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

<span class="kw">reconsider </span><font color="Maroon" title="c1">i0</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D1" title="INT_1:def.1">INT_1:def 1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c2">Q</font> =  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>);<br/>
<span class="kw">set </span><font color="Maroon" title="c3">T</font> = <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>)</span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> (<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>);<br/>
<span class="kw">set </span><font color="Maroon" title="c4">S</font> =  <a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">G</font> =  <a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">A</font> =  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> (<a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a>,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a>);<br/>
<a NAME="E2:41_1_1"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
 <a href="lang1.html#K1" title="LANG1:func.1">Terminals</a> <span class="p1">(<span class="default"><a href="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> (<a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a>,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="freealg.html#T3" title="FREEALG:th.3">FREEALG:3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">q0</font> = <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c1">i0</font> as    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>) <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T8" title="FUNCT_2:th.8">FUNCT_2:8</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">s</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>) ex <font color="Olive" title="b2">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>)</span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>) ex <font color="Olive" title="b3">e</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>)</span>)</span>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>) st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">s</font>,<span class="p2">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">v</font>,<font color="Olive" title="b3">e</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> (<span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b1">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b3">e</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b1">s</font></span>)</span>) );<br/>
<a NAME="E4:41_1_1"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
 <a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <span class="p1">(<span class="default"><a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> (<a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a>,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="freealg.html#K12" title="FREEALG:func.12">FreeGenSetNSG</a> (<a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a>,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_000.html#T70" title="AOFA_000:th.70">AOFA_000:70</a></span>;<br/>
<a NAME="E5:41_1_1"/><span class="lab"><font color="Green" title="E28">A3</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>)</span>)</span>,<span class="p2">(<span class="default"><a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <span class="p3">(<span class="default"><a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> (<a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a>,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a>)</span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/> ex <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">x</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="c8">x</font> <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"><span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>)</span>)</span>,<span class="p2">(<span class="default"><a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <span class="p3">(<span class="default"><a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> (<a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a>,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a>)</span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> implies  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">x</font>,<font color="Olive" title="b1">y</font>] ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:41_1_1_1"/>
<font color="Maroon" title="c8">x</font> <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"><span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>)</span>)</span>,<span class="p2">(<span class="default"><a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <span class="p3">(<span class="default"><a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> (<a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a>,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a>)</span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">s</font>, <font color="Maroon" title="c10">I</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:41_1_1_1"/><span class="lab"><font color="Green" title="E29">A4</font></span>: 
<font color="Maroon" title="c9">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>)
 <span class="kw">and </span><br/><a NAME="E4:41_1_1_1"/><span class="lab"><font color="Green" title="E30">A5</font></span>: 
<font color="Maroon" title="c10">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <span class="p1">(<span class="default"><a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> (<a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a>,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:41_1_1_1"/><span class="lab"><font color="Green" title="E31">A6</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">s</font>,<font color="Maroon" title="c10">I</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">s</font> = <font color="Maroon" title="c9">s</font> as    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>) <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41_1_1_1"><span class="lab"><font color="Green" title="E29">A4</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">a</font> being   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> (<a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a>,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E8:41_1_1_1"/><span class="lab"><font color="Green" title="E32">A7</font></span>: 
<font color="Maroon" title="c10">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c12">a</font>
 <span class="kw">and </span><br/><a NAME="E9:41_1_1_1"/><span class="lab"><font color="Green" title="E33">A8</font></span>: 
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K1" title="LANG1:func.1">Terminals</a> <span class="p1">(<span class="default"><a href="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> (<a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a>,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41_1_1"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E4:41_1_1_1"><span class="lab"><font color="Green" title="E30">A5</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">v</font>, <font color="Maroon" title="c14">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E11:41_1_1_1"/><span class="lab"><font color="Green" title="E34">A9</font></span>: 
<font color="Maroon" title="c13">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>)</span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>)
 <span class="kw">and </span><br/><a NAME="E12:41_1_1_1"/><span class="lab"><font color="Green" title="E35">A10</font></span>: 
<font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>)</span>)</span>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>)
 <span class="kw">and </span><br/><a NAME="E13:41_1_1_1"/><span class="lab"><font color="Green" title="E36">A11</font></span>: 
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">v</font>,<font color="Maroon" title="c14">e</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41_1_1"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E9:41_1_1_1"><span class="lab"><font color="Green" title="E33">A8</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">reconsider </span><font color="Maroon" title="c15">e</font> = <font color="Maroon" title="c14">e</font> as    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>)</span>)</span>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>) <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:41_1_1_1"><span class="lab"><font color="Green" title="E35">A10</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">v</font> = <font color="Maroon" title="c13">v</font> as    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>)</span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>) <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:41_1_1_1"><span class="lab"><font color="Green" title="E34">A9</font></span></a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c17">y</font> = <font color="Maroon" title="c11">s</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c16">v</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">e</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">s</font></span>)</span>); <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="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c17">y</font>] )</span><br/>

<span class="kw">thus </span><a NAME="E16:41_1_1_1"/>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T8" title="FUNCT_2:th.8">FUNCT_2:8</a></span>; <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="c8">x</font>,<font color="Maroon" title="c17">y</font>]</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c11">s</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>)</span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>) ex <font color="Olive" title="b2">e</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>)</span>)</span>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>) st <br/>( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">s</font>,<span class="p2">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">e</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">e</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">s</font></span>)</span>) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c16">v</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">e</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>)</span>)</span>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>) st <br/>( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">s</font>,<span class="p2">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">v</font>,<font color="Olive" title="b1">e</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c16">v</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">e</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">s</font></span>)</span>) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c15">e</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="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">s</font>,<span class="p2">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">v</font>,<font color="Maroon" title="c15">e</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c16">v</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">e</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">s</font></span>)</span>) )</span><br/>

<span class="kw">thus </span><a NAME="E17:41_1_1_1"/>
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">s</font>,<span class="p2">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">v</font>,<font color="Maroon" title="c15">e</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c17">y</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c11">s</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c16">v</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">e</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">s</font></span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:41_1_1_1"><span class="lab"><font color="Green" title="E31">A6</font></span></a>, <a class="txt" href="#E8:41_1_1_1"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="txt" href="#E13:41_1_1_1"><span class="lab"><font color="Green" title="E36">A11</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">consider </span><font color="Maroon" title="c8">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E7:41_1_1"/><span class="lab"><font color="Green" title="E29">A12</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>)</span>)</span>,<span class="p2">(<span class="default"><a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <span class="p3">(<span class="default"><a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> (<a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a>,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a>)</span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>) )
 <span class="kw">and </span><br/><a NAME="E8:41_1_1"/><span class="lab"><font color="Green" title="E30">A13</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>)</span>)</span>,<span class="p2">(<span class="default"><a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <span class="p3">(<span class="default"><a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> (<a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a>,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a>)</span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_1.html#S5" title="FUNCT_1:sch.5">FUNCT_1:sch 5</a>(<a class="txt" href="#E5:41_1_1"><span class="lab"><font color="Green" title="E28">A3</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c9">g</font> = <font color="Maroon" title="c8">g</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>)</span>)</span>,<span class="p2">(<span class="default"><a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <span class="p3">(<span class="default"><a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> (<a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a>,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a>)</span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:41_1_1"><span class="lab"><font color="Green" title="E29">A12</font></span></a>, <a class="ref" href="funct_2.html#T2" title="FUNCT_2:th.2">FUNCT_2:2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">f</font> being    <a href="aofa_000.html#M1" title="AOFA_000:mode.1">ExecutionFunction</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> (<a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a>,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a>), <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>),<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>)</span>)</span> <a href="aofa_i00.html#K1" title="AOFA_I00:func.1">\</a> (<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)<span class="kw"> such that </span><br/><a NAME="E11:41_1_1"/><span class="lab"><font color="Green" title="E31">A14</font></span>: 
<font color="Maroon" title="c10">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>)</span>)</span>,<span class="p2">(<span class="default"><a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <span class="p3">(<span class="default"><a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> (<a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a>,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a>)</span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font>
 <span class="kw">and </span><br/><a NAME="E12:41_1_1"/>
 for <font color="Olive" title="b1">s</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>)<br/>  for <font color="Olive" title="b2">C</font>, <font color="Olive" title="b3">I</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> (<a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a>,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a>)</span>)</span>  st  not <font color="Maroon" title="c10">f</font> <a href="aofa_000.html#R4" title="AOFA_000:pred.4">iteration_terminates_for</a> <font color="Olive" title="b3">I</font> <a href="aofa_000.html#K13" title="AOFA_000:func.13">\;</a> <font color="Olive" title="b2">C</font>,<font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">C</font>) holds <br/><font color="Maroon" title="c10">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><a href="aofa_000.html#K16" title="AOFA_000:func.16">while</a> (<font color="Olive" title="b2">C</font>,<font color="Olive" title="b3">I</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_000.html#T91" title="AOFA_000:th.91">AOFA_000:91</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c10">f</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">s</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>)<br/>  for <font color="Olive" title="b2">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>)</span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>)<br/>  for <font color="Olive" title="b3">e</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>)</span>)</span>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>) holds  <font color="Maroon" title="c10">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">v</font>,<font color="Olive" title="b3">e</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> (<span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b1">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b3">e</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Olive" title="b1">s</font></span>)</span>)</span><br/>

<span class="kw">let </span><font color="Maroon" title="c11">s</font> be    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</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">v</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>)</span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>)<br/>  for <font color="Olive" title="b2">e</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>)</span>)</span>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>) holds  <font color="Maroon" title="c10">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c11">s</font>,<span class="p1">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">e</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">e</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">s</font></span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c12">v</font> be    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>)</span>)</span>,<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">  for <font color="Olive" title="b1">e</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>)</span>)</span>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>) holds  <font color="Maroon" title="c10">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c11">s</font>,<span class="p1">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">v</font>,<font color="Olive" title="b1">e</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c12">v</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">e</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">s</font></span>)</span>)</span><br/><span class="kw">let </span><font color="Maroon" title="c13">e</font> be    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>)</span>)</span>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</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="c10">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c11">s</font>,<span class="p1">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">v</font>,<font color="Maroon" title="c13">e</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c12">v</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">e</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">s</font></span>)</span>)</span><br/>





<span class="kw">set </span><font color="Maroon" title="c14">I</font> =  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">v</font>,<font color="Maroon" title="c13">e</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<a NAME="E13:41_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">v</font>,<font color="Maroon" title="c13">e</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<a NAME="E14:41_1_1"/><span class="kw">then </span>
 <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">v</font>,<font color="Maroon" title="c13">e</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <span class="p1">(<span class="default"><a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> (<a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a>,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41_1_1"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E2:41_1_1"><span class="lab"><font color="Green" title="E26">A1</font></span></a></span>;<br/>
<a NAME="E15:41_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A15</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">s</font>,<span class="p2">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">v</font>,<font color="Maroon" title="c13">e</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></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"><span class="p2">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>)</span>)</span>,<span class="p2">(<span class="default"><a href="aofa_000.html#K18" title="AOFA_000:func.18">ElementaryInstructions</a> <span class="p3">(<span class="default"><a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> (<a href="aofa_000.html#K19" title="AOFA_000:func.19">ECIW-signature</a>,<a href="aofa_i00.html#K26" title="AOFA_I00:func.26">INT-ElemIns</a>)</span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">s9</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>), <font color="Maroon" title="c16">v9</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>)</span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>), <font color="Maroon" title="c17">e9</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>)</span>)</span>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>)<span class="kw"> such that </span><br/><a NAME="E17:41_1_1"/><span class="lab"><font color="Green" title="E33">A16</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">s</font>,<span class="p2">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">v</font>,<font color="Maroon" title="c13">e</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">s9</font>,<span class="p2">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">v9</font>,<font color="Maroon" title="c17">e9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E18:41_1_1"/><span class="lab"><font color="Green" title="E34">A17</font></span>: 
<font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">s</font>,<span class="p2">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">v</font>,<font color="Maroon" title="c13">e</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">s9</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c16">v9</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c15">s9</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">e9</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c15">s9</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:41_1_1"><span class="lab"><font color="Green" title="E30">A13</font></span></a></span>;<br/>
<a NAME="E19:41_1_1"/>
 <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">v</font>,<font color="Maroon" title="c13">e</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">v9</font>,<font color="Maroon" title="c17">e9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:41_1_1"><span class="lab"><font color="Green" title="E33">A16</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<a NAME="E20:41_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A18</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">v</font>,<font color="Maroon" title="c13">e</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">v9</font>,<font color="Maroon" title="c17">e9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_4.html#T4" title="TREES_4:th.4">TREES_4:4</a></span>;<br/>
<a NAME="E21:41_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A19</font></span>: 
<font color="Maroon" title="c12">v</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c16">v9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<a NAME="E22:41_1_1"/><span class="lab"><font color="Green" title="E37">A20</font></span>: 
<font color="Maroon" title="c13">e</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c17">e9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:41_1_1"><span class="lab"><font color="Green" title="E35">A18</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<a NAME="E23:41_1_1"/>
<font color="Maroon" title="c11">s</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c15">s9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:41_1_1"><span class="lab"><font color="Green" title="E33">A16</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E24:41_1_1"/>
<font color="Maroon" title="c10">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c11">s</font>,<span class="p1">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">v</font>,<font color="Maroon" title="c13">e</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c12">v</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">e</font> <a href="aofa_i00.html#K21" title="AOFA_I00:func.21">.</a> <font color="Maroon" title="c11">s</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:41_1_1"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="txt" href="#E15:41_1_1"><span class="lab"><font color="Green" title="E32">A15</font></span></a>, <a class="txt" href="#E18:41_1_1"><span class="lab"><font color="Green" title="E34">A17</font></span></a>, <a class="txt" href="#E21:41_1_1"><span class="lab"><font color="Green" title="E36">A19</font></span></a>, <a class="txt" href="#E22:41_1_1"><span class="lab"><font color="Green" title="E37">A20</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</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/>


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