<?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">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>, <font color="Maroon">h</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>assume </b><a NAME="E1:176"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> holds <br/> ( <font color="Olive">s</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">h</font> iff <font color="Olive">s</font> <a href="modelc_1.html#R3" target="_self">|=</a>  <a href="modelc_1.html#K61" target="_self">Foax</a> <font color="Maroon">g</font>,<font color="Maroon">f</font>,<font color="Maroon">h</font> )
 ;<br/>

<b>let </b><font color="Maroon">s0</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font>;<br/>

<b>assume </b><a NAME="E2:176"/>
<font color="Maroon">s0</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/>

<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="E4:176"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">s0</font>
 <b>and </b><br/><a NAME="E5:176"/><i><font color="Green">E50</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/>
<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="E7:176"/><i><font color="Green">E52</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="E8:176"/><i><font color="Green">E53</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><a class="txt" href="modelc_1.html#E45"><i><font color="Green">Lemma135</font></i></a></i>;<br/>
<a NAME="E9:176"/><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#R1">&lt;=</a> <font color="Maroon">m</font> holds <br/><font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K2" target="_self">k_nat</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K10">-</a> <font color="Olive">j</font></span>)</span></span>)</span> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">h</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="176_1"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> implies <font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K2" target="_self">k_nat</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">h</font> );<br/>
<a NAME="E1:176_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="176_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:176_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E2:176_1_1"/><i><font color="Green">E56</font></i>: 
 <a href="modelc_1.html#K2" target="_self">k_nat</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K10">-</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:176_1_1"/>
<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>  <a href="modelc_1.html#K61" target="_self">Foax</a> <font color="Maroon">g</font>,<font color="Maroon">f</font>,<font color="Maroon">h</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E47"><i><font color="Green">Lemma137</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E4:176_1_1"/>
<font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K2" target="_self">k_nat</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K10">-</a> 0</span>)</span></span>)</span> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="ref" href="modelc_1.html#D23" target="_self">Def23</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:176_1"/><i><font color="Green">E57</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="Maroon">S<sub>1</sub></font>[<font color="Olive">j</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">j</font> <a href="binop_2.html#K23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="176_1_2"><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="E1:176_1_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">j</font>]
 ;<br/>

<b>set </b><font color="Maroon">j1</font> = <font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1;<br/>
<a NAME="E2:176_1_2"/>
( <font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> implies <font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K2" target="_self">k_nat</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">h</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="176_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:176_1_2_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<b>set </b><font color="Maroon">k</font> = <font color="Maroon">m</font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">m</font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="modelc_1.html#D24" target="_self">Def24</a>, <a class="ref" href="nat_1.html#T61">NAT_1:61</a></i>;<br/>
<a NAME="E3:176_1_2_1"/><i><font color="Green">E60</font></i>: 
 <a href="modelc_1.html#K2" target="_self">k_nat</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:176_1_2_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">h</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="modelc_1.html#D24" target="_self">Def24</a>, </i>;<br/>
<b>set </b><font color="Maroon">pai1</font> =  <a href="modelc_1.html#K57" target="_self">PathShift</a> <font color="Maroon">pai</font>,<font color="Maroon">k</font>;<br/>
<i><font color="Green">E62</font></i>: <span class="p1">(<span class="default"><a href="modelc_1.html#K57" target="_self">PathShift</a> <font color="Maroon">pai</font>,<font color="Maroon">k</font></span>)</span> <a href="funct_2.html#K3">.</a> 0 = 
<font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 0</span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font>

;<br/>
<a NAME="E6:176_1_2_1"/>
<span class="p1">(<span class="default"><a href="modelc_1.html#K57" target="_self">PathShift</a> <font color="Maroon">pai</font>,<font color="Maroon">k</font></span>)</span> <a href="funct_2.html#K3">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E7:176_1_2_1"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> <a href="modelc_1.html#R3" target="_self">|=</a>  <a href="modelc_1.html#K27" target="_self">EX</a> <font color="Maroon">h</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E8:176_1_2_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="176_1_2_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:176_1_2_1_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E2:176_1_2_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E3:176_1_2_1_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:176_1_2_1"/><b>then </b>
<font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E46"><i><font color="Green">Lemma136</font></i></a></i>;<br/>
<a NAME="E10:176_1_2_1"/><b>then </b>
<font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> <a href="modelc_1.html#R3" target="_self">|=</a>  <a href="modelc_1.html#K55" target="_self">Fax</a> <font color="Maroon">f</font>,<font color="Maroon">h</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:176_1_2_1"/><b>then </b>
<font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> <a href="modelc_1.html#R3" target="_self">|=</a>  <a href="modelc_1.html#K61" target="_self">Foax</a> <font color="Maroon">g</font>,<font color="Maroon">f</font>,<font color="Maroon">h</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E12:176_1_2_1"/>
<font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K2" target="_self">k_nat</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="ref" href="modelc_1.html#D23" target="_self">Def23</a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:176_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">j</font> <a href="binop_2.html#K23">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:176_1"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">j</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E4:176_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#R1">&lt;=</a> <font color="Maroon">m</font> holds <br/><font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K2" target="_self">k_nat</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K10">-</a> <font color="Olive">j</font></span>)</span></span>)</span> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">h</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:176"/>
<font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K2" target="_self">k_nat</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">h</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E48"><i><font color="Green">Lemma138</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:176"/>
<font color="Maroon">s0</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">h</font>
 <b>by </b><i>, <a class="txt" href="modelc_1.html#E44"><i><font color="Green">Lemma134</font></i></a></i>;<br/>


</div>
