<?xml version="1.0"?>
<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>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2">Valuations_in</a> <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="cqc_lang.html#NM2">CQC-variable_list</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="substut1.html#NM1">CQC_Substitution</a>;<br/>











<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="sublemma.html#K4" target="_self">Sub_P</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>;<br/>
<a NAME="E1:20"/>
 <a href="sublemma.html#K4" target="_self">Sub_P</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="ref" href="substut1.html#T9">SUBSTUT1:9</a></i>;<br/>
<a NAME="E2:20"/><b>then </b><i><font color="Green">E13</font></i>: 
<span class="p1">(<span class="default"><a href="sublemma.html#K4" target="_self">Sub_P</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="substut1.html#K19">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="sublemma.html#K5" target="_self">CQC_Subst</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>;<br/>
<a NAME="E3:20"/><i><font color="Green">E14</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="qc_lang1.html#D8">QC_LANG1:def 8</a></i>;<br/>
<a NAME="E4:20"/><i><font color="Green">E15</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="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p3">(<span class="default"><a href="sublemma.html#K3" target="_self">Val_S</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="sublemma.html#K4" target="_self">Sub_P</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#K8">*'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="valuat_1.html#K8">*'</a> <span class="p2">(<span class="default"><a href="sublemma.html#K5" target="_self">CQC_Subst</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:20_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:20_1"/><i><font color="Green">E17</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="sublemma.html#E1:20_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:20_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="sublemma.html#E1:20_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="qc_lang1.html#D8">QC_LANG1:def 8</a></i>;<br/>
<a NAME="E4:20_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> as   <a href="qc_lang1.html#NM3">bound_QC-variable</a> <b>by </b><i><a class="ref" href="sublemma.html#T5" target="_self">Th5</a></i>;<br/>
<div><i><font color="Green">E18</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_1_1"/><i><font color="Green">E19</font></i>: 
not <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:20_1_1"/><b>then </b><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K5" target="_self">CQC_Subst</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="sublemma.html#E3:20"><i><font color="Green">E14</font></i></a>, <a class="txt" href="sublemma.html#E2:20_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="substut1.html#D3">SUBSTUT1:def 3</a></i>;<br/><a NAME="E3:20_1_1"/><i><font color="Green">E21</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="valuat_1.html#K8">*'</a> <span class="p2">(<span class="default"><a href="sublemma.html#K5" target="_self">CQC_Subst</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K5" target="_self">CQC_Subst</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="sublemma.html#E2:20_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="valuat_1.html#D8">VALUAT_1:def 8</a></i>;<br/><a NAME="E4:20_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K3" target="_self">Val_S</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="sublemma.html#K4" target="_self">Sub_P</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="sublemma.html#E2:20"><i><font color="Green">E13</font></i></a>, <a class="txt" href="sublemma.html#E1:20_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="sublemma.html#T12" target="_self">Th12</a></i>;<br/><b>hence </b><a NAME="E5:20_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p3">(<span class="default"><a href="sublemma.html#K3" target="_self">Val_S</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="sublemma.html#K4" target="_self">Sub_P</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#K8">*'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="valuat_1.html#K8">*'</a> <span class="p2">(<span class="default"><a href="sublemma.html#K5" target="_self">CQC_Subst</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="sublemma.html#E2:20_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="sublemma.html#E2:20_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="sublemma.html#E3:20_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="valuat_1.html#D8">VALUAT_1:def 8</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_1_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:20_1_2"/><b>then </b><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K5" target="_self">CQC_Subst</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K4" target="_self">Sub_P</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="substut1.html#K19">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sublemma.html#E2:20"><i><font color="Green">E13</font></i></a>, <a class="txt" href="sublemma.html#E3:20"><i><font color="Green">E14</font></i></a>, <a class="txt" href="sublemma.html#E2:20_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="substut1.html#D3">SUBSTUT1:def 3</a></i>;<br/><a NAME="E3:20_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K3" target="_self">Val_S</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="sublemma.html#K4" target="_self">Sub_P</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" target="_self">Val_S</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="sublemma.html#K4" target="_self">Sub_P</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="sublemma.html#E2:20"><i><font color="Green">E13</font></i></a>, <a class="txt" href="sublemma.html#E1:20_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="sublemma.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E4:20_1_2"/><b>then </b><i><font color="Green">E21</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K3" target="_self">Val_S</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="sublemma.html#K4" target="_self">Sub_P</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2">@</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K4" target="_self">Sub_P</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="substut1.html#K19">`2</a> </span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 ;<br/><a NAME="E5:20_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K4" target="_self">Sub_P</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="substut1.html#K19">`2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sublemma.html#E2:20"><i><font color="Green">E13</font></i></a>, <a class="txt" href="sublemma.html#E1:20_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="substut1.html#D2">SUBSTUT1:def 2</a></i>;<br/><a NAME="E6:20_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K3" target="_self">Val_S</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="sublemma.html#K4" target="_self">Sub_P</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2">@</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K4" target="_self">Sub_P</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="substut1.html#K19">`2</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sublemma.html#E4:20_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E7:20_1_2"/><b>then </b><i><font color="Green">E22</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K3" target="_self">Val_S</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="sublemma.html#K4" target="_self">Sub_P</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K4" target="_self">Sub_P</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="substut1.html#K19">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="substut1.html#D2">SUBSTUT1:def 2</a></i>;<br/><a NAME="E8:20_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p3">(<span class="default"><a href="sublemma.html#K3" target="_self">Val_S</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="sublemma.html#K4" target="_self">Sub_P</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#K8">*'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K3" target="_self">Val_S</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="sublemma.html#K4" target="_self">Sub_P</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="sublemma.html#E2:20_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="valuat_1.html#D8">VALUAT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E9:20_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p3">(<span class="default"><a href="sublemma.html#K3" target="_self">Val_S</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="sublemma.html#K4" target="_self">Sub_P</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#K8">*'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="valuat_1.html#K8">*'</a> <span class="p2">(<span class="default"><a href="sublemma.html#K5" target="_self">CQC_Subst</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="sublemma.html#E2:20_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="sublemma.html#E2:20_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="sublemma.html#E7:20_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="valuat_1.html#D8">VALUAT_1:def 8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:20_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p3">(<span class="default"><a href="sublemma.html#K3" target="_self">Val_S</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="sublemma.html#K4" target="_self">Sub_P</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#K8">*'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="valuat_1.html#K8">*'</a> <span class="p2">(<span class="default"><a href="sublemma.html#K5" target="_self">CQC_Subst</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="sublemma.html#E6:20_1"><i><font color="Green">E18</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:20"/><i><font color="Green">E16</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p3">(<span class="default"><a href="sublemma.html#K3" target="_self">Val_S</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><a href="sublemma.html#K4" target="_self">Sub_P</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#K8">*'</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>
 
<b>by </b><i><a class="ref" href="valuat_1.html#D8">VALUAT_1:def 8</a></i>;<br/>
<a NAME="E6:20"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="valuat_1.html#K8">*'</a> <span class="p2">(<span class="default"><a href="sublemma.html#K5" target="_self">CQC_Subst</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="valuat_1.html#D8">VALUAT_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E7:20"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K3" target="_self">Val_S</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><a href="sublemma.html#K4" target="_self">Sub_P</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#K8">*'</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="valuat_1.html#K8">*'</a> <span class="p1">(<span class="default"><a href="sublemma.html#K5" target="_self">CQC_Subst</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="sublemma.html#E4:20"><i><font color="Green">E15</font></i></a>, <a class="txt" href="sublemma.html#E5:20"><i><font color="Green">E16</font></i></a>, <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/>


</div>
