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

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="qc_lang1.html#K5">QC-pred_symbols</a> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</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#K5">QC-pred_symbols</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; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="cqc_lang.html#NM2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6">the_arity_of</a> <font color="Olive">b<sub>1</sub></font> : <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1" target="_self">|-</a> <font color="Olive">b<sub>1</sub></font> <a href="henmodel.html#K3" target="_self">!</a> <font color="Olive">b<sub>2</sub></font> </span>}</span>  );<br/>
<a NAME="E1:23_1_1"/><i><font color="Green">E16</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#K5">QC-pred_symbols</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] &amp; <font color="Maroon">S<sub>1</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>
 
;<br/>
<a NAME="E2:23_1_1"/><i><font color="Green">E17</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#K5">QC-pred_symbols</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>1</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="23_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:23_1_1_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K5">QC-pred_symbols</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K5">QC-pred_symbols</a>  <b>by </b><i><a class="txt" href="henmodel.html#E1:23_1_1_1"><i><font color="Green">E18</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:23_1_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="cqc_lang.html#NM2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6">the_arity_of</a> <font color="Maroon">c<sub>4</sub></font> : <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1" target="_self">|-</a> <font color="Maroon">c<sub>4</sub></font> <a href="henmodel.html#K3" target="_self">!</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 ;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>take </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font>;<br/>

<b>thus </b><a NAME="E5:23_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="cqc_lang.html#NM2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6">the_arity_of</a> <font color="Maroon">c<sub>6</sub></font> : <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1" target="_self">|-</a> <font color="Maroon">c<sub>6</sub></font> <a href="henmodel.html#K3" target="_self">!</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  )
 <b>by </b><i><a class="txt" href="henmodel.html#E4:23_1_1_1"><i><font color="Green">E19</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:23_1_1"/><i><font color="Green">E18</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K5">QC-pred_symbols</a>  &amp; ( 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#K5">QC-pred_symbols</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>3</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="henmodel.html#E1:23_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="henmodel.html#E2:23_1_1"><i><font color="Green">E17</font></i></a>);<br/></i>
<a NAME="E5:23_1_1"/><i><font color="Green">E19</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#K5">QC-pred_symbols</a> <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="margrel1.html#K3">relations_on</a> <a href="henmodel.html#K2" target="_self">HCar</a>  st <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> holds <br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> iff ex <font color="Olive">b<sub>4</sub></font> being  <a href="cqc_lang.html#NM2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6">the_arity_of</a> <font color="Olive">b<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1" target="_self">|-</a> <font color="Olive">b<sub>1</sub></font> <a href="henmodel.html#K3" target="_self">!</a> <font color="Olive">b<sub>4</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2"><b>proof </b></a><div class="add">

<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#K5">QC-pred_symbols</a> ;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="margrel1.html#K3">relations_on</a> <a href="henmodel.html#K2" target="_self">HCar</a> ;<br/>



<b>assume </b><a NAME="E1:23_1_1_2"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E2:23_1_1_2"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> implies ex <font color="Olive">b<sub>1</sub></font> being  <a href="cqc_lang.html#NM2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6">the_arity_of</a> <font color="Maroon">c<sub>4</sub></font> st <br/>( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1" target="_self">|-</a> <font color="Maroon">c<sub>4</sub></font> <a href="henmodel.html#K3" target="_self">!</a> <font color="Olive">b<sub>1</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1_1_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E2:23_1_1_2_1"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="henmodel.html#E1:23_1_1_2"><i><font color="Green">E20</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K5">QC-pred_symbols</a> <b> such that </b><br/><a NAME="E4:23_1_1_2_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="cqc_lang.html#NM2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6">the_arity_of</a> <font color="Maroon">c<sub>7</sub></font> : <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1" target="_self">|-</a> <font color="Maroon">c<sub>7</sub></font> <a href="henmodel.html#K3" target="_self">!</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  )
 <b>by </b><i><a class="txt" href="henmodel.html#E4:23_1_1"><i><font color="Green">E18</font></i></a></i>;<br/>
<b>thus </b><a NAME="E5:23_1_1_2_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="cqc_lang.html#NM2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6">the_arity_of</a> <font color="Maroon">c<sub>4</sub></font> st <br/>( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1" target="_self">|-</a> <font color="Maroon">c<sub>4</sub></font> <a href="henmodel.html#K3" target="_self">!</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="henmodel.html#E2:23_1_1_2_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="henmodel.html#E4:23_1_1_2_1"><i><font color="Green">E22</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E3:23_1_1_2"/>
( ex <font color="Olive">b<sub>1</sub></font> being  <a href="cqc_lang.html#NM2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6">the_arity_of</a> <font color="Maroon">c<sub>4</sub></font> st <br/>( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1" target="_self">|-</a> <font color="Maroon">c<sub>4</sub></font> <a href="henmodel.html#K3" target="_self">!</a> <font color="Olive">b<sub>1</sub></font> ) implies <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">c<sub>7</sub></font> being   <a href="cqc_lang.html#NM2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6">the_arity_of</a> <font color="Maroon">c<sub>4</sub></font><b> such that </b><a NAME="E1:23_1_1_2_2"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1" target="_self">|-</a> <font color="Maroon">c<sub>4</sub></font> <a href="henmodel.html#K3" target="_self">!</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/>

<a NAME="E2:23_1_1_2_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K5">QC-pred_symbols</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="cqc_lang.html#NM2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6">the_arity_of</a> <font color="Olive">b<sub>1</sub></font> : <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1" target="_self">|-</a> <font color="Olive">b<sub>1</sub></font> <a href="henmodel.html#K3" target="_self">!</a> <font color="Olive">b<sub>2</sub></font> </span>}</span>  )
 
<b>by </b><i><a class="txt" href="henmodel.html#E4:23_1_1"><i><font color="Green">E18</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:23_1_1_2_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="henmodel.html#E1:23_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="henmodel.html#E1:23_1_1_2_2"><i><font color="Green">E21</font></i></a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E6:23_1_1"/><i><font color="Green">E20</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#K5">QC-pred_symbols</a> <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="margrel1.html#K3">relations_on</a> <a href="henmodel.html#K2" target="_self">HCar</a>  holds <br/>(  not <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="margrel1.html#R1">=</a>  <a href="margrel1.html#K4">empty_rel</a> <a href="henmodel.html#K2" target="_self">HCar</a>  or  <a href="qc_lang1.html#K6">the_arity_of</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K5">the_arity_of</a> <font color="Olive">b<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_3"><b>proof </b></a><div class="add">

<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#K5">QC-pred_symbols</a> ;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="margrel1.html#K3">relations_on</a> <a href="henmodel.html#K2" target="_self">HCar</a> ;<br/>



<b>assume </b><a NAME="E1:23_1_1_3"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>assume </b><a NAME="E2:23_1_1_3"/><i><font color="Green">E22</font></i>: 
not <font color="Maroon">c<sub>5</sub></font> <a href="margrel1.html#R1">=</a>  <a href="margrel1.html#K4">empty_rel</a> <a href="henmodel.html#K2" target="_self">HCar</a> 
 ;<br/>

<a NAME="E3:23_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="margrel1.html#T32">MARGREL1:32</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:23_1_1_3"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K5">QC-pred_symbols</a> <b> such that </b><br/><a NAME="E7:23_1_1_3"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="cqc_lang.html#NM2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6">the_arity_of</a> <font color="Maroon">c<sub>7</sub></font> : <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1" target="_self">|-</a> <font color="Maroon">c<sub>7</sub></font> <a href="henmodel.html#K3" target="_self">!</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  )
 <b>by </b><i><a class="txt" href="henmodel.html#E4:23_1_1"><i><font color="Green">E18</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="cqc_lang.html#NM2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6">the_arity_of</a> <font color="Maroon">c<sub>7</sub></font><b> such that </b><br/><a NAME="E9:23_1_1_3"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1" target="_self">|-</a> <font color="Maroon">c<sub>7</sub></font> <a href="henmodel.html#K3" target="_self">!</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="henmodel.html#E1:23_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="henmodel.html#E5:23_1_1_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="henmodel.html#E7:23_1_1_3"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E10:23_1_1_3"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a> 
 
<b>by </b><i><a class="ref" href="cqc_lang.html#D5">CQC_LANG:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="henmodel.html#K2" target="_self">HCar</a>  <b>by </b><i><a class="txt" href="henmodel.html#E9:23_1_1_3"><i><font color="Green">E25</font></i></a>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E12:23_1_1_3"/>
 <a href="margrel1.html#K5">the_arity_of</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="henmodel.html#E2:23_1_1_3"><i><font color="Green">E22</font></i></a>, <a class="txt" href="henmodel.html#E5:23_1_1_3"><i><font color="Green">E23</font></i></a>, <a class="ref" href="margrel1.html#D11">MARGREL1:def 11</a></i>;<br/>
<b>hence </b><a NAME="E13:23_1_1_3"/>
 <a href="qc_lang1.html#K6">the_arity_of</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K5">the_arity_of</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="henmodel.html#E7:23_1_1_3"><i><font color="Green">E24</font></i></a>, <a class="txt" href="henmodel.html#E9:23_1_1_3"><i><font color="Green">E25</font></i></a>, <a class="ref" href="qc_lang1.html#D8">QC_LANG1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:23_1_1"/>
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="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="margrel1.html#K3">relations_on</a> <a href="henmodel.html#K2" target="_self">HCar</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:23_1_1_4"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:23_1_1_4"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K5">QC-pred_symbols</a> <b> such that </b><br/><a NAME="E5:23_1_1_4"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="cqc_lang.html#NM2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6">the_arity_of</a> <font color="Maroon">c<sub>6</sub></font> : <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1" target="_self">|-</a> <font color="Maroon">c<sub>6</sub></font> <a href="henmodel.html#K3" target="_self">!</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  )
 <b>by </b><i><a class="txt" href="henmodel.html#E4:23_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="henmodel.html#E3:23_1_1_4"><i><font color="Green">E21</font></i></a></i>;<br/>
<a NAME="E6:23_1_1_4"/>
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> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <a href="henmodel.html#K2" target="_self">HCar</a>  <a href="finseq_1.html#K13">*</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:23_1_1_4_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="cqc_lang.html#NM2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6">the_arity_of</a> <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E3:23_1_1_4_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1" target="_self">|-</a> <font color="Maroon">c<sub>6</sub></font> <a href="henmodel.html#K3" target="_self">!</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="henmodel.html#E3:23_1_1_4"><i><font color="Green">E21</font></i></a>, <a class="txt" href="henmodel.html#E5:23_1_1_4"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E4:23_1_1_4_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a> 
 
<b>by </b><i><a class="ref" href="cqc_lang.html#D5">CQC_LANG:def 5</a></i>;<br/>
<a NAME="E5:23_1_1_4_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="henmodel.html#K2" target="_self">HCar</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E6:23_1_1_4_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <a href="henmodel.html#K2" target="_self">HCar</a>  <a href="finseq_1.html#K13">*</a> 
 <b>by </b><i><a class="txt" href="henmodel.html#E3:23_1_1_4_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:23_1_1_4"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <a href="henmodel.html#K2" target="_self">HCar</a>  <a href="finseq_1.html#K13">*</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E8:23_1_1_4"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="henmodel.html#K2" target="_self">HCar</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> holds <br/> <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="henmodel.html#K2" target="_self">HCar</a> ;<br/>

<b>assume </b><a NAME="E1:23_1_1_4_2"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="cqc_lang.html#NM2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6">the_arity_of</a> <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E3:23_1_1_4_2"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1" target="_self">|-</a> <font color="Maroon">c<sub>6</sub></font> <a href="henmodel.html#K3" target="_self">!</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="henmodel.html#E3:23_1_1_4"><i><font color="Green">E21</font></i></a>, <a class="txt" href="henmodel.html#E5:23_1_1_4"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="cqc_lang.html#NM2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6">the_arity_of</a> <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E5:23_1_1_4_2"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1" target="_self">|-</a> <font color="Maroon">c<sub>6</sub></font> <a href="henmodel.html#K3" target="_self">!</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="henmodel.html#E3:23_1_1_4"><i><font color="Green">E21</font></i></a>, <a class="txt" href="henmodel.html#E5:23_1_1_4"><i><font color="Green">E22</font></i></a>, <a class="txt" href="henmodel.html#E1:23_1_1_4_2"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E6:23_1_1_4_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K6">the_arity_of</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="henmodel.html#E3:23_1_1_4_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="qc_lang1.html#D8">QC_LANG1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E7:23_1_1_4_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="henmodel.html#E5:23_1_1_4_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="qc_lang1.html#D8">QC_LANG1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:23_1_1_4"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="margrel1.html#K3">relations_on</a> <a href="henmodel.html#K2" target="_self">HCar</a> 
 <b>by </b><i><a class="txt" href="henmodel.html#E7:23_1_1_4"><i><font color="Green">E23</font></i></a>, <a class="ref" href="margrel1.html#D8">MARGREL1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:23_1_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="margrel1.html#K3">relations_on</a> <a href="henmodel.html#K2" target="_self">HCar</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="qc_lang1.html#K5">QC-pred_symbols</a> , <a href="margrel1.html#K3">relations_on</a> <a href="henmodel.html#K2" target="_self">HCar</a>  <b>by </b><i><a class="txt" href="henmodel.html#E4:23_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="valuat_1.html#M1">interpretation</a> of  <a href="henmodel.html#K2" target="_self">HCar</a>  <b>by </b><i><a class="txt" href="henmodel.html#E6:23_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="valuat_1.html#D10">VALUAT_1:def 10</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E11:23_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K5">QC-pred_symbols</a> <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="margrel1.html#K3">relations_on</a> <a href="henmodel.html#K2" target="_self">HCar</a>  st <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="margrel1.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> holds <br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> iff ex <font color="Olive">b<sub>4</sub></font> being  <a href="cqc_lang.html#NM2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6">the_arity_of</a> <font color="Olive">b<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1" target="_self">|-</a> <font color="Olive">b<sub>1</sub></font> <a href="henmodel.html#K3" target="_self">!</a> <font color="Olive">b<sub>4</sub></font> ) )
 <b>by </b><i><a class="txt" href="henmodel.html#E5:23_1_1"><i><font color="Green">E19</font></i></a></i>;<br/>


</div>
