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

<span class="kw">let </span><font color="Maroon" title="c1">V</font> be   <a href="modelc_1.html#NM3" title="MODELC_1:NM.3">CTLModel</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Kai</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="modelc_1.html#K19" title="MODELC_1:func.19">atomic_WFF</a>, the <a href="modelc_1.html#U3" title="MODELC_1:sel.3">BasicAssign</a> of <font color="Maroon" title="c1">V</font><br/>  for <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b3">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font> st <font color="Olive" title="b3">f</font> <a href="modelc_1.html#R2" title="MODELC_1:pred.2">is-PreEvaluation-for</a> <font color="Olive" title="b2">n</font>,<font color="Olive" title="b1">Kai</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">Kai</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="modelc_1.html#K19" title="MODELC_1:func.19">atomic_WFF</a>, the <a href="modelc_1.html#U3" title="MODELC_1:sel.3">BasicAssign</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b2">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font> st <font color="Olive" title="b2">f</font> <a href="modelc_1.html#R2" title="MODELC_1:pred.2">is-PreEvaluation-for</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c2">Kai</font></span><br/>



<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font> st <font color="Olive" title="b1">f</font> <a href="modelc_1.html#R2" title="MODELC_1:pred.2">is-PreEvaluation-for</a> $1,<font color="Maroon" title="c2">Kai</font>;<br/>
<a NAME="E1:64"/><span class="lab"><font color="Green" title="E52">A1</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">h</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E3:64_1"/><span class="lab"><font color="Green" title="E53">A2</font></span>: 
<font color="Maroon" title="c4">h</font> <a href="modelc_1.html#R2" title="MODELC_1:pred.2">is-PreEvaluation-for</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">Kai</font>
 ;<br/>
<a NAME="E4:64_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="modelc_1.html#K20" title="MODELC_1:func.20">GraftEval</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">Kai</font>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c3">k</font>,<span class="p1">(<span class="default"><a href="modelc_1.html#K18" title="MODELC_1:func.18">CastCTLformula</a> $1</span>)</span>);<br/>
<a NAME="E1:64_1_1"/><span class="lab"><font color="Green" title="E54">A3</font></span>: 
 for <font color="Olive" title="b1">H</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a>  holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">H</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">H</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a>  implies <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">H</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_1_1_1"/><span class="lab"><font color="Green" title="E55">A4</font></span>: 
<font color="Maroon" title="c5">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">H</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">H</font> = <font color="Maroon" title="c5">H</font> as   <a href="modelc_1.html#NM1" title="MODELC_1:NM.1">CTL-formula</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1_1"><span class="lab"><font color="Green" title="E55">A4</font></span></a>, <a class="ref" href="modelc_1.html#T1" target="_self" title="MODELC_1:th.1">Th1</a></span>;<br/>
<a NAME="E3:64_1_1_1"/><span class="lab"><font color="Green" title="E56">A5</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">H</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_1.html#K20" title="MODELC_1:func.20">GraftEval</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">Kai</font>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c6">H</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1_1"><span class="lab"><font color="Green" title="E55">A4</font></span></a>, <a class="ref" href="modelc_1.html#D24" target="_self" title="MODELC_1:def.24">Def24</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:64_1_1_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V2" title="MODELC_1:attr.2">atomic</a>  ) or (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V3" title="MODELC_1:attr.3">negative</a>  ) or (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V4" title="MODELC_1:attr.4">conjunctive</a>  ) or (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V5" title="MODELC_1:attr.5">ExistNext</a>  ) or (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V6" title="MODELC_1:attr.6">ExistGlobal</a>  ) or (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V7" title="MODELC_1:attr.7">ExistUntill</a>  ) or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#T2" target="_self" title="MODELC_1:th.2">Th2</a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:64_1_1_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">H</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font></span><br/><div class="add"><a NAME="E2:64_1_1_1_1_1"/><span class="kw">then </span>
 <a href="modelc_1.html#K20" title="MODELC_1:func.20">GraftEval</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">Kai</font>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c6">H</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#D28" target="_self" title="MODELC_1:def.28">Def28</a></span>;<br/><span class="kw">hence </span><a NAME="E3:64_1_1_1_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">H</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1_1"><span class="lab"><font color="Green" title="E55">A4</font></span></a>, <a class="txt" href="#E3:64_1_1_1"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:64_1_1_1_1_2"/><span class="lab"><font color="Green" title="E57">A6</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V2" title="MODELC_1:attr.2">atomic</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">H</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font></span><br/><div class="add"><a NAME="E2:64_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K19" title="MODELC_1:func.19">atomic_WFF</a> 
 ;<br/><a NAME="E3:64_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">Kai</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="modelc_1.html#U3" title="MODELC_1:sel.3">BasicAssign</a> of <font color="Maroon" title="c1">V</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></span>;<br/><a NAME="E4:64_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">Kai</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 ;<br/><span class="kw">hence </span><a NAME="E5:64_1_1_1_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">H</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1_1_1"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="txt" href="#E1:64_1_1_1_1_2"><span class="lab"><font color="Green" title="E57">A6</font></span></a>, <a class="ref" href="modelc_1.html#D28" target="_self" title="MODELC_1:def.28">Def28</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:64_1_1_1_1_3"/><span class="lab"><font color="Green" title="E57">A7</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V3" title="MODELC_1:attr.3">negative</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">H</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font></span><br/><div class="add"><a NAME="E2:64_1_1_1_1_3"/>
 <a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#T1" target="_self" title="MODELC_1:th.1">Th1</a></span>;<br/><a NAME="E3:64_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A8</font></span>: 
<font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></span>;<br/><a NAME="E4:64_1_1_1_1_3"/>
 <a href="modelc_1.html#K20" title="MODELC_1:func.20">GraftEval</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">Kai</font>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c6">H</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="robbins1.html#U1" title="ROBBINS1:sel.1">Compl</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1_1_1_3"><span class="lab"><font color="Green" title="E57">A7</font></span></a>, <a class="ref" href="modelc_1.html#D28" target="_self" title="MODELC_1:def.28">Def28</a></span>;<br/><span class="kw">hence </span><a NAME="E5:64_1_1_1_1_3"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">H</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1_1_1"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="txt" href="#E3:64_1_1_1_1_3"><span class="lab"><font color="Green" title="E58">A8</font></span></a>, <a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:64_1_1_1_1_4"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V4" title="MODELC_1:attr.4">conjunctive</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">H</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font></span><br/><div class="add"><a NAME="E2:64_1_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A9</font></span>: 
 <a href="modelc_1.html#K20" title="MODELC_1:func.20">GraftEval</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">Kai</font>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c6">H</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#D28" target="_self" title="MODELC_1:def.28">Def28</a></span>;<br/><a NAME="E3:64_1_1_1_1_4"/>
 <a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#T1" target="_self" title="MODELC_1:th.1">Th1</a></span>;<br/><a NAME="E4:64_1_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A10</font></span>: 
<font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></span>;<br/><a NAME="E5:64_1_1_1_1_4"/>
 <a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#T1" target="_self" title="MODELC_1:th.1">Th1</a></span>;<br/><a NAME="E6:64_1_1_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></span>;<br/><a NAME="E7:64_1_1_1_1_4"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_1_1_1_1_4"><span class="lab"><font color="Green" title="E58">A10</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E8:64_1_1_1_1_4"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">H</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1_1_1"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="txt" href="#E2:64_1_1_1_1_4"><span class="lab"><font color="Green" title="E57">A9</font></span></a>, <a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_1_5"><span class="kw">suppose </span></a><a NAME="E1:64_1_1_1_1_5"/><span class="lab"><font color="Green" title="E57">A11</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V5" title="MODELC_1:attr.5">ExistNext</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">H</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font></span><br/><div class="add"><a NAME="E2:64_1_1_1_1_5"/>
 <a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#T1" target="_self" title="MODELC_1:th.1">Th1</a></span>;<br/><a NAME="E3:64_1_1_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A12</font></span>: 
<font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></span>;<br/><a NAME="E4:64_1_1_1_1_5"/>
 <a href="modelc_1.html#K20" title="MODELC_1:func.20">GraftEval</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">Kai</font>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c6">H</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="modelc_1.html#U4" title="MODELC_1:sel.4">EneXt</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1_1_1_5"><span class="lab"><font color="Green" title="E57">A11</font></span></a>, <a class="ref" href="modelc_1.html#D28" target="_self" title="MODELC_1:def.28">Def28</a></span>;<br/><span class="kw">hence </span><a NAME="E5:64_1_1_1_1_5"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">H</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1_1_1"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="txt" href="#E3:64_1_1_1_1_5"><span class="lab"><font color="Green" title="E58">A12</font></span></a>, <a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_1_6"><span class="kw">suppose </span></a><a NAME="E1:64_1_1_1_1_6"/><span class="lab"><font color="Green" title="E57">A13</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V6" title="MODELC_1:attr.6">ExistGlobal</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">H</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font></span><br/><div class="add"><a NAME="E2:64_1_1_1_1_6"/>
 <a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#T1" target="_self" title="MODELC_1:th.1">Th1</a></span>;<br/><a NAME="E3:64_1_1_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A14</font></span>: 
<font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></span>;<br/><a NAME="E4:64_1_1_1_1_6"/>
 <a href="modelc_1.html#K20" title="MODELC_1:func.20">GraftEval</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">Kai</font>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c6">H</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="modelc_1.html#U5" title="MODELC_1:sel.5">EGlobal</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1_1_1_6"><span class="lab"><font color="Green" title="E57">A13</font></span></a>, <a class="ref" href="modelc_1.html#D28" target="_self" title="MODELC_1:def.28">Def28</a></span>;<br/><span class="kw">hence </span><a NAME="E5:64_1_1_1_1_6"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">H</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1_1_1"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="txt" href="#E3:64_1_1_1_1_6"><span class="lab"><font color="Green" title="E58">A14</font></span></a>, <a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_1_7"><span class="kw">suppose </span></a><a NAME="E1:64_1_1_1_1_7"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V7" title="MODELC_1:attr.7">ExistUntill</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">H</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font></span><br/><div class="add"><a NAME="E2:64_1_1_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A15</font></span>: 
 <a href="modelc_1.html#K20" title="MODELC_1:func.20">GraftEval</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">Kai</font>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c6">H</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="modelc_1.html#U6" title="MODELC_1:sel.6">EUntill</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#D28" target="_self" title="MODELC_1:def.28">Def28</a></span>;<br/><a NAME="E3:64_1_1_1_1_7"/>
 <a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#T1" target="_self" title="MODELC_1:th.1">Th1</a></span>;<br/><a NAME="E4:64_1_1_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A16</font></span>: 
<font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></span>;<br/><a NAME="E5:64_1_1_1_1_7"/>
 <a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#T1" target="_self" title="MODELC_1:th.1">Th1</a></span>;<br/><a NAME="E6:64_1_1_1_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></span>;<br/><a NAME="E7:64_1_1_1_1_7"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_1_1_1_1_7"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E8:64_1_1_1_1_7"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">H</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1_1_1"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="txt" href="#E2:64_1_1_1_1_7"><span class="lab"><font color="Green" title="E57">A15</font></span></a>, <a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_1_8"><span class="kw">suppose </span></a><a NAME="E1:64_1_1_1_1_8"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">H</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font></span><br/><div class="add"><a NAME="E2:64_1_1_1_1_8"/><span class="kw">then </span>
 <a href="modelc_1.html#K20" title="MODELC_1:func.20">GraftEval</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">Kai</font>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c6">H</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#D28" target="_self" title="MODELC_1:def.28">Def28</a></span>;<br/><span class="kw">hence </span><a NAME="E3:64_1_1_1_1_8"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">H</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1_1"><span class="lab"><font color="Green" title="E55">A4</font></span></a>, <a class="txt" href="#E3:64_1_1_1"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c5">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E3:64_1_1"/><span class="lab"><font color="Green" title="E55">A17</font></span>: 
 for <font color="Olive" title="b1">H</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a>  holds <br/><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">H</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S2" title="FUNCT_2:sch.2">FUNCT_2:sch 2</a>(<a class="txt" href="#E1:64_1_1"><span class="lab"><font color="Green" title="E54">A3</font></span></a>);<br/></span>
<span class="kw">take </span>
<font color="Maroon" title="c5">f</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="modelc_1.html#R2" title="MODELC_1:pred.2">is-PreEvaluation-for</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1,<font color="Maroon" title="c2">Kai</font></span><br/>

<a NAME="E4:64_1_1"/><span class="lab"><font color="Green" title="E56">A18</font></span>: 
 for <font color="Olive" title="b1">H</font> being   <a href="modelc_1.html#NM1" title="MODELC_1:NM.1">CTL-formula</a>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">H</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">H</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">H</font> be   <a href="modelc_1.html#NM1" title="MODELC_1:NM.1">CTL-formula</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_1_1_2"/><span class="lab"><font color="Green" title="E57">A19</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font></span><br/>

<a NAME="E2:64_1_1_2"/><span class="lab"><font color="Green" title="E58">A20</font></span>: 
<font color="Maroon" title="c6">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#T1" target="_self" title="MODELC_1:th.1">Th1</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> = 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">H</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1_1"><span class="lab"><font color="Green" title="E55">A17</font></span></a></span>
<br/>.= 
 <a href="modelc_1.html#K20" title="MODELC_1:func.20">GraftEval</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">Kai</font>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c6">H</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_1_1_2"><span class="lab"><font color="Green" title="E58">A20</font></span></a>, <a class="ref" href="modelc_1.html#D24" target="_self" title="MODELC_1:def.24">Def24</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:64_1_1_2"/>
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1_2"><span class="lab"><font color="Green" title="E57">A19</font></span></a>, <a class="ref" href="modelc_1.html#D28" target="_self" title="MODELC_1:def.28">Def28</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:64_1_1"/>
 for <font color="Olive" title="b1">H</font> being   <a href="modelc_1.html#NM1" title="MODELC_1:NM.1">CTL-formula</a>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">H</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/>( ( <font color="Olive" title="b1">H</font> is  <a href="modelc_1.html#V2" title="MODELC_1:attr.2">atomic</a>  implies <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Kai</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">H</font> ) &amp; ( <font color="Olive" title="b1">H</font> is  <a href="modelc_1.html#V3" title="MODELC_1:attr.3">negative</a>  implies <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="robbins1.html#U1" title="ROBBINS1:sel.1">Compl</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Olive" title="b1">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Olive" title="b1">H</font> is  <a href="modelc_1.html#V4" title="MODELC_1:attr.4">conjunctive</a>  implies <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Olive" title="b1">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Olive" title="b1">H</font></span>)</span></span>)</span>) ) &amp; ( <font color="Olive" title="b1">H</font> is  <a href="modelc_1.html#V5" title="MODELC_1:attr.5">ExistNext</a>  implies <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="modelc_1.html#U4" title="MODELC_1:sel.4">EneXt</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Olive" title="b1">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Olive" title="b1">H</font> is  <a href="modelc_1.html#V6" title="MODELC_1:attr.6">ExistGlobal</a>  implies <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="modelc_1.html#U5" title="MODELC_1:sel.5">EGlobal</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Olive" title="b1">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Olive" title="b1">H</font> is  <a href="modelc_1.html#V7" title="MODELC_1:attr.7">ExistUntill</a>  implies <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="modelc_1.html#U6" title="MODELC_1:sel.6">EUntill</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Olive" title="b1">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Olive" title="b1">H</font></span>)</span></span>)</span>) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">H</font> be   <a href="modelc_1.html#NM1" title="MODELC_1:NM.1">CTL-formula</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( ( <font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V2" title="MODELC_1:attr.2">atomic</a>  implies <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Kai</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> ) &amp; ( <font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V3" title="MODELC_1:attr.3">negative</a>  implies <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="robbins1.html#U1" title="ROBBINS1:sel.1">Compl</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V4" title="MODELC_1:attr.4">conjunctive</a>  implies <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V5" title="MODELC_1:attr.5">ExistNext</a>  implies <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="modelc_1.html#U4" title="MODELC_1:sel.4">EneXt</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V6" title="MODELC_1:attr.6">ExistGlobal</a>  implies <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="modelc_1.html#U5" title="MODELC_1:sel.5">EGlobal</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V7" title="MODELC_1:attr.7">ExistUntill</a>  implies <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="modelc_1.html#U6" title="MODELC_1:sel.6">EUntill</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>) ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:64_1_1_3"/><span class="lab"><font color="Green" title="E57">A21</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V2" title="MODELC_1:attr.2">atomic</a>  implies <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Kai</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> ) &amp; ( <font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V3" title="MODELC_1:attr.3">negative</a>  implies <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="robbins1.html#U1" title="ROBBINS1:sel.1">Compl</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V4" title="MODELC_1:attr.4">conjunctive</a>  implies <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V5" title="MODELC_1:attr.5">ExistNext</a>  implies <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="modelc_1.html#U4" title="MODELC_1:sel.4">EneXt</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V6" title="MODELC_1:attr.6">ExistGlobal</a>  implies <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="modelc_1.html#U5" title="MODELC_1:sel.5">EGlobal</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V7" title="MODELC_1:attr.7">ExistUntill</a>  implies <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="modelc_1.html#U6" title="MODELC_1:sel.6">EUntill</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>) ) )</span><br/>

<a NAME="E2:64_1_1_3"/><span class="lab"><font color="Green" title="E58">A22</font></span>: 
<font color="Maroon" title="c6">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#T1" target="_self" title="MODELC_1:th.1">Th1</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E59">A23</font></span>: <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> = 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">H</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1_1"><span class="lab"><font color="Green" title="E55">A17</font></span></a></span>
<br/>.= 
 <a href="modelc_1.html#K20" title="MODELC_1:func.20">GraftEval</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">Kai</font>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c6">H</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_1_1_3"><span class="lab"><font color="Green" title="E58">A22</font></span></a>, <a class="ref" href="modelc_1.html#D24" target="_self" title="MODELC_1:def.24">Def24</a></span>
;<br/>
<a NAME="E4:64_1_1_3"/><span class="lab"><font color="Green" title="E60">A24</font></span>: 
( <font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V3" title="MODELC_1:attr.3">negative</a>  implies <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="robbins1.html#U1" title="ROBBINS1:sel.1">Compl</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:64_1_1_3_2"/><span class="lab"><font color="Green" title="E61">A25</font></span>: 
<font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V3" title="MODELC_1:attr.3">negative</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="robbins1.html#U1" title="ROBBINS1:sel.1">Compl</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span></span><br/>

<a NAME="E2:64_1_1_3_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="modelc_1.html#E47"><span class="lab"><font color="Green" title="E40">Lm22</font></span></a></span>;<br/>
<a NAME="E3:64_1_1_3_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1_3"><span class="lab"><font color="Green" title="E57">A21</font></span></a>, <a class="txt" href="modelc_1.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/>
<a NAME="E4:64_1_1_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A26</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_3_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:64_1_1_3_2_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1_3"><span class="lab"><font color="Green" title="E57">A21</font></span></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_3_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:64_1_1_3_2_1_1"/><span class="lab"><font color="Green" title="E63">A27</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="robbins1.html#U1" title="ROBBINS1:sel.1">Compl</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:64_1_1_3_2_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> = 
<font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_1_1"><span class="lab"><font color="Green" title="E56">A18</font></span></a></span>
<br/>.= 
 the <a href="robbins1.html#U1" title="ROBBINS1:sel.1">Compl</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="txt" href="#E1:64_1_1_3_2"><span class="lab"><font color="Green" title="E61">A25</font></span></a>, <a class="txt" href="#E1:64_1_1_3_2_1_1"><span class="lab"><font color="Green" title="E63">A27</font></span></a>, <a class="ref" href="modelc_1.html#D27" target="_self" title="MODELC_1:def.27">Def27</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:64_1_1_3_2_1_1"/>
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="robbins1.html#U1" title="ROBBINS1:sel.1">Compl</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_1_1"><span class="lab"><font color="Green" title="E56">A18</font></span></a>, <a class="txt" href="#E4:64_1_1_3_2"><span class="lab"><font color="Green" title="E62">A26</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_3_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:64_1_1_3_2_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="robbins1.html#U1" title="ROBBINS1:sel.1">Compl</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span></span><br/><div class="add"><span class="kw">then </span> <a href="modelc_1.html#K20" title="MODELC_1:func.20">GraftEval</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">Kai</font>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c6">H</font>) = 
 the <a href="robbins1.html#U1" title="ROBBINS1:sel.1">Compl</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1_3_2"><span class="lab"><font color="Green" title="E61">A25</font></span></a>, <a class="ref" href="modelc_1.html#D28" target="_self" title="MODELC_1:def.28">Def28</a></span>
<br/>.= 
 the <a href="robbins1.html#U1" title="ROBBINS1:sel.1">Compl</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_1_1"><span class="lab"><font color="Green" title="E56">A18</font></span></a>, <a class="txt" href="#E4:64_1_1_3_2"><span class="lab"><font color="Green" title="E62">A26</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E3:64_1_1_3_2_1_2"/>
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="robbins1.html#U1" title="ROBBINS1:sel.1">Compl</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1_1_3"><span class="lab"><font color="Green" title="E59">A23</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E5:64_1_1_3"/><span class="lab"><font color="Green" title="E61">A28</font></span>: 
( <font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V7" title="MODELC_1:attr.7">ExistUntill</a>  implies <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="modelc_1.html#U6" title="MODELC_1:sel.6">EUntill</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_3_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:64_1_1_3_3"/><span class="lab"><font color="Green" title="E62">A29</font></span>: 
<font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V7" title="MODELC_1:attr.7">ExistUntill</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="modelc_1.html#U6" title="MODELC_1:sel.6">EUntill</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>)</span><br/>

<a NAME="E2:64_1_1_3_3"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="modelc_1.html#E48"><span class="lab"><font color="Green" title="E41">Lm23</font></span></a></span>;<br/>
<a NAME="E3:64_1_1_3_3"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1_3"><span class="lab"><font color="Green" title="E57">A21</font></span></a>, <a class="txt" href="modelc_1.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/>
<a NAME="E4:64_1_1_3_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A30</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E5:64_1_1_3_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1_3_3"><span class="lab"><font color="Green" title="E62">A29</font></span></a>, <a class="txt" href="modelc_1.html#E48"><span class="lab"><font color="Green" title="E41">Lm23</font></span></a></span>;<br/>
<a NAME="E6:64_1_1_3_3"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1_3"><span class="lab"><font color="Green" title="E57">A21</font></span></a>, <a class="txt" href="modelc_1.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/>
<a NAME="E7:64_1_1_3_3"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E8:64_1_1_3_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A31</font></span>: 
<font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_1_1"><span class="lab"><font color="Green" title="E56">A18</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_3_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:64_1_1_3_3_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1_3"><span class="lab"><font color="Green" title="E57">A21</font></span></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_3_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:64_1_1_3_3_1_1"/><span class="lab"><font color="Green" title="E65">A32</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="modelc_1.html#U6" title="MODELC_1:sel.6">EUntill</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>)</span><br/><div class="add"><a NAME="E2:64_1_1_3_3_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> = 
<font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_1_1"><span class="lab"><font color="Green" title="E56">A18</font></span></a></span>
<br/>.= 
 the <a href="modelc_1.html#U6" title="MODELC_1:sel.6">EUntill</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="txt" href="#E1:64_1_1_3_3"><span class="lab"><font color="Green" title="E62">A29</font></span></a>, <a class="txt" href="#E1:64_1_1_3_3_1_1"><span class="lab"><font color="Green" title="E65">A32</font></span></a>, <a class="ref" href="modelc_1.html#D27" target="_self" title="MODELC_1:def.27">Def27</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:64_1_1_3_3_1_1"/>
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="modelc_1.html#U6" title="MODELC_1:sel.6">EUntill</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_1_1"><span class="lab"><font color="Green" title="E56">A18</font></span></a>, <a class="txt" href="#E8:64_1_1_3_3"><span class="lab"><font color="Green" title="E64">A31</font></span></a>, <a class="txt" href="#E4:64_1_1_3_3"><span class="lab"><font color="Green" title="E63">A30</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_3_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:64_1_1_3_3_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="modelc_1.html#U6" title="MODELC_1:sel.6">EUntill</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>)</span><br/><div class="add"><span class="kw">then </span> <a href="modelc_1.html#K20" title="MODELC_1:func.20">GraftEval</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">Kai</font>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c6">H</font>) = 
 the <a href="modelc_1.html#U6" title="MODELC_1:sel.6">EUntill</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1_3_3"><span class="lab"><font color="Green" title="E62">A29</font></span></a>, <a class="ref" href="modelc_1.html#D28" target="_self" title="MODELC_1:def.28">Def28</a></span>
<br/>.= 
 the <a href="modelc_1.html#U6" title="MODELC_1:sel.6">EUntill</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_1_1"><span class="lab"><font color="Green" title="E56">A18</font></span></a>, <a class="txt" href="#E8:64_1_1_3_3"><span class="lab"><font color="Green" title="E64">A31</font></span></a>, <a class="txt" href="#E4:64_1_1_3_3"><span class="lab"><font color="Green" title="E63">A30</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E3:64_1_1_3_3_1_2"/>
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="modelc_1.html#U6" title="MODELC_1:sel.6">EUntill</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1_1_3"><span class="lab"><font color="Green" title="E59">A23</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E6:64_1_1_3"/><span class="lab"><font color="Green" title="E62">A33</font></span>: 
( <font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V4" title="MODELC_1:attr.4">conjunctive</a>  implies <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_3_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:64_1_1_3_4"/><span class="lab"><font color="Green" title="E63">A34</font></span>: 
<font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V4" title="MODELC_1:attr.4">conjunctive</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>)</span><br/>

<a NAME="E2:64_1_1_3_4"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="modelc_1.html#E48"><span class="lab"><font color="Green" title="E41">Lm23</font></span></a></span>;<br/>
<a NAME="E3:64_1_1_3_4"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1_3"><span class="lab"><font color="Green" title="E57">A21</font></span></a>, <a class="txt" href="modelc_1.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/>
<a NAME="E4:64_1_1_3_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A35</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E5:64_1_1_3_4"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1_3_4"><span class="lab"><font color="Green" title="E63">A34</font></span></a>, <a class="txt" href="modelc_1.html#E48"><span class="lab"><font color="Green" title="E41">Lm23</font></span></a></span>;<br/>
<a NAME="E6:64_1_1_3_4"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1_3"><span class="lab"><font color="Green" title="E57">A21</font></span></a>, <a class="txt" href="modelc_1.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/>
<a NAME="E7:64_1_1_3_4"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E8:64_1_1_3_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A36</font></span>: 
<font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_1_1"><span class="lab"><font color="Green" title="E56">A18</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_3_4_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:64_1_1_3_4_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1_3"><span class="lab"><font color="Green" title="E57">A21</font></span></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_3_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:64_1_1_3_4_1_1"/><span class="lab"><font color="Green" title="E66">A37</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>)</span><br/><div class="add"><a NAME="E2:64_1_1_3_4_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> = 
<font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_1_1"><span class="lab"><font color="Green" title="E56">A18</font></span></a></span>
<br/>.= 
 the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="txt" href="#E1:64_1_1_3_4"><span class="lab"><font color="Green" title="E63">A34</font></span></a>, <a class="txt" href="#E1:64_1_1_3_4_1_1"><span class="lab"><font color="Green" title="E66">A37</font></span></a>, <a class="ref" href="modelc_1.html#D27" target="_self" title="MODELC_1:def.27">Def27</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:64_1_1_3_4_1_1"/>
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_1_1"><span class="lab"><font color="Green" title="E56">A18</font></span></a>, <a class="txt" href="#E8:64_1_1_3_4"><span class="lab"><font color="Green" title="E65">A36</font></span></a>, <a class="txt" href="#E4:64_1_1_3_4"><span class="lab"><font color="Green" title="E64">A35</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_3_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:64_1_1_3_4_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>)</span><br/><div class="add"><span class="kw">then </span> <a href="modelc_1.html#K20" title="MODELC_1:func.20">GraftEval</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">Kai</font>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c6">H</font>) = 
 the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1_3_4"><span class="lab"><font color="Green" title="E63">A34</font></span></a>, <a class="ref" href="modelc_1.html#D28" target="_self" title="MODELC_1:def.28">Def28</a></span>
<br/>.= 
 the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_1_1"><span class="lab"><font color="Green" title="E56">A18</font></span></a>, <a class="txt" href="#E8:64_1_1_3_4"><span class="lab"><font color="Green" title="E65">A36</font></span></a>, <a class="txt" href="#E4:64_1_1_3_4"><span class="lab"><font color="Green" title="E64">A35</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E3:64_1_1_3_4_1_2"/>
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1_1_3"><span class="lab"><font color="Green" title="E59">A23</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E7:64_1_1_3"/><span class="lab"><font color="Green" title="E63">A38</font></span>: 
( <font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V6" title="MODELC_1:attr.6">ExistGlobal</a>  implies <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="modelc_1.html#U5" title="MODELC_1:sel.5">EGlobal</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_3_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:64_1_1_3_5"/><span class="lab"><font color="Green" title="E64">A39</font></span>: 
<font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V6" title="MODELC_1:attr.6">ExistGlobal</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="modelc_1.html#U5" title="MODELC_1:sel.5">EGlobal</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span></span><br/>

<a NAME="E2:64_1_1_3_5"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="modelc_1.html#E47"><span class="lab"><font color="Green" title="E40">Lm22</font></span></a></span>;<br/>
<a NAME="E3:64_1_1_3_5"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1_3"><span class="lab"><font color="Green" title="E57">A21</font></span></a>, <a class="txt" href="modelc_1.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/>
<a NAME="E4:64_1_1_3_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A40</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_3_5_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:64_1_1_3_5_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1_3"><span class="lab"><font color="Green" title="E57">A21</font></span></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_3_5_1_1"><span class="kw">suppose </span></a><a NAME="E1:64_1_1_3_5_1_1"/><span class="lab"><font color="Green" title="E66">A41</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="modelc_1.html#U5" title="MODELC_1:sel.5">EGlobal</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:64_1_1_3_5_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> = 
<font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_1_1"><span class="lab"><font color="Green" title="E56">A18</font></span></a></span>
<br/>.= 
 the <a href="modelc_1.html#U5" title="MODELC_1:sel.5">EGlobal</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="txt" href="#E1:64_1_1_3_5"><span class="lab"><font color="Green" title="E64">A39</font></span></a>, <a class="txt" href="#E1:64_1_1_3_5_1_1"><span class="lab"><font color="Green" title="E66">A41</font></span></a>, <a class="ref" href="modelc_1.html#D27" target="_self" title="MODELC_1:def.27">Def27</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:64_1_1_3_5_1_1"/>
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="modelc_1.html#U5" title="MODELC_1:sel.5">EGlobal</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_1_1"><span class="lab"><font color="Green" title="E56">A18</font></span></a>, <a class="txt" href="#E4:64_1_1_3_5"><span class="lab"><font color="Green" title="E65">A40</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_3_5_1_2"><span class="kw">suppose </span></a><a NAME="E1:64_1_1_3_5_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="modelc_1.html#U5" title="MODELC_1:sel.5">EGlobal</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span></span><br/><div class="add"><span class="kw">then </span> <a href="modelc_1.html#K20" title="MODELC_1:func.20">GraftEval</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">Kai</font>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c6">H</font>) = 
 the <a href="modelc_1.html#U5" title="MODELC_1:sel.5">EGlobal</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1_3_5"><span class="lab"><font color="Green" title="E64">A39</font></span></a>, <a class="ref" href="modelc_1.html#D28" target="_self" title="MODELC_1:def.28">Def28</a></span>
<br/>.= 
 the <a href="modelc_1.html#U5" title="MODELC_1:sel.5">EGlobal</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_1_1"><span class="lab"><font color="Green" title="E56">A18</font></span></a>, <a class="txt" href="#E4:64_1_1_3_5"><span class="lab"><font color="Green" title="E65">A40</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E3:64_1_1_3_5_1_2"/>
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="modelc_1.html#U5" title="MODELC_1:sel.5">EGlobal</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1_1_3"><span class="lab"><font color="Green" title="E59">A23</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E8:64_1_1_3"/><span class="lab"><font color="Green" title="E64">A42</font></span>: 
( <font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V5" title="MODELC_1:attr.5">ExistNext</a>  implies <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="modelc_1.html#U4" title="MODELC_1:sel.4">EneXt</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_3_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:64_1_1_3_6"/><span class="lab"><font color="Green" title="E65">A43</font></span>: 
<font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V5" title="MODELC_1:attr.5">ExistNext</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="modelc_1.html#U4" title="MODELC_1:sel.4">EneXt</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span></span><br/>

<a NAME="E2:64_1_1_3_6"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="modelc_1.html#E47"><span class="lab"><font color="Green" title="E40">Lm22</font></span></a></span>;<br/>
<a NAME="E3:64_1_1_3_6"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1_3"><span class="lab"><font color="Green" title="E57">A21</font></span></a>, <a class="txt" href="modelc_1.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/>
<a NAME="E4:64_1_1_3_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A44</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_3_6_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:64_1_1_3_6_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1_3"><span class="lab"><font color="Green" title="E57">A21</font></span></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_3_6_1_1"><span class="kw">suppose </span></a><a NAME="E1:64_1_1_3_6_1_1"/><span class="lab"><font color="Green" title="E67">A45</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="modelc_1.html#U4" title="MODELC_1:sel.4">EneXt</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:64_1_1_3_6_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> = 
<font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_1_1"><span class="lab"><font color="Green" title="E56">A18</font></span></a></span>
<br/>.= 
 the <a href="modelc_1.html#U4" title="MODELC_1:sel.4">EneXt</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="txt" href="#E1:64_1_1_3_6"><span class="lab"><font color="Green" title="E65">A43</font></span></a>, <a class="txt" href="#E1:64_1_1_3_6_1_1"><span class="lab"><font color="Green" title="E67">A45</font></span></a>, <a class="ref" href="modelc_1.html#D27" target="_self" title="MODELC_1:def.27">Def27</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:64_1_1_3_6_1_1"/>
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="modelc_1.html#U4" title="MODELC_1:sel.4">EneXt</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_1_1"><span class="lab"><font color="Green" title="E56">A18</font></span></a>, <a class="txt" href="#E4:64_1_1_3_6"><span class="lab"><font color="Green" title="E66">A44</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_3_6_1_2"><span class="kw">suppose </span></a><a NAME="E1:64_1_1_3_6_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="modelc_1.html#U4" title="MODELC_1:sel.4">EneXt</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span></span><br/><div class="add"><span class="kw">then </span> <a href="modelc_1.html#K20" title="MODELC_1:func.20">GraftEval</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">Kai</font>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c4">h</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c6">H</font>) = 
 the <a href="modelc_1.html#U4" title="MODELC_1:sel.4">EneXt</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1_3_6"><span class="lab"><font color="Green" title="E65">A43</font></span></a>, <a class="ref" href="modelc_1.html#D28" target="_self" title="MODELC_1:def.28">Def28</a></span>
<br/>.= 
 the <a href="modelc_1.html#U4" title="MODELC_1:sel.4">EneXt</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_1_1"><span class="lab"><font color="Green" title="E56">A18</font></span></a>, <a class="txt" href="#E4:64_1_1_3_6"><span class="lab"><font color="Green" title="E66">A44</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E3:64_1_1_3_6_1_2"/>
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="modelc_1.html#U4" title="MODELC_1:sel.4">EneXt</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1_1_3"><span class="lab"><font color="Green" title="E59">A23</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E9:64_1_1_3"/>
( <font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V2" title="MODELC_1:attr.2">atomic</a>  implies <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Kai</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1_3_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:64_1_1_3_7"/><span class="lab"><font color="Green" title="E65">A46</font></span>: 
<font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V2" title="MODELC_1:attr.2">atomic</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Kai</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_3_7_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:64_1_1_3_7_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1_3"><span class="lab"><font color="Green" title="E57">A21</font></span></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_3_7_1_1"><span class="kw">suppose </span></a><a NAME="E1:64_1_1_3_7_1_1"/><span class="lab"><font color="Green" title="E66">A47</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Kai</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font></span><br/><div class="add"><a NAME="E2:64_1_1_3_7_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> = 
<font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_1_1"><span class="lab"><font color="Green" title="E56">A18</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c2">Kai</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="txt" href="#E1:64_1_1_3_7"><span class="lab"><font color="Green" title="E65">A46</font></span></a>, <a class="txt" href="#E1:64_1_1_3_7_1_1"><span class="lab"><font color="Green" title="E66">A47</font></span></a>, <a class="ref" href="modelc_1.html#D27" target="_self" title="MODELC_1:def.27">Def27</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:64_1_1_3_7_1_1"/>
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Kai</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_3_7_1_2"><span class="kw">suppose </span></a><a NAME="E1:64_1_1_3_7_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Kai</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:64_1_1_3_7_1_2"/>
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Kai</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1_1_3"><span class="lab"><font color="Green" title="E59">A23</font></span></a>, <a class="txt" href="#E1:64_1_1_3_7"><span class="lab"><font color="Green" title="E65">A46</font></span></a>, <a class="ref" href="modelc_1.html#D28" target="_self" title="MODELC_1:def.28">Def28</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E10:64_1_1_3"/>
( ( <font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V2" title="MODELC_1:attr.2">atomic</a>  implies <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Kai</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> ) &amp; ( <font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V3" title="MODELC_1:attr.3">negative</a>  implies <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="robbins1.html#U1" title="ROBBINS1:sel.1">Compl</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V4" title="MODELC_1:attr.4">conjunctive</a>  implies <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>) ) &amp; ( <font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V5" title="MODELC_1:attr.5">ExistNext</a>  implies <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="modelc_1.html#U4" title="MODELC_1:sel.4">EneXt</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V6" title="MODELC_1:attr.6">ExistGlobal</a>  implies <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="modelc_1.html#U5" title="MODELC_1:sel.5">EGlobal</a> of <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K15" title="MODELC_1:func.15">the_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c6">H</font> is  <a href="modelc_1.html#V7" title="MODELC_1:attr.7">ExistUntill</a>  implies <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="modelc_1.html#U6" title="MODELC_1:sel.6">EUntill</a> of <font color="Maroon" title="c1">V</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K16" title="MODELC_1:func.16">the_left_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K17" title="MODELC_1:func.17">the_right_argument_of</a> <font color="Maroon" title="c6">H</font></span>)</span></span>)</span>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:64_1_1_3"><span class="lab"><font color="Green" title="E60">A24</font></span></a>, <a class="txt" href="#E6:64_1_1_3"><span class="lab"><font color="Green" title="E62">A33</font></span></a>, <a class="txt" href="#E8:64_1_1_3"><span class="lab"><font color="Green" title="E64">A42</font></span></a>, <a class="txt" href="#E7:64_1_1_3"><span class="lab"><font color="Green" title="E63">A38</font></span></a>, <a class="txt" href="#E5:64_1_1_3"><span class="lab"><font color="Green" title="E61">A28</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:64_1_1"/>
<font color="Maroon" title="c5">f</font> <a href="modelc_1.html#R2" title="MODELC_1:pred.2">is-PreEvaluation-for</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1,<font color="Maroon" title="c2">Kai</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#D27" target="_self" title="MODELC_1:def.27">Def27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:64_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:64"/><span class="lab"><font color="Green" title="E53">A48</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c3">v0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E2:64_2"/><span class="lab"><font color="Green" title="E54">A49</font></span>: 
<font color="Maroon" title="c3">v0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">f</font> = <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c3">v0</font>;<br/>
<a NAME="E3:64_2"/><span class="lab"><font color="Green" title="E55">A50</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c3">v0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/>
<a NAME="E4:64_2"/><span class="lab"><font color="Green" title="E56">A51</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c3">v0</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">v0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/>
<a NAME="E5:64_2"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">v0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_2"><span class="lab"><font color="Green" title="E54">A49</font></span></a>, <a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">f</font> = <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c3">v0</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_2"><span class="lab"><font color="Green" title="E55">A50</font></span></a>, <a class="txt" href="#E4:64_2"><span class="lab"><font color="Green" title="E56">A51</font></span></a>, <a class="ref" href="funct_2.html#T2" title="FUNCT_2:th.2">FUNCT_2:2</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">f</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="modelc_1.html#R2" title="MODELC_1:pred.2">is-PreEvaluation-for</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c2">Kai</font></span><br/>

<span class="kw">thus </span><a NAME="E7:64_2"/>
<font color="Maroon" title="c5">f</font> <a href="modelc_1.html#R2" title="MODELC_1:pred.2">is-PreEvaluation-for</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c2">Kai</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="modelc_1.html#E56"><span class="lab"><font color="Green" title="E47">Lm24</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:64"/>
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E2:64"><span class="lab"><font color="Green" title="E53">A48</font></span></a>, <a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E52">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:64"/>
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b2">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="modelc_1.html#K13" title="MODELC_1:func.13">CTL_WFF</a>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font> st <font color="Olive" title="b2">f</font> <a href="modelc_1.html#R2" title="MODELC_1:pred.2">is-PreEvaluation-for</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c2">Kai</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
