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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(),  <a href="funct_2.html#NM1">Function</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>(),   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> ] means ( ( <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K11" target="_self">VERUM</a>  implies <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) &amp; ( <font color="Maroon">a<sub>3</sub></font> is <a href="qc_lang1.html#V2" target="_self">atomic</a> implies <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">a<sub>3</sub></font>) ) &amp; ( <font color="Maroon">a<sub>3</sub></font> is <a href="qc_lang1.html#V3" target="_self">negative</a> implies <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K17" target="_self">the_argument_of</a> <font color="Maroon">a<sub>3</sub></font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon">a<sub>3</sub></font> is <a href="qc_lang1.html#V4" target="_self">conjunctive</a> implies <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" target="_self">the_left_argument_of</a> <font color="Maroon">a<sub>3</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" target="_self">the_right_argument_of</a> <font color="Maroon">a<sub>3</sub></font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon">a<sub>3</sub></font> is <a href="qc_lang1.html#V5" target="_self">universal</a> implies <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">a<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" target="_self">the_scope_of</a> <font color="Maroon">a<sub>3</sub></font></span>)</span></span>)</span>) ) );<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="funct_2.html#NM1">Function</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>(),  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <a href="qc_lang1.html#K11" target="_self">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a>  st  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>2</sub></font> holds <br/>( ( <font color="Olive">b<sub>1</sub></font> is <a href="qc_lang1.html#V2" target="_self">atomic</a> implies <font color="Maroon">a<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) &amp; ( <font color="Olive">b<sub>1</sub></font> is <a href="qc_lang1.html#V3" target="_self">negative</a> implies <font color="Maroon">a<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K17" target="_self">the_argument_of</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span>) ) &amp; ( <font color="Olive">b<sub>1</sub></font> is <a href="qc_lang1.html#V4" target="_self">conjunctive</a> implies <font color="Maroon">a<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" target="_self">the_left_argument_of</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" target="_self">the_right_argument_of</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span>) ) &amp; ( <font color="Olive">b<sub>1</sub></font> is <a href="qc_lang1.html#V5" target="_self">universal</a> implies <font color="Maroon">a<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" target="_self">the_scope_of</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span>) ) ) ) );<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>() st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">a<sub>1</sub></font>];<br/>
<a NAME="E1:46_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>1</sub></font> = <a href="qc_lang1.html#K8" target="_self">QC-WFF</a>  <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">F<sub>2</sub></font>() as   <a href="funct_2.html#NM1">Function</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="ref" href="funcop_1.html#T57">FUNCOP_1:57</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>1</sub></font>
;<br/>

<b>thus </b><a NAME="E2:46_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <a href="qc_lang1.html#K11" target="_self">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>

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

<b>assume </b><a NAME="E3:46_1_1"/><i><font color="Green">E38</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/>

<a NAME="E4:46_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="qc_lang1.html#T34" target="_self">Th34</a></i>;<br/>
<b>hence </b><a NAME="E5:46_1_1"/>
( ( <font color="Maroon">c<sub>2</sub></font> is <a href="qc_lang1.html#V2" target="_self">atomic</a> implies <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>2</sub></font>) ) &amp; ( <font color="Maroon">c<sub>2</sub></font> is <a href="qc_lang1.html#V3" target="_self">negative</a> implies <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K17" target="_self">the_argument_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon">c<sub>2</sub></font> is <a href="qc_lang1.html#V4" target="_self">conjunctive</a> implies <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" target="_self">the_left_argument_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" target="_self">the_right_argument_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon">c<sub>2</sub></font> is <a href="qc_lang1.html#V5" target="_self">universal</a> implies <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" target="_self">the_scope_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>) ) )
 <b>by </b><i><a class="txt" href="qc_lang1.html#E3:46_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:46_1"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>given </b><font color="Maroon">c<sub>2</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><a NAME="E1:46_1_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>]
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> ,   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()] means ( (  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> ) &amp; (  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">a<sub>1</sub></font>] ) );<br/>
<a NAME="E2:46_1_2"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_1_2_1_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 or (  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K11" target="_self">VERUM</a>  ) or (  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="qc_lang1.html#V2" target="_self">atomic</a> ) or (  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="qc_lang1.html#V3" target="_self">negative</a> ) or (  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="qc_lang1.html#V4" target="_self">conjunctive</a> ) or (  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="qc_lang1.html#V5" target="_self">universal</a> ) )
 </span><b>by </b><i><a class="ref" href="qc_lang1.html#T33" target="_self">Th33</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_2_1_1_1_1"><b>case </b></a><a NAME="E1:46_1_2_1_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>;<br/><b>thus </b><a NAME="E2:46_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_2_1_1_1_2"><b>case </b></a><a NAME="E1:46_1_2_1_1_1_2"/><i><font color="Green">E41</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K11" target="_self">VERUM</a>  )
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">F<sub>2</sub></font>();<br/><b>thus </b><a NAME="E2:46_1_2_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>]
 <b>by </b><i><a class="txt" href="qc_lang1.html#E1:46_1_2_1_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="qc_lang1.html#T51" target="_self">Th51</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_2_1_1_1_3"><b>case </b></a><a NAME="E1:46_1_2_1_1_1_3"/><i><font color="Green">E41</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="qc_lang1.html#V2" target="_self">atomic</a> )
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>3</sub></font>);<br/><b>thus </b><a NAME="E2:46_1_2_1_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>]
 <b>by </b><i><a class="txt" href="qc_lang1.html#E1:46_1_2_1_1_1_3"><i><font color="Green">E41</font></i></a>, <a class="ref" href="qc_lang1.html#T51" target="_self">Th51</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_2_1_1_1_4"><b>case </b></a><a NAME="E1:46_1_2_1_1_1_4"/><i><font color="Green">E41</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="qc_lang1.html#V3" target="_self">negative</a> )
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K17" target="_self">the_argument_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>);<br/><b>thus </b><a NAME="E2:46_1_2_1_1_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>]
 <b>by </b><i><a class="txt" href="qc_lang1.html#E1:46_1_2_1_1_1_4"><i><font color="Green">E41</font></i></a>, <a class="ref" href="qc_lang1.html#T51" target="_self">Th51</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_2_1_1_1_5"><b>case </b></a><a NAME="E1:46_1_2_1_1_1_5"/><i><font color="Green">E41</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="qc_lang1.html#V4" target="_self">conjunctive</a> )
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" target="_self">the_left_argument_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" target="_self">the_right_argument_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>);<br/><b>thus </b><a NAME="E2:46_1_2_1_1_1_5"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>]
 <b>by </b><i><a class="txt" href="qc_lang1.html#E1:46_1_2_1_1_1_5"><i><font color="Green">E41</font></i></a>, <a class="ref" href="qc_lang1.html#T51" target="_self">Th51</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_2_1_1_1_6"><b>case </b></a><a NAME="E1:46_1_2_1_1_1_6"/><i><font color="Green">E41</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="qc_lang1.html#V5" target="_self">universal</a> )
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" target="_self">the_scope_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>);<br/><b>thus </b><a NAME="E2:46_1_2_1_1_1_6"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>]
 <b>by </b><i><a class="txt" href="qc_lang1.html#E1:46_1_2_1_1_1_6"><i><font color="Green">E41</font></i></a>, <a class="ref" href="qc_lang1.html#T51" target="_self">Th51</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:46_1_2_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( (  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> ) &amp; (  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 implies <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>] ) )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E4:46_1_2"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a>  holds <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="qc_lang1.html#E2:46_1_2"><i><font color="Green">E40</font></i></a>);<br/></i>
<b>take </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font>;<br/>

<b>thus </b><a NAME="E5:46_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:46_1_2_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <a href="qc_lang1.html#K11" target="_self">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>()
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_1_2_2_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <a href="qc_lang1.html#K11" target="_self">VERUM</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 or  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <a href="qc_lang1.html#K11" target="_self">VERUM</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:46_1_2_2_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <a href="qc_lang1.html#K11" target="_self">VERUM</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:46_1_2_2_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <a href="qc_lang1.html#K11" target="_self">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i><a class="txt" href="qc_lang1.html#E1:46_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="qc_lang1.html#E4:46_1_2"><i><font color="Green">E41</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:46_1_2_2_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <a href="qc_lang1.html#K11" target="_self">VERUM</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:46_1_2_2_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <a href="qc_lang1.html#K11" target="_self">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i><a class="txt" href="qc_lang1.html#E4:46_1_2"><i><font color="Green">E41</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> ;<br/>

<b>assume </b><a NAME="E2:46_1_2_2"/><i><font color="Green">E42</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<b>thus </b><a NAME="E3:46_1_2_2"/>
( <font color="Maroon">c<sub>5</sub></font> is <a href="qc_lang1.html#V2" target="_self">atomic</a> implies <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>5</sub></font>) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2_2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_2_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_1_2_2_2_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 or  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_2_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:46_1_2_2_2_1_1_1"/><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:46_1_2_2_2_1_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="qc_lang1.html#E2:46_1_2_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/><a NAME="E3:46_1_2_2_2_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="qc_lang1.html#E4:46_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="qc_lang1.html#E1:46_1_2_2_2_1_1_1"><i><font color="Green">E43</font></i></a></i>;<br/><b>hence </b><a NAME="E4:46_1_2_2_2_1_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> is <a href="qc_lang1.html#V2" target="_self">atomic</a> implies <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>5</sub></font>) )
 <b>by </b><i><a class="txt" href="qc_lang1.html#E1:46_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="qc_lang1.html#E2:46_1_2_2_2_1_1_1"><i><font color="Green">E44</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_2_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:46_1_2_2_2_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:46_1_2_2_2_1_1_2"/>
( <font color="Maroon">c<sub>5</sub></font> is <a href="qc_lang1.html#V2" target="_self">atomic</a> implies <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>5</sub></font>) )
 <b>by </b><i><a class="txt" href="qc_lang1.html#E4:46_1_2"><i><font color="Green">E41</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:46_1_2_2_2"/>
( <font color="Maroon">c<sub>5</sub></font> is <a href="qc_lang1.html#V2" target="_self">atomic</a> implies <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>5</sub></font>) )
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:46_1_2_2"/>
( <font color="Maroon">c<sub>5</sub></font> is <a href="qc_lang1.html#V3" target="_self">negative</a> implies <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K17" target="_self">the_argument_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:46_1_2_2_3"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="qc_lang1.html#V3" target="_self">negative</a>
 ;<br/>

<a NAME="E2:46_1_2_2_3"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K17" target="_self">the_argument_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="qc_lang1.html#E2:46_1_2_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="qc_lang1.html#T45" target="_self">Th45</a></i>;<br/>
<a NAME="E3:46_1_2_2_3"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K17" target="_self">the_argument_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K17" target="_self">the_argument_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="qc_lang1.html#E4:46_1_2"><i><font color="Green">E41</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2_2_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_2_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_1_2_2_3_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 or  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_2_2_3_1_1_1"><b>suppose </b></a><a NAME="E1:46_1_2_2_3_1_1_1"/><i><font color="Green">E45</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:46_1_2_2_3_1_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="qc_lang1.html#E2:46_1_2_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/><a NAME="E3:46_1_2_2_3_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="qc_lang1.html#E4:46_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="qc_lang1.html#E1:46_1_2_2_3_1_1_1"><i><font color="Green">E45</font></i></a></i>;<br/><b>hence </b><a NAME="E4:46_1_2_2_3_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K17" target="_self">the_argument_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>)
 <b>by </b><i><a class="txt" href="qc_lang1.html#E1:46_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="qc_lang1.html#E1:46_1_2_2_3"><i><font color="Green">E43</font></i></a>, <a class="txt" href="qc_lang1.html#E3:46_1_2_2_3"><i><font color="Green">E44</font></i></a>, <a class="txt" href="qc_lang1.html#E2:46_1_2_2_3_1_1_1"><i><font color="Green">E46</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_2_2_3_1_1_2"><b>suppose </b></a><a NAME="E1:46_1_2_2_3_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:46_1_2_2_3_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K17" target="_self">the_argument_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>)
 <b>by </b><i><a class="txt" href="qc_lang1.html#E4:46_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="qc_lang1.html#E1:46_1_2_2_3"><i><font color="Green">E43</font></i></a>, <a class="txt" href="qc_lang1.html#E3:46_1_2_2_3"><i><font color="Green">E44</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:46_1_2_2_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K17" target="_self">the_argument_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>)
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E5:46_1_2_2"/>
( <font color="Maroon">c<sub>5</sub></font> is <a href="qc_lang1.html#V4" target="_self">conjunctive</a> implies <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" target="_self">the_left_argument_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" target="_self">the_right_argument_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2_2_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:46_1_2_2_4"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="qc_lang1.html#V4" target="_self">conjunctive</a>
 ;<br/>

<a NAME="E2:46_1_2_2_4"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" target="_self">the_left_argument_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="qc_lang1.html#E2:46_1_2_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="qc_lang1.html#T46" target="_self">Th46</a></i>;<br/>
<a NAME="E3:46_1_2_2_4"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K18" target="_self">the_left_argument_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K18" target="_self">the_left_argument_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="qc_lang1.html#E4:46_1_2"><i><font color="Green">E41</font></i></a></i>;<br/>
<a NAME="E4:46_1_2_2_4"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" target="_self">the_right_argument_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="qc_lang1.html#E2:46_1_2_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="qc_lang1.html#E1:46_1_2_2_4"><i><font color="Green">E43</font></i></a>, <a class="ref" href="qc_lang1.html#T46" target="_self">Th46</a></i>;<br/>
<a NAME="E5:46_1_2_2_4"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K19" target="_self">the_right_argument_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K19" target="_self">the_right_argument_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="qc_lang1.html#E4:46_1_2"><i><font color="Green">E41</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2_2_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_2_2_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_1_2_2_4_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 or  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_2_2_4_1_1_1"><b>suppose </b></a><a NAME="E1:46_1_2_2_4_1_1_1"/><i><font color="Green">E46</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:46_1_2_2_4_1_1_1"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="qc_lang1.html#E2:46_1_2_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/><a NAME="E3:46_1_2_2_4_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="qc_lang1.html#E4:46_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="qc_lang1.html#E1:46_1_2_2_4_1_1_1"><i><font color="Green">E46</font></i></a></i>;<br/><b>hence </b><a NAME="E4:46_1_2_2_4_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" target="_self">the_left_argument_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" target="_self">the_right_argument_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>)
 <b>by </b><i><a class="txt" href="qc_lang1.html#E1:46_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="qc_lang1.html#E1:46_1_2_2_4"><i><font color="Green">E43</font></i></a>, <a class="txt" href="qc_lang1.html#E3:46_1_2_2_4"><i><font color="Green">E44</font></i></a>, <a class="txt" href="qc_lang1.html#E5:46_1_2_2_4"><i><font color="Green">E45</font></i></a>, <a class="txt" href="qc_lang1.html#E2:46_1_2_2_4_1_1_1"><i><font color="Green">E47</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_2_2_4_1_1_2"><b>suppose </b></a><a NAME="E1:46_1_2_2_4_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:46_1_2_2_4_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" target="_self">the_left_argument_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" target="_self">the_right_argument_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>)
 <b>by </b><i><a class="txt" href="qc_lang1.html#E4:46_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="qc_lang1.html#E1:46_1_2_2_4"><i><font color="Green">E43</font></i></a>, <a class="txt" href="qc_lang1.html#E3:46_1_2_2_4"><i><font color="Green">E44</font></i></a>, <a class="txt" href="qc_lang1.html#E5:46_1_2_2_4"><i><font color="Green">E45</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:46_1_2_2_4"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" target="_self">the_left_argument_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" target="_self">the_right_argument_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>)
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E6:46_1_2_2"/>
( <font color="Maroon">c<sub>5</sub></font> is <a href="qc_lang1.html#V5" target="_self">universal</a> implies <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" target="_self">the_scope_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2_2_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:46_1_2_2_5"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="qc_lang1.html#V5" target="_self">universal</a>
 ;<br/>

<a NAME="E2:46_1_2_2_5"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" target="_self">the_scope_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="qc_lang1.html#E2:46_1_2_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="qc_lang1.html#T47" target="_self">Th47</a></i>;<br/>
<a NAME="E3:46_1_2_2_5"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K21" target="_self">the_scope_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K21" target="_self">the_scope_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="qc_lang1.html#E4:46_1_2"><i><font color="Green">E41</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2_2_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_2_2_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_1_2_2_5_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 or  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_2_2_5_1_1_1"><b>suppose </b></a><a NAME="E1:46_1_2_2_5_1_1_1"/><i><font color="Green">E45</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:46_1_2_2_5_1_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="qc_lang1.html#E2:46_1_2_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/><a NAME="E3:46_1_2_2_5_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="qc_lang1.html#E4:46_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="qc_lang1.html#E1:46_1_2_2_5_1_1_1"><i><font color="Green">E45</font></i></a></i>;<br/><b>hence </b><a NAME="E4:46_1_2_2_5_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" target="_self">the_scope_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>)
 <b>by </b><i><a class="txt" href="qc_lang1.html#E1:46_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="qc_lang1.html#E1:46_1_2_2_5"><i><font color="Green">E43</font></i></a>, <a class="txt" href="qc_lang1.html#E3:46_1_2_2_5"><i><font color="Green">E44</font></i></a>, <a class="txt" href="qc_lang1.html#E2:46_1_2_2_5_1_1_1"><i><font color="Green">E46</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_2_2_5_1_1_2"><b>suppose </b></a><a NAME="E1:46_1_2_2_5_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:46_1_2_2_5_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" target="_self">the_scope_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>)
 <b>by </b><i><a class="txt" href="qc_lang1.html#E4:46_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="qc_lang1.html#E1:46_1_2_2_5"><i><font color="Green">E43</font></i></a>, <a class="txt" href="qc_lang1.html#E3:46_1_2_2_5"><i><font color="Green">E44</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:46_1_2_2_5"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" target="_self">the_scope_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>)
 ;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E3:46_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="qc_lang1.html#E1:46_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="qc_lang1.html#E2:46_1"><i><font color="Green">E38</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>() st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>2</sub></font>, <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Olive">b<sub>1</sub></font></span>)</span>] holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> ) );<br/>
<a NAME="E4:46_1"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a>  &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>] holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:46_1_3"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> 
 <b>and </b><br/><a NAME="E2:46_1_3"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>]
 <b>and </b><br/><a NAME="E3:46_1_3"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>]
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> <b> such that </b><br/><a NAME="E5:46_1_3"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E6:46_1_3"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>() st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>, <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>] holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="qc_lang1.html#E2:46_1_3"><i><font color="Green">E41</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E8:46_1_3"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>5</sub></font>, <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>]
 <b>by </b><i><a class="txt" href="qc_lang1.html#E3:46_1"><i><font color="Green">E39</font></i></a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>2</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="qc_lang1.html#E6:46_1_3"><i><font color="Green">E44</font></i></a>, <a class="txt" href="qc_lang1.html#E8:46_1_3"><i><font color="Green">E45</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="qc_lang1.html#E3:46_1_3"><i><font color="Green">E42</font></i></a>, <a class="txt" href="qc_lang1.html#E5:46_1_3"><i><font color="Green">E43</font></i></a>, <a class="txt" href="qc_lang1.html#E8:46_1_3"><i><font color="Green">E45</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E5:46_1"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a>  holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:46_1_4"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a>  ;<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E4:46_1_4"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>3</sub></font>, <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>]
 <b>by </b><i><a class="txt" href="qc_lang1.html#E3:46_1"><i><font color="Green">E39</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>2</sub></font>
;<br/>

<b>thus </b><a NAME="E5:46_1_4"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>();<br/>

<b>assume </b><a NAME="E6:46_1_4"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>4</sub></font>, <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>]
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>5</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> ] means (  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> implies <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<div><i><font color="Green">E44</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> ;<br/><b>thus </b><a NAME="E1:46_1_4_1"/>
( <font color="Maroon">c<sub>5</sub></font> is <a href="qc_lang1.html#V2" target="_self">atomic</a> implies <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon">c<sub>5</sub></font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_4_1_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:46_1_4_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="qc_lang1.html#V2" target="_self">atomic</a>
 <b>and </b><br/><a NAME="E2:46_1_4_1_1"/><i><font color="Green">E46</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>5</sub></font>)
<b>by </b><i><a class="txt" href="qc_lang1.html#E4:46_1_4"><i><font color="Green">E42</font></i></a>, <a class="txt" href="qc_lang1.html#E1:46_1_4_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="qc_lang1.html#E2:46_1_4_1_1"><i><font color="Green">E46</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="qc_lang1.html#E6:46_1_4"><i><font color="Green">E43</font></i></a>, <a class="txt" href="qc_lang1.html#E1:46_1_4_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="qc_lang1.html#E2:46_1_4_1_1"><i><font color="Green">E46</font></i></a></i>
;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E2:46_1_4_1"/>
<font color="Maroon">S<sub>5</sub></font>[ <a href="qc_lang1.html#K11" target="_self">VERUM</a> ]
 <b>by </b><i><a class="txt" href="qc_lang1.html#E4:46_1_4"><i><font color="Green">E42</font></i></a>, <a class="txt" href="qc_lang1.html#E6:46_1_4"><i><font color="Green">E43</font></i></a></i>;<br/><b>thus </b><a NAME="E3:46_1_4_1"/>
( <font color="Maroon">c<sub>5</sub></font> is <a href="qc_lang1.html#V3" target="_self">negative</a> &amp; <font color="Maroon">S<sub>5</sub></font>[ <a href="qc_lang1.html#K17" target="_self">the_argument_of</a> <font color="Maroon">c<sub>5</sub></font>] implies <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon">c<sub>5</sub></font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_4_1_2"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:46_1_4_1_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="qc_lang1.html#V3" target="_self">negative</a>
 <b>and </b><br/><a NAME="E2:46_1_4_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">S<sub>5</sub></font>[ <a href="qc_lang1.html#K17" target="_self">the_argument_of</a> <font color="Maroon">c<sub>5</sub></font>]
 <b>and </b><br/><a NAME="E3:46_1_4_1_2"/><i><font color="Green">E47</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<a NAME="E4:46_1_4_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K17" target="_self">the_argument_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="qc_lang1.html#E1:46_1_4_1_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="qc_lang1.html#T45" target="_self">Th45</a></i>;<br/>
<b>hence </b><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K17" target="_self">the_argument_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>)
<b>by </b><i><a class="txt" href="qc_lang1.html#E4:46_1_4"><i><font color="Green">E42</font></i></a>, <a class="txt" href="qc_lang1.html#E1:46_1_4_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="qc_lang1.html#E2:46_1_4_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="qc_lang1.html#E3:46_1_4_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="qc_lang1.html#E6:46_1_4"><i><font color="Green">E43</font></i></a>, <a class="txt" href="qc_lang1.html#E1:46_1_4_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="qc_lang1.html#E3:46_1_4_1_2"><i><font color="Green">E47</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div><b>thus </b><a NAME="E4:46_1_4_1"/>
( <font color="Maroon">c<sub>5</sub></font> is <a href="qc_lang1.html#V4" target="_self">conjunctive</a> &amp; <font color="Maroon">S<sub>5</sub></font>[ <a href="qc_lang1.html#K18" target="_self">the_left_argument_of</a> <font color="Maroon">c<sub>5</sub></font>] &amp; <font color="Maroon">S<sub>5</sub></font>[ <a href="qc_lang1.html#K19" target="_self">the_right_argument_of</a> <font color="Maroon">c<sub>5</sub></font>] implies <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon">c<sub>5</sub></font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_4_1_3"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:46_1_4_1_3"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="qc_lang1.html#V4" target="_self">conjunctive</a>
 <b>and </b><br/><a NAME="E2:46_1_4_1_3"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">S<sub>5</sub></font>[ <a href="qc_lang1.html#K18" target="_self">the_left_argument_of</a> <font color="Maroon">c<sub>5</sub></font>]
 <b>and </b><br/><a NAME="E3:46_1_4_1_3"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">S<sub>5</sub></font>[ <a href="qc_lang1.html#K19" target="_self">the_right_argument_of</a> <font color="Maroon">c<sub>5</sub></font>]
 <b>and </b><br/><a NAME="E4:46_1_4_1_3"/><i><font color="Green">E48</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<a NAME="E5:46_1_4_1_3"/><i><font color="Green">E49</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" target="_self">the_left_argument_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="qc_lang1.html#E1:46_1_4_1_3"><i><font color="Green">E45</font></i></a>, <a class="ref" href="qc_lang1.html#T46" target="_self">Th46</a></i>;<br/>
<a NAME="E6:46_1_4_1_3"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" target="_self">the_right_argument_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="qc_lang1.html#E1:46_1_4_1_3"><i><font color="Green">E45</font></i></a>, <a class="ref" href="qc_lang1.html#T46" target="_self">Th46</a></i>;<br/>
<b>hence </b><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" target="_self">the_left_argument_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" target="_self">the_right_argument_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>)
<b>by </b><i><a class="txt" href="qc_lang1.html#E4:46_1_4"><i><font color="Green">E42</font></i></a>, <a class="txt" href="qc_lang1.html#E1:46_1_4_1_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="qc_lang1.html#E2:46_1_4_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="qc_lang1.html#E3:46_1_4_1_3"><i><font color="Green">E47</font></i></a>, <a class="txt" href="qc_lang1.html#E4:46_1_4_1_3"><i><font color="Green">E48</font></i></a>, <a class="txt" href="qc_lang1.html#E5:46_1_4_1_3"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="qc_lang1.html#E6:46_1_4"><i><font color="Green">E43</font></i></a>, <a class="txt" href="qc_lang1.html#E1:46_1_4_1_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="qc_lang1.html#E4:46_1_4_1_3"><i><font color="Green">E48</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div><b>thus </b><a NAME="E5:46_1_4_1"/>
( <font color="Maroon">c<sub>5</sub></font> is <a href="qc_lang1.html#V5" target="_self">universal</a> &amp; <font color="Maroon">S<sub>5</sub></font>[ <a href="qc_lang1.html#K21" target="_self">the_scope_of</a> <font color="Maroon">c<sub>5</sub></font>] implies <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon">c<sub>5</sub></font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_4_1_4"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:46_1_4_1_4"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="qc_lang1.html#V5" target="_self">universal</a>
 <b>and </b><br/><a NAME="E2:46_1_4_1_4"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">S<sub>5</sub></font>[ <a href="qc_lang1.html#K21" target="_self">the_scope_of</a> <font color="Maroon">c<sub>5</sub></font>]
 <b>and </b><br/><a NAME="E3:46_1_4_1_4"/><i><font color="Green">E47</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<a NAME="E4:46_1_4_1_4"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" target="_self">the_scope_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="qc_lang1.html#E1:46_1_4_1_4"><i><font color="Green">E45</font></i></a>, <a class="ref" href="qc_lang1.html#T47" target="_self">Th47</a></i>;<br/>
<b>hence </b><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" target="_self">the_scope_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>)
<b>by </b><i><a class="txt" href="qc_lang1.html#E4:46_1_4"><i><font color="Green">E42</font></i></a>, <a class="txt" href="qc_lang1.html#E1:46_1_4_1_4"><i><font color="Green">E45</font></i></a>, <a class="txt" href="qc_lang1.html#E2:46_1_4_1_4"><i><font color="Green">E46</font></i></a>, <a class="txt" href="qc_lang1.html#E3:46_1_4_1_4"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="qc_lang1.html#E6:46_1_4"><i><font color="Green">E43</font></i></a>, <a class="txt" href="qc_lang1.html#E1:46_1_4_1_4"><i><font color="Green">E45</font></i></a>, <a class="txt" href="qc_lang1.html#E3:46_1_4_1_4"><i><font color="Green">E47</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E8:46_1_4"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a>  holds <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="qc_lang1.html#S2" target="_self">QC_LANG1:sch 2</a>(<a class="txt" href="qc_lang1.html#E7:46_1_4"><i><font color="Green">E44</font></i></a>);<br/></i>
<b>hence </b><a NAME="E9:46_1_4"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>1</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E7:46_1"/><i><font color="Green">E42</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> 
 <b>and </b><br/><a NAME="E8:46_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a>  holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(<a class="txt" href="qc_lang1.html#E4:46_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="qc_lang1.html#E5:46_1"><i><font color="Green">E41</font></i></a>);<br/></i>
<a NAME="E9:46_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:46_1_5"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:46_1_5"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> 
 <b>and </b><br/><a NAME="E4:46_1_5"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="qc_lang1.html#E7:46_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> <b> such that </b><br/><a NAME="E6:46_1_5"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E7:46_1_5"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>() st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>, <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>] holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="qc_lang1.html#E8:46_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="qc_lang1.html#E3:46_1_5"><i><font color="Green">E44</font></i></a>, <a class="txt" href="qc_lang1.html#E4:46_1_5"><i><font color="Green">E45</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E9:46_1_5"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>5</sub></font>, <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>]
 <b>by </b><i><a class="txt" href="qc_lang1.html#E3:46_1"><i><font color="Green">E39</font></i></a></i>;<br/>
<a NAME="E10:46_1_5"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="qc_lang1.html#E7:46_1_5"><i><font color="Green">E46</font></i></a>, <a class="txt" href="qc_lang1.html#E9:46_1_5"><i><font color="Green">E47</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:46_1_5"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="txt" href="qc_lang1.html#E7:46_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>2</sub></font>
;<br/>

<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E12:46_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>3</sub></font>, <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <a href="qc_lang1.html#K11" target="_self">VERUM</a> </span>)</span>]
 <b>by </b><i><a class="txt" href="qc_lang1.html#E3:46_1"><i><font color="Green">E39</font></i></a></i>;<br/>
<a NAME="E13:46_1"/>
<font color="Maroon">S<sub>4</sub></font>[ <a href="qc_lang1.html#K11" target="_self">VERUM</a> ,<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <a href="qc_lang1.html#K11" target="_self">VERUM</a> ]
 
<b>by </b><i><a class="txt" href="qc_lang1.html#E8:46_1"><i><font color="Green">E43</font></i></a></i>;<br/>
<b>hence </b><a NAME="E14:46_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <a href="qc_lang1.html#K11" target="_self">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i><a class="txt" href="qc_lang1.html#E12:46_1"><i><font color="Green">E44</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> ;<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> <b> such that </b><br/><a NAME="E16:46_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E17:46_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>() st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>, <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>] holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="qc_lang1.html#E8:46_1"><i><font color="Green">E43</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E19:46_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>6</sub></font>, <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>]
 <b>by </b><i><a class="txt" href="qc_lang1.html#E3:46_1"><i><font color="Green">E39</font></i></a></i>;<br/>
<a NAME="E20:46_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="qc_lang1.html#E16:46_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="qc_lang1.html#E17:46_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="qc_lang1.html#E19:46_1"><i><font color="Green">E47</font></i></a></i>;<br/>
<b>hence </b><a NAME="E21:46_1"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="qc_lang1.html#V2" target="_self">atomic</a> implies <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>4</sub></font>) )
 <b>by </b><i><a class="txt" href="qc_lang1.html#E16:46_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="qc_lang1.html#E19:46_1"><i><font color="Green">E47</font></i></a></i>;<br/>

<a NAME="E22:46_1"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:46_1_6"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>

<b>thus </b><a NAME="E2:46_1_6"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <a href="qc_lang1.html#K11" target="_self">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i><a class="txt" href="qc_lang1.html#E19:46_1"><i><font color="Green">E47</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> ;<br/>

<b>assume </b><a NAME="E3:46_1_6"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<a NAME="E4:46_1_6"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="qc_lang1.html#E1:46_1_6"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:46_1_6"/>
( ( <font color="Maroon">c<sub>8</sub></font> is <a href="qc_lang1.html#V2" target="_self">atomic</a> implies <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>8</sub></font>) ) &amp; ( <font color="Maroon">c<sub>8</sub></font> is <a href="qc_lang1.html#V3" target="_self">negative</a> implies <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K17" target="_self">the_argument_of</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon">c<sub>8</sub></font> is <a href="qc_lang1.html#V4" target="_self">conjunctive</a> implies <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" target="_self">the_left_argument_of</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" target="_self">the_right_argument_of</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon">c<sub>8</sub></font> is <a href="qc_lang1.html#V5" target="_self">universal</a> implies <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" target="_self">the_scope_of</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>) ) )
 <b>by </b><i><a class="txt" href="qc_lang1.html#E16:46_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="qc_lang1.html#E19:46_1"><i><font color="Green">E47</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E23:46_1"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="qc_lang1.html#V3" target="_self">negative</a> implies <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K17" target="_self">the_argument_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_7"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:46_1_7"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="qc_lang1.html#V3" target="_self">negative</a>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="qc_lang1.html#K17" target="_self">the_argument_of</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K17" target="_self">the_argument_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>;<br/>
<a NAME="E2:46_1_7"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K17" target="_self">the_argument_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="qc_lang1.html#E1:46_1_7"><i><font color="Green">E50</font></i></a>, <a class="ref" href="qc_lang1.html#T45" target="_self">Th45</a></i>;<br/>
<a NAME="E3:46_1_7"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>6</sub></font>, <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K17" target="_self">the_argument_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>]
 
<b>by </b><i><a class="txt" href="qc_lang1.html#E22:46_1"><i><font color="Green">E49</font></i></a></i>;<br/>
<a NAME="E4:46_1_7"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K17" target="_self">the_argument_of</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>() st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>2</sub></font>, <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Olive">b<sub>1</sub></font></span>)</span>] holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K17" target="_self">the_argument_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 
<b>by </b><i><a class="txt" href="qc_lang1.html#E8:46_1"><i><font color="Green">E43</font></i></a></i>;<br/>
<a NAME="E5:46_1_7"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K17" target="_self">the_argument_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K17" target="_self">the_argument_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="qc_lang1.html#E3:46_1_7"><i><font color="Green">E51</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:46_1_7"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K17" target="_self">the_argument_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>)
 <b>by </b><i><a class="txt" href="qc_lang1.html#E16:46_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="qc_lang1.html#E19:46_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="qc_lang1.html#E20:46_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="qc_lang1.html#E1:46_1_7"><i><font color="Green">E50</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E24:46_1"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="qc_lang1.html#V4" target="_self">conjunctive</a> implies <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" target="_self">the_left_argument_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" target="_self">the_right_argument_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_8"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:46_1_8"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="qc_lang1.html#V4" target="_self">conjunctive</a>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="qc_lang1.html#K18" target="_self">the_left_argument_of</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" target="_self">the_left_argument_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="qc_lang1.html#K19" target="_self">the_right_argument_of</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> =  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" target="_self">the_right_argument_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>;<br/>
<a NAME="E2:46_1_8"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" target="_self">the_left_argument_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="qc_lang1.html#E1:46_1_8"><i><font color="Green">E50</font></i></a>, <a class="ref" href="qc_lang1.html#T46" target="_self">Th46</a></i>;<br/>
<a NAME="E3:46_1_8"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>6</sub></font>, <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" target="_self">the_left_argument_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>]
 
<b>by </b><i><a class="txt" href="qc_lang1.html#E22:46_1"><i><font color="Green">E49</font></i></a></i>;<br/>
<a NAME="E4:46_1_8"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" target="_self">the_right_argument_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="qc_lang1.html#E1:46_1_8"><i><font color="Green">E50</font></i></a>, <a class="ref" href="qc_lang1.html#T46" target="_self">Th46</a></i>;<br/>
<a NAME="E5:46_1_8"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>6</sub></font>, <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" target="_self">the_right_argument_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>]
 
<b>by </b><i><a class="txt" href="qc_lang1.html#E22:46_1"><i><font color="Green">E49</font></i></a></i>;<br/>
<a NAME="E6:46_1_8"/><i><font color="Green">E53</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K18" target="_self">the_left_argument_of</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>() st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>2</sub></font>, <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Olive">b<sub>1</sub></font></span>)</span>] holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K18" target="_self">the_left_argument_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 
<b>by </b><i><a class="txt" href="qc_lang1.html#E8:46_1"><i><font color="Green">E43</font></i></a></i>;<br/>
<a NAME="E7:46_1_8"/><i><font color="Green">E54</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K19" target="_self">the_right_argument_of</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>() st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>2</sub></font>, <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Olive">b<sub>1</sub></font></span>)</span>] holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K19" target="_self">the_right_argument_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 
<b>by </b><i><a class="txt" href="qc_lang1.html#E8:46_1"><i><font color="Green">E43</font></i></a></i>;<br/>
<a NAME="E8:46_1_8"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K18" target="_self">the_left_argument_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K18" target="_self">the_left_argument_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="qc_lang1.html#E3:46_1_8"><i><font color="Green">E51</font></i></a>, <a class="txt" href="qc_lang1.html#E6:46_1_8"><i><font color="Green">E53</font></i></a></i>;<br/>
<a NAME="E9:46_1_8"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K19" target="_self">the_right_argument_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K19" target="_self">the_right_argument_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="qc_lang1.html#E5:46_1_8"><i><font color="Green">E52</font></i></a>, <a class="txt" href="qc_lang1.html#E7:46_1_8"><i><font color="Green">E54</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:46_1_8"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" target="_self">the_left_argument_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" target="_self">the_right_argument_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>)
 <b>by </b><i><a class="txt" href="qc_lang1.html#E16:46_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="qc_lang1.html#E19:46_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="qc_lang1.html#E20:46_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="qc_lang1.html#E1:46_1_8"><i><font color="Green">E50</font></i></a>, <a class="txt" href="qc_lang1.html#E8:46_1_8"><i><font color="Green">E55</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E25:46_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="qc_lang1.html#V5" target="_self">universal</a>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="qc_lang1.html#K21" target="_self">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" target="_self">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>;<br/>
<a NAME="E26:46_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" target="_self">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="qc_lang1.html#E25:46_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="qc_lang1.html#T47" target="_self">Th47</a></i>;<br/>
<a NAME="E27:46_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>6</sub></font>, <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" target="_self">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>]
 
<b>by </b><i><a class="txt" href="qc_lang1.html#E22:46_1"><i><font color="Green">E49</font></i></a></i>;<br/>
<a NAME="E28:46_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K21" target="_self">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>() st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>2</sub></font>, <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Olive">b<sub>1</sub></font></span>)</span>] holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K21" target="_self">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 
<b>by </b><i><a class="txt" href="qc_lang1.html#E8:46_1"><i><font color="Green">E43</font></i></a></i>;<br/>
<a NAME="E29:46_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K21" target="_self">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K21" target="_self">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="qc_lang1.html#E27:46_1"><i><font color="Green">E51</font></i></a></i>;<br/>
<b>hence </b><a NAME="E30:46_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" target="_self">the_scope_of</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>)
 <b>by </b><i><a class="txt" href="qc_lang1.html#E16:46_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="qc_lang1.html#E19:46_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="qc_lang1.html#E20:46_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="qc_lang1.html#E25:46_1"><i><font color="Green">E50</font></i></a></i>;<br/>


</div>
