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

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">P9</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K5" title="QC_LANG1:func.5">QC-pred_symbols</a>  st <br/>( <font color="Olive" title="b1">P9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">ll</font> where <font color="Olive" title="b2">ll</font> is   <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Olive" title="b1">P9</font> : <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">P9</font> <a href="henmodel.html#K2" title="HENMODEL:func.2">!</a> <font color="Olive" title="b2">ll</font> </span>}</span>  );<br/>
<span class="kw">set </span><font color="Maroon" title="c2">A</font> =  <a href="qc_lang1.html#K5" title="QC_LANG1:func.5">QC-pred_symbols</a> ;<br/>
<a NAME="E1:23_1_1"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K5" title="QC_LANG1:func.5">QC-pred_symbols</a>  holds <br/> ex <font color="Olive" title="b2">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K5" title="QC_LANG1:func.5">QC-pred_symbols</a>  implies  ex <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">b</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:23_1_1_1"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K5" title="QC_LANG1:func.5">QC-pred_symbols</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">b</font>]</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">a</font> = <font color="Maroon" title="c3">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K5" title="QC_LANG1:func.5">QC-pred_symbols</a>  ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:23_1_1_1"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">ll</font> where <font color="Olive" title="b1">ll</font> is   <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Maroon" title="c4">a</font> : <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c4">a</font> <a href="henmodel.html#K2" title="HENMODEL:func.2">!</a> <font color="Olive" title="b1">ll</font> </span>}</span> 
 ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">b</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">b</font>]</span><br/>

<span class="kw">take </span><font color="Maroon" title="c6">P9</font> = <font color="Maroon" title="c4">a</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">P9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">ll</font> where <font color="Olive" title="b1">ll</font> is   <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Maroon" title="c6">P9</font> : <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c6">P9</font> <a href="henmodel.html#K2" title="HENMODEL:func.2">!</a> <font color="Olive" title="b1">ll</font> </span>}</span>  )</span><br/>

<span class="kw">thus </span><a NAME="E5:23_1_1_1"/>
( <font color="Maroon" title="c6">P9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">ll</font> where <font color="Olive" title="b1">ll</font> is   <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Maroon" title="c6">P9</font> : <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c6">P9</font> <a href="henmodel.html#K2" title="HENMODEL:func.2">!</a> <font color="Olive" title="b1">ll</font> </span>}</span>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23_1_1_1"><span class="lab"><font color="Green" title="E15">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c3">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E3:23_1_1"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K5" title="QC_LANG1:func.5">QC-pred_symbols</a>  &amp; ( for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K5" title="QC_LANG1:func.5">QC-pred_symbols</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="classes1.html#S1" title="CLASSES1:sch.1">CLASSES1:sch 1</a>(<a class="txt" href="#E1:23_1_1"><span class="lab"><font color="Green" title="E14">A1</font></span></a>);<br/></span>
<a NAME="E4:23_1_1"/><span class="lab"><font color="Green" title="E16">A4</font></span>: 
for <font color="Olive" title="b1">P</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K5" title="QC_LANG1:func.5">QC-pred_symbols</a> <br/> for <font color="Olive" title="b2">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="margrel1.html#K3" title="MARGREL1:func.3">relations_on</a> <a href="henmodel.html#K1" title="HENMODEL:func.1">HCar</a>   st <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> holds <br/>for <font color="Olive" title="b3">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">r</font> iff  ex <font color="Olive" title="b4">ll</font> being  <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Olive" title="b1">P</font> st <br/>( <font color="Olive" title="b3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">ll</font> &amp; <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">P</font> <a href="henmodel.html#K2" title="HENMODEL:func.2">!</a> <font color="Olive" title="b4">ll</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">P</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K5" title="QC_LANG1:func.5">QC-pred_symbols</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="margrel1.html#K3" title="MARGREL1:func.3">relations_on</a> <a href="henmodel.html#K1" title="HENMODEL:func.1">HCar</a>   st <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> holds <br/>for <font color="Olive" title="b2">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">r</font> iff  ex <font color="Olive" title="b3">ll</font> being  <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Maroon" title="c4">P</font> st <br/>( <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">ll</font> &amp; <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c4">P</font> <a href="henmodel.html#K2" title="HENMODEL:func.2">!</a> <font color="Olive" title="b3">ll</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">r</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="margrel1.html#K3" title="MARGREL1:func.3">relations_on</a> <a href="henmodel.html#K1" title="HENMODEL:func.1">HCar</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> implies for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">r</font> iff  ex <font color="Olive" title="b2">ll</font> being  <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Maroon" title="c4">P</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">ll</font> &amp; <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c4">P</font> <a href="henmodel.html#K2" title="HENMODEL:func.2">!</a> <font color="Olive" title="b2">ll</font> ) ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:23_1_1_2"/><span class="lab"><font color="Green" title="E17">A5</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">r</font> iff  ex <font color="Olive" title="b2">ll</font> being  <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Maroon" title="c4">P</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">ll</font> &amp; <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c4">P</font> <a href="henmodel.html#K2" title="HENMODEL:func.2">!</a> <font color="Olive" title="b2">ll</font> ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">r</font> iff  ex <font color="Olive" title="b1">ll</font> being  <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Maroon" title="c4">P</font> st <br/>( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">ll</font> &amp; <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c4">P</font> <a href="henmodel.html#K2" title="HENMODEL:func.2">!</a> <font color="Olive" title="b1">ll</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E2:23_1_1_2"/>
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">r</font> implies  ex <font color="Olive" title="b1">ll</font> being  <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Maroon" title="c4">P</font> st <br/>( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">ll</font> &amp; <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c4">P</font> <a href="henmodel.html#K2" title="HENMODEL:func.2">!</a> <font color="Olive" title="b1">ll</font> ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">ll</font> being  <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Maroon" title="c4">P</font> st <br/>( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">ll</font> &amp; <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c4">P</font> <a href="henmodel.html#K2" title="HENMODEL:func.2">!</a> <font color="Olive" title="b1">ll</font> ) implies <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">r</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:23_1_1_2_1"/><span class="lab"><font color="Green" title="E18">A6</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">ll</font> being  <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Maroon" title="c4">P</font> st <br/>( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">ll</font> &amp; <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c4">P</font> <a href="henmodel.html#K2" title="HENMODEL:func.2">!</a> <font color="Olive" title="b1">ll</font> )</span><br/>

<a NAME="E2:23_1_1_2_1"/>
 ex <font color="Olive" title="b1">P9</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K5" title="QC_LANG1:func.5">QC-pred_symbols</a>  st <br/>( <font color="Olive" title="b1">P9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">ll</font> where <font color="Olive" title="b2">ll</font> is   <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Olive" title="b1">P9</font> : <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">P9</font> <a href="henmodel.html#K2" title="HENMODEL:func.2">!</a> <font color="Olive" title="b2">ll</font> </span>}</span>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1_1"><span class="lab"><font color="Green" title="E15">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:23_1_1_2_1"/>
 ex <font color="Olive" title="b1">ll</font> being  <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Maroon" title="c4">P</font> st <br/>( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">ll</font> &amp; <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c4">P</font> <a href="henmodel.html#K2" title="HENMODEL:func.2">!</a> <font color="Olive" title="b1">ll</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1_2"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="txt" href="#E1:23_1_1_2_1"><span class="lab"><font color="Green" title="E18">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E3:23_1_1_2"/>
(  ex <font color="Olive" title="b1">ll</font> being  <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Maroon" title="c4">P</font> st <br/>( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">ll</font> &amp; <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c4">P</font> <a href="henmodel.html#K2" title="HENMODEL:func.2">!</a> <font color="Olive" title="b1">ll</font> ) implies <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">r</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><font color="Maroon" title="c7">ll</font> being   <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Maroon" title="c4">P</font><span class="kw"> such that </span><a NAME="E2:23_1_1_2_2"/><span class="lab"><font color="Green" title="E18">A7</font></span>: 
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">ll</font> &amp; <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c4">P</font> <a href="henmodel.html#K2" title="HENMODEL:func.2">!</a> <font color="Maroon" title="c7">ll</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">r</font></span><br/>

<a NAME="E3:23_1_1_2_2"/>
 ex <font color="Olive" title="b1">P9</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K5" title="QC_LANG1:func.5">QC-pred_symbols</a>  st <br/>( <font color="Olive" title="b1">P9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">l</font> where <font color="Olive" title="b2">l</font> is   <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Olive" title="b1">P9</font> : <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">P9</font> <a href="henmodel.html#K2" title="HENMODEL:func.2">!</a> <font color="Olive" title="b2">l</font> </span>}</span>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1_1"><span class="lab"><font color="Green" title="E15">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:23_1_1_2_2"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1_2"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="txt" href="#E2:23_1_1_2_2"><span class="lab"><font color="Green" title="E18">A7</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div><span class="kw">end;</span></div>
<a NAME="E5:23_1_1"/><span class="lab"><font color="Green" title="E17">A8</font></span>: 
for <font color="Olive" title="b1">P</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K5" title="QC_LANG1:func.5">QC-pred_symbols</a> <br/> for <font color="Olive" title="b2">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="margrel1.html#K3" title="MARGREL1:func.3">relations_on</a> <a href="henmodel.html#K1" title="HENMODEL:func.1">HCar</a>   holds <br/>(  not <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> or <font color="Olive" title="b2">r</font> <a href="margrel1.html#R1" title="MARGREL1:pred.1">=</a>  <a href="margrel1.html#K4" title="MARGREL1:func.4">empty_rel</a> <a href="henmodel.html#K1" title="HENMODEL:func.1">HCar</a>  or  <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K5" title="MARGREL1:func.5">the_arity_of</a> <font color="Olive" title="b2">r</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">P</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K5" title="QC_LANG1:func.5">QC-pred_symbols</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="margrel1.html#K3" title="MARGREL1:func.3">relations_on</a> <a href="henmodel.html#K1" title="HENMODEL:func.1">HCar</a>   holds <br/>(  not <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> or <font color="Olive" title="b1">r</font> <a href="margrel1.html#R1" title="MARGREL1:pred.1">=</a>  <a href="margrel1.html#K4" title="MARGREL1:func.4">empty_rel</a> <a href="henmodel.html#K1" title="HENMODEL:func.1">HCar</a>  or  <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K5" title="MARGREL1:func.5">the_arity_of</a> <font color="Olive" title="b1">r</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">r</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="margrel1.html#K3" title="MARGREL1:func.3">relations_on</a> <a href="henmodel.html#K1" title="HENMODEL:func.1">HCar</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> or <font color="Maroon" title="c5">r</font> <a href="margrel1.html#R1" title="MARGREL1:pred.1">=</a>  <a href="margrel1.html#K4" title="MARGREL1:func.4">empty_rel</a> <a href="henmodel.html#K1" title="HENMODEL:func.1">HCar</a>  or  <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K5" title="MARGREL1:func.5">the_arity_of</a> <font color="Maroon" title="c5">r</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:23_1_1_3"/><span class="lab"><font color="Green" title="E18">A9</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">r</font> <a href="margrel1.html#R1" title="MARGREL1:pred.1">=</a>  <a href="margrel1.html#K4" title="MARGREL1:func.4">empty_rel</a> <a href="henmodel.html#K1" title="HENMODEL:func.1">HCar</a>  or  <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K5" title="MARGREL1:func.5">the_arity_of</a> <font color="Maroon" title="c5">r</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">P9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K5" title="QC_LANG1:func.5">QC-pred_symbols</a> <span class="kw"> such that </span><br/><a NAME="E3:23_1_1_3"/><span class="lab"><font color="Green" title="E19">A10</font></span>: 
( <font color="Maroon" title="c6">P9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">ll</font> where <font color="Olive" title="b1">ll</font> is   <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Maroon" title="c6">P9</font> : <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c6">P9</font> <a href="henmodel.html#K2" title="HENMODEL:func.2">!</a> <font color="Olive" title="b1">ll</font> </span>}</span>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1_1"><span class="lab"><font color="Green" title="E15">A3</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:23_1_1_3"/><span class="lab"><font color="Green" title="E20">A11</font></span>: 
not <font color="Maroon" title="c5">r</font> <a href="margrel1.html#R1" title="MARGREL1:pred.1">=</a>  <a href="margrel1.html#K4" title="MARGREL1:func.4">empty_rel</a> <a href="henmodel.html#K1" title="HENMODEL:func.1">HCar</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K5" title="MARGREL1:func.5">the_arity_of</a> <font color="Maroon" title="c5">r</font></span><br/>

<a NAME="E5:23_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="margrel1.html#T32" title="MARGREL1:th.32">MARGREL1:32</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:23_1_1_3"/><span class="lab"><font color="Green" title="E21">A12</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">ll9</font> being   <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Maroon" title="c6">P9</font><span class="kw"> such that </span><br/><a NAME="E9:23_1_1_3"/><span class="lab"><font color="Green" title="E22">A13</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">ll9</font>
 <span class="kw">and </span><br/><a NAME="E10:23_1_1_3"/>
<font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c6">P9</font> <a href="henmodel.html#K2" title="HENMODEL:func.2">!</a> <font color="Maroon" title="c8">ll9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1_1_3"><span class="lab"><font color="Green" title="E18">A9</font></span></a>, <a class="txt" href="#E7:23_1_1_3"><span class="lab"><font color="Green" title="E21">A12</font></span></a>, <a class="txt" href="#E3:23_1_1_3"><span class="lab"><font color="Green" title="E19">A10</font></span></a></span>;<br/>
<a NAME="E11:23_1_1_3"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">ll9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">a</font> = <font color="Maroon" title="c7">a</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="henmodel.html#K1" title="HENMODEL:func.1">HCar</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:23_1_1_3"><span class="lab"><font color="Green" title="E22">A13</font></span></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<a NAME="E13:23_1_1_3"/>
 <a href="margrel1.html#K5" title="MARGREL1:func.5">the_arity_of</a> <font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23_1_1_3"><span class="lab"><font color="Green" title="E20">A11</font></span></a>, <a class="txt" href="#E7:23_1_1_3"><span class="lab"><font color="Green" title="E21">A12</font></span></a>, <a class="ref" href="margrel1.html#D11" title="MARGREL1:def.11">MARGREL1:def 11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:23_1_1_3"/>
 <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K5" title="MARGREL1:func.5">the_arity_of</a> <font color="Maroon" title="c5">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1_1_3"><span class="lab"><font color="Green" title="E19">A10</font></span></a>, <a class="txt" href="#E9:23_1_1_3"><span class="lab"><font color="Green" title="E22">A13</font></span></a>, <a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:23_1_1"/>
for <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font> holds <br/><font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="margrel1.html#K3" title="MARGREL1:func.3">relations_on</a> <a href="henmodel.html#K1" title="HENMODEL:func.1">HCar</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font> implies <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="margrel1.html#K3" title="MARGREL1:func.3">relations_on</a> <a href="henmodel.html#K1" title="HENMODEL:func.1">HCar</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:23_1_1_4"/>
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="margrel1.html#K3" title="MARGREL1:func.3">relations_on</a> <a href="henmodel.html#K1" title="HENMODEL:func.1">HCar</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:23_1_1_4"/><span class="lab"><font color="Green" title="E18">A14</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">and </span><br/><a NAME="E4:23_1_1_4"/><span class="lab"><font color="Green" title="E19">A15</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">P9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K5" title="QC_LANG1:func.5">QC-pred_symbols</a> <span class="kw"> such that </span><br/><a NAME="E6:23_1_1_4"/><span class="lab"><font color="Green" title="E20">A16</font></span>: 
( <font color="Maroon" title="c6">P9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">ll</font> where <font color="Olive" title="b1">ll</font> is   <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Maroon" title="c6">P9</font> : <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c6">P9</font> <a href="henmodel.html#K2" title="HENMODEL:func.2">!</a> <font color="Olive" title="b1">ll</font> </span>}</span>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1_1"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E3:23_1_1_4"><span class="lab"><font color="Green" title="E18">A14</font></span></a></span>;<br/>
<a NAME="E7:23_1_1_4"/>
for <font color="Olive" title="b1">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="henmodel.html#K1" title="HENMODEL:func.1">HCar</a>  <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="henmodel.html#K1" title="HENMODEL:func.1">HCar</a>  <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:23_1_1_4_1"/>
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="henmodel.html#K1" title="HENMODEL:func.1">HCar</a>  <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">ll</font> being   <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Maroon" title="c6">P9</font><span class="kw"> such that </span><br/><a NAME="E3:23_1_1_4_1"/><span class="lab"><font color="Green" title="E21">A17</font></span>: 
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">ll</font>
 <span class="kw">and </span><br/><a NAME="E4:23_1_1_4_1"/>
<font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c6">P9</font> <a href="henmodel.html#K2" title="HENMODEL:func.2">!</a> <font color="Maroon" title="c8">ll</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23_1_1_4"><span class="lab"><font color="Green" title="E19">A15</font></span></a>, <a class="txt" href="#E6:23_1_1_4"><span class="lab"><font color="Green" title="E20">A16</font></span></a></span>;<br/>
<a NAME="E5:23_1_1_4_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">ll</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<a NAME="E6:23_1_1_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">ll</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="henmodel.html#K1" title="HENMODEL:func.1">HCar</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:23_1_1_4_1"/>
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="henmodel.html#K1" title="HENMODEL:func.1">HCar</a>  <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1_1_4_1"><span class="lab"><font color="Green" title="E21">A17</font></span></a>, <a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:23_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A18</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="henmodel.html#K1" title="HENMODEL:func.1">HCar</a>  <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E9:23_1_1_4"/>
for <font color="Olive" title="b1">fin</font>, <font color="Olive" title="b2">fin9</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="henmodel.html#K1" title="HENMODEL:func.1">HCar</a>   st <font color="Olive" title="b1">fin</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Olive" title="b2">fin9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">b</font> holds <br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">fin</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">fin9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_4_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">fin</font>, <font color="Maroon" title="c8">fin9</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="henmodel.html#K1" title="HENMODEL:func.1">HCar</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">fin</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c8">fin9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">b</font> implies  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">fin</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">fin9</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:23_1_1_4_2"/><span class="lab"><font color="Green" title="E22">A19</font></span>: 
<font color="Maroon" title="c7">fin</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">b</font>
 <span class="kw">and </span><br/><a NAME="E2:23_1_1_4_2"/><span class="lab"><font color="Green" title="E23">A20</font></span>: 
<font color="Maroon" title="c8">fin9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">fin</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">fin9</font></span><br/>

<a NAME="E3:23_1_1_4_2"/>
 ex <font color="Olive" title="b1">ll</font> being  <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Maroon" title="c6">P9</font> st <br/>( <font color="Maroon" title="c7">fin</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">ll</font> &amp; <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c6">P9</font> <a href="henmodel.html#K2" title="HENMODEL:func.2">!</a> <font color="Olive" title="b1">ll</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23_1_1_4"><span class="lab"><font color="Green" title="E19">A15</font></span></a>, <a class="txt" href="#E6:23_1_1_4"><span class="lab"><font color="Green" title="E20">A16</font></span></a>, <a class="txt" href="#E1:23_1_1_4_2"><span class="lab"><font color="Green" title="E22">A19</font></span></a></span>;<br/>
<a NAME="E4:23_1_1_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A21</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">fin</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Maroon" title="c6">P9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>;<br/>
<a NAME="E5:23_1_1_4_2"/>
 ex <font color="Olive" title="b1">ll9</font> being  <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Maroon" title="c6">P9</font> st <br/>( <font color="Maroon" title="c8">fin9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">ll9</font> &amp; <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c6">P9</font> <a href="henmodel.html#K2" title="HENMODEL:func.2">!</a> <font color="Olive" title="b1">ll9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23_1_1_4"><span class="lab"><font color="Green" title="E19">A15</font></span></a>, <a class="txt" href="#E6:23_1_1_4"><span class="lab"><font color="Green" title="E20">A16</font></span></a>, <a class="txt" href="#E2:23_1_1_4_2"><span class="lab"><font color="Green" title="E23">A20</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:23_1_1_4_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">fin</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">fin9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23_1_1_4_2"><span class="lab"><font color="Green" title="E24">A21</font></span></a>, <a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E10:23_1_1_4"/>
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="margrel1.html#K3" title="MARGREL1:func.3">relations_on</a> <a href="henmodel.html#K1" title="HENMODEL:func.1">HCar</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:23_1_1_4"><span class="lab"><font color="Green" title="E21">A18</font></span></a>, <a class="ref" href="margrel1.html#D8" title="MARGREL1:def.8">MARGREL1:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:23_1_1"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="margrel1.html#K3" title="MARGREL1:func.3">relations_on</a> <a href="henmodel.html#K1" title="HENMODEL:func.1">HCar</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">f</font> = <font color="Maroon" title="c3">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="qc_lang1.html#K5" title="QC_LANG1:func.5">QC-pred_symbols</a> ,<span class="p1">(<span class="default"><a href="margrel1.html#K3" title="MARGREL1:func.3">relations_on</a> <a href="henmodel.html#K1" title="HENMODEL:func.1">HCar</a> </span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1_1"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">f</font> = <font color="Maroon" title="c4">f</font> as    <a href="valuat_1.html#M1" title="VALUAT_1:mode.1">interpretation</a> of  <a href="henmodel.html#K1" title="HENMODEL:func.1">HCar</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23_1_1"><span class="lab"><font color="Green" title="E17">A8</font></span></a>, <a class="ref" href="valuat_1.html#D10" title="VALUAT_1:def.10">VALUAT_1:def 10</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">f</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">P</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K5" title="QC_LANG1:func.5">QC-pred_symbols</a> <br/> for <font color="Olive" title="b2">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="margrel1.html#K3" title="MARGREL1:func.3">relations_on</a> <a href="henmodel.html#K1" title="HENMODEL:func.1">HCar</a>   st <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">P</font> <a href="margrel1.html#R1" title="MARGREL1:pred.1">=</a> <font color="Olive" title="b2">r</font> holds <br/>for <font color="Olive" title="b3">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">r</font> iff  ex <font color="Olive" title="b4">ll</font> being  <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Olive" title="b1">P</font> st <br/>( <font color="Olive" title="b3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">ll</font> &amp; <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">P</font> <a href="henmodel.html#K2" title="HENMODEL:func.2">!</a> <font color="Olive" title="b4">ll</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E10:23_1_1"/>
for <font color="Olive" title="b1">P</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K5" title="QC_LANG1:func.5">QC-pred_symbols</a> <br/> for <font color="Olive" title="b2">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="margrel1.html#K3" title="MARGREL1:func.3">relations_on</a> <a href="henmodel.html#K1" title="HENMODEL:func.1">HCar</a>   st <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">P</font> <a href="margrel1.html#R1" title="MARGREL1:pred.1">=</a> <font color="Olive" title="b2">r</font> holds <br/>for <font color="Olive" title="b3">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">r</font> iff  ex <font color="Olive" title="b4">ll</font> being  <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Olive" title="b1">P</font> st <br/>( <font color="Olive" title="b3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">ll</font> &amp; <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">P</font> <a href="henmodel.html#K2" title="HENMODEL:func.2">!</a> <font color="Olive" title="b4">ll</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23_1_1"><span class="lab"><font color="Green" title="E16">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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