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

<b>let </b><font color="Maroon">H</font> be   <a href="modelc_1.html#NM1">CTL-formula</a>;<br/><b>let </b><font color="Maroon">S</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">R</font> be  <a href="partfun1.html#V1">total</a>  <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">S</font>,<font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">s</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">BASSIGN</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K32" target="_self">ModelSP</a> <font color="Maroon">S</font></span>)</span>;<br/><b>let </b><font color="Maroon">kai</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="modelc_1.html#K19" target="_self">atomic_WFF</a> ,the <a href="modelc_1.html#U6" target="_self">BasicAssign</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/>











<a NAME="E1:138"/>
( <font color="Maroon">s</font>,<font color="Maroon">kai</font> <a href="modelc_1.html#R4" target="_self">|=</a>  <a href="modelc_1.html#K11" target="_self">EG</a> <font color="Maroon">H</font> iff <font color="Maroon">s</font> <a href="modelc_1.html#R3" target="_self">|=</a>  <a href="modelc_1.html#K24" target="_self">Evaluate</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K11" target="_self">EG</a> <font color="Maroon">H</font></span>)</span>,<font color="Maroon">kai</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E2:138"/><b>then </b><i><font color="Green">E27</font></i>: 
( <font color="Maroon">s</font>,<font color="Maroon">kai</font> <a href="modelc_1.html#R4" target="_self">|=</a>  <a href="modelc_1.html#K11" target="_self">EG</a> <font color="Maroon">H</font> iff <font color="Maroon">s</font> <a href="modelc_1.html#R3" target="_self">|=</a>  <a href="modelc_1.html#K28" target="_self">EG</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K24" target="_self">Evaluate</a> <font color="Maroon">H</font>,<font color="Maroon">kai</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:138"/><i><font color="Green">E28</font></i>: 
(  ex <font color="Olive">pai</font> being   <a href="modelc_1.html#M1" target="_self">inf_path</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">pai</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; ( 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="Olive">pai</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="modelc_1.html#R3" target="_self">|=</a>  <a href="modelc_1.html#K24" target="_self">Evaluate</a> <font color="Maroon">H</font>,<font color="Maroon">kai</font> ) ) implies  ex <font color="Olive">pai</font> being   <a href="modelc_1.html#M1" target="_self">inf_path</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">pai</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; ( 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="Olive">pai</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>,<font color="Maroon">kai</font> <a href="modelc_1.html#R4" target="_self">|=</a> <font color="Maroon">H</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="138_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">pai</font> being    <a href="modelc_1.html#M1" target="_self">inf_path</a> of <font color="Maroon">R</font><b> such that </b><a NAME="E1:138_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &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>  <a href="modelc_1.html#K24" target="_self">Evaluate</a> <font color="Maroon">H</font>,<font color="Maroon">kai</font> ) )
 ;<br/>

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

<a NAME="E2:138_1"/>
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>,<font color="Maroon">kai</font> <a href="modelc_1.html#R4" target="_self">|=</a> <font color="Maroon">H</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="138_1_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:138_1_1"/>
<font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="modelc_1.html#R3" target="_self">|=</a>  <a href="modelc_1.html#K24" target="_self">Evaluate</a> <font color="Maroon">H</font>,<font color="Maroon">kai</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:138_1_1"/>
<font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>,<font color="Maroon">kai</font> <a href="modelc_1.html#R4" target="_self">|=</a> <font color="Maroon">H</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:138_1"/>
( <font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; ( 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>,<font color="Maroon">kai</font> <a href="modelc_1.html#R4" target="_self">|=</a> <font color="Maroon">H</font> ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:138"/>
(  ex <font color="Olive">pai</font> being   <a href="modelc_1.html#M1" target="_self">inf_path</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">pai</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; ( 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="Olive">pai</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>,<font color="Maroon">kai</font> <a href="modelc_1.html#R4" target="_self">|=</a> <font color="Maroon">H</font> ) ) implies  ex <font color="Olive">pai</font> being   <a href="modelc_1.html#M1" target="_self">inf_path</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">pai</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; ( 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="Olive">pai</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="modelc_1.html#R3" target="_self">|=</a>  <a href="modelc_1.html#K24" target="_self">Evaluate</a> <font color="Maroon">H</font>,<font color="Maroon">kai</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="138_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">pai</font> being    <a href="modelc_1.html#M1" target="_self">inf_path</a> of <font color="Maroon">R</font><b> such that </b><a NAME="E1:138_2"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &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>,<font color="Maroon">kai</font> <a href="modelc_1.html#R4" target="_self">|=</a> <font color="Maroon">H</font> ) )
 ;<br/>

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

<a NAME="E2:138_2"/>
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>  <a href="modelc_1.html#K24" target="_self">Evaluate</a> <font color="Maroon">H</font>,<font color="Maroon">kai</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="138_2_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:138_2_1"/>
<font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>,<font color="Maroon">kai</font> <a href="modelc_1.html#R4" target="_self">|=</a> <font color="Maroon">H</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:138_2_1"/>
<font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="modelc_1.html#R3" target="_self">|=</a>  <a href="modelc_1.html#K24" target="_self">Evaluate</a> <font color="Maroon">H</font>,<font color="Maroon">kai</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:138_2"/>
( <font color="Maroon">pai</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; ( 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>  <a href="modelc_1.html#K24" target="_self">Evaluate</a> <font color="Maroon">H</font>,<font color="Maroon">kai</font> ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:138"/>
( <font color="Maroon">s</font>,<font color="Maroon">kai</font> <a href="modelc_1.html#R4" target="_self">|=</a>  <a href="modelc_1.html#K11" target="_self">EG</a> <font color="Maroon">H</font> iff  ex <font color="Olive">pai</font> being   <a href="modelc_1.html#M1" target="_self">inf_path</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">pai</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; ( 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="Olive">pai</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>,<font color="Maroon">kai</font> <a href="modelc_1.html#R4" target="_self">|=</a> <font color="Maroon">H</font> ) ) )
 <b>by </b><i>, , </i>;<br/>


</div>
