Maroon" title="c12">q9</font>
 ;<br/><a NAME="E12:43_1_1_1_4"/><span class="lab"><font color="Green" title="E35">A25</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c11">p9</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c11">p9</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c12">q9</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c8">s</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/><a NAME="E13:43_1_1_1_4"/><span class="lab"><font color="Green" title="E36">A26</font></span>: 
 <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c7">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c11">p9</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c12">q9</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:43_1_1_1_4"><span class="lab"><font color="Green" title="E34">A24</font></span></a>, <a class="ref" href="finseq_1.html#T32" onmouseover="rs('finseq_1/T32')" onmouseout="rh()">FINSEQ_1:32</a></span>;<br/><a NAME="E14:43_1_1_1_4"/><span class="kw">then </span>
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c9">p</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c10">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c11">p9</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c12">q9</font></span>)</span></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c8">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E4:43_1_1_1_4"><span class="lab"><font color="Green" title="E31">A21</font></span></a>, <a class="ref" href="finseq_1.html#T32" onmouseover="rs('finseq_1/T32')" onmouseout="rh()">FINSEQ_1:32</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A27</font></span>: <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c9">p</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c10">q</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c11">p9</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c12">q9</font></span>)</span></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c8">s</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T33" onmouseover="rs('finseq_1/T33')" onmouseout="rh()">FINSEQ_1:33</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c11">p9</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c12">q9</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c8">s</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T32" onmouseover="rs('finseq_1/T32')" onmouseout="rh()">FINSEQ_1:32</a></span>
;<br/><a NAME="E16:43_1_1_1_4"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c11">p9</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c12">q9</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c8">s</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43_1_1_1_4"><span class="lab"><font color="Green" title="E32">A22</font></span></a>, <a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>;<br/><a NAME="E17:43_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A28</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c11">p9</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c6">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:43_1_1_1_4"><span class="lab"><font color="Green" title="E35">A25</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E18:43_1_1_1_4"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c10">q</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c9">p</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c10">q</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/><a NAME="E19:43_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A29</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c10">q</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c6">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43_1_1_1_4"><span class="lab"><font color="Green" title="E33">A23</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E20:43_1_1_1_4"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c9">p</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c9">p</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c10">q</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/><a NAME="E21:43_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A30</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c9">p</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c6">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43_1_1_1_4"><span class="lab"><font color="Green" title="E33">A23</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E22:43_1_1_1_4"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c9">p</font></span>)</span> <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> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c11">p9</font></span>)</span> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c11">p9</font></span>)</span> <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> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c9">p</font></span>)</span> )
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">a</font>, <font color="Maroon" title="c14">b</font>, <font color="Maroon" title="c15">c</font>, <font color="Maroon" title="c16">d</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E24:43_1_1_1_4"/><span class="lab"><font color="Green" title="E41">A31</font></span>: 
( ( <font color="Maroon" title="c13">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c9">p</font> &amp; <font color="Maroon" title="c14">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c11">p9</font> ) or ( <font color="Maroon" title="c13">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c11">p9</font> &amp; <font color="Maroon" title="c14">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c9">p</font> ) )
 <span class="kw">and </span><br/><a NAME="E25:43_1_1_1_4"/><span class="lab"><font color="Green" title="E42">A32</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">a</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="c14">b</font> &amp; <font color="Maroon" title="c13">a</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c15">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">b</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c16">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:43_1_1_1_4"><span class="lab"><font color="Green" title="E37">A27</font></span></a></span>;<br/><a NAME="E26:43_1_1_1_4"/>
 ex <font color="Olive" title="b1">t</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <font color="Maroon" title="c13">a</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:43_1_1_1_4"><span class="lab"><font color="Green" title="E42">A32</font></span></a>, <a class="ref" href="finseq_1.html#T47" onmouseover="rs('finseq_1/T47')" onmouseout="rh()">FINSEQ_1:47</a></span>;<br/><a NAME="E27:43_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A33</font></span>: 
 <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c11">p9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="txt" href="#E2:43_1"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="txt" href="#E24:43_1_1_1_4"><span class="lab"><font color="Green" title="E41">A31</font></span></a>, <a class="txt" href="#E21:43_1_1_1_4"><span class="lab"><font color="Green" title="E40">A30</font></span></a>, <a class="txt" href="#E17:43_1_1_1_4"><span class="lab"><font color="Green" title="E38">A28</font></span></a></span>;<br/><a NAME="E28:43_1_1_1_4"/><span class="kw">then </span>
 <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c12">q9</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c8">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:43_1_1_1_4"><span class="lab"><font color="Green" title="E37">A27</font></span></a>, <a class="ref" href="finseq_1.html#T33" onmouseover="rs('finseq_1/T33')" onmouseout="rh()">FINSEQ_1:33</a></span>;<br/><span class="kw">hence </span><a NAME="E29:43_1_1_1_4"/>
 <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c7">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="txt" href="#E2:43_1"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="txt" href="#E4:43_1_1_1_4"><span class="lab"><font color="Green" title="E31">A21</font></span></a>, <a class="txt" href="#E13:43_1_1_1_4"><span class="lab"><font color="Green" title="E36">A26</font></span></a>, <a class="txt" href="#E27:43_1_1_1_4"><span class="lab"><font color="Green" title="E43">A33</font></span></a>, <a class="txt" href="#E19:43_1_1_1_4"><span class="lab"><font color="Green" title="E39">A29</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_5"><span class="kw">suppose </span></a><a NAME="E1:43_1_1_1_5"/>
<font color="Maroon" title="c6">F</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c7">G</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">x</font> being   <a href="qc_lang1.html#NM5" title="QC_LANG1:NM.5">bound_QC-variable</a> of <font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c10">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon" title="c1">A</font><span class="kw"> such that </span><br/><a NAME="E3:43_1_1_1_5"/><span class="lab"><font color="Green" title="E30">A34</font></span>: 
<font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K15" title="QC_LANG1:func.15">All</a> (<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">p</font>)
 ;<br/><a NAME="E4:43_1_1_1_5"/><span class="lab"><font color="Green" title="E31">A35</font></span>: 
 <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c10">p</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1_1_1_5"><span class="lab"><font color="Green" title="E30">A34</font></span></a>, <a class="ref" href="finseq_1.html#T32" onmouseover="rs('finseq_1/T32')" onmouseout="rh()">FINSEQ_1:32</a></span>;<br/><span class="kw">then </span> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c6">F</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c10">p</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c10">p</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T40" onmouseover="rs('finseq_1/T40')" onmouseout="rh()">FINSEQ_1:40</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c10">p</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c10">p</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T40" onmouseover="rs('finseq_1/T40')" onmouseout="rh()">FINSEQ_1:40</a></span>
;<br/><a NAME="E6:43_1_1_1_5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c10">p</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c6">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E7:43_1_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A36</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c10">p</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c6">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E8:43_1_1_1_5"/>
<span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_1_1_1_5"><span class="lab"><font color="Green" title="E31">A35</font></span></a>, <a class="ref" href="finseq_1.html#T41" onmouseover="rs('finseq_1/T41')" onmouseout="rh()">FINSEQ_1:41</a></span>;<br/><a NAME="E9:43_1_1_1_5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43_1"><span class="lab"><font color="Green" title="E28">A5</font></span></a></span>;<br/><a NAME="E10:43_1_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c7">G</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang1.html#T12" target="_self" onmouseover="rs('qc_lang1/T12')" onmouseout="rh()">Th12</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">x9</font> being   <a href="qc_lang1.html#NM5" title="QC_LANG1:NM.5">bound_QC-variable</a> of <font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c12">p9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon" title="c1">A</font><span class="kw"> such that </span><br/><a NAME="E12:43_1_1_1_5"/><span class="lab"><font color="Green" title="E33">A37</font></span>: 
<font color="Maroon" title="c7">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K15" title="QC_LANG1:func.15">All</a> (<font color="Maroon" title="c11">x9</font>,<font color="Maroon" title="c12">p9</font>)
 ;<br/><span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c10">p</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c11">x9</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c12">p9</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c8">s</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E3:43_1_1_1_5"><span class="lab"><font color="Green" title="E30">A34</font></span></a>, <a class="txt" href="#E12:43_1_1_1_5"><span class="lab"><font color="Green" title="E33">A37</font></span></a>, <a class="ref" href="finseq_1.html#T32" onmouseover="rs('finseq_1/T32')" onmouseout="rh()">FINSEQ_1:32</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c11">x9</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c12">p9</font></span>)</span></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c8">s</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T32" onmouseover="rs('finseq_1/T32')" onmouseout="rh()">FINSEQ_1:32</a></span>
;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A38</font></span>: <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c10">p</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c11">x9</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c12">p9</font></span>)</span></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c8">s</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1_1_1_5"><span class="lab"><font color="Green" title="E30">A34</font></span></a>, <a class="txt" href="#E4:43_1_1_1_5"><span class="lab"><font color="Green" title="E31">A35</font></span></a>, <a class="ref" href="finseq_1.html#T33" onmouseover="rs('finseq_1/T33')" onmouseout="rh()">FINSEQ_1:33</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c11">x9</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c12">p9</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c8">s</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T32" onmouseover="rs('finseq_1/T32')" onmouseout="rh()">FINSEQ_1:32</a></span>
;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A39</font></span>: <font color="Maroon" title="c9">x</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c11">x9</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c12">p9</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c8">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T41" onmouseover="rs('finseq_1/T41')" onmouseout="rh()">FINSEQ_1:41</a></span>
<br/>.= 
<font color="Maroon" title="c11">x9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T41" onmouseover="rs('finseq_1/T41')" onmouseout="rh()">FINSEQ_1:41</a></span>
;<br/><a NAME="E16:43_1_1_1_5"/><span class="kw">then </span>
 <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c10">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c12">p9</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c8">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:43_1_1_1_5"><span class="lab"><font color="Green" title="E34">A38</font></span></a>, <a class="ref" href="finseq_1.html#T33" onmouseover="rs('finseq_1/T33')" onmouseout="rh()">FINSEQ_1:33</a></span>;<br/><span class="kw">hence </span><a NAME="E17:43_1_1_1_5"/>
 <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c7">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="txt" href="#E2:43_1"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="txt" href="#E3:43_1_1_1_5"><span class="lab"><font color="Green" title="E30">A34</font></span></a>, <a class="txt" href="#E12:43_1_1_1_5"><span class="lab"><font color="Green" title="E33">A37</font></span></a>, <a class="txt" href="#E15:43_1_1_1_5"><span class="lab"><font color="Green" title="E35">A39</font></span></a>, <a class="txt" href="#E7:43_1_1_1_5"><span class="lab"><font color="Green" title="E32">A36</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E9:43_1"/>
 <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c7">G</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:43"/><span class="lab"><font color="Green" title="E25">A40</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S4" onmouseover="rs('nat_1/S4')" onmouseout="rh()">NAT_1:sch 4</a>(<a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E24">A1</font></span></a>);<br/></span>
<span class="kw">thus </span><a NAME="E3:43"/>
(  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c3">G</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c4">s</font> implies  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">@</a> <font color="Maroon" title="c3">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E25">A40</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
