<?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">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>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="E1:166"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">s0</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">g</font>
 ;<br/>

<b>assume </b><a NAME="E2:166"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font>  st <font color="Olive">s</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">g</font> holds <br/><font color="Olive">s</font> <a href="modelc_1.html#R3" target="_self">|=</a>  <a href="modelc_1.html#K27" target="_self">EX</a> <font color="Maroon">g</font>
 ;<br/>

<a NAME="E3:166"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">s0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
;<br/>
<a NAME="E4:166"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">p</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="orders_1.html#K1">BOOL</a> <font color="Maroon">S</font> holds <br/><font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="orders_1.html#T4">ORDERS_1:4</a></i>;<br/>
<b>consider </b><font color="Maroon">Choice</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:166"/><i><font color="Green">E53</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">Choice</font> <a href="hidden.html#R1">=</a>  <a href="orders_1.html#K1">BOOL</a> <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E7:166"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">p</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="orders_1.html#K1">BOOL</a> <font color="Maroon">S</font> holds <br/><font color="Maroon">Choice</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">p</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E39"><i><font color="Green">Lemma129</font></i></a>, <a class="ref" href="wellord2.html#T28">WELLORD2:28</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> : <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a<sub>1</sub></font>,<font color="Olive">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> </span>}</span> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="modelc_1.html#K3" target="_self">UnivF</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a<sub>1</sub></font>) <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span></span>)</span>,<font color="Maroon">Choice</font>,<font color="Maroon">s0</font>;<br/>
<a NAME="E8:166"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> holds <br/><font color="Maroon">H<sub>2</sub></font>(<font color="Olive">x</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="166_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:166_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/>

<b>set </b><font color="Maroon">Y</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>) <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="166_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:166_1_1"/>
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>) <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Choice</font> or  not <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>) <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Choice</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="166_1_1_1"><b>suppose </b></a><a NAME="E1:166_1_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>) <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Choice</font>
 ;<br/><div class="add"><a NAME="E2:166_1_1_1"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">x</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">Choice</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>) <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="modelc_1.html#D23" target="_self">Def23</a>, </i>;<br/><a NAME="E3:166_1_1_1"/><b>then </b>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">x</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>) <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="modelc_1.html#D23" target="_self">Def23</a>, <a class="txt" href="modelc_1.html#E40"><i><font color="Green">Lemma130</font></i></a>, <a class="ref" href="modelc_1.html#D21" target="_self">Def21</a></i>;<br/><a NAME="E4:166_1_1_1"/><b>then </b>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">x</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>)
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">z</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E6:166_1_1_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">x</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 ;<br/><b>thus </b><a NAME="E7:166_1_1_1"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">x</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E44"><i><font color="Green">Lemma134</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="166_1_1_2"><b>suppose </b></a><a NAME="E1:166_1_1_2"/>
not <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x</font>) <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Choice</font>
 ;<br/><div class="add"><a NAME="E2:166_1_1_2"/><b>then </b>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">x</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">s0</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:166_1_1_2"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">x</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon">h</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">S</font><b> such that </b><br/><a NAME="E10:166"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> holds <br/><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">x</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S2">FUNCT_2:sch 2</a>(<a class="ref" href="modelc_1.html#D22" target="_self">Def22</a>);<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> = <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="modelc_1.html#K31" target="_self">|**</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K2" target="_self">k_nat</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s0</font>;<br/>
<a NAME="E11:166"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">H<sub>3</sub></font>(<font color="Olive">n</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
;<br/>
<b>consider </b><font color="Maroon">pai</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">S</font><b> such that </b><br/><a NAME="E13:166"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">pai</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S2">FUNCT_2:sch 2</a>(<a class="txt" href="modelc_1.html#E44"><i><font color="Green">Lemma134</font></i></a>);<br/></i>
<i><font color="Green">E61</font></i>: <font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> 0 = 
<font color="Maroon">H<sub>3</sub></font>(0)
<b>by </b><i><a class="txt" href="modelc_1.html#E45"><i><font color="Green">Lemma135</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="modelc_1.html#K31" target="_self">|**</a> 0</span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s0</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s0</font>
<b>by </b><i><a class="ref" href="funct_7.html#T86">FUNCT_7:86</a></i>
<br/>.= 
<font color="Maroon">s0</font>
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/>
<a NAME="E15:166"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</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="166_3"><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>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">g</font> );<br/>
<a NAME="E1:166_3"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="166_3_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">Y</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">s0</font>) <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span>;<br/>
<a NAME="E1:166_3_1"/>
<font color="Maroon">s0</font> <a href="modelc_1.html#R3" target="_self">|=</a>  <a href="modelc_1.html#K27" target="_self">EX</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E36"><i><font color="Green">Lemma126</font></i></a>, <a class="txt" href="modelc_1.html#E37"><i><font color="Green">Lemma127</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">pai01</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="E3:166_3_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">pai01</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="E4:166_3_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">pai01</font> <a href="funct_2.html#K3">.</a> 1 <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#E19:166_3_1"><i><font color="Green">E100</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">x</font> = <font color="Maroon">pai01</font> <a href="funct_2.html#K3">.</a> 1;<br/>
<a NAME="E5:166_3_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">pai01</font> <a href="funct_2.html#K3">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">pai01</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default">0 <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E11:166_3_1"><i><font color="Green">E68</font></i></a></i>;<br/>
<a NAME="E6:166_3_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">pai01</font> <a href="funct_2.html#K3">.</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">s0</font>)
 
<b>by </b><i/>;<br/>
<a NAME="E7:166_3_1"/>
<font color="Maroon">pai01</font> <a href="funct_2.html#K3">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:166_3_1"/><b>then </b>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">s0</font>) <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E9:166_3_1"/><b>then </b>
not <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">s0</font>) <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E10:166_3_1"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">s0</font>) <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">S</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E11:166_3_1"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">s0</font>) <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="orders_1.html#K1">BOOL</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#D2">ORDERS_1:def 2</a></i>;<br/>
<a NAME="E12:166_3_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">s0</font>) <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Choice</font>
 
<b>by </b><i><a class="ref" href="modelc_1.html#D24" target="_self">Def24</a>, <a class="txt" href="modelc_1.html#E40"><i><font color="Green">Lemma130</font></i></a>, <a class="ref" href="orders_1.html#D2">ORDERS_1:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">y</font> = <font color="Maroon">Choice</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">s0</font>) <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span></span>)</span>;<br/>
<a NAME="E13:166_3_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">Choice</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">s0</font>) <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">s0</font>) <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="modelc_1.html#D21" target="_self">Def21</a></i>;<br/>
<a NAME="E14:166_3_1"/><b>then </b>
<font color="Maroon">Choice</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">s0</font>) <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">s0</font>)
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">t</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E16:166_3_1"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">Choice</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">s0</font>) <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">s0</font>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 ;<br/>
<a NAME="E17:166_3_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">s1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E19:166_3_1"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">s1</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">g</font> )
 ;<br/>
<font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> 1 = 
<font color="Maroon">H<sub>3</sub></font>(1)
<b>by </b><i><a class="txt" href="modelc_1.html#E45"><i><font color="Green">Lemma135</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="modelc_1.html#K31" target="_self">|**</a> 1</span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s0</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s0</font>
<b>by </b><i><a class="ref" href="funct_7.html#T72">FUNCT_7:72</a></i>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">s0</font>)
<b>by </b><i><a class="ref" href="modelc_1.html#D23" target="_self">Def23</a></i>
<br/>.= 
<font color="Maroon">s1</font>
<b>by </b><i>, , , <a class="ref" href="modelc_1.html#D26" target="_self">Def26</a></i>
;<br/>
<b>hence </b><a NAME="E21:166_3_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 <b>by </b><i><a class="txt" href="modelc_1.html#E36"><i><font color="Green">Lemma126</font></i></a>, <a class="txt" href="modelc_1.html#E46"><i><font color="Green">Lemma136</font></i></a>, , <a class="ref" href="modelc_1.html#D26" target="_self">Def26</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:166_3"/><i><font color="Green">E101</font></i>: 
for <font color="Olive">k</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">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="binop_2.html#K23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="166_3_2"><b>proof </b></a><div class="add">

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

<b>set </b><font color="Maroon">k1</font> = <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1;<br/>
<b>set </b><font color="Maroon">k2</font> = <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1;<br/>
<b>set </b><font color="Maroon">p0</font> = <font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font>;<br/>
<b>set </b><font color="Maroon">p1</font> = <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>;<br/>
<b>set </b><font color="Maroon">p2</font> = <font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon">Y1</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font>) <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">Y2</font> = <font color="Maroon">H<sub>1</sub></font>(<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="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span>;<br/>
<b>assume </b><a NAME="E1:166_3_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<a NAME="E2:166_3_2"/><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#K27" target="_self">EX</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E37"><i><font color="Green">Lemma127</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">pai01</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="E4:166_3_2"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">pai01</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E5:166_3_2"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">pai01</font> <a href="funct_2.html#K3">.</a> 1 <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">g</font>
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">x1</font> = <font color="Maroon">pai01</font> <a href="funct_2.html#K3">.</a> 1;<br/>
<a NAME="E6:166_3_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">pai01</font> <a href="funct_2.html#K3">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">pai01</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default">0 <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:166_3_2"/><b>then </b><i><font color="Green">E104</font></i>: 
<font color="Maroon">pai01</font> <a href="funct_2.html#K3">.</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font>)
 
<b>by </b><i/>;<br/>
<a NAME="E8:166_3_2"/>
<font color="Maroon">pai01</font> <a href="funct_2.html#K3">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font>
 
<b>by </b><i/>;<br/>
<a NAME="E9:166_3_2"/><b>then </b>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font>) <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="modelc_1.html#D24" target="_self">Def24</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E10:166_3_2"/><b>then </b>
not <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font>) <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E11:166_3_2"/><b>then </b><i><font color="Green">E105</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font>) <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">S</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E12:166_3_2"/><b>then </b><i><font color="Green">E106</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font>) <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="orders_1.html#K1">BOOL</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#D2">ORDERS_1:def 2</a></i>;<br/>
<a NAME="E13:166_3_2"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font>) <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Choice</font>
 
<b>by </b><i>, <a class="txt" href="modelc_1.html#E40"><i><font color="Green">Lemma130</font></i></a>, <a class="ref" href="orders_1.html#D2">ORDERS_1:def 2</a></i>;<br/>
<a NAME="E14:166_3_2"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">s0</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="modelc_1.html#K31" target="_self">|**</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E38"><i><font color="Green">Lemma128</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<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> = 
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1)
<b>by </b><i><a class="txt" href="modelc_1.html#E45"><i><font color="Green">Lemma135</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="modelc_1.html#K31" target="_self">|**</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s0</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="modelc_1.html#K31" target="_self">|**</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s0</font>
<b>by </b><i><a class="ref" href="funct_7.html#T73">FUNCT_7:73</a></i>
<br/>.= 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="modelc_1.html#K31" target="_self">|**</a> <font color="Maroon">k</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s0</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">k</font>)
<b>by </b><i/>
<br/>.= 
<font color="Maroon">h</font> <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">k</font></span>)</span>
<b>by </b><i><a class="txt" href="modelc_1.html#E45"><i><font color="Green">Lemma135</font></i></a></i>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font>)
<b>by </b><i><a class="ref" href="modelc_1.html#D23" target="_self">Def23</a></i>
<br/>.= 
<font color="Maroon">Choice</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font>) <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
;<br/>
<a NAME="E16:166_3_2"/><b>then </b>
<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="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font>) <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="modelc_1.html#D21" target="_self">Def21</a></i>;<br/>
<a NAME="E17:166_3_2"/><b>then </b>
<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="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">q1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E19:166_3_2"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">q1</font> <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> &amp; <font color="Maroon">q1</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">g</font> )
 ;<br/>
<a NAME="E20:166_3_2"/>
<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>  <a href="modelc_1.html#K27" target="_self">EX</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E37"><i><font color="Green">Lemma127</font></i></a>, <a class="ref" href="modelc_1.html#D26" target="_self">Def26</a></i>;<br/>
<b>then consider </b><font color="Maroon">pai02</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="E22:166_3_2"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">pai02</font> <a href="funct_2.html#K3">.</a> 0 <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>and </b><br/><a NAME="E23:166_3_2"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">pai02</font> <a href="funct_2.html#K3">.</a> 1 <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">g</font>
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">x2</font> = <font color="Maroon">pai02</font> <a href="funct_2.html#K3">.</a> 1;<br/>
<a NAME="E24:166_3_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">pai02</font> <a href="funct_2.html#K3">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">pai02</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default">0 <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i/>;<br/>
<a NAME="E25:166_3_2"/><b>then </b><i><font color="Green">E112</font></i>: 
<font color="Maroon">pai02</font> <a href="funct_2.html#K3">.</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<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><a class="ref" href="modelc_1.html#D27" target="_self">Def27</a></i>;<br/>
<a NAME="E26:166_3_2"/>
<font color="Maroon">pai02</font> <a href="funct_2.html#K3">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font>
 
<b>by </b><i/>;<br/>
<a NAME="E27:166_3_2"/><b>then </b>
<font color="Maroon">pai02</font> <a href="funct_2.html#K3">.</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<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="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="modelc_1.html#D28" target="_self">Def28</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E28:166_3_2"/><b>then </b>
not <font color="Maroon">H<sub>1</sub></font>(<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="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E29:166_3_2"/><b>then </b><i><font color="Green">E113</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<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="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">S</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E30:166_3_2"/><b>then </b><i><font color="Green">E114</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<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="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="orders_1.html#K1">BOOL</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#D2">ORDERS_1:def 2</a></i>;<br/>
<a NAME="E31:166_3_2"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<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="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Choice</font>
 
<b>by </b><i>, <a class="txt" href="modelc_1.html#E40"><i><font color="Green">Lemma130</font></i></a>, <a class="ref" href="orders_1.html#D2">ORDERS_1:def 2</a></i>;<br/>
<a NAME="E32:166_3_2"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">s0</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="modelc_1.html#K31" target="_self">|**</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E38"><i><font color="Green">Lemma128</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> = 
<font color="Maroon">H<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1)
<b>by </b><i><a class="txt" href="modelc_1.html#E45"><i><font color="Green">Lemma135</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="modelc_1.html#K31" target="_self">|**</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s0</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="modelc_1.html#K31" target="_self">|**</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s0</font>
<b>by </b><i><a class="ref" href="funct_7.html#T73">FUNCT_7:73</a></i>
<br/>.= 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="modelc_1.html#K31" target="_self">|**</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s0</font></span>)</span>
<b>by </b><i><a class="txt" href="modelc_1.html#E55"><i><font color="Green">Lemma154</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1)
<b>by </b><i/>
<br/>.= 
<font color="Maroon">h</font> <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> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="modelc_1.html#E45"><i><font color="Green">Lemma135</font></i></a></i>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<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><a class="ref" href="modelc_1.html#D23" target="_self">Def23</a></i>
<br/>.= 
<font color="Maroon">Choice</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<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="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="modelc_1.html#E54"><i><font color="Green">Lemma153</font></i></a>, </i>
;<br/>
<a NAME="E34:166_3_2"/><b>then </b>
<font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<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="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="modelc_1.html#D21" target="_self">Def21</a></i>;<br/>
<a NAME="E35:166_3_2"/><b>then </b>
<font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<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><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">q2</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E37:166_3_2"/><i><font color="Green">E117</font></i>: 
( <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">q2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 ;<br/>
<b>thus </b><a NAME="E38:166_3_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1]
 <b>by </b><i><a class="ref" href="modelc_1.html#D26" target="_self">Def26</a>, <a class="txt" href="modelc_1.html#E56"><i><font color="Green">Lemma155</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:166_3"/>
for <font color="Olive">k</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">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="modelc_1.html#E48"><i><font color="Green">Lemma138</font></i></a>, );<br/></i>
<b>hence </b><a NAME="E4:166_3"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">g</font> )
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">pai</font> = <font color="Maroon">pai</font> as    <a href="modelc_1.html#M1" target="_self">inf_path</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">pai</font>
;<br/>

<b>thus </b><a NAME="E17:166"/>
( <font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">s0</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</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#E46"><i><font color="Green">Lemma136</font></i></a>, <a class="txt" href="modelc_1.html#E47"><i><font color="Green">Lemma137</font></i></a></i>;<br/>


</div>
