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

<b>let </b><font color="Maroon" title="c1">S</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">R</font> being <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">BASSIGN</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K32" title="MODELC_1:func.32">ModelSP</a> <font color="Maroon" title="c1">S</font></span>)</span><br/> for <font color="Olive" title="b3">g</font> being  <a href="modelc_1.html#NM2" title="MODELC_1:NM.2">Assign</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K50" title="MODELC_1:func.50">CTLModel</a> <font color="Olive" title="b1">R</font>,<font color="Olive" title="b2">BASSIGN</font></span>)</span><br/> for <font color="Olive" title="b4">s0</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b4">s0</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Olive" title="b3">g</font> &amp; ( for <font color="Olive" title="b5">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b5">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Olive" title="b3">g</font> holds <br/><font color="Olive" title="b5">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K27" title="MODELC_1:func.27">EX</a> <font color="Olive" title="b3">g</font> ) holds <br/> ex <font color="Olive" title="b5">pai</font> being   <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Olive" title="b1">R</font> st <br/>( <font color="Olive" title="b5">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">s0</font> &amp; ( for <font color="Olive" title="b6">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Olive" title="b5">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b6">n</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Olive" title="b3">g</font> ) )</span><br/><b>let </b><font color="Maroon" title="c2">R</font> be  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">BASSIGN</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K32" title="MODELC_1:func.32">ModelSP</a> <font color="Maroon" title="c1">S</font></span>)</span><br/> for <font color="Olive" title="b2">g</font> being  <a href="modelc_1.html#NM2" title="MODELC_1:NM.2">Assign</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K50" title="MODELC_1:func.50">CTLModel</a> <font color="Maroon" title="c2">R</font>,<font color="Olive" title="b1">BASSIGN</font></span>)</span><br/> for <font color="Olive" title="b3">s0</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b3">s0</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Olive" title="b2">g</font> &amp; ( for <font color="Olive" title="b4">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b4">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Olive" title="b2">g</font> holds <br/><font color="Olive" title="b4">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K27" title="MODELC_1:func.27">EX</a> <font color="Olive" title="b2">g</font> ) holds <br/> ex <font color="Olive" title="b4">pai</font> being   <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c2">R</font> st <br/>( <font color="Olive" title="b4">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">s0</font> &amp; ( for <font color="Olive" title="b5">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Olive" title="b4">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">n</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Olive" title="b2">g</font> ) )</span><br/><b>let </b><font color="Maroon" title="c3">BASSIGN</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K32" title="MODELC_1:func.32">ModelSP</a> <font color="Maroon" title="c1">S</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">g</font> being  <a href="modelc_1.html#NM2" title="MODELC_1:NM.2">Assign</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K50" title="MODELC_1:func.50">CTLModel</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font></span>)</span><br/> for <font color="Olive" title="b2">s0</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b2">s0</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Olive" title="b1">g</font> &amp; ( for <font color="Olive" title="b3">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b3">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Olive" title="b1">g</font> holds <br/><font color="Olive" title="b3">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K27" title="MODELC_1:func.27">EX</a> <font color="Olive" title="b1">g</font> ) holds <br/> ex <font color="Olive" title="b3">pai</font> being   <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c2">R</font> st <br/>( <font color="Olive" title="b3">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">s0</font> &amp; ( for <font color="Olive" title="b4">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Olive" title="b3">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">n</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Olive" title="b1">g</font> ) )</span><br/><b>let </b><font color="Maroon" title="c4">g</font> be   <a href="modelc_1.html#NM2" title="MODELC_1:NM.2">Assign</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K50" title="MODELC_1:func.50">CTLModel</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">s0</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b1">s0</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font> &amp; ( for <font color="Olive" title="b2">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b2">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font> holds <br/><font color="Olive" title="b2">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K27" title="MODELC_1:func.27">EX</a> <font color="Maroon" title="c4">g</font> ) holds <br/> ex <font color="Olive" title="b2">pai</font> being   <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c2">R</font> st <br/>( <font color="Olive" title="b2">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s0</font> &amp; ( for <font color="Olive" title="b3">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Olive" title="b2">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">n</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font> ) )</span><br/><b>let </b><font color="Maroon" title="c5">s0</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">s0</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font> &amp; ( for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b1">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font> holds <br/><font color="Olive" title="b1">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K27" title="MODELC_1:func.27">EX</a> <font color="Maroon" title="c4">g</font> ) implies  ex <font color="Olive" title="b1">pai</font> being   <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c2">R</font> st <br/>( <font color="Olive" title="b1">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s0</font> &amp; ( for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Olive" title="b1">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">n</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font> ) ) )</span><br/>









<b>assume </b><a NAME="E1:166"/><i><font color="Green" title="E126">A1</font></i>: 
<font color="Maroon" title="c5">s0</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Olive" title="b1">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font> &amp; not <font color="Olive" title="b1">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K27" title="MODELC_1:func.27">EX</a> <font color="Maroon" title="c4">g</font> ) or  ex <font color="Olive" title="b1">pai</font> being   <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c2">R</font> st <br/>( <font color="Olive" title="b1">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s0</font> &amp; ( for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Olive" title="b1">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">n</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font> ) ) )</span><br/>

<b>assume </b><a NAME="E2:166"/><i><font color="Green" title="E127">A2</font></i>: 
for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b1">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font> holds <br/><font color="Olive" title="b1">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K27" title="MODELC_1:func.27">EX</a> <font color="Maroon" title="c4">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">pai</font> being   <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c2">R</font> st <br/>( <font color="Olive" title="b1">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s0</font> &amp; ( for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Olive" title="b1">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">n</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font> ) )</span><br/>

<a NAME="E3:166"/><i><font color="Green" title="E128">A3</font></i>: 
<font color="Maroon" title="c5">s0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font>
 
;<br/>
<a NAME="E4:166"/>
for <font color="Olive" title="b1">p</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> <font color="Maroon" title="c1">S</font> holds <br/><font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="orders_1.html#T4" title="ORDERS_1:th.4">ORDERS_1:4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">Choice</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E6:166"/><i><font color="Green" title="E129">A4</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">Choice</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> <font color="Maroon" title="c1">S</font>
 <b>and </b><br/><a NAME="E7:166"/><i><font color="Green" title="E130">A5</font></i>: 
for <font color="Olive" title="b1">p</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> <font color="Maroon" title="c1">S</font> holds <br/><font color="Maroon" title="c6">Choice</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p</font>
 <b>by </b><i><a class="ref" href="wellord2.html#T28" title="WELLORD2:th.28">WELLORD2:28</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> : <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">$1,<font color="Olive" title="b1">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> </span>}</span> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="modelc_1.html#K3" title="MODELC_1:func.3">UnivF</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>($1) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">Choice</font>,<font color="Maroon" title="c5">s0</font>;<br/>
<a NAME="E8:166"/><i><font color="Green" title="E131">A6</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font> holds <br/><font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">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" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font> implies <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font> )</span><br/>

<b>assume </b><a NAME="E1:166_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font></span><br/>

<b>set </b><font color="Maroon" title="c8">Y</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">x</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">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" title="c7">x</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">Choice</font> or  not <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">x</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">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" title="E132">A7</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">x</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">Choice</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font></span><br/><div class="add"><a NAME="E2:166_1_1_1"/><b>then </b>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Choice</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">x</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="modelc_1.html#D3" target="_self" title="MODELC_1:def.3">Def3</a></i>;<br/><a NAME="E3:166_1_1_1"/><b>then </b>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">x</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E6:166"><i><font color="Green" title="E129">A4</font></i></a>, <a class="txt" href="#E7:166"><i><font color="Green" title="E130">A5</font></i></a>, <a class="txt" href="#E1:166_1_1_1"><i><font color="Green" title="E132">A7</font></i></a></i>;<br/><a NAME="E4:166_1_1_1"/><b>then </b>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c7">x</font>)
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E6:166_1_1_1"/><i><font color="Green" title="E133">A8</font></i>: 
( <font color="Maroon" title="c9">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">x</font>) &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> )
 ;<br/><b>thus </b><a NAME="E7:166_1_1_1"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font>
 <b>by </b><i><a class="txt" href="#E6:166_1_1_1"><i><font color="Green" title="E133">A8</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="c7">x</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">Choice</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font></span><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" title="c7">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s0</font>
 <b>by </b><i><a class="ref" href="modelc_1.html#D3" target="_self" title="MODELC_1:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E3:166_1_1_2"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c7">h</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E10:166"/><i><font color="Green" title="E132">A9</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font> holds <br/><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">x</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S2" title="FUNCT_2:sch.2">FUNCT_2:sch 2</a>(<a class="txt" href="#E8:166"><i><font color="Green" title="E131">A6</font></i></a>);<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="modelc_1.html#K31" title="MODELC_1:func.31">|**</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K2" title="MODELC_1:func.2">k_nat</a> $1</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">s0</font>;<br/>
<a NAME="E11:166"/><i><font color="Green" title="E133">A10</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">n</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font>
 
;<br/>
<b>consider </b><font color="Maroon" title="c8">pai</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E13:166"/><i><font color="Green" title="E134">A11</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon" title="c8">pai</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">n</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S2" title="FUNCT_2:sch.2">FUNCT_2:sch 2</a>(<a class="txt" href="#E11:166"><i><font color="Green" title="E133">A10</font></i></a>);<br/></i>
<i><font color="Green" title="E135">A12</font></i>: <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<font color="Maroon">H<sub>3</sub></font>( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> )
<b>by </b><i><a class="txt" href="#E13:166"><i><font color="Green" title="E134">A11</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="modelc_1.html#K31" title="MODELC_1:func.31">|**</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">s0</font>
<b>by </b><i><a class="ref" href="modelc_1.html#D2" target="_self" title="MODELC_1:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">s0</font>
<b>by </b><i><a class="ref" href="funct_7.html#T86" title="FUNCT_7:th.86">FUNCT_7:86</a></i>
<br/>.= 
<font color="Maroon" title="c5">s0</font>
<b>by </b><i><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>
;<br/>
<a NAME="E15:166"/><i><font color="Green" title="E136">A13</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">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#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default">$1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1 <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font> );<br/>
<a NAME="E1:166_3"/><i><font color="Green" title="E137">A14</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<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" title="c9">Y</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">s0</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span>;<br/>
<a NAME="E1:166_3_1"/>
<font color="Maroon" title="c5">s0</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K27" title="MODELC_1:func.27">EX</a> <font color="Maroon" title="c4">g</font>
 
<b>by </b><i><a class="txt" href="#E1:166"><i><font color="Green" title="E126">A1</font></i></a>, <a class="txt" href="#E2:166"><i><font color="Green" title="E127">A2</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">pai01</font> being    <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c2">R</font><b> such that </b><br/><a NAME="E3:166_3_1"/><i><font color="Green" title="E138">A15</font></i>: 
<font color="Maroon" title="c10">pai01</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s0</font>
 <b>and </b><br/><a NAME="E4:166_3_1"/><i><font color="Green" title="E139">A16</font></i>: 
<font color="Maroon" title="c10">pai01</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1 <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font>
 <b>by </b><i><a class="ref" href="modelc_1.html#T14" target="_self" title="MODELC_1:th.14">Th14</a></i>;<br/>
<b>set </b><font color="Maroon" title="c11">x</font> = <font color="Maroon" title="c10">pai01</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1;<br/>
<a NAME="E5:166_3_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">pai01</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">pai01</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 
<b>by </b><i><a class="ref" href="modelc_1.html#D39" target="_self" title="MODELC_1:def.39">Def39</a></i>;<br/>
<a NAME="E6:166_3_1"/><b>then </b><i><font color="Green" title="E140">A17</font></i>: 
<font color="Maroon" title="c10">pai01</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">s0</font>)
 
<b>by </b><i><a class="txt" href="#E3:166_3_1"><i><font color="Green" title="E138">A15</font></i></a></i>;<br/>
<a NAME="E7:166_3_1"/>
<font color="Maroon" title="c10">pai01</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">g</font>
 
<b>by </b><i><a class="txt" href="#E4:166_3_1"><i><font color="Green" title="E139">A16</font></i></a></i>;<br/>
<a NAME="E8:166_3_1"/><b>then </b>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">s0</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="#E6:166_3_1"><i><font color="Green" title="E140">A17</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E9:166_3_1"/><b>then </b>
not <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">s0</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E10:166_3_1"/><b>then </b><i><font color="Green" title="E141">A18</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">s0</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/>
<a NAME="E11:166_3_1"/><b>then </b><i><font color="Green" title="E142">A19</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">s0</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> <font color="Maroon" title="c1">S</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#D2" title="ORDERS_1:def.2">ORDERS_1:def 2</a></i>;<br/>
<a NAME="E12:166_3_1"/><i><font color="Green" title="E143">A20</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">s0</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">Choice</font>
 
<b>by </b><i><a class="txt" href="#E6:166"><i><font color="Green" title="E129">A4</font></i></a>, <a class="txt" href="#E10:166_3_1"><i><font color="Green" title="E141">A18</font></i></a>, <a class="ref" href="orders_1.html#D2" title="ORDERS_1:def.2">ORDERS_1:def 2</a></i>;<br/>
<b>set </b><font color="Maroon" title="c12">y</font> = <font color="Maroon" title="c6">Choice</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">s0</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span>;<br/>
<a NAME="E13:166_3_1"/><i><font color="Green" title="E144">A21</font></i>: 
<font color="Maroon" title="c6">Choice</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">s0</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">s0</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E7:166"><i><font color="Green" title="E130">A5</font></i></a>, <a class="txt" href="#E11:166_3_1"><i><font color="Green" title="E142">A19</font></i></a></i>;<br/>
<a NAME="E14:166_3_1"/><b>then </b>
<font color="Maroon" title="c6">Choice</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">s0</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">s0</font>)
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">t</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E16:166_3_1"/><i><font color="Green" title="E145">A22</font></i>: 
( <font color="Maroon" title="c6">Choice</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">s0</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">t</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">s0</font>,<font color="Maroon" title="c13">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> )
 ;<br/>
<a NAME="E17:166_3_1"/>
<font color="Maroon" title="c13">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">g</font>
 
<b>by </b><i><a class="txt" href="#E13:166_3_1"><i><font color="Green" title="E144">A21</font></i></a>, <a class="txt" href="#E16:166_3_1"><i><font color="Green" title="E145">A22</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">s1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E19:166_3_1"/><i><font color="Green" title="E146">A23</font></i>: 
( <font color="Maroon" title="c13">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">s1</font> &amp; <font color="Maroon" title="c14">s1</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font> )
 ;<br/>
<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1 = 
<font color="Maroon">H<sub>3</sub></font>(1)
<b>by </b><i><a class="txt" href="#E13:166"><i><font color="Green" title="E134">A11</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="modelc_1.html#K31" title="MODELC_1:func.31">|**</a> 1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">s0</font>
<b>by </b><i><a class="ref" href="modelc_1.html#D2" target="_self" title="MODELC_1:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c7">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">s0</font>
<b>by </b><i><a class="ref" href="funct_7.html#T72" title="FUNCT_7:th.72">FUNCT_7:72</a></i>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">s0</font>)
<b>by </b><i><a class="txt" href="#E10:166"><i><font color="Green" title="E132">A9</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c14">s1</font>
<b>by </b><i><a class="txt" href="#E12:166_3_1"><i><font color="Green" title="E143">A20</font></i></a>, <a class="txt" href="#E16:166_3_1"><i><font color="Green" title="E145">A22</font></i></a>, <a class="txt" href="#E19:166_3_1"><i><font color="Green" title="E146">A23</font></i></a>, <a class="ref" href="modelc_1.html#D3" target="_self" title="MODELC_1:def.3">Def3</a></i>
;<br/>
<b>hence </b><a NAME="E21:166_3_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <b>by </b><i><a class="txt" href="#E1:166"><i><font color="Green" title="E126">A1</font></i></a>, <a class="txt" href="#E14:166"><i><font color="Green" title="E135">A12</font></i></a>, <a class="txt" href="#E16:166_3_1"><i><font color="Green" title="E145">A22</font></i></a>, <a class="txt" href="#E19:166_3_1"><i><font color="Green" title="E146">A23</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E2:166_3"/><i><font color="Green" title="E138">A24</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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" title="c9">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<b>set </b><font color="Maroon" title="c10">k1</font> = <font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/>
<b>set </b><font color="Maroon" title="c11">k2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/>
<b>set </b><font color="Maroon" title="c12">p0</font> = <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">k</font>;<br/>
<b>set </b><font color="Maroon" title="c13">p1</font> = <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c14">p2</font> = <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c15">Y1</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">k</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c16">Y2</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">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" title="c9">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<a NAME="E2:166_3_2"/><b>then </b>
<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">k</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K27" title="MODELC_1:func.27">EX</a> <font color="Maroon" title="c4">g</font>
 
<b>by </b><i><a class="txt" href="#E2:166"><i><font color="Green" title="E127">A2</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">pai01</font> being    <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c2">R</font><b> such that </b><br/><a NAME="E4:166_3_2"/><i><font color="Green" title="E139">A25</font></i>: 
<font color="Maroon" title="c17">pai01</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">k</font>
 <b>and </b><br/><a NAME="E5:166_3_2"/><i><font color="Green" title="E140">A26</font></i>: 
<font color="Maroon" title="c17">pai01</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1 <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font>
 <b>by </b><i><a class="ref" href="modelc_1.html#T14" target="_self" title="MODELC_1:th.14">Th14</a></i>;<br/>
<b>set </b><font color="Maroon" title="c18">x1</font> = <font color="Maroon" title="c17">pai01</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1;<br/>
<a NAME="E6:166_3_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">pai01</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">pai01</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 
<b>by </b><i><a class="ref" href="modelc_1.html#D39" target="_self" title="MODELC_1:def.39">Def39</a></i>;<br/>
<a NAME="E7:166_3_2"/><b>then </b><i><font color="Green" title="E141">A27</font></i>: 
<font color="Maroon" title="c17">pai01</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">k</font>)
 
<b>by </b><i><a class="txt" href="#E4:166_3_2"><i><font color="Green" title="E139">A25</font></i></a></i>;<br/>
<a NAME="E8:166_3_2"/>
<font color="Maroon" title="c17">pai01</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">g</font>
 
<b>by </b><i><a class="txt" href="#E5:166_3_2"><i><font color="Green" title="E140">A26</font></i></a></i>;<br/>
<a NAME="E9:166_3_2"/><b>then </b>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">k</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="#E7:166_3_2"><i><font color="Green" title="E141">A27</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E10:166_3_2"/><b>then </b>
not <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">k</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E11:166_3_2"/><b>then </b><i><font color="Green" title="E142">A28</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">k</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/>
<a NAME="E12:166_3_2"/><b>then </b><i><font color="Green" title="E143">A29</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">k</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> <font color="Maroon" title="c1">S</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#D2" title="ORDERS_1:def.2">ORDERS_1:def 2</a></i>;<br/>
<a NAME="E13:166_3_2"/><i><font color="Green" title="E144">A30</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">k</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">Choice</font>
 
<b>by </b><i><a class="txt" href="#E6:166"><i><font color="Green" title="E129">A4</font></i></a>, <a class="txt" href="#E11:166_3_2"><i><font color="Green" title="E142">A28</font></i></a>, <a class="ref" href="orders_1.html#D2" title="ORDERS_1:def.2">ORDERS_1:def 2</a></i>;<br/>
<a NAME="E14:166_3_2"/><i><font color="Green" title="E145">A31</font></i>: 
<font color="Maroon" title="c5">s0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="modelc_1.html#K31" title="MODELC_1:func.31">|**</a> <font color="Maroon" title="c9">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:166"><i><font color="Green" title="E128">A3</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1)
<b>by </b><i><a class="txt" href="#E13:166"><i><font color="Green" title="E134">A11</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="modelc_1.html#K31" title="MODELC_1:func.31">|**</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">s0</font>
<b>by </b><i><a class="ref" href="modelc_1.html#D2" target="_self" title="MODELC_1:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="modelc_1.html#K31" title="MODELC_1:func.31">|**</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">s0</font>
<b>by </b><i><a class="ref" href="funct_7.html#T73" title="FUNCT_7:th.73">FUNCT_7:73</a></i>
<br/>.= 
<font color="Maroon" title="c7">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="modelc_1.html#K31" title="MODELC_1:func.31">|**</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">s0</font></span>)</span>
<b>by </b><i><a class="txt" href="#E14:166_3_2"><i><font color="Green" title="E145">A31</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c7">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c9">k</font>)
<b>by </b><i><a class="ref" href="modelc_1.html#D2" target="_self" title="MODELC_1:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c7">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">k</font></span>)</span>
<b>by </b><i><a class="txt" href="#E13:166"><i><font color="Green" title="E134">A11</font></i></a></i>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">k</font>)
<b>by </b><i><a class="txt" href="#E10:166"><i><font color="Green" title="E132">A9</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">Choice</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">k</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E13:166_3_2"><i><font color="Green" title="E144">A30</font></i></a>, <a class="ref" href="modelc_1.html#D3" target="_self" title="MODELC_1:def.3">Def3</a></i>
;<br/>
<a NAME="E16:166_3_2"/><b>then </b>
<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">k</font>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E7:166"><i><font color="Green" title="E130">A5</font></i></a>, <a class="txt" href="#E12:166_3_2"><i><font color="Green" title="E143">A29</font></i></a></i>;<br/>
<a NAME="E17:166_3_2"/><b>then </b>
<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">g</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c19">q1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E19:166_3_2"/><i><font color="Green" title="E146">A32</font></i>: 
( <font color="Maroon" title="c19">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c19">q1</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font> )
 ;<br/>
<a NAME="E20:166_3_2"/>
<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K27" title="MODELC_1:func.27">EX</a> <font color="Maroon" title="c4">g</font>
 
<b>by </b><i><a class="txt" href="#E2:166"><i><font color="Green" title="E127">A2</font></i></a>, <a class="txt" href="#E19:166_3_2"><i><font color="Green" title="E146">A32</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c20">pai02</font> being    <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c2">R</font><b> such that </b><br/><a NAME="E22:166_3_2"/><i><font color="Green" title="E147">A33</font></i>: 
<font color="Maroon" title="c20">pai02</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E23:166_3_2"/><i><font color="Green" title="E148">A34</font></i>: 
<font color="Maroon" title="c20">pai02</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1 <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font>
 <b>by </b><i><a class="ref" href="modelc_1.html#T14" target="_self" title="MODELC_1:th.14">Th14</a></i>;<br/>
<b>set </b><font color="Maroon" title="c21">x2</font> = <font color="Maroon" title="c20">pai02</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1;<br/>
<a NAME="E24:166_3_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">pai02</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">pai02</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 
<b>by </b><i><a class="ref" href="modelc_1.html#D39" target="_self" title="MODELC_1:def.39">Def39</a></i>;<br/>
<a NAME="E25:166_3_2"/><b>then </b><i><font color="Green" title="E149">A35</font></i>: 
<font color="Maroon" title="c20">pai02</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
 
<b>by </b><i><a class="txt" href="#E22:166_3_2"><i><font color="Green" title="E147">A33</font></i></a></i>;<br/>
<a NAME="E26:166_3_2"/>
<font color="Maroon" title="c20">pai02</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">g</font>
 
<b>by </b><i><a class="txt" href="#E23:166_3_2"><i><font color="Green" title="E148">A34</font></i></a></i>;<br/>
<a NAME="E27:166_3_2"/><b>then </b>
<font color="Maroon" title="c20">pai02</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E25:166_3_2"><i><font color="Green" title="E149">A35</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E28:166_3_2"/><b>then </b>
not <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E29:166_3_2"/><b>then </b><i><font color="Green" title="E150">A36</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/>
<a NAME="E30:166_3_2"/><b>then </b><i><font color="Green" title="E151">A37</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="orders_1.html#K1" title="ORDERS_1:func.1">BOOL</a> <font color="Maroon" title="c1">S</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#D2" title="ORDERS_1:def.2">ORDERS_1:def 2</a></i>;<br/>
<a NAME="E31:166_3_2"/><i><font color="Green" title="E152">A38</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">Choice</font>
 
<b>by </b><i><a class="txt" href="#E6:166"><i><font color="Green" title="E129">A4</font></i></a>, <a class="txt" href="#E29:166_3_2"><i><font color="Green" title="E150">A36</font></i></a>, <a class="ref" href="orders_1.html#D2" title="ORDERS_1:def.2">ORDERS_1:def 2</a></i>;<br/>
<a NAME="E32:166_3_2"/><i><font color="Green" title="E153">A39</font></i>: 
<font color="Maroon" title="c5">s0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="modelc_1.html#K31" title="MODELC_1:func.31">|**</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:166"><i><font color="Green" title="E128">A3</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<font color="Maroon">H<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1)
<b>by </b><i><a class="txt" href="#E13:166"><i><font color="Green" title="E134">A11</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="modelc_1.html#K31" title="MODELC_1:func.31">|**</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">s0</font>
<b>by </b><i><a class="ref" href="modelc_1.html#D2" target="_self" title="MODELC_1:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="modelc_1.html#K31" title="MODELC_1:func.31">|**</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">s0</font>
<b>by </b><i><a class="ref" href="funct_7.html#T73" title="FUNCT_7:th.73">FUNCT_7:73</a></i>
<br/>.= 
<font color="Maroon" title="c7">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="modelc_1.html#K31" title="MODELC_1:func.31">|**</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">s0</font></span>)</span>
<b>by </b><i><a class="txt" href="#E32:166_3_2"><i><font color="Green" title="E153">A39</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c7">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1)
<b>by </b><i><a class="ref" href="modelc_1.html#D2" target="_self" title="MODELC_1:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c7">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E13:166"><i><font color="Green" title="E134">A11</font></i></a></i>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
<b>by </b><i><a class="txt" href="#E10:166"><i><font color="Green" title="E132">A9</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">Choice</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E31:166_3_2"><i><font color="Green" title="E152">A38</font></i></a>, <a class="ref" href="modelc_1.html#D3" target="_self" title="MODELC_1:def.3">Def3</a></i>
;<br/>
<a NAME="E34:166_3_2"/><b>then </b>
<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E7:166"><i><font color="Green" title="E130">A5</font></i></a>, <a class="txt" href="#E30:166_3_2"><i><font color="Green" title="E151">A37</font></i></a></i>;<br/>
<a NAME="E35:166_3_2"/><b>then </b>
<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c22">q2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E37:166_3_2"/><i><font color="Green" title="E154">A40</font></i>: 
( <font color="Maroon" title="c22">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c22">q2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> )
 ;<br/>
<b>thus </b><a NAME="E38:166_3_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <b>by </b><i><a class="txt" href="#E19:166_3_2"><i><font color="Green" title="E146">A32</font></i></a>, <a class="txt" href="#E37:166_3_2"><i><font color="Green" title="E154">A40</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:166_3"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E1:166_3"><i><font color="Green" title="E137">A14</font></i></a>, <a class="txt" href="#E2:166_3"><i><font color="Green" title="E138">A24</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:166_3"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon" title="c8">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c9">pai</font> = <font color="Maroon" title="c8">pai</font> as    <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c2">R</font> <b>by </b><i><a class="ref" href="modelc_1.html#D39" target="_self" title="MODELC_1:def.39">Def39</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">pai</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s0</font> &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c9">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font> ) )</span><br/>

<b>thus </b><a NAME="E17:166"/>
( <font color="Maroon" title="c9">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s0</font> &amp; ( for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c9">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">g</font> ) )
 <b>by </b><i><a class="txt" href="#E14:166"><i><font color="Green" title="E135">A12</font></i></a>, <a class="txt" href="#E15:166"><i><font color="Green" title="E136">A13</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
