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

<b>let </b><font color="Maroon">S</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">R</font> be  <a href="partfun1.html#V1">total</a>  <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">S</font>,<font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">s</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">BASSIGN</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K32" target="_self">ModelSP</a> <font color="Maroon">S</font></span>)</span>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="modelc_1.html#NM2">Assign</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K50" target="_self">CTLModel</a> <font color="Maroon">R</font>,<font color="Maroon">BASSIGN</font></span>)</span>, <font color="Maroon">g</font> be   <a href="modelc_1.html#NM2">Assign</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K50" target="_self">CTLModel</a> <font color="Maroon">R</font>,<font color="Maroon">BASSIGN</font></span>)</span>;<br/>









<b>set </b><font color="Maroon">V</font> =  <a href="modelc_1.html#K50" target="_self">CTLModel</a> <font color="Maroon">R</font>,<font color="Maroon">BASSIGN</font>;<br/>
<a NAME="E1:129"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">f</font> <a href="modelc_1.html#K29" target="_self">EU</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="modelc_1.html#K45" target="_self">EUntill_0</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>,<font color="Maroon">R</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E30"><i><font color="Green">Lemma93</font></i></a></i>;<br/>
<a NAME="E2:129"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">s</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">f</font> <a href="modelc_1.html#K29" target="_self">EU</a> <font color="Maroon">g</font> implies  ex <font color="Olive">pai</font> being   <a href="modelc_1.html#M1" target="_self">inf_path</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">pai</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp;  ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> holds <br/><font color="Olive">pai</font> <a href="funct_2.html#K3">.</a> <font color="Olive">j</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">f</font> ) &amp; <font color="Olive">pai</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">g</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="129_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:129_1"/>
<font color="Maroon">s</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">f</font> <a href="modelc_1.html#K29" target="_self">EU</a> <font color="Maroon">g</font>
 ;<br/>

<a NAME="E2:129_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="modelc_1.html#K33" target="_self">Fid</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K45" target="_self">EUntill_0</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>,<font color="Maroon">R</font></span>)</span>,<font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:129_1"/><b>then </b>
 <a href="modelc_1.html#K44" target="_self">EUntill_univ</a> <font color="Maroon">s</font>,<span class="p1">(<span class="default"><a href="modelc_1.html#K33" target="_self">Fid</a> <font color="Maroon">f</font>,<font color="Maroon">S</font></span>)</span>,<span class="p1">(<span class="default"><a href="modelc_1.html#K33" target="_self">Fid</a> <font color="Maroon">g</font>,<font color="Maroon">S</font></span>)</span>,<font color="Maroon">R</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="modelc_1.html#E27"><i><font color="Green">Lemma90</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">pai</font> being    <a href="modelc_1.html#M1" target="_self">inf_path</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E5:129_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E6:129_1"/><i><font color="Green">E52</font></i>: 
 ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> holds <br/><span class="p1">(<span class="default"><a href="modelc_1.html#K33" target="_self">Fid</a> <font color="Maroon">f</font>,<font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Olive">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) &amp; <span class="p1">(<span class="default"><a href="modelc_1.html#K33" target="_self">Fid</a> <font color="Maroon">g</font>,<font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 <b>by </b><i><a class="txt" href="modelc_1.html#E26"><i><font color="Green">Lemma89</font></i></a></i>;<br/>
<a NAME="E7:129_1"/><i><font color="Green">E53</font></i>: 
 ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> holds <br/><font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Olive">j</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">f</font> ) &amp; <font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">g</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="129_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E2:129_1_1"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> holds <br/><span class="p1">(<span class="default"><a href="modelc_1.html#K33" target="_self">Fid</a> <font color="Maroon">f</font>,<font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Olive">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>and </b><br/><a NAME="E3:129_1_1"/><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><a href="modelc_1.html#K33" target="_self">Fid</a> <font color="Maroon">g</font>,<font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i/>;<br/>
<a NAME="E4:129_1_1"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> holds <br/><font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Olive">j</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="129_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:129_1_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E2:129_1_1_1"/>
<span class="p1">(<span class="default"><a href="modelc_1.html#K33" target="_self">Fid</a> <font color="Maroon">f</font>,<font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:129_1_1_1"/>
<font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">j</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">m</font>
;<br/>

<b>thus </b><a NAME="E5:129_1_1"/>
( ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> holds <br/><font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Olive">j</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">f</font> ) &amp; <font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">g</font> )
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">pai</font>
;<br/>

<b>thus </b><a NAME="E8:129_1"/>
( <font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp;  ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> holds <br/><font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Olive">j</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">f</font> ) &amp; <font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">g</font> ) )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:129"/>
(  ex <font color="Olive">pai</font> being   <a href="modelc_1.html#M1" target="_self">inf_path</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">pai</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp;  ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> holds <br/><font color="Olive">pai</font> <a href="funct_2.html#K3">.</a> <font color="Olive">j</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">f</font> ) &amp; <font color="Olive">pai</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">g</font> ) ) implies <font color="Maroon">s</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">f</font> <a href="modelc_1.html#K29" target="_self">EU</a> <font color="Maroon">g</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="129_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:129_2"/><i><font color="Green">E58</font></i>: 
 ex <font color="Olive">pai</font> being   <a href="modelc_1.html#M1" target="_self">inf_path</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">pai</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp;  ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> holds <br/><font color="Olive">pai</font> <a href="funct_2.html#K3">.</a> <font color="Olive">j</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">f</font> ) &amp; <font color="Olive">pai</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">g</font> ) )
 ;<br/>

<a NAME="E2:129_2"/>
 ex <font color="Olive">pai</font> being   <a href="modelc_1.html#M1" target="_self">inf_path</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">pai</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp;  ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> holds <br/><span class="p1">(<span class="default"><a href="modelc_1.html#K33" target="_self">Fid</a> <font color="Maroon">f</font>,<font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">pai</font> <a href="funct_2.html#K3">.</a> <font color="Olive">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) &amp; <span class="p1">(<span class="default"><a href="modelc_1.html#K33" target="_self">Fid</a> <font color="Maroon">g</font>,<font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">pai</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="129_2_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">pai</font> being    <a href="modelc_1.html#M1" target="_self">inf_path</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E2:129_2_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E3:129_2_1"/><i><font color="Green">E60</font></i>: 
 ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> holds <br/><font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Olive">j</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">f</font> ) &amp; <font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">g</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:129_2_1"/><i><font color="Green">E61</font></i>: 
 ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> holds <br/><span class="p1">(<span class="default"><a href="modelc_1.html#K33" target="_self">Fid</a> <font color="Maroon">f</font>,<font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Olive">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) &amp; <span class="p1">(<span class="default"><a href="modelc_1.html#K33" target="_self">Fid</a> <font color="Maroon">g</font>,<font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="129_2_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E2:129_2_1_1"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> holds <br/><font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Olive">j</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E3:129_2_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">g</font>
 <b>by </b><i/>;<br/>
<a NAME="E4:129_2_1_1"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> holds <br/><span class="p1">(<span class="default"><a href="modelc_1.html#K33" target="_self">Fid</a> <font color="Maroon">f</font>,<font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Olive">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="129_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:129_2_1_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E2:129_2_1_1_1"/>
<font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">j</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">f</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:129_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="modelc_1.html#K33" target="_self">Fid</a> <font color="Maroon">f</font>,<font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">m</font>
;<br/>

<b>thus </b><a NAME="E5:129_2_1_1"/>
( ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> holds <br/><span class="p1">(<span class="default"><a href="modelc_1.html#K33" target="_self">Fid</a> <font color="Maroon">f</font>,<font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Olive">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) &amp; <span class="p1">(<span class="default"><a href="modelc_1.html#K33" target="_self">Fid</a> <font color="Maroon">g</font>,<font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">pai</font>
;<br/>

<b>thus </b><a NAME="E5:129_2_1"/>
( <font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp;  ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> holds <br/><span class="p1">(<span class="default"><a href="modelc_1.html#K33" target="_self">Fid</a> <font color="Maroon">f</font>,<font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Olive">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) &amp; <span class="p1">(<span class="default"><a href="modelc_1.html#K33" target="_self">Fid</a> <font color="Maroon">g</font>,<font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:129_2"/><b>then </b>
 <a href="modelc_1.html#K44" target="_self">EUntill_univ</a> <font color="Maroon">s</font>,<span class="p1">(<span class="default"><a href="modelc_1.html#K33" target="_self">Fid</a> <font color="Maroon">f</font>,<font color="Maroon">S</font></span>)</span>,<span class="p1">(<span class="default"><a href="modelc_1.html#K33" target="_self">Fid</a> <font color="Maroon">g</font>,<font color="Maroon">S</font></span>)</span>,<font color="Maroon">R</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="modelc_1.html#E26"><i><font color="Green">Lemma89</font></i></a></i>;<br/>
<a NAME="E4:129_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="modelc_1.html#K33" target="_self">Fid</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="modelc_1.html#K29" target="_self">EU</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i>, <a class="txt" href="modelc_1.html#E27"><i><font color="Green">Lemma90</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:129_2"/>
<font color="Maroon">s</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">f</font> <a href="modelc_1.html#K29" target="_self">EU</a> <font color="Maroon">g</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:129"/>
( <font color="Maroon">s</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">f</font> <a href="modelc_1.html#K29" target="_self">EU</a> <font color="Maroon">g</font> iff  ex <font color="Olive">pai</font> being   <a href="modelc_1.html#M1" target="_self">inf_path</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">pai</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp;  ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> holds <br/><font color="Olive">pai</font> <a href="funct_2.html#K3">.</a> <font color="Olive">j</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">f</font> ) &amp; <font color="Olive">pai</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">g</font> ) ) )
 <b>by </b><i/>;<br/>


</div>
