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

<b>let </b><font color="Maroon">V</font> be    <a href="modelc_1.html#L2" target="_self">CTLModelStr</a> ;<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 <font color="Maroon">V</font>;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b> ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="modelc_1.html#K13" target="_self">CTL_WFF</a> ,the <a href="modelc_1.html#U5" target="_self">Assignations</a> of <font color="Maroon">V</font> st <font color="Olive">f</font> <a href="modelc_1.html#R2" target="_self">is-PreEvaluation-for</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">Kai</font>;<br/>
<a NAME="E1:57"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">v0</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:57_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">v0</font> <a href="hidden.html#R2">in</a> the <a href="modelc_1.html#U5" target="_self">Assignations</a> of <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E3:57_1"/><i><font color="Green">E50</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v0</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> the <a href="modelc_1.html#U5" target="_self">Assignations</a> of <font color="Maroon">V</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E4:57_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>set </b><font color="Maroon">f</font> = <a href="modelc_1.html#K13" target="_self">CTL_WFF</a>  <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">v0</font>;<br/>
<a NAME="E4:57_1"/><i><font color="Green">E52</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K13" target="_self">CTL_WFF</a>  <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">v0</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="modelc_1.html#K13" target="_self">CTL_WFF</a>  &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K13" target="_self">CTL_WFF</a>  <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">v0</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v0</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E5:57_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K13" target="_self">CTL_WFF</a>  <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">v0</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="modelc_1.html#U5" target="_self">Assignations</a> of <font color="Maroon">V</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E2:57"><i><font color="Green">E53</font></i></a>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <a href="modelc_1.html#K13" target="_self">CTL_WFF</a>  <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">v0</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="modelc_1.html#K13" target="_self">CTL_WFF</a> ,the <a href="modelc_1.html#U5" target="_self">Assignations</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="funct_2.html#T4">FUNCT_2:4</a>, </i>;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E7:57_1"/>
<font color="Maroon">f</font> <a href="modelc_1.html#R2" target="_self">is-PreEvaluation-for</a> 0,<font color="Maroon">Kai</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E13"><i><font color="Green">Lemma46</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:57"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="binop_2.html#K23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:57_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<b>consider </b><font color="Maroon">h</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="modelc_1.html#K13" target="_self">CTL_WFF</a> ,the <a href="modelc_1.html#U5" target="_self">Assignations</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:57_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">h</font> <a href="modelc_1.html#R2" target="_self">is-PreEvaluation-for</a> <font color="Maroon">k</font>,<font color="Maroon">Kai</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E2:57"><i><font color="Green">E53</font></i></a></i>;<br/>
<a NAME="E4:57_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_1"><b>proof </b></a><div class="add">

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="modelc_1.html#K20" target="_self">GraftEval</a> <font color="Maroon">V</font>,<font color="Maroon">Kai</font>,<font color="Maroon">h</font>,<font color="Maroon">h</font>,<font color="Maroon">k</font>,<span class="p1">(<span class="default"><a href="modelc_1.html#K18" target="_self">CastCTLformula</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E1:57_2_1"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">H</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">H</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K13" target="_self">CTL_WFF</a>  holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">H</font>) <a href="hidden.html#R2">in</a> the <a href="modelc_1.html#U5" target="_self">Assignations</a> of <font color="Maroon">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:57_2_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">H</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K13" target="_self">CTL_WFF</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon">H</font> = <font color="Maroon">H</font> as   <a href="modelc_1.html#NM1">CTL-formula</a> <b>by </b><i>, <a class="txt" href="modelc_1.html#E1:57_2_1"><i><font color="Green">E56</font></i></a></i>;<br/>
<a NAME="E3:57_2_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">H</font>) <a href="hidden.html#R1">=</a>  <a href="modelc_1.html#K20" target="_self">GraftEval</a> <font color="Maroon">V</font>,<font color="Maroon">Kai</font>,<font color="Maroon">h</font>,<font color="Maroon">h</font>,<font color="Maroon">k</font>,<font color="Maroon">H</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E1:57_2_1"><i><font color="Green">E56</font></i></a>, </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_2_1_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 or (  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">H</font> is <a href="modelc_1.html#V2" target="_self">atomic</a> ) or (  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">H</font> is <a href="modelc_1.html#V3" target="_self">negative</a> ) or (  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">H</font> is <a href="modelc_1.html#V4" target="_self">conjunctive</a> ) or (  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">H</font> is <a href="modelc_1.html#V5" target="_self">ExistNext</a> ) or (  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">H</font> is <a href="modelc_1.html#V6" target="_self">ExistGlobal</a> ) or (  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">H</font> is <a href="modelc_1.html#V7" target="_self">ExistUntill</a> ) or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:57_2_1_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1
 ;<br/><div class="add"><a NAME="E2:57_2_1_1_1_1"/><b>then </b>
 <a href="modelc_1.html#K20" target="_self">GraftEval</a> <font color="Maroon">V</font>,<font color="Maroon">Kai</font>,<font color="Maroon">h</font>,<font color="Maroon">h</font>,<font color="Maroon">k</font>,<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:57_2_1_1_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">H</font>) <a href="hidden.html#R2">in</a> the <a href="modelc_1.html#U5" target="_self">Assignations</a> of <font color="Maroon">V</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E1:57_2_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a>, <a class="txt" href="modelc_1.html#E1:57_2_1_1"><i><font color="Green">E57</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:57_2_1_1_1_2"/><i><font color="Green">E59</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">H</font> is <a href="modelc_1.html#V2" target="_self">atomic</a> )
 ;<br/><div class="add"><a NAME="E2:57_2_1_1_1_2"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">H</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K19" target="_self">atomic_WFF</a> 
 ;<br/><a NAME="E3:57_2_1_1_1_2"/>
<font color="Maroon">Kai</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font> <a href="hidden.html#R2">in</a> the <a href="modelc_1.html#U6" target="_self">BasicAssign</a> of <font color="Maroon">V</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E1:57_2_1_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E4:57_2_1_1_1_2"/><b>then </b>
<font color="Maroon">Kai</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font> <a href="hidden.html#R2">in</a> the <a href="modelc_1.html#U5" target="_self">Assignations</a> of <font color="Maroon">V</font>
 ;<br/><b>hence </b><a NAME="E5:57_2_1_1_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">H</font>) <a href="hidden.html#R2">in</a> the <a href="modelc_1.html#U5" target="_self">Assignations</a> of <font color="Maroon">V</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E3:57_2_1_1"><i><font color="Green">E58</font></i></a>, , <a class="txt" href="modelc_1.html#E1:57_2_1_1"><i><font color="Green">E57</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_1_1_3"><b>suppose </b></a><a NAME="E1:57_2_1_1_1_3"/><i><font color="Green">E61</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">H</font> is <a href="modelc_1.html#V3" target="_self">negative</a> )
 ;<br/><div class="add"><a NAME="E2:57_2_1_1_1_3"/>
 <a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Maroon">H</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K13" target="_self">CTL_WFF</a> 
 <b>by </b><i/>;<br/><a NAME="E3:57_2_1_1_1_3"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="modelc_1.html#U5" target="_self">Assignations</a> of <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E4:57_2_1_1_1_3"/>
 <a href="modelc_1.html#K20" target="_self">GraftEval</a> <font color="Maroon">V</font>,<font color="Maroon">Kai</font>,<font color="Maroon">h</font>,<font color="Maroon">h</font>,<font color="Maroon">k</font>,<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> the <a href="modelc_1.html#U8" target="_self">Not</a> of <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="modelc_1.html#E3:57_2_1_1"><i><font color="Green">E58</font></i></a>, </i>;<br/><b>hence </b><a NAME="E5:57_2_1_1_1_3"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">H</font>) <a href="hidden.html#R2">in</a> the <a href="modelc_1.html#U5" target="_self">Assignations</a> of <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a>, <a class="txt" href="modelc_1.html#E1:57_2_1_1"><i><font color="Green">E57</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_1_1_4"><b>suppose </b></a><a NAME="E1:57_2_1_1_1_4"/><i><font color="Green">E63</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">H</font> is <a href="modelc_1.html#V4" target="_self">conjunctive</a> )
 ;<br/><div class="add"><a NAME="E2:57_2_1_1_1_4"/>
 <a href="modelc_1.html#K16" target="_self">the_left_argument_of</a> <font color="Maroon">H</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K13" target="_self">CTL_WFF</a> 
 <b>by </b><i/>;<br/><a NAME="E3:57_2_1_1_1_4"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K16" target="_self">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="modelc_1.html#U5" target="_self">Assignations</a> of <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E4:57_2_1_1_1_4"/>
 <a href="modelc_1.html#K17" target="_self">the_right_argument_of</a> <font color="Maroon">H</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K13" target="_self">CTL_WFF</a> 
 <b>by </b><i/>;<br/><a NAME="E5:57_2_1_1_1_4"/><b>then </b>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K17" target="_self">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="modelc_1.html#U5" target="_self">Assignations</a> of <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E6:57_2_1_1_1_4"/><b>then </b><i><font color="Green">E65</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="modelc_1.html#K16" target="_self">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="modelc_1.html#K17" target="_self">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="modelc_1.html#U5" target="_self">Assignations</a> of <font color="Maroon">V</font>,the <a href="modelc_1.html#U5" target="_self">Assignations</a> of <font color="Maroon">V</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><a NAME="E7:57_2_1_1_1_4"/>
 <a href="modelc_1.html#K20" target="_self">GraftEval</a> <font color="Maroon">V</font>,<font color="Maroon">Kai</font>,<font color="Maroon">h</font>,<font color="Maroon">h</font>,<font color="Maroon">k</font>,<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> the <a href="modelc_1.html#U7" target="_self">And</a> of <font color="Maroon">V</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" target="_self">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" target="_self">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="modelc_1.html#E3:57_2_1_1"><i><font color="Green">E58</font></i></a>, </i>;<br/><b>hence </b><a NAME="E8:57_2_1_1_1_4"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">H</font>) <a href="hidden.html#R2">in</a> the <a href="modelc_1.html#U5" target="_self">Assignations</a> of <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a>, <a class="txt" href="modelc_1.html#E1:57_2_1_1"><i><font color="Green">E57</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_1_1_5"><b>suppose </b></a><a NAME="E1:57_2_1_1_1_5"/><i><font color="Green">E66</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">H</font> is <a href="modelc_1.html#V5" target="_self">ExistNext</a> )
 ;<br/><div class="add"><a NAME="E2:57_2_1_1_1_5"/>
 <a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Maroon">H</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K13" target="_self">CTL_WFF</a> 
 <b>by </b><i/>;<br/><a NAME="E3:57_2_1_1_1_5"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="modelc_1.html#U5" target="_self">Assignations</a> of <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E4:57_2_1_1_1_5"/>
 <a href="modelc_1.html#K20" target="_self">GraftEval</a> <font color="Maroon">V</font>,<font color="Maroon">Kai</font>,<font color="Maroon">h</font>,<font color="Maroon">h</font>,<font color="Maroon">k</font>,<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> the <a href="modelc_1.html#U9" target="_self">EneXt</a> of <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="modelc_1.html#E3:57_2_1_1"><i><font color="Green">E58</font></i></a>, </i>;<br/><b>hence </b><a NAME="E5:57_2_1_1_1_5"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">H</font>) <a href="hidden.html#R2">in</a> the <a href="modelc_1.html#U5" target="_self">Assignations</a> of <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a>, <a class="txt" href="modelc_1.html#E1:57_2_1_1"><i><font color="Green">E57</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_1_1_6"><b>suppose </b></a><a NAME="E1:57_2_1_1_1_6"/><i><font color="Green">E68</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">H</font> is <a href="modelc_1.html#V6" target="_self">ExistGlobal</a> )
 ;<br/><div class="add"><a NAME="E2:57_2_1_1_1_6"/>
 <a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Maroon">H</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K13" target="_self">CTL_WFF</a> 
 <b>by </b><i/>;<br/><a NAME="E3:57_2_1_1_1_6"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="modelc_1.html#U5" target="_self">Assignations</a> of <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E4:57_2_1_1_1_6"/>
 <a href="modelc_1.html#K20" target="_self">GraftEval</a> <font color="Maroon">V</font>,<font color="Maroon">Kai</font>,<font color="Maroon">h</font>,<font color="Maroon">h</font>,<font color="Maroon">k</font>,<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> the <a href="modelc_1.html#U10" target="_self">EGlobal</a> of <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="modelc_1.html#E3:57_2_1_1"><i><font color="Green">E58</font></i></a>, </i>;<br/><b>hence </b><a NAME="E5:57_2_1_1_1_6"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">H</font>) <a href="hidden.html#R2">in</a> the <a href="modelc_1.html#U5" target="_self">Assignations</a> of <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a>, <a class="txt" href="modelc_1.html#E1:57_2_1_1"><i><font color="Green">E57</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_1_1_7"><b>suppose </b></a><a NAME="E1:57_2_1_1_1_7"/><i><font color="Green">E70</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">H</font> is <a href="modelc_1.html#V7" target="_self">ExistUntill</a> )
 ;<br/><div class="add"><a NAME="E2:57_2_1_1_1_7"/>
 <a href="modelc_1.html#K16" target="_self">the_left_argument_of</a> <font color="Maroon">H</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K13" target="_self">CTL_WFF</a> 
 <b>by </b><i/>;<br/><a NAME="E3:57_2_1_1_1_7"/><b>then </b><i><font color="Green">E99</font></i>: 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K16" target="_self">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="modelc_1.html#U5" target="_self">Assignations</a> of <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E4:57_2_1_1_1_7"/>
 <a href="modelc_1.html#K17" target="_self">the_right_argument_of</a> <font color="Maroon">H</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K13" target="_self">CTL_WFF</a> 
 <b>by </b><i/>;<br/><a NAME="E5:57_2_1_1_1_7"/><b>then </b>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K17" target="_self">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="modelc_1.html#U5" target="_self">Assignations</a> of <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E6:57_2_1_1_1_7"/><b>then </b><i><font color="Green">E100</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="modelc_1.html#K16" target="_self">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="modelc_1.html#K17" target="_self">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="modelc_1.html#U5" target="_self">Assignations</a> of <font color="Maroon">V</font>,the <a href="modelc_1.html#U5" target="_self">Assignations</a> of <font color="Maroon">V</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><a NAME="E7:57_2_1_1_1_7"/>
 <a href="modelc_1.html#K20" target="_self">GraftEval</a> <font color="Maroon">V</font>,<font color="Maroon">Kai</font>,<font color="Maroon">h</font>,<font color="Maroon">h</font>,<font color="Maroon">k</font>,<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> the <a href="modelc_1.html#U11" target="_self">EUntill</a> of <font color="Maroon">V</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" target="_self">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" target="_self">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="modelc_1.html#E3:57_2_1_1"><i><font color="Green">E58</font></i></a>, </i>;<br/><b>hence </b><a NAME="E8:57_2_1_1_1_7"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">H</font>) <a href="hidden.html#R2">in</a> the <a href="modelc_1.html#U5" target="_self">Assignations</a> of <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a>, <a class="txt" href="modelc_1.html#E1:57_2_1_1"><i><font color="Green">E57</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_1_1_8"><b>suppose </b></a><a NAME="E1:57_2_1_1_1_8"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1
 ;<br/><div class="add"><a NAME="E2:57_2_1_1_1_8"/><b>then </b>
 <a href="modelc_1.html#K20" target="_self">GraftEval</a> <font color="Maroon">V</font>,<font color="Maroon">Kai</font>,<font color="Maroon">h</font>,<font color="Maroon">h</font>,<font color="Maroon">k</font>,<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:57_2_1_1_1_8"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">H</font>) <a href="hidden.html#R2">in</a> the <a href="modelc_1.html#U5" target="_self">Assignations</a> of <font color="Maroon">V</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E1:57_2_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a>, <a class="txt" href="modelc_1.html#E1:57_2_1_1"><i><font color="Green">E57</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="modelc_1.html#K13" target="_self">CTL_WFF</a> ,the <a href="modelc_1.html#U5" target="_self">Assignations</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E3:57_2_1"/><i><font color="Green">E101</font></i>: 
for <font color="Olive">H</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">H</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K13" target="_self">CTL_WFF</a>  holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">H</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S2">FUNCT_2:sch 2</a>(<a class="txt" href="modelc_1.html#E3:57_2"><i><font color="Green">E55</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<a NAME="E4:57_2_1"/><i><font color="Green">E102</font></i>: 
for <font color="Olive">H</font> being  <a href="modelc_1.html#NM1">CTL-formula</a>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">H</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">H</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">H</font> be   <a href="modelc_1.html#NM1">CTL-formula</a>;<br/>

<b>assume </b><a NAME="E1:57_2_1_2"/><i><font color="Green">E103</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1
 ;<br/>

<a NAME="E2:57_2_1_2"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">H</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K13" target="_self">CTL_WFF</a> 
 
<b>by </b><i/>;<br/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font> = 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">H</font>)
<b>by </b><i>, <a class="txt" href="modelc_1.html#E1:57_2_1"><i><font color="Green">E56</font></i></a></i>
<br/>.= 
 <a href="modelc_1.html#K20" target="_self">GraftEval</a> <font color="Maroon">V</font>,<font color="Maroon">Kai</font>,<font color="Maroon">h</font>,<font color="Maroon">h</font>,<font color="Maroon">k</font>,<font color="Maroon">H</font>
<b>by </b><i>, </i>
;<br/>
<b>hence </b><a NAME="E4:57_2_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:57_2_1"/>
<font color="Maroon">f</font> <a href="modelc_1.html#R2" target="_self">is-PreEvaluation-for</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">Kai</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:57_2_1_3"/>
for <font color="Olive">H</font> being  <a href="modelc_1.html#NM1">CTL-formula</a>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">H</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 holds <br/>( ( <font color="Olive">H</font> is <a href="modelc_1.html#V2" target="_self">atomic</a> implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Kai</font> <a href="funct_1.html#K1">.</a> <font color="Olive">H</font> ) &amp; ( <font color="Olive">H</font> is <a href="modelc_1.html#V3" target="_self">negative</a> implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">H</font> <a href="hidden.html#R1">=</a> the <a href="modelc_1.html#U8" target="_self">Not</a> of <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Olive">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Olive">H</font> is <a href="modelc_1.html#V4" target="_self">conjunctive</a> implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">H</font> <a href="hidden.html#R1">=</a> the <a href="modelc_1.html#U7" target="_self">And</a> of <font color="Maroon">V</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" target="_self">the_left_argument_of</a> <font color="Olive">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" target="_self">the_right_argument_of</a> <font color="Olive">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Olive">H</font> is <a href="modelc_1.html#V5" target="_self">ExistNext</a> implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">H</font> <a href="hidden.html#R1">=</a> the <a href="modelc_1.html#U9" target="_self">EneXt</a> of <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Olive">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Olive">H</font> is <a href="modelc_1.html#V6" target="_self">ExistGlobal</a> implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">H</font> <a href="hidden.html#R1">=</a> the <a href="modelc_1.html#U10" target="_self">EGlobal</a> of <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Olive">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Olive">H</font> is <a href="modelc_1.html#V7" target="_self">ExistUntill</a> implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">H</font> <a href="hidden.html#R1">=</a> the <a href="modelc_1.html#U11" target="_self">EUntill</a> of <font color="Maroon">V</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" target="_self">the_left_argument_of</a> <font color="Olive">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" target="_self">the_right_argument_of</a> <font color="Olive">H</font></span>)</span></span>)</span> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">H</font> be   <a href="modelc_1.html#NM1">CTL-formula</a>;<br/>

<b>assume </b><a NAME="E1:57_2_1_3_1"/><i><font color="Green">E105</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1
 ;<br/>

<a NAME="E2:57_2_1_3_1"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">H</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K13" target="_self">CTL_WFF</a> 
 
<b>by </b><i/>;<br/>
<i><font color="Green">E107</font></i>: <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font> = 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">H</font>)
<b>by </b><i>, <a class="txt" href="modelc_1.html#E1:57_2_1"><i><font color="Green">E56</font></i></a></i>
<br/>.= 
 <a href="modelc_1.html#K20" target="_self">GraftEval</a> <font color="Maroon">V</font>,<font color="Maroon">Kai</font>,<font color="Maroon">h</font>,<font color="Maroon">h</font>,<font color="Maroon">k</font>,<font color="Maroon">H</font>
<b>by </b><i>, </i>
;<br/>
<a NAME="E4:57_2_1_3_1"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">H</font> is <a href="modelc_1.html#V2" target="_self">atomic</a> implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Kai</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_1_3_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:57_2_1_3_1_2"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">H</font> is <a href="modelc_1.html#V2" target="_self">atomic</a>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_3_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_2_1_3_1_2_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T26">NAT_1:26</a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_3_1_2_1_1"><b>suppose </b></a><a NAME="E1:57_2_1_3_1_2_1_1"/><i><font color="Green">E110</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:57_2_1_3_1_2_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then </b><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font> = 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">Kai</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font>
<b>by </b><i><a class="txt" href="modelc_1.html#E1:57_2"><i><font color="Green">E54</font></i></a>, , , </i>
;<br/><b>hence </b><a NAME="E4:57_2_1_3_1_2_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Kai</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_3_1_2_1_2"><b>suppose </b></a><a NAME="E1:57_2_1_3_1_2_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:57_2_1_3_1_2_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Kai</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E5:57_2_1_3_1"/><i><font color="Green">E111</font></i>: 
( <font color="Maroon">H</font> is <a href="modelc_1.html#V3" target="_self">negative</a> implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> the <a href="modelc_1.html#U8" target="_self">Not</a> of <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_1_3_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:57_2_1_3_1_3"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">H</font> is <a href="modelc_1.html#V3" target="_self">negative</a>
 ;<br/>

<a NAME="E2:57_2_1_3_1_3"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Maroon">H</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:57_2_1_3_1_3"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Maroon">H</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:57_2_1_3_1_3"/><b>then </b><i><font color="Green">E113</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Maroon">H</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_3_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_2_1_3_1_3_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T26">NAT_1:26</a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_3_1_3_1_1"><b>suppose </b></a><a NAME="E1:57_2_1_3_1_3_1_1"/><i><font color="Green">E114</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:57_2_1_3_1_3_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then </b><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font> = 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font>
<b>by </b><i/>
<br/>.= 
the <a href="modelc_1.html#U8" target="_self">Not</a> of <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="modelc_1.html#E1:57_2"><i><font color="Green">E54</font></i></a>, , , </i>
;<br/><b>hence </b><a NAME="E4:57_2_1_3_1_3_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> the <a href="modelc_1.html#U8" target="_self">Not</a> of <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_3_1_3_1_2"><b>suppose </b></a><a NAME="E1:57_2_1_3_1_3_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1
 ;<br/><div class="add"><b>then </b> <a href="modelc_1.html#K20" target="_self">GraftEval</a> <font color="Maroon">V</font>,<font color="Maroon">Kai</font>,<font color="Maroon">h</font>,<font color="Maroon">h</font>,<font color="Maroon">k</font>,<font color="Maroon">H</font> = 
the <a href="modelc_1.html#U8" target="_self">Not</a> of <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
the <a href="modelc_1.html#U8" target="_self">Not</a> of <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
;<br/><b>hence </b><a NAME="E3:57_2_1_3_1_3_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> the <a href="modelc_1.html#U8" target="_self">Not</a> of <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E6:57_2_1_3_1"/><i><font color="Green">E115</font></i>: 
( <font color="Maroon">H</font> is <a href="modelc_1.html#V4" target="_self">conjunctive</a> implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> the <a href="modelc_1.html#U7" target="_self">And</a> of <font color="Maroon">V</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" target="_self">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" target="_self">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_1_3_1_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:57_2_1_3_1_4"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">H</font> is <a href="modelc_1.html#V4" target="_self">conjunctive</a>
 ;<br/>

<a NAME="E2:57_2_1_3_1_4"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K16" target="_self">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:57_2_1_3_1_4"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K16" target="_self">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:57_2_1_3_1_4"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K16" target="_self">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E5:57_2_1_3_1_4"/><b>then </b><i><font color="Green">E117</font></i>: 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K16" target="_self">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K16" target="_self">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:57_2_1_3_1_4"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K17" target="_self">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:57_2_1_3_1_4"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K17" target="_self">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:57_2_1_3_1_4"/><b>then </b><i><font color="Green">E118</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K17" target="_self">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_3_1_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_2_1_3_1_4_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T26">NAT_1:26</a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_3_1_4_1_1"><b>suppose </b></a><a NAME="E1:57_2_1_3_1_4_1_1"/><i><font color="Green">E119</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:57_2_1_3_1_4_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then </b><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font> = 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font>
<b>by </b><i/>
<br/>.= 
the <a href="modelc_1.html#U7" target="_self">And</a> of <font color="Maroon">V</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" target="_self">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" target="_self">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="modelc_1.html#E1:57_2"><i><font color="Green">E54</font></i></a>, , , </i>
;<br/><b>hence </b><a NAME="E4:57_2_1_3_1_4_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> the <a href="modelc_1.html#U7" target="_self">And</a> of <font color="Maroon">V</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" target="_self">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" target="_self">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_3_1_4_1_2"><b>suppose </b></a><a NAME="E1:57_2_1_3_1_4_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1
 ;<br/><div class="add"><b>then </b> <a href="modelc_1.html#K20" target="_self">GraftEval</a> <font color="Maroon">V</font>,<font color="Maroon">Kai</font>,<font color="Maroon">h</font>,<font color="Maroon">h</font>,<font color="Maroon">k</font>,<font color="Maroon">H</font> = 
the <a href="modelc_1.html#U7" target="_self">And</a> of <font color="Maroon">V</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" target="_self">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" target="_self">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
the <a href="modelc_1.html#U7" target="_self">And</a> of <font color="Maroon">V</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" target="_self">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" target="_self">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>
<b>by </b><i>, , </i>
;<br/><b>hence </b><a NAME="E3:57_2_1_3_1_4_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> the <a href="modelc_1.html#U7" target="_self">And</a> of <font color="Maroon">V</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" target="_self">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" target="_self">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E7:57_2_1_3_1"/><i><font color="Green">E120</font></i>: 
( <font color="Maroon">H</font> is <a href="modelc_1.html#V5" target="_self">ExistNext</a> implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> the <a href="modelc_1.html#U9" target="_self">EneXt</a> of <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_1_3_1_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:57_2_1_3_1_5"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">H</font> is <a href="modelc_1.html#V5" target="_self">ExistNext</a>
 ;<br/>

<a NAME="E2:57_2_1_3_1_5"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Maroon">H</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:57_2_1_3_1_5"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Maroon">H</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:57_2_1_3_1_5"/><b>then </b><i><font color="Green">E122</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Maroon">H</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_3_1_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_2_1_3_1_5_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T26">NAT_1:26</a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_3_1_5_1_1"><b>suppose </b></a><a NAME="E1:57_2_1_3_1_5_1_1"/><i><font color="Green">E123</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:57_2_1_3_1_5_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then </b><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font> = 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font>
<b>by </b><i/>
<br/>.= 
the <a href="modelc_1.html#U9" target="_self">EneXt</a> of <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="modelc_1.html#E1:57_2"><i><font color="Green">E54</font></i></a>, , , </i>
;<br/><b>hence </b><a NAME="E4:57_2_1_3_1_5_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> the <a href="modelc_1.html#U9" target="_self">EneXt</a> of <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_3_1_5_1_2"><b>suppose </b></a><a NAME="E1:57_2_1_3_1_5_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1
 ;<br/><div class="add"><b>then </b> <a href="modelc_1.html#K20" target="_self">GraftEval</a> <font color="Maroon">V</font>,<font color="Maroon">Kai</font>,<font color="Maroon">h</font>,<font color="Maroon">h</font>,<font color="Maroon">k</font>,<font color="Maroon">H</font> = 
the <a href="modelc_1.html#U9" target="_self">EneXt</a> of <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
the <a href="modelc_1.html#U9" target="_self">EneXt</a> of <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
;<br/><b>hence </b><a NAME="E3:57_2_1_3_1_5_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> the <a href="modelc_1.html#U9" target="_self">EneXt</a> of <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E8:57_2_1_3_1"/><i><font color="Green">E159</font></i>: 
( <font color="Maroon">H</font> is <a href="modelc_1.html#V6" target="_self">ExistGlobal</a> implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> the <a href="modelc_1.html#U10" target="_self">EGlobal</a> of <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_1_3_1_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:57_2_1_3_1_6"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">H</font> is <a href="modelc_1.html#V6" target="_self">ExistGlobal</a>
 ;<br/>

<a NAME="E2:57_2_1_3_1_6"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Maroon">H</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:57_2_1_3_1_6"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Maroon">H</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:57_2_1_3_1_6"/><b>then </b><i><font color="Green">E161</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Maroon">H</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_3_1_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_2_1_3_1_6_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T26">NAT_1:26</a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_3_1_6_1_1"><b>suppose </b></a><a NAME="E1:57_2_1_3_1_6_1_1"/><i><font color="Green">E162</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:57_2_1_3_1_6_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then </b><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font> = 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font>
<b>by </b><i/>
<br/>.= 
the <a href="modelc_1.html#U10" target="_self">EGlobal</a> of <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="modelc_1.html#E1:57_2"><i><font color="Green">E54</font></i></a>, , , </i>
;<br/><b>hence </b><a NAME="E4:57_2_1_3_1_6_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> the <a href="modelc_1.html#U10" target="_self">EGlobal</a> of <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_3_1_6_1_2"><b>suppose </b></a><a NAME="E1:57_2_1_3_1_6_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1
 ;<br/><div class="add"><b>then </b> <a href="modelc_1.html#K20" target="_self">GraftEval</a> <font color="Maroon">V</font>,<font color="Maroon">Kai</font>,<font color="Maroon">h</font>,<font color="Maroon">h</font>,<font color="Maroon">k</font>,<font color="Maroon">H</font> = 
the <a href="modelc_1.html#U10" target="_self">EGlobal</a> of <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
the <a href="modelc_1.html#U10" target="_self">EGlobal</a> of <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
;<br/><b>hence </b><a NAME="E3:57_2_1_3_1_6_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> the <a href="modelc_1.html#U10" target="_self">EGlobal</a> of <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E9:57_2_1_3_1"/>
( <font color="Maroon">H</font> is <a href="modelc_1.html#V7" target="_self">ExistUntill</a> implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> the <a href="modelc_1.html#U11" target="_self">EUntill</a> of <font color="Maroon">V</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" target="_self">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" target="_self">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_1_3_1_7"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:57_2_1_3_1_7"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">H</font> is <a href="modelc_1.html#V7" target="_self">ExistUntill</a>
 ;<br/>

<a NAME="E2:57_2_1_3_1_7"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K16" target="_self">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:57_2_1_3_1_7"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K16" target="_self">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:57_2_1_3_1_7"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K16" target="_self">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E5:57_2_1_3_1_7"/><b>then </b><i><font color="Green">E164</font></i>: 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K16" target="_self">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K16" target="_self">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:57_2_1_3_1_7"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K17" target="_self">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:57_2_1_3_1_7"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K17" target="_self">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:57_2_1_3_1_7"/><b>then </b><i><font color="Green">E165</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K17" target="_self">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_3_1_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_2_1_3_1_7_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T26">NAT_1:26</a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_3_1_7_1_1"><b>suppose </b></a><a NAME="E1:57_2_1_3_1_7_1_1"/><i><font color="Green">E166</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:57_2_1_3_1_7_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then </b><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font> = 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font>
<b>by </b><i/>
<br/>.= 
the <a href="modelc_1.html#U11" target="_self">EUntill</a> of <font color="Maroon">V</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" target="_self">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" target="_self">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="modelc_1.html#E1:57_2"><i><font color="Green">E54</font></i></a>, , , </i>
;<br/><b>hence </b><a NAME="E4:57_2_1_3_1_7_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> the <a href="modelc_1.html#U11" target="_self">EUntill</a> of <font color="Maroon">V</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" target="_self">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" target="_self">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_3_1_7_1_2"><b>suppose </b></a><a NAME="E1:57_2_1_3_1_7_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1
 ;<br/><div class="add"><b>then </b> <a href="modelc_1.html#K20" target="_self">GraftEval</a> <font color="Maroon">V</font>,<font color="Maroon">Kai</font>,<font color="Maroon">h</font>,<font color="Maroon">h</font>,<font color="Maroon">k</font>,<font color="Maroon">H</font> = 
the <a href="modelc_1.html#U11" target="_self">EUntill</a> of <font color="Maroon">V</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" target="_self">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" target="_self">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
the <a href="modelc_1.html#U11" target="_self">EUntill</a> of <font color="Maroon">V</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" target="_self">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" target="_self">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>
<b>by </b><i>, , </i>
;<br/><b>hence </b><a NAME="E3:57_2_1_3_1_7_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> the <a href="modelc_1.html#U11" target="_self">EUntill</a> of <font color="Maroon">V</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" target="_self">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" target="_self">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E10:57_2_1_3_1"/>
( ( <font color="Maroon">H</font> is <a href="modelc_1.html#V2" target="_self">atomic</a> implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Kai</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font> ) &amp; ( <font color="Maroon">H</font> is <a href="modelc_1.html#V3" target="_self">negative</a> implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> the <a href="modelc_1.html#U8" target="_self">Not</a> of <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Maroon">H</font> is <a href="modelc_1.html#V4" target="_self">conjunctive</a> implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> the <a href="modelc_1.html#U7" target="_self">And</a> of <font color="Maroon">V</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" target="_self">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" target="_self">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Maroon">H</font> is <a href="modelc_1.html#V5" target="_self">ExistNext</a> implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> the <a href="modelc_1.html#U9" target="_self">EneXt</a> of <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Maroon">H</font> is <a href="modelc_1.html#V6" target="_self">ExistGlobal</a> implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> the <a href="modelc_1.html#U10" target="_self">EGlobal</a> of <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" target="_self">the_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Maroon">H</font> is <a href="modelc_1.html#V7" target="_self">ExistUntill</a> implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> the <a href="modelc_1.html#U11" target="_self">EUntill</a> of <font color="Maroon">V</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" target="_self">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" target="_self">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span></span>)</span> ) )
 <b>by </b><i>, , , , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:57_2_1_3"/>
<font color="Maroon">f</font> <a href="modelc_1.html#R2" target="_self">is-PreEvaluation-for</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">Kai</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:57_2_1"/>
<font color="Maroon">f</font> <a href="modelc_1.html#R2" target="_self">is-PreEvaluation-for</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1,<font color="Maroon">Kai</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:57_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:57"/>
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">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E4:57"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="modelc_1.html#K13" target="_self">CTL_WFF</a> ,the <a href="modelc_1.html#U5" target="_self">Assignations</a> of <font color="Maroon">V</font> st <font color="Olive">f</font> <a href="modelc_1.html#R2" target="_self">is-PreEvaluation-for</a> <font color="Olive">n</font>,<font color="Maroon">Kai</font>
 ;<br/>


</div>
